Recurrent-Neural-Network-Based Multivariable Adaptive Control for a Class of Nonlinear Dynamic Systems With Time-Varying Delay

At the beginning, an approximate nonlinear autoregressive moving average (NARMA) model is employed to represent a class of multivariable nonlinear dynamic systems with time-varying delay. It is known that the disadvantages of robust control for the NARMA model are as follows: 1) suitable control parameters for larger time delay are more sensitive to achieving desirable performance; 2) it only deals with bounded uncertainty; and 3) the nominal NARMA model must be learned in advance. Due to the dynamic feature of the NARMA model, a recurrent neural network (RNN) is online applied to learn it. However, the system performance becomes deteriorated due to the poor learning of the larger variation of system vector functions. In this situation, a simple network is employed to compensate the upper bound of the residue caused by the linear parameterization of the approximation error of RNN. An e-modification learning law with a projection for weight matrix is applied to guarantee its boundedness without persistent excitation. Under suitable conditions, the semiglobally ultimately bounded tracking with the boundedness of estimated weight matrix is obtained by the proposed RNN-based multivariable adaptive control. Finally, simulations are presented to verify the effectiveness and robustness of the proposed control.

I. Introduction

It is well known that most real-world dynamic systems are (highly) nonlinear. However, the linearization of such systems around the equilibrium states may yield linear models that are mathematically tractable. It is also known that a conventionally designed linear controller may not achieve an adequate performance over a variety of operating regimes, especially when the system is highly nonlinear [1]. Although a linear adaptive control problem with unknown system parameters can deal with this difficult situation, its effectiveness is limited. It is also known that a robust controller design based on a nominal system is not enough to stabilize the system with large uncertainty [2]. There are some nonlinear multivariable systems that can be modeled by an interconnected nonlinear matrix gain and a linear dynamic system, e.g., Wiener model, Hammerstein model, and hysteresis model [3], [4]. Furthermore, a nonlinear autoregressive moving average (NARMA) model is a generalized representation of input-output behavior of finite-dimensional nonlinear discrete dynamic system. Comparing with a nonlinear state space representation of dynamic systems, the NARMA model does not require a state estimator, which is easier for system identification, and can represent a wider class of nonlinear dynamic systems with time-varying delay for the controller design [5], [6]. Motivated by these, in this paper, a neural controller design for a class of unknown and multivariable NARMA models with time-varying delay is addressed.