Brief paperParameter convergence and minimal internal model with an adaptive output regulation problem☆
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: where and are the states, is the output, the input, the tracking error, the uncertain parameter, and the exogenous signal. The signal represents either the disturbance or reference input or both and is generated by the exosystem where represents the unknown parameters. It is assumed that all the functions in system (1.1) are sufficiently smooth and , and for all . 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 are distinct with zero real parts for all .
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 and amplitudes and phase angles on the initial condition .
Briefly, our problem can be stated as follows: For a given, design a controller with feedback, such that, for all, the trajectory of the closed-loop system starting from any initial state of the plant and the controller exists and is bounded for all, and furthermore, the tracking errorapproaches 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 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 for all .
Assumption 2.2 There exists a sufficiently smooth function with , such that, for all , , ,
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 and are real-valued continuous functions, there exist sufficiently smooth functions , , such that, for each , ,
Define a coordinate transformation and , where the functions ’s are defined recursively as follows:
Convergence of to and the minimal internal model
In this section, we will ascertain conditions under which the unknown parameter vector will converge to as tends to infinity. For this purpose, let us first introduce two lemmas.
Lemma 4.1 Letbe continuously differentiable function andbe bounded piecewise continuous function. Further assume there exist positive constantssuch that, for any unit row vectorof dimension, and any,Thenifand ■
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: where and are unknown parameters with . The exosystem (1.2) is given with and . 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 and 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.
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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.