Data-based Fault Tolerant Control for Affine Nonlinear Systems Through Particle Swarm Optimized Neural Networks

Haowei Lin,Bo Zhao,,Derong Liu,,and Cesare Alippi,

Abstract—In this paper,a data-based fault tolerant control(FTC)scheme is investigated for unknown continuous-time(CT)affine nonlinear systems with actuator faults. First,a neural network(NN) identifier based on particle swarm optimization(PSO) is constructed to model the unknown system dynamics. By utilizing the estimated system states, the particle swarm optimized critic neural network (PSOCNN) is employed to solve the Hamilton-Jacobi-Bellman equation(HJBE) more efficiently.Then,a data-based FTC scheme, which consists of the NN identifier and the fault compensator, is proposed to achieve actuator fault tolerance. The stability of the closed-loop system under actuator faults is guaranteed by the Lyapunov stability theorem. Finally,simulations are provided to demonstrate the effectiveness of the developed method.

I.Introduction

MODERN complex control systems always require optimal control but ensuring closed-loop stability is a difficult task,as accurate dynamics are hardly modelled.Traditional optimal control theory built upon dynamic programming and Pontryagin’s maximum principle finds the optimal objective by optimizing self-defined cost functions.However,those methodologies operate off-line and require availability of equations describing the system in advance[1].Adaptive dynamic programming (ADP)[2]–[5]is an approximate optimal control approach emerging in the field of intelligent control.Similar to reinforcement learning(RL)[6]–[9],ADP uses two main algorithms named policy iteration(PI)[10],[11]and value iteration(VI)[12]to achieve policy evaluation and policy improvement iteratively.ADP aims at adaptively learning the optimal control strategy by constructing a critic neural network(NN)approximating the solution of the Hamilton-Jacobi-Bellman equation(HJBE)[13].Based on the NN approximator,the optimal control is obtained forward-in-time and the`curse of dimensionality' is conquered[14].ADP is a well-known advanced and effective method for optimal control in both the theoretical research and real-world applications[15]–[18].Extensive efforts have been dedicated to developing ADP approaches for nonlinear systems.

In real systems,it is unavoidable to experience faults in actuators,sensors,or other system parts[19],[20].In particular,actuator faults would cause severe damages as faults cannot be accommodated by a pre-designed controller.In order to solve this problem and integrate fault tolerance ability at actuator level,robust control strategies should be considered at the control design phase[21].

There are some ADP-based control algorithms that consider both optimization and fault tolerant abilities.In[22],a fault tolerant control(FTC)algorithm based on PI was developed for nonlinear systems.The solution of the HJBE was achieved by using the NN approximation.In order to solve the actuator fault problem,a fault compensator,which did not require fault detection and isolation abilities,was designed,and the closedloop system with actuator faults was guaranteed to be stable.Zhaoet al.[23]developed an ADP-based actuator FTC scheme by designing a fault observer for nonlinear systems.The key idea is that the FTC problem was regarded as an optimization problem by considering the fault estimate in the design of the loss function.Wuet al.[24]considered the actuator failure in the tracking control task and developed an optimal adaptive compensation control based on the estimation of actuator failure coefficients.These studies require the availability of system equations.The topic on FTC for discrete-time systems with unknown dynamics has attracted considerable attention[25]–[27].The RL-based adaptive tracking FTC was studied for M IMO(multi-input and multi-output)discrete-time systems in[25]and[26].Based on the actor-critic NN structure,systems affected by abrupt faults at actuator level could be maintained stable.The proposed strategy[25]required a lower computational load and fewer learning parameters as it estimated the Euclidean norm of unknown weights of NNs instead of updating the NN weights directly. A model-free FTC strategy was proposed in[27]for single-input single-output systems.The original system is transformed into a model-free data form.By designing an NN approximator to learn the sensor fault, the FTC strategy is reconstructed based on the optimality criterion.For ADP-based FTC of unknown continuous-time(CT)systems,Zhanget al.[28]proposed a fuzzy FTC strategy based on RL for systems whose dynamics was partially unknown.They designed a new performance index function which reflects four types of actuator failures.Then,based on the constructed fuzzy-augmented dynamics, the control policy which achieved the tracking goal and stabilized the closed-loop system under actuator failures was obtained.However,this methodology is applicable only to partially unknown fuzzy systems.We finally comment that there are few ADP-based FTC schemes for completely unknown CT nonlinear systems.

As we know,the gradient-based critic NN(GDCNN)methods are widely used to solve HJBEs in order to achieve approximate optima.To train the critic NN with the gradientbased(GD)learning algorithm,one starts with random initial weights and updates them by moving along the direction of gradient descent.It means that the GD algorithm provides a tractable way for local hill climbing on the landscape of the critic NN weight parameter space.However,when initialized at a low hill in the parameter space,the GD algorithm may be trapped by unsatisfactory local optimization, resulting in inefficient HJBE solutions.One may avoid this problem by training the critic NN more than one session or applying specific prior knowledge to choose a good initial parameter.In this paper,we propose a particle swarm optimization(PSO)method to solve this problem.

PSO is a stochastic optimization algorithm where each particle has a virtual position that represents a possible solution to the optimization problem[29]–[31].In the training phase,a set of particles are initialized and evolve to search the optimal solution associated with the particle characterized by the best fitness value.PSO has multiple initial positions and relies on the global heuristic search principle,which increases the probability to avoid and even jump out of local optimums.Recently,a better performance for NN based methods has been achieved by integrating PSO into NNs.Martinet al.[32]developed the PSO-trained NN to solve the electrical impedance tomography problem.It was shown that the PSOtrained NN converged faster compared to the GD algorithm.Daset al.[33]considered PSO-trained NNs in channel equalization problems.The proposed equalizer performs better than other NN-based equalizers in noisy conditions.Chanet al.[34] presented a short-term traffic flow forecast algorithm based on PSO and artificial NNs,which required simple NNs and contained memory.

Motivated by the above analysis,this paper develops an ADP approach based on PSO and NNs to achieve actuator fault tolerance of unknown CT affine nonlinear systems.The main contributions are:

1)The proposed data-based FTC algorithm deals with completely unknown CT nonlinear systems, rather than known or partially unknown systems(as in[23]and[28]).Moreover,the dynamics of the unknown systems are approximated by the PSO-trained nonlinear NN identifier based on available measurements,and hence making the method effective in real applications.

2)The HJBE is solved through the particle swarm optimized critic NN(PSOCNN)instead of the general GDCNN;in this way the HJBE is solved with a high successful rate.

3)The presented data-based FTC strategy provides an online fault tolerant control which is shown to be optimal.

The rest of this paper is organized as follows.In Section II,the problem statement for faulty nonlinear CT systems is presented.In Section III,an NN identifier is constructed to estimate the system dynamics.Then, the data-based FTC through the PSONNs is developed based on the adaptive fault estimation.In Section IV,two simulation examples are provided to demonstrate the effectiveness of the proposed method.Finally,Section V concludes the present paper.

II.Problem Statement

III.Data-based Fault Tolerant Control Methodology Based on PSONNs

V.Conclusions

A data-based FTC algorithm exploiting PSONNs is developed for unknown CT affine nonlinear systems characterized by actuator faults.By constructing a PSOtrained NN identifier,the unknown system dynamics are obtained.Then,the PSOCNN is proposed to approximate the solution of the HJBE for the optimal control.In order to tolerate actuator faults in unknown nonlinear systems,the data-based FTC law is derived by an adaptive compensator.Simulation results show that the proposed data-based FTC algorithm can guarantee the stability of the closed-loop systems with actuator faults.Furthermore,the PSOCNN is better in producing a good solution for the HJBE than that of GDCNN.To the best of our knowledge,the unknown system should be modeled first to estimate the system states and the control matrix before constructing the FTC scheme,which means that the NN identifier is trained off-line and the proposed control scheme cannot be used for non-affine systems.In future work,we will focus on developing an online PSOCNN-based FTC scheme for unknown non-affine nonlinear systems.