Abstract
The main difficulty in the traditional nonlinear $H_{\infty }$ control design lies in how to solve the nonlinear partial differential Hamilton-Jacobi-Isaacs equation (HJIE), especially for nonlinear time-varying systems. In this study, a novel HJIE-embedded DNN $H_{\infty }$ control scheme is proposed to be efficiently trained for nonlinear $H_{\infty }$ stabilization and tracking control designs of nonlinear dynamic systems with the external disturbance. The proposed DNN-based $H_{\infty }$ control approach not only capitalizes on the availability of theoretical partial differential HJIE but also reduces the amount of empirical data and the complexity to train HJIE-embedded DNN. We have shown that the proposed DNN-based $H_{\infty }$ control scheme can approach the theoretical result of $H_{\infty }$ robust control when the training error approaches zero and the asymptotic stability is also guaranteed if the nonlinear time-varying system is free of external disturbance. The proposed method could be easily extended to DNN-based $H_{\infty }$ reference tracking control of nonlinear systems for more practical applications. Finally, two examples, including $({i})$ an $H_{\infty }$ stabilization of nonlinear time-varying system and $(ii)$ an $H_{\infty }$ unmanned aerial vehicle (UAV) reference tracking control system, are proposed to illustrate the design procedure and to demonstrate the effectiveness of our DNN-based $H_{\infty }$ method.
Highlights
D EEP neural network (DNN) is an information processing model inspired by biological neural systems and enables us to perform tasks by learning from big data for a large variety of applications
(III) By the proposed Hamilton Jacobi Issac equation (HJIE)-embedded DNN approach, the priori expert knowledge of nonlinear dynamic models and theoretical robust H∞ control results, which have been acquired over decades, can complement the traditional pure big data-driven deep learning approaches to facilitate the use of deep learning schemes for application to robust H∞ control system designs of complex nonlinear time-varying systems, which can not be solved by conventional methods
With the nonlinear timevarying dynamic model as well as H∞ control u∗(t) and the worst-case disturbance v∗(t), we generate trajectories by nonlinear time-varying system model as input to DNN and train weighting and bias parameters of neurons in the hidden units of DNN via Adam learning algorithm by the HJIE error ε(θi(t)) so that DNN is a big-data driven scheme for solving the conventional classification and recognition problem, and a dynamic model-based scheme to be with much potential for solving system control design in nonlinear systems and filter design in nonlinear signal processing systems
Summary
D EEP neural network (DNN) is an information processing model inspired by biological neural systems and enables us to perform tasks by learning from big data for a large variety of applications. The proposed HJIE-embedded DNN H∞ robust control design can significantly reduce the amount of training data and training time when compared with the traditional deep learning approaches based on big data. The DNN-based control scheme can be shown to approach the H∞ robust control of nonlinear timevarying system by Adam learing algorithm and can be extended to the design of H∞ reference tracking control on nonlinear dynamic systems with external disturbance. (III) By the proposed HJIE-embedded DNN approach, the priori expert knowledge of nonlinear dynamic models and theoretical robust H∞ control results, which have been acquired over decades, can complement the traditional pure big data-driven deep learning approaches to facilitate the use of deep learning schemes for application to robust H∞ control system designs of complex nonlinear time-varying systems, which can not be solved by conventional methods.
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