nonlinear adaptive model predictive controller for a flexible

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1

Nonlinear Adaptive Model Predictive Controller fora Flexible Manipulator: An Experimental Study

Santanu Kumar Pradhan, Member, IEEE, and Bidyadhar Subudhi, Senior Member, IEEE

Abstract— This paper presents a new nonlinear adaptive modelpredictive controller (NAMPC) for tip position control of aflexible manipulator (FLM) when subjected to handle differentpayloads. The proposed adaptive control strategy consists ofan online FLM dynamics identifier in the form of a nonlinearautoregressive moving average with exogenous input model and acontrol signal generator based on the above identified dynamics.The effectiveness of the proposed control algorithm is thenverified by comparing its performance with that of a self-tuningcontroller (STC) and a nonlinear adaptive controller (NAC). Themetrics of the controller performance are chosen as tip trajectorytracking accuracy and fast suppression of tip deflection. Fromthe simulation and experimental results, it is observed thatthe proposed controller exhibits superior performance comparedwith the STC and NAC.

Index Terms— Flexible manipulator, nonlinear autoregressivemoving average with exogenous input (NARMAX), nonlinearmodel predictive control, self-tuning control, trajectory tracking.

NOMENCLATURE

ypi Redefined tip trajectory of the i th link.θri Desired tip trajectory of the i th link.Yi NARMAX model output of the

i th link.ei Tip trajectory error of the i th link.ξi Prediction error of the i th link.TDL Time-delay unit.T Sampling time.Pk Covariance matrix of the prediction error.� Forgetting factor.

J1 Quadratic performance index.

J2 Minimum variance-based performance index.Wi NARMAX parameters.�i NARMAX regressor vector.di Delay operator.� Difference operator.�i Expectation operator.Ki Proportional gain.TD Derivative time.

Manuscript received July 16, 2013; revised November 17, 2013; acceptedNovember 22, 2013. Manuscript received in final form December 6, 2013.Recommended by Associate Editor F. Caccavale.

S. K. Pradhan is with the Department of Electrical and ElectronicsEngineering, Birla Institute of Technology, Ranchi 835215, India (e-mail:[email protected]).

B. Subudhi is with the Department of Electrical Engineering, National Insti-tute of Technology, Rourkela 769008, India (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCST.2013.2294545

Ti Integral time.Z Z-Transform variable

I. INTRODUCTION

T IP position control of a FLM, i.e., manipulator with thinand light weight links is challenging due to distributed

link flexibility, which makes the system nonminimum phase,underactuated and infinite dimensional [1], [2]. In the presenceof external disturbances like change in payload, the perfor-mance of the FLM deteriorates further.

As the manipulator is expected to maneuver with unantic-ipated payload at the end effector, thus payload variabilityis also an important concern. Furthermore, due to suddenchange in payload, there may be large variation in manipulatorparameters and that in turn adds further complexities to theFLM dynamics. Hence, the torque applied to the actuatorsof an FLM to control the tip position and its deflection withchanges in payload should be adaptive in nature [3]–[6].

A great deal of research has been directed in the past todesign an identified model-based adaptive controller for anFLM, for instance, a PID control law has been developed in [7]for a single-link FLM using an autoregressive moving average(ARMA) model with recursive least-square (RLS) algorithmto estimate the parameters of the model. A simple decoupledself-tuning control law comprising of the estimation of link’snatural frequency for a single-link FLM under variable payloadis proposed in [8]. In [9], an adaptive pole placement controllaw has been proposed using a finite-dimensional ARMAmodel of a single-link FLM to control the tip trajectoryand tip deflection under unknown payload. The limitation ofthe above methods is that a linear model is considered toestimate the FLM dynamics, which is complex and nonlinear.Thus, the motivation of this paper is to propose a nonlinearidentified model-based adaptive controller, which will capturethe change in FLM dynamics due to change in payload and itwill be incorporated in the tuning of the controller parametersadaptively.

Identification of the nonlinear dynamics of the FLM isaccomplished in two steps. First, the model structure is chosenas a nonlinear ARMAX model and then the parameters ofthis model are identified. The FLM dynamics can be rep-resented as a NARMAX model if discrete-time differenceequation representation of the FLM is stable in bounded-input bounded-output sense according to [10]. The NARMAXmodel is a better modeling paradigm as compared with thestructure consisting of nonlinear time series representationlike Volterra and Hammerstein model, which needs a largenumber of parameters to describe nonlinear system dynamics,

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2 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY

thus making large representation and hence involve withmore computational time [10]–[13]. In addition, there havebeen several examples where NARMAX model-based systemidentification has been successfully accomplished for verycomplex nonlinear systems, for example, [11] used NARMAXmodel for representing nonlinear ankle dynamics systemsand [12] for a rigid manipulator dynamics. In addition, softcomputing approaches, such as differential evolution-basedneural network, have been used for system identification usinga NARMAX model in [13]. The generated NARMAX modelis then used to design adaptive controllers. These algorithmshave been employed to different systems, for example, acontinuous stirred-tank reactor in [14], a pilot-scale levelplus temperature control system in [15], a predictive controlstrategy for nonlinear NOx decomposition process in thermalpower plants in [16], and a state-space self-tuning control foran active fault-tolerant pulsewidth-modulation (PWM) trackerfor unknown nonlinear stochastic hybrid systems [17].

This paper describes how to control the tip trajectory whilequickly suppressing its deflection when subjected to handlevariable payloads for a two-link FLM (TLFM) using a modelpredictive control (MPC). In MPC, the current control actionis obtained by solving a finite-horizon open-loop optimalcontrol problem online, at each sampling instant, where theoptimization yields an optimal control sequence. Only the firstcontrol in this sequence is applied to the plant. MPC attractedattention of many researchers to exploit it as one of the mostrecommended advanced control algorithms [18], [19]. Thiscontrol concept has been applied successfully to control manycomplex systems, such as inverse unstable systems, open-loopunstable systems, and variable dead time processes [20]–[25].From the results obtained in [26]–[28], it can be observedthat MPC provides robustness with respect to modeling errors,over and under parameterization, and sensor noise for an FLMsystem but, restricted to a linear model. Hence, an attempt ismade here to develop a nonlinear MPC using a NARMAXmodel of the TLFM. Furthermore, NAMPC is made adaptivewith respect to change in the payload mass by estimatingthe parameters NARMAX of the TLFM model online. Theperformances of the proposed controller are also comparedwith a nonlinear self-tuning controller (STC) and popular NACusing both simulation and experimental studies.

The rest of this paper is organized as follows. Section IIpresents the dynamic model of the TLFM. Section IIIdescribes online identification of the TLFM dynamics usingNARMAX model. The description of the proposed NAMPCalong with the STC algorithm for the studied TLFM isalso described in Section IV. The simulation results are pre-sented in Section V. The experimental results are discussedin Section VI. The concluding remarks are presented inSection VII.

II. DYNAMIC MODEL OF THE TLFM

The schematic diagram of a planar TLFM is shown in Fig. 1,where τi is the actuated torque of the i th link, θi is the jointangle of the i th joint, and di (li , t) represents the deflectionalong i th link. The outer free end of the TLFM is attached with

Fig. 1. Schematic diagram of a planar TLFM.

payload mass, M p. δi and δi being the modal displacementand modal velocity for the i th link, respectively, and the actualoutput vector ypi is considered as the output for i th link insteadof θi to avoid the difficulty of nonminimum phase behavior ofthe FLM

ypi = θi +[

di (li , t)

li

]. (1)

Then, considering the system energy and the Lagrangianformulation approach along with the assumed mode method(AMM), the dynamics of the TLFM are derived. The dynamicmodel is developed using the Lagrangian approach, which isdefined as

d

d t

∂ L

∂ qi− ∂ L

∂ qi= τi (2)

where

L lagrangian expressed as difference between total kineticenergy KT and total potential energy UT of the TLFM;

τi generalized force at the i th joint;qi generalized coordinate of the i th link.

The i th generalized coordinate qi comprises of joint angles,joint velocities, and modal coordinates. The total kineticenergy of the i th link can be expressed as KTi = (total kineticenergy due to i th joint) + (total kinetic energy due to i thlink) + (total kinetic energy due to payload Mp) and in theabsence of gravity. The links are modeled as Euler–Bernoullibeams with deformation di (li , t) satisfying the i th link partialdifferential equation

(E I )i∂4di ( li , t)

∂ l4i

+ ρi∂2di ( li , t)

∂ t2i

= 0 (3)

where

ρi density of the i th link (i =1, 2);(EI)i flexural rigidity of the i th link;li length of the i th link;

The solution of (3) can be obtained by applying properboundary conditions at the base and the end of each link.Inertia–Inertia boundary condition (often referred to as thepseudopinned) and, it is locked in the vertical direction butfree to move in the angular direction with the help of a rotary


PRADHAN AND SUBUDHI: NAMPC FOR A FLEXIBLE MANIPULATOR 3

Fig. 2. Clamped-inertia boundary conditions.

actuator mounted on the base that does not provide a torqueto the link. Considering that each link is clamped at the end,the mass of the link is negligible compared with the mass ofthe payload, as shown in Fig. 2, it can be found that

(EI)i∂2di ( li , t)

∂ l2i

= −Jeqi

d2

dt2

(∂di ( li , t)

∂ li

)

(EI)i∂3di ( li , t)

∂ l3i

= Meqi

d2

dt2 (di ( li , t)) (4)

where Jeqi and Meqi are the mass and moment of iner-tia at the end of i th link. Since, (3) is a partialdifferential equation with respect to time and space coordinate.A finite-dimensional expression for di (li , t) can be representedusing the AMM [29]–[31] as

di ( li , t) =n∑

j=1

ϕi j ( li ) δi j (t) (5)

whereϕij j th mode shapes (spatial coordinate) of the i th link;δij j th modal coordinates (time coordinate) of the i th link;N number of assume modes.Substituting the expression for di (li , t) from (5) into the

solution of (3) yields a space eigenfunction of the form

δi j (t) = e jωi j t . (6)

The dynamic model of the TLFM used in designing theadaptive controller for the physical setup is derived based onthe AMM with clamped-mass shape functions given by (6) isrewritten as

ϕi ( li ) = C1, i sin (βi , li )+ C2, i cos (βi , li )

+C3, i sin h (βi , li )+ C4, i cos h (βi , li ) (7)

where ωi is the natural frequency of the i th link. On applyingthe boundary condition shown in Fig. 2, the constant coeffi-cients in (4) can be determined using (5) and from Fig. 4(b),one obtains (6). The solutions for βij are obtained from (7) asfollows:

det [ f (βi , li )] = 0 (8)

which gives a transcendental equation given as

(1 + cos (βi , li ) cos h (βi , li ))

− Meqiβi

ρi(sin (βi , li ) cos h (βi , li )− cos (βi , li ) sin h (βi , li ))

+ Jeqiβ3i

ρi(sin (βi , li ) cos h (βi , li )+ cos (βi , li ) sin h (βi , li ))

+ Meqi Jeqiβ4i

ρ2i

(1 − cos (βi , li ) cos h (βi , li )) = 0 (9)

where Meqi is the equivalent mass of i th link; Jeqi is theequivalent inertia of i th link.

Consider the schematic diagram of the TLFM shown inFig. 1. Let A1i be the cross-sectional area of the i th link.

In addition, let us assume a point p1(l1, t) on link 1 as

p1 ( l1, t) = R1 ( θ1)

[l1

ϕ1 ( l1) δ1 ( t)

](10)

where

R1 ( θ1) =[

cos θ1 − sin θ1sin θ1 cos θ1

].

Hence, kinetic energy due to link 1 is given as

(KTl

)1 = ρ1

2

∫ (Al1 pT

1 p1

)dl1. (11)

Similarly, for link 2

p2 ( l2, t) = R1 (θ1)

([l1

d1 ( l1, t)

]+ R2 (θ2)

[l2

d2 ( l2, t)

])

(12)

where

R2 ( θ2) =[

cos θ2 − sin θ2

sin θ2 cos θ2

], θ2 = θ2 + d1 ( l1, t) .

Therefore, kinetic energy due to link 2 is given by

(KTl

)2 = ρ2

2

∫ (Al2 pT

2 p2

)dl2. (13)

The kinetic energy due to i th joint is

(KTh

)i = Jhi θ

2i

2+ mhi pT

i pi

2. (14)

The kinetic energy due to payload mass Mp

KT p =Jp

2∑i=1

y2pi

2+ Mp pT

2 p2

2(15)

where

Mhi mass of i th joint;Jhi equivalent inertia of i th joint;Mp payload mass;Jp moment of inertia due to payload mass.

Thus, the total kinetic energy can be obtained as

(KT )i = 1

2

⎡⎢⎢⎢⎢⎢⎢⎢⎣

ρ1∫ (

Al1 pT1 p1

)dl1

+ρ2∫ (

Al2 pT2 p2

)dl2

+Jhi θ2i + mhi pT

i pi

+Jp

2∑i=1

y2pi

+ Mp pT2 p2

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(16)

and the total potential energy is given as

(UT )i =2∑

i=1

li∫0

(EI)i (li )

[d2 di (li , t)

d l2i

]2

d li . (17)

As a result, taking qi as the generalized coordinates, i.e.,qi = [

θi , θi , δi , δi]T

ii along with total kinetic energy and



total potential energy from (16) and (17), respectively, theLagrangian equation given in (2) can be rewritten as

d

d t

∂[( KT )i − (UT )i

]∂ qi

− ∂[( KT )i − (UT )i

]∂ qi

= τi (18)

where τi is given by

τi =[

I2×202×2

]ui . (19)

Hence, a finite solution to the link deformation as well as to(3) is obtained. As a result, using the Lagrangian equation in(2), the dynamic model of the TLFM is

M (θi , δi )

[θi

δi

]+

[c1

(θi , δi , θi , δi

)c2


)]

+ K[

0δi

]+ D

[0δi

]=

[τi

0

](20)

where

θi , θi joint angle and velocity of the i th joint;M inertia matrix;c1,c2 vectors containing of Coriolis and Centrifugal

forces;K stiffness matrix;D damping matrix.

(20) can be rewritten in state-space form as

x = fi (x)+ gi (x) ui (21)

with x as the state vector, i.e., x = [θi , θi , δi , δi

]T, and

fi (x) = M (θi , δi )−1

×(

−[

c1(θi , δi , θi , δi

)c2


)]

− K[

0δi

]− D

[0δi

])

gi (x) = M (θi , δi )−1 ui =

[τi

0

].

The discrete-time approximation of x(t) can be obtained usingforward difference approximation as

x (t) ∼= 1

T[x (kT + T )− x (kT )] (22)

where T is the sampling time and x(kT ) is the state vector.Similarly, the discrete-time representation of (21) using (22)in nonlinear form can be expressed as

Y (k) = Fni

[yi (k − 1) , yi (k − 2) , . . . , yi

(k − Ny

)ui (k − 1) , ui (k − 2) , . . . , ui (k − Nu )

](23)

where yi (k) denotes the tip position of the i th link, ui (k) isthe input to the i th joint, Ny and Nu represent the maximumdelay in the output and input vectors, respectively, and Fn [·]represents a multi-input, multi-output (MIMO) nonlinear mapof the TLFM input–output behavior.

Fig. 3. Structure of the NARMAX model of a planar TLFM.

Fig. 4. Structure for the estimation of NARMAX parameters for TLFM.

III. IDENTIFICATION OF THE TLFM USING

NARMAX MODEL

A noise term ξi (k) is added to (23) for representing theTLFM dynamics in form of NARMAX model. Hence, theNARMAX model for the TLFM can be written as

Y (k)=Fni

[yi (k − 1) , yi (k − 2) , . . . , yi

(k − Ny

),

ui (k − 1) , ui (k − 2) , . . . , ui (k − Nu) , ξi (k)

].

(24)

The structure for discrete-time nonlinear MIMO representationof TLFM as a NARMAX model is shown in Fig. 3, where

Yi (k) nonlinear autoregressive (NAR) vector;Ui (k) exogenous (X) vector;ξi (k) moving average (MA) variable vector;



Fi nonlinear function with i th input and output;N order of the nonlinearity.

Ny and Nu order of NAR, MA, and X, respectively.The NARMAX model (24) can be rewritten using the online

estimated parameters in its regressor form as

Y (k) = φTi(k) w (k)+ ξi (k) (25)

where

φTi(k)

=

⎡⎢⎢⎢⎢⎢⎢⎣

yi (k−1) , . . . , yi(k−ny

), y2

i (k−1) , . . . , y2i

(k−ny

),

ui (k−1) , . . . , ui (k−nu) , u2i (k−1) , . . . , u2

i

(k−ny

),

ξi (k) , yi (k−1) ui (k−1) , . . . , yi(k−ny

)ui (k−nu) ,

y2i (k − 1) ui (k − 1) , . . . , y2

pi2(k − ny

)u2

i (k − nu)T

y2i

(k − ny

)ui (k − nu) , y2

i (k − 1) u2i (k − 1) , . . . ,

yi(k − ny

)u2

i (k − nu) , y2i (k − 1) u2

i (k − 1) , . . . ,

⎤⎥⎥⎥⎥⎥⎥⎦

T

w

= [w1, w2, w3, w4, w5, w6, w7

]w1 link-1 inertia;w2 link-2 inertia;w3 hub-1 inertia;w4 hub-2 inertia;w5 link-1 equivalent mass;w6 link-2 equivalent mass;w7 total coupling mass between the links.

Fig. 4 shows the structure for the online estimation ofthe NARMAX model parameters where an ZOH is usedto discretize the continuous signals. The estimated value ofparameters w (k) of the NARMAX model is w (k), which canbe estimated using an RLS algorithm as

w (k) = w (k − 1)+ Pi (k − 1) φi (k)

λ+ φTi (k) Pi (k − 1) φi (k)

ξi (k) (26)

Pi (k) = 1

λ

{Pi (k − 1)− Pi (k − 1) φi (k) φT

i (k) Pi (k − 1)

λ+ φTi (k) Pi (k − 1) φi (k)

}

(27)

Yi (k) = φTi (k) w (k − 1)+ ξi (k) (28)

where λ is the forgetting factor and Pi (k) is the covariancematrix.

IV. PROPOSED ADAPTIVE CONTROL STRATEGIES

A. Nonlinear Adaptive MPC

To achieve desired tip trajectory tracking and damping oftip deflection for a variable payload, an NAMPC is proposed.

Fig. 5 shows the basic nonlinear MPC concept, where Np

is the prediction horizon and Nc is the control horizon. Thebasic idea of the proposed NAMPC to is to predict the vectorof future tip position using (25), i.e., NARMAX model inreal time, as shown in Fig. 3, so that the norm of future tiptrajectory error vector is minimized over a specific number offuture torque inputs. The j -step-ahead predicted NARMAXoutput can be constructed with the available sequence of past

Fig. 5. Basic nonlinear MPC concept.

torque inputs, past tip position outputs, and noise at samplingtime T as

S (k) =

⎡⎢⎢⎢⎣

Y (k + 1)Y (k + 2)

...

Y (k + j)

⎤⎥⎥⎥⎦ . (29)

Equation (29) can be represented in a more generalized formas

S (k) = Gφ (k)+ HU (k) (30)

where

G =

⎡⎢⎢⎢⎣

G11 G12 · · · G1N y

0 G22 · · · G2N y...

......

...0 0 0 GNy−1 N y

⎤⎥⎥⎥⎦

H =

⎡⎢⎢⎢⎣

H1H2...

HNu

⎤⎥⎥⎥⎦

U (k) =

⎡⎢⎢⎢⎣

ui (k − 1)ui (k − 2)

...ui (k − Nu )

⎤⎥⎥⎥⎦ .

The predicted output (30) consists of control inputs U(k)(present values) and U(k − 1),…,U(k-Nu) (past values). Thenonlinear functions Gφ (k) and H U (k) can be written inparametric form as

Gφ (k) =N j∑

k=1

[(wi f jkψi jk

)φ (k)

](31)

HU (k) =N jl∑k=1

[(wig jlkψi jlk

)(φ (k))

]. (32)

Hence, the j -step ahead prediction for the TLFM-NARMAXmodel (25) is rewritten using the parametric representation of



Fig. 6. Structure of the proposed adaptive NAMPC.

Gφ (k) and H U (k) as

Y (k)=N j∑

k=1

[(wi f jkψi jk

)(φ (k))

]+N jl∑k=1

[(wig jlkψi jlk

)(φ (k))

].

(33)

Equation (15) can be rewritten as

Y (k) =N j∑

k=1

wi f jk

N jl∑k=1

wig jlk

[(ψi jk

) (ψi jlk

)(φ (k))

] = wipψip

(34)

where

wip =N j∑

k=1

wi f jk

N jl∑k=1

wig jlk

[(ψi jk

) (ψi jlk

)(φ (k))

]

ψip =N j∑

k=1

N jl∑k=1

[(ψi jk

) (ψi jlk

)(φ (k))

]

wi f jk = [w1, w3, w5, w7

]wig jlk = [

w2, w4, w6, w7]

ψi jk = [yi (k − 1) , . . . , yi

(k − ny

)y2

i (k − 1) , . . . , y2i

(k − ny

) ]T andψi jlk = [

ui (k − 1) , . . . , ui (k − nu)

u2i (k − 1) , . . . , u2

i

(k − ny

) ]T.

The structure of the proposed NAMPC is shown in Fig. 6.Using the predicted output in (34) along with control inputU(k) and the desired tip trajectory R(k) for i th link, a costfunction J1 is defined that minimizes the tip position errorR (k) − Y (k) subjected to simultaneously minimizing thecontrol input U(k), given as

J1 =[

R (k)− Y (k)]T

KQ

[R (k)− Y (k)

]+ U T (k) K RU (k) (35)

where

R (k) =[θr1 (k) , θr1 (k + 1) , · · · θr1

(k + Np

)θr2 (k) , θr2 (k + 1) , · · · θr2

(k + Np

)]

U (k) =[

u1 (k) , u1 (k + 1) , · · · u1 (k + Nc − 1)u2 (k) , u2 (k + 1) , · · · u2 (k + Nc − 1)

]

KQ = diag[qi · · · qiNp

]

K R = diag[ri · · · riNC

].

The optimal control sequence over the prediction horizon Ny

can be obtained by minimizing the cost function J1 withrespect to the control input U(k). This can be achieved bysetting ∂ J1/∂U = 0. Taking the derivative of the cost function(35) with respect to the control input, one obtains

∂ J1

∂U=

[(R (k)− Y (k)

)TKQ

∂

∂U

(R (k)− Y (k)

)]

+[∂

∂U

(R (k)− Y (k)

)]T

KQ

+[

U T ∂

∂UK RU

]T

+[∂

∂UU

]T

K RU. (36)

Solving for the partial derivatives with respect to U(k) gives

∂

∂U

(R (k)− Y (k)

)= − ∂

∂UY (k) = −KY (37)

∂

∂U(K RU (k)) = K R (38)

and

∂

∂U

(U (k)T

)= I. (39)

Using (19)–(21) in (18) gives

∂ J1

∂U= −2K T

YKQ

(R (k)− Y (k)

)+ 2K T

R U (k) . (40)

Setting ∂ J1/∂U to zero for minimizing the cost function withrespect to U(k), and substituting Y (k) from (33) in (40), wehave

0 = −2K TY

KQ(R (k)− wipψip

) + 2K TR U (k) . (41)

Hence, the control input U(k) can be obtained as

U (k) =(

K TR

)−1 [(K T

YKQ

) (R (k)− wipψip

)]. (42)

Define a constant KU as

KU =(

K TR

)−1 [K T

YKQ

](43)

and the difference operator � = 1 − q−1 is used to calculatethe change in control input �U(k), where q−1 denotes one-sample delay operator. Now, using (43) in (42), (42) can berewritten in terms of �U(k) as

�U (k) = �KU(R (k)− wipψip

)U (k) = �KU

(R (k)− wipψip

) + U (k − 1) . (44)

The desired adaptive torque to the actuator of the i th jointis given by (44). The proposed algorithm of the NAMPC isshown in Fig. 7.

B. Self-Tuning Controller

A multivariable STC has three main elements such as:1) a control law generator in terms of multivariable differenceequation; 2) an online parameters estimator that uses measuredsystem output and input values; and 3) an algorithm that relatesthe estimated parameters and control parameters [32]. The



Fig. 7. Algorithm for the proposed adaptive NAMPC.

Fig. 8. Structure of the STC controller.

STC algorithm for the TLFM is described as follows. Let usrepresent NARMAX output as

Y (k)= 1

A(z−1

)[(bui (k−Nu) φ (k))+C

(z−1

)ξi (k)

](45)

where the polynomials A(z−1) and C(z−1) and parametersassociated with ui (k − Nu) are given for i th link as

A(

z−1)

= 1 + ai,1z−1 + ai,2z−2

C(

z−1)

= 1 + ci,1z−1 + ci,2z−2

b = [b1, · · · , bn]T .

The discrete-time multivariable PID control law is givenby

�ui (k)= Ki

⎡⎢⎢⎣

ei (k)− ei (k−1)

+ T2TI

{ei (k)− ei (k−1)}+ TD

T {ei (k)− ei (k−1)+ ei (k − 2)}

⎤⎥⎥⎦ (46)

where �ui (k) = ui (k)−ui (k −1) and ei (k) = ypi (k)−θri(k).Furthermore, (46) can be rewritten as

�ui (k) = Ki

⎡⎣

(1 + T

TI+ TD

T

)ei (k)−

(1 + 2TD

T

)

ei (k − 1)+ TDT ei (k − 2)

⎤⎦

⇒ �ui (k) =⎡⎣Ki

(1+ T

TI+ TD

T

)

−Ki

(1+ 2TD

T

)z−1+ Ki TD

T z−2

⎤⎦ ei (k) . (47)

Let Li (z−1) be a polynomial defined as

Li

(z−1

)= Ki

(1 + T

TI+ TD

T

)− Ki

(1 + 2TD

T

)z−1

+ Ki TD

Tz−2. (48)

Then, using (48) in (47), we have

�ui (k) = Li

(z−1

) (ypi − θri

). (49)

Equation (49) can be rewritten as

Li(z−1)(φT

i (k)w(k)) +�ui (k)− Li

(z−1)θri (k) = 0. (50)

The polynomial Li (z−1) for the STC law is tuned usingminimum variance control law. Fig. 8 shows the structureof the proposed NARMAX-based STC for TLFM, and thealgorithm of the STC is shown in Fig. 9. To tune the PIDparameters based on the principle of minimum variance, aperformance index J2 is considered as follows:

J2 =∑

�i[Qi

(z−1)(φT

i

(k)w(k)) + �i�ui − Ri

(z−1)θri

](51)

where � is the expectation operator, �i is the weightingfactor with respect to the control input in (47), and Qi (z−1)is the user-defined weighting polynomial with respect to thepredicted input of the form

Qi(z−1) = 1 + qi,1z−1 + qi,2z−2. (52)



Fig. 9. Algorithm for STC.

The control law minimizing the performance index J2 in(51) can be obtained as

Fi(z−1)(φT

i (k)w(k)) + {

Ei(z−1)Ci

(z−1) + �i

}�ui

−Qi(z−1)θri = 0 (53)

where Ei (z−1) and Fi (z−1) are found out by solving thefollowing Diophantine equation:

Qi(z−1) = Ai

(z−1)Ei

(z−1) + z−

(−di+1

)Fi

(z−1) (54)

where

Ei(z−1) = 1 + ei,1z−1 + · · · + ei,kmi z−di

Fi(z−1) = fi,0 + fi,1z−1 + fi,2z−2. (55)

Let Ei (z−1)+ Ci (z−1)+�i be defined as vi in (35) and nowmultiplying by v−1

i (34) becomes

Fi(z−1

)vi

(φT

i (k)w (k))

+�ui − Ri(z−1

)vi

θri = 0. (56)

Let Ri (z−1) = Fi (z−1), then (56) can be rewritten as

Fi(z−1

)vi

(φT

i (k)w (k))

+�ui − Fi(z−1

)vi

θri = 0. (57)

Equating (57) with (50) leads to

Fi(z−1

)vi

= Li(z−1) (58)

and based on (50), (51), and (58), the PID parameters Ki , TD,and Ti can be calculated as follows:

Ki = −(

fi,1 + 2 fi,2)

vi(59)

TIi = − fi,1 + 2 fi,2

fi,0 + fi,1 + fi,2T (60)

TDi = − fi,2

fi,1 + 2 fi,2T . (61)

V. RESULTS AND DISCUSSION

The simulation and experimentation studies were conductedto verify the efficacies of the proposed NAMPC. The NAMPCand two other existing controllers, namely STC and NAC,a regressor-based NAC [10], were applied to the TLFMavailable in Advanced Control and Robotics Research Lab-oratory, National Institute of Technology, Rourkela. Beforeimplementing the control algorithms, the derived model ofthe TLFM was validated, as presented in the following. Thephysical parameters of the TLFM are given in ref [33].

A. Model Validation

Open-loop responses of the TLFM were obtained throughboth numerical simulations using MATLAB/SIMULINK andexperiments on actual TLFM by exciting bang-bang torqueinputs to the TLFM are shown in Fig. 10. They are symmetricand their values are 0.042 Nm for Joint 1 and 0.008 Nm forJoint 2, respectively.

Joint 1 position response is shown in Fig. 11 from which itis observed that the joint responses has a maximum amplitudeof 10° and minimum amplitude of −5°. Joint 2 positionresponse is shown in Fig. 12. The maximum and minimumjoint-position amplitudes are 9.8° and −4°, respectively. InFig. 13, Link 2 tip deflection trajectory responses obtainedfrom the experiments are compared with that of obtained fromthe simulation. Fig. 13 shows that the maximum tip deflectionof 1.2 mm is noted in the experimental results, whereas thesimulation model shows maximum deflection of 1 mm.

From Figs. 10–13, it can be observed that the deriveddynamic model of the TLFM closely matches with the actualresponse of the physical TLFM system.

B. Evaluation of Control Performances

To validate the tip trajectory tracking performances ofthe NAMPC, the desired trajectory vector for two jointsθri (t) i = 1, 2 are chosen as

θri (t) = θi (t)+[6

t5

t5d

− 15t4

t4d

+ 10t3

t3d

](θt (t)− θi (t)) (62)



Fig. 10. Torque profile for (Joint 1 and 2).

Fig. 11. Joint 1 position.

Fig. 12. Joint 2 position.

where θri (t) = [θr1, θr2]T and θr (0) = {0, 0} are the initialpositions of the links, θt (0) = {π/4, π/6} are the finalpositions for Link 1 and 2, td is the time taken to reach thefinal position, which is taken 4 s, and the total simulation timeis set as 8 s. The control parameters are given in Table I.

Fig. 13. Link 2 tip deflection.

TABLE I

CONTROLLER PARAMETERS FOR TLFM

C. Simulation Results for Controller Performance With anInitial Payload of 0.157 kg

The performances of the proposed NAMPC and two existingadaptive controllers, namely STC and NAC adaptive con-trollers, while carrying a nominal payload of 0.157 kg arecompared in Figs. 14–17. Fig. 14 shows the tip deflectiontrajectory of Link 2 carrying a nominal payload 0.157 kg fromwhich it is observed that the maximum tip deflection amplitudeyielded by the STC is 0.09 mm, whereas 0.12 in the case of theNAC. However, NAMPC yields least maximum tip deflectionamplitude, i.e., 0.01 mm.

Fig. 15 shows the tip trajectory tracking error curve forLink 2 with a nominal payload of 0.157 kg. This figure showsthat there is a maximum tip tracking error amplitude of 0.6 mmin the case of NAC, 0.21 mm in the case of STC but NAMPCyields the least tracking error, i.e., 0.08 mm.

Figs. 16 and 17 show the control torque profiles generatedby NAC, STC, and NAMPC for Joint 1 and 2, respectively.From the joint toque trajectories for Joint 1 shown in Fig. 16,it is observed that the control torque generated by the NAC is1.03 Nm at 2 s, 0.86 Nm for STC, whereas on the other hand,NAMPC generates a smooth control input with maximumamplitude of 0.32 Nm at 2 s and then it becomes almost zeroat 4 s. For Joint 2, as shown in Fig. 17, the control input in thecase of NAC is 1.29 Nm at 3 s, 0.9 Nm at 3 s in the case ofSTC whereas for NAMPC, it is 0.32 Nm. Thus, it is clear thatNAMPC needs less control excitation for handling a payloadof 0.157 kg compared with NAC and STC.



Fig. 14. Simulation results for comparison of Link 2 tip deflectionperformances (0.157 kg): NAC, STC, and NAMPC.

Fig. 15. Simulation results for tip trajectory tracking errors (Link 2)(0.157 kg): NAC, STC, and NAMPC.

Fig. 16. Simulation results for torque profiles (Joint 1) (0.157 kg): NAC,STC, and NAMPC.

D. Simulation Results for Controller Performance Withan Additional Payload of 0.3 kg

To test the adaptive performance of the proposed NAMPC,an additional payload of 0.3 kg is now attached to theexisting initial payload of 0.157 kg making the overall payload


Fig. 18. Simulation results for comparison of Link 2 tip deflectionperformances (0.457 kg): NAC, STC, and NAMPC.

Fig. 19. Simulation results for tip trajectory tracking errors (Link 2)(0.457 kg): NAC, STC, and NAMPC.

0.457 kg. Performances of NAC, NAMPC, and STC for a0.457-kg payload are compared in Figs. 18–21.

Fig. 18 shows the tip deflections profiles of Link 2. FromFig. 18, it is observed that, due to change in payload, themaximum tip deflection amplitude is 0.16 mm in the case





of NAC and this controller fails to damp out the deflections.The maximum tip deflection amplitude is 0.08 mm for STCwhereas NAMPC damps out the tip deflection within 4 s with amaximum tip deflection of 0.01 mm. Fig. 19 compares the tiptrajectory tracking error curves for Link 2, which shows thatthere is a maximum tip tracking error amplitude of 1.63 mmin the case of NAC, 0.7 mm in the case of STC, whereas inthe case of NAMPC, it is 0.2 mm.

Fig. 20 shows the control torques generated by STC, NAC,and NAMPC for Joint 1 from this it is observed that thecontrol input generated by the STC and NAC have maximumamplitudes of 1.05 and 1.26 Nm, respectively, while on otherhand, for NAMPC, it is 0.4 Nm only. Fig. 21 shows the controltoques generated by the controllers for Joint 2. The maximumcontrol input is 1.75 Nm for NAC and 1.6 Nm for STC,whereas for NAMPC, it is 0.46 Nm. Thus, it is noted that theproposed NAMPC needs almost one-fourth torque comparedwith NAC and STC controllers.

VI. EXPERIMENTAL RESULTS

In this section, the experimental TLFM hardware setup withthe sensors, actuators, and digital processor are described in

Fig. 22. Photograph of the experimental setup of the TLFM.

detail. The setup has two links, two joints, and an end effecterto carry the payload. There is an arrangement available forconnecting payload at the end effector [33]. The photograph ofthe experimental setup is shown in Fig. 22. The hardware andsoftware components for the experimental setup are describedin the following.

A. Hardware Components

The hardware components of the TLFM experimental setupare shown in Fig. 23. It constitutes a data acquisition board(DAQ), two-channel linear current amplifier, a personal com-puter (PC) with Intel(R) core (TM) 2 DUO E7400 processorand operates at 2.8-GHz clock cycle, an interface board, aTLFM with digital optical encoders, and two strain gaugesat the base of each link. TLFM is provided with one pair offlexible links.

The TLFM is driven by two dc servo motors located atthe bottom of the hub and between the joint of two links.They are permanent magnet, brush-type dc servo motorsthat generate a torque τ i = Kti × Ii(t), where Ii(t) is themotor current for i th joint whereas Kt1 = 0.119 Nm/A andKt2 = 0.0234 Nm/A.

B. Software Components

Fig. 24 shows the interfacing of signals for the TLFMsetup, which works on MS Windows operating system withthe MATLAB/Simulink 2007a software. There is a provisionof real-time target logic code builder to interface the Simulinkmodel. The MATLAB code for controller is built up in thereal-time setup using the real-time target logic code in Clanguage.

C. Experimental Results for Controller PerformanceWith an Initial Payload of 0.157 kg

The experimental results for a nominal payload of 0.157 kgare shown through Figs. 25–28. The tip deflection responsesobtained by NAC, STC, and NAMPC with a nominal payloadof 0.157 kg is shown in Fig. 25 for Link 2. From Fig. 25,it is observed that the maximum tip deflection amplitudeyielded NAMPC, STC, and NAC are −0.13, −0.32, and−0.38 mm, respectively. Fig. 26 shows the tip trajectorytracking error curves for Link 2, which reveals that the tracking



Fig. 23. Schematic diagram of the experimental setup showing each hardware and payload arrangement.

Fig. 24. Interfacing of signals for the TLFM setup.

error is 0.66 mm for NAC, 0.25 mm for STC, whereas it is0.11 mm in the case of the NAMPC. Fig. 27 shows the controltorque profiles generated by NAC, STC, and NAMPC forJoint 1.

From Fig. 27, it is observed that NAMPC generates controltorque, which has a maximum amplitude of 0.3 Nm andreduces to almost zero after 4 s, whereas in the case of NAC,the maximum value of the torque signal is 1.4 Nm and itbecomes 0.7 Nm till the tip attains its final position, but in

the case of STC, the maximum amplitude is 1.1 Nm and thiseventually reduces to zero after 4 s with sustained oscillations.On the other hand, the torque profiles for Joint 2 (Fig. 28), theNAMPC yields a maximum torque of 0.5 Nm and reduces toalmost zero after 4 s whereas in the case of STC, the maximuminput control torque is −1.4 Nm that reduces to zero after 6 s.NAC generates a torque of maximum amplitude 2.3 Nm andit then reduces to a value of 0.5 Nm when the tip attains itsfinal position.



Fig. 25. Experimental results for comparison of Link 2 tip deflectionperformances (0.157 kg): NAC, STC, and NAMPC.

Fig. 26. Experimental results for tip trajectory tracking errors (Link 2)(0.157 kg): NAC, STC, and NAMPC.

Fig. 27. Experimental results for torque profiles (Joint 1) (0.157 kg): NAC,STC, and NAMPC.

D. Experimental Results for Controller PerformanceWith an Additional Payload of 0.3 kg

An additional payload of 0.3 kg was added making theoverall payload as 0.457 kg. Performances of NAMPC,


Fig. 29. Experimental results for comparison of Link 2 tip deflectionperformances (0.457 kg): NAC, STC, and NAMPC.

Fig. 30. Experimental results for tip trajectory tracking errors (Link 2)(0.457 kg): NAC, STC, and NAMPC.

STC, and NAC with 0.457-kg payload were compared inFigs. 29–32. Fig. 29 shows the tip deflections pro-files of Link 2. This figure shows that STC and NACyield a maximum deflection of −0.45 and −0.55 mm,





respectively, and damp out the deflection within 7 s, respec-tively, whereas NAMPC yields a maximum deflection of0.3 mm and takes 4 s to damp out tip deflection. Fig. 30shows the tip trajectory tracking error curve for Link 2. Fig. 30shows that the maximum tracking error is 1 mm for STC,1.95 mm for NAC, whereas it is 0.27 mm in case of theNAMPC.

Figs. 31 and 32 show the control torque profiles generatedby NAC, STC, and NAMPC for Joint 1 and 2, respectively.From Joint 1 torque signals shown in Fig. 31, it is observedthat the maximum control input generated by NAC andSTC are 1.5 and 0.9 Nm, respectively, whereas the NAMPCgenerates smooth control signal with a maximum value of0.5 Nm. From Fig. 32, it is observed that the maximum controlsignal generated by the three controllers, NAC, STC, andNAMPC, are 2.08, 1.9, and 0.65 Nm, respectively. Thus, it canbe concluded from Figs. 31 and 32 that NAMPC needs lesscontrol excitation to control tip position and suppress the tipdeflection with an additional payload of 0.3 kg compared withSTC and NAC.

VII. CONCLUSION

The paper has presented a new NAMPC to control tiptrajectory and suppression of tip deflection for a TLFMwhile handling variable payloads. The design of the proposedNAMPC is based on the online identified NARMAX model.An STC, NAC, and the proposed NAMPC have been appliedsuccessfully to a flexible robot setup in the laboratory. Fromthe simulation and experimental results, it is established thatthe proposed NAMPC generates appropriate adaptive torque tocontrol tip trajectory tracking and suppression of tip deflectionfor the TLFM compared with STC and NAC, when the manip-ulator is asked to handle a variable payload. The reason forits superiority is because NAMPC generates optimal controlsequence by optimum tuning of its control parameters in realtime adaptively.

REFERENCES

[1] H. Geniele, R. V. Patel, and K. Khorasani, “End-point control of aflexible-link manipulator: An experimental study,” IEEE Trans. ControlSyst. Technol., vol. 5, no. 6, pp. 556–570, Nov. 1997.

[2] M. E. Stieber, M. McKay, G. Vukovich, and E. Petriu, “Vision-based sensing and control for space robotics applications,” IEEE Trans.Instrum. Meas., vol. 48, no. 4, pp. 807–812, Aug. 1999.

[3] A. Jnifene and W. Andrews, “Experimental study on active vibrationcontrol of a single-link flexible manipulator using tools of fuzzy logicand neural networks,” IEEE Trans. Instrum. Meas., vol. 54, no. 3,pp. 1200–1208, Jun. 2005.

[4] S. Kamalasadan and A. A. Ghandakly, “A neural network paralleladaptive controller for dynamic system control,” IEEE Trans. Instrum.Meas., vol. 56, no. 5, pp. 1786–1796, Oct. 2007.

[5] Q. Zhou, P. Shi, J. Lu, and S. Xu, “Adaptive output feedback fuzzytracking control for a class of nonlinear systems,” IEEE Trans. FuzzySyst., vol. 19, no. 5, pp. 972–982, Oct. 2011.

[6] P. Shi, Q. Zhou, S. Xu, and H. Li, “Adaptive output feedback control fornonlinear time-delay systems by fuzzy approximation approach,” IEEETrans. Fuzzy Syst., vol. 21, no. 2, pp. 301–313, Apr. 2012.

[7] M. R. Rokui and K. Khorsani, “Experimental results on discrete timenonlinear adaptive tracking control of a flexible-link manipulator,” IEEETrans. Syst. Man Cybern., vol. 30, no. 1, pp. 151–164, Feb. 2000.

[8] S. Yurkovich and A. P. Tzes, “Experiments in identification and controlof flexible-link manipulators,” IEEE Control Syst. Mag., vol. 10, no. 2,pp. 41–46, Feb. 1990.

[9] V. Feliu, K. S. Rattan, and B. H. Brown, “Adaptive control of a single-link flexible manipulator,” IEEE Control Syst. Mag., vol. 10, no. 2,pp. 29–33, Feb. 1990.

[10] J. H. Yang, F. L. Lian, and L. C. Fu, “Nonlinear adaptive control forflexible-link manipulators,” IEEE Trans. Robot. Autom., vol. 13, no. 1,pp. 140–148, Feb. 1997.

[11] S. Chen and S. A. Billings, “Representations of nonlinear systems:The NARMAX model,” Int. J. Control, vol. 49, no. 3, pp. 1013–1032,1989.

[12] S. Kukreja, L. H. L. Galiana, and R. E. Kearney, “NARMAX represen-tation and identification of ankle dynamics,” IEEE Trans. Biomed. Eng.,vol. 50, no. 1, pp. 70–81, Jan. 2003.

[13] J. S. H. Tsai, C. T. Yang, C. C. Kuang, S. M. Guo, L. S. Shieh, andC. W. Chen, “NARMAX model-based state-space self-tuning controlfor nonlinear stochastic hybrid systems,” J. Appl. Math. Model., vol. 31,no. 1, pp. 3030–3054, 2010.

[14] B. Subudhi and D. Jena, “A differential evolution based neural networkapproach to nonlinear system identification,” Appl. Soft Comput., vol. 1,no. 1, pp. 861–871, 2011.

[15] T. Yamamoto and S. L. Shah, “Design and experimental evaluation ofa multivariable self-tuning PID controller,” IEEE Proc. Control TheoryAppl., vol. 151, no. 5, pp. 645–652, Sep. 2004.

[16] H. Peing, W. Gui, H. Shioya, and R. A. Zou, “Predictive control strategyfor nonlinear NOx decomposition process in thermal power plants,”IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 36, no. 5,pp. 904–921, Sep. 2006.



[17] C. T. Wang, J. S. H. Tsai, C. W. Chen, Y. Lin, S. M. Guo, andL. S. Shieh, “An active fault-tolerant PWM Tracker for unknownnonlinear stochastic hybrid systems: NARMAX model and OKID basedstate-space self-tuning control,” J. Control Sci. Eng., vol. 1, pp. 1–22,Jan. 2010.

[18] D. W. Clarke, C. Mohtadi, and P. S. Tuffs, “Generalized predictivecontrol—Part I. The basic algorithm,” Automatica, vol. 23, no. 2,pp. 137–148, 1987.

[19] C. E. Garcia, D. M. Prett, and M. Morari, “Model predictive con-trol: Theory and practice—A survey,” Automatica, vol. 25, no. 3,pp. 335–348, 1989.

[20] M. Morari and J. H. Lee, “Model predictive control: Past, presentand future,” Comput. Chem. Eng., vol. 23, no. 4, pp. 667–682,1999.

[21] D. Nagrath, V. Prasad, and B. W. Bequette, “A model predictiveformulation for control of open-loop unstable cascade systems,” Chem.Eng. Sci., vol. 57, no. 3, pp. 365–378, 2002.

[22] R. Srhridhar and D. J. Cooper, “A novel tuning strategy for multivariablemodel predictive control,” ISA Trans., vol. 36, no. 4, pp. 273–280,1997.

[23] J. Yang, S. Li, X. Chen, and Q. Li, “Disturbance rejection of dead-time processes using disturbance observer and model predictive control,”Chem. Eng. Res. Des., vol. 89, no. 2, pp. 125–135, 2011.

[24] A. A. Patwardhan, G. T. Wright, and T. F. Edgar, “Nonlinear modelpredictive control of a distributed-parameter systems,” Chem. Eng. Sci.,vol. 47, no. 4, pp. 721–735, 1992.

[25] C.-H. Lu and C.-C. Tsai, “Adaptive predictive control with recurrentneural network for industrial processes: An application to temperaturecontrol of a variable-frequency oil-cooling machine,” IEEE Trans. Ind.Electron., vol. 55, no. 3, pp. 1366–1375, Mar. 2008.

[26] M. Hassan, R. Dubay, C. Li, and R. Wang, “Active vibration control of aflexible one-link manipulator using a multivariable predictive controller,”Mechatronics, vol. 17, pp. 311–323, Jan. 2007.

[27] P. Boscariol, A. Gasparetto, and V. Zanotto, “Model predictive controlof a flexible links mechanism,” J. Intell. Robot Syst., vol. 58, no. 2,pp. 125–147, 2010.

[28] P. Boscariol, A. Gasparetto, and V. Zanotto, “Vibration reductionin a single-link flexible mechanism trough the synthesis of anMPC controller,” in Proc. IEEE Int. Conf. Mechatron., Apr. 2009,pp. 1–6.

[29] A. de Luca and B. Siciliano, “Closed-form dynamic model of planarmultilink lightweight robots,” IEEE Trans. Syst. Man Cybern., vol. 21,no. 4, pp. 826–839, Jul./Aug. 1991.

[30] B. Subudhi and A. S. Morris, “Dynamic modeling, simulation andcontrol of a manipulator with flexible links and joints,” Robot. Auto.Syst., vol. 41, no. 4, pp. 257–270, 2002.

[31] H. Asada, Z.-D. Ma, and H. Tokumaru, “Inverse dynamics of flexiblerobot arms: Modeling and computation for trajectory control,” ASME J.Dyn. Syst., Meas., Control, vol. 112, no. 2, pp. 177–185, 1990.

[32] M. Agarwal and D. E. Seborg, “Self-tuning controllers for nonlinearsystems,” Automatica, vol. 23, no. 1, pp. 2009–2014, 1987.

[33] S. K. Pradhan and B. Subudhi, “Real-time adaptive control of a flexiblemanipulator using reinforcement learning,” IEEE Trans. Autom. Sci.Eng., vol. 9, no. 2, pp. 237–2749, Apr. 2012.

Santanu Kumar Pradhan (S’12–M’13) receivedthe M.Tech. degree in energy system from IIT,Roorkee, India, in 2009, and the Ph.D. degree inelectrical engineering from the National Institute ofTechnology, Rourkela, India, in 2013.

He is an Assistant Professor with the Depart-ment of Electrical and Electronics Engineering, BirlaInstitute of Technology, Ranchi, India. His currentresearch interests include adaptive control, nonlinearsystem identification, and robotics.

Bidyadhar Subudhi (M’94–SM’08) received thebachelor’s degree in electrical engineering from theNational Institute of Technology, Rourkela, India,the M.Tech. degree in control and instrumentationfrom IIT, Delhi, India, in 1988 and 1994, respec-tively, and the Ph.D. degree in control system engi-neering from the University of Sheffield, Sheffield,U.K., in 2003.

Currently he serves as a Professor in the Depart-ment of Electrical Engineering at National Instituteof Technology Rourkela, India. His research interests

include system identification and adaptive control, networked control system,control of flexible and under water robots.

nonlinear adaptive model predictive controller for a flexible

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