Nonlinear adaptive control using multiple identification models

doi:10.1016/j.sysconle.2007.12.007

Systems & Control Letters

Volume 57, Issue 7, July 2008, Pages 578-584

https://doi.org/10.1016/j.sysconle.2007.12.007 Get rights and content

Abstract

A multiple model adaptive controller is proposed for nonlinear systems in parametric-strict-feedback form. By running in parallel multiple identification models and designing a suitable switching scheme, some models close to the real plant can be selected quickly, so that transient performance can be improved significantly. Global asymptotic stability of the closed-loop switching system is proved. A simulation example is given to demonstrate the effectiveness of the proposed multiple model adaptive controller.

Introduction

Adaptive control employing switching has attracted much attention in the control community [1], [2], [5], [6], [7], [10], [12], [15], [16], [17], [18], [21], [22], [23]. It may be traced back to [11]. While in [1], [2], [4], [7], [12], [15], [22], [23], the main purpose is to design stable adaptive controllers for linear or nonlinear uncertain systems, in [16], [17], [18], [19], the main purpose is to improve transient performance of adaptive control systems. A well-known problem for adaptive control is that transient performance may be quite bad, especially when there exist large initial estimation errors. The basic idea of [16] is running in parallel multiple identification models, and by designing a suitable switching scheme based on their identification errors, some model close to the real plant can be selected quickly (by switching), so that transient performance can be improved. Such a multiple model adaptive controller, by nature, is indirect, since parameter identification plays a crucial role. The approach has been applied to general linear systems in both continuous and discrete cases [17], [18]. Extensive simulations and practical applications have demonstrated that it can improve transient performance significantly.

Designing multiple model adaptive controllers for nonlinear systems is, of course, very interesting. However, the closed-loop stability is much more difficult to analyze (and ensure). The problem is considered for a simple nonlinear system in [19]. To ensure closed-loop stability, a direct (Lyapunov-based) parameter update law (instead of parameter identification algorithm) is used, but to apply the certainty-equivalence-based switching scheme [16], [17], [18], the direct approach is viewed as indirect by a special artifice. A quite different nonlinear multiple model adaptive control is proposed in [3], which, however, requires the condition of persistence of excitation, so that unknown parameters can be calculated at the very beginning.

In this paper, we will consider multiple model adaptive control of nonlinear systems in parametric-strict-feedback form. After formulating the control problem in Section 2, we present the nonlinear multiple model adaptive controller design in Section 3, and analyze the stability of closed-loop switching system in Section 4. A simulation example is given in Section 5, and finally the paper is concluded in Section 6.

Section snippets

Problem formulation

We will consider multiple model adaptive control of the following nonlinear systems in parametric-strict-feedback (PSF) form [9] ${\dot{x}}_{i} = x_{i + 1} + φ_{i} {(x_{1}, \dots, x_{i})}^{T} θ, i = 1, 2, \dots, n - 1$ ${\dot{x}}_{n} = β (x) u + φ_{n} {(x)}^{T} θ$ $y = x_{1}$ where $θ$ is the unknown parameter vector belonging to a (large) compact set $S$ , and $φ_{i} (x_{1}, \dots, x_{i})$ ’s and $β (x)$ are known smooth functions with $β (x) \neq 0$ . The main purpose is to improve transient performance in the presence of large parametric uncertainties, and meanwhile, ensure the stability of the closed-loop switching

Design of nonlinear multiple model adaptive controller

Our nonlinear multiple model adaptive controller contains three components: a nonlinear parametrized controller $C (\hat{θ}), N$ parallelly-operating identification models ${I_{j}}_{j = 1}^{N}$ , and a switching scheme determining on-line which the identification model should be switched into connection with the parametrized controller.¹ For improving transient performance, it is necessary

Stability analysis

Theorem 1

Suppose that the nonlinear multiple model adaptive controller proposed in Section 3is applied to the nonlinear PSF plant(1). Then for any initial conditions, all closed-loop states $x$ , $Ω_{0}$ , $Ω$ and ${\hat{θ}}_{j}$ , $j = 1, \dots, N$ , are bounded on $[0, \infty)$ and furthermore, asymptotic tracking is achieved, i.e., $y (t) \to y_{r} (t)$ as $t \to \infty$ .

Proof

Since the time intervals between successive switchings of $I_{c}$ can never be less than $T_{min}$ , and since for any fixed $I_{c}$ , no finite time escape phenomenon is possible, so the solution of the

Simulation example

Consider the following nonlinear system in parametric-strict-feedback form ${\dot{x}}_{1} = x_{2} + θ^{(1)} x_{1} + θ^{(2)} x_{1}^{2}$ ${\dot{x}}_{2} = u$ where $θ^{(1)} \in [0, 5]$ and $θ^{(2)} \in [0, 40]$ are unknown parameters. The output $y = x_{1}$ is to asymptotically track the reference signal $y_{r} (t) = sin 2 t$ .

In simulation, the parametrized controller is designed according to (2) with $c_{1} = c_{2} = 4$ , $κ_{1} = κ_{2} = g_{2} = 0.1$ . The parameter update laws are given by (10) with $υ = 0$ , $Γ = 5$ , but since in (31) the unknown parameters only appear in the first equation, we implement the following

Conclusions

In this paper, we propose a multiple model adaptive controller for nonlinear systems in parametric-strict-feedback form. By running in parallel multiple identification models and designing a suitable switching scheme, some model close to the real plant can be selected quickly, so that transient performance can be improved significantly. Global asymptotic stability of the closed-loop switching system is proved. A simulation example is given to demonstrate the effectiveness of our proposed

References (23)

A.G. Aghdam et al.
Pseudo-decentralized switching control
Automatica
(2003)
L.B. Freidovich et al.
Logic-based switching for control of minimum-phase nonlinear systems
Systems Control Lett.
(2005)
J.P. Hespanha et al.
Supervision of integral-input-to-state stabilizing controllers
Automatica
(2002)
J.P. Hespanha et al.
Overcoming the limitations of adaptive control by means of logic-based switching
Systems Control Lett.
(2003)
B. Martensson
The order of any stabilizing regulator is sufficient information for adaptive stabilization
Systems Control Lett.
(1985)
Z. Qu
Robust control of nonlinear uncertain systems under generalized matching condition
Automatica
(1993)
X. Ye
Switching adaptive output-feedback control of nonlinearly parametrized systems
Automatica
(2005)
B.D.O. Anderson et al.
Multiple model adaptive control, part 1: Finite controller converings
Int. J. Robust Nonlinear Control
(2000)
M.K. Ciliz et al.
Increased transient performance for the adaptive control of feedback linearizable systems using multiple models
Int. J. Control
(2006)
J.P. Hespanha et al.
Multiple model adaptive control, part 2: Switching
Int. J. Robust Nonlinear Control
(2001)

Z.P. Jiang et al.

A unifying framework for global regulation via nonlinear output feedback: From ISS to iISS

IEEE Trans. Automatic Control AC-49

(2004)

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^☆: The work is supported by the National Natural Science Foundation of China under Grants 60474010 and 60525316, and National Basic Research Program of China under Grant 2006CB705400.

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Nonlinear adaptive control using multiple identification models☆

Abstract

Introduction

Section snippets

Problem formulation

Design of nonlinear multiple model adaptive controller

Stability analysis

Simulation example

Conclusions

Automatica

Systems Control Lett.

Automatica

Systems Control Lett.

Systems Control Lett.

Automatica

Automatica

Multiple model adaptive control, part 1: Finite controller converings

Int. J. Robust Nonlinear Control

Increased transient performance for the adaptive control of feedback linearizable systems using multiple models

Int. J. Control

Multiple model adaptive control, part 2: Switching

Int. J. Robust Nonlinear Control

A unifying framework for global regulation via nonlinear output feedback: From ISS to iISS

IEEE Trans. Automatic Control AC-49