real-time implementation of neuro adaptive observer-based robust backstepping controller for twin...

J Control Autom Electr Syst (2014) 25:137–150DOI 10.1007/s40313-013-0098-y

Real-Time Implementation of Neuro Adaptive Observer-BasedRobust Backstepping Controller for Twin Rotor Control System

Bhanu Pratap · Shubhi Purwar

Received: 11 August 2013 / Revised: 20 November 2013 / Accepted: 12 December 2013 / Published online: 31 December 2013© Brazilian Society for Automatics–SBA 2013

Abstract In this paper, a robust backstepping controllerbased on the neuro adaptive observer for the twin rotormultiple-input-multiple-output (MIMO) system is designedand implemented in real time. The twin rotor MIMO system(TRMS) belongs to a class of nonlinear uncertain system hav-ing unstable, coupled dynamics. Nonlinearities of the TRMSare estimated using Chebyshev neural network. A tuningscheme based on Lyapunov theory of stability is developedwhich can guarantee the boundedness of tracking error andweights of the neural network. The proposed observer-basedcontrol guarantees the stability of the closed-loop adaptivesystem and the tracking errors converge to small residual setsin the presence of constraints on the control input. The effec-tiveness of the proposed observer-based robust controller isillustrated through simulation and experimental results. Thereal time implementation has been carried out on the real-time TRMS using MATLAB real-time tool box and Advan-tech PCI1711 card.

Keywords Backstepping technique · Chebyshevneural network · Nonlinear coupled systems · Observer-based controller · Twin rotor MIMO system

B. Pratap (B)Department of Electrical Engineering, National Instituteof Technology, Kurukshetra, Indiae-mail: [email protected]

S. PurwarDepartment of Electrical Engineering, M. N. National Instituteof Technology, Allahabad, Indiae-mail: [email protected]

1 Introduction

In the past decade, control design of nonlinear systems hasattracted an ever increasing interest. There have been signif-icant research efforts on intelligent control (Ge and Zhang2004; Ge et al. 1999; He et al. 1998), sliding mode control(Elmali and Olgac 1992; Byungkook and Woonchul 1998),robust adaptive control (Yao and Tomizuka 2001; Haddadet al. 2003; Lee and Lee 2004; Kwan and Lewis 2000),and backstepping control (Kwan and Lewis 2000; Zhanget al. 2000; Gong and Yao 2001; Huang and Chen 2004;Wang and Huang 2005). To enhance the control performanceof unknown/uncertain nonlinear systems, different kinds oftechniques can be integrated, utilizing respective advantagesin the control system design.

The modeling and control of the TRMS (2006) has gaineda lot of attention because the dynamics of the TRMS anda helicopter are similar in certain aspects (Khan and Iqbal2003, 2004; Kim et al. 2006). Due to unstable, nonlineardynamics and high coupling effect between two propellers,the control problem of the TRMS has been considered as achallenging research topic. In Wen and Lu (2008), a decou-pling control of TRMS using robust deadbeat control tech-nique is designed. The system is decoupled into two SISOsystems, and the cross couplings are considered as distur-bances. A robust deadbeat control scheme is applied to thetwo SISO systems and a controller is designed for each ofthem. This design is evaluated in simulations, and the finalresult is tested in real-time. Comparing with PID controllers,this method is easy to follow, and the results show that thisscheme has less overshoot, shorter settling time and is morerobust to cross-coupling disturbances. Feedback lineariza-tion controller has been proposed for TRMS in Sacki et al.(1999) and Mustafa and Iqbal (2004). Controller design forTRMS via exact state feedback linearization is reported in

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138 J Control Autom Electr Syst (2014) 25:137–150

Mustafa and Iqbal (2004). In this paper, the idea is to dividethe dynamics of the system into two subsystems, exact statefeedback linearization of subsystem-1 is done and desiredstate for subsystem-2 is obtained. Then a servo controller isdesigned to track the desired state assuming all the states areavailable for measurement. In Juang et al. (2008) PID controlusing presearched genetic algorithms for TRMS that utilizesevolutionary computation is presented. The presented controlscheme includes PID controllers with independent input. Inorder to reduce total error and control energy, all parametersof the controller are obtained by a real-value-type geneticalgorithm (RGA) with a system performance index as thefitness function. The system performance index was appliedto the integral of time multiplied by the square error criterionto build a suitable fitness function in the RGA. The work ofJuang et al. (2008) has been extended in Juang et al. (2011)where fuzzy compensators are applied to the PID controllers.Xilinx Spartan II SP200 FPGA (Field Programmable GateArray) is employed to construct a hardware-in-the-loop sys-tem for real-time control. In Juang et al. (2008), a compari-son of classical and intelligent control schemes for TRMS ispresented. In classical control, three of the most popular con-troller design techniques are utilized in this study. These arethe Ziegler–Nichols PID rule, the gain margin and phase mar-gin rule, and the pole placement method. Intelligent controldesign based on fuzzy logic system and genetic algorithm isgiven to improve the tracking accuracy of the TRMS. In Taoet al. (2010), a fuzzy sliding and fuzzy integral-sliding con-troller (FSFISC) is investigated to position the pitch and yawangles of TRMS. With the coupling effects, which are consid-ered as the uncertainties, the highly coupled nonlinear TRMSis pseudo decomposed into a horizontal subsystem and a ver-tical subsystem. The FSFISC consists of a fuzzy sliding con-troller and fuzzy integral-sliding controller for the horizontaland the vertical subsystem, respectively. The performancecomparisons with the PID approach using a modified RGAare provided to show that the FSFISC has better performancein the aspects of error and control indices. In Tao et al. (2010),the complex nonlinear functions of the TRMS are representedas proportional combinations of linear functions. Based onthis representation, the fuzzy model of the TRMS is obtained.A parallel distributed fuzzy LQR controller is designed forthe fuzzy model of the TRMS. Adaptive nonlinear modelinversion controls for TRMS are detailed in Rahideh et al.(2012). The development of an adaptive dynamic nonlinearmodel inversion control law for 1-DOF TRMS utilizing artifi-cial neural networks and genetic algorithms is reported. Sim-ulation results are presented to demonstrate the efficacy of thegiven technique. The experimental results are presented forthe same model inversion control law for 1-DOF TRMS. InMondal and Mahanta (2012), an adaptive second-order slid-ing mode controller for the TRMS is presented. The nonlinearmodel of TRMS is transformed into a quasi-linear parameter

varying (LPV) system and real-time LPV state observer andstate feedback controller is designed in Rotondo et al. (2013)using LPV pole placement method. A nonlinear autopilotcontrol for 2-DOF helicopter is proposed in Kaloust et al.(1997). In Tee et al. (2008), an adaptive neural network con-trol for helicopters in vertical flight is presented. Most of themdeal with simulation studies except for Rahideh et al. (2012)and Rotondo et al. (2013). Moreover, all state variables areassumed to be measurable which is practically not feasible.One of the solutions is to design of observer. As the separa-tion principle does not hold for nonlinear systems, designingan observer-based controller is complex and difficult. In theliterature, the observer-based controller design for TRMShas not received much attention. Based on linearized model,proportional-integral-differential (PID) controllers are pre-sented for TRMS in Tao et al. (2010), Rahideh et al. (2012),and Kaloust et al. (1997). These linearized results only pro-vide local stability and are liable to be unstable with unstruc-tured perturbations and uncertainties. Moreover, tuning thegains of PID controllers is noted to be tedious. In literature,most of the papers deal with simulation studies where all statevariables are assumed to be measurable which is practicallynot feasible. Controller design techniques for nonlinear sys-tems generally require the knowledge of true/estimated val-ues of all the state variables. One of the solutions is to designan observer. As the separation principle does not hold fornonlinear systems, designing an observer-based controller iscomplex and difficult. In the literature, the observer-basedcontroller design for TRMS has not received much attention.

The observer-based controller design for nonlinear sys-tems problem becomes more challenging in the presence ofunknown nonlinearities. The growing need of the industryfor tackling complex problems and the capability of NNsfor approximating functions and dynamical systems Funa-hashi (1989) and Hornick et al. (1989) have motivated NN-based identification and control approaches. An observerfor flexible-joint manipulators using NN is proposed inAbdollahi et al. (2006). Most of the observer-based con-troller design techniques using NN are based on multilayerfeed-forward networks such as multilayer perceptron (MLP)trained with backpropogation or more efficient variations ofthis algorithm Ge and Zhang (2003) and Sun et al. (2001). Asan alternative to MLP, there has been considerable interest inradial basis function (RBF) networks, primarily because ofits simpler structure. A RBF network has been proposed forobserver-based controller design of nonlinear systems Ge etal. (1999). In these networks, choosing an appropriate set ofRBF centers for effective learning still remains a problem.The observer-based controllers in Ge et al. (1999), Leu et al.(1999), and Leu et al. (2005) are designed for single inputsingle output nonlinear system.

CNN is a functional link neural network (FLN) whoseinput is generated by use of a subset of Chebyshev

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polynomial. The back-propagation learning algorithm is usedin the CNNs. CNN has been shown to be capable to approxi-mate any continuous functions over a compact set to arbitraryaccuracy (Namatame and Ueda 1992; Lee and Jeng 1998;Patra and Kot 2002). CNN’s have been applied for attitudecontrol of a single spacecraft (Zou et al. 2010, 2011). Simi-larly, Purwar et al. (2007) and Purwar et al. (2008) establishthe efficacy of CNN in the areas of on-line system identifica-tion and tracking controller for robot manipulators, respec-tively. The real-time implementation of CNN observer forTRMS is reported in Shaik et al. (2011). The proposed paperutilizes the observer implemented in Shaik et al. (2011) fordesigning a robust backstepping controller for the TRMS. Anexperimental evaluation of three different kinds of real-timestate observers for TRMS is presented in Pratap and Purwar(2013).

Motivated by the above observations, an adaptiveobserver-based robust backstepping controller (RBC) forTRMS is presented in this paper. To the best of our knowl-edge there is no literature available on the real-time imple-mentation of observer-based nonlinear controller for TRMS.Thus, this work is novel from application and experimentalpoint of view. The proposed observer-based controller doesnot necessitate exact knowledge of the system nonlinearities.The CNN is used for estimating the unknown nonlinearitiesof the TRMS. The adaptation laws for the CNN weights aresuch that they guarantee the stability of the overall scheme.The magnitude of the tracking error depends mainly on theCNN, feedback functions to be used for the weight adaptationlaw and other design parameters. The experimental resultsobtained reveal that the proposed control strategy gives goodtracking performance.

The remainder of the paper is arranged as follows. InSect. 2, the preliminaries comprising of modeling of TRMSwith system parameters and structure of CNN is presented.The problem statement is introduced in the Sect. 3. The neuroadaptive observer is presented in Sect. 4. NAO-based robustadaptive backstepping controller is design in Sect. 5. Sec-tion 6 validates the performance of the proposed observer-based controller through experimental results. Finally, con-clusions are given in the Sect. 7.

2 Preliminaries

2.1 Modeling of TRMS

The TRMS mechanical unit has two rotors placed on a beamtogether with a counterbalance whose arm with a weight atits end is fixed to the beam at the pivot and it determinesa stable equilibrium position as shown in the Fig. 1. Thebeam is pivoted on its base in such a way that it can rotatefreely both in the horizontal and vertical planes. Either the

Fig. 1 The twin rotor MIMO system

horizontal or the vertical degree of freedom (DOF) can berestricted to 1 degree of freedom using nylon screws foundnear pivot point. At both ends of the beam there are rotors(the main and tail rotors) driven by dc motors. The main rotorproduces a lifting force allowing the beam to rise verticallymaking a rotation around the pitch axis. While, the tail rotoris used to make the beam turn left or right around the yawaxis. The whole unit is attached to the tower allowing forsafe helicopter control experiments. Apart from the mechan-ical units, the electrical unit (placed under the tower) playsan important role for TRMS control. It allows for measuredsignals transfer to the PC and control signal application viaan I/O card. The mechanical and electrical units provide acomplete control system setup. This device is a multivari-able, nonlinear and strongly coupled system, with degrees offreedom on the pitch and yaw angle denoted by ψ and ϕ,respectively.

The state of the beam is described by four process vari-ables: horizontal and vertical angles measured by positionsensors fitted at the pivot, and two corresponding angularvelocities. Two additional state variables are the momentumof the dc motors. In a normal helicopter, the aerodynamicforce is controlled by changing the angle of attack. The lab-oratory setup in Fig. 1 is so constructed that the angle ofattack is fixed. The aerodynamic force is controlled by vary-ing the speed of the rotors. Therefore, the control inputs arethe supply voltage of the dc motors. A change in the voltagevalue results in a change in the rotation speed of the propeller.This further results change in the corresponding position ofthe beam. The system parameters of the TRMS are given inTable 1.

The complete dynamics of the TRMS system (TRMS2006; Shaik et al. 2011) represented in the state-space formas follows:

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Table 1 TRMS system parameters

Parameters Values Parameters Values

I1 = Moment of inertia of vertical rotor 6.8 × 10−2 kg − m2 k1 = Motor 1 gain 1.1

I2 = Moment of inertia of horizontal rotor 2 × 10−2 kg − m2 k2 = Motor 2 gain 0.8

a1 = Static characteristic parameter 0.0135 T11 = Motor 1 denominatorparameter

1.1

b1 = Static characteristic parameter 0.0924 T10 = Motor 1 denominatorparameter

1

a2 = Static characteristic parameter 0.02 T21 = Motor 2 denominator 1

b2 = Static characteristic parameter 0.09 T20 = Motor 1 denominatorparameter

1

Mg = Gravity momentum 0.32 N − m Tp = Cross reaction momentumparameter

2

B1ψ = Friction momentum function parameter 6 × 10−3 N − m − s/rad T0 = Cross reaction momentumparameter

3.5

B1ϕ = Friction momentum function parameter 1 × 10−1 N − m − s/rad kc = Cross reaction momentumgain

−0.2

kgy = Gyroscopic momentum parameter 0.05 s/rad u1 and u2= Input voltage applied tomain and tail rotor are bounded

±2.5 V

d

dtψ = ψ (1a)

d

dtψ = a1

I1τ 2

1 + b1

I1τ1 − Mg

I1sin(ψ)− B1ψ

I1ψ

+ 0.0326

2I1sin(2ψ)ϕ2 − kgy

I1cos(ψ)ϕ(a1τ

21 + b1τ1)

(1b)

d

dtϕ = ϕ (1c)

d

dtϕ = a2

I2τ 2

2 + b2

I2τ2− B1ϕ

I2ϕ− 1.75

I2kc(a1τ

21 + b1τ1) (1d)

d

dtτ1 = −T10

T11τ1 + k1

T11u1 (1e)

d

dtτ2 = −T20

T21τ2 + k2

T21u2 (1f)

y = [ψ ϕ

]T(1g)

whereψ is the pitch (elevation) angle, ϕ is the yaw (azimuth)angle, τ1 is the momentum of main rotor, τ2 is the momentumof tail rotor, u1 is the voltage applied to main rotor, u2 is thevoltage applied to tail rotor.

2.2 Structure of Neural Network

The Neural Network structure used in this paper is a singlelayer Chebyshev neural network. CNN is a functional linknetwork (FLN) based on Chebyshev polynomials. One wayto approximate a function by a polynomial is to use a trun-cated power series. The power series expansion represents

the function with very small error near the point of expan-sion, but the error increases rapidly as we employ it at pointsfarther away. The computational economy to be gained byChebyshev series increases when the power series is slowlyconvergent. Therefore, Chebyshev series are frequently usedfor approximations to functions and are much more efficientthan other power series of the same degree. Among orthog-onal polynomials, the Chebyshev polynomials occupy animportant place, since, in the case of a broad class of func-tions, expansions in Chebyshev polynomials converge morerapidly than expansions in other set of polynomials. Hence,we consider the Chebyshev polynomials as basis functionsfor the neural network.

The Chebyshev polynomials can be generated by the fol-lowing recursive formula (Lee and Jeng 1998)

Ti+1(x)=2x Ti (x)−Ti−1(x), T0(x) = 1, T1(x)= x (2)

where Ti (x) is a Chebyshev polynomial, i is the order ofChebyshev polynomials chosen and here x is scalar quantity.T1(x) can have several definitions that includes x, 2x, 2x−1,and 2x + 1. In this paper, T1(x) is chosen as x .

For example the structure of the CNN with two inputs andtwo outputs are given in Fig. 2. An enhance pattern using theChebyshev polynomials is obtained as

φ(x) = [1 T1(x1) T2(x1) . . . T1(x2) T2(x2) . . .]T

(3)

where Ti (x j ), is a Chebyshev polynomial, i is the order of theselected Chebyshev polynomial and j = 1, 2. φ(x) denotesthe Chebyshev polynomial basis function.

Referring to Fig. 2, the architecture of the CNN consistsof two parts (Lee and Jeng 1998); namely, numerical trans-formation part and learning part. Numerical transformation

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ChebyshevPolynomial

Basis

1x

2x

ˆ TW1y

2y

Σ

Σ

Fig. 2 Structure of CNN

deals with the input to the hidden layer by approximatetransformable method. The transformation is the functionalexpansion of the input pattern comprising of a finite set ofChebyshev polynomials. As a result the Chebyshev polyno-mial basis can be viewed as a new input vector. The learningpart is a functional-link neural network based on Chebyshevpolynomials.

Based on the approximation property of CNN Purwar et al.(2007), a continuous nonlinear function y(x) can be approx-imated by CNN as

y = W T φ + ε (4)

where ε is the bounded CNN approximation error, W is theoptimal weight matrix.

The CNN is a single-layered neural network, and in gen-eral, its learning is fast (Namatame and Ueda 1992; Lee andJeng 1998). The output of the CNN is given by

ˆy = [ ˆy1 ˆy2]T = ˆW T φ (5)

where ˆW is the estimate of the optimal weight W .

3 Problem Statement

The TRMS described by the state space representation in (1)can be rewritten as

x = Ax + f (x, u)+ τdis

y = Cx (6)

where x = [ψ ψ ϕ ϕ τ1 τ2

]T = [x1 x2 x3 x4 x5 x6

]T

is the state vector, u = [u1 u2

]Tis the input vector, and

y = [x1 x3

]Tis the output vector, τdis ∈ �6 represents

the disturbances which are bounded. A, C , and f (x, u) aregiven by

A =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 1 0 0 0 0

0 − B1ψI1

0 0 b1I1

00 0 0 1 0 0

0 0 0 − B1ϕI2

−1.75 kcb1I2

b2I2

0 0 0 0 − T10T11

0

0 0 0 0 0 − T20T21

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

C =[

1 0 0 0 0 00 0 1 0 0 0

]

f (x, u) =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0a1I1

x25 − Mg

I1sin(x1)+ 0.0326

2I1sin(2x1)x2

4

− kgyI1

cos(x1)x4(a1x25 + b1x5)

0a2I2

x26 − 1.75

I2kca1x2

5k1T11

u1k2T21

u2

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

≡

⎡

⎢⎢⎢⎢⎢⎢⎣

0g1

0g2

f5

f6

⎤

⎥⎥⎥⎥⎥⎥⎦

where A comprises the linear terms of the TRMS andf (x, u) ∈ �6 represents the unknown nonlinearities whichare approximated using CNN.

The objective of this paper is to design and implementin real time, neuro adaptive observer-based robust back-stepping controller for TRMS, that forces the plant output

y = [y1 y2

]Tto track a specified smooth reference trajec-

tory yd = [yd1 yd2

]Ti.e., lim

t→∞(y − yd) = 0, subjected to

the constraint |ui | ≤ 2.5 for i = 1, 2.

4 Neuro Adaptive Observer Design

In order to guarantee that the solution to the differential equa-tion described by Eq. (6) exists and is unique for any initialcondition x0 ∈ X and u ∈ U we impose the following mildassumptions (Purwar et al. 2007):

Assumption 1 For any u ∈ U and any finite initial conditionx0 and any finite T > 0, we have‖x(T )‖ < ∞ and‖y(T )‖ <∞. The symbol ‖•‖ denotes the Euclidean norm, i.e., givena vector v = (v1, v2, . . . , vn), the Euclidean norm of v isgiven by

‖v‖ =√√√√

n∑

k=1

v2k

Assumption 2 The vector fields f : X × U → �6 is con-tinuous with respect to their arguments, and satisfies a localLipschitz condition so that the solution x of the differentialequation (6) is unique for any initial condition x0 ∈ X andu ∈ U .

The neuro adaptive observer is given by Shaik et al. (2011)

˙x = Ax + f (x, u)+ L[y − y

]

y = Cx (7)

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where x denotes the state of the observer, f (x, u) is theestimate of f (x, u), which is estimated using CNN. and

L =[

L11 L21 L31 L41 L51 L61

L12 L22 L32 L42 L52 L62

]T

is the observer

gain selected such that Ao = (A − LC) is a Hurwitz matrix.The nonlinear function f can be approximated as f using

CNN is given by

f = W Tf φ f (x, u)+ ε f (x, u) (8)

where ε f is the bounded CNN approximation error, W f isthe optimal weight, and φ f (x, u) is the basis function.

f = W Tf φ f (x, u) (9)

where W f is the estimate of the W f and φ f (x, u) is the basisfunction.

Define the observer error as

x = x − x (10)

Differentiating (10) and using (8) and (9) gives

˙x = Ao x + W Tf φ f (x, u)− W T

f φ f (x, u)+ ε f + τdis

˙x = Ao x + W Tf φ f (x, u)+ W T

f

[φ f (x, u)− φ f (x, u)

]

+ ε f + τdis (11)

where W f = W f − W f .

Theorem 1 Consider the TRMS (6) and NAO (7) satisfyingAssumptions 1 and 2. If the weights of the CNN are updatedaccording to adaptation law (Shaik et al. 2011)

˙W f = −η1φ f (x, u){

yT C A−1o

}− ρ1 ‖y‖ W f (12)

where η1 is the learning rate and ρ1 is damping coefficient,then the state estimation error x = x − x , weight errorW f = W f − W f and output estimation error y = y − y areuniformly ultimately bounded (UUB).

Proof The detailed proof of the above theorem is given inShaik et al. (2011), where the positive definitive Lyapunovfunction is chosen as

Vo = 1

2x T P x + 1

2tr

{W T

f W f

}(13)

The following condition on x guarantees the negative semi-definiteness of Vo

‖x‖ > 2

λmin Q

[‖P‖ωF + (ρ1 ‖C‖ − k2

1)k22

]≡ bx

and ρ1 ≥ k21

‖C‖ (14)

In fact Vo is negative definite outside the ball with radius bx

described as χ = { x | ‖x‖ > bx } and x is UUB.

5 NAO-Based Robust Backstepping Controller DesignAnd Stability Analysis

NAO (7) is rewritten in the following form for the implemen-tation of the robust backstepping controller.

˙X1 = X2 + L1 y˙X2 = A1 X2 + A2 X3 + g(X)+ L2 y˙X3 = A3 X3 + B1u + L3 y

y = X1 (15)

where X1 = [x1 x3

]T, X2 = [

x2 x4]T, X3 =

[x5 x6

]T,u =[

u1 u2]T, A1, A2, A3, B1, L1, L2, L3

and g(X) are given by

A1 =[

− B1ψI1

0

0 − B1ϕI2

]

, A2 =[

b1I1

0

− 1.75I2

kcb1b2I2

]

,

A3 =[

− T10T11

0

0 − T20T21

]

, B1 =[

k1T11

0

0 k2T21

]

L1 =[

L11 L12

L31 L32

], L2 =

[L21 L22

L41 L42

], L3 =

[L51 L52

L61 L62

]

g(X)=

⎡

⎢⎢⎢⎣

a1I1

x25 − Mg

I1sin(x1)+ 0.0326

2I1sin(2x1)x2

4

− kgyI1

cos(x1)x4(a1 x25 + b1 x5)

a2I2

x26 − 1.75

I2kca1 x2

5

⎤

⎥⎥⎥⎦

=[

g1

g2

]

where g(X) is assumed to be unknown nonlinearities whichis estimated by CNN.

The robust adaptive backstepping controller is designedin the following three steps Huang and Chen (2004):Step 1 Define the sliding surfaces

S1 = X1 − X1d , S2 = X2 − X2d and S3 = X3 − X3d (16)

where X1d represents the desired value of state X1 and S1

can be rewritten as

S1 = X1 + X1 − X1d

where X1 = X1 − X1. Let X1d = yd and using (15) and(16), the time derivative of S1 can be found as

S1 = S2 + X2d + L1 y + ˙X1 − yd (17)

where X2d can be regarded as a virtual control to stabilizethe dynamic system represented in (17) by selecting

X2d = −L1 y − ˙X1 + yd − c1S1

γ1(18)

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where c1 ∈ �2×2 is positive definite diagonal matrix, γ1 >

0 is the thickness of the boundary layer of the first slidingsurface.

Substituting X2d from (18), (17) becomes

S1 = S2 − c1S1

γ1(19)

Differentiating (18)

X2d =−L1 ˙y− ¨X1+ yd − c1

γ1(X2+L1 y + ˙X1− yd) (20)

Let,

X2d = X ′2d − d(y, X) (21)

where X ′2d = yd − c1

γ1(X2 + L1 y − yd) is known and

d(y, X) = L1 ˙y + ¨X1 + c1γ1

˙X1 is the unknown term estimatedusing CNN.Step 2 Using (15), (16), and (21), the time derivative of S2

is given as

S2 = A1 X2+ A2(S3+X3d)+ g+L2 y − X ′2d + d (22)

To stabilize (22) X3d is chosen as

X3d = A−12

[−A1 X2− g−L2 y+ X ′

2d −d−β1S1 − c2S2

γ2

]

(23)

where c2 ∈ �2×2 is positive definite diagonal matrix, β1 >

0, γ2 > 0, and d is estimates of unknown nonlinearity dgiven in (21).Using (21) and (23), (22) becomes

S2 = A2S3 + (d − d)− β1S1 − c2S2

γ2(24)

Substituting X ′2d from (21) and using (16), (23) is expanded

as

X3d = A−12

[−g−d+ yd + c1

γ1yd −

{c1

γ1+ c2

γ2+ A1

}X2

−{

c1

γ1L1+L2

}y−β1(X1−yd)+ c2

γ2X2d

](25)

Now X3d can obtained by differentiating (25)

X3d = A−12

[− ˙g− ˙d+ ...

y d + c1

γ1yd −

{c1

γ1+ c2

γ2+ A1

} ˙X2

−{

c1

γ1L1+L2

}˙y−β1(

˙X1− yd)+ c2

γ2X2d

](26)

Substituting X2d from (21) and ˙X2 from (15), (26) becomes

X3d = A−12

[− ˙g − ˙d + ...

y d +{

c1

γ1+ c2

γ2

}yd

−{

c1

γ1L1 + L2

}˙y −

{c1

γ1+ c2

γ2+ A1

}

× (A1 X2 + A2 X3 + g + L2 y)

−{

c1

γ1

c2

γ2+ β1

}(X2 + L1 y − yd)− c2

γ2d

](27)

Let,

X3d = X ′3d − h(y, X) (28)

where X ′3d is known and h(y, X) is unknown and are given

by

X ′3d = A−1

2

[...y d −

{c1

γ1+ c2

γ2+ A1

}(A1 X2+ A2 X3+L2 y)

+{

c1

γ1+ c2

γ2

}yd −

{c1

γ1

c2

γ2+ β1

}(X2 + L1 y− yd)

]

(29)

and

h(y, X) = A−12

[˙g + ˙d +

{c1

γ1+ c2

γ2+ A1

}g

+{

c1

γ1L1 − L2

}˙y{

c1

γ1L1 − L2

}˙y + c2

γ2d

].

(30)

The unknown term h(y, X) is estimated using CNN.Step 3 Using (15), (16), and (28), the time derivative of S3

is given by

S3 = A3 X3 + B1u + L3 y − X3d (31)

Substituting X3d in (31) gives

S3 = A3 X3 + B1u + L3 y − X ′3d + h(y, X) (32)

Stabilize (32) by choosing

u = B−11

[−A3 X3−L3 y+ X ′

3d −h−β2 A2S2−c3S3

γ3+us

]

(33)

where c3 ∈ �2×2 is positive definite diagonal matrix, β2 >

0, γ3 > 0, and h is estimates of h given in (30). A robust termus = −K msgn(S3) is chosen for the disturbance rejection,K ∈ �2×2 > 0, m = diag

[|S31|α , |sS32|α], with 0 <

α < 1 and S3 = [S31 S32

]T. Substituting (33) in (32), gives

S3 = (h − h)− β2 A2S2 − c3S3

γ3− K msgn(S3) (34)

Note that, the term β1S1 in (23) and β2 A2S2 in (33) compen-sate the effect of the coupling due to S2 in (19) and A2S3 in(24), respectively.

The nonlinear functions d and h can be approximated asd and h using CNN are given by

d = W Td φd(y, X)+ εd (35)

h = W Th φh(y, X)+ εh (36)

123


where εd , εh are the bounded CNN approximation errors,Wd , Wh , are the optimal weights and φd , φh , are the basisfunctions.

d = W Td φd(y, X) (37)

h = W Th φh(y, X) (38)

where Wd and Wh are the estimates of the Wd and Wh , respec-tively.

Multiplying both sides of (19) by β1β2 and of (24) by β2,and using (35–38), (19), (24), and (34) are rewritten as

β1β2 S1 = β1β2

{S2 − c1

S1

γ1

}(39)

β2 S2 = β2

{W T

d φd + A2S3 − β1S1 − c2S2

γ2+ εd

}(40)

S3 = W Th φh − β2 A2S2 − c3

S3

γ3− K m sgn(S3)+ εh

(41)

where W Td = W T

d − W Td and W T

h = W Th − W T

h . Define

S = [S1 S2 S3

]T, ε = [

0 εd εh]T

,

W T =diag{

0 W Td W T

h

}, β=diag

{β1β2 I2 β2 I2 I2

},

K =diag{β1β2

c1γ1β2

c2γ2

(c3γ3

+ K m‖S3‖

)}, φ=[

0 φd φh]T,

H =⎡

⎣0 β1β2 I2 0

−β1β2 I2 0 β2 A2

0 −β2 A2 0

⎤

⎦ .

Thus, the error dynamics (39–41) can be expressed in termsof the above quantities as

β S = β(W Tφ + ε)+ H S − K S. (42)

Note that, H S denote the coupling between the error dynam-ics in (42). The matrix H is skew-symmetric. The closed-loop stability analysis and the weight tuning algorithms willbe discussed in the next subsection.

Two standard assumptions, which are quite common inthe neural networks literature are given below (Kwan andLewis 2000):

Assumption 3 The optimal weights W f , Wd , and Wh arebounded by known positive values so that∥∥W f

∥∥

F ≤ WF (43)

‖Wd‖F ≤ WD and ‖Wh‖F ≤ WH

or equivalently

‖W‖F ≤ WC (44)

where W = diag{

0 Wd Wh}

and WC is known. We onlyneed to know that ideal weights exist to prove the conver-gence analysis. The exact value of the ideal weights need not

be known. The symbol ‖•‖F denotes the Frobenius norm,i.e., given a matrix A, the Frobenius norm is given by∥∥ A

∥∥2F = tr( AT A) =

∑

i, j

a2i j

Assumption 4 The desired trajectory yd and its derivativesup to third order are bounded.

The CNN weights are updated according to adaptationlaws

˙Wd = η2φd ST2 − ρη2 ‖S2‖ Wd

˙Wh = η3φh ST3 − ρη3 ‖S3‖ Wh

or in generalized form, ˙W = ηφST − ρη ‖S‖ W (45)

where W = diag{

0 Wd Wh

}is the estimate of W , η =

diag[

0 η2 I η3 I]

is the learning rate and ρ is damping coef-ficient. Based on the above assumptions, the stability analysisis given in the next section.

Theorem 2 Consider the TRMS (6) under the Assumptions1–4, the NAO (7), the observer-based controller (33) andCNN weight adaptation laws (12) and (45), guarantee that allthe signals in the close loop system are uniformly ultimatelybounded.

Proof Consider the Lyapunov function

V = 1

2x T P x + 1

2tr

{W T

f W f

}

︸︷︷︸Vo

+ 1

2STβS + 1

2tr

{βW T η−1W

}

︸︷︷︸Vc

≡ Vo + Vc (46)

where P is a positive definite matrix satisfying

ATo P + P Ao = −Q

where Q is positive definite matrix. Now differentiating (46)gives

V = 1

2

{x T P ˙x + ˙xT Px

}− tr

{W T

f˙W f

}

︸︷︷︸Vo

+ STβ S − tr{βW T η−1 ˙W

}

︸︷︷︸Vc

≡ Vo + Vc (47)

Substituting (11), (12), (42) and (45) in (47) gives

V = x T P[W T

f φ f (x)+ ω(t)]

+ x T P Ao x

+ tr[W T

f η1φ f (x){

yT C A−1o

}+ W T

f ρ1 ‖y‖ W f

]

− ST K S + ρ ‖S‖ tr{βW T W

}+ STβε (48)

where ω(t) = W Tf

[φ f (x)− φ f (x)

] + ε f

123


Define l1 = η1CT C A−1o , (48) becomes

V = −1

2x T Qx + x T P

[W T

f φ f (x)+ ω(t)]

+ tr[W T

f φ f (x)xT l1 + W T

f ρ1 ‖Cx‖ (W f − W f )]

− ST K S + ρ ‖S‖ tr{βW T (W − W )

}+ STβε (49)

Assume that the upper bounds of φ f (x), ω(t) and βε are∥∥φ f (x)

∥∥ ≤ φF , ‖ω(t)‖ ≤ ωF , and ‖βε‖ ≤ εC

Now applying the following inequalities (Abdollahi et al.2006; Lewis et al. 1999) to (49)

tr[W T

f (W f − W f )]

≤ WF

∥∥∥W f

∥∥∥ −

∥∥∥W f

∥∥∥

2

tr[W T

f φ f (x)xT l1

]≤ φF

∥∥∥W f

∥∥∥ ‖x‖ ‖l1‖

tr[βW T (W − W )

]≤ ‖β‖

(WC

∥∥∥W∥∥∥ −

∥∥∥W∥∥∥

2)

Now we can express (49) as

V ≤ −1

2λmin Q ‖x‖2 + ‖x‖ ‖P‖

[∥∥∥W f

∥∥∥φF + ωF

]

+ φF

∥∥∥W f

∥∥∥ ‖x‖ ‖l1‖

+[

WF

∥∥∥W f

∥∥∥ −∥∥∥W f

∥∥∥2]ρ1 ‖Cx‖

− λmin K ‖S‖2 + ρ ‖S‖{‖β‖

(WC

∥∥∥W

∥∥∥ −

∥∥∥W

∥∥∥

2)}

+ εC ‖S‖ (50)

Define k1 = ‖l1‖2

Adding and subtracting k21

∥∥∥W f

∥∥∥2 ‖x‖

V ≤ −1

2λmin Q ‖x‖2+‖x‖

[‖P‖ωF −

∥∥∥W f

∥∥∥(ρ1 ‖C‖−k2

1

)

+∥∥∥W f

∥∥∥ (‖P‖φF + φF ‖l1‖ + WFρ1 ‖C‖)

− k21

∥∥∥W f

∥∥∥2]

− ‖S‖[λmin K ‖S‖ + ρ ‖β‖

(∥∥∥W∥∥∥

2

− WC

∥∥∥W∥∥∥)

− εC

](51)

Define k2 = ‖P‖φF +φF ‖l1‖+WFρ1‖C‖2(ρ1‖C‖−k2

1)

Adding and subtracting k22 ‖x‖ and completing the square

for the term inside the square bracket in (52) yields

V ≤ − 1

2λmin Q ‖x‖2

+ ‖x‖[‖P‖ωF +

(ρ1 ‖C‖ − k2

1

)k2

2

−(ρ1 ‖C‖ − k2

1

) (k2 −

∥∥∥W f

∥∥∥)2 − (

k1∥∥W f

∥∥)2]

− ‖S‖[λmin K ‖S‖ + ρ ‖β‖

(∥∥∥W∥∥∥ − WC/2

)2

− ρ ‖β‖ (WC/2)2 − εC

](52)

The following conditions are guarantee the negative semi-definiteness of V

‖x‖ > 2

λmin Q

[‖P‖ωF +

(ρ1 ‖C‖ − k2

1

)k2

2

]≡ bx ,

ρ1 ≥ k21

‖C‖ (53)

and

‖S‖ > 1

λmin K

[ρ ‖β‖ (WC/2)

2 + εC

]≡ bs (54)

Thus, V is negative outside the outside the ball with radius bx

described as χ = { x | ‖x‖ > bx } and the ball with radius bs

described as Us ≡ { S| ‖S‖ ≤ bs}, where bs and bs are posi-tive constants. According to the standard Lyapunov theoremextension Pratap and Purwar (2013), this demonstrates that

‖x‖ , ‖S‖ ,∥∥∥W f

∥∥∥,∥∥∥W f

∥∥∥ and∥∥∥W

∥∥∥ are uniformly ultimately

bounded (UUB). The close loop system with the observer andcontroller is shown in Fig. 3. Blocks with dotted lines arepresenting TRMS plant, adaptive observer, and robust adap-tive backstepping controller. The dotted blocks are showingTRMS plant, NAO and robust backstepping controller. TheCNN-1 block has been used to estimate the unknown non-linearity f . Similarly CNN-2 and CNN-3 have been usedto estimate the unknown nonlinearity d and h, respectively.Which are utilized to design and implement the observer-based control effort u.

Fig. 3 Block diagram of NAObased RBC

( )18neq TRMS

CNN 2 CNN 3 CNN 1

C

C

( )23neq ( )33neq

( )7neq

, , ,d d d dy y y y, ,d d dy y y,d dy yContol effort u

2dX 3dX

2X 3X2S 3S

f x y

y

y

x

d h+−

Plant

Controller Observer

+ +− −

+−dy

1S

123


Fig. 4 TRMS with digital computer

6 Experimental Results

The complete set up of TRMS with digital computer is shownin Fig. 4. The TRMS consists of four main elements, PCwith a clocked control algorithm, A/D and D/A converters–serving as an interface between the PC and external envi-ronment, the controlled process and sensors. PC I 1711 Labis a universal Feedback unit having two blocks, Feedbackencoder block and Feedback DAC block. For the TRMS twoencoders are used thus the values of theψ and ϕ are returnedin Feedback encoder block. There are three parameters forthis block: sample time, i.e. 0.001 sec, channel one and chan-nel two offsets. Channel one refers to the first encoder outputψ and channel two to the second encoder outputϕ. The digitalinput value given to PC I 1711 is converted to analog out-put by Feedback DAC block. The proposed observer-basedrobust controller is implemented in real-time using MAT-LAB real-time tool box.

A detailed experimental study of the design and imple-mentation of observer based controller is carried out. Thedesign parameters chosen in the experiments are based onthe simulation studies.

For the experimental study, the desired trajectory yd formain and tail rotor are chosen as

y1d = 0.13 {sin(0.0225π t)+ sin(0.0675π t)

+ sin(0.1125π t)}

y2d = 0.1 {sin(0.0225π t)+ sin(0.0675π t)

+ sin(0.1125π t)+ 0.35}The initial conditions of the plant and observer are[

0 0 0 0 0 0]

and[

0.05 0 0.05 0 0 0]

respec-tively. CNN is used to approximate the unknown nonlinear-ities in the system. The tuning of the NN weights is doneonline.

Furthermore, to take care of uncertainties and faster con-vergence of error to zero, observer gain L is chosen high.

The closed-loop poles of the observer system are chosen as−5, −10, −20, −30, −40, −50, which are in the open lefthalf of the s plane and also provide adequate response time.The observer gain L is obtained as,

L =[

79.6 1841.6 0.6 14.5 7759.7 −2652.1−3.9 −148.7 68.6 922.8 −580 1019.6

]T

The design parameters of NAO based robust backsteppingcontroller are chosen as,

c1 =[

0.5 00 0.5

], c2 =

[0.1 00 0.1

],

c3 =[

0.2 00 0.2

], K =

[25 00 25

]

γ1 = γ2 = γ3 = 0.1, α = 0.8, β1 = β2 = 0.5, ρ =0.5, η1 = 300, η2 = 1, and η3 = 5

The initial weights of the neural network are selected aszeroes. The inputs to the CNN-1 are x and u. Every inputis expanded to 2 terms each thereby resulting in a total 17terms [refer to (3)]. Thus the basis function φ f (x, u) has adimension of (17 × 1). The output of CNN-1 is f (x, u) has adimension of (6 × 1). Thus the dimension of W f is (17 × 6)resulting in total 102 weights to be updated.

The inputs to CNN-2 are y and X2. Every input isexpanded to 2 terms each thereby resulting in a total 9 terms.

0 20 40 60 80 100

-0.2

-0.1

0

0.1

0.2

0.3

Time (sec)

Pitc

h T

rack

ing

(rad

)

desired trajectorypitch angle

Fig. 5 Pitch angle tracking (x1 and y1d )

0 20 40 60 80 100

-0.2

-0.1

0

0.1

0.2

0.3

Time (sec)

Yaw

Tra

ckin

g (r

ad)

desired trajectoryyaw angle

Fig. 6 Yaw angle tracking (x3 and y2d )

123


0 20 40 60 80 100-0.2

-0.1

0

0.1

0.2

Time (sec)

Pitc

h T

rack

ing

Err

or (

rad)

pitch tracking error

Fig. 7 Pitch angle tracking error (x1 − y1d )

0 20 40 60 80 100-0.2

-0.15

-0.1

-0.05

0

0.05

Time (sec)

Yaw

Tra

ckin

g E

rror

(ra

d)

yaw tracking error

Fig. 8 Yaw angle tracking error (x3 − y2d )

Thus the basis function φd(y, X) has a dimension of (9 × 1).The output of CNN-2 is d(y, X) has a dimension of (2 × 1).Thus the dimension of Wd is (9 × 2) resulting in total 18weights to be updated.

0 20 40 60 80 1000

1

2

3

4x 10

-3

Time (sec)

CN

N W

eigh

ts |

| Wca

pf ||

norm of CNN weights

Fig. 11 Norm of weights(∥∥∥W f

∥∥∥)

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

Time (sec)

CN

N W

eigh

ts ||

Wca

pd ||

norm of CNN weights

Fig. 12 Norm of weights(∥∥∥Wd

∥∥∥)

The inputs to CNN-3 are y and X . Every input is expandedto 2 terms each thereby resulting in a total 17 terms. Thusthe basis function φh(y, X) has a dimension of (17 × 1). The

Fig. 9 Pitch control effort (u1)

0 10 20 30 40 50 60 70 80 90 100-3

-2

-1

0

1

2

3

Time (sec)

Pitc

h C

ontr

ol E

ffor

t (vo

lt)

Fig. 10 Yaw control effort (u2)

0 10 20 30 40 50 60 70 80 90 100-3

-2

-1

0

1

2

3

Time (sec)

Yaw

Con

trol

Eff

ort (

volt)

123


0 20 40 60 80 1000

5

10

15

20

25

Time (sec)

CN

N W

eigh

ts ||

Wca

ph ||

norm of CNN weights

Fig. 13 Norm of weights(∥∥∥Wh

∥∥∥)

0 20 40 60 80 100

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time (sec)

Pitc

h T

rack

ing

(rad

)

desired trajectorypitch angle

Fig. 14 Pitch angle tracking (x1 and y1d )

0 20 40 60 80 100

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time (sec)

Yaw

Tra

ckin

g (r

ad)

desired trajectoryyaw angle

Fig. 15 Yaw angle tracking (x3 and y2d )

output of CNN-3 is h(y, X) has a dimension of (2 × 1). Thusthe dimension of Wh is (17 × 2) resulting in total 34 weightsto be updated.

Figures 5 and 6 shows that the outputs pitch and yawangles follow the desired trajectories y1d and y2d with a smallerror despite the unknown system dynamics. The trackingerrors for pitch and yaw angles are shown in Figs. 7 and8 respectively. In addition, Figs. 9 and 10, indicate that theactual control efforts u1 and u2 are within the limits ±2.5 asstated in section III. From Figs. 11, 12 and 13, it can seenthat the CNN weights W f , Wd , and Wh are bounded.

0 20 40 60 80 100-0.04

-0.03

-0.02

-0.01

0

0.01

Time (sec)

Pitc

h T

rack

ing

Err

or (

rad)

pitch tracking error

Fig. 16 Pitch angle tracking error (x1 − y1d )

0 20 40 60 80 100-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

Time (sec)

Yaw

Tra

ckin

g E

rror

(ra

d)

yaw tracking error

Fig. 17 Yaw angle tracking error (x3 − y2d )

0 20 40 60 80 100-3

-2

-1

0

1

2

3

Time (sec)

Pitc

h C

ontr

ol E

ffor

t (vo

lt)

pitch control effort

Fig. 18 Pitch control effort (u1)

A detailed simulation study of the observer-based con-troller has been carried out also. The design parameters forthe experimental results are chosen on the basis of simula-tion studies. The simulation and experimental analysis arecarried out for different sets of initial conditions. Howeverthe results are presented for a specific case. The simula-tion results are presented below in Figs. 14, 15, 16, 17, 18and 19.

The tracking of actual and desired pitch and yaw angle areillustrated in Figs. 14 and 15. The tracking error between theactual and observed pitch and yaw angle are shown in Figs. 16and 17. The actuator outputs u1 and u2 given in Figs. 18 and

123


0 20 40 60 80 100-3

-2

-1

0

1

2

3

Time (sec)

Yaw

Con

trol

Eff

ort (

volt)

yaw control effort

Fig. 19 Yaw control effort (u2)

19 are within the limit ±2.5V. The detailed simulation studyof the proposed observer shows reliable performance andacceptable computation time. On the basis of simulation andexperimental results it can be concluded that the proposedobserver-based controller shows reliable performance andacceptable computation time.

7 Conclusion

Real-time implementation of a nonlinear robust backstep-ping controller based on the neuro adaptive observer for theTRMS is presented in this paper. The proposed observer-based controller does not necessitate exact knowledge ofthe system nonlinearities. The unknown nonlinearities ofthe TRMS are estimated using CNN, which learns throughthe weight adaptation law derived from Lyapunov theoryof stability. The magnitude of the tracking error dependsmainly on the CNN, feedback functions to be used forthe weight adaptation law and other design parameters.The proposed observer based control guarantees the sta-bility of the closed-loop adaptive system and the track-ing errors converge to small residual sets in the presenceof constraints on the control input. The simulation stud-ies has been done for the tuning of design parameters(L , c1, c2, c3, K , γ1, γ2, γ3, α, β1, β2, ρ, η1, η2, andη3) which are chosen after several trials. Thus, simulationsas well as experimental results are demonstrated to show thevalidity of the proposed observer-based adaptive control sys-tem. Finally, simulations as well as experimental results aredemonstrated to show the validity of the proposed observer-based adaptive control system. The real time implementationhas been carried out on the real-time TRMS using MATLABreal-time tool box and Advantech PC I 1711 card.

Acknowledgments The authors acknowledge the contribution ofDepartment of Science and Technology, Government of India, NewDelhi, India through Project SR/S3/EECE/004/2008.

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