Elsevier

Automatica

Volume 45, Issue 5, May 2009, Pages 1306-1311
Automatica

Brief paper
Parameter convergence and minimal internal model with an adaptive output regulation problem

https://doi.org/10.1016/j.automatica.2009.01.003Get rights and content

Abstract

The parameter convergence of nonlinear adaptive control systems is an important yet not well addressed issue. In this paper, using the global robust output regulation problem of output feedback systems with unknown exosystems as a platform, we will show that the well known persistency of excitation (PE) condition still guarantees the convergence of the estimated parameter vector to the true parameter vector. Moreover, the PE condition will be satisfied if the internal model is minimal in certain sense.

Introduction

PE is a well known condition arising in adaptive control of linear systems, e.g., in Boyd and Sastry (1983) and Narendra and Annaswamy (1989). This condition guarantees the exponential convergence to zero of the estimation error of some unknown parameter vector in a given plant. This result is established by applying a quadratic Lyapunov function to the closed-loop system. Under the PE condition, the closed-loop system is exponentially stable. As the estimation error is part of the state vector of the closed-loop system, the PE condition guarantees that the unknown parameter will converge to the true parameter exponentially. In what follows, this approach of establishing the parameter convergence will be called the Lyapunov argument.

There have also been some recent efforts in connecting the PE condition to the parameter convergence issue in nonlinear systems using adaptive control based on Lyapunov methods, e.g., Serrani, Isidori, and Marconi (2001). In this paper, we will develop an alternative approach to address the parameter convergence issue in a nonlinear adaptive control system. This approach utilizes a special case of Lemma 2 of Yuan and Wonham (1977) (also see Lemma 1 of Ortega and Fradkov (1993)). Hopefully, this alternative approach is more straightforward and more thorough.

To be more specific, we will use the global robust output regulation problem of output feedback systems subject to unknown exosystems as a platform. The approach can also be extended to the global or semi-global robust output regulation of other nonlinear systems with unknown exosystems. The output feedback systems are described as follows: ż=F(w)z+G(y,v,w)y+D1(v,w)ẏ=H(w)z+K(y,v,w)y+b(w)ξ1+D2(v,w)ξ̇i=λiξi+ξi+1,i=1,,r2ξ̇r1=λr1ξr1+ue=yq(v,w) where col(z,y)Rn and ξ=col(ξ1,,ξr1)Rr1 are the states, yR is the output, uR the input, eR the tracking error, wRp the uncertain parameter, and vRq the exogenous signal. The signal v represents either the disturbance or reference input or both and is generated by the exosystem v̇=A1(σ)v where σSR represents the unknown parameters. It is assumed that all the functions in system (1.1) are sufficiently smooth and D1(0,w)=0,D2(0,w)=0, and q(0,w)=0 for all wRp. The system described by (1.1), (1.2) is obtained by performing dynamic extension and coordinate transformation on the class of nonlinear systems in output feedback form (Huang, 2004). Therefore, for convenience, we will also call the system described by (1.1), (1.2) as an output feedback system.

To have our problem well posed, we assume the following:

Assumption 1.1

All the eigenvalues of A1(σ) are distinct with zero real parts for all σS.

As a result, the general solution of the exosystem is a sum of finitely many sinusoidal functions with their frequencies depending on the eigenvalues of A1(σ) and amplitudes and phase angles on the initial condition v0=v(0).

Briefly, our problem can be stated as follows: For a givenS, design a controller with feedbackcol(e,ξ), such that, for allcol(v0,w,σ)Rq+p×S, the trajectory of the closed-loop system starting from any initial state of the plant and the controller exists and is bounded for allt0, and furthermore, the tracking errore(t)approaches zero asymptotically.

Some special version of the above problem is studied in Ding, 2003, Ding, 2006. Also, the global robust output regulation problem for a class of large-scale systems was studied in Ye and Huang (2003). The large-scale system in Ye and Huang (2003) includes the output feedback system as a special case when the number of the subsystems is equal to one. When the exosystem is unknown, the adaptive control technique has to be adopted to handle a parameter vector which depends on the unknown parameter σ and whose dimension is equal to that of the internal model employed. It is important to know whether or not the estimated parameter vector will converge to the true parameter vector. However, this critical issue was not addressed in Ye and Huang (2003). In this paper, we will address this parameter convergence problem. We will also introduce the concept of minimal internal model which is an internal model with its dimension being no more than twice as many as the sinusoids in the steady-state input of the system. Then we will show that the estimated parameter vector will converge to the true parameter vector if the minimal internal model is employed.

It should be noted that the output regulation problem for some other classes of nonlinear systems with uncertain exosystem has also been studied in Chen and Huang (2002), Nikiforov (1998), Priscoli, Marconi, and Isidori (2006) and Serrani et al. (2001).

Section snippets

Problem formulation and preliminaries

Like in Ye and Huang (2003), the output regulation of (1.1) as described above can be converted into an adaptive regulation problem of an augmented system defined in (2.7). To introduce this conversion, let us list a few additional assumptions.

Assumption 2.1

b(w)>0 for all wRp.

Assumption 2.2

There exists a sufficiently smooth function z(v,w,σ) with z(0,0,0)=0, such that, for all vRq, wRp, σS,z(v,w,σ)vA1(σ)v=F(w)z(v,w,σ)+G(q(v,w),v,w)q(v,w)+D1(v,w).

Remark 2.1

As explained in Chen and Huang (2005), Assumption 2.1, Assumption 2.2

Solvability of the problem

Let us first introduce a few inequalities and notations to be used later. Since G̃(x1,μ) and K̃(x1,μ) are real-valued continuous functions, there exist sufficiently smooth functions qi(μ)1,ai(x1)1, i=1,2, such that, for each μRq+p, x1R, |G̃(x1,μ)x1|2q1(μ)a1(x1)x12|K̃(x1,μ)x1|2q2(μ)a2(x1)x12.

Define a coordinate transformation x̄1=x1 and x̄i+1=xi+1αi,i=1,,r, where the functions αi’s are defined recursively as follows: α1(x1,k,η,Ψˆ)=kρ(x̄1)x̄1+Ψˆηαi(x1,,xi,k,η,Ψˆ,bˆ)=λi1xi+αi1kk̇+j=2

Convergence of Ψˆ to Ψσ and the minimal internal model

In this section, we will ascertain conditions under which the unknown parameter vector Ψˆ will converge to Ψσ as t tends to infinity. For this purpose, let us first introduce two lemmas.

Lemma 4.1

Letg:[0,)Rnbe continuously differentiable function andf:[0,)Rnbe bounded piecewise continuous function. Further assume there exist positive constantsϵ,t0,T0such that, for any unit row vectorcof dimensionn, and anytt0,1T0tt+T0|cf(s)|dsϵ.Thenlimtg(t)=0iflimtġ(t)=0andlimtgT(t)f(t)=0. 

Remark 4.1

Condition (4.1)

An example

A practical example demonstrating the application of the result of this paper can be found in Liu and Huang (2008). Here, for the purpose of illustrating the design methodology, we consider a worked example in the form of (1.1) as follows: ż1=z1+sin2(yv1)yż2=z2+yẏ=z2w1yw2y3+bξ1ξ̇1=ξ1+ue=yv1 where w1,w2 and b are unknown parameters with b>0. The exosystem (1.2) is given with A1(σ)=[0σσ0] and σS={σ>0}. It can be verified that all assumptions needed by Theorem 3.1, Theorem 4.1 are

Conclusion

In this paper, we have addressed the parameter convergence issue associated with the output regulation problem of output feedback systems with unknown exosystems. An interesting point about our approach is that it only depends on the characteristics of the signals g(t) and f(t) without assuming that the signals are governed by some time varying linear systems as assumed in the literature of adaptive control of linear systems. Thus, the result on the convergence of the estimated parameter is not

Lu Liu received the B.S. degree from Northwestern Polytechnical University, Xi’an, China, in 2003, and the M.S. degree in 2005, Ph.D. degree in 2008, both from the Chinese University of Hong Kong. She is currently a postdoctoral fellow in the Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong. Her research interests include robust and adaptive nonlinear control, smart materials and biomedical devices.

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Lu Liu received the B.S. degree from Northwestern Polytechnical University, Xi’an, China, in 2003, and the M.S. degree in 2005, Ph.D. degree in 2008, both from the Chinese University of Hong Kong. She is currently a postdoctoral fellow in the Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong. Her research interests include robust and adaptive nonlinear control, smart materials and biomedical devices.

Zhiyong Chen received the B.S. degree from the University of Science and Technology of China, and the M.S. and Ph.D. degrees from the Chinese University of Hong Kong, in 2000, 2002, and 2005 respectively. He worked as a research associate at the University of Virginia during 2005–2006. He is now a lecturer at the University of Newcastle. His research interests include biological and nonlinear systems and control.

Jie Huang is now a professor with Department of Mechanical and Automation Engineering, Chinese University of Hong Kong. His research interests include nonlinear control theory and applications, neural networks, and flight guidance and control.

This paper has not been presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Antonio Loria under the direction of Editor Andrew R. Teel. The work of the second author was partially supported by the Australian Research Council under grant No. DP0878724. The work of the third author was substantially supported by the Research Grants Council of the Hong Kong Special Administration Region under grant No. CUHK412006.

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