linear experiment #10: vibration controlvibration control laboratory – student handout 1....

Linear Motion Servo Plant: AMD-2

Linear Experiment #10:Vibration Control

Active Mass Damper - Two Floors(AMD-2)

Student Handout

Vibration Control Laboratory – Student Handout

Table of Contents1. Objectives............................................................................................................................12. Prerequisites.........................................................................................................................23. References............................................................................................................................24. Experimental Setup..............................................................................................................3

4.1. Main Components........................................................................................................34.2. Wiring..........................................................................................................................3

5. Controllers Design Specifications.......................................................................................45.1. Excitation Mode: PV Controller Design Specifications..............................................45.2. Active Mass Damping (AMD) Mode..........................................................................4

5.2.1. State-Feedback Design: Pole Locations...............................................................55.2.2. Full-Order State Observer Design: Pole Locations..............................................5

6. Pre-Lab Assignments...........................................................................................................66.1. Assignment #1: Proportional-Velocity (PV) Controller Design..................................66.2. AMD-2 System Representation and Notations............................................................86.3. Assignment #2: Determination of the AMD-2 System's Linear Equations Of Motion(EOM).................................................................................................................................96.4. Assignment #3: AMD-2 State-Space Representation................................................106.5. Full-Order Observer...................................................................................................11

7. In-Lab Procedure................................................................................................................137.1. Experimental Setup And Wiring................................................................................137.2. Real-Time Implementation Of The AMD-2 Switching-Mode Controller.................13

7.2.1. Objectives...........................................................................................................137.2.2. Experimental Procedure.....................................................................................13

Appendix A. Nomenclature...................................................................................................24Appendix B. AMD-2 Equations Of Motion (EOM)..............................................................28

Document Number: 571 ! Revision: 02 ! Page: i


1. ObjectivesAs illustrated in Figure 1, the purpose of the AMD-2experiment is to design a control system that dampens thevibration of a bench-scale building-like tall two-story structureusing an Active Mass Damper (AMD) mounted at the top. TheActive Mass Damper � Two-Floor (AMD-2) experiment canbe used in earthquake mitigation studies and to investigateControl-Structure Interaction (CSI). It is conceptually similarto active mass dampers used to suppress vibrations in tallstructures (e.g. high-rise buildings) and to protect not onlyagainst earthquakes but also, for example, strong winds (e.g.hurricanes).

The Active Mass Damper � Two-Floor (AMD-2) plant isillustrated in Figure 1 and is fully described in Reference [1].This laboratory takes advantage that the dynamics of the activemass (i.e. cart) are tightly coupled to these of the building-likestructure to which it is attached. Therefore, the active mass caneither be used to excite or to dampen the flexible structurevibration. The purpose of the AMD-2 laboratory is to design aswitching-mode control system that first excites one of the twovibration modes of the two-story structure and then dampensthe structure oscillation of both floors.

During the course of this experiment, you will become familiarwith the design of Proportional-plus-Velocity (PV) positioncontroller to drive the linear cart (i.e. active mass) such that itexcites the flexible structure natural vibrations. Then avibration reduction control strategy will be designed in order todampen the structure oscillations. Such a AMD control strategywill be implemented using a full-state feedback law based on afull-order observer. Pole placement will be used to tune thecontrol scheme.

Figure 1 AMD-2 Experiment

At the end of the session, you should know the following:How to mathematically model the IP01- or IP02-based linear servo plant from firstprinciples in order to obtain the open-loop transfer function, in the Laplace domain.How to design and tune a Proportional-Velocity (PV) position controller to meet therequired design specifications.

Document Number: 571 ! Revision: 02 ! Page: 1


How to mathematically model the Active Mass Damper � Two-Floor (AMD-2) plantusing the Lagrange's method and to obtain its state-space representation.How to design and tune a full-order state-observer based on the structure's twoacceleration feedback signals.How to design and tune a state-feedback controller satisfying the closed-loop system'sdesired design specifications.How to implement in real-time the total AMD-2 mode-switching (between vibrationexcitation and active mass damping) control scheme and evaluate its actualperformance.

2. PrerequisitesTo successfully carry out this laboratory, the prerequisites are:

i) To be familiar with your Active Mass Damper � Two-Floor (AMD-2) maincomponents (e.g. actuator, sensors), your data acquisition card (e.g. Q8, MultiQ), andyour power amplifier (e.g. UPM), as described in References [1], [3], [4], and [5].

ii) To have successfully completed the pre-laboratory described in Reference [2].Students are therefore expected to be familiar in using WinCon to control and monitorthe plant in real-time and in designing their controller through Simulink, as detailed inReference [6].

iii) To have successfully completed the laboratory described in Reference [7].iv) To be familiar with the complete wiring of your Active Mass Damper � Two-Floor

(AMD-2) plant, as per dictated in Reference [1].v) To be familiar with the design theory of full-order state observers, as described for

example in Reference [8].

3. References[1] Active Mass Damper – Two Floors (AMD-2) User Manual.[2] IP01 and IP02 – Linear Experiment #0: Integration with WinCon – Student Handout.[3] IP01 and IP02 User Manual.[4] Data Acquisition Card User Manual.[5] Universal Power Module User Manual.[6] WinCon User Manual.[7] IP01 and IP02 - Linear Experiment #1: PV Position Control – Student Handout.[8] P. R. Bélanger. Control Engineering: A Modern Approach. 1995, Saunders College

Publishing.



4. Experimental Setup

4.1. Main ComponentsTo setup this experiment, the following hardware and software are required:

Power Module: Quanser UPM 1503 / 2405, or equivalent.

Data Acquisition Board: Quanser Q8 / MultiQ PCI / MQ3, or equivalent.

Active Mass Damper Plant: Quanser Active Mass Damper � Two-Floor (AMD-2),as represented in Figure 1, above.

Real-Time Control Software: The WinCon-Simulink-RTX configuration, asdetailed in Reference [6], or equivalent.

For a complete and detailed description of the main components comprising this setup,please refer to the manuals corresponding to your configuration.

4.2. WiringTo wire up the system, please follow the default wiring procedure for your Active MassDamper � Two-Floor (AMD-2) system, as fully described in Reference [1]. When you areconfident with your connections, you can power up the UPM.



5. Controllers Design SpecificationsThe Active Mass Damper � Two-Floor (AMD-2) experiment essentially consists of aswitching-mode controller using a pre-defined logic sequence to alternate between a modeof repeatable structural excitation and the Active Mass Damping (AMD) mode to stiffen thestructure. Each mode is achieved through its own closed-loop control scheme. However, the power amplifier (e.g. UPM) should not go into saturation in any case. Alsoboth controllers control effort, which is proportional to the motor input voltage Vm, shouldstay within the system's physical limitations.

5.1. Excitation Mode: PV Controller Design SpecificationsThe (self-)excitation mode uses the active mass to generate a repeatable disturbance to theAMD-2's building-like structure. In this case, the linear cart (i.e. active mass) control loopconsists of a Proportional-Velocity (PV) position controller.

In the present laboratory (i.e. the pre-lab and in-lab sessions), you will design andimplement a control strategy based on the Proportional-Velocity (PV) control scheme, inorder for your linear cart system to satisfy the following performance closed-looprequirements:

1. The cart position Percent Overshoot, PO, should be less than 10%, i.e.: ≤ PO 10 [ ]"%" [1a]

2. The time to first peak should be less than 150 ms, i.e.: ≤ tp 0.15 [ ]s [1b]

5.2. Active Mass Damping (AMD) ModeIn the Active Mass Damping (AMD) mode, the linear cart is controlled by using a state-feedback law based on a full-order observer. This time around, the purpose of the controlstrategy is to dampen the AMD-2 top and middle floor oscillations (created in the excitationmode) by stiffening the structure.

The first AMD closed-loop design requirement concerns the second floor deflection (xf2)behaviour in terms of its settling time, ts2, and its decayed amplitude of oscillation inresponse to an initial deflection of around ±10 millimeters. Both modes of vibration shouldbe considered. This specification is formulated below:



≤ ts2 1.5 [ ]s for ≤ xf2 2.5 [ ]mm [2a]

Similarly, the second and last AMD design requirement is expressed with regard to theADM-2 first floor deflection (xf1) in terms of its settling time, ts1, and its decayed oscillationamplitude in response to an initial deflection of around ±10 millimeters. Both modes ofvibration should be considered. This specification is formulated below:

≤ ts1 1.5 [ ]s for ≤ xf1 3.0 [ ]mm [2b]

The particular pole placements presented in the following by Equations [3] and [4] werechosen for the AMD closed-loop system to meet the specifications stated above.

5.2.1. State-Feedback Design: Pole LocationsThe AMD-2 full-state feedback gain vector, K, should be determined such that the closed-loop poles (i.e. eigenvalues) due to the state-feedback law are placed at the followinglocations:

, , , , ,− + 6 15 j − − 6 15 j − + 7 10 j − − 7 10 j − + 16 13 j − − 16 13 j [3]

5.2.2. Full-Order State Observer Design: Pole LocationsHowever in order to be based on a full-state feedback (i.e. all structural displacements andvelocities) law, the AMD-2's active structural control strategy should be based on a stateobserver. This is explained by the fact that the AMD-2 floors deflections and velocities arenot measured directly, only their accelerations are. This particular design for the AMD-2plant is due to full-scale real-life applications where floor deflections and velocities aredifficult to measure directly. However, they can be estimated from accelerationmeasurements, since accelerometers provide an affordable and reliable way of sensing abuilding dynamic behaviour.

The AMD-2 full-order observer gain matrix, G, should be determined such that the closed-loop poles (i.e. eigenvalues) due to the observer error dynamics are placed at the followinglocations:

, , , , ,-40 -45 -50 -55 -60 -65 [4]

Comparing Equations [3] and [4], it can be noted that the dynamics of the observer error(i.e. rate at which the estimation error goes to zero) are at least six times faster (i.e. fartherto the left in the s-plane) than the plant itself. This should ensure that the estimator does notinterfere with the plant's dynamics.



6. Pre-Lab Assignments

6.1. Assignment #1: Proportional-Velocity (PV) ControllerDesignIn this design procedure for the PV controller, the two-floor flexible structure on top ofwhich the linear cart is mounted is ignored. The system is considered to consist of the IP01-or IP02-based linear servo plant alone. A schematic of the AMD-2 linear cart system's inputand output is represented in Figure 2, below.

Figure 2 The AMD-2 Linear Cart Input and Output

1. Based on your previous work from Reference [7], write down the open-loop transferfunction of your IP01- or IP02-based system, Gc(s), as defined below:

= ( )Gc s( )xc s( )Vm s [5]

2. The Proportional-Velocity (PV) position controller implemented in this lab for your lin-ear servo plant introduces two corrective terms: one is proportional (by Kp) to the cart po-sition error while the other is proportional (by Kv) to the cart velocity. Equation [6], be-low, expresses the resulting PV control law:

= ( )Vm t − Kp ( ) − ( )xc_r t ( )xc t Kv

d

dt ( )xc t [6]

Quickly re-iterate your previous work from Pre-Lab Assignment #3 (i.e. the PV



Controller Design Section) of Reference [7] in order to determine the two PV controller'sgains, Kp and Kv, as functions of the second-order system's characteristic parameters ωnc

and ζc.

3. Using the two Hint formulae provided below, express ωnc and ζc as functions of the twoPV design specifications previously defined, PO and tp.Hint formula #1:

= PO 100 eeee

−πζ

c

− 1 ζc

2 [7]

Hint formula #2:

= tpπ

ωnc − 1 ζ c2 [8]

4. Determine the values of ωnc and ζc corresponding to the desired PV design specifications,as defined previously. Determine, then, from your results the numerical values of Kp andKv satisfying the desired time requirements of your closed-loop PV-plus-cart system.



6.2. AMD-2 System Representation and NotationsA schematic of the Active Mass Damper � Two-Floor (AMD-2) plant is represented inFigure 3, below. The AMD-2 scaled building is a Two-Degree-Of-Freedom (2-DOF)flexible structure.

Figure 3 Schematic of the AMD-2 Plant



The AMD-2 system's nomenclature is provided in Appendix A. As illustrated in Figure 3,the positive direction of horizontal displacement is towards the right when facing thesystem.

For small floor deflection angles, both AMD-2 floors are modelled as standard linearspring-mass systems, as represented in Figure 3 above. Both AMD-2 floors' linear stiffnessconstants, Kf1 and Kf2, for small angular structure oscillations, are given in Reference [1].Kf1 and Kf2 model the lateral stiffnesses of the structure. In the presented modellingapproach, the structure viscous damping coefficients, Bf1 and Bf2, are neglected.

6.3. Assignment #2: Determination of the AMD-2 System'sLinear Equations Of Motion (EOM)The determination of the AMD-2's two-story-structure-plus-cart equations of motion isderived in Appendix B. If Appendix B has not been supplied with this handout, derive thesystem's equations of motion following the system's schematic and notations previouslydefined and illustrated in Figure 3. Also, put the resulting EOM under the following format:

= ∂∂2

t2 xc

∂

∂2

t2 xc , , , , , ,xc xf1 xf2 ∂∂t xc ∂

∂t xf1 ∂

∂t xf2 Fc [9]

and:

= ∂∂2

t2 xf1

∂

∂2

t2 xf1 , , , , , ,xc xf1 xf2 ∂∂t xc ∂

∂t xf1 ∂

∂t xf2 Fc [10]

and:

= ∂∂2

t2 xf2

∂

∂2

t2 xf2 , , , , , ,xc xf1 xf2 ∂∂t xc ∂

∂t xf1 ∂

∂t xf2 Fc [11]

Hint #1:By neglecting the Coulomb (a.k.a. static) friction of the cart system, the three EOM shouldbe linear. They represent two coupled spring-mass-damper systems.

Hint #2:You can use the method of your choice to model the system's dynamics. However, themodelling developed in Appendix B uses the energy-based Lagrangian approach. In thiscase, since the system has three Degrees-Of-Freedom (DOF), there should be threeLagrangian coordinates (a.k.a. generalized coordinates). The chosen three coordinates arenamely: xc, xf1, and xf2. Also, the input to the system is defined to be Fc, the linear forceapplied by the motorized cart.



6.4. Assignment #3: AMD-2 State-Space RepresentationIn order to design and implement a state-feedback controller for our system, a state-spacerepresentation of that system needs to be derived. Moreover, it is reminded that state-spacematrices, by definition, represent a set of linear differential equations that describe the sys-tem's dynamics. Since the three EOM of the AMD-2, as found in Assignment #2, should al-ready be linear, they can directly be written under the state-space matrices representation.

Answer the following questions:

1. Determine from the system's three equations of motion, the state-space representation ofour AMD-2 plant. That is to say, determine the state-space matrices A, B, C, and Dverifying the following relationships:

= ∂∂t X + A X B U and = Y + C X D U [12]

where X is the system's state vector. In practice, X is often chosen to include thegeneralized coordinates as well as their first-order time derivatives. In our case, X isdefined such that its transpose is as follows:

= X T

, , , , ,( )xc t ( )xf1 t ( )xf2 t d

dt ( )xc t d

dt ( )xf1 t d

dt ( )xf2 t [13]

Furthermore, it is reminded that the system's measured output vector is:

= YT

, ,( )xc t

dd2

t2 ( )xf1 tdd2

t2 ( )xf2 t [14]

Also in Equation [12], the input U is set in a first time to be Fc, the driving force of thelinear motorized cart. Thus we have:

= U Fc [15]As a remark, it can be seen from Equations [14] and [15] that the AMD-2 systemconsists of three outputs for one input.

2. Evaluate the system's state-space representation A, B, C, and D previously found, andnumerically transform the state-space matrices for the case where the system's input U isequal to the linear cart's DC motor voltage Vm, instead of the linear force Fc. The system'sinput U can now be expressed by:

= U Vm [16]Hint #1:In order to convert the previously found force equation state-space representation tovoltage input, it is reminded that the driving force, Fc, generated by the DC motor andacting on the cart through the motor pinion has already been determined in previous



laboratories. As shown for example in Equation [B.9] of Reference [7], Fc can beexpressed by:

= Fc − + Kg

2Kt Km

d

dt ( )xc t

Rm rmp2

Kg Kt Vm

Rm rmp

[17]

Hint #2:Evaluate the force equation [17] and the state-space matrices by using the modelparameter values given in Reference [1]. Ask your laboratory instructor what systemconfiguration you are going to use in your in-lab session. In case no additionalinformation is provided, assume that the two additional masses are mounted on top of thelinear cart.

3. Calculate the open-loop poles from the system's state-space representation, as previouslyevaluated. Is it stable? What is the type of the system? What can you infer regarding thesystem's dynamic behaviour? Do you see the need for a closed-loop controller? Explain.Hint:The characteristic equation of the open-loop system can be expressed as shown below:

= ( )det − s I A 0 [18]where det() is the determinant function, s is the Laplace operator, and I the identitymatrix. Therefore, the system's open-loop poles can be seen as the eigenvalues of thestate-space matrix A.

6.5. Full-Order ObserverSince not all the states contained in X, as defined in Equation [13], can be directly measured(e.g. xf1 and xf2), a state observer needs to be built to estimate them. The system's variablesdirectly measured are expressed in the output vector defined in Equation [14]. This sectionpresents the design of a full-order observer.

The full-order observer structure is defined as follows:

= ∂∂t Xo + + A Xo B U G ( ) − Y Yo [19]

and:

= Yo + C Xo D U [20]



It can be seen from Equations [19] and [20] that the state observer has for inputs the AMD-2 system's input(s) and output(s) and calculates as outputs the states estimates. The observeris basically a replica of the actual plant with a corrective term (Y-Yo) multiplied by the ob-server gain matrix, G.

The obtained estimated state vector can then be used for state-feedback law, as expressedbelow:

= ( ) = U Vm −K Xo [21]

Let us define the estimation error vector as follows:

= Xe − X Xo [22]

Using Equations [12], [19], [20], and [22], the estimation error dynamically behaves accord-ingly to the following relationship:

= ∂∂t Xe ( ) − A G C Xe [23]

Therefore from Equation [23], the estimation error will asymptotically go to zero if and onlyif (A-GC) is stable, that is to say iff G is determined such as (A-GC) has all its eigenvaluesin the left-hand plane. However, a theorem shows that if (A,C) is observable, then (A-GC)can always be made stable by a proper choice of G. Therefore, it is possible to estimate thestate(s) of a system if and only if that system is observable. Please refer to your in-classnotes as required. To this matter Reference [8], for example, also provides additional infor-mation.

It should be noted that in the case of the AMD-2 system, (A,C) is observable. The observ-ability matrix is defined as follows:

= Wo [ ], , , , ,C C A C A 2 C A 3 C A 4 C A 5 T[24]

By definition, a system is observable iff its observability matrix has full rank (i.e. number ofstates). For the AMD-2 plant, it can be determined that Wo has a rank of 6, which is fullrank. Therefore, the AMD-2 system is observable and a state observer can be designed.



7. In-Lab Procedure

7.1. Experimental Setup And WiringEven if you do not configure the experimental setup entirely yourself, you should be at leastcompletely familiar with it and understand it. If in doubt, refer to References [1], [4], [5],and/or [6].

The first task upon entering the lab is to ensure that the complete system is wired as fullydescribed in Reference [1]. You should be familiar with the complete wiring and connec-tions of your Active Mass Damper � Two-Floor (AMD-2) system. If you are still unsure ofthe wiring, please ask for assistance from the Teaching Assistant assigned to the lab. Whenyou are confident with your connections, you can power up the UPM. You are now ready tobegin the lab.

7.2. Real-Time Implementation Of The AMD-2 Switching-Mode Controller

7.2.1. ObjectivesTo implement in real-time with WinCon the previously designed PV positioncontroller in order to command your actual AMD-2 linear servo plant.To design through pole placement a full-order observer for the actual AMD-2 system.To design through pole placement a state-feedback law for the actual AMD-2 systemby using the obtained estimated state vector.To implement in real-time with WinCon the previously determined observer-basedstate-feedback to dampen the vibration of the AMD-2 structure by appropriatelydriving the active mass on top of it.

7.2.2. Experimental ProcedurePlease follow the steps described below:

Step1. If you have not done so yet, you can start-up Matlab now. Depending on your sys-tem configuration, open the Simulink model file of name type q_amd2_ZZ.mdl orq_amd2_e_ZZ.mdl, where ZZ stands for either for 'mq3', 'mqp', 'q8', or 'nie'. Ask theTA assigned to this lab if you are unsure which Simulink model is to be used in the



lab. You should obtain a diagram similar to the one shown in Figure 4, below.

Figure 4 Real-Time Implementation of the AMD-2 Controller

Step2. In order to use your actual AMD-2 system, the controller diagram directly inter-faces with your system hardware, as shown in Figure 5, below. However, it should benoted that accelerometers provide absolute acceleration measurements. Therefore inorder to obtain the desired relative accelerations of both floors, the following relation-ships are applied for coordinate transformation:

= dd2

t2 ( )xf1 t −

d

d2

t2 ( )Xf1 t

d

d2

t2 ( )Xg t [25]

and:

= dd2

t2 ( )xf2 t −

d

d2

t2 ( )Xf2 t

d

d2

t2 ( )Xf1 t [26]



Figure 5 Interface Subsystem to the Actual AMD-2 Plant Using the MultiQ-PCI Card

Step3. To familiarize yourself with the diagram, it is suggested that you open the modelsubsystems to get a better idea of their composing blocks as well as take note of theI/O connections. The real-time model, shown in Figure 4, implements a switching-mode controller. A pre-defined Control Mode Switching Sequence alternatively con-trols the linear cart using either an observer-based Proportional-Velocity (PV) positioncontrol or state-feedback law. In short, the PV controller is mostly used in the excita-tion mode to self-excite the vibration in the building-like structure. On the other hand,the state-feedback law is designed to dampen the structure oscillation by driving theactive mass (i.e. cart). Both control loops are the actual implementations of the pre-laboratory assignments previously carried out.

Step4. Opening the Disturbance Setpoint Sequence sub-system should show a diagramsimilar to Figure 6. This diagram generates the setpoint for the cart position to followin order to best excite the AMD-2's flexible structure. The two modes of vibration ofthe flexible structure are excited in turns. First, a mostly first-mode type of response istriggered by applying one full period of a 6-cm sine wave at 1.3 Hz. Then this is fol-lowed by one full period of a 3.5-cm sine wave at 4.4 Hz in order to excite a mostlysecond-mode type of structural response. The self-excitation sequence repeats itselfevery 23.6 seconds.



Figure 6 Self-Excitation Mode: Cart Position Setpoint Generation

Step5. Figure 4, above, also includes a subsystem named PV Position Controller, whichimplements your linear cart PV controller's two feedback loops. This PV controller isbased on the observed cart position and velocity.

Figure 7 Diagram used for the Real-Time Implementation of the AMD-2 Cart PV Position Controller



Step6. When the Control Mode Switching Sequence block outputs "2" instead of "1", thatturns on the Active Mass Damping (AMD) capabilities of the linear cart located ontop of the structure. The AMD behaviour is achieved by implementing a full-order-observer-based state-feedback control loop, as illustrated in Figure 4 above by the or-ange blocks. The full-order state observer, previously defined by Equations [19] and[20], is implemented in the subsystem called Full-Order Observer and depicted inFigure 8, below.

Figure 8 Actual AMD-2 Full-Order State Observer

Step7. Before being able to run the actual control loops, the different controller gainsmust be initialized in the Matlab workspace, since they are to be used by the Simulinkcontroller diagram. Start by running the Matlab script called setup_lab_amd2.m.However, ensure beforehand that the CONTROLLER_TYPE flag is set to 'MANUAL'.This file initializes all the AMD-2 model parameters, user-defined parameters, andAMD-2's state-space matrices, as defined in the pre-lab section. First, the two PV con-troller gains, Kp and Kv, calculated in pre-lab Assignment #1 and satisfying the desiredtime requirements must be entered in the Matlab workspace. To assign Kp and Kv,type their values in the Matlab command window by following the Matlab notationsused for the controller gains as presented in Table A.2 of Appendix A. Second, thestate-feedback gain vector, K, must be calculated and entered in the Matlab work-space. Use Matlab to carry out the pole-placement calculations satisfying the designrequirement expressed by Equation [3]. Third and last, the observer gain matrix, G,must also be calculated and entered in the Matlab workspace. Use Matlab to carry out



the pole-placement calculations satisfying the design requirement expressed by Equa-tion [4]. Refer to your in-class notes regarding the full-order observer design theory asneeded.Hint:Pole-placement calculations can be achieved in Matlab by using the function 'place()'or 'acker()'.CAUTION:Have your lab assistant check your controller gain values. Do not proceed to thenext step without his or her approval.

Step8. You are now ready to go ahead with compiling and running your actual switch-ing-mode controller for the AMD-2 system. First, compile the real-time code corre-sponding to your diagram, by using the WinCon | Build option from the Simulinkmenu bar. After successful compilation and download to the WinCon Client, youshould see the green START button available on the WinCon Server window. You arenow in a position to use WinCon Server to run in real-time your actual closed-loopsystem.CAUTION:Before starting your actual controller, manually move the linear cart (located onthe top floor) to the middle of the track (i.e. mid-stroke position). Also make surethat the AMD-2 "ground" floor is properly clamped/mounted to a table.



Step9. You can now run your Active Mass Damper experiment on the actual AMD-2plant by clicking on the START/STOP button of the WinCon Server window. Thisshould start the AMD-2 control sequence, as illustrated by real data in Figure 9,below.

Figure 9 AMD-2 Actual Experiment

As previously mentioned, the AMD-2's linear cart is commanded by a switching-mode controller. In excitation mode (i.e. the Control Mode Switching Sequence out-puts "1", the AMD is OFF), the cart is subject to the PV position controller. In ActiveMass Damping (AMD) mode (i.e. the Control Mode Switching Sequence outputs "2",the AMD is ON), the cart is subject to an observer-based state-feedback controller.The Control Mode Switching Sequence is set up to generate both first-mode and sec-ond-mode types of response of the AMD-2 flexible structure. Both self-excitation andcontrol mode switching sequences repeat themselves every 23.57 seconds. In Figure 9,Excitation #1 and Excitation #2 both trigger a mostly first-mode type of response,while Excitation #3 and Excitation #4 activate more the second-mode type of oscilla-tions. These initial and repeatable mass movements cause a vibration in the AMD-2flexible structure which persists at its natural rate if the AMD mode/controller is notswitched on, as depicted in Figure 9. The AMD mode is only turned on once every



two structural excitations. This causes the first and second floor vibrations to dampenout quickly, as also shown in Figure 9. Such a mode-switching logic for the AMDcontroller allows the observation of the flexible structure natural behaviour versus itsactively damped (or controlled) behaviour.

Step10. In order to observe the system's real-time responses from the actual system, openthe following WinCon Scopes: xf2_ddot (m/s^2), xf1_ddot (m/s^2), xf2 (mm), xf1(mm), and PV Position Controller/PV Control: xc (mm). You should now be able tomonitor on-the-fly, as the system goes through the synchronized excitation/AMD-control sequences, the measured second and first floor accelerations, the controllermode, the estimated second and first floor deflections, and the linear cart setpoint andactual position, respectively. Furthermore, you can also open the sink Vm (V) in aWinCon Scope. This allows you to monitor on-line the actual commanded motorvoltage, which is proportional to the control effort spent, sent to the power amplifier.Hint #1: To open a WinCon Scope, click on the Scope button of the WinCon Server windowand choose the display that you want to open (e.g. xf2_ddot (m/s^2)) from the selec-tion list.Hint #2:For a good visualization of the actual system responses, you should set the WinConscope buffer to 24.57 seconds (i.e. the value of the Matlab workspace parameter te9).Do so by using the Update | Buffer... menu item from the desired WinCon scope.

Step11. Over a complete sequence run, your actual AMD-2's second and first floordeflections and control mode should look similar to the ones displayed in Figures 10and 11, below. Corresponding to the same experimental run, the time records of theAMD-2's cart measured position and excitation setpoint are depicted in Figure 12,below. The associated commanded cart motor voltage is illustrated in Figure 13,below. In agreement with the principles of a linear closed-loop control system, it canbe seen in Figure 13 that the actual command voltage input never goes into saturation.



Figure 10 AMD-2 Second Floor Estimated Deflection And Control Mode

Figure 11 AMD-2 First Floor Estimated Deflection



Figure 12 AMD-2 Excitation Setpoint And Actual Cart Position Response

Figure 13 AMD-2 Cart Actual Command Voltage: Vm (V)



Step12. Compare the structure's actively damped and natural behaviours in terms ofsettling times and amplitude of both floor deflections (xf1 and xf2) and/or accelerations(xf1_ddot and xf2_ddot). Assess the actual performance of your Active Mass Dampingcontroller. Measure the settling times of both floor deflection responses. Are thedesign specifications satisfied? Explain.Hint:In order to accurately measure a signal amplitude and time values from your WinConScope plot, you can first select Freeze Plot from the WinCon Scope Update menu andthen reduce the window's time interval by opening the Set Time Interval input boxthrough the Scope's Axis | Time... menu item. You should now be able to scrollthrough your plotted data. Alternatively, you can also save your Scope trace(s) to aMatlab file for further data processing. Do so by using the File | Save selection listfrom the WinCon Scope menu bar.

Step13. If your AMD-2 responses do not meet the desired design specifications, reviewyour PV and/or observer and/or state-feedback gain calculations and/or alter theclosed-loop pole locations until they do. If you are still unable to achieve the requiredperformance level, ask your T.A. for advice.

Step14. Include in your lab report your final values for Kp, Kv, K, and G as well as theresulting plots of the actual system responses (i.e. both floor accelerations, bothestimated deflections, control mode, cart position, and command voltage). Ensure toproperly document all your results and observations.

Step15. You can move on and begin your report for this lab.



Appendix A. NomenclatureTable A.1, below, provides a complete listing of the symbols and notations used in theActive Mass Damper � Two-Floor (AMD-2) mathematical modelling, as presented in thislaboratory. The numerical values of the system parameters can be found in Reference [1].

Symbol Description Matlab / SimulinkNotation

Mf1 First Floor Mass Mf1Mf2 Second Floor Mass (With Rack) Mf2Kf1 First Floor Linear Stiffness Constant (Relative To The

Ground)Kf1

Kf2 Second Floor Linear Stiffness Constant (Relative To TheFirst Floor)

Kf2

Mc Total Mass of the Cart System McRm Cart Motor Armature Resistance RmKt Cart Motor Torque Constant KtKm Cart Back-ElectroMotive-Force (EMF) Constant Km

Jm Cart Rotor Moment of Inertia JmKg Cart Planetary Gearbox Gear Ratio Kgηg Cart Planetary Gearbox Efficiency Eff_gηm Cart Motor Efficiency Eff_mrmp Cart Motor Pinion Radius r_mpBeq Equivalent Viscous Damping Coefficient, as seen at the

Motor PinionBeq

Vm Cart Motor Armature Voltage VmFc Cart Driving Force Produced by the Motor (i.e. Control

Force)Fc

xc Cart Linear Position Relative To The Second Floor xc

∂∂t xc Cart Linear Velocity Relative To The Second Floor xc_dot




xf1 First Floor Linear Deflection Relative To The Ground xf1

∂∂t xf1 First Floor Linear Velocity Relative To The Ground xf1_dot

∂∂2

t2 xf1 First Floor Linear Acceleration Relative To The Ground xf1_ddot

xf2 Second Floor Linear Deflection Relative To The First Floor xf2

∂∂t xf2 Second Floor Linear Velocity Relative To The First Floor xf2_dot

∂∂2

t2 xf2Second Floor Linear Acceleration Relative To The FirstFloor xf2_ddot

∂∂2

t2 Xg Ground Absolute Acceleration Xg_ddot

∂∂2

t2 Xf1 First Floor Absolute Acceleration Xf1_ddot

∂∂2

t2 Xf2 Second Floor Absolute Acceleration Xf2_ddot

VT Total Potential Energy of the AMD-2 System

Ttc Cart's Translational Kinetic EnergyTrc Cart Rotor's Rotational Kinetic EnergyTtf1 Structure First Floor's Translational Kinetic EnergyTtf2 Structure Second Floor's Translational Kinetic EnergyTT Total Kinetic Energy of the AMD-2 SystemQxc Generalized Force Applied on the Generalized Coordinate

xc

Qxf1 Generalized Force Applied on the Generalized Coordinatexf1




Qxf2 Generalized Force Applied on the Generalized Coordinatexf2

Table A.1 AMD-2 Model Nomenclature

Table A.2, below, provides a complete listing of the symbols and notations used in the PVposition controller design, as used in this laboratory.


PO Cart Percent Overshoot POtp Cart Peak Time tpωnc Cart Undamped Natural Frequency wn_cζc Cart Damping Ratio zeta_cKp Cart Proportional Gain KpKv Cart Velocity Gain Kv

xc_r Cart Desired Position (i.e. Reference Signal) xc_rt Continuous Times Laplace Operator

Table A.2 PV Cart Controller Nomenclature

Table A.3, below, provides a complete listing of the symbols and notations used in the full-order observer and state-feedback controller design, as presented in this laboratory.


A, B, C, D State-Space Matrices of the AMD-2 System A, B, C, DX Actual State Vector XY Actual Output Vector YU System Input UK State-Feedback Gain Vector K




Xo Estimated State Vector XoYo Estimated Output Vector YoG Full-Order State Observer Gain Matrix G

Wo Observer's Observability Matrix WoXe State Vector Estimation Error Xe

Table A.3 Control Loop Nomenclature



Appendix B. AMD-2 Equations OfMotion (EOM)

This Appendix derives the general dynamic equations of the Active Mass Damper � Two-Floor (AMD-2) system. The Lagrange's method is used to obtain the dynamic model of thesystem. In this approach, the single input to the system is considered to be Fc. Furthermore,it is reminded that the reference frame used is defined in Figure 3, on page 8.

To carry out the Lagrange's approach, the Lagrangian of the system needs to be determined.This is done through the calculation of the system's total potential and kinetic energies.

Let us first calculate the system's total potential energy VT. The potential energy in a systemis the amount of energy that that system, or system element, has due to some kind of workbeing, or having been, done to it. It is usually caused by its vertical displacement fromnormality (gravitational potential energy) or by a spring-related sort of displacement (elasticpotential energy). Here, there is no gravitational potential energy since both AMD-2 cartand two-story structure are each assumed to stay at a constant elevation (i.e. no verticaldisplacement from normality), for small angular structure oscillations. Both AMD-2 floorsare modelled as linear spring-mass systems. Therefore, the AMD-2's total potential energyis only due to its total elastic potential energy. It results that the total potential energy of theAMD-2 plant can be fully expressed as:

= VT + 12 Kf1 ( )xf1 t

2 12 Kf2 ( )xf2 t

2[B.1]

It can be seen from Equation [B.1] that the total potential energy can be expressed in termsof the system's generalized coordinates alone.

Let us now determine the system's total kinetic energy TT. The kinetic energy measures theamount of energy in a system due to its motion. Here, the total kinetic energy is the sum ofthe translational and rotational kinetic energies arising from the motorized linear cart (sincethe cart's direction of translation is orthogonal to that of the rotor's rotation) and thetranslational kinetic energy of the flexible structure's two floors. In other words, the totalkinetic energy of the AMD-2 system can be formulated as below:

= TT + + + Ttc Trc Ttf1 Ttf2 [B.2] First, the translational kinetic energy of the motorized cart can be expressed as a function ofits centre of gravity's linear velocity, as shown by the following equation:

= Ttc12 Mc

+ +

d

dt ( )xc t

d

dt ( )xf1 t

d

dt ( )xf2 t

2

[B.3]



Second, the rotational kinetic energy due to the cart's DC motor can be characterized by:

= Trc12

Jm Kg2

d

dt ( )xc t

2

rmp2

[B.4]

Third, the structure first floor's translational kinetic energy can be characterized as follows:

= Ttf112 Mf1

d

dt ( )xf1 t

2

[B.5]

Fourth and last, the structure second floor's translational kinetic energy can be expressed bythe following relationship:

= Ttf212 Mf2

+

d

dt ( )xf1 t

d

dt ( )xf2 t

2

[B.6]

Therefore by replacing Equations [B.3], [B.4], [B.5], and [B.6] into Equation [B.2], thesystem's total kinetic energy results to be such as:

TT12

+ Mc

Jm Kg2

rmp2

d

dt ( )xc t

2

Mc

+

d

dt ( )xf1 t

d

dt ( )xf2 t

d

dt ( )xc t + =

12 ( ) + + Mc Mf1 Mf2

d

dt ( )xf1 t

2

( ) + Mc Mf2

d

dt ( )xf2 t

d

dt ( )xf1 t + +

12 ( ) + Mc Mf2

d

dt ( )xf2 t

2

+

[B.7]

It can be seen from Equation [B.7] that the total kinetic energy can be expressed in terms ofthe generalized coordinates' first-time derivatives.

Let us now consider the Lagrange's equations for our system. By definition, the threeLagrange's equations, resulting from the previously-defined three generalized coordinates,xc xf1 and xf2, have the following formal formulations:



= −

∂ ∂

∂

t ddt ( )xc t

L

∂

∂xc

L Qxc [B.8]

and:

= −

∂ ∂

∂

t ddt ( )xf1 t

L

∂

∂xf1

L Qxf1 [B.9]

and:

= −

∂ ∂

∂

t ddt ( )xf2 t

L

∂

∂xf2

L Qxf2 [B.10]

In Equations [B.8], [B.9], and [B.10], above, L is called the Lagrangian and is defined to beequal to:

= L − TT VT [B.11]

For our system, the generalized forces can be defined as follows:

= ( )Qxct − ( )Fc t Beq

d

dt ( )xc t [B.12]

and: = ( ) = ( )Qxf1

t ( )Qxf2t 0 [B.13]

It should be noted that the (nonlinear) Coulomb friction applied to the linear cart has beenneglected. Furthermore, the viscous damping forces applied to the structure two floors havealso been neglected.

Calculating Equation [B.12] results in a more explicit expression for the first Lagrange'sequation, such that:

+ ( ) + Mc rmp

2Jm Kg

2

d

d2

t2 ( )xc t

rmp2 Mc

+

d

d2

t2 ( )xf1 t

d

d2

t2 ( )xf2 t =

− Fc Beq

d

dt ( )xc t

[B.14]



Likewise, calculating Equation [B.9] also results in a more explicit form for the secondLagrange's equation, as shown below:

Kf1 ( )xf1 t Mc

d

d2

t2 ( )xc t ( ) + + Mc Mf1 Mf2

d

d2

t2 ( )xf1 t + +

( ) + Mc Mf2

d

d2

t2 ( )xf2 t + 0 = [B.15]

Finally, calculating Equation [B.10] results as well in a more explicit form for the thirdLagrange's equation, as expressed underneath:

= + + + Kf2 ( )xf2 t Mc

d

d2

t2 ( )xc t ( ) + Mc Mf2

d

d2

t2 ( )xf1 t ( ) + Mc Mf2

d

d2

t2 ( )xf2 t 0 [B.16]

Solving the set of the three Lagrange's equations, as previously expressed in Equations[B.14], [B.15], and [B.16], for the second-order time derivative of the three Lagrangiancoordinates results in the three following equations:

dd2

t2 ( )xc t =

rmp2

− + + − Mc Fc Mc Beq

d

dt ( )xc t Mc Kf2 ( )xf2 t Mf2 Fc Mf2 Beq

d

dt ( )xc t

+ + Mc rmp2 Mf2 Jm Kg

2 Mc Jm Kg2 Mf2

[B.17]

and:

= dd2

t2 ( )xf1 t − Kf2 ( )xf2 t

Mf1

Kf1 ( )xf1 tMf1

[B.18]

and:



dd2

t2 ( )xf2 t Mf1 Mc Fc rmp2

Mf1 Mc Beq

d

dt ( )xc t rmp

2Mf1 Mc rmp

2Kf2 ( )xf2 t− + −

=

Mc rmp2 Mf2 Kf1 ( )xf1 t Mf1 Jm Kg

2 Kf2 ( )xf2 t Jm Kg2 Mc Kf1 ( )xf1 t + − +

Jm Kg2 Mc Kf2 ( )xf2 t Jm Kg

2 Mf2 Kf1 ( )xf1 t Mc rmp2 Mf2 Kf2 ( )xf2 t − + −

Jm Kg2

Mf2 Kf2 ( )xf2 t − Mf1 ( ) + + Mc rmp2

Mf2 Jm Kg2

Mc Jm Kg2

Mf2 ( )

[B.19]

Equations [B.17], [B.18], and [B.19] represent the Equations Of Motion (EOM) of thesystem. It can be noticed, in the case of the AMD-2 system, that the EOM are linear.

As a remark, if both Beq and Jm are neglected, Equations [B.17], [B.18], and [B.19] become:

= dd2

t2 ( )xc t+ Mc Kf2 ( )xf2 t ( )+ Mc Mf2 Fc

Mc Mf2[B.20]

and:

= dd2

t2 ( )xf1 t − Kf2 ( )xf2 t

Mf1

Kf1 ( )xf1 tMf1

[B.21]

and:

= dd2

t2 ( )xf2 t− − Mf2 Kf1 ( )xf1 t Kf2 ( )+ Mf1 Mf2 ( )xf2 t Mf1 Fc

Mf1 Mf2[B.22]


linear experiment #10: vibration controlvibration control laboratory – student handout 1....

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