Adaptive MNN control for a class of non-affine NARMAX systems with disturbances

doi:10.1016/j.sysconle.2004.02.016

Systems & Control Letters

Volume 53, Issue 1, September 2004, Pages 1-12

https://doi.org/10.1016/j.sysconle.2004.02.016 Get rights and content

Abstract

In this paper, adaptive multi-layer neural network (MNN) control is developed for a class of discrete-time non-affine nonlinear systems in nonlinear auto regressive moving average with eXogenous inputs (NARMAX) model. By using implicit function theorem, the existence of the implicit desired feedback control (IDFC) is proved. MNNs are used as the emulator of the desired feedback control. Projection algorithms are used to guarantee the boundedness of the neural network (NN) weights, which removes the need of persistent exciting (PE) condition for parameter convergence. Simulation results show the effectiveness of the proposed control scheme.

Introduction

Over the past few years, adaptive control for continuous-time nonlinear system has been studied extensively. In general, these methods cannot be directly applied to discrete-time systems due to some technical difficulties, such as the lack of applicability of Lyapunov techniques [9]. Furthermore, due to the fact that discrete-time Lyapunov differences are quadratic in the state first difference, while for continuous-time systems the Lyapunov derivative is linear in the state derivative, discrete-time adaptive control design is far more complex than continuous-time design. Therefore, it is academically challenging and practically interesting to develop adaptive control scheme for discrete-time nonlinear systems, as plants are controlled at discrete-time instant. Most techniques for estimation and control of unknown nonlinear systems involve the use of a model based on past history of input–output data. These techniques depend largely on the accuracy of these models and the availability of sufficient historical input–output data. It is usually easier to identify discrete-time models and use these data as a basis for designing discrete-time control systems for computer implementation. This observation motivates us to study the control problems of discrete-time models. One of the most popular discrete-time model is nonlinear autoregressive moving average with eXogenous inputs (NARMAX) model, which has been studied in previous literature [1], [2], [5], [6]. In this paper, we investigate a class of non-affine NARMAX systems by using multi-layer neural network (MNN) approximation, for which, because input is in non-affine form, feedback linearization method cannot be implemented.

In recent years, there has been increasing interest in the application of neural networks (NNs) to process modelling and control. In [3], [14], the authors showed how artificial NNs, including radial basis function networks, may be used as universal function approximators, which inspired the use of NNs as emulators to approximate unknown nonlinear functions to construct stable adaptive controller. Frequently used single-layer NNs include high-order neural networks (HONN) [10], radial basis function neural networks (RBF) [13], etc. For MNNs, its universal approximation abilities, parallel distributed processing abilities, learning and adaptation abilities make it one of the most popular tools in function approximation. In [11], [12], MNNs were effectively used in nonlinear discrete-time system identification and control. But the MNN-based tracking control in those works was based on the realization that the linearization of a system around an equilibrium point is “well behaved” (controllable, observable, etc.). In this paper, we proposed a MNN control scheme which removes the restriction and is applicable to a wide class of nonlinear non-affine discrete-time systems. Firstly, the existence of the implicit desired feedback control (IDFC) control, which can drive the system output to track the desired trajectory, is proved. Then MNNs are used to emulate the IDFC control. Though single-layer NNs are also applicable to construct the stable adaptive control, MNNs are used in this paper for the following main reasons: (i) tuning of single-layer NNs has been studied extensively; (ii) MNNs make single-layer NN as a special case, if the hidden layers are fixed; and (iii) there is the “curse of dimensionality” problem for the frequently used RBF NNs, and MNNs are very good alternative in reducing the problem.

The main contributions of this paper are that (i) a discrete-time projection algorithm is proposed by extending the continuous-time projection algorithm used in [7], [8], [16]; (ii) MNNs are used to emulate the desired feedback control of non-affine discrete-time systems, which is not only a challenging topic but also of academic interest, and (iii) semi-global uniformly ultimate boundedness (SGUUB) stability is proposed for a class of non-affine NARMAX systems in the presence of bounded disturbances, for which, feedback linearization method cannot be implemented.

This paper is organized as follows. In Section 2, the NARMAX model is proposed. Preliminaries about MNNs and projection algorithms are presented in Section 3, respectively. Controller design procedure and stability analysis are presented in Section 4. Finally, numerical simulation is carried out in Section 5.

Section snippets

System dynamics

Consider the following τ-step ahead non-affine NARMAX model [1] $y(k+τ)=f(y ̄_{k}, u ̄_{k−1}, d ̄_{k+τ−1},u(k)),$ where $y ̄_{k} =[y(k),…,y(k−n+1)]^{T}$ , $u ̄_{k−1} =[u(k−1),…,u(k−n+1)]^{T}$ and $d ̄_{k+τ−1} =[d(k+τ−1),…,d(k)]^{T}$ . Sequences ${y(k)}, {u(k)}$ and {d(k)} represent system outputs, inputs and disturbances, respectively. τ denotes the system delay, or the relative degree of the system.

Assumption 1

The unknown nonlinear function f(·) is continuous and differentiable.

Assumption 2

System output y(k) can be measured and its initial values are assumed to remain

MNNs and function approximation

Because the IDFC input $u^{∗} (k)$ is a continuous function on the compact set $Ω_{z}$ , we know that there exists an integer l (the number of hidden neurons) and ideal constant weight matrices $W^{∗} ∈ R^{l+1}$ and $V^{∗} =[v_{1}^{∗},…,v_{l}^{∗}]∈ R^{(n+1)×l}$ , such that $u^{∗} (k)=u^{∗} (z)=W^{∗T} S(V^{∗T} z ̄)+ε_{u} (z),∀z∈Ω_{z},$ where $z ̄ =[z^{T},1]^{T}$ and $z∈ R^{n}$ denotes the input vector, $S(V^{∗T} z ̄)=[s(v^{∗}_{1}^{T} z ̄),…,s(v^{∗}_{l}^{T} z ̄),1]^{T}$ with $s(∗)$ denotes the sigmoid function [4]. In this paper, we choose s(x)=1/(1+e^−x) as the activation function. The derivative of s(x) with respect

Controller design

Considering the universal approximate ability of MNN, in this section, we use MNN to approximate the IDFC control. The use of MNNs in discrete-time nonlinear system control is not only challenging but also of academic interest.

At first, considering the MNNs, neural weights adaptation laws and projection algorithms used in this paper, we have the following lemma.

Lemma 3

Considering the projection algorithms we used, on the compact set $Ω_{z}$ , the estimate NN weights $W ̂, V ̂$ , and the weight approximation errors

Simulation studies

Consider the following non-affine continuous stirred tank reactors (CSTR) system [4]: $x ̇_{1} =1−x_{1} −a_{0} x_{1} e^{−10^{4}/x_{2}},$ $x ̇_{2} =350−x_{2} +a_{1} x_{1} e^{−10^{4}/x_{2}} + a_{3} u(1− e^{−a_{2}/u})(350−x_{2}),$ $y=x_{1},$ where $a_{1} =1.44×10^{13}, a_{2} =6.987×10^{2}$ and a₃=0.01. (Detailed definition can be found in [4].) The major challenge of this control problem is that the plant does not assume the customary control affine system structure because the control input u appears nonlinearly.

The control objective is to make the output y(t) track the set-point step change

Conclusion

In this paper, an adaptive MNN control scheme was proposed for a class of discrete-time non-affine nonlinear systems in NARMAX form. The existence of IDFC, which can drive the system output to track desired trajectory, was proved by using implicit function theorem. Based on the input–output model, MNNs were used to emulate the IDFC. Lyapunov stability techniques and projection algorithms were used to develop the control scheme and adaptive learning laws. The proposed controller guarantees the

References (16)

S.S. Ge et al.
Adaptive NN control for a class of strict-feedback discrete-time nonlinear systems
Automatica
(2003)
J.Q. Gong et al.
Neural network adaptive robust control of nonlinear systems in semi-strict feedback form
Automatica
(2001)
G.C. Goodwin et al.
A parameter estimation perspective of continuous time model reference adaptive control
Automatica
(1987)
S.A. Billings et al.
A prediction-error and stepwise regression algorithm for nonlinear systems
Internat. J. Control
(1986)
S. Chen et al.
Representations of non-linear systemsthe narmax model
Internat. J. Control
(1989)
G. Cybenko
Approximation by superpositions of a sigmoidal function
Math. Control Signals Systems
(1989)
S.S. Ge et al.
Stable Adaptive Neural Network Control
(2001)
S.S. Ge et al.
Adaptive NN control for a class of discrete-time nonlinear systems
Internat. J. Control
(2003)

There are more references available in the full text version of this article.

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Adaptive MNN control for a class of non-affine NARMAX systems with disturbances

Abstract

Introduction

Section snippets

System dynamics

MNNs and function approximation

Controller design

Simulation studies

Conclusion

Automatica

Automatica

Automatica

A prediction-error and stepwise regression algorithm for nonlinear systems

Internat. J. Control

Representations of non-linear systemsthe narmax model

Internat. J. Control

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Math. Control Signals Systems

Stable Adaptive Neural Network Control

Adaptive NN control for a class of discrete-time nonlinear systems

Internat. J. Control