Elsevier

Neurocomputing

Volume 453, 17 September 2021, Pages 512-523
Neurocomputing

Neural networks optimized learning control of state constraints systems

https://doi.org/10.1016/j.neucom.2020.10.034Get rights and content

Abstract

An adaptive neural networks (NN) backstepping optimized tracking learning control approach is presented for nonlinear strict-feedback systems with state constraints. In the control design, neural networks are used to learn the unknown nonlinear dynamics, and a NN state identifier is proposed. By constructing Barrier Lyapunov functions and optimal Barrier type cost functions for identifier error dynamic systems and the subsystems of the identifier systems, and under the actor-critic architecture, the virtual and actual optimal controllers are constructed by the backstepping recursive control design algorithm. The proposed adaptive NN optimal control scheme can guarantee that the closed-loop system is uniformly ultimately bounded (UUB) and the states of the controlled system are ensured not to transgress their constrained sets. Moreover, the proposed optimized controller can guarantee that the system output can track the given reference signal. The effectiveness of the proposed control method is verified by a numerical example.

Introduction

There are several methods to handle the nonlinear systems where adaptive neural network and fuzzy control are two hot research topics [1], [2], [3], [4], [5], [6], [7], [8]. During the past decades, the adaptive backstepping learning control design of strict feedback nonlinear systems has brought about much attention, and some representative research results have been reported, for example, [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. Refs. [6], [7], [8], [9] investigated the adaptive NN/fuzzy backstepping control problem for single-input and single-output (SISO) strict-feedback nonlinear systems. The authors in [10], [11], [12], [13], [14], [15], [16] developed adaptive NN/fuzzy backstepping control schemes for multi-input and multi-output (MIMO) nonlinear systems or interconnected large-scale nonlinear systems. Although a great progress has been made in adaptive NN/fuzzy backstepping control for strict-feedback nonlinear systems, the control problems on output or state constraint are not been considered in the above mentioned results. The output or state constraints are ubiquitous in physical systems, e.g., robot systems, aerospace system, and industrial control systems. Violation of these constraints during operation may result in performance degradation, hazards or system damage. Therefore, the control design for strict-feedback nonlinear systems with output or state constraints is very important. Recently, some adaptive NN/fuzzy control approaches have been proposed for strict-feedback nonlinear systems with output or state constraints [17], [18], [19], [20], [21], [22], [23]. The authors in [17], [18], [19], [20] investigated the adaptive NN/fuzzy output constraint control problem for strict feedback nonlinear systems, and the authors in [21], [22], [23] proposed adaptive NN/fuzzy control methods for strict feedback nonlinear systems with state constraints. It should be noted that the above mentioned adaptive neural network and fuzzy backstepping constraint control methods do not address the optimal control problem.

It is well known that optimal control problem is one of key issues in modern control theory. As for nonlinear systems, the optimal control problem eventually boils down to solve the equation of Hamilton–Jacobi-Bellman (HJB). However, since the HJB equation is a nonlinear partial differential equation, it is very difficult to obtain its closed-loop form solution. More recently, some adaptive neural networks or fuzzy optimal control schemes have been proposed nonlinear systems [24], [25], [26], [27], [28], [29], [30], [31], [32]. However, the controlled systems in [24], [25], [26], [27], [28], [29], [30], [31], [32] are not strict feedback nonlinear systems. Besides, the control design methods in [24], [25], [26], [27], [28], [29], [30], [31], [32] are not based on the backstepping design principle. The authors in [33], [34], [35], [36], [37], [38] proposed the adaptive NN/fuzzy backstepping optimal control methods for different types of strict feedback nonlinear systems. However, the above adaptive NN/fuzzy optimal control approaches all do not study the state constraint optimal control problem. To the best of our knowledge, there are no results on adaptive NN/fuzzy optimal control for strict feedback nonlinear systems will full states constraints.

Motivated by the above observations, this paper investigates the adaptive NN backstepping optimal learning tracking control problem for a class of strict-feedback nonlinear systems with state constraints and unknown nonlinear dynamics. NNs are used to approximate the unknown nonlinear functions, critic and actor networks, and an adaptive NN state identifier is constructed. By constructing Barrier Lyapunov functions and optimal Barrier type cost functions, and employing the actor-critic architecture, the virtual and actual optimal controllers are developed by the backstepping recursive control design algorithm. The proposed adaptive NN optimal control scheme can guarantee that the closed-loop system is uniformly ultimately bounded and the system output can tracking the given reference signal. Moreover, the states of the controlled system are ensured not to transgress their constrained sets. The main contributions of this paper are summarized in the following two aspects:

  • 1.

    This paper first investigates the NN backstepping optimal learning tracking control problem for strict-feedback nonlinear systems with unknown nonlinear dynamics. The proposed control scheme not only can guarantee the closed-loop system to be UUB, but also achieve the tracking control objective. Although [38] investigated the NN backstepping optimal control problem for strict-feedback nonlinear systems, the nonlinear dynamics in [38] are known.

  • 2.

    This paper first studies the adaptive NN backstepping optimal constraint control problem for strict-feedback nonlinear systems, and a NN state feedback optimal control method has been proposed, and the proposed method can guarantee that the states of the controlled system not to transgress their constrained sets. Although [33], [34], [35], [36], [37], [38] also investigated the intelligent optimal control problem, they do not consider the state constrained problem.

The rest of this work is organized as follows. Section 2 exhibits preliminaries for SISO strict-feedback nonlinear system. The neural network state identifier design and based-identifier optimized controller and stability analysis is exhibited in Section 4. Section 5 gives simulation results. Section 6 concludes this paper.

Section snippets

Nonlinear control problem

Consider the following SISO strict-feedback nonlinear systemẋi=xi+1+fi(x¯i),1in-1ẋn=u+fn(x¯n)y=x1where x¯n=[x1,x2,,xn]TRn is the system state vector, uR is the control input, yR is the output of system, fi(x¯i)(i=1,2,,n) with fi(0¯i)=0 are unknown smooth functions. xi+1+fi(x¯i)(1in-1) and u+fn(x¯n) are assumed Lipschitz continuous and stabilizable on the sets containing origin. All the states are supposed to be constrained in the compact sets, i.e., xi<kci, where kci is a known

NN controller design and stability analysis

In this section, a NN adaptive backstepping optimized controller is proposed based on the actor-critic architecture, and the stability analysis of the closed-loop system is given.

Simulation example

In this section, a numerical example is given to illustrate the effectiveness of the proposed control scheme.

Consider the following strict-feedback nonlinear system:ẋ1=-sin2(2x1)+x2ẋ2=1-(1+sin(x1)cos(x2))2+uwhere x1 and x2R are the system states and uR is control input, f1(x1)=-sin2(2x1) and f2(x1,x2)=1-(1+sin(x1)cos(x2))2, the reference signal is ym=sint.

There exist six RBF neural networks in the control system. Neural networks Ŵ1,fTφ1,f(x1) contains five nodes with centers evenly spaced

Conclusion

In this study, a NN learning optimized tracking control approach has been proposed for a class of strict-feedback nonlinear systems with state constraints. The neural networks have been used to learn the unknown nonlinear dynamics, and a NN state identifier has been proposed. By constructing Barrier Lyapunov functions and optimal Barrier type cost functions for identifier error dynamic systems and the subsystems of the identifier systems, and adopting the backstepping recursive control design

CRediT authorship contribution statement

Xiaomei Li: Writing - original draft, Validation. Yongming Li: Methodology, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work is supported in part by the National Nature Science Foundation of China (No: 52070091), the Social Science Planning Foundation of Liaoning Province (No: L19BJY028), the Innovation Talent Support Plan Project of Liaoning Province in 2019.

Xiaomei Li received the B.S degree in technology economic management and the M.S degree in Business management from Liaoning University of Technology, Jinzhou, china, and received the PH.D degree in technology economic from Liaoning University, Shenyang, China. She is a professor and a Doctoral supervisor who had been as a scholar to visit Edith Cowan University in Australia 2015. Her current research interests include technology innovation and input-output efficiency evaluation for industry

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  • Cited by (0)

    Xiaomei Li received the B.S degree in technology economic management and the M.S degree in Business management from Liaoning University of Technology, Jinzhou, china, and received the PH.D degree in technology economic from Liaoning University, Shenyang, China. She is a professor and a Doctoral supervisor who had been as a scholar to visit Edith Cowan University in Australia 2015. Her current research interests include technology innovation and input-output efficiency evaluation for industry and enterprise.

    Yongming Li received the B.S. degree and the M.S. degree in Applied Mathematics from Liaoning University of Technology, Jinzhou, China, in 2004 and 2007, respectively. He received the Ph.D degree in Transportation Information Engineering & Control from Dalian Maritime University, Dalian, China in 2014. He is currently a Professor in the College of Science, Liaoning University of Technology. His current research interests include adaptive control, fuzzy control and neural networks control for nonlinear systems.

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