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

Abstract

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.

Adaptive Recurrent Neural Network Enhanced Variable Structure Control for Nonlinear Discrete Mimo Systems

“…ð Þ∈ℜ nÂn are respectively the nominal vector and matrix functions, which are continuous and known; F 2 X ð Þ is nonsingular for all X(k) [10,14]; ΔF 1 (X, k) ∈ ℜ n and ΔF 2 (X, k) ∈ ℜ n × n are the uncertain vector and matrix functions, respectively; and Ω(k) is the external disturbances, which can be relatively bounded. Remark 1.…”

“…As the lumped uncertainties Δ 2 (Z, k) are huge, the DVSC fails to obtain an acceptable result. In this situation, it is approximated by a RNN [10], i.e.,…”

“…where Δ 1 is shown in (7b) and e Δ 2 ¼ Δ 2 ÀΔ 2 : Based on the similar simplification of [10], the change rate of Lyapunov function becomes…”

“…Furthermore, a nonlinear autoregressive moving average (NARMA) model is a generalized representation of the input–output behavior of a finite‐dimensional nonlinear discrete dynamic system. Compared with a nonlinear state space representation of dynamic systems (e.g., , 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 .…”

“…The first category is that all the system functions are unknown. In this situation, all system functions must be learned on‐line to design an indirect or direct adaptive control. Besides the poor transient response, they need a significant amount of computation to deal with these unknown dynamic control problems .…”