universit Ë degl i studi di p a via

UNIVERSITÀ DEGLI STUDI DI PAVIADIPARTIMENTO DI INFORMATICA E SISTEMISTICA

Modeling, Identification and MultivariableControl of a Flexible Arm

Luca Bossi

UNIVERSITÀ DEGLI STUDI DI PAVIADIPARTIMENTO DI INFORMATICA E SISTEMISTICA

Dottorato di Ricerca in Ingegneria Elettronica, Informatica ed Elettrica

Modeling, Identification and MultivariableControl of a Flexible Arm

Ph.D. dissertation by

Luca Bossi

Advisor

Prof. Lalo Magni

2006-2009

A Emiliana e Carlo

ii

Acknowledgement

This thesis is the product of three years of research that I have carried outin the Department of Informatica e Sistemistica at the University of Pavia,starting from a collaboration with the Department of Meccanica Strutturalewhere I performed the experimental tests.

All my gratitude goes to my supervisor, Prof. Lalo Magni, whose en-couragement, guidance, support from the initial to the final level enabledme to develop an understanding of the subject.

I also would like to thank Prof. Giovanni Mimmi which promoted thisproject allowing the availability of the experimental equipment, and Prof.Carlo Rottenbacher who had a crucial role in my work.

I am particularly grateful to Prof. Riccardo Scattolini, Prof. FrancescoCastelli Dezza, Prof. Franco Bernelli, Prof. Paolo Rocco, Prof. AlessandroReali, Dr. Gianluca De Felici, Dr. Luca Capisani, Dr. Giovanni Bonandrinifor their valuable contributes to the completion of this project.

A special thanks for his teachings to Prof. Tarunraj Singh, who receivedme at University of New York at Buffalo.

Lastly, thanks to Dr. Davide Raimondo, Dr. Riccardo Porreca, Dr.Matteo Rubagotti, Dr. Ravi Kumar for their useful tips.

Luca Bossi

Contents

1 Introduction 1

2 Experimental setup 6

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Flexible arm mechanical structure . . . . . . . . . . . . . . 6

2.2.1 New support design . . . . . . . . . . . . . . . . . . 10

2.3 Actuation and sensing devices . . . . . . . . . . . . . . . . . 11

2.3.1 Potentiometer . . . . . . . . . . . . . . . . . . . . . . 13

2.3.2 Strain gauges . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Control system . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Flexible arm modeling 19

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Pseudo-clamped approach . . . . . . . . . . . . . . . . . . . 21

3.3.1 Boundary value problem . . . . . . . . . . . . . . . . 24

3.3.2 Eigenvalues problem solution . . . . . . . . . . . . . 26

3.3.3 Orthogonality conditions . . . . . . . . . . . . . . . . 30

3.3.4 Assumed modes method . . . . . . . . . . . . . . . . 32

3.4 Pseudo-pinned approach . . . . . . . . . . . . . . . . . . . . 34

3.5 Friction model . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.6 State space representation . . . . . . . . . . . . . . . . . . . 41

4 Identification 43

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Inertia terms computation . . . . . . . . . . . . . . . . . . . 44

4.2.1 Motor-fork subsystem inertia . . . . . . . . . . . . . 44

iv Contents

4.2.2 Payload inertia . . . . . . . . . . . . . . . . . . . . . 50

4.3 Young modulus characterization . . . . . . . . . . . . . . . . 53

4.4 Relevant modes identification . . . . . . . . . . . . . . . . . 54

4.5 Damping identification . . . . . . . . . . . . . . . . . . . . . 56

4.6 Friction identification . . . . . . . . . . . . . . . . . . . . . . 59

4.7 Model validation for control . . . . . . . . . . . . . . . . . . 60

4.7.1 Open loop validation . . . . . . . . . . . . . . . . . . 60

4.7.2 Single Input Single Output closed-loop validation . . 61

4.7.3 Multivariable control oriented validation experiments 65

4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5 Control 70

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2 PD regulation . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.3 Theoretical background . . . . . . . . . . . . . . . . . . . . 75

5.4 MPC implementation . . . . . . . . . . . . . . . . . . . . . . 77

5.5 Kalman filter design . . . . . . . . . . . . . . . . . . . . . . 80

5.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.7 Experimental results . . . . . . . . . . . . . . . . . . . . . . 96

5.7.1 Software Real-Time . . . . . . . . . . . . . . . . . . 96

5.7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . 97

6 Stochastic MPC 105

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2 Stochastic MPC formulation . . . . . . . . . . . . . . . . . . 107

6.3 Deterministic reformulation of MPC . . . . . . . . . . . . . 109

6.4 Stochastic MPC with guaranteed stability . . . . . . . . . . 111

6.5 Numerical implementation . . . . . . . . . . . . . . . . . . . 115

6.6 A simulation example . . . . . . . . . . . . . . . . . . . . . 117

6.6.1 Control design parameters . . . . . . . . . . . . . . . 117

Contents v

6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.8.1 Notation, basic definitions and available results . . . 121

6.8.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Conclusions 131

Bibliography 132

List of Figures

2.1 Experimental test bed . . . . . . . . . . . . . . . . . . . . . 7

2.2 Experimental test-bed: Autodesk inventor model . . . . . . 9

2.3 Pneumatic system . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Support with regulations . . . . . . . . . . . . . . . . . . . . 11

2.5 Potentiometer . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6 Potentiometer look-up table. Blue dots: experimental data.Red line: least squares data fitting. . . . . . . . . . . . . . . 15

2.7 Wheatstone bridge . . . . . . . . . . . . . . . . . . . . . . . 15

2.8 Strain gauges . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.9 Servo system . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.10 Control logic . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Single link flexible arm: schematic representation . . . . . . 21

3.2 Flexible arm in the pseudo-clamped reference frame . . . . . 22

3.3 Flexible arm in the pseudo-pinned reference frame . . . . . 35

3.4 Friction model . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1 Fork partitioning . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2 Part F2: reference frame for inertia computation . . . . . . 47

4.3 Payload partitioning . . . . . . . . . . . . . . . . . . . . . . 51

4.4 Resonance frequency shift . . . . . . . . . . . . . . . . . . . 54

4.5 Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . 55

4.6 Oscillations for strain gauge look-up table identification. Redline: deflection measure provided by the laser acquisition[mm]. Blue line: deflection measure provided by the straingauge [Volts] . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.7 Strain gauge look-up table. Blue dots: experimental data.Green line: least square fitting . . . . . . . . . . . . . . . . 57

viii List of Figures

4.8 Arm free oscillation for damping identification . . . . . . . . 59

4.9 SISO validation test. Black dotted line: sinusoidal positionreference at 0.2 Hz. θhub(t) response. Green dashed line:simulation. Blue line: experimental behavior. . . . . . . . . 62

4.10 SISO validation test. Sinusoidal position reference at 0.2 Hz.Applied torque τ(t). Green dashed line: simulation. Blueline: experimental behavior. . . . . . . . . . . . . . . . . . 62

4.11 SISO validation test. Sinusoidal position reference at 0.2 Hz.Deflection response w(L, t). Green dashed line: simulation.Blue line: experimental behavior. . . . . . . . . . . . . . . 63

4.12 SISO validation test. Black dotted line: sinusoidal positionreference at 0.4 Hz. θhub(t) response. Green dashed line:simulation. Blue line: experimental behavior. . . . . . . . . 63

4.13 SISO validation test. Sinusoidal position reference at 0.4 Hz.Applied torque τ(t). Green dashed line: simulation. Blueline: experimental behavior. . . . . . . . . . . . . . . . . . 64

4.14 SISO validation test. Sinusoidal position reference at 0.4 Hz.Deflection response w(L, t). Green dashed line: simulation.Blue line: experimental behavior. . . . . . . . . . . . . . . 64

4.15 LQ control scheme with state observer and integral action . 65

4.16 MIMO validation test. θhub(t) response. Black dotted line:reference. Green dashed line: simulation. Blue line: experi-mental behavior. . . . . . . . . . . . . . . . . . . . . . . . . 67

4.17 MIMO validation test. Applied torque τ(t). Green dashedline: simulation. Blue line: experimental behavior. . . . . . 67

4.18 MIMO validation test. Deflections response w(L, t). Greendashed line: simulation. Blue line: experimental behavior. 68

5.1 Rest-to-rest manoeuvre: slow PD regulation. Kp = 0.0006,Kd = 0.001. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72



List of Figures ix

5.4 Rest-to-rest manoeuvre: fast PD regulation. Kp = 0.05,Kd = 0.03. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73



5.7 Comparison 1. Simulation comparison: θhub. Blue dashedline: unconstrained MPC. Red line: LQR. . . . . . . . . . . 84

5.8 Comparison 2. Simulation comparison: θhub. Black dottedline: step reference. Blue dashed line: constrained MPC.Red line: LQR. . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.9 Comparison 2. Simulation comparison: applied torque. Bluedashed line: constrained MPC. Red line: LQR. . . . . . . . 85

5.10 Comparison 2. Simulation comparison: deflections. Bluedashed line: constrained MPC. Red line: LQR. Black dashedline: MPC constraints. . . . . . . . . . . . . . . . . . . . . . 86

5.11 Sampling time variation. Constrained MPC: motor position.Black dotted line: step reference. Blue line: Ts = 0.03 [s].Green dash dotted line: Ts = 0.015 [s]. Red line: Ts = 0.005[s]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.12 Sampling time variation. Constrained MPC: ideal torque.Blue dashed line: Ts = 0.03 [s]. Green dash dotted line:Ts = 0.015 [s]. Red line: Ts = 0.005 [s]. Black dashed line:MPC constraints. . . . . . . . . . . . . . . . . . . . . . . . . 87

5.13 Sampling time variation. Constrained MPC: applied torque.Blue dashed line: Ts = 0.03 [s]. Green dash dotted line:Ts = 0.015 [s]. Red line: Ts = 0.005 [s]. . . . . . . . . . . . . 87

5.14 Sampling time variation. Constrained MPC: deflection. Bluedashed line: Ts = 0.03 [s]. Green dash dotted line: Ts =0.015 [s]. Red line: Ts = 0.005 [s]. Black dashed line: MPCconstraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

x List of Figures

5.15 Sampling time variation. Constrained MPC: deflection zoom.Blue dashed line: Ts = 0.03 [s]. Green dash dotted line:Ts = 0.015 [s]. Red line: Ts = 0.005 [s]. Black dashed line:MPC constraints. . . . . . . . . . . . . . . . . . . . . . . . . 89


5.17 Comparison 3. Simulation comparison: ideal torque. Bluedashed line: constrained MPC. Red line: LQR. Black dashedline: MPC constraints. . . . . . . . . . . . . . . . . . . . . . 90

5.18 Comparison 3. Simulation comparison: applied torque. Bluedashed line: constrained MPC. Red line: LQR. . . . . . . . 91

5.19 Comparison 3. Simulation comparison: deflections. Bluedashed line: constrained MPC. Red line: LQR. . . . . . . . 91


5.21 Comparison 4. Simulation comparison: deflections. Bluedashed line: constrained MPC. Red line: LQR. . . . . . . . 93

5.22 Comparison 4. Simulation comparison: applied torque. Bluedashed line: constrained MPC. Red line: LQR. Black dashedline: MPC constraints. . . . . . . . . . . . . . . . . . . . . . 93


5.24 Comparison 5. Simulation comparison: applied torque. Bluedashed line: constrained MPC. Red line: LQR. Black dashedline: MPC constraints. . . . . . . . . . . . . . . . . . . . . . 95

5.25 Comparison 5. Simulation comparison: deflections. Bluedashed line: constrained MPC. Red line: LQR. Black dashedline: MPC constraints. . . . . . . . . . . . . . . . . . . . . . 95

5.26 Comparison 1, 2. Experimental comparison: θhub. Blackdotted line: step reference. Blue dashed line: constrainedMPC. Green line: unconstrained MPC. Red line: LQR. . . . 98

List of Figures xi

5.27 Comparison 1, 2. Experimental comparison: applied torque.Blue dashed line: constrained MPC. Green line: uncon-strained MPC. Red line: LQR. Black dashed line: constraints 99

5.28 Comparison 1, 2. Experimental comparison: deflections.Blue dashed line: constrained MPC. Green line: uncon-strained MPC. Red line: LQR. Black dashed line: MPCconstraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.29 Comparison 3. Experimental comparison: θhub. Black dottedline: step reference. Blue dashed line: constrained MPC. Redline: LQR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.30 Comparison 3. Experimental comparison: deflections. Bluedashed line: constrained MPC. Red line: LQR. Black dashedline: constraints . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.31 Comparison 3. Experimental comparison: applied torque.Blue dashed line: constrained MPC. Red line: LQR. Blackdashed line: MPC constraints. . . . . . . . . . . . . . . . . . 101

5.32 Comparison 4. Experimental comparison: θhub. Black dottedline: step reference. Blue dashed line: constrained MPC. Redline: LQR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.33 Comparison 4. Experimental comparison: deflections. Bluedashed line: constrained MPC. Red line: LQR. Black dashedline: constraints. . . . . . . . . . . . . . . . . . . . . . . . . 102

5.34 Comparison 4. Experimental comparison: applied torque.Blue dashed line: constrained MPC. Red line: LQR. Blackdashed line: MPC constraints. . . . . . . . . . . . . . . . . . 102

5.35 90 Degrees manoeuvre. Experimental comparison: appliedtorque. Black dotted line: step reference. Blue dashed line:constrained MPC. Red line: LQR. . . . . . . . . . . . . . . 103

5.36 90 Degrees manoeuvre. Experimental comparison: appliedtorque. Blue dashed line: constrained MPC. Red line: LQR.Black dashed line: MPC constraints. . . . . . . . . . . . . . 103

5.37 90 Degrees manoeuvre. Experimental comparison: appliedtorque. Blue dashed line: constrained MPC. Red line: LQR.Black dashed line: MPC constraints. . . . . . . . . . . . . . 104

xii List of Figures

6.1 Hub angle θhub. Blue dashed line: stochastic MPC. Red line:nominal MPC. Black dashed line: MPC constraints. . . . . 118

6.2 Ideal torque. Blue dashed line: Stochastic MPC. Red con-tinuous line: Nominal MPC. Green line: additive torque dis-turbance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.3 Real applied torque. Blue dashed line: Stochastic MPC. Redcontinuous line: Nominal MPC. . . . . . . . . . . . . . . . . 119

6.4 Simulation comparison. Deflection. Blue dashed line:stochastic MPC, Red line: nominal MPC. Black dashed line:MPC constraints. . . . . . . . . . . . . . . . . . . . . . . . . 120

6.5 Exit code of Algorithm 1. . . . . . . . . . . . . . . . . . . . 121

List of Tables

2.1 Hexcel foils characteristics . . . . . . . . . . . . . . . . . . . 8

2.2 Beam characteristics . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Kevlar beam dimensions . . . . . . . . . . . . . . . . . . . . 8

2.4 Tip characteristics . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Brushless motor characteristics . . . . . . . . . . . . . . . . 12

2.6 Servostar characteristics . . . . . . . . . . . . . . . . . . . . 13

2.7 Potentiometer characteristics . . . . . . . . . . . . . . . . . 14

2.8 Strain gauges conditioning switch characteristics . . . . . . 16

2.9 Acquisition board characteristics . . . . . . . . . . . . . . . 18

4.1 Directly measurable plant parameters . . . . . . . . . . . . . 44

4.2 Fork parts measures . . . . . . . . . . . . . . . . . . . . . . 45

4.3 Fork parts volumes and masses . . . . . . . . . . . . . . . . 46

4.4 Fork parts CoM coordinates and distance from rotation axis 47

4.5 Moments of inertia of the fork parts . . . . . . . . . . . . . 49

4.6 Payload parts measures . . . . . . . . . . . . . . . . . . . . 50

4.7 Payload parts volumes and masses . . . . . . . . . . . . . . 50

4.8 CoM coordinates in (Xp,Yp,Zp) . . . . . . . . . . . . . . . 51

4.9 Arm elements measures . . . . . . . . . . . . . . . . . . . . 52

4.10 Young modulus characterization . . . . . . . . . . . . . . . . 53

4.11 Model parameters . . . . . . . . . . . . . . . . . . . . . . . . 60

4.12 Repeatability experiments analysis . . . . . . . . . . . . . . 61

4.13 Closed-loop validation variability . . . . . . . . . . . . . . . 68

5.1 Comparison 1: controllers parameters . . . . . . . . . . . . . 83



xiv List of Tables



6.1 Exit code Algorithm 1 . . . . . . . . . . . . . . . . . . . . . 120

Chapter 1

Introduction

Lightweight flexible manipulators have been a widely investigated topic inthe field of mechatronic systems. They represent an attractive alternativeto heavy and bulky robots in a wide spectrum of applications because oftheir high payload-to-weight ratio and lower energy consumption. On theother hand lightweight manipulators are subject to elastic deformations withconsequent complex dynamical beahviour. This challenging topic has beendeeply investigated and many prototypes of flexible manipulators have beendeveloped in the research centers throughout the world. Although there ex-ists also research finalized to get a technological improvement in industrialor surgical field, majority of the research behind those prototypes is con-nected to space applications, [Book 1993]. The International Space Stationhas pushed great involvement in this area gathering synergies through thewhole world. The Candarm 2 and the European Robotic Arm are significantexamples. Canadarm 2 has a length of 17.6 [m], while European RoboticArm has a length of 11.3 [m]. Those manipulators work on different partsof the International Space Station. The material used to manufacture thelink is the carbon fiber that in an arm of that size gives as result a flex-ible structure, with consequent oscillations of the arm tip. According toBook definition in [Book 1993], a structure is considered flexible when "de-flections too large to complete a task persist too long to allow the task tobe completed". A way to "solve" this problem is to perform sufficientlyslow maneuvers. When the maximum acceptable time for task completionbecome smaller or task content larger, effective control algorithms need tobe developed. To reach the current technological level in this area a hugeamount of strategies have been developed, handling challenging problemsof increasing difficulty that are traditionally classified according to two dif-ferent criteria. The first classification depends on manipulator structurecomplexity, defined by the following categories:

• planar single link flexible manipulators;

2 Chapter 1. Introduction

• planar multilink manipulators;

• multilink manipulators presenting deformations along the three di-mensions.

The second classification is based on the aim of the control task to beperformed.

• End effector regulation problem. It consists in the achievement of adesired position in an optimal time w.r.t. the residual vibration atthe end of the manouvre.

• Rest-to-rest motion in a desired fixed time. In this case, the goalincludes also a constraint on the rest-to-rest time.

• End effector trajectory tracking. The end-effector of the arm mustfollow a desired trajectory in the operative space.

It is proved that a traditional PD with feedback of the motor position sta-bilizes the system. Nevertheless this strategy is not suitable for the controltasks mentioned above, since oscillations of the end-effector at the end ofthe maneuver arise. Trying to solve this problem by using a feedback onthe end-effector position, the link results to be actuated at one end andthe feedback measure is taken at the other end. Such system is called noncolocated and it is characterized by the presence of a time delay and non-minimum phase [Cannon & Schmitz 1984]. Remarkably, a PD controllercan’t stabilize the non-colocated system, due to the presence of real zerosin the positive half-plane in the transfer function from motor torque to theend-effector position. Trajectory tracking also become a difficult task dueto non causal solutions of the inverse dynamic [Bayo et al. 1989]. Anotherissue concerns the infinite dimensionality of the problem: theoretically thedynamic behavior of a distributed system need an infinite number of de-grees of freedom to be described, even if in real applications a finite modelapproximation is enough, since high vibration modes own negligible energycontent [Meirovitch 1967]. Problem arises when restrictive appoximationsare adopted, i.e. when high modes neglected in the model are excited dur-ing the control task. In this case instability phenomena could appear. Thisis the so-called spillover effect [Balas 1978]. Moreover, shortcomings in theconstruction of a mechanical device could produce important effects on the

3

behavior. An example is the presence of clearance, typically in the jointgear. Friction too has important implication on the dynamic behavior ofthe manipulators. Static or Coulomb friction are the most common, butseldom are included in the model equations, since viscous or linear frictionis easier to model. Anyway some progress has been made in identificationand compensation of Coulomb friction. Majority of models used are ableto describe manipulator’s dynamics in case of small oscillations. Only inrecent results new models have been developed trying to describe non-lineardynamics in order to reproduce the system behaviour at wider level of ve-locities and deformations.To solve the control problem many solutions have been developed dur-ing the years such as closed loop algorithm including Input Shap-ing feedforward action [Mohamed & Tokhi 2003], feedback linearization[Wang & Vidyasagar 1991], strain feedback [Mohamed et al. 2005], passiv-ity based approach [Pereira et al. 2007], adaptive control [Yuan et al. 1989]and regulation schemes based on singular perturbation approach[Siciliano & Book 1988], [Bascetta & Rocco 2006]. Recently active vibra-tion suppression techniques using smart materials gained most attentionand interesting works have been presented [Hassan et al. 2007]. Remark-able review work on control issues is [Benosman & Le Vey 2004] whilea panoramic view of develeped modeling techiniques can be found in[Dwivedy & Eberhard 2006].

Our interst in this field is focused on the exploration of the effectivenessof multivariable control techniques based on the optimization of a stagecost. They require an accurate model, but give the possibility to reach thecontrol objectives just finding suitable values of the weights of the stagecost. In particular Linear Quadratic Regulator (LQR) control and ModelPredictive Control (MPC) are considered here. LQR control can’t handleconstraints on state or input variables; anyway it is possible to try to satisfythe constraints modifying opportunely the weights. Nevertheless, conser-vative solutions may occur. The MPC strategy overcomes this limitation,since it can explicitly handle constraints. Traditionally applied to controlMIMO systems characterized by slow dynamics, the MPC presents manyadvantages that could be exploited also by faster systems such as a flexi-ble arm. In fact, since the main limitation of this algorithm is due to itscomputational burden, strong efforts to set up fast MPC algorithms areincreasing. Papers presenting fast MPC alghortims appeared recently ac-

4 Chapter 1. Introduction

compained with MATLAB code [Wang & Boyd 2008]. Moreover, hardwarecomputational capabilities are increasing. For these reasons it is very in-teresting to test the effectiveness of MPC on real fast applications. Sinceit can handle constraints, optimal performance can be reached also in pres-ence of actuator saturations and moreover constraints on the maximumoscillation during the maneuver can be specified. This feature can be use-ful, since smaller displacements imply a structure stress reduction. More-over, for large displacements, oscillations may present a non linear behavior,while keeping displacements within a certain range, a linear approximationis more reliable and the system can be better controlled. For these rea-sons the flexible manipulator can be considered a good platform to test theeffectiveness of the MPC technique. To the best of our knowledge the lit-erature in this field is focused on simulation results [Boscariol et al. 2009a][Boscariol et al. 2009b]. An experimental result concerning the active vi-bration suppression approach has been obtained in [Hassan et al. 2007].

Model based approach for flexible manipulators control is affected bythe following intrinsic limitations:

• the complexity of the control algorithm increases significantly withthe system order;

• the stability of the closed loop system is sensitive to model parameteruncertainties, changes in the robot payload, high-order unmodeleddynamics as the control bandwidth is raised (spillover effects).

Nevertheless, it is well known that MPC strategies own inherent robustnessproperties, so that the main dynamics of a system can be approximatedwith enough accuracy with low order models. Hence the aim of this thesisis to highlight the effectiveness of multivariable approaches, through theachievement of the following aims:

• build an accurate simulator of a single link flexible apparatus validatedon experimental data, necessary for the application of a model basedcontrol algorithm;

• compare in simulation LQR and MPC control schemes to show theperformance improvement introduced by handling constraints;

• test the effectiveness of the MPC scheme on the experimental plant;

5

• give a methodological contribution in the field of robust MPC, de-veloping an innovative MPC algorithm capable to explicitly handledisturbancies and to guarantee stability properties.

The single link planar manipulator considered in this thesis is a part ofthe plant designed in the context of an ASI (Italian Space Agency) multi-objective research contract [Bernelli-Zazzera et al. 2001]. The goal of theoverall research was the design and realization of an experimental device forthe validation of control techniques applied to flexible articulated systems.The robotic arm is suspended on a suitable air-pad, floating on a glassplanar surface to reduce friction effects. Such designed plant works in thesame conditions of a system operating in a microgravity environement, inwhich the experimental tests shall be carried out. In fact, the presence ofthe air pad counteracts the effect of the gravity field, supporting the wholeweight of the arm, without adding friction with the table.

The thesis is organized as follows: Chapter 2 gives a detailed descrip-tion of the experimental apparatus. A linear mathematical model, derivedby means of the assumed mode methods and then complemented with anonlinear friction model is presented in Chapter 3. Then, in Chapter 4, theparameters identification procedure are described and carried out. In par-ticular suitable experiments are designed in order to identify the parametersof the model that cannot be measured or analytically computed. The identi-fied model is then validated. The closed loop validation has been done usingtwo control strategies. The first one is a single input (position error) singleoutput (motor torque) controller synthesized in the frequency domain andthe second one is a multi input (position error and deflection) single output(motor torque) controller based on an LQR control law complemented withan observer and an integral action. Simulation and experimental controlresults are reported in Chapter 5. Chapter 6 presents an innovative robustMPC strategy. Theoretical results and implementation concerning stabil-ity issues are discussed in detail. The algorithm is applied to the flexiblearm verifying its effectiveness in simulation. Finally, conclusions and futureworks are presented.

Chapter 2

Experimental setup

Contents1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 71.2 Flexible arm mechanical structure . . . . . . . . . . 7

1.2.1 New support design . . . . . . . . . . . . . . . . . . 111.3 Actuation and sensing devices . . . . . . . . . . . . . 12

1.3.1 Potentiometer . . . . . . . . . . . . . . . . . . . . . . 131.3.2 Strain gauges . . . . . . . . . . . . . . . . . . . . . . 15

1.4 Control system . . . . . . . . . . . . . . . . . . . . . . 17

2.1 Introduction

The experimental device used for the experimental tests (Figure 2.1) is apart of the TEMSRAD (Testbed for Microgravity Simulation in RoboticArm Dynamics) [Mimmi et al. 2008]. It consists of a flexible robotic armdriven by a brushless servomotor, operating in a working space compatiblewith the volume of a standard Express Pallet Adapter (EPA) designed for onboard experiments on the International Space Station (ISS). In this chaptera description of the plant is reported in detail. First the structure of themanipulator only in its mechanical parts is introduced, then the actuationand sensing devices are described, and finally the acquisition and controlsystem is presented.

2.2 Flexible arm mechanical structure

In Figure 2.2 an Autodesk Inventor model shows more clearly the variousparts that compose the structure. The arm is made of an aluminium fork

8 Chapter 2. Experimental setup

and a composite laminate material beam.

Figure 2.1: Experimental test bed

The beam constitutes the flexible part of the structure. This is realizedin composite material whose matrix has been obtained overlapping 3 foils ofHexcel K49-285-F161-188 pasted together with epossidic resin. The cross-section of the obtained rectangular beam 45× 2 [mm] results to be empty.In Tables 2.1, 2.2, 2.3, 2.4 are listed rispectively the elastic characteristics ofthe single foil, the elastic characteristics of the beam, the dimensions of thebeam and the dimensions of the tip. E, G and ν are respectively the Youngmodulus, the shear modulus and the Poisson modulus. The subscripts x andy, are defined respectively as the direction of the longitudinal axis of thebeam and as the one that is parallel to the longest side of the cross-section.Ex, Ey, Gxy have been computed through the application of the laminationtheory. Concluding, the link is very flexible in the operating plane; on thecontrary it can be considered rigid in the other directions.

The thickness of every foil is equal to 0.25 [mm], hence the compositefoil thickness results to be 0.75 [mm].

The joint is formed by a brushless motor, pinned to the table. A re-solver and a potentiometer are mounted on the motor axis. The connectionelement between the motor hub and the kevlar link is an aluminium fork.

2.2. Flexible arm mechanical structure 9

Feature ValueEx 28.50 [GPa]Ey 27.00 [GPa]νxy 0.05Gxy 1.96 [GPa]ρ 1400 [kg/m3]

Table 2.1: Hexcel foils characteristics

Feature ValueEx 21.69 [GPa]Ey 20.68 [GPa]Gxy 7.8 [Gpa]νxy 0.346

Table 2.2: Beam characteristics

Feature ValueLength 387 [mm]Height 45 [mm]Width 2 [mm]Cross-section area 68.5 [mm2]Volume 26509.5 [mm3]Density 1400 [kg/m3]Mass 0.037 [kg]

Table 2.3: Kevlar beam dimensions

Feature ValueVolume 57271 [mm3]Density 2710 [kg/m3]Mass 0.155 [kg]

Table 2.4: Tip characteristics


Figure 2.2: Experimental test-bed: Autodesk inventor model

The whole structure is placed on the glass table of dimensions 2.00 × 2.00[m] that is the support surface for the tip of the arm. An aluminium air-pad support is mounted at the end of the link that connect the air-pad tothe beam through a steel sphere. In this way the gravity is compensatedand the arm can move with negligible friction over the glass table. Suchdesigned structure reproduces the test situation in a micro-gravity environ-ment, where the only force acting on the beam is the torque supplied by themotor. The presence of the air-pad should also be able to reduce the tor-sional components of the vibrations. The air pad, together with its supportand the sphere constitute the payload of the arm.

As it can be observed in Figure 2.3 a silicon pipe connects the air-tankto the airpad; the pipe has been selected with opportune weight and size notto affect the dynamical characteristics of the arm. The air of the pneumaticsystem is hold at a pressure of 2 [bar] by means of a manometer placed on theinput of the tank (Figure 2.3b), that is supplied with air by a compressor.Excluding the silicon pipe, all the other cables (connecting potentiometerand strain gauges) come down from a purpose-made structure fixed to thetable. In this way the cables displacement and torsion during the manouvres

2.2. Flexible arm mechanical structure 11

a) Air-pad with support b) Air tank

Figure 2.3: Pneumatic system

is minimized, in order to avoid interaction with the system dynamics.

2.2.1 New support design

Some preliminar open loop tests performed on the plant, highlighted thepresence of disalignement of the motor axis w.r.t. the vertical axis withconsequent asymmetric behavior. For this reason the following physicalmodification has been introduced: the arm support that constraints it tothe table has been redesigned to allow a tight regulation of the motor axisorientation. The new support is made of two parts: the fixed part is analuminium block with two screws that connect it to the other block. Thisblock, that is an evolution of the original support, can be rotated acting onthe screws.

The effectiveness of the regulation has been verified through a series ofsuitable manouvres.

The sensible reduction of the motor axis disalignement determines asubstantial improvement of the mechanical behavior of the manipulator,


Figure 2.4: Support with regulations

opening the possibility to start with experimental tests finalized to themodel parameters identification.

2.3 Actuation and sensing devices

The robotic arm is actuated with a Kollmorgen 713RBH brushless motor.The brushless motors are synchronous motors supplied with a three-phasesystem. The name get its origin from its fundamental design characteristic,that is the absence of brushes, which are used in the continuous current mo-tor to realize the electrical connection for the supply of the rotor windings.In the brushless motor the rotor is composed by permanent magnets to cre-ate the magnetic flux. Cavities are present in the stator, where the supplywindings are lodged. All these components are ermetically sealed such thatthe only accessible parts are the stator clamps and the motor hub. Theapplication field of these devices is mainly restricted to low power (tipicallyless than 50 W) due to the magnets size limitation. Moreover, due to thevirtual absence of rotoric losses, these motors don’t need forced cooling andfor this reason they are particularly suitable for aerospace applications orfor use in polluted environements.

The brushless motor is supplied through a three-phase current sys-tem that must be synchronous and in-phase with the back-electromotiveforces generated by the induction field of the rotor. The trend of the back-electromotive forces is related to the structure of the magnets: these can be

2.3. Actuation and sensing devices 13

designed to produce either a sinusoidal or trapeziodal trend. In both casesit is needed to know, at each sampling time, the angular position of therotor to be able to reconstruct the value of the back-electromotive forcesneeded by the amplifier in order to provide the right signal for motor driv-ing. To this aim, a resolver is mounted on motor axis. The frequency of thegenerated signals is related to the rotation velocity of the motor throughthe relation

n =60 · f

pp(2.1)

where n is the rotation velocity expressed in [rpm], f is the supply-voltagefrequency and pp is the number of polar pair of the motor. Currents andvoltages are required to be in-phase to obtain the maximum mean power andhence a constant supplied torque. In fact if a phase displacement betweenthe supply voltage and the relative current occurred, the torque value woulddecrease. The Table 2.5 lists the parameters of the motor.

MODELLO RBEH-0713-800Max power [W] 141Max velocity [rpm] 20000Max torque [Nm] 0.597Pair poles 3Weight [Kg] 0.344Torque sensitivity kt [Nm/A] 0.0611Current at Cont. Torque Icont [A] 3.51Inductance [mH] 1.10Resistence [Ω] 1.7

Table 2.5: Brushless motor characteristics

The motor is driven by a sinusoidal digital servoamplifier Danhaer Mo-tion Servostar S606, capable to produce a maximum rated current of 6A.An internal current loop realized by the servo amplifier, with parameterstuned by the manufacturer, allows the torque control of the motor. In Table2.6 are reported the main Servostar characteristics.


Servostar 606-ASRated supply voltage [V] 3× 230 ..... 480 V, 50 HzRated installed load for S1 operation [kVA] 4Rated DC-link voltage [V] 260 - 675Rated output current [Arms] 6Peak output current (5 sec) [Arms] 12Clock frequency of the output stage [kHz] 8 (16 with VDCmax=400 V )Aux. power supply [V] 24 (−0% +15%Aux. power supply [A] 3Analog input speed mode ±10 V

Table 2.6: Servostar characteristics

2.3.1 Potentiometer

A potentiometer mounted on the motor hub measures the angle position ofthe motor shaft.

a) Electric scheme b) Physical device

Figure 2.5: Potentiometer

In Table 2.7 the characteristics of the chosen potentiometer WALL 305by Novotechnik are reported.

The main problem using this device is due to the sensibility to theelectric noise induced by the motors. This problem has been significantlyreduced using shielded cables and separating the masses of the motor fromthe ones of the acquisition system. The potentiometer voltage signal y has

2.3. Actuation and sensing devices 15

Modello WALL 305Max supply voltage 35 VElectric angle 340

Mechanic angle (360)Resistive element Conductive plasticMax resistence 5 kΩIndipendent linearity ±2%Mass 8%Operative temperature −25÷+75CResolution 0.03

Supply 5 V

Table 2.7: Potentiometer characteristics

a linear relation with the angle rotation x expressed by the equation

y = θ1x + θ2

with identified values of θ1 = 68.1027 and θ2 = −109.0188. The resultinglookup table is represented in Figure 2.6

2.3.2 Strain gauges

In order to take deformation measures, the link is equipped with straingauges, which by means of a conditiong switch provide two voltage signalsproportional to the local deformations of the beam, that can be related tothe deflection.

The sensitivity of the strain gauge is expressed by the gage factor k,defined by the ratio:

k =∆RR∆ll

where ∆R indicates the resistence variation, R the nominal resistence ofthe strain gauge at rest. Analogously ∆l is the length variation and lis the nominal length. The strain gauges are mounted on the beam inthe Wheatstone bridge configuration (Figure 2.7) which allows to removethe effect of the temperature on resistivity and hence on the sensitivity(k = k(T )) of the strain gauge. They are positioned one close to the motor


1.5 2 2.5 3 3.5 4

0

20

40

60

80

100

120

140

160

180

Potentiometer measure [V]

Mo

tor

An

gle

[D

eg

ree

s]

Figure 2.6: Potentiometer look-up table. Blue dots: experimental data.Red line: least squares data fitting.

Figure 2.7: Wheatstone bridge

and the other in the middle of the link (Figure 2.8).

The conditioning switch is the Scout 55, which amplifies the signal andgives the possibility to make some signal preprocessing before the acquisi-tion, if needed. Its characteristics are reported in Table 2.8.

2.4. Control system 17

Figure 2.8: Strain gauges

Model HBM SCOUT55Supply voltage 230 V 50-60 HzMax power 10 VA

Table 2.8: Strain gauges conditioning switch characteristics

2.4 Control system

The scheme in Figure 2.9 represents the components of the servo while thescheme in Figure 2.10 shows in a qualitative way the control logic of thebrushless motor realized through the servo-drive amplifier Servostar606-AS.It is programmable through RS232 with the software "DRIVE" provided bythe manufacturer. For our tests it has been programmed in order to workin torque control mode; in other words an analogic voltage signal is usedto command the desired armature current given a known factor defined bythe user in the software DRIVE settings. It is chosen 10[V]

3.51[A] . Moreover thearmature current is proportional to the torque through the torque sensitivityfactor kt. Finally, exploiting this relation, the torque actuator saturationhas been computed as

τmax = kticont = 0.0611 · 3.51 = 0.214[Nm]


Figure 2.9: Servo system

Figure 2.10: Control logic

2.4. Control system 19

.

To this first current loop another external loop realizes a position loop.This second loop is implemented through a notebook running MATLABReal-Time Workshop, which communicates with the external world troughthe acquisition board National Instruments DAQCard-6036E. Its character-istics are reported in Table 2.9. Real Time Workshop translates in C-codethe developed Simulink control scheme and loads the code on the Real TimeWindows Target such that it is possible to perform an easy real-time im-plementation of control strategies.

To this aim a plant model is required first. Next chapter illustratesthe followed mathematical approach to obtain the dynamic equation of theconsidered system.

Modello DAQCard-6036EBus PCI, PCMCIAAnalog Inputs 16 SE/8 DIResolution 16 bitSampling Rate 200 kS/sInput Range ±0.05 to ±10 VAnalog Outputs 2Resolution 16 bitOutput kS/sOutput Range ±10 VDigital I/O 8Counter/Timers 2, 24-bitTriggers Digital

Table 2.9: Acquisition board characteristics

Chapter 3

Flexible arm modeling

Contents2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Pseudo-clamped approach . . . . . . . . . . . . . . . 21

2.3.1 Boundary value problem . . . . . . . . . . . . . . . . 24

2.3.2 Eigenvalues problem solution . . . . . . . . . . . . . 26

2.3.3 Orthogonality conditions . . . . . . . . . . . . . . . 30

2.3.4 Assumed modes method . . . . . . . . . . . . . . . . 31

2.4 Pseudo-pinned approach . . . . . . . . . . . . . . . . 33

2.5 Friction model . . . . . . . . . . . . . . . . . . . . . . 38

2.6 State space representation . . . . . . . . . . . . . . . 40

3.1 Introduction

In this chapter an analytical technique for the modeling of the single linkflexible manipulator is presented. The mechanical device is a continuos bodywith a theoretical infinite number of degrees of freedom. For this reasonsome simplifications are introduced in order to develop a finite dimensionallinear model, more suitable for simulation and control applications. Themodel is obtained applying the assumed mode method according to the ap-proach presented in [Bellezza et al. 1990], in which the exact modal shapesare computed. Finally, adding both damping and friction terms a completemodel of the considered plant is obtained and a state-space representationis derived.

22 Chapter 3. Flexible arm modeling

3.2 Assumptions

The robotic arm described in the previous chapter, can be sketched as shownin Figure 3.1. The presented modeling technique requires some preliminarassumptions that allow to obtain a linear model of the system. In particular,according to the Eulero-Bernoulli beam theory:

1. the link is a slender beam with uniform geometric characteristics andan homogeneous mass distribution;

2. the link is flexible along the longitudinal direction, but it is consideredstiff w.r.t.:

• the flexural components due to the gravity;

• the axial forces:

• the torsions;

3. only elastic deformations are considered (small deflections);

4. nonlinear deformations and internal frictions are neglected.

The chosen modeling approach [Bellezza et al. 1990] consists in finding theexact modal shapes of the system. Refering to Figure 3.1, let define (X0,Y0)the fixed reference frame, (Xc,Yc) the pseudo-clamped reference frame and(Xp,Yp) the pseudo-pinned reference frame, where Xc is tangent to thebeam in the origin while Xp pass through the centre of mass of the arm. Theprefix pseudo underlines the fact that the modal shapes of this system donot correspond to the modal shape of a pinned or clamped beam, but theyare an intermediate solution, since the inertia J0 of the joint is considered.It is easy to figure out and it will be also shown in the next sections, thatthe modal shapes of the system tend to coincide to the pinned or clampedones, as the inertia J0 tends respectively to zero or infinity. θhub(t) is therotation angle of the pseudo-clamped reference frame w.r.t. the fixed one,θtip(t) is the angle between the fixed reference frame and the line passingthrough the point P and finally α(t) is the angle between the pseudo-pinnedreference frame and the fixed one.

3.3. Pseudo-clamped approach 23

Figure 3.1: Single link flexible arm: schematic representation

3.3 Pseudo-clamped approach

Let be considered the pseudo-clamped approach. With reference to Figure3.2 let be defined the time variable t and the spatial coordinate x thatidentify a generic point P on the beam in its undeformed configuration,while x′ is the projection of the point P on the axis Xc. θP (x, t) is theangle between X0 and the line passing through P . The vector r(x, t) isthe vector position of P . Then let be defined wc(x, t) the deflection of thelink, i.e. the distance between the point P of the beam in its deflectedconfiguration from the position of the same point P when the beam is inits undeformed configuration; it is straightforward to derive the definitionof wc(x′, t). Other physical parameters of the system are: the linear massdensity of the link ρ, the link length L, the payload mass Mp, the payloadinertia Jp, the motor inertia J0, the flexural stiffness of the link, given bythe product between the Young Modulus E and the cross-section inertiamomentum I. According to the Hamilton Principle [Meirovitch 1967] the


Figure 3.2: Flexible arm in the pseudo-clamped reference frame

system equations are derived from the following variational condition∫ t2

t1

(δT (t)− δU(t) + δW (t))dt = 0 (3.1)

where δT (t) and δU(t) are respectively the variations of kinetic and po-tential energy while δW (t) = τ(t)δθhub(t) represents the virtual work per-formed by the actuator to supply the torque τ(t). First the energy termsof the system have to be computed. Looking at the picture of Figure 3.2 itis possible to do some trigonometric considerations:

wc(x′, t) = x′ tan(θP (x, t)− θhub(t)) (3.2)


θP (x, t)− θhub(t) = arctanwc(x′, t)

x′(3.3)

From the following approximations:

x′ ≈ x (3.4)

andwc(x′, t) ≈ wc(x, t) (3.5)

holdsarctan

wc(x′, t)x′

≈ arctanwc(x, t)

x(3.6)

The deflection wc(x, t) is assumed to be small w.r.t. x, i.e. wc(x,t)x is small.

Thereforearctan

wc(x, t)x

≈ wc(x, t)x

(3.7)

θP (x, t) ≈ θhub(t) +wc(x, t)

x(3.8)

The position vector r of P in the reference frame (X0,Y0) is:

r(x, t) =(

x0(x, t)y0(x, t)

)=

(x′ cos θhub(t)− wc(x′, t) sin θhub(t)x′ sin θhub(t) + wc(x′, t) cos θhub(t)

)(3.9)

and by means of the approximations (3.4) and (3.5), it results:

r(x, t) =(

x0(x, t)y0(x, t)

)≈

(x cos θhub(t)− wc(x, t) sin θhub(t)x sin θhub(t) + wc(x, t) cos θhub(t)

)(3.10)

and the velocity r(x, t):

r(x, t) =[

x0(x, t)y0(x, t)

]≈ (3.11)

≈[−xθhub(t) sin θhub(t)− wc(x, t) sin θhub(t)− wc(x, t)θhub(t) cos θhub(t)xθhub(t) cos θhub(t) + wc(x, t) cos θhub(t)− wc(x, t)θhub(t) sin θhub(t)

]

(3.12)The kinetic energy T is given by the sum of the motor, beam and thepayload contributions Th, Tl and Tp. The motor motion is purely rotational,while the payload one is both rotational and traslational. The contributionof the cross-section moment of inertia I is neglected in the kinetic energy


computation.

Th =12J0θ

2hub(t)

Tb =12ρ

∫ L

0|r(x, t)|2dx

=12ρ

∫ L

0[x2

0(x, t) + y20(x, t)]dx (3.13)

Tp =12Mp|r(L, t)|2 +

12Jp

[θP (L, t)

]2

=12Mp[x2

0(L, t) + y20(L, t)] +

12Jp

[θP (L, t)

]2(3.14)

Substituting (3.10) in (3.13) yields

Tb ≈12ρ

∫ L

0[x2θ2

hub(t) + w2c (x, t) + θ2

hub(t)w2c (x, t) + 2xθhub(t)wc(x, t)]dx

Using (3.10) derivative w.r.t. t togheter with (3.8) the (3.14) become

Tp ≈12Mp[L2θ2

hub(t) + w2c (L, t) + 2Lwc(L, t)θhub(t) + θ2

hub(t)w2c (L, t)]+

+12Jp

[θhub(t) +

∂

∂t

(wc(x, t)

x

∣∣∣∣x=L

)]2

The total kinetic energy term is given by

T = Th + Tb + Tp (3.15)

while the potential energy term is

U =12EI

∫ L

0(w′′c (x, t))2dx (3.16)

3.3.1 Boundary value problem

The variational calculus is used to find the solution of the integral (3.1)[Morris & Taylor 1996], obtaining:

EIwcIV (x, t) + ρ(wc(x, t) + xθhub(t)) = 0 (3.17)

τ(t)− J θhub(t)− µ(t) = 0 (3.18)


where

µ(t) = ρ

∫ L

0xwc(x, t)dx + MpLwc(L, t) + Jpw

′c(L, t) (3.19)

represents the inertial torque due to the flexibility of the link, τ(t) is theinput torque and J is the inertial torque defined as:

J = J0 + Jp + ρ

∫ L

0x2dx + MpL

2 (3.20)

where J0 is the motor inertia and Jp is the payload inertia. The associatedboundary conditions are:

wc(0, t) = 0wc

′(0, t) = 0EIwc

′′(L, t) = −Jp(θhub(t) + w′c(L, t))EIwc

′′′(L, t) = Mp(Lθhub(t) + wc(L, t))

(3.21)

It is to remark that beyond the geometric conditions wc(0, t) = 0 andw′c(0, t) = 0, the other two boundary conditions express the balance ofthe momenta and of the shear forces acting on the free end of the link.Solving the equation (3.18) w.r.t. θhub(t) yields

θhub(t) =τ(t)− µ(t)

J(3.22)

Then substituting (3.22) in (3.17) and in (3.21) it results

EIwcIV (x, t) + ρwc(x, t) + ρx

τ(t)− µ(t)J

= 0 (3.23)

with the boundary conditions

wc(0, t) = 0wc

′(0, t) = 0EIwc

′′(L, t) = −Jp

(τ(t)−µ(t)

J + w′c(L, t))

EIwc′′′(L, t) = Mp

(L τ(t)−µ(t)

J + wc(L, t))

(3.24)

The obtained system of partial differential equations together with theboundary conditions is called boundary value problem. To solve the prob-


lem in an exact way the modal analysis is used. It allows to decouple theequations by means of a change of coordinates, making easier the solutionof the problem. The modal analysis of continuous systems can be appliedonly if it is possible to represent the solution as a separated variables solu-tion. In fact assuming a separated variables solution, the boundary valueproblem can be rewritten as an eignevalue problem. The eigenvalue problemis defined as a boundary value problem in which the differential equationsof motion and the boundary conditions are homogeneous and they dependon a parameter that is called eigenvalue, usually indicated with the greekletter λ. Moreover a nontrivial solution must be obtained only for a certainvalue of the parameter λ. These condition are verified for this problem andthe mathematical steps which lead to the final solution are presented in thenext sections.

3.3.2 Eigenvalues problem solution

Assuming separate variables solution

wc(x, t) = φc(x)δ(t) (3.25)

Applying (3.25) to (3.19) it is obtained

µ(t) = δ(t)[ρ

∫ L

0xφc(x)dx + MpLφc(L) + Jpφ

′c(L)

](3.26)

and defining

µ0 = ρ

∫ L


′c(L) (3.27)

(3.26) can be written as a separated variables expression

µ(t) = µ0δ(t) (3.28)


The free vibration solution can be obtained setting the input torque to zero.Substituting (3.25), (3.28) in (3.23) yields

EIφIVc (x)δ(t) + ρφc(x)δ(t)− xρµ0

Jδ(t) = 0

EIφIVc (x)

ρφc(x)− xρµ0J

= − δ(t)δ(t)

! ω2

The first term depends only on x, while the second one is a function onlyof the variable t. Therefore x and t are independent variables and theboundary values problem has solution only if both the terms are equal to aconstant that we define ω2. On the base of these considerations it is possibleto reformulate the problem as

EIφIVc (x)− ρω2φc(x) + xω2 ρµ0

J= 0 (3.29)

δ(t) + ω2δ(t) = 0 (3.30)

with the associated boundary conditions

φc(0) = 0φ′c(0) = 0EIφ′′c (L) = −Jpω2(φ′c(L)− µ0

J )EIφ′′′c (L) = Mpω2(µ0L

J − φc(L))

(3.31)

The values of the parameter ω2 that satisfy (3.29) and in (3.30) are calledcharacteristic values or eigenvalues and the associated non trivial solutionsφc(x) are named eigenfunctions. The equation (3.29) is an homogeneousdifferential equation, thus its solution can be determined unless a constantmultiplying factor. The solution is

φc(x) = φc(x) + Fx (3.32)

where F is a constant and φc(x) is the general integral of (3.29)

φc(x) = A sin(βx) + B cos(βx) + C sinh(βx) + D cosh(βx) (3.33)

with β is such that

β4 =ρω2

EI. (3.34)


F , as function of A, B, C, D and β, can be obtained substituting in (3.29)the value of φc(x) expressed by (3.32). Thus

EI(φc(x) + Fx)IV − ρω2(φc(x) + Fx) + xρ

Jω2µ0 = 0

EIφIVc (x)− ρω2φc(x)− ρω2Fx + x

ρ

Jω2µ0 = 0

Carrying out the fourth derivative of the function (3.33) is easily obtained

φIVc (x) = β4[Asin(βx) + Bcos(βx) + Csinh(βx) + Dcosh(βx)]

φIVc (x) =

ρω2

EIφc(x)

henceF =

1J

µ0 (3.35)

Substituting (3.32) in (3.27) and expliciting µ0 from the (3.35) it is possibleto see that the following equality holds:

µ0 = FJ =ρ

∫ L

0xφc(x)dx + F

(ρ

∫ L

0x2dx + MpL

2 + Jp

)+

+ Mplφc(L) + Jpψ′h(L)

Using (3.20), F can be expressed in function of only A, B, C, D and β as

FJ = ρ

∫ L

0xφc(x)dx + F (J − J0) + Mplφc(L) + Jpψ

′h(L)

F =1J0

[ρ

∫ L


′c(L)

]

Then the boundary conditions (3.31) are exploited to obtain four equationsin A, B, C, D. Computing the derivatives of φc(x) and substituting themin the boundary conditions, leads to the following linear system of four


equations and four unknown variables.

B + D = 0β(A + C) + F = 0EIβ2(−As−Bc + Csh + Dch)− ω2Jpβ(Ac−Bs + Cch + Dsh + F ) =

= −Jp

Jµ0ω

2

EIβ3(−Ac + Bs + Cch + Dsh)− ω2Mp(As + Bc + Csh + Dch + FL) =

= −MpL

Jµ0ω

2

where, for the ease of the reader, the following notation has been adopted.c = cos(βL), s = sin(βL), ch = cosh(βL) e sh = sinh(βL). The fourboundary conditions determine univocally the shape of the solution, unlessthe amplitude. Moreover they generate a characteristic equation that isobtained imposing to zero the determinant of the coefficients matrix. Thiscondition is needed to exclude the trivial solution. Simplifying, the followingequation is obtained

csh − sch − 2Mp

ρ βssh +−2Jp

ρ β3cch − J0ρ β3(1 + cch)− (3.36)

Mp

ρ2 β4(J0 + Jp)(csh − sch) + J0Jp

ρ2 β6(csh + sch) +

−J0JpMp

ρ3 β7(1− cch) = 0

Since the link has finite length, the solution of the characteristic equation(3.36) consists of an infinite numerable succession of βi to which the eigen-values ω2

i and the natural frequencies ωi are associated. To each eigenvalueand i.e. to each natural frequency corresponds an eigenfunction Γiφci(x),where the multiplying constant Γi represents an arbitrary amplitude while

φc(x) = [φc1(x),φc2(x), ...]

represents the vector of the modal shapes but a constant. The functionsΓiφci(x) represent the natural modes of vibration of the system which afterbeing normalized become the normal vibration modes. It is worth to remarkthat if Mp = Jp = 0 the inertia J0 of the constraint tends towards infinitethe equation (3.36) is reduced to the characteristic equation of the clamped-free case:

1 + c + ch = 0


Furthermore, if Mp = Jp = 0 and the inertia J0 tends towards to zerothe equation (3.36) leads to the pinned-free case beam for which holds(wc(0, t) = w′′c (0, t) = w′′c (L, t) = w′′′c (L, t)).

csh − sch = 0

It can be seen from (3.30) and (3.32) that at each solution βi of the char-acteristic equation and hence at each natural frequency ωi, correspond re-spectively different modal shapes φci(x) and elastic variables δi(t).

3.3.3 Orthogonality conditions

Multiplying (3.29), evaluated for a particular eigenfunction φcs(x), for an-other different eigenfunction φcr(x), with s, r = 1, 2, ... and integrating onbeam length, adopting the simplified notation φ := φ(x), it results

EI

∫ L

0φIV

cs φcrdx = ρω2s

∫ L

0φcsφcrdx−

∫ L

0

ρxω2s

Jφcrµsdx

The integration for parts of the left term yields

EI[φ′′′csφcr]L0 − EI

∫ L

0φ′′′csφcr

′dx =ρω2s

∫ L

0φcsφcrdx+

− µs

∫ L

0

ρxω2s

Jφcrdx

EI[φ′′′csφcr]L0 − EI[φ′′csφ

′cr]

L0 + EI

∫ L

0φ′′csφ

′′crdx =ρω2

s

∫ L

0φcsφcrdx+

− µs

∫ L

0

ρxω2s

Jφcrdx


Substituting the boundary conditions (3.31) is obtained

ω2s

(−Mpφcs(L)φcr(L) +

MpL

Jµsφcr(L)− Jpφ

′cs(L)φ′cr(L) +

Jp

Jµsφ

′cr(L)

)+

+ EI

∫ L

0φ′′csφ

′′crdx = ρω2

s

∫ L

0φcsφcrdx− µsρ

∫ L

0

xω2s

Jφcrdx

ω2sµs

(MpL

Jφcr(L) +

Jp

Jφ′cr(L) + ρ

∫ L

0

x

Jφcrdx

)=

ω2s

(ρ

∫ L

0φcsφcrdx + Mpφcs(L)φcr(L) + Jpφ

′cs(L)φ′cr(L)

)− EI

∫ L

0φ′′csφ

′′crdx

(3.37)

Recalling the equation (3.27)

µi = ρ

∫ L

0xφci(x)dx + MpLφci(L) + Jpφ

′ci(L) (3.38)

it is easy to observe that (3.37) can be rewritten as

ωs

(ρ

∫ L

0φcsφcrdx + Mpφcs(L)φcr(L) + Jpφcs

′(L)φ′cr(L)− µsµr

J

)=

EI

∫ L

0φ′′csφ

′′crdx (3.39)

and doing the same thing but inverting the roles of φcs and φcr and sub-stracting the resulting expression from (3.39) it is obtained

(ωs−ωr)(

ρ

∫ L


′cs(L)φ′cr(L))− µsµr

J

)= 0

and since ωs $= ωr

ρ

∫ L



J= 0

From this it is possible to write the following orthonormality condition

ρ

∫ L



J) = ∆rs

(3.40)


where ∆ is the kronecker delta defined as:

∆rs =

1, r = s0, r $= s

3.3.4 Assumed modes method

The assumed modes method [Meirovitch 1967] is based on the assumptionthat the response of the system in free vibration can be described by a linearcombination of admissible functions φc(x), multiplied by time-dependentgeneralized coordinates, i.e. a truncation of the expansion series of eigen-functions to a finite number N i.e.

w(x, t) =N∑

i=1

φci(x)δi(t) (3.41)

The kinetic energy expression is given by

Tb =12

∫ L

0w(x, t)2dx +

12Jbθ

2hub(t) + θhubρ

∫ L

0xw(x, t)dx

T0 =12J0θ

2hub(t)

Tp =12Mp(θL + w(L, t))2 +

12Jpθ

2 +12Jpw

′(L, t) + Jpθw′(L, t)

Then the expression of the kinetic and potential energy for a discrete systemare recalculated. Provided the kinetic and potential energy expressions interms of a set of generalized coordinates, it is possible to derive the equationsof motion applying the Lagrangian method. Hence, simplifying the notationcalling θhub(t) = δ0, δi(t) = δi, δj(t) = δj and x = φc0, and rewriting the


expression of kinetic energy for each contribution

Tb =12

∫ L

0

∑

i

∑

j

δiδjφciφcjdx +12Jbδ

20 + δ0ρ

∫ L

0φc0

∑

i

δi

T0 =12J0δ

20

Tp =12Mp(δ2

0L2 +

∑

i

∑

j

δiδjφci(L)φcj(L)) + Jpδ0

∑

i

∑

j

φ′ci(L)φ′cj(L)δiδj

+ Mpδ0L∑

i

∑

j

δiδjφci(L)φcj(L) +12Jpδ

20 +

12Jp

∑

i

∑

j

φ′ci(L)φ′cj(L)δiδj

with i,j = 1, ...,N , it is easy to see that the expression of the total kineticenergy of the system, given by (3.15) can be written in the compact form

12

∑

i

∑

j

δiδjmij (3.42)

where mij , recalling (3.40) is ∆ij + µiµj

J , i, j = 1, ...,N , while m00 = J andm0j = mi0 = µi. Writing the expression (3.42) in matricial form it results

T =12qT Mq

where q is the vector of the lagrangian coordinates[

δ1 δ2 · · · δn

]and

M is

M =

J µ1 µ2 · · · µN

µ1 m11µ1µ2

J · · · µ1µ2J

µ2µ1µ2

J m22 · · · · · ·...

...... . . . µN−1µN

JµN

µNµ1J · · · µNµN−1

J mNN

In an analog way it is possible to obtain the term relative to the potentialenergy and the term due to the work performed by the motor. In particularorthogonality conditions can be expressed also in terms of the elastic forcesas

EI

∫ L

0

∂2φci(x)∂x2

∂2φcj(x)∂x2

= ∆ijkii i, j = 1, 2, ...


Then the potential energy term can be rewritten as follow

V =12

N∑

i=0

N∑

j=0

δiδjkii∆ij

with k00

V =12qT Kq

with K

K =

0 0 · · · 00 k11 · · · · · ·...

... . . . 00 · · · 0 kNN

while the energy term due to the motor torque is

W = τ(t)N∑

i=0

φ′ci(0)δi

and solving the Lagrangian

d

dt

∂L

∂qi− ∂L

∂qi=

∂W

∂qi

where L = T − V , neglecting the term relative to Coriolis forces, the equa-tions of motion are

Mq + Kq = fTτ(t)

where fT = [ 1 0 · · · 0 ] [Spong et al. 2006].

3.4 Pseudo-pinned approach

Let now consider the pseudo-pinned approach (Figure 3.3). Definingwp(x, t) the deflection in this reference frame, the problem is reformulatedas follows

EIwIVp (x, t) + ρ(wp(x, t) + xα(t)) = 0 (3.43)

τ(t)− Jα(t)− ν(t) = 0 (3.44)

3.4. Pseudo-pinned approach 37

Figure 3.3: Flexible arm in the pseudo-pinned reference frame

where

ν(t) = ρ

∫ L

0xwp(x, t)dx + J0w

′p(0, t) + MpLwp(L, t) + Jpw

′p(L, t) (3.45)

with the associated boundary conditions

wp(0, t) = 0EIwp

′′(0, t) = J0[α(t) + w′p(0, t)

]− τ(t)

EIwp′′(L, t) = −Jp

[α(t) + w′p(L, t)

]

EIwp′′′(L, t) = Mp [Lα(t) + wp(L, t)]

(3.46)

It is possible to compute the rotation angle solving (3.44) for α(t):

α(t) =τ(t)− ν(t)

J(3.47)


The substitution of the obtained result in the equation of flexibility (3.43)and in the boundary conditions (3.46) yields

EIwpIV (x, t) + ρwp(x, t) + ρx

τ(t)− ν(t)J

= 0 (3.48)

wp(0, t) = 0EIwp

′′(0, t) = J0

[τ(t)−ν(t)

J + w′p(0, t)]− τ(t)

EIwp′′(L, t) = −Jp

[τ(t)−ν(t)

J + w′p(L, t)]

EIwp′′′(L, t) = Mp

[L τ(t)−ν(t)

J + wp(L, t)]

(3.49)

Assuming wp(x, t) = φp(x)δ(t) and exploiting the solution derived from(3.30) it results

ν(t) = ν0δ(t) (3.50)

where

ν0 = ρ

∫ L

0xφp(x)dx + J0φ

′p(0) + MpLφp(L) + Jpφ

′p(L) (3.51)

Thus, the problem formulation in the spatial coordinates become

EIφIVp (x)− ρω2φp(x) + ρω2 ν0

Jx = 0 (3.52)

with boundary conditions

φp(0) = 0EIφ′′p(0) = J0ω2(ν0

J − φ′p(0))EIφ′′p(L) = −Jpω2(ν0

J − φ′p(L))EIφ′′′p (L) = Mpω2(Lν0 − φp(L))

(3.53)

Even in this representation, an ordinary differential equation, depending oneignevalue ω2, which solutions φp(x) are the eigenfunctions of the problem.It is also known that the eigenvalues must be the same of the pseudo-clampedapproach since in both cases the same physical system is represented.Analogously to the pseudo clamped approach the solution is

φp(x) = φp(x) + Fx (3.54)

3.4. Pseudo-pinned approach 39

where

φp(x) = A sin(βx) + B cos(βx) + C sinh(βx) + D cosh(βx)

Substituting (3.54) in (3.52) the value of F is obtained:

F =1J

ν0 (3.55)

Then, the substitution of (3.54) in (3.51) yields:

ν0 = ρ

∫ L

0x(φp(x)+Fx)dx+J0(φ′p(0)+F )+MpL(φp(L)+Fl)+Jp(φ′p(L)+F )

(3.56)Collecting F and applying (3.20),the equations (3.55) and (3.56) becomerespictively

ν0 = FJ

ν0 = FJ + ρ

∫ L

0xφp(x)dx + J0φ


′p(L)

from which immediately derives

ρ

∫ L

0xφp(x)dx + J0φ


′p(L) = 0 (3.57)

Hence, it is easy to see that F can assume any value. For simplicity itis possible to choose F = 0 and as a consequence ν0 = 0, ν(t) = 0 andφp(x) = φp(x) and (3.43) become

EIwpIV (x, t) + ρ(wp(x, t) + x

τ(t)J

) = 0

τ(t)− Jα(t) = 0 (3.58)


Now assuming a separated variable solution and following analogous stepsof the clamped approach, first the orthonormality conditions are derived

ρ

∫ L

0φpi(x)φpj(x)dx + J0φ

′(0)φ′pj(0) + Mpφpi(L)φpj(L) + Jpφ′pi(L)φ′pj(L) =

= ∆ij

EI

∫ L

0

∂2φpi(x)∂x2

∂2φpi(x)∂x2

dx = ∆ijω2i i, j = 1, 2, ...

to finally arrive to the dynamic equations of the pinned case

Mq + Kq = fTτ(t)

where M and K are diagonal matrices

M =

J 0 · · · 0

0 1 . . . ...... . . . . . . 00 · · · 0 1

K =

0 0 · · · 0

0 ω21

. . . ...... . . . . . . 00 · · · 0 ω2

N

and fT = [ 1 φ′1(0) · · · φ′N (0) ]. These can be written as

Jα(t) = τ(t)δi(t) + ω2

i δi(t) = φ′pi(0)τ(t) i = 1, . . . ,N(3.59)

Some interesting properties hold

1. during the free motion (τ(t) = 0) the system can be considered as asum of a rigid component α(t) and a set of harmonic oscillations δi(t)

2. the vibration modes are excited with weighted relevance according toφ′pi(0)

3. the flexural stiffness of the arm is expressed by the natural frequenciesωi of the system.

Two equivalent linear model of the same physical system have been derived.Since the physical characteristics of the system are the same in both ap-proaches, also the solutions of the characteristic equation are same. In fact

3.5. Friction model 41

it is easy to see that the characteristic equation is the same too. Moreoverthrough simple geometric consideration it is straightforward to obtain:

θhub(t) = α(t) +N∑

i=1

φ′pi(0)δi(t) (3.60)

φci(x) = φpi(x)− xφ′pi(0)

The model presented so far do not consider the dissipative effects of thesystem: the internal friction of the beam and the internal friction of themotor, the friction between the air-pad and the table and the air resistance.We will take into account these effects in the next sections, first buildinga friction model and then introducing the damping ratio ξ in the dynamicequations of the system before obtaining a state-space representation of thesystem. Note that the damping ratio ξ represents only the dissipation effectsdue to the internal frictions of the beam. However, it does not representthe damping of the overall structure, also due to other friction sources suchas the friction in the joint. As it will be clarified in the model identificationsection this interaction between motor friction and system damping, drivethe choice to design particular experiments to identify both damping andfriction model parameters.

3.5 Friction model

Two classical friction model [Olsson et al. 1998], used as base to obtain ourmodel, are recalled here.Stribeck Friction Model

F =

F (v) if |v| $= 0Fe if |Fe| < Fs, |v| = 0Fssgn(Fe) otherwise

where F is the friction force, F (v) is a function of the velocity, Fe is theexternal force applied to the system, Fs is the force due to static frictioneffect (stiction).Karnopp friction model define a deadzone to solve problems that mayoccurr in simulations, due to the hard task of identify exactly the zerovelocity point. Let be defined d the deadzone threshold, where the range


[−d, d] is the interval in which the velocity is considered to be zero. In otherwords, the velocity may change, being different from zero, but the outputof the block is maintained at zero within the deadzone. Introducing thisvariant the friction model become

F =

F (v) if |v| > dFe if |Fe| < Fs, |v| < dFssgn(Fe) otherwise

Choosing F (v) = Fcsgn(v), then, assuming that for values of velocitiesaround zero, a positive velocity indicates a positive Fe, and substitutingforce with torque and linear velocities with angular velocities, the frictionmodel results to be

τf =

τcsgn(θhub) if |θhub| > dτe if |τ | < τs, |θhub| < dτssgn(θhub) otherwise

(3.61)

or in a compact form, easier to be graphically represented

τf =

τcsgn(θhub) if |θhub| > dsat(τe)sgn(θhub) otherwise

wheresat(τe) =

|τe| if |τe| < τs

τs otherwise

where τf is the overall friction torque, τs the static friction, τc the Coulombfriction, d the velocity dead-zone.

Figure 3.4: Friction model

3.6. State space representation 43

3.6 State space representation

A state-space representation of the model is derived in this section startingfrom the dynamic equations (3.59) obtained by means of the pseudo-pinnedapproach. Including the damping term 2ξωiδi(t), the nonlinear effect of thesaturation τmin ≤ τ(t) ≤ τmax due to the limit on the torque supplied bythe motor, and the friction effect, it results

Jα(t) = τ(t)− τf (t)δi(t) + 2ξωiδi(t) + ω2

i δi(t) = φ′pi(0)(τ(t)− τf (t)) i = 1, . . . ,N

Solving the first equation for α(t) and the second equation for δi(t), anddefining the linearizing input u(t) = τ(t)− τf (t) it is obtained

α(t) = 1J u(t)

δi(t) = −ω2i δi(t)− 2ξωiδi(t) + φ′pi(0)u(t) i = 1, . . . ,N

Defining the state and the output vectors

χ =[

α δ1 · · · δn α δ1 · · · δn

]′

y =[

θhub wc(l, ·)]′

the system equations can be rewritten in the following state - space repre-sentation

χ(t) = Aχ(t) + Bu(t)y(t) = Cχ(t)

(3.62)

where

A =

0 0 0 1 0 · · · 0

0 0 · · · 0... 1 · · · 0

...... . . . ...

...... . . . ...

0 0 · · · 0 0 0 · · · 10 0 · · · 0 0 0 · · · 00 −ω2

1 · · · 0 0 −2ξ1ω1 · · · 0...

... . . . ......

... . . . ...0 0 · · · −ω2

n 0 0 · · · −2ξnωn


B =[

0 0 · · · 0 1J φ′p1(0) · · · φ′pn(0)

]

where the dimensions of the matrices, states and input depend on thenumber of the N considered vibration modes. In particular, A ∈R(2N+2)×(2N+2), x(t) ∈ R2N+2, B ∈ R(2N+2)×1, u(t) ∈ R1. Recalling theexpressions (3.41) and (3.60)

θhub(t) = α(t) +n∑

i=1

φ′pi(0)δi(t)

wc(l, t) =n∑

i=1

φci(l)δi(t)

It is easy to see that the matrices C e D of the model result to be

C =[

1 φ′p1(0) · · · φ′pn(0) 0 0 · · · 00 φc1(l) · · · φcn(l) 0 0 · · · 0

]D =

000...00

Th matrix C ∈ R2×(2N+2) while D ∈ R(N+2)×1.

Chapter 4

Identification

Contents3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 433.2 Inertia terms computation . . . . . . . . . . . . . . . 44

3.2.1 Motor-fork subsystem inertia . . . . . . . . . . . . . 443.2.2 Payload inertia . . . . . . . . . . . . . . . . . . . . . 50

3.3 Young modulus characterization . . . . . . . . . . . 533.4 Relevant modes identification . . . . . . . . . . . . . 543.5 Damping identification . . . . . . . . . . . . . . . . . 583.6 Friction identification . . . . . . . . . . . . . . . . . . 593.7 Model validation for control . . . . . . . . . . . . . . 60

3.7.1 Open loop validation . . . . . . . . . . . . . . . . . . 613.7.2 Single Input Single Output closed-loop validation . . 613.7.3 Multivariable control oriented validation experiments 62

3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 69

4.1 Introduction

The analytical modeling technique presented so far has been used to obtainthe system of equations which describe the dynamic behaviour of the arm.In this chapter the parameters of the model are identified. In particular themass of the payload mp, the beam length L, are easily measurable from theplant; the linear mass density of the beam ρ, the cross-section moment ofinertia I, the inertia moment of the payload Jp and the joint inertia momentJ0 are computed starting from the geometrical and physical characteristicsof the system, while the Young modulus E, the damping ratio ξ and the

46 Chapter 4. Identification

friction model parameters are identified on the base of experimental data,obtained with suitable experiments performed either on the single parts ofthe plant or on the overall plant.

4.2 Inertia terms computation

4.2.1 Motor-fork subsystem inertia

In the theoretical model the inertia term J0 represents the inertia of themotor but things are a bit different when considering our system: the motoris attached to the kevlar link by means of an aluminium fork having nonnegligible mass and dimensions. This additional part is not included in themodel description presented so far, nevertheless it can be easily consideredas an additional inertia term Jf such that results:

J0 = Jhub + Jf

where Jhub is provided in the datasheet of the motor. Its value is reportedin Table 4.1 together with the other parameters that can be immediatelyobtained through direct measures on the plant or that are given in technicaldatasheets. The fork has a non regular shape (Figure 4.1). F1, F3, F5, F6

Parameter ValueLink length L 0.387 [m]Link cross-section area A 6.45× 10−5 [m2]Link mass density ρlink 1400 [kg/m3]Aluminium mass density ρal 2710 [kg/m3]Motor inertia Jhub 1.41× 10−6 [kg ·m2]Payload mass Mp 0.163 [kg]Link cross-section inertia I 2.954× 10−11 [m4]

Table 4.1: Directly measurable plant parameters

and F7 are parallelepipeds, while F2 and F4 are prisms having as basea triangle rectangle. The fork is placed in a system of coordinates axes(X,Y ,Z). The Table 4.2 reports the measures in metres of the individualparts γ that compose the fork; aγ indicates the measures of the edges parallelto the X-axis, bγ indicates the edges parallel to the Y -axis and with cγ the

4.2. Inertia terms computation 47

Figure 4.1: Fork partitioning

ones parallel to the Z-axis. Then the volumes Vγ and the masses mγ can

γ aγ [m] bγ [m] cγ [m]F1 0.01 0.087 0.015F2 0.01 0.0395 0.015F3 0.01 0.056 0.055F4 0.01 0.0395 0.015F5 0.01 0.087 0.015F6 0.01 0.04 0.015F7 0.01 0.04 0.015

Table 4.2: Fork parts measures

be easily computed:

Vγ = aγbγcγ −πr2

1cγ

2for parts F1 and F6, where r1 = 0.005 [m] is the radius of the upper cavity,

Vγ = aγbγcγ −πr2

2cγ

2


for parts F5 and F7, where r2 = 0.003 [m] is the radius of the lower cavity,

Vγ = aγbγcγ − asbscs

for part F3, where as = 0.0035 [m], bs = 0.05 [m], cs = 0.051 [m] are thedimensions of the groove where the beam is attached to the fork, and

Vγ =aγbγcγ

2

for parts F2 and F4. Screws are not considered. For the masses it is used

mγ = ρalVγ

In Table 4.3 the mγ and Vγ computed values for each part are reported.Table 4.4 shows the coordinates (xγ , yγ , zγ) of the centres of mass and their

γ Vγ [m3] mγ [kg]F1 1.2461× 10−5 0.033769F2 2.9625× 10−6 0.0080284F3 2.1875× 10−5 0.059281F4 2.9625× 10−6 0.0080284F5 1.2838× 10−5 0.034791F6 5.7879× 10−6 0.014664F7 5.4699× 10−6 0.015685fork 6.2698× 10−5 0.1742

Table 4.3: Fork parts volumes and masses

distances rZγ from the fork rotation axis (Z-axis), computed through the

formula:rZγ =

√x2

γ + y2γ (4.1)

The baricentric inertia moment of the part γ around the axis parallel to thefork rotation axis is represented by the symbol Jγ . For the parallelepipedshape parts it is used:

Jγ =112

mγ(a2

γ + b2γ

)(4.2)

Parts F2 and F4 are triangular base prisms. The formula for inertia iscomputed as follows. A generic triangular base prisms of dimensions a, b,


γ xγ [m] yγ [m] zγ [m] rZγ [m]

F1 −0.005 0.0235 0.0775 0.024026F2 −0.005 0.080167 0.0750 0.080322F3 −0.005 0.0785 0.0425 0.078659F4 −0.005 0.08016 0.0100 0.080322F5 −0.005 0.0235 0.0075 0.024026F6 0.005 0 0.0775 0.005F7 0.005 0 0.0075 0.005

Table 4.4: Fork parts CoM coordinates and distance from rotation axis

c is placed in the reference frame (X,Y ,Z), such that the prism centre ofmass (CoM) coincides with the axes origin, the Z-axis is the solid rotationaxis and a rectangular face is parallel to the plane (X,Y ), while the otherrectangular face is parallel to the plane (X,Z) (Figure 4.2). The line be-

Figure 4.2: Part F2: reference frame for inertia computation

longing to the plane (Y ,Z) passing by the points A(2b3 , −c

3 ) and B(−b3 , 2c

3 )is z = − c

by + c3 .


The formula to compute the inertia of an homogeneous rigid body is

J =∫

Vρr(x, y, z)2dV

where ρ is the density and r(x, y, z) the distance from the rotation axis (inthis case Z, with r(x, y, z)2 = x2 + y2. Defining dV = dxdydz it results

Jtbp = ρ

∫

V(x2 + y2)dxdydz

where the subscript tbp stands for triangular base prism. To compute Jtbp,x is integrated between −a

2 and a2 , y between − b

3 and 2b3 , z between − c

3and the line z = − c

ay + c3

Jtbp = ρ

∫ − cb y+ c

3

− c3

∫ 2b3

− b3

∫ a2

−a2

(x2 + y2)dxdydz

integration on z yields

Jtbp = ρ

∫ 2b3

− b3

∫ a2

−a2

[(x2 + y2)z]−cb y+ c

3− c

3dxdy

= ρ

∫ 2b3

− b3

∫ a2

−a2

(x2 + y2)(−c

by +

2c

3

)dxdy

= ρ

∫ 2b3

− b3

∫ a2

−a2

(−c

bx2y − c

by3 +

2c

3x2 +

2c

3y2

)dxdy

while integration on x

Jtbp = ρ

∫ 2b3

− b3

[−c

b

x3

3y − c

bxy3 +

2c

3x3

3+

2c

3xy2

]a2

−a2

dy

= ρ

∫ 2b3

− b3

[− c

3b

(a3

8+

a3

8

)y − c

b

(a

2+

a

2

)y3

]dy+

+ ρ

∫ 2b3

− b3

[+

2c

9

(a3

8+

a3

8

)+

2c

3

(a

2+

a

2

)y2

]dy

= ρ

∫ 2b3

− b3

(−a3c

12by − ac

by3 +

a3c

18+

2ac

3y2

)dy


γ Jγ [Kg ·m2] Jγ [Kg ·m2]F1 2.1581× 10−5 4.1075× 10−5

F2 7.6281× 10−7 5.2559× 10−5

F3 1.5986× 10−5 3.8277× 10−4

F4 7.6281× 10−7 5.2559× 10−5

F5 2.2234× 10−5 4.2317× 10−5

F6 2.0774× 10−6 2.4439× 10−6

F7 2.2221× 10−6 2.6142× 10−6

Table 4.5: Moments of inertia of the fork parts

and integration on y

Jtbp = ρ

[−a3c

12b

y2

2− ac

b

y4

4+

a3c

18y +

2ac

3y3

3

] 2b3

− b3

= ρ

[−a3c

24b

(4b2

9− b2

9

)− ac

4b

(16b4

81− b4

81

)]+

+ ρ

[a3c

18

(2b

3+

b

3

)+

2ac

9

(8b3

27+

b3

27

)]

= ρ

(−a3c

24b

3b2

9− ac

4b

15b4

81+

a3c

18b +

2ac

99b3

27

)

= ρ

(−a3bc

2413− ab3c

4527

+a3bc

18+

2ab3c

913

)

= ρabc

(− 1

72a2 − 5

108b2 +

118

a2 +227

b2

)

= ρabc

(−1 + 4

72a2 +

−5 + 8108

b2

)

= ρabc

(124

a2 +136

b2

)

= ρabc

12

(12a2 +

13b2

)

Being ρabc2 = m it is possible to write

Jtbp =16m

(12a2 +

13b2

)(4.3)


Then the Huygens-Steiner theorem is applied in order to compute the inertiaJγ of each part refered to a rotation axis parallel to the baricentric axis,being distant from it of the distance r,

Jγ = Jγ + mγr2 (4.4)

Summing the moments of inertia of each part it is possible to computeanalytically the inertia moment of the fork Jfork, that will be finally addedto the the motor inertia Jhub to obtain the total inertia J0.

Jfork = 5.7634× 10−4 [kg ·m2]J0 = Jfork + Jhub = 5.7775× 10−4 [kg ·m2]

4.2.2 Payload inertia

Analogous steps are followed with regard to the payload moment of inertiacomputation. It has been divided in three parallelepipeds. The cylindricair-pad and the steel sphere are considered too (see Figure 4.3). The baseof the air-pad has a diameter da of 0.025 [m], the height is 0.013 [m] andthe mass ma of 0.0153 [Kg]. The sphere has a diameter ds of 0.012 [m] anda mass ms of 0.0071 [kg]. In Table 4.6 the measures in metres of the partsof the payload with parallelepiped shape are reported.The Table 4.7 shows masses and volumes of payload components. Both

γ aγ [m] bγ [m] cγ [m]P1 0.015 0.04 0.012P2 0.012 0.04 0.09P3 0.012 0.052 0.01

Table 4.6: Payload parts measures

γ Vγ [m3] mγ [kg]P1 7.2× 10−6 0.019512P2 4.32× 10−5 0.11707P3 6.24× 10−6 0.01691

Table 4.7: Payload parts volumes and masses


Figure 4.3: Payload partitioning

the coordinates w.r.t. the axes Xp e Yp of the centres of mass of the variousparts that compose the payload and the coordinates of the centres of massof the payload are reported in Table 4.8.

γ xγ [m] yγ [m]P1 0.0045 0.02P2 0.006 0.02P3 0.006 0.066

air-pad 0.006 0.08sphere 0.006 0.08

Table 4.8: CoM coordinates in (Xp,Yp,Zp)

The inertia moment of each part w.r.t. the axis passing through its CoM


is computed as follows

JP1 =112

m1(a21 + b2

1) = 2.9674× 10−6 [kg ·m2]

JP2 =112

m2(a22 + b2

2) = 1.7014× 10−5 [kg ·m2]

JP3 =112

m3(a23 + b2

3) = 4.0134× 10−6 [kg ·m2]

Jair =12ma(

da

2)2 = 1.1953× 10−6 [kg ·m2]

Jsph =25ms(

ds

2)2 = 1.0162× 10−7 [kg ·m2]

then, the baricentric inertia obtained, resulting.

Jp = 1.1606× 10−4 [kg ·m2]

The total system inertia J is given by

J = J0 + Jp + ρ

∫ D1+L

D1

x2dx + Mp(D1 + L + D2)2 =

J0 + Jp + ρ

((D1 + L)3

3− D3

1

3

)+ Mp(D1 + L + D2)2

where ρ = 0.0903 [kg/m] is the linear density of the link, product betweenthe link cross-section area A and its density ρlink. It results J = 0.0493[kg ·m2]. The values of D1, D2, L (see Figure 2.2) used in the computationare reported in Table 4.9 below.

Parameter Value [m]L 0.387D1 0.1065D2 0.0321

Table 4.9: Arm elements measures

4.3. Young modulus characterization 55

Applied Mass [Kg] Displacement [m] Young Modulus E [GPa]0.005 0.023 6.68670.010 0.029 6.40810.015 0.034 6.40810.020 0.039 6.40810.025 0.044 6.4081

Table 4.10: Young modulus characterization

4.3 Young modulus characterization

The identification of the Young modulus E requires a beam test. In factthe theoretical estimation of this parameter may be quite different from thereal one because the composite material link has peculiar characteristicsstrongly dependent on the manufacturing process. The composite materialsare also subject to ageing that produces link stiffness loss. The beam testis performed as follows: the link is clamped at one end and increasing masspayloads are hanged up at the free end of the link. Starting from therelationship between the applied force F and the displacement ∆ we canfind the Young modulus by the following equation derived by structuralmechanic [Gurtin 1981]

E =Fl3

3I∆where F = mg is the force acting on the end point of the beam; m and gare respectively the total applied mass and the gravity acceleration, withm = m1 + m2 where m1 is the weight of the applied mass reported inTable 4.10, while m2 = 0.019 [kg] is the weight of a thin plate, insertedin the cavity of the link, in correspondence of the link surface where theair-pad support is screwed, to avoid the beam break in that point. Theobtained values, summarized in Table 4.10, highlight a quite linear elasticitymodel of the beam in the range of displacements where the tests have beenperformed satisfying the linearity assumptions introduced in the previouschapter. Nevertheless different experiments made on the overall systemshow the presence of a nonlinear behavior of the system for very smalland very large values of the beam deflections. Different sweep signals withincreasing amplitude are applied to the motor as torque references. As canbe seen in Figure 4.4 the wider is the produced displacement, the lower is


!

Figure 4.4: Resonance frequency shift

the frequency at which the resonance peak occurs. In conclusion the mostsuitable choice seemed to be E = 6.408 [GPa] since it better describes thevibrations dynamic behavior in an intermidiate range of displacement.

4.4 Relevant modes identification

Through the knowledge of the parameters identified so far it is possibleto compute the natural frequencies of the system. In particular, findingthe first three solutions β1, β2, β3 of the characteristic equation (3.36) andapplying the relation

ωi = β2i

√EI

ρ, i = 1, ...

that results from (3.34), it is obtained ω1 = 45.65 [rad · s−1], ω2 = 114.47[rad · s−1] and ω2 = 266.47 [rad · s−1] corresponding to f1 = 7.26 [Hz] thefirst one, f2 = 18.21 [Hz] the second one and f2 = 42.41 [Hz] the last

4.4. Relevant modes identification 57

0 5 10 15 20 25 30 35

10−5

10−4

10−3

10−2

10−1

Frequency [Hz]

Am

pli

tud

e

Spectrum estimate

No Airpad, Central

Airpad, Central

Airpad, Base

No Airpad, Base

Figure 4.5: Spectral analysis

one. Anyway it is important to look for experimental evidences to checkthe reliability of these theoretical values and to understand which modesof vibration are actually excited, within a certain range of solicitation fre-quencies. To this aim, the Matlab System Identification Toolbox has beenused, to analyze and compare the results of the experimental tests with theanalytical ones. In the frequency response of a linear system with oscillatingresponse the amplitude of the output signal is expected to be maximum incorrespondence of the natural frequencies. This is the so called resonancepeak. For the test performed to find out the relevant modes of vibration, asinuosoidal sweep signal has been applied as motor torque reference. Thesweep frequency increases linearly during the time, passing from 1 to 50[Hz] in 30 seconds. The oscillations were measured by means of the straingauges. The spectral analysis of the strain-gauges measurements is shownin Figure 4.5. The terms "Central" and "Base" reported in the legendof the figure indicate respectively "Signal acquired by the strain-gaugesmounted in the central position of the beam" and "Signal acquired by thestrain-gauges mounted in the position of the beam close to the motor fork".Looking at the figure, it is possible to recognize the first mode of vibration,


exactly at the same frequency resulting from the analytical computation,while we see another relevant component at about 30 [Hz], detected by the"Base" strain-gauge, that is very different from the second analytical fre-quency. To better understand this phenomenon a second experiment hasbeen performed without the airpad at the end of the arm to excite possibletorsional vibration modes. From the spectral analysis reported in Figure4.5 it is possible to observe that the presence of the air pad reduces theentity of the peak at 30 [Hz]. This experiments highlights that the observedpeak is due to a torsional mode that is not described by the model thattakes in account only flexural vibrations. It is also possible to see that the"Base" strain-gauge detect a peak between 16 and 18 [Hz], very close to f2.Hence the spectral analysis confirms the reliability of the model. Anywaythe energetic contents of the second mode is very low and can be neglected.In view of these experimental results we decided to consider only the firstflexural mode of vibration. Since only one vibration mode is assumed, themeasure of the displacement of the tip, is obtained finding its relation withthe link deformation. To this aim the following identification experiment isperformed. With the motor clamped, and given an initial displacement tothe tip, the tip oscillation measures in [mm], easily obtained starting fromlaser measures, and the deformation measures, obtained through the straingauge, are acquired (Figure 4.6). Hence it is possible to plot one signalversus the other (Figure fig:lookupesten) and observing that the relation isof the form

y = θ1x + θ2

the parameters θ1 = 7.506 and θ2 = 0 are identified by means of a linearregression.

4.5 Damping identification

To identify the damping ξ included in the model equation (3.62) the log-arithmic decay method is used [Rao 2003], which validity holds for linearsystems having a damping ξ < 1. A short description of this technique isreported here to introduce the identification process. Let be considered aone d.o.f. damped linear system in free oscillation, governed by the followingdynamic equation

x(t) + 2ξω0x(t) + ω20x(t) = 0 (4.5)

4.5. Damping identification 59

0 2 4 6 8 10−40

−30

−20

−10

0

10

20

30

40

50

Time [s]

Sig

na

l a

mp

litu

de

Figure 4.6: Oscillations for strain gauge look-up table identification. Redline: deflection measure provided by the laser acquisition [mm]. Blue line:deflection measure provided by the strain gauge [Volts]

−5 0 5−40

−30

−20

−10

0

10

20

30

40

Strain Gauge measure [V]

Defl

ecti

on

[m

m]

Figure 4.7: Strain gauge look-up table. Blue dots: experimental data.Green line: least square fitting


where x(t) is the trajectory of the oscillating body, function of the time,ω0 is the natural frequency and ξ the damping ratio. If ξ < 1, the generalintegral of the (4.5) can be written in the form

x(t) = Xe−ξω0t cos(ωdt + ϕ) (4.6)

where X represents the initial condition and ωd = ω0

√1− ξ2. The solution

(4.6) owns infinite maxima and minima alternatively positive and negative;the time interval intercurring between two subsequent maxima or minimais equal to the period T = 2π

ωd. Defining η

η = lnx(t∗i )

x(t∗i + T )= ln

Xe−ξω0t∗i cos(ωdt∗i + ϕ)Xe−ξω0(t∗i +T ) cos[ωd(t∗i + T ) + ϕ]

(4.7)

where x(t∗i ) and x(t∗i + T ) are subsequent peaks of x(t). Due to the period-icity of the cosine function the equation (4.7) become

η = lne−ξω0t∗i

e−ξω0(t∗i +T )= ln(eξω0T ) = ξω0T = ξω0

2π

ωd=

2πξ√1− ξ2

Assuming ξ ' 1, it is possible to write

η = lnx(t∗i )

x(t∗i + T )≈ 2πξ (4.8)

In this way a relation between the the damping ratio ξ and available datais provided. In the real case nonlinearities may occur, thus the value of ηmay change depending on the pairs of considered contiguous peaks. Henceit is convenient to compute a mean value. In order to exploit this techniquea simple experiment is designed: applying an initial displacement to thetip of the beam, constraining the motor, the beam starts to oscillate atabout 1[Hz]. The base strain gauge signal is shown in Figure 4.8. Throughthe (4.8) it is possible to derive the logarithmic decay η for every peakspairs and derive a mean value of the damping ξ of the system. It resultsξ = 0.034.

4.6. Friction identification 61

0 5 10 15−8

−6

−4

−2

0

2

4

6

8

Time [s]

Am

plitu

de [

V]

Figure 4.8: Arm free oscillation for damping identification

4.6 Friction identification

To obtain the friction model of the plant, described in (3.61), three pa-rameters should be identified: static friction τs, the velocity threshold dand coulomb static friction τc. The first parameter is identified applyingincreasing torque till a joint motion is produced. The needed torque valueto move the motor is the value assigned to τs. The value d is usually verysmall and difficult to estimate. We have arbitrarily chosen it equal to 0.001[degrees/s]. The procedure to identify τc follows the sequent steps. Firstan experiment similar to the one made for the identification of the damp-ing factor ξ, but without clamping the motor, is carried out. Based on itan equivalent damping ratio ς of the whole plant is identified. Then weperform several simulations for different increasing values of τc. For eachsimulation, ς is computed by means of the logarithmic decay applied to thedeflection responses obtained in simulation. Then the damping values areplotted on the y-axis vs the correspondent τc values on the x-axis. Then,interpolating the graph, a function γ(τc) is obtained. Finally, starting from


Parameter Value Parameter Valueξ 0.034 E 6.4081[GPa]I 2.95E−11 [m4] J0 0.0016 [kgm2]Jp 1.73E−5 [kgm2] L 0.387 [m]τs 0.065 [Nm] ρ 0.09 [kg/m]τc 0.0055 [Nm] mp 0.155 [kg]

τmax 0.214 [Nm] τmin −0.214 [Nm]

Table 4.11: Model parameters

the knowledge of ς, we can find an estimation τc s.t. γ(τc) = ς.

4.7 Model validation for control

Once the single components of the model have been identified, it is necessaryto validate the whole model, useful for the synthesis of feedback control laws.It is well known that this requires a good model in a particular range offrequencies while it is not necessary to have a very precise model at low orhigh frequencies. The first validation experiments have been done in openloop in order to verify the repeatability of the experiments. The closedloop validation has been done using two control strategies. The first oneis a single input (position error) single output (motor torque) controller,synthesized in the frequency domain, while the second one is a multi input(position error and deflection) single output (motor torque) controller basedon an LQ control law complemented with an observer and an integral action.

4.7.1 Open loop validation

Well suited bang-bang torque profiles, designed on the base of the systemmodel without friction, are applied several times to the motor in order toobtain hub rotation of 20, 45, 60, 90 and 120 respectively. Table 4.12summarizes the obtained results. In particular, the asymptotic value of therotation obtained in simulation with the full model (also with the frictionmodel), the mean value of the accomplished experimental rotation and itsstandard deviation (SD) are reported. Looking at the results of these exper-imental tests, it is important to notice that the standard deviations obtained

4.7. Model validation for control 63

Designed Model Actual Hub Rotation Actual HubHub Rotation Hub rotation Mean Value Rotation SD

20 18.84 10.49 0.2545 42.70 29.15 0.4560 58.25 51.60 0.4490 85.00 69.00 0.46120 112.00 67.62 2.01

Table 4.12: Repeatability experiments analysis

are very small while the mean value of the real rotation differs significantlyfrom the simulations; moreover a nonlinear behavior with respect to theamplitude of the hub rotation is highlighted. This information suggests tosynthesize a regulator with a high gain in order to obtain a small regulationasymptotic error.

4.7.2 Single Input Single Output closed-loop validation

A standard position control scheme is implemented using only the hub po-sition measurement as feedback signal. The controller, synthesized in thefrequency domain, is given by

R(s) =5(s + 1)

(s + 5)(s + 300)

The validation has been performed applying sinusoidal position referenceswith different frequencies and amplitudes. In particular Figures 4.9 - 4.14show the system response to a sinusoidal position reference signal, having40 amplitude at 0.2 Hz and 0.4 Hz frequency respectively. Theblue line is obtained via simulation while the green line is the experimentalresponse. Looking at the figures, we can observed that the model is capableto capture with good approximation the dynamics of the rigid mode as wellas the flexible one. Moreover the torque required by the regulator in thereal plant and in the simulated one are comparable, with only very smalldeviations due to the uncertainty of the friction model.


0 1 2 3 4 540

60

80

100

120

140

θhub

Time [s]

Figure 4.9: SISO validation test. Black dotted line: sinusoidal positionreference at 0.2 Hz. θhub(t) response. Green dashed line: simulation. Blueline: experimental behavior.

0 1 2 3 4 5−0.2

−0.1

0

0.1

0.2Actual Torque [Nm]

Time [s]

Figure 4.10: SISO validation test. Sinusoidal position reference at 0.2 Hz.Applied torque τ(t). Green dashed line: simulation. Blue line: experimentalbehavior.


0 1 2 3 4 5−40

−20

0

20

40Deflection [mm]

Time [s]

Figure 4.11: SISO validation test. Sinusoidal position reference at 0.2 Hz.Deflection response w(L, t). Green dashed line: simulation. Blue line: ex-perimental behavior.

0 1 2 3 4 50

50

100

150

θhub

Time [s]

Figure 4.12: SISO validation test. Black dotted line: sinusoidal positionreference at 0.4 Hz. θhub(t) response. Green dashed line: simulation. Blueline: experimental behavior.


0 1 2 3 4 5−0.4

−0.2

0

0.2


Time [s]

Figure 4.13: SISO validation test. Sinusoidal position reference at 0.4 Hz.Applied torque τ(t). Green dashed line: simulation. Blue line: experimentalbehavior.

0 1 2 3 4 5−100

−50

0

50

100Deflection [mm]

Time [s]

Figure 4.14: SISO validation test. Sinusoidal position reference at 0.4 Hz.Deflection response w(L, t). Green dashed line: simulation. Blue line: ex-perimental behavior.


4.7.3 Multivariable control oriented validation experiments

A second closed-loop validation is performed using also the strain gaugemeasure. To this aim a regulator based on the Linear Quadratic (LQ) opti-mal control law has been adopted. In Figure 4.15 the overall control scheme

!

Figure 4.15: LQ control scheme with state observer and integral action

is reported. In view of the model gain uncertainties stressed by the openloop validation, an integral action on the position error has been introducedin order to guarantee an asymptotic zero-error regulation. Starting from themodel (3.62), but neglecting the presence of the friction, i.e. u(t) = τ(t)the following system has been derived:

.χ(t) = Aχ(t) + Bτ(t) + Bθθ

0hub(t), t ≥ 0, χ(0) = χ0

whereχ =

[α δ1 α δ1 v

]′

A =[

A 0−C1 0

], B =

[B0

], Bθ =

[01

]

v is the state of the integrator, C1 is the first line of the matrix C andθ0hub(t) is the reference signal for θhub(t). The cost function to be minimized

isJ (χ0 , τ(·)) =

∫ ∞

0

(χ (λ)′Qχ (λ) + τ(λ)′Rτ(λ)

)dλ


where

Q =

50 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 0.05

, R = 1000

so that the control law isτ(t) = −Kχ(t)

whereK = R−1B′P

and P is the only positive definite solution of steady Riccati equation

0 = PA + A′P + Q− PBR−1B′P .

Moreover since the plant is equipped only with two transducers that furnisha measure proportional to the motor rotation and a measure proportionalto the end-effector displacement from the neutral axis, the state of systemis not completely accessible. Then the following state observer is used:

.χ(t) = Aχ(t) + Bτ(t)− L (y(t)− Cχ(t))

where χ(t) is an estimation of the state of the system χ(t) and L is chosensuch that the eigenvalues of the matrix A + LC are less faster than theslower eigenvalues of the matrix A − BK. In particular the vector of theeigenvalues of A + LC is equal to r =

[−5 −5 −10 −10

]. Then the

overall regulator is given by.χ(t) = Aχ(t) + Bτ(t)− L (y(t)− Cχ(t))v(t) = C1χ(t)− θ0

hub(t)

τ(t) = −K

[χ(t)v(t)

]

In Figures 4.17 4.16 4.18 a step response of 120 on the hub is reported.In Table 4.13 the mean value and the standard deviation of the asymptoticvalue of the hub position for several experiments are reported. From themean value it is possible to notice that in spite of the presence of the integralaction, a small error after 5 seconds remains. This is due to the small


0 1 2 3 4 5−50

0

50

100

150

θhub

Time [s]

Figure 4.16: MIMO validation test. θhub(t) response. Black dotted line:reference. Green dashed line: simulation. Blue line: experimental behavior.

0 1 2 3 4 5−0.1

−0.05

0

0.05

0.1


Time [s]

Figure 4.17: MIMO validation test. Applied torque τ(t). Green dashedline: simulation. Blue line: experimental behavior.


0 1 2 3 4 5−40

−20

0

20Deflection [mm]

Time [s]

Figure 4.18: MIMO validation test. Deflections response w(L, t). Greendashed line: simulation. Blue line: experimental behavior.

Designed Model Actual Hub Rotation Actual HubHub Rotation Hub rotation Mean Value Rotation SD

120 120.00 121.1 0.3

Table 4.13: Closed-loop validation variability

value of the element (5,5) of the matrix Q that penalizes the integral state.However, this small error is negligible while an increase of the penalty onthe integral action increases the arm oscillations on the arm. Moreover, theSD is reduced with respect to the same experiment made in open-loop (seeTable 4.12).

4.8 Conclusion

All parameters necessary to build a simulator for the plant have been identi-fied. Their values are summarized in Table 4.11. Such obtained model hasbeen succesfully validated through suitable experiments. Such obtainedmodel can be used both for multivariable control tuning and it is accurateenough to be employed in the computation of the control law of a Model

4.8. Conclusion 71

Predictive Control algorithm.

Chapter 5

Control

Contents4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 714.2 PD regulation . . . . . . . . . . . . . . . . . . . . . . 724.3 Theoretical background . . . . . . . . . . . . . . . . . 784.4 MPC implementation . . . . . . . . . . . . . . . . . . 804.5 Kalman filter design . . . . . . . . . . . . . . . . . . . 864.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . 884.7 Experimental results . . . . . . . . . . . . . . . . . . 97

4.7.1 Software Real-Time . . . . . . . . . . . . . . . . . . 974.7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.1 Introduction

This chapter gives experimental evidence of the capabilities of the ModelPredictive Control (MPC) strategy applied in the field of flexible arm con-trol. In particular to prove the effectiveness of this technique, its perfor-mances are compared with Linear Quadratic Regulator (LQR). In the firstsection a Proportional Derivative controller (PD) is presented to introducethe control problem. Then, the LQR and the MPC are presented in theirdiscrete formulations and a well-known theoretical result is recalled to gen-eralize the validity of the presented results to every possible tuning of theregulators. The following section presents different tests performed in simu-lation, obtained varying some parameters of the regulators. It foreshadowsthe expected experimental results highlighting the advantage of the MPCover the LQR control. Finally an ad hoc platform for the online implemen-tation of the MPC is designed and the obtained experimental results areshown and discussed.

74 Chapter 5. Control

5.2 PD regulation

The Proportional Derivative (PD) controller is one of the most used algo-rithms in the industrial world. It requires a simple model and it can beeasily tuned also with automatic procedures, resulting suitable for a hugevariety of applications. For this reason it can be considered as starting pointeven in the cases in which it revealed unable to reach the control aims. Tothis concern the application of this algorithm to the flexible arm model isconsidered in this section, to give an example of the control problem thatpushes the investigations of alternative control strategies.

An industrial realization of the PD control with feedback on the motorposition θhub(t) is implemented in the Simulink model. The control lawimplemented by the regulator is

τ(t) = Kp(θ0hub(t)− θhub(t))−Kd

d

dtθhub(t)

where Kp and Kd are the controllers gains of the proportional action and thederivative action respectively, θ0

hub(t) is the position reference, and τ(t) thetorque generated by the controller. Two different controllers are synthesizedin order to obtain two closed loop systems with different bandwidth. For thesimulations a 180 [Degrees] step reference is applied. The simulated systemresponse of the slowest closed loop system is shown in Figures 5.1, 5.2. It ischaracterized by small deflections but the settling time of the motor positionis strongly penalized. To the contrary, the response of the fast closed loopsystem reported in Figures 5.4, 5.5 presents fast motor positioning but largepersisting oscillations of the end-effector.

These results highlight that the fast end-effector positioning task haveto cope with contrasting performance requirements that cannot be achievedusing a colocated PD control. On the other hand as the literature resultsput in evidence a feedback on the end effector position would lead to anon-minimum phase system, difficult to control. Moreover additional hard-ware is required to obtain a reliable measure of the end effector position,with possible limitation to the bandwidth, for example if a camera is usedto provide the feedback. These considerations suggest to investigate theeffectiveness of multivariable techniques.

5.2. PD regulation 75

0 1 2 3 4 50

50

100

150

θhub

Time [s]

Figure 5.1: Rest-to-rest manoeuvre: slow PD regulation. Kp = 0.0006,Kd = 0.001.

0 1 2 3 4 5−30

−20

−10

0

10Deflection [mm]

Time [s]



0 1 2 3 4 5−0.05

0

0.05

0.1


Time [s]


0 1 2 3 4 50

50

100

150

200

θhub

Time [s]

Figure 5.4: Rest-to-rest manoeuvre: fast PD regulation. Kp = 0.05, Kd =0.03.

5.2. PD regulation 77

0 1 2 3 4 5−100

−50

0

50

100Deflection [mm]

Time [s]


0 1 2 3 4 5−0.4

−0.2

0

0.2


Time [s]



5.3 Theoretical background

Let be considered the following discrete-time system

x(k + 1) = Ax(k) + Bu(k)y(k) = Cx(k)

(5.1)

with x ∈ Rn, u ∈ Rm, u ∈ Rq and A, B, C matrices with suitable dimen-sions. The LQR control problem is defined as

arg minu(k)

J(x(k), k, ·) (5.2)

with

J(x(k), k, ·) =∞∑

k=0

x(k)′Qx(k) + u(k)′Ru(k) Q ≥ 0, R > 0

where Q = Q′ ≥ 0, R = R′ > 0 are the weighting matrices having suitabledimensions, to be chosen as design parameters. The resulting feedbak law,solution of (5.3) is

u(k) = −Kx(k)

where

K = (R + B′PB)−1B′PA

and P is the solution of the steady Riccati equation

P = A′PA + Q−A′PB(R + B′PB)−1BPA (5.3)

Considering the same dynamic system described by the equations (5.1), thestage cost for the MPC is

J(x(k),u(·), k) =N−1∑

i=0

(‖x′(k + i)‖2Q + ‖u(k + i)‖2R

)+ ‖x(k + N)‖2S (5.4)

where Q = Q′ ≥ 0, R = R′ > 0, S = S′ ≥ 0 are suitable dimensionsmatrices. The positive integer N is usually defined prediction horizon. Thecontrol problem solution consists in the determination of the sequence of

5.3. Theoretical background 79

control variables U(k)=u(k), u(k + 1), ..., u(k + N − 1) that satisfies

minU(k)

J(x(k),u(·), k).

It is given by

u0(k + i) = −K(i)x(k + i) i = 0, 1, ...,N − 1

whereK(i) = (R + B′P (i + 1)B)−1B′P (i + 1)A

with

P (i) = Q+A′P (i+1)A−A′P (i+1)B(R+B′P (i+1)B)−1B′P (i+1)A (5.5)

where P (i + N) = S. This is a closed loop solution since the value of thecontrol variable u(k + i) depends on the state x(k + i). It can be obtainedfrom the Hamilton− Jacobi−Bellman equation.Receding Horizon strategy. A time invariant control law can be ob-tained applying the Receding Horizon (RH) principle which works as fol-lows: at instant k the optimal sequence U(k) on the finite horizon [k, k+N ]is computed, but only the first element of the vector U(k) is applied as in-put to the system. At the next instant k + 1 the solution is recomputedover the prediction horizon [k + 1, k + N + 1] and so on for every instantsk during the runtime. Through the RH approach a time-invariant controllaw always based on the current state x(k) is obtained.

u(k) = −K(0)x(k)

K(0) = (R + B′P (1)B)−1B′P (1)′A

For this unconstrained case, it is easy to see that choosing the final weightS = P , the equation (5.5), coincides with (5.3), hence the MPC controllaw is the same of the LQR one on infinite horizon. Otherwise, choosingany arbitrary final weight S, but choosing the LQR state weight matrix asQ = Q + P (1) − P (0), the two control laws are still equivalent. In facttaking the equation (5.5)

P (i) = Q + A′P (i + 1)A−A′P (i + 1)B(R + B′P (i + 1)B)−1B′P (i + 1)A


and adding and substracting P (i + 1) at the left term it results:

P (i + 1)− P (i + 1) + P (i) = Q + A′P (i + 1)A−+ A′P (i + 1)B(R + B′P (i + 1)B)−1B′P (i + 1)A

P (i + 1) = Q + P (i + 1)− P (i) + A′P (i + 1)A−+ A′P (i + 1)B(R + B′P (i + 1)B)−1B′P (i + 1)A

and takingQ = Q + P (i + 1)− P (i)

it results

P (i + 1) =Q + A′P (i + 1)A−A′P (i + 1)B(R + B′P (i + 1)B)−1B′P (i + 1)A

that is the so called fake Riccati equation. These results lead respectivelyto the following considerations.

1. Choosing opportunely the final weight S, the MPC reaches the sameperformances of the LQR synthesized using the same weighting ma-trices Q and R. Moreover the MPC own the capability to handleconstraints. In other words MPC is expected to improve LQR perfor-mance.

2. The MPC can be also used as tool to generate non diagonal weightingmatrices for the LQR a result can also be viewed as a way to choosea non diagonal matrix Q for the LQR which otherwise would be verydifficult to findin order to obtain better performances.

5.4 MPC implementation

It is presented at this time an open loop solution that allows to take intoaccount explicit constraints on both input and state variables. Recallinghere the discrete-time system (5.1)

x(k + 1) = Ax(k) + u(k)y(k) = Cx(k)

5.4. MPC implementation 81

The expression of the impulse response of the system, according to theLagrangian equation is

x(k + i) = Aix(k) +i−1∑

i=0

Ai−j−1Bu(k + j) (5.6)

and defining

A =

AA2

...AN−1

AN

,B =

B 0 0 · · · 0 0AB B 0 · · · 0 0· · · · · · · · · · · · · · · · · ·

AN−2B AN−3B AN−4B · · · B 0AN−1B AN−2B AN−3B · · · AB B

X(k) =

x(k + 1)x(k + 2)

...x(K + N − 1)

x(K + N)

,U(k) =

u(k)u(k + 1)

...u(k + N − 2)u(k + N − 1)

the expression (5.6) can be written in matricial form as

X(k) = Ax(k) + BU(k) (5.7)

The optimization problem to solve is

minU(k)

J(x(k),u(·), k) = minU(k)

J(x(k),u(·), k)) (5.8)

where J(x(k),u(·), k) is the cost function (5.4) written in the compact way

J(x(k),u(·), k) = X ′(k)QX(k) + U ′(k)RU(k) (5.9)

with

Q =

Q 0 · · · 0 00 Q · · · 0 0

· · · · · · . . . · · · · · ·0 · · · 0 Q 00 · · · 0 0 S

,R =

R 0 · · · 0 00 R · · · 0 0

· · · · · · . . . · · · · · ·0 · · · 0 R 00 · · · 0 0 R


with suitable dimensions. Substituting system dynamic equation (5.7) in(5.9) yields

J(x(k),u(·), k) = (Ax(k) + BU(k))′Q(Ax(k) + BU(k)) + U ′(k)RU(k)(5.10)

= x′(k)A′QAx(k) + 2x′(k)A′QBU(k) + U ′(k)(B′QB +R)U(k) (5.11)

The control sequence U(k) can be found in an analytical way. The equa-tion (5.11) is a positive definite quadratic form, hence its minimum can becomputed deriving on U(k) and imposing the derivative to zero. In thisway it is possible to obtain the vector of the control variables, within theconsidered prediction horizon

U(k) = −(B′QB +R)−1B′QAx(k) = −Kx(k).

This can be viewed as an open loop solution since it depends on the pre-diction of the future state value based on the current state. This openloop approach allows to include constraints in the MPC formulation. Inparticular it is possible to compute the value of the control sequence U(k)using a suitable optimization routine that solves the quadratic programmingproblem:

x∗ = arg minx

0.5x′Hx + f ′x (5.12)

subject toAvx = bv, linear equality constraintsLvx ≤ kv, linear inequality constraints

lv ≤ x ≤ uv, bound constraints

where

x = U(k) H = 2B′QB +R f = 2x(k)′A′QB

Lv =[−BB

]kv =

[xbnd +Ax(k)xbnd −Ax(k)

]

lv = umin uv = umax

where xbnd is a vector in which the i-th element, where i = k · j specify thevalue of the bound of the j-th state at the instant k. Lv and kv are obtained

5.5. Kalman filter design 83

expliciting the following constraints on the state prediction at every samplewithin the horizon [

−X(k)X(k)

]≤

[xbnd

xbnd

]

that using (5.7) results

Ax(k) + BU(k) ≥ −xbnd

Ax(k) + BU(k) ≤ xbnd

that is

−BU(k) ≤ xbnd +Ax(k)BU(k) ≤ xbnd −Ax(k)

which is exactlyLvU(k) ≤ kv.

5.5 Kalman filter design

Since the system state is not completely accessible a Kalman filter is de-signed. A short introduction to the Kalman prediction and filtering is pre-sented here. In case of discrete time systems it is possible to develop theKalman theory to develop the predictor, if the estimation of the state x(k)depends on the output y and u till the instant k− 1, or the filter, when theestimation of the state x(k) depends on the output y and u till the instantk.

Let be considered the system

x(k + 1) = Ax(k) + Bu(k) + vx(k)y(k) = Cx(k) + vy(k)

(5.13)

in which v =[

vx

vy

]is a white noise gaussian noise with expected value

null and covariance matrix V i.e.

E[v(k1)v′(k2)] = V δ(k1 − k2)


. where

V =[

Q 00 R

], Q ∈ Rn,n, R ∈ Rp,p, Q ≥ 0, R > 0

Moreover it is assumed that the initial state x(0) is a gaussian randomvariable with expected value x0 and covariance matrix

E[(x(0)− x0)(x(0)− x0)′] = P0 ≥ 0

and that x(0), v(k) and vy(k) are uncorrelated i.e.

E[x(0)v′(k)] = 0, ∀k ≥ 0

Defining x(k|k−1) and x(k|k−1) the estimation of the state depending onthe knowledge of input and output respectively at the instant k and k − 1,it is possible to derive the following equation of the discrete kalman filter

x(k + 1|k + 1) = Ax(k|k) + Bu(k) + L(k + 1)[y(k + 1)− C(Ax(k|k) + Bu(k))]

where

L(k) = P (k|k − 1)C ′[CP (k|k − 1)C ′ + R

]−1

P (k|k − 1) = AP (k − 1|k − 1)A′ + Q

P (k|k) = P (k|k − 1)− L(k)CP (k|k − 1)

P (0|0) = P0

with P (k|k) is the covariance of the estimation error e(k|k) = x(k)− x(k|k).Convergence of L(k) to L and P (k|k− 1) to ¯P can be proved under certainconditions. The MATLAB command kalman is used to compute the filter,choosing suitable weights.

5.6 Simulations

The simulations presented in this section are performed in order to obtaindifferent comparisons between LQR and MPC. They are based on the modelpresented in Chapter 3 described by the system equations (3.62), identifiedand validated in the previous chapter, discretized and complemented with

5.6. Simulations 85

the Kalman filter. For the tests a 180 [Degrees] rest-to-rest manoeuvre isperformed and both the input and the outputs of the system are plotted.The controllers parameters are reported in Table 5.1. Qdiag, Rdiag are thevectors whose elements constitute the diagonal of the diagonal matrices ofthe weights, Q and R respectively, which are the same for both the MPCand the LQR. Ts is the chosen sampling time.

Comparison 1 emphasizes the equivalency of the LQR and the lin-ear unconstrained MPC algorithm tuned as summarized in Table 5.1. InFigure 5.7 it is possible to observe that the plot lines coincide exactly. Itis important to notice that the ideal torque computed by the regulatorsis different from the actual applied torque due to the presence of thesaturation. This effect can be avoided by the constrained MPC, as shownin Figures 5.17, 5.18, relative to the Comparison 3.

Comparison 2 highlights the performance improvement introduced bythe constrained MPC defined in Table 5.2. MPC capabilities to reduceoscillations without compromising the rest-to-rest time are shown, even ifthe constraints are not always satisfied (see Figures 5.8, 5.10). The con-straints violation can be due both to the presence of friction and to thesampling time. In order to clarify the observed phenomenon three furthersimulations are performed varying the sampling time, eliminating the fric-tion effects from the model. In this way it is possible to observe the systemresponse in the nominal linear case. The results are reported in Figures5.11-5.15. Figure 5.15 highlights the effect of the discretization: the fig-ure shows clearly that the deflection stays within the specified bounds incorrespondence of the sampling instants, while the constraints are violatedbetween them. The discretization effect can be reduced decreasing the sam-pling time till the violation disappears. Concerning friction effect, Figure5.14 shows that the constraint violation for positive values of deflection wascompletely due to the non linear effect of the friction. In fact, if the sim-ulation is performed using the linear model the violation disappears. Thisobservation constitutes a stimulus to develop MPC algorithms that considerexplicitly the presence of disturbances. This topic is presented in the nextchapter.

Comparison 3 shows that the presence of saturations can affect crit-ically the LQR performance. In the same conditions the MPC, handlingexplicitly the constraints on the controlled variable, mantaining high per-formance standards. This can be observed very clearly looking at Figures


Parameters ValuesRdiag [1000]Qdiag [100, 1, 1, 1]Ts 0.03Kalman filter ValuesRdiag [1, 1]Qdiag [100, 100, 100, 100]MPC Parameters ValuesNc 40N 40S P

Table 5.1: Comparison 1: controllers parameters

5.16, 5.19. Parameters adopted are in Table 5.3.

Comparison 4 differs from Comparison 3 since state constraints areincluded in the MPC to reduce deflections. From this comparison it ispossible to observe that even reducing significantly the oscillation duringthe manoeuvre the rest-to-rest time for the MPC is lower than the LQRone (see Figures 5.20, 5.22). Controller parameters can be found in Table5.4

Comparison 5 stress the effect of the saturations. The weights aretuned in order to require a more aggressive control action. LQR controlperformance become completely unsatisfactory, while MPC is still able toproduce the desired behavior, that is lower reduced rest-to-rest time. Thiscan be observed very clearly looking at Figures 5.23, 5.25. Parameters arereported in Table 5.5.

In the next section experimental results will prove the effectiveness ofthe adopted strategy in even if applied on line on the real plant.

5.6. Simulations 87

0 2 4−200

−100

0

100

θhub

0 2 4−0.5

0

0.5

1Ideal Torque [Nm]

0 2 4−0.2

0

0.2

0.4


Time [s]0 2 4

−100

−50

0

50Deflection [mm]

Time [s]

Figure 5.7: Comparison 1. Simulation comparison: θhub. Blue dashed line:unconstrained MPC. Red line: LQR.

Parameters ValuesRdiag [1000]Qdiag [100, 1, 1, 1]Ts 0.03Kalman filter ValuesRdiag [1, 1]Qdiag [100, 100, 100, 100]MPC Parameters ValuesNc 40N 40S PConstraint 41 −0.214 < τ(t) < 0.214Constraint 42 −0.002 < χ2(t) < 0.002

(−0.028 < w(L, t) < 0.028)



0 1 2 3 4 5−200

−150

−100

−50

0

50

θhub

Time [s]

Figure 5.8: Comparison 2. Simulation comparison: θhub. Black dotted line:step reference. Blue dashed line: constrained MPC. Red line: LQR.

0 1 2 3 4 5−0.4

−0.2

0

0.2


Time [s]

Figure 5.9: Comparison 2. Simulation comparison: applied torque. Bluedashed line: constrained MPC. Red line: LQR.

5.6. Simulations 89

0 1 2 3 4 5−100

−50

0

50Deflection [mm]

Time [s]

Figure 5.10: Comparison 2. Simulation comparison: deflections. Bluedashed line: constrained MPC. Red line: LQR. Black dashed line: MPCconstraints.

0 1 2 3 4 5−200

−150

−100

−50

0

50

θhub

Time [s]

Figure 5.11: Sampling time variation. Constrained MPC: motor position.Black dotted line: step reference. Blue line: Ts = 0.03 [s]. Green dashdotted line: Ts = 0.015 [s]. Red line: Ts = 0.005 [s].


0 1 2 3 4 5−0.4

−0.2

0

0.2

0.4Ideal Torque [Nm]

Time [s]

Figure 5.12: Sampling time variation. Constrained MPC: ideal torque. Bluedashed line: Ts = 0.03 [s]. Green dash dotted line: Ts = 0.015 [s]. Red line:Ts = 0.005 [s]. Black dashed line: MPC constraints.

0 1 2 3 4 5−0.4

−0.2

0

0.2


Time [s]

Figure 5.13: Sampling time variation. Constrained MPC: applied torque.Blue dashed line: Ts = 0.03 [s]. Green dash dotted line: Ts = 0.015 [s].Red line: Ts = 0.005 [s].

5.6. Simulations 91

0 1 2 3 4 5−40

−20

0

20

40Deflection [mm]

Time [s]

Figure 5.14: Sampling time variation. Constrained MPC: deflection. Bluedashed line: Ts = 0.03 [s]. Green dash dotted line: Ts = 0.015 [s]. Red line:Ts = 0.005 [s]. Black dashed line: MPC constraints.


0.15 0.2 0.25 0.3

−35

−30

−25

Deflection [mm]

Time [s]

Figure 5.15: Sampling time variation. Constrained MPC: deflection zoom.Blue dashed line: Ts = 0.03 [s]. Green dash dotted line: Ts = 0.015 [s].Red line: Ts = 0.005 [s]. Black dashed line: MPC constraints.

Parameters ValuesRdiag [100]Qdiag [100, 1, 1, 1]Ts 0.03Kalman filter ValuesRdiag [1, 1]Qdiag [100, 100, 100, 100]MPC Parameters ValuesNc 40N 40S PConstraint 41 −0.214 < τ(t) < 0.214


5.6. Simulations 93

0 1 2 3 4 5−200

−150

−100

−50

0

50

θhub

Time [s]

Figure 5.16: Comparison 3. Simulation comparison: θhub. Black dottedline: step reference. Blue dashed line: constrained MPC. Red line: LQR.

0 1 2 3 4 5−1

0

1

2

3Ideal Torque [Nm]

Time [s]

Figure 5.17: Comparison 3. Simulation comparison: ideal torque. Bluedashed line: constrained MPC. Red line: LQR. Black dashed line: MPCconstraints.


0 1 2 3 4 5−0.4

−0.2

0

0.2


Time [s]

Figure 5.18: Comparison 3. Simulation comparison: applied torque. Bluedashed line: constrained MPC. Red line: LQR.

0 1 2 3 4 5−100

−50

0

50

100

150Deflection [mm]

Time [s]

Figure 5.19: Comparison 3. Simulation comparison: deflections. Bluedashed line: constrained MPC. Red line: LQR.

5.6. Simulations 95

Parameters ValuesRdiag [100]Qdiag [100, 1, 1, 1]Ts 0.03Kalman filter ValuesRdiag [1, 1]Qdiag [100, 100, 100, 100]MPC Parameters ValuesNc 40N 40S PConstraint 41 −0.214 < τ(t) < 0.214Constraint 42 −0.003 < χ2(t) < 0.003

(−0.043 < w(L, t) < 0.043)


0 1 2 3 4 5−200

−150

−100

−50

0

50

θhub

Time [s]



0 1 2 3 4 5−0.4

−0.2

0

0.2


Time [s]

Figure 5.21: Comparison 4. Simulation comparison: deflections. Bluedashed line: constrained MPC. Red line: LQR.

0 1 2 3 4 5−100

−50

0

50

100Deflection [mm]

Time [s]

Figure 5.22: Comparison 4. Simulation comparison: applied torque. Bluedashed line: constrained MPC. Red line: LQR. Black dashed line: MPCconstraints.

5.6. Simulations 97

Parameters ValuesRdiag [1]Qdiag [100, 1, 1, 1, 0.05]Ts 0.03Kalman filter ValuesRdiag [1, 1]Qdiag [100, 100, 100, 100]MPC Parameters ValuesNc 40N 40S PConstraint 41 −0.004 < χ2(t) < 0.004

(−0.057 < w(L, t) < 0.057)Constraint 42 −0.214 < τ(t) < 0.214


0 1 2 3 4 5−200

−100

0

100

θhub

Time [s]



0 1 2 3 4 5−0.4

−0.2

0

0.2


Time [s]

Figure 5.24: Comparison 5. Simulation comparison: applied torque. Bluedashed line: constrained MPC. Red line: LQR. Black dashed line: MPCconstraints.

0 1 2 3 4 5−200

−100

0

100

200Deflection [mm]

Time [s]

Figure 5.25: Comparison 5. Simulation comparison: deflections. Bluedashed line: constrained MPC. Red line: LQR. Black dashed line: MPCconstraints.

5.7. Experimental results 99

5.7 Experimental results

It is important to point out that for the aims of this thesis, the on-lineimplementation of the MPC algorithm must be identical to the simula-tion. The commercial toolboxes are often based on suboptimal solutions ofthe QP problem (i.e. MPT toolbox) or integrate peculiarities that makethem very difficult to compare with the theoretical results (i.e. MPC Tool-box). Anyway some trials has been performed using the MPC toolbox. Itis compatible with the Real Time Windows Target (RTWT) and hence astraightforward application on the plant is allowed. Nevertheless severaltrials evidenced instability of the system (MATLAB/Windows crash), justadding constraints on the states or increasing the control horizon. Exclud-ing this possibility another strategy has been attempted to keep MATLABReal Time Windows Target. In fact, it is possible to implement any kindof controller compatible with RTWT, writing a MATLAB C-S function.In MATLAB exist many tools (i.e. MATLAB Legacy Tool, MATLAB S-function Builder) that make easier to the user the conversion of existingC code in C-S function blocks. The problem of this strategy is that theavailable C source code of the Quadratic Programming routine need to linkmathematical libraries that can not be linked by the Watcom C compilerused by MATLAB Real Time Target during the application building pro-cess. After more research, a suitable solution has been found and developed.It is described in the next section. Its main advantage consists in the pos-sibility to keep MATLAB as computational engine for the implementationof the MPC algorithm.

5.7.1 Software Real-Time

The developed application for the on-line implementation of the MPC algo-rithm is a C++ application interfaced with MATLAB through the MAT-LAB Engine libraries. In runtime phase the following tasks are executed:

• connection with MATLAB;

• reading of the input analog channels of the I/O board to get themeasures from the plant;

• state reconstruction through the kalman filter;


• execution in the MATLAB environment, at each sampling instant,of the quadratic programming routine used to find the MPC controlsequence;

• result retrievement from MATLAB workspace;

• writing of the output analog channel of the I/O board;

It is easy to figure out that this program structure is particularly suitable toexplore the capabilities of the MPC algorithm, since all the computationalpart is delegated to MATLAB. The QP routine used is reliable, freeware,and fast and the C-MEX free source code is available. The software has beendeveloped and compiled in Visual Studio 2005 and run on a Win32 archi-tecture. The critical issue in the implementation of this code for a real-timeapplication is represented by the execution time required by the QuadraticProgramming routine that solves at each sampling time the optimizationproblem. Before proceeding with the experiments it has been verified thatwith a sampling time of 0.03 [s] it is possible to respect the computationalconstraints within a prediction horizon of 40 steps. For these experimentssome modifications in the hardware are introduced: a faster PC mountingthe I/O board RTI-DAC4 PCI by Inteco has been considered more suitablein this phase due to the increasing amount of the computational burden.

5.7.2 Results

The experimental responses are obtained performing on the real plant thesame tests presented in Section 5.6, from Comparison 1 to Comparison 4reported in Figures 5.26-5.34. The effectiveness of the MPC shown in thesimulation results is essentially confirmed by these experimental results. Itis important to remark the following:

• friction model uncertainty yields consistent performance degradationin term of settling time, if small weights on the control variable arechosen (see Figure 5.26);

• the LQR turns out to be more sensitive to the presence of saturationthan in the simulation tests. This fact underlines the benefit dueto the possibility given by the MPC to introduce constraints on thecontrol variable (see Figure 5.29);


• the constraints are not always satisfied due to model uncertainty. Any-way MPC prove its actual effectiveness, being able to visibly reducethe oscillations, also in faster manoeuvres (see Figures 5.28, 5.34);

• at the end of the manoeuvre persisting small entity oscillations areinduced by non zero motor torque, due to model uncertainty.

Finally an additional test has been performed reducing the manoeuvre am-plitude (see Figures 5.35-5.37), using the same controllers parameters ofComparison 4. In this case the performance degradation due to the actua-tor saturation is not evident, but it is much more evident the capability ofthe MPC to reduce the deflection amplitude.

0 1 2 3 4 5

0

50

100

150

θhub

Time [s]

Figure 5.26: Comparison 1, 2. Experimental comparison: θhub. Blackdotted line: step reference. Blue dashed line: constrained MPC. Greenline: unconstrained MPC. Red line: LQR.


0 1 2 3 4 5−0.4

−0.2

0

0.2


Time [s]

Figure 5.27: Comparison 1, 2. Experimental comparison: applied torque.Blue dashed line: constrained MPC. Green line: unconstrained MPC. Redline: LQR. Black dashed line: constraints

0 1 2 3 4 5−100

−50

0

50

100Deflection [mm]

Time [s]

Figure 5.28: Comparison 1, 2. Experimental comparison: deflections. Bluedashed line: constrained MPC. Green line: unconstrained MPC. Red line:LQR. Black dashed line: MPC constraints.


0 1 2 3 4 5−100

0

100

200

300

θhub

Time [s]

Figure 5.29: Comparison 3. Experimental comparison: θhub. Black dottedline: step reference. Blue dashed line: constrained MPC. Red line: LQR.

0 1 2 3 4 5−0.4

−0.2

0

0.2


Time [s]

Figure 5.30: Comparison 3. Experimental comparison: deflections. Bluedashed line: constrained MPC. Red line: LQR. Black dashed line: con-straints


0 1 2 3 4 5−100

−50

0

50

100Deflection [mm]

Time [s]

Figure 5.31: Comparison 3. Experimental comparison: applied torque.Blue dashed line: constrained MPC. Red line: LQR. Black dashed line:MPC constraints.

0 1 2 3 4 5−100

0

100

200

300

θhub

Time [s]

Figure 5.32: Comparison 4. Experimental comparison: θhub. Black dottedline: step reference. Blue dashed line: constrained MPC. Red line: LQR.


0 1 2 3 4 5−0.4

−0.2

0

0.2


Time [s]

Figure 5.33: Comparison 4. Experimental comparison: deflections. Bluedashed line: constrained MPC. Red line: LQR. Black dashed line: con-straints.

0 1 2 3 4 5−100

−50

0

50

100Deflection [mm]

Time [s]

Figure 5.34: Comparison 4. Experimental comparison: applied torque.Blue dashed line: constrained MPC. Red line: LQR. Black dashed line:MPC constraints.


0 1 2 3 4 5−50

0

50

100

θhub

Time [s]

Figure 5.35: 90 Degrees manoeuvre. Experimental comparison: appliedtorque. Black dotted line: step reference. Blue dashed line: constrainedMPC. Red line: LQR.

0 1 2 3 4 5−0.4

−0.2

0

0.2


Time [s]

Figure 5.36: 90 Degrees manoeuvre. Experimental comparison: appliedtorque. Blue dashed line: constrained MPC. Red line: LQR. Black dashedline: MPC constraints.


0 1 2 3 4 5−100

−50

0

50

100Deflection [mm]

Time [s]

Figure 5.37: 90 Degrees manoeuvre. Experimental comparison: appliedtorque. Blue dashed line: constrained MPC. Red line: LQR. Black dashedline: MPC constraints.

Chapter 6

Stochastic MPC

Contents5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1035.2 Stochastic MPC formulation . . . . . . . . . . . . . . 1055.3 Deterministic reformulation of MPC . . . . . . . . . 1075.4 Stochastic MPC with guaranteed stability . . . . . 1105.5 Numerical implementation . . . . . . . . . . . . . . . 1135.6 A simulation example . . . . . . . . . . . . . . . . . . 115

5.6.1 Control design parameters . . . . . . . . . . . . . . . 1155.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 1205.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.8.1 Notation, basic definitions and available results . . . 1205.8.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.1 Introduction

In recent years, many robust Model Predictive Control (MPC) meth-ods for dynamic systems subject to bounded noise have been de-veloped and many algorithms are nowadays available guaranteeingfundamental properties of the resulting closed-loop system, such asthe Input-to-State Stability (ISS), see e.g. [Magni & Scattolini 2007],[Limon et al. 2009]. These algorithms are usually based either on an open-loop approach, where the effect of noise is projected over the prediction hori-zon [Michalska & Mayne 1993], [Chisci et al. 2001], [Limon et al. 2002],[Mayne et al. 2005], [Pin et al. 2008] or on a closed-loop one, com-puted through the solution of a suitable min-max optimization prob-lem [Scokaert & Mayne 1998], [Magni et al. 2003], [Fontes & Magni 2003],

110 Chapter 6. Stochastic MPC

[Magni et al. 2006], [Lazar et al. 2008], [Raimondo et al. 2009]. In bothcases the resultant control law is very conservative, taking into accounta worst-case scenario. In order to reduce conservativeness, as well asto consider probabilistic and possibly unbounded disturbances, some re-cent papers have reformulated the MPC problem into a stochastic frame-work, see [Primbs & Sung 2009] and the papers quoted there. Amongthem, two fundamental contributions for linear systems can be found in[Yan & Bitmead 2005] and [Primbs & Sung 2009] for additive and multi-plicative noises, respectively. Specifically, in [Yan & Bitmead 2005] a con-strained optimization problem is solved by resorting to an open-loop controllaw. The considered probabilistic constraints on the state are reformulatedby exploiting the gaussian distribution of the prediction error. However,stability results can not be achieved due to the unboundedness of the noisedistribution. In [Primbs & Sung 2009], constraints on the future expectedvalues of linear and quadratic combinations of the state and of the controlinput over the considered prediction horizon are included in the optimiza-tion problem. Then, a SemiDefinite Programming (SDP) reformulation ofthe original problem is provided and stability is proven in probability terms.

In this presented approach, the system under control is assumed to bedescribed by a linear model subject to an additive stochastic bounded noise.The considered MPC algorithm consists in minimizing a quadratic cost un-der probabilistic constraints on the input and state variables. In order toreformulate the optimization problem in terms of a SDP one, these con-straints are first transformed into a deterministic form by means of theChebyshev-Cantelli inequality, as in [Locatelli et al. 1983]. Then, some ad-ditional stability constraints are included in the problem formulation and aconstraints relaxation procedure is defined to guarantee that, for boundeddisturbances, the resulting closed-loop system is globally Input to StateStable (ISS) and the feasibility problem has always a solution. Finally, theMPC algorithm is tested in simulation on the flexible arm model. Theachieved results are discussed to witness the potentialities of the proposedapproach.

The chapter is organized as follows. In Section 6.2 the MPC problem isdefined in the stochastic framework, while in Section 6.3 it is reformulated asan equivalent deterministic one. In Section 6.4 the MPC control algorithm ismodified in order to guarantee ISS for limited disturbances. The numericalimplementation is described in Section 6.5 and the simulation example is

6.2. Stochastic MPC formulation 111

reported in Section 6.6. Some conclusions close the paper. The adoptednotation, together with some useful definitions and results already availablefrom the literature are reported in the Appendix, which also contains theproofs of the main results of this chapter.

6.2 Stochastic MPC formulation

Consider the following discrete-time linear system

x(k + 1) = Ax(k) + Bu(k) + Fw(k), k ≥ t, x(t) = x (6.1)

where x(k) ∈ Rn is the state, u(k) ∈ Rm is the input and w(k) ∈ Rp

is a zero-mean white noise with variance W . Perfect state information isassumed, together with the stabilizability of the pair (A,B). Since the noisew is stochastic, the state turns out to be a stochastic process with mean x(k)and covariance matrix X(k). Moreover, assuming that the input variablewill be computed with a feedback control law, it is a stochastic processas well. For these reasons constraints on the state and input variablesare imposed in a probabilistic sense, i.e. we allow a (small) probability ofviolating constraints. In other words, at any time instant k, we considerprobabilistic constraints of the form

P(bTr x(k + j) ≥ xmax

r ) ≤ px(r), j > 0, r = 1, . . . ,nr

P(cTs u(k + j) ≥ umax

s ) ≤ pu(s), j ≥ 0, s = 1, . . . ,ns (6.2)

where P(ϕ) is the probability of ϕ, br and cs are constant vectors,xmax

r ,umaxs are the considered bounds for the state and control variables

and px, pu are design parameters. In view of (6.2), the parameters br, cs,xmax

r , umaxs define polytopes denoted by X and U respectively.

In the proposed MPC algorithm, the cost function to be minimized at timet with respect to the sequence of inputs u(t) . . . u(t + N − 1) (denoted byu[t,t+N−1]) is

J(x,u[t,t+N−1]) := E

[t+N−1∑

i=t

(xT (i)Qx(i) +

uT (i)Ru(i))

+ xT (t + N)Sx(t + N)

](6.3)


where Q, R and S are positive definite, symmetric matrices of appropriatesize and the positive integer N is the adopted prediction horizon. Theminimization of (6.3) must be performed subject to the constraints (6.2)and to the additional terminal constraint

P(x(t + N) /∈ Xf ) ≤ pxf (6.4)

where Xf is a set containing the origin and pxf is a design parameter.

The control variables over the prediction horizon are assumed to havethe following expression

u(k) = u(k) + K(k)(x(k)− x(k)

)k = t, . . . , t + N − 1 (6.5)

therefore, they are composed by an open-loop part, i.e. the term u(i),typical of deterministic MPC formulations, and by a closed-loop onewhich depends on the gain K(i), as in classical robust MPC, see e.g.[Mayne et al. 2000]. Note that, at the initial time k = t, the state is knownso that x(t) = x(t) and matrix K(t) can be set to zero. A control law withthe structure of (6.5) has already been considered in [Primbs & Sung 2009].

To sum up, by defining

u[t,t+N−1] = [u(t), . . . ,u(t + N − 1)]

andK[t,t+N−1] = [K(t), . . . ,K(t + N − 1)]

the stochastic optimization problem is:

Problem P 1 Minimize the performance index (6.3) with respect tou[t,t+N−1] and K[t,t+N−1] subject to the system dynamics (6.1), to (6.5)and to the probabilistic constraints (6.2), (6.4).

Once all the elements of the optimization problem have been stated,i.e. the cost function to be minimized, the constraints and the structureof the control law, at least conceptually it is possible to compute at anytime instant t the optimal control sequence uo

[t,t+N−1]. Then, according tothe Receding Horizon (RH) principle, only the first element of this vector,i.e. uo(t), is used and the whole procedure is repeated at the next timeinstant. In so doing, one implicitly defines the stochastic MPC control

6.3. Deterministic reformulation of MPC 113

law. As a matter of fact, some further elaboration is required to transformthe stochastic optimization problem with probabilistic constraints into adeterministic and more tractable one.

6.3 Deterministic reformulation of MPC

By resorting to standard arguments, see e.g. [Yan & Bitmead 2005], themean of the state along the prediction horizon evolves according to

x(k + 1) = Ax(k) + Bu(k), k ≥ t, x(t) = x (6.6)

while the state covariance obeys the following expression

X(k + 1) = (A + BK(k))X(k)(A + BK(k))T ++FWF T , k ≥ t, X(t) = 0 (6.7)

Then, the cost (6.3) can be written in the following form

J(x,u[t,t+N−1]) = Jm(x,u[t,t+N−1]) + Jv(K[t,t+N−1]) (6.8)

where

Jm(x,u[t,t+N−1]) =t+N−1∑

i=t

(xT (i)Qx(i) + uT (i)Ru(i)

)+

+xT (t + N)Sx(t + N) (6.9)

and

Jv(K[t,t+N−1]) =

=t+N−1∑

i=t

(trQX(i)+ trKT (i)RK(i)X(i)

)+

+trSX(t + N) (6.10)

The probabilistic constraints (6.2) can be transformed into deter-ministic ones by means of the following well-known result, see e.g.[Grimmett & Stirzaker 2001]:


Theorem 1 (Cantelli’s inequality) Let y be a (scalar) random variablewith mean y and variance σ2

y. Then for every R + α ≥ 0 it holds that

P(y ≥ y + α) ≤σ2

y

σ2y + α2

Now, consider a constraint of the type (6.2) (for simplicity, we drop thetime index, the indices r, s and make reference to state constraints only,since input constraints can be treated in the same way)

P(bT x ≥ xmax) ≤ p (6.11)

Given δx ≥ 0 such that

bT x ≤ xmax − δx (6.12)

by Cantelli’s inequality, it holds that

P(bT x ≥ xmax) ≤ P(bT x ≥ bT x + δx) ≤ bT Xb

bT Xb + δx2

Therefore, if1− p

pbT Xb ≤ δx2 (6.13)

then (6.11) is satisfied. It follows that the original probabilistic constraint(6.11) can be replaced by the two deterministic constraints (6.12), (6.13)involving the mean and the covariance of x. By applying this procedure toall the constraints (6.2), the original stochastic optimization problem canbe completely transformed into a deterministic one, which can be solvedby standard methods. Note also that, by combining (6.13) and (6.12), oneobtains

bT x ≤ xmax −√

bT Xb

√1− p

p(6.14)

Constraint (6.14) is very similar to the restricted constraints considered inseveral works in the deterministic framework see e.g. [Chisci et al. 2001],[Limon et al. 2002], [Pin et al. 2008]. In fact the evolution of the mean xdepends on the open loop term u(k) (see (6.5)) and is obtained with thenominal dynamics (see (6.6)), while the term that depends on the variancetakes into account the effect of the disturbance along the optimization hori-

6.4. Stochastic MPC with guaranteed stability 115

zon (see (6.7)). The main advantage of this stochastic approach is thatthe restriction of the constraints can be optimized on-line by choosing thematrices K(i), i = t, ..., t + N − 1.

Although the Cantelli’s inequality allows us to transform the originalprobabilistic constraints (6.2), (6.4) into a deterministic one, still (6.14) isdifficult to handle in view of its nonlinearity with respect to X. For thisreason, and in order to place the optimization problem into a linear setting,it is advisable to resort to a linearization procedure. To this end, lettingδx = εxmax, ε ∈ [0, 1], be the linearization point, standard computationsallow one to reformulate (6.14) as follows

bT x ≤ (1− 0.5ε)xmax − 1− p

2εxmaxpbT Xb (6.15)

where ε is an additional tuning knob available in the algorithm implemen-tation.

Remark 1 Note that, when the considered probability in (6.11) is p = 1,constraint (6.14) is linear and can be directly used in the algorithm imple-mentation instead of its linearized form (6.15).

The original optimization problem P1 can now be reformulated as fol-lows:

Problem P 2 Minimize the performance index (6.8)-(6.10) with respectto u[t,t+N−1] and K[t,t+N−1] subject to the system dynamics (6.6), (6.7) andto the probabilistic constraints (6.2), (6.4) reformulated in a deterministicform according to (6.15).

Although the deterministic Problem P2 can now be solved with stan-dard optimization techniques, the resulting MPC control law does not guar-antee stability. Then, stability constraints must be introduced.

6.4 Stochastic MPC with guaranteed stability

In deterministic MPC, the closed-loop stability can be proven by takingthe optimal cost as a Lyapunov function and by properly choosing the


terminal penalty, i.e. the matrix S, and the terminal region Xf , see e.g.[Mayne et al. 2000]. Specifically, letting KLQ be the solution of the infinitehorizon linear quadratic (LQ) control problem for the nominal system (w =0) with state and input matrices Q and R, in (6.3) take S = P where P isthe solution of the Lyapunov equation

ATc PAc − P = −Qc (6.16)

where Ac = A + BKLQ and Qc = Q + KTLQRKLQ. Moreover, choose the

terminal region Xf so that

KLQx ∈ U , ∀x ∈ Xf

Acx ∈ Xf , ∀x ∈ Xf(6.17)

Notably, a procedure for the computation of the terminal region Xf canbe found in [Kolmanovsky & Gilbert 1998]. In the stochastic frameworkconsidered in this paper, the above choices do not automatically solve thefeasibility and stability problems, so that additional constraints must beincluded. To this end, define the optimal average cost

V (x(t)) := Jm(x(t),uo[t,t+N−1]) (6.18)

where uo is the mean of the optimal input uo and consider the additionalconstraints

∆V (t) := V (x(t))− V (x(t− 1)) ≤≤ −λQ

min|x(t− 1)|2 + |x(t− 1)|C ′1|Fw(t− 1)|+

+ C ′2|Fw(t− 1)|2, ∀t (6.19)

andV (x(t)) ≤ λP

max|x(t)|2, ∀t (6.20)

where C′1, C

′2 are computable positive constants (see equations (6.40) and

(6.41) in the Appendix), λQmin is the minimum eigenvalue of the matrix Q

and λPmax is the maximum eigenvalue of the matrix P .

Inequality (6.19) forces a bound on the variation of the optimal averagecost V (x) along the trajectories of the closed-loop system, while (6.20) isan additional stability constraint.

Assuming that the design parameters S and Xf satisfy (6.16) and (6.17),

6.4. Stochastic MPC with guaranteed stability 117

the MPC problem P2 can now be reformulated as follows.

Problem P 3 Minimize the performance index (6.8)-(6.10) with respectto u[t,t+N−1] and K[t,t+N−1] subject to the system dynamics (6.6), (6.7), tothe probabilistic constraints (6.2), (6.4) reformulated according to (6.15),and to the stability constraints (6.19), (6.20).

In the stochastic formulation considered here it can happen that at timet + 1 the system does not satisfy the constraints (6.15), (6.19), (6.20) eventhough a feasible control sequence exists at time t. Therefore, we proposeto resort to a constraint relaxation method: if at an instant t problem P3becomes infeasible, we gradually raise the probability of violating the stateand control constraints, until feasibility is recovered, as described by thefollowing algorithm.

Algorithm 1

step -1 At the initial time k = 0 define the multiplicative parameter µ > 1;

step 0 at any time instant, set px(r), r = 1, ...,nr and pu(s), s = 1, ...,ns

equal to the original problem data;

step 1 solve problem P3. If the problem is feasible, set ExitFlag=0 andgoto step 9, otherwise goto step 2;

step 2 define a new optimization problem P3′ by removing from problem

P3 the terminal constraint (6.4). If P3′ is feasible, set ExitFlag=1

and goto step 9, otherwise goto step 3;

step 3 set px(r) = min1,µpx(r), r = 1, ...,nr, pu(s) = min1,µpu(s),s = 1, ...,ns and pmin = min

r=1,...,nr,s=1,...,ns

px(r), pu(s). If pmin < 1

goto step 4, otherwise goto step 5;

step 4 if P3′ is feasible with the probabilities px(r), r = 1, ...,nr, pu(s),

s = 1, ...,ns computed at the previous step, set ExitFlag=2 and gotostep 9, otherwise goto step 3;

step 5 from P3′ define a new optimization problem P3

′′ by substitutingthe constraints (6.15) with the constraints (6.14) on the mean with


δx = 0 (recall that pmin = 1 at this stage of the algorithm and Remark1). If P3

′′ is feasible, set ExitFlag=3 and goto step 9, otherwise gotostep 6;

step 6 define a new optimization problem P3′′′ by removing from problem

P3′′ all the state constraints. If P3

′′′ is feasible, set ExitFlag=4 andgoto step 9, otherwise goto step 7;

step 7 define a new optimization problem P3iv by removing from problemP3

′′′ all the constraints on the control mean u(t + i), i > 0. If P3iv

is feasible, set ExitFlag=5 and goto step 9, otherwise goto step 8;

step 8 define a new optimization problem P3v by removing from problemP3iv the constraint on u(t). Set ExitFlag=6 and goto step 9;

step 9 Solve the resulting optimization problem, compute the optimal con-trol sequences uo

[t,t+N−1] and Ko[t,t+N−1] and stop.

Once the optimal control sequences have been computed with Algorithm1, the closed-loop control law

u(k) = κMPC(x(k),V (x(k − 1))) (6.21)

is obtained by applying the Receding Horizon principle. Note that thiscontrol law is a function of V (x(k − 1)) in view of the constraint (6.19).

Concerning Algorithm 1, some remarks are in order. First, the stabilityconstraints (6.19), (6.20) are always maintained in the problem formula-tion, while in step 8 all the state and control constraints are completelyremoved. Second, in step 8 of the algorithm it is implicitly assumed thatthe corresponding optimization problem is feasible. Concerning this crucialpoint, the pair u[t,t+N−1], K[t,t+N−1] is called feasible for Algorithm 1 if itis feasible for problem P3v.

Lemma 1 If there exists a feasible pair u[t,t+N−1], K[t,t+N−1] at time t,then there exist a feasible pair u[t+1,t+N ], K[t+1,t+N ] at time t + 1.

It is now possible to state the main stability result, under the followingassumption.

6.5. Numerical implementation 119

Assumption 1 The noise acting on the system is bounded:

|w| < wmax (6.22)

Theorem 2 Under Assumption 1, the closed-loop system (6.1), (6.21) isInput-to-State Stable.

6.5 Numerical implementation

In this section the control algorithm is formulated as a SDP. To this end,first note that the mean and the variance of the state at time t = k, . . . , k+Ndepend on the optimization variables u(i), i = k, . . . , k + N − 1, and K(i),i = k + 1, . . . , k + N − 1. Specifically, the dependence of the mean on u(i)is linear, as shown in (6.6); while the variance is governed by equation (6.7)which is nonlinear with respect to K(i). As shown in [Primbs & Sung 2009],it is possible to reformulate the nonlinear equality (6.7) with a linear in-equality and to resort to Schur complements. Since X(i) > 0 ∀i and defin-ing the new variable G(i) := K(i)X(i) we obtain the constraint (∗ denotestranspositions)

X(i + 1) ∗ ∗

(AX(i) + BG(i))T X(i) ∗(FW )T 0 W

≥ 0 (6.23)

Note that now also the covariance matrices X(i) must be considered as opti-mization variables. Finally, the input constraints depend on the covariancematrix of the input, denoted by U , which clearly obeys the relation

U(i) = K(i)X(i)KT (i) (6.24)

Thus we can pass to Schur complements and obtain[

U(i) G(i)GT (i) X(i)

]≥ 0 (6.25)

The cost function can be treated exactly as shown in [Primbs & Sung 2009],thus we just report the final results. The cost function (6.8) can be replaced


by the following (linear) expression

Jp(x(t), Γ[t,t+N ]) =t+N−1∑

i=t

Tr(MΓ(i)) + Tr(SΓ(t + N)) (6.26)

where M := diag(Q,R) and the variables Γ(i) are symmetric matrices sub-ject to the following constraints

Γ(i) ∗ ∗[

X(i) GT (i)]

X(i) ∗[xT (i) uT (i)

]0 1

≥ 0 (6.27)

for i = t + 1, . . . , t + N − 1, while for i = t we have the constraint[

Γ(t) ∗[xT (t) uT (t)

]1

]≥ 0 (6.28)

and, for i = t + N :[

Γ(t + N)−X(t + N) ∗xT (t + N) 1

]≥ 0 (6.29)

Finally, by defining

Vm(x(t), Φ[t,t+N ]) =t+N−1∑

i=t

Tr(MΦ(i)) + Tr(SΦ(t + N)) (6.30)

where the variables Φ(i) are symmetric matrices subject to the followingconstraints [

Φ(i) ∗[xT (i) uT (i)

]1

]≥ 0 (6.31)

for i = t, . . . , t + N − 1, while for i = t + N :[

Φ(t + N) ∗xT (t + N) 1

]≥ 0 (6.32)

6.6. A simulation example 121

the stability constraints (6.19) and (6.20) can be trasformed in the followinglinear inequality constraint

Vm(x(t), Φ[t,t+N ])

≤ min

V (x(t− 1))− λQmin|x(t− 1)|2

+|x(t− 1)|C ′1|Fw(t− 1)|

+C ′2|Fw(t− 1)|2,λP

max|x(t)|2

(6.33)

thus Problem P3 can be reformulated in a SDP form.

Problem P 4 Minimize the performance index (6.26)-(6.29) with respectto u[t,t+N−1], G[t+1,t+N−1], x[t+1,t+N−1], Γ[t,t+N ], X[t,t+N ], Φ[t,t+N ] subjectto the system dynamics (6.6), (6.23), to the probabilistic constraints (6.2),(6.4) reformulated according to (6.15), (6.24), (6.25) and to the stabilityconstraints (6.30)-(6.33).

6.6 A simulation example

6.6.1 Control design parameters

The stochastic MPC algorithm proposed in this paper has been used by firstdiscretizing the model (3.62) with the sampling time Ts = 0.02. Moreover, ithas been assumed that the noise is an additive torque w with covariance 4∗10−4. The considered prediction horizon is N = 20, while the probabilisticconstraints are chosen to satisfy the state and input limitations

P(χ1(i) ≥ 0.5236) ≤ 0.6 i = t + 1, . . . , t + NP(χ1(i) ≤ −π) ≤ 0.6 i = t + 1, . . . , t + N

P(χ2(i) ≥ 0.0021) ≤ 0.6 i = t + 1, . . . , t + NP(χ2(i) ≤ −0.0021) ≤ 0.6 i = t + 1, . . . , t + N

P(χ4(i) ≥ 0.02) ≤ 0.6 i = t + 1, . . . , t + NP(χ4(i) ≤ −0.02) ≤ 0.6 i = t + 1, . . . , t + NP(τ(i) ≥ 0.124) ≤ 0.6 i = t, . . . , t + N − 1

P(τ(i) ≤ −0.124) ≤ 0.6 i = t, . . . , t + N − 1


The matrix KLQ has been computed with weighting matrices Q =diag50, 1, 1, 1 and R = 100, while the constraints (6.15) are with ε = 0.2and the multiplicative parameter in Algorithm 1 is µ = 1.1.

0 1 2 3 4 5

−100

−50

0

50

θhub

Time [s]

Figure 6.1: Hub angle θhub. Blue dashed line: stochastic MPC. Red line:nominal MPC. Black dashed line: MPC constraints.

Figures 6.1-6.3 compare the input and state trajectories computed withthe stochastic MPC algorithm to those obtained with a nominal MPC withthe same tuning parameters. The blue dashed line is the evolution obtainedwith stochastic MPC while the red continuous line is the evolution obtainedwith a nominal MPC. The trajectory of the motor angle is reported in Fig-ure 6.1. Nominal MPC results in a greater overshoot which affects theperformance. Figure 6.4 shows the time evolution of the link deflections.As it can be observed, the stochastic controller offers better robustness (thenominal controller violates one of the constraints). In Figure 6.2 it is shownthe evolution of the torque computed by the nominal and stochastic con-trollers, together with the adopted realization of the random noise. Theapplied torque, that is the ideal one plus the disturbance, is reported inFigure 6.3.To better appreciate the characteristics of the proposed MPC algorithm,in Figure 6.5 the histograms of the exit codes of Algorithm 1 (summarizedin Table 6.1) are reported. It is apparent that the constraints relaxation

6.6. A simulation example 123

0 1 2 3 4 5

−0.2

−0.1

0

0.1

0.2

0.3Ideal Torque [Nm]

Time [s]

Figure 6.2: Ideal torque. Blue dashed line: Stochastic MPC. Red continuousline: Nominal MPC. Green line: additive torque disturbance.

0 1 2 3 4 5

−0.2

−0.1

0

0.1

0.2

0.3Applied Torque [Nm]

Time [s]

Figure 6.3: Real applied torque. Blue dashed line: Stochastic MPC. Redcontinuous line: Nominal MPC.


0 1 2 3 4 5−40

−20

0

20

40Deflection [mm]

Time [s]

Figure 6.4: Simulation comparison. Deflection. Blue dashed line: stochasticMPC, Red line: nominal MPC. Black dashed line: MPC constraints.

procedure implemented in Algorithm 1 is often required to compute a fea-sible solution of the optimization problem. However, the constraint on thecurrent control input (Exit code 6 ) is never removed.

Exit code Description0 Feasibility of Problem P31 Remove final region constraint2 Relaxed constraints violation probability3 Remove variance constraint4 Remove constraints on state variables5 Remove constraints on u(t + i), i > 06 Remove constraints on u(t)

Table 6.1: Exit code Algorithm 1

6.7. Conclusions 125

0 1 2 3 4 5 60

100

200

300

400

500

600

700

800

Exit Code

# In

sta

nces

Figure 6.5: Exit code of Algorithm 1.

6.7 Conclusions

In this chapter, a MPC controller for constrained discrete-time systemsaffected by a stochastic disturbance is presented. The optimization problemis reformulated as a deterministic one so that only a SDP problem mustbe solved on-line. An algorithm for the constraints relaxation is proposedin order to always guarantee the existence of a feasible solution. Finallystability constraints are introduced in the optimization problem to obtainthe ISS property of the closed-loop system when a limited disturbance isconsidered.

6.8 Appendix

6.8.1 Notation, basic definitions and available results

Let R, R+, Z, Z+ denote the real, the non-negative real, the integer andthe non-negative integer numbers, respectively. For any vector x ∈ Rn,|x| denotes the 2-norm, while for any matrix A ∈ Rn,n, |A| stands for theinduced 2-norm. The set of signals ψ taking values in some subset Ψ ⊆ Rm

is denoted by MΨ, while ||ψ|| := maxψ∈Ψ|ψ|. The symbol id represents


the identity function from R to R, while γ1 γ2 is the composition oftwo functions γ1 and γ2 from R to R. Given a set A ⊆ Rn, d(ζ,A) :=inf|η − ζ|, η ∈ A is the point-to-set distance from ζ ∈ Rn to A. Thedifference between two given sets A ⊆ Rn and B ⊆ A is denoted by A\B :=x : x ∈ A,x /∈ B. Given a closed set A ⊆ Rn, ∂A denotes the border ofA.

Definition 6.8.1 (K-function) A function γ : R+ /→ R+ is a K-functionif it is continuous, strictly increasing and such that γ(0) = 0.

Definition 6.8.2 (K∞-function) A function γ : R+ /→ R+ is a K∞-function if it is a K-function and γ(x) → +∞ as x → +∞. The symbolE[ψ] denotes the expected value of ψ.

Definition 6.8.3 (KL-function) A function β : R+ × Z+ /→ R+ is aKL-function if for every t ≥ 0 β(·, t) is a K-function, for every s ≥ 0 β(s, ·)is decreasing and β(s, t) → 0 as t →∞.

Consider the following discrete-time dynamic system:

x(k + 1) = F (x(k),w(k)), k ≥ t, x(t) = x (6.34)

where F (0, 0) = 0, x(k) ∈ Rn is the state, w(k) ∈ W is the input (dis-turbance), limited in a compact set W ⊂ Rr containing the origin. Thetransient of the system (6.34) with initial state x and input w is denotedby x(k, x,w), k ≥ t.

Definition 6.8.4 (ISS) System (6.34) is Input-to-State Stable (ISS) ifthere exists a KL-function β and a K-function γ such that

|x(k, x,w)| ≤ β(|x|, k) + γ(||w||) ∀k ≥ t, ∀x ∈ Rn

with w ∈MW .

Definition 6.8.5 (Robust Positively Invariant Set) A set Ξ ⊆ Rn isa robust positively invariant set for system (6.34) if F (x,w) ∈ Ξ, ∀x ∈ Ξand ∀w ∈W.

6.8. Appendix 127

Definition 6.8.6 (ISS in Ξ) System (6.34) is Input-to-State Stable (ISS)in a compact set including the origin as an interior point Ξ ⊆ Rn if Ξ is arobust positively invariant set for (6.34) and there exists a KL-function βand a K-function γ such that

|x(k, x,w)| ≤ β(|x|, k) + γ(||w||) ∀k ≥ t, ∀x ∈ Ξ.

with w ∈MW .

Definition 6.8.7 (ISS-Lyapunov function in Ξ) A function V : Rn /→R+ is an ISS-Lyapunov function in Ξ for system (6.34) if the followingrelations hold:

• Ξ is a compact robust positively invariant set for (6.34) including theorigin as an interior point.

• There exist two K∞-functions, α1 and α2, and a compact set Ω ⊆ Ξincluding the origin as an interior point such that

V (x) ≥ α1(|x|) ∀x ∈ ΞV (x) ≤ α2(|x|) ∀x ∈ Ω

• Defining ∆V (x,w) = V (F (x,w))−V (x), ∀x ∈ Ξ, ∀w ∈W there exista K∞-function α3 and a K-function σ such that

∆V (x,w) ≤ −α3(|x|) + σ(|w|) ∀x ∈ Ξ, ∀w ∈W

• There exist a K∞-function ρ (with ρ such that id−ρ is a K∞-function)and a suitable constant c > 0, such that, there exist a nonempty com-pact set D ⊂ IΩ := x : x ∈ Ω, d(x, ∂Ω) > c defined as

D := x : V (x) ≤ b(||w||) (6.35)

where b = α−14 ρ−1 σ, with α4 = α3 α−1

2 .

Then, the following sufficient condition for regional ISS of system (6.34) canbe stated.

Theorem 3 [Magni et al. 2006] If (6.34) admits an ISS-Lyapunov func-tion in Ξ, then it is ISS in Ξ and limk→∞ d(x(k, x,w),D) = 0 ∀x ∈ Ξ.


6.8.2 Proofs

Proof of Lemma 1 Proof: First note that feasibility is related to thesatisfaction of the stability constraints (6.19), (6.20), which, in view of thedefinition of V (x) through (6.18), depends only on u[t,t+N−1] (and not onK[t,t+N−1]).Denote by Et[·] expectation at time t and consider the optimal solution atime t of the unconstrained optimization problem with cost function (6.9)that is given by u[t,t+N−1] = [KLQEt[x(t)], ...,KLQEt[x(t + N − 1)] in viewof the choice S = P in the terminal cost. By optimality

V (x(t)) ≥ Jm(x(t), u[t,t+N−1]) (6.36)

Now consider the control sequence at time t + 1 with mean given byu[t+1,t+N ] = [KLQEt+1[x(t + 1)], ...,KLQEt+1[x(t + N)] . Then we have

Jm(x(t + 1), u[t+1,t+N ]) =

=t+N−1∑

i=t+1

(Et+1[xT (i)]QEt+1[x(i)] +

+ Et+1[xT (i)]KTLQRKLQEt+1[xT (i)]

)+

+ Et+1[xT (t + N)]QEt+1[x(t + N)] ++ Et+1[xT (t + N)]KT

LQRKLQEt+1[x(t + N)] ++ Et+1[xT (t + N + 1)]PEt+1[x(t + N + 1)]

In view of (6.36)

Jm(x(t + 1), u[t+1,t+N ])− V (x(t))≤ Jm(x(t + 1), u[t+1,t+N ])− Jm(x(t), u[t,t+N−1]) =

−(Et[xT (t)]QcEt[x(t)]

)+

+t+N−1∑

i=t+1

(Et+1[xT (i)]QcEt+1[x(i)]−

− Et[xT (i)]QcEt[x(i)])

++ Et+1[xT (t + N)]QcEt+1[x(t + N)] +− Et[xT (t + N)]PEt[x(t + N)] ++ Et+1[xT (t + N + 1)]PEt+1[x(t + N + 1)] (6.37)

6.8. Appendix 129

Moreover, it holds that

Et[x(i)] = Ai−tc Et[x(t)], i > t

andEt+1[x(i)] = Et[x(i)] + Ai−t−1Fw(t), i > t

so that recalling that Et[x(t)] = x(t)

Et+1[xT (i)]QcEt+1[x(i)]− Et[xT (i)]QcEt[x(i)] == (Et[x(i)] + Ai−t−1Fw(t))T Qc(Et[x(i)] ++ Ai−t−1Fw(t))− Et[xT (i)]QcEt[x(i)] == Et[xT (i)]QcEt[x(i)] + 2Et[xT (i)]QcA

i−t−1Fw(t) ++ (Ai−t−1Fw(t))T QcA

i−t−1Fw(t)−− Et[xT (i)]QcEt[x(i)] == 2Et[xT (i)]QcA

i−t−1Fw(t) ++ (Ai−t−1Fw(t))T QcA

i−t−1Fw(t)≤ 2LAc (i− t) |x(t)|LQcLA (i− t− 1) |Fw(t)|++ LQcLA (2(i− t− 1)) |Fw(t)|2

where LQc = |Qc|, LAc(i) = |Aic| and LA(i) = |Ai|. It then follows that

t+N−1∑

i=t+1

(Et+1[xT (i)]QcEt+1[x(i)]− Et[xT (i)]QcEt[x(i)]

)

≤t+N−1∑

i=t+1

2LAc(i− t)|x(t)|LQcLi−t−1A |Fw(t)|

+t+N−1∑

i=t+1

LQcLA (2(i− t− 1)) |Fw(t)|2 =

= |x(t)|(

t+N−1∑

i=t+1

2LAc (i− t) LQcLA (i− t− 1)

)|Fw(t)|

+

(LQc

t+N−1∑

i=t+1

LA (2(i− t− 1))

)|Fw(t)|2


and we also have

Et+1[xT (t + N)]PEt+1[x(t + N)]−− Et[xT (t + N)]PEt[x(t + N)]≤ 2LAc(N)|x(t)|LP LA(N − 1)|Fw(t)|++ LP LA (2(N − 1)) |Fw(t)|2 (6.38)

where LP = |P |. Thus, summing and subtracting the term Et+1[xT (t +N)]PEt+1[x(t + N)] in equation (6.37) we get

Jm(x(t + 1), u[t+1,t+N ])− V (x(t))≤ −

(xT (t)Qcx(t)

)+

|x(t)|(

t+N−1∑

i=t+1

2LAc (i− t)LQcLA (i− t− 1)

)|Fw(t)|

+

(LQc

t+N−1∑

i=t+1

LA (2(i− t− 1))

)|Fw(t)|2 +

+ Et+1[xT (t + N)]QcEt+1[x(t + N)] ++ Et+1[xT (t + N + 1)]PEt+1[x(t + N + 1)]−Et+1[xT (t + N)]PEt+1[x(t + N)] ++Et+1[xT (t + N)]PEt+1[x(t + N)]−Et[xT (t + N)]PEt[x(t + N)]

Moreover in view of (6.38) and (6.16)

Jm(x(t + 1), u[t+1,t+N ])− V (x(t))

≤ −(λQc

min|x(t)|2)

+

|x(t)|(

t+N−1∑

i=t+1

2LAc (i− t) LQcLA (i− t− 1)

)|Fw(t)| +

+

(LQc

t+N−1∑

i=t+1

LA (2(i− t− 1))

)|Fw(t)|2

+2LAc(N)|x(t)|LP LA (N − 1) |Fw(t)|+LP LA (2(N − 1)) |Fw(t)|2

6.8. Appendix 131

and then noting that λQcmin ≥ λQ

min

Jm(x(t + 1), u[t+1,t+N ])− V (x(t))

≤ −(λQ

min|x(t)|2)

++|x(t)|C ′

1|Fw(t)|+ C ′2|Fw(t)|2 (6.39)

where

C ′1 =

(t+N−1∑

i=t+1

2LAc(i− t)LQcLA (i− t− 1)

)

+2LAc(N)LP LA(N − 1) (6.40)

C ′2 =

(LQc

t+N−1∑

i=t+1

LA (2(i− t− 1))

)

+LP LA (2(N − 1)) (6.41)

so that u[t+1,t+N ] fullfilles constraint (6.19).

As for the constraint (6.20), note that

Jm(x(t + 1), u[t+1,t+N ]) = x(t + 1)′Px(t + 1)≤ λP

max|x(t + 1)|2

so that it can be concluded that u[t+1,t+N ] is a feasible sequence and theLemma is proven.

Proof of Theorem 2

Proof: First we show that the closed-loop system (6.1), (6.21) is Input-to-State Stable in

Ξ := x : V (x) ≤ b(wmax)

whereb(|w|) :=

λPmax

λQmin

(1 + δ)(xMC1|w|+ C2|w|2) (6.42)

Ci = |F |C ′i (i = 1, 2), δ is an arbitrary positive constant,

xM :=λP

max

(λQmin)2

(1 + δ)C1wmax + C2w2max (6.43)


and wmax := wmax + c, c > 0 being an arbitrary positive constant.

It is first shown that |x| ≤ xM , for every x ∈ Ξ. In fact, from (6.42)and (6.43), we have

b(wmax) = λQmin

xM − C2w2max

C1wmax(xMC1wmax + C2w

2max)

so that

b(wmax) = λQminx2

M + λQmin

C2w2max

C1wmax(xM − xMC1wmax)−

− λQmin

(C2w2max)2

C1wmax

Without loss of generality we can suppose that C1wmax > 1, then b(wmax) ≤λQ

minx2M . Moreover, for every x ∈ Rn it holds that λQ

min|x|2 ≤ V (x), so thatthe region Ξ is bounded. In fact, for every x ∈ Ξ

λQmin|x|

2 ≤ V (x) ≤ b ≤ x2MλQ

min

so that|x| ≤ xM

To see that Ξ is also a robust positively invariant set for equation (6.19), inview of (6.20) for every x ∈ Ξ

∆V ≤ −λQmin|x|

2 + xMC1wmax + C2w2max =

= −λQmin

λPmax

λPmax|x|2 + xMC1wmax + C2w

2max

≤ −λQmin

λPmax

V (x) + xMC1wmax + C2w2max

and thus, from (6.42) for every x(t) ∈ Ξ

V (x(t + 1)) ≤ V (x(t))(1− λQmin

λPmax

) + xMC1wmax + C2w2max

≤ b(wmax)(1− λQmin

λPmax

) + b(wmax)λQ

min

λPmax(1 + δ)

≤ b(wmax)

6.8. Appendix 133

and thus x(t+1) ∈ Ξ. We conclude that V (x) is an ISS-Lyapunov functionin Ξ, with the functions of Definition 6.8.7 given by:

Ω = Ξα1 = α3 = λQ

min|x|2α2 = λP

max|x|2σ = xMC1|w|+ C2|w|2ρ = |x|/(1 + δ)

By Theorem 3 we conclude that the system is Input-to-State Stable in Ξ.

In oreder to prove global stability, note that if x ∈ Ξ then ISS in Ξimplies that

|x(k, x,w)| ≤ βr(|x|, k) + γ(||w||) ∀k ≥ t. (6.44)

for some KL-function βr and K-function γ. Now suppose that x /∈ Ξ: thanit can be shown that λP

max|x|2 ≥ b(wmax). Thus, if |x| ≤ xM then by (6.19)


2 + xMC1wmax + C

≤ −λQmin

λPmax

b(wmax) + xMC1wmax + C

where C := C2w2max. By (6.42) the last expression becomes

∆V ≤ −(1 + δ)(xMC1wmax + C) ++ xMC1wmax + C

= −δ(xMC1wmax + C)≤ −δ(|x|C1wmax + C)

If instead |x| > xM then by (6.19)


2 + |x|C1wmax + C := −f(|x|)

where f(·) is a quadratic equation with positive quadratic coefficient andnegative constant coefficient. Thus, since f(xM ) > 0 then f(|x|) > 0 if|x| > xM . It follows that for every x ∈ Rn there exists a finite time instantt∗ such that x(k, x,w) ∈ Ξ, ∀k ≥ t∗ and that there exists a K∞-function α3


such that, for every x /∈ Ξ

∆V ≤ −α3(|x|) ≤ −α3(α−12 (V (x)))

with α2(|x|) = λPmax|x|2, see (6.20). By a standard comparison principle,

see [Jiang & Wang 2002], it follows that there exists a KL-function β suchthat

V (x(k)) ≤ β(V (x), k), ∀k < t∗

and thus|x(k)| ≤ β(|x|, k), ∀k < t∗ (6.45)

where β = α−11 β with α1 = λQ

min|x|2. By combining (6.45) and (6.44) weconclude that

|x(k, x,w)| ≤ β(|x|, k) + βr(|x|, k) + γ(||w||) ∀k ≥ t.

Conclusions

In this work the Model Predictive Control is applied for the first time tocontrol the motion of a flexible arm. The obtained results give experimen-tal evidence of the capability of MPC to maximize the utilization of theactuator effort, maintaining under control the arm oscillations also duringthe maneuver. These features differentiate the MPC from any other con-troller synthesized on linear models, for which the only possibilities to satisfyinput and output constraints can be obtained through the controller detun-ing, with consequent performance degradation. The experimental resultsare a significant contribution of this work and have been obtained throughthe development of a suitable software solution that allows to interface theoptimization algorithm with the sensors and the actuator. A further contri-bution of this thesis consists in the development of a MPC algorithm withISS properties with respect to stochastic additive disturbances. This choicemake the algorithm suitable for a wider class of application with respect tostandard stochastic MPC algorithms. In order to limit the computationaleffort the original stochastic problem has been reformulated as a determin-istic one and the constraints have been trasformed so that an SDP problemmust be solved.

In the future, more work could be done in order to understand thepossible improvement achievable with a reduction of the sampling time.Moreover, based on the observation that one of the most effective strategyused in the field of vibration suppression is represented by reference pre-shaping, MPC controller can be thought as an automatic tool to give, inclosed loop, a shaped position reference for an existing position control loop.This possibility provides good performance in many applications where theMPC is used to generate the set-point for another faster regulator withina gerarchic scheme [Scattolini 2009]. Final results could be extended totrajectory control of multilink robotic arms.

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