Available online at www.sciencedirect.com

Mathematics and Computers in Simulation 79 (2009) 1745–1753

Neural network-based robust adaptive control of nonlinearsystems with unmodeled dynamics

Dan Wang a,∗, Jialiang Huang b,c, Weiyao Lan d, Xiaoqiang Li b

a Marine Engineering College, Dalian Maritime University, Dalian 116026, PR Chinab Dalian Maritime University, Dalian 116026, PR China

c Jimei University, Xiamen 361021, PR Chinad Department of Automation, Xiamen University, Xiamen 361005, PR China

Received 10 May 2007; received in revised form 2 September 2008; accepted 18 September 2008Available online 30 September 2008

Abstract

A neural network-based robust adaptive control design scheme is developed for a class of nonlinear systems represented byinput–output models with an unknown nonlinear function and unmodeled dynamics. By on-line approximating the unknownnonlinear functions and unmodeled dynamics by radial basis function (RBF) networks, the proposed approach does not require theunknown parameters to satisfy the linear dependence condition. It is proved that with the proposed control law, the closed-loopsystem is stable and the tracking error converges to zero in the presence of unmodeled dynamics and unknown nonlinearity. Asimulation example is presented to demonstrate the method.© 2008 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords: Adaptive control; Neural networks; Nonlinear control; Robustness; Unmodeled dynamics

1. Introduction

In the past two decades, a great deal of progress in the feedback control of nonlinear systems has been achieved. Atthe same time, neural network (NN)-based adaptive control techniques are extensively studied. Motivated by the devel-opments in the two areas, in this paper, a neural network-based robust adaptive control approach for a class of nonlinearsystems with uncertain nonlinearities and unmodeled dynamics is studied. Adaptive control of nonlinear systems withparameter uncertainty and uncertain nonlinearities are studied by many researchers [11,1,15,20,16]. Adaptive controlfor nonlinear systems with unmodeled dynamics has drawn great research attention [23,2,4–9,13,14]. In [11], adap-tive control for a class of nonlinear systems is studied. The system under consideration is single-input–single-output,input–output linearizable, minimum phase, and modeled by an input–output model of the form of an n th-order dif-ferential equation. The uncertain nonlinear functions of the model depend linearly on constant unknown parameters.This is a wide class of nonlinear systems which includes as a special case the nonlinear systems treated in [15] and[10] for output feedback adaptive control and the linear systems treated in the traditional adaptive control literature,e.g., [3,17,22]. The dynamics of the system is extended by adding a series of integrator at the input side and the

∗ Corresponding author. Tel.: +86 411 8472 8286; fax: +86 411 8472 8286.E-mail address: dwangdl@gmail.com (D. Wang).

0378-4754/$32.00 © 2008 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.matcom.2008.09.002

1746 D. Wang et al. / Mathematics and Computers in Simulation 79 (2009) 1745–1753

augmented system is represented by a state-space model, where the states are the input, the output, and a numberof their derivatives. In the work of [11], a semiglobal controller is designed that achieves asymptotic output trackingfor reference signals which are bounded and have bounded derivatives up to the n th order. However, the adaptivecontroller is not robust to unmodeled dynamics. Improvements are reported in a later paper [1]. In which the resultsof [11] are extended to the case with bounded disturbance, but an upper bound on the disturbance must be known. Ina recent paper [13], a robust adaptive controller for a class of nonlinear systems represented by input–output modelscontaining unmodeled dynamics is presented. In the design of the adaptive controller, it is not necessary to know theupper bound of the disturbances. The systems considered in [11,1,13] have a restriction of linear dependence on theunknown parameters. The linear dependence condition is removed in [14]. Instead, a smooth nonlinear function mustbe known such that an unknown nonlinear function in the system is bounded by the product of the known nonlinearfunction and an unknown constant.

Neural network techniques have been found to be particularly useful for controlling nonlinear systems with non-linearly parameterized uncertainty or totally unknown nonlinear functions. One of the major neural network-basedadaptive approaches is based on the Lyapunovs stability theory which gives an adaptive control law with guaranteedstability properties of the closed-loop system. Representative work can be found in, to just name a few, [24,12,5,25] andthe references therein. Inspired by these works, in this paper, we further study the adaptive control problem for the sameclass of nonlinear systems considered in [11,1,13,14] and expand the results to a more general case by using neuralnetwork techniques. By on-line approximating the unmodeled dynamics by radial basis function (RBF) networks, wewill combine the feedback linearization technique and the neural network-based adaptive control technique to developa robust adaptive controller design method. The above-mentioned restriction in [14] will be removed. In addition,we will prove that the closed-loop system is stable and the tracking error converges to zero. An example is used todemonstrate our proposed approach.

2. Problem formulation and preliminaries

We consider a single-input–single-output nonlinear system described by

y(n) = f (y, y, . . . , y(n−1), u, u, . . . , u(m−1)) + 1

γu(m) + �(y, y, . . . , y(n−1), u, u, . . . , u(m−1)) (2.1)

where y is the output; u is the control; y(i) is the ith derivative of y; f is an unknown smooth nonlinear function; �(·)represents the uncertain nonlinearity and the unmodeled dynamics; and γ is an unknown constant parameter, but thesign of γ is known. Without loss of generality, we assume that γ > 0.

Let

x1 = y, x2 = y(1), . . . , xn = y(n−1)

z1 = u, z2 = u(1), . . . , zm = u(m−1) (2.2)

System (2.1) can be represented by

xi = xi+1, 1 ≤ i ≤ n − 1

xn = f (x, z) + 1

γv + �(x, z)

zi = zi+1; 1 ≤ i ≤ m − 1

zm = v

(2.3)

where v = u(m) is the control input for the augmented system (2.3), and x = [x1, . . . , xn]T , z = [z1, . . . , zm]T . Theinitial states of the integrators are chosen such that z(0) ∈ Z0, a compact subset of Rm. We assume that �(x, z) iscontinuous and satisfies �(0, z) = 0, and the reference signal yr(t) is bounded with bounded derivatives up to the n thorder and y(n)

r (t) is piecewise continuous. Denote

Yr = [yr, y(1)r , . . . , y(n−1)

r ]T

(2.4)

Our objective is to design a robust adaptive state feedback controller for (2.3) such that the closed-loop system isstable and the output y(t) of the system tracks the reference signal yr(t) asymptotically in the presence of unmodeleddynamics and the unknown nonlinear function.

D. Wang et al. / Mathematics and Computers in Simulation 79 (2009) 1745–1753 1747

Since the control coefficient γ is a constant, (2.3) is input–output linearizable by full state feedback for all x. It alsoguarantees that for every x, there is a globally defined normal form for (2.3). Let

ζi = zi − γxn−m+i, 1 ≤ i ≤ m (2.5)

From (2.3), we have

ζi = ζi+1, 1 ≤ i ≤ m − 1

ζm = −γf (x, z) − γ�(x, z)|zi=ζi+γxn−m+i

(2.6)

together with the first n state equations of (2.3), (2.6) define the global normal form. Setting x = 0 in (2.6) results inthe zero dynamics of (2.3). We further make the following assumption which includes a minimum-phase condition[11].

Assumption 1. The system (2.6) has the property that for any z(0) ∈ Z0 and any bounded x(t), the state ζ(t) is bounded.

From Assumption 1, we conclude that z is bounded.

Remark 1. In [14], it is assumed that f is unknown but a smooth nonlinear function f must be known for controldesign which satisfies:

|f (y, y, . . . , y(n−1), u, u, . . . , u(m−1))| ≤ βf (y, y, . . . , y(n−1), u, u, . . . , u(m−1)) (2.7)

where β > 0 is an unknown constant. In this paper, we do not need a known f , i.e., no prior knowledge of f can beused for control design. In the case, there is actually no difference between f and �(x, z)—both are totally unknownfor control law design. So we will treat them together as one unknown term later. We use a similar system description(2.1) to that used in [14] so that it is easy to see the relationship between this work and the existing results.

In this work, we assume that the state (x, z) of system (2.3) is available for feedback. Let e1 = x1 − yr, e2 =x2 − yr,. . ., en = xn − y(n−1)

r , e = [e1, e2, . . . , en]T . From (2.3), we get

e = Ae + b[f (e + Yr, z) + 1

γv − y(n)

r + �(e + Yr, z)]

z = Az + bv

(2.8)

where

A =

⎡⎢⎢⎢⎢⎣

0 1 . . . 0...

......

0 0 . . . 1

0 0 . . . 0

⎤⎥⎥⎥⎥⎦ b =

⎡⎢⎢⎢⎢⎣

0...

0

1

⎤⎥⎥⎥⎥⎦ (2.9)

and A, b have the same forms as A and b but with different sizes. Let Ac = A − bK, where K is chosen so that Ac isHurwitz. Then, we have

e = Ace + b[Ke + f (e + Yr, z) + 1

γv − y(n)

r + �(e + Yr, z)] (2.10)

In this paper, we employ RBF network to approximate the unknown nonlinear terms in the system. Before introducingour control design method, let us first recall the approximation property of the RBF neural networks [18,21]. The RBFneural networks take the form θT ξ(x) where θ ∈ RN for some integer N is called weight vector, and ξ(x) ∈ Rn is avector valued function defined in Rn. Denote the components of ξ(x) by ρi(x), i = 1, . . . , N, with N being the numberof neural network nodes and ρi(x) a basis function. In this work, ρi(x) is chosen as the commonly used Gaussianfunctions, which have the following form:

ρj(x) = 1√2πσ

exp

(−‖x − ζj‖2

2σ2

), σ ≥ 0, j = 1, . . . , N (2.11)

1748 D. Wang et al. / Mathematics and Computers in Simulation 79 (2009) 1745–1753

where ζj ∈ Rn, j = 1, . . . , N, are constant vectors called the center of the basis function, and σ is a real numbercalled the width of the basis function. According to the approximation property of the RBF networks [18,19,21], givena continuous real-valued function f : → R with ∈ Rn a compact set, and any δm > 0, by appropriately choosingσ, ζj ∈ Rn, j = 1, . . . , N, for some sufficiently large integer N, there exists an ideal weight vector θ∗ ∈ RN such thatthe RBF network θ∗T ξ(x) can approximate the given function f with the approximation error bounded by δm, i.e.,

f (x) = θ∗Tξ(x) + δ∗, x ∈ (2.12)

with |δ∗| ≤ δm, where δ∗ represents the network reconstruction error, i.e.,

δ∗ = f (x) − θ∗Tξ(x) (2.13)

Since θ∗ is unknown, we need to estimate θ∗ on-line. We will use the notation θ to denote the estimation of θ∗ anddevelop an adaptive law to update the parameter θ.

Remark 2. In our proposed approach, RBF neural networks are used to approximate the unstructured uncertaintiesin the system. However, the RBF neural networks can be simply replaced by any linearly parameterized networks suchas fuzzy systems, polynomial and wavelet networks [5].

3. Robust adaptive control design

In this work, f in (2.1) and (2.3) is totally unknown. In this case, there is no difference between f and �. So, theywill be treated as one unknown nonlinear function. Let

F = f (e + Yr, z) + �(e + Yr, z) (3.1)

Then, (2.10) becomes

e = Ace + b

[Ke + F + 1

γv − y(n)

r

](3.2)

Given a compact set (e+Yr,z) ∈ Rn+m which includes Z0 as a subset, let θ∗ and δ∗ be such that for any (e +Yr, z) ∈ (e+Yr,z):

F = θ∗Tξ(e + Yr, z) + δ∗ (3.3)

with |δ∗| ≤ δm.The following robust adaptive control law is proposed to solve the given tracking problem:

v = γ[−Ke − θT ξ + y(n)r − Sign(eT Pb)δm] (3.4)

where θ is the estimate of θ∗ and is updated as follows:

˙θ = �eT Pbξ (3.5)

with any constant matrix � = �T > 0. P is a matrix satisfying:

PAc + ATc P = −Q, Q = QT > 0 (3.6)

γ is the estimate of γ and is updated as follows:

˙γ = −eT Pb(−Ke − θT ξ + y(n)r − Sign(eT Pb)δm) (3.7)

Adaptive law (3.5) is used to on-line tune θ, i.e., to tune the RBF neural networks to approximate the unknownterms in the system in the way of (3.3).

Theorem 1. For any z(0) ∈ Z0, x(0) and Yr satisfying (e + Yr, z) ∈ (e+Yr,z), the proposed robust adaptive statefeedback controller (3.4) and adaptive laws (3.5) and (3.7) guarantee that the closed-loop system is stable and theoutput y(t) of the given system (2.3) converges to the reference signal yr(t) in the presence of unmodeled dynamics andunknown nonlinearity.

D. Wang et al. / Mathematics and Computers in Simulation 79 (2009) 1745–1753 1749

Proof. Let

θ:=θ∗ − θ, γ:=γ − γ

Consider the Lyapunov function candidate:

L = 1

2eT Pe + 1

2γγ2 + 1

2θT �−1θ (3.8)

Taking the time derivative of V along the solutions of (3.2), (3.4), (3.5) and (3.7) yields:

L = 1

2[eT Pe + eT Pe] − 1

γγ ˙γ − θT �−1 ˙

θ

= 1

2

{Ace + b

[Ke + F + γ

γ(−Ke − θT ξ + y(n)

r − Sign(eT Pb)δm) − y(n)r

]}T

Pe

+ 1

2eT P

{Ace + b

[Ke + F + γ

γ(−Ke − θT ξ + y(n)

r − Sign(eT Pb)δm) − y(n)r

]}− 1

γγ ˙γ − θT �−1 ˙

θ

= 1

2eT (AT

c P + PAc)e + eT Pb

[Ke + F + γ − γ

γ(−Ke − θT ξ + y(n)

r − Sign(eT Pb)δm) − y(n)r

]− 1

γγ ˙γ − θT �−1 ˙

θ

= − 1

2eT Qe + eT Pb(F − θT ξ − Sign(eT Pb)δm) − eT Pb

γ

γ(−Ke − θT ξ + y(n)

r − Sign(eT Pb)δm) − 1

γγ ˙γ − θT �−1 ˙

θ

= − 1

2eT Qe + eT Pb(θT ξ + δ∗ − Sign(eT Pb)δm) − γ

γ[eT Pb(−Ke − θT ξ + y(n)

r − Sign(eT Pb)δm) + ˙γ] − θT �−1 ˙θ

= − 1

2eT Qe + θT (eT Pbξ − �−1 ˙

θ) + eT Pbδ∗ − eT Pb Sign(eT Pb)δm

= − 1

2eT Qe + eT Pbδ∗ − eT Pb Sign(eT Pb)δm

≤ − 1

2eT Qe

≤ 0.

(3.9)

This proves that the closed-loop system is stable and the tracking error converges to zero. This concludes theproof. �

Remark 3. Due to the sign function used in the control law, there may be chattering phenomena in control input.However, since δm is very small in general, the chattering will not be significant. Chattering can be weaken by reducingthe value of δm. In fact, it is easy to show that when the sign function term is dropped the closed-loop system is stillstable with bounded tracking error.

Remark 4. In this paper, only state feedback control is considered. Output feedback control is extensively studied inthe literature (see, for example, [1,11,13,14]). It is not difficult but interesting to extend our result to output feedbackcase by combining the output feedback techniques reported in the literature.

4. Example

Consider a nonlinear system:

y(3) = f (y, y, y(2), u) + u + � (4.1)

where f is an unknown smooth nonlinear function. � is unmodeled dynamics. The objective is to design a robustadaptive controller such that y tracks yr(t) = 0.1 sin(t). Let

x1 = y, x2 = y, x3 = y(2)

z1 = u, v = u(4.2)

1750 D. Wang et al. / Mathematics and Computers in Simulation 79 (2009) 1745–1753

Fig. 1. The output of closed-loop system.

We have

x1 = x2

x2 = x3

x3 = f (y, y, y(2), u) + v + �

z1 = v

(4.3)

Since no information about f is available for controller design, existing control design methods cannot solve thisproblem. We have to use the neural network-based approach proposed in Sections 2 and 3 of this paper. The RBF neuralnetwork described in Section 2 with the basis function given in (2.11) is applied to on-line approximate the unknownterms in the system (4.3). We consider the centers for Gaussian functions to be evenly spaced in a regular lattice inR4. Employing five nodes for each input dimension and centered at from −8 to +8, respectively, we end up withN = 54 = 625 nodes for the RBF neural network. Then, the control law and update laws for the given control problemcan be designed according to (3.4)–(3.7). The design parameters are chosen as σ = 1, � = 0.08I and δm = 0.01. K isdesigned to make Ac be Hurwitz as required in Section 2. Choosing the eigenvalues of Ac are [−3; −4; −5] and bypole placement, we obtain K = [60, 49, 12]. Letting Q = I and Solving PAc + AT

c P = −I, P is obtained as follows:

P =

⎡⎢⎣

0.5643 −0.5 −0.9048

−0.5 0.9048 −0.5

−0.9048 −0.5 6.5238

⎤⎥⎦ (4.4)

As observed in Fig. 1, the output of the closed-loop system tracks the reference input fairly good. After a shorttransient process the output tracks the reference input at a high precision. Fig. 2 gives the control input v for theaugmented system. Comparing with the simulation results presented in [14], in addition to that certain informationabout f is needed for control law design in [14], the control signal has very sever chattering in the beginning of thesimulation (Fig. 2 b in [14]). As stated in Remark 3, due to the sign function used in the control law, there may bechattering phenomena in our case. However, for appropriately chosen design parameters, the chattering cannot besignificant. As can be found in Fig. 2, there is no chattering in our simulation with the parameters given above. For acomparison, we change δm = 0.01 to δm = 0.12 and keep all the other parameters unchanged to conduct the simulationagain. Fig. 3 is obtained in this scenario and some chattering phenomena can be seen there.

D. Wang et al. / Mathematics and Computers in Simulation 79 (2009) 1745–1753 1751

Fig. 2. The control input v for the augmented system.

The example used in the simulation is taken from [13] with proper modifications. To simulate the controlled plant,the following plant dynamics given in [13] are used:

f = α0(u + y − y(2)) + 2α1(yy + y2 + yy(2)) (4.5)

where α0 = α1 = 1. The unmodeled dynamics � is given by

ω = −ω + y2 + y2 + 0.5, � = 2ω (4.6)

Remark 5. One major difference between our proposed design approach and previous results, for example, [13], isthat our control design does not need any information about f given in (4.5) while the design method proposed in [13]requires some of the information. In [13], the structure of f, i.e., (4.5) is used in control law and the only unknowns are

Fig. 3. The control input v for the augmented system (with chattering).

1752 D. Wang et al. / Mathematics and Computers in Simulation 79 (2009) 1745–1753

Fig. 4. Tracking error.

α0 and α1. In our case, f is totally unknown for control design and the information given in (4.5) is used for simulatingthe plant only. The method given in [13] cannot solve this kind of problems.

Fig. 4 presents the tracking error obtained in our simulation. The simulation was conducted using the followinginitial conditions: x(0) = [1, 0, 0]. Thus, the result is comparable with the tracking error presented in Fig. 1a in[13]. It can be found that in [13] the tracking error in the steady state is not ignorable but in our simulation result,perfect steady state performance is achieved. In addition, the tracking error converges faster in our result than it doesin [13].

5. Conclusions

Combining the feedback linearization technique and the neural network-based adaptive control technique, in thiswork, we developed a robust adaptive controller design method for a class of nonlinear systems represented byinput–output models. Using RBF networks to on-line approximate the unmodeled dynamics, some restriction inexisting results is removed. We proved that the closed-loop system is stable and the tracking error converges to zero.An example is used to demonstrate our proposed approach.

Acknowledgments

The work described in this paper was partially supported by National Nature Science Foundation of China, underGrant 60674037 and 60704042, the Scientific Research Foundation for the Returned Overseas Chinese Scholars, StateEducation Ministry, China, and Program for Liaoning Excellent Talents in University.

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