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**Adaptive control of a class of nonlinear discrete-time systems using neural networks.**
*(English)*
Zbl 0925.93461

Let \(f(\cdot)\) and \(g(\cdot)\) be smooth functions of real variables \(y_{k-i}\) and \(u_{k-d-i},i= 0,1,\dots,m-1\). The nonlinear discrete-time system is described by the equation \(y_{k+1} =f(\cdot)+g(\cdot)u_{k-d+1}\), where \(m\leq n\), \(y\) is the output, \(u\) is the input, \(d\) is the relative degree of the system. It is shown in the article that this system can be linearized by means of a feedback control. K. S.Narendra and K. Parthasarathy [IEEE Trans. Neural Networks 1, 4–27 (1990)] and F.-C. Chen [IEEE Contr. Syst. Mag., Special Issue on Neural Networks for Control Systems (1990)] developed the idea of applying multilayer neural networks to adaptive control of such systems. It has been shown by some authors [e.g. K. Funahashi, Neural Networks 2, 183–192 (1989)] that the majority of nonlinear functions can be approximated by multilayer neural networks to any desired accuracy, but these results are existence results only.

The present paper is devoted to constructively modelling the unknown system and to generating the feedback control by a neural network of the form \(y^*_{k+1}= \widehat f_{d-1} [{\mathbf x}(k),{\mathbf w}]+\widehat{g}_{d-1}[{\mathbf x}(k),{\mathbf v}]u_k\), where \({\mathbf w},{\mathbf v}\) are vectors of unknown weights. Denote \(\Theta={{\mathbf w}\choose {\mathbf v}}\). The procedure of updating the weights of the neural network is based on the error \(e^*_{k+1}=y^*_{k+1}-y_{k+1}\) between the required \(y_{k+1}\) and model \(y^*_{k+1}\) outputs. The updating rule is \(\Theta(k+1)=\Theta(k)-\frac{1}{r_k}\;D(e^*_{k+1}){\mathbf J}_{k-d+1}\), where \({\mathbf J}_{k-d+1}=[(\partial y^*_{k+1}/\partial\Theta)|_{\Theta(k)}]^T\) is the Jacobian matrix, computed by the backpropagation algorithm, \(r_k = 1+ {\mathbf J}{\neg}T_{k-d+1}{\mathbf J}_{k-d+1}\), and \(D(e)\) is a dead-zone function, defined by \(D(e)=0\) if \(|e|\leq d_0\), \(D(e)=e-d_0\) if \(e>d_0\), \(D(e)=e+d_0\) if \(e<-d_0\).

The convergence of the procedure is proved by a theorem that states that for any bounded initial conditions, if the network weights are close to the correct ones, then the tracking error between the plant outputs and the reference command will converge to a bounded ball whose radius is determined by a dead-zone nonlinearity centered at the origin. The results of computer simulations are also presented.

The present paper is devoted to constructively modelling the unknown system and to generating the feedback control by a neural network of the form \(y^*_{k+1}= \widehat f_{d-1} [{\mathbf x}(k),{\mathbf w}]+\widehat{g}_{d-1}[{\mathbf x}(k),{\mathbf v}]u_k\), where \({\mathbf w},{\mathbf v}\) are vectors of unknown weights. Denote \(\Theta={{\mathbf w}\choose {\mathbf v}}\). The procedure of updating the weights of the neural network is based on the error \(e^*_{k+1}=y^*_{k+1}-y_{k+1}\) between the required \(y_{k+1}\) and model \(y^*_{k+1}\) outputs. The updating rule is \(\Theta(k+1)=\Theta(k)-\frac{1}{r_k}\;D(e^*_{k+1}){\mathbf J}_{k-d+1}\), where \({\mathbf J}_{k-d+1}=[(\partial y^*_{k+1}/\partial\Theta)|_{\Theta(k)}]^T\) is the Jacobian matrix, computed by the backpropagation algorithm, \(r_k = 1+ {\mathbf J}{\neg}T_{k-d+1}{\mathbf J}_{k-d+1}\), and \(D(e)\) is a dead-zone function, defined by \(D(e)=0\) if \(|e|\leq d_0\), \(D(e)=e-d_0\) if \(e>d_0\), \(D(e)=e+d_0\) if \(e<-d_0\).

The convergence of the procedure is proved by a theorem that states that for any bounded initial conditions, if the network weights are close to the correct ones, then the tracking error between the plant outputs and the reference command will converge to a bounded ball whose radius is determined by a dead-zone nonlinearity centered at the origin. The results of computer simulations are also presented.

Reviewer: A.N.Karkishchenko (Taganrog)