Elsevier

Neurocomputing

Volume 493, 7 July 2022, Pages 474-485
Neurocomputing

Adaptive fuzzy command filtering control for nonlinear MIMO systems with full state constraints and unknown control direction

https://doi.org/10.1016/j.neucom.2021.12.091Get rights and content

Abstract

In this paper, the command-filter-based adaptive control strategy is proposed for a class of unknown nonlinear multiple input multiple output (MIMO) systems with full state constraints and unknown control direction, whose unknown nonlinear function is approximated by the fuzzy logic system (FLS). First, the fuzzy logic system (FLS) act as the universal approximator of the unknown nonlinear function and the command-filtered backstepping control method is utilized to handle the difficulty induced by the explosion of differentiation, as well as the compensating signals are designed to make up for the command filter error. Besides, the appropriate barrier Lyapunov function and the Nussbaum-type functions are employed to deal with the constraints violation and the unknown direction control gains, respectively. It is proved that the practical tracking performance of the system can be achieved with the designed protocols and all signals in the closed-loop system are semiglobal uniformly ultimately bounded. Finally, the simulation example is performed to verify the effectiveness of the proposed control strategy.

Introduction

Because of the widespread applications of the nonlinear multiple input multiple output (MIMO) systems, considerable attentions have been obtained in the past few years [1], [2], [3]. It is worthy that the tracking problem of nonlinear MIMO system has been widely regarded as a hot topic owing to its important application prospect in practical engineering systems [4], [5], [6]. In the context of this topic, a lot of effective approaches have been proposed. For instance, the authors in [5] proposed the novel output-constrained tracking control strategy for a class of nonlinear systems subjected to the actuator faults, wherein the desire reference signals were unknown in advance. Among them, the adaptive control has attracted much attention due to its unique flexibility in updating the control parameters [7], [8], [9].

Note that the main drawback of traditional adaptive control is the accurate prediction of the system’s dynamic behavior. In practical engineering applications, most of the systems, however, are often difficult or even impossible to obtain directly as result of complicated and volatile environment [10]. That is to say, the traditional adaptive control method can not be applied directly in the investigation of the systems with unknown nonlinear dynamic behavior. Fortunately, the adaptive neural/fuzzy control strategy have given some insightful light on this problem owing to the fact that the neural networks (NNs) and fuzzy logical systems (FLSs) can approximate any nonlinear function with a desired error over a compact set. Combining with the backstepping method, many remarkable outcomes have been emerged [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. In [11], a novel adaptive neural controller was designed to cope with distance-based formation control problem for nonlinear multi-agent systems with unknown state time-delay. The authors in [13] addressed the problem of output tracking for a class of nonlinear strict-feedback systems with a direct adaptive fuzzy control approach. In [16], the fuzzy adaptive finite-time containment control scheme was proposed to deal with nonlinear multi-agent systems with input delay, wherein the state observer was used to obtain unmeasurable state. In general, adaptive fuzzy and neural network backstepping control provide two effective methodologies to solve nonlinear systems with uncertainties. It is worthy to note that the differentiation explosion problem of the virtual controller has been neglected. Recently, several new techniques have been utilized to deal with the drawback. For example, the dynamic surface control method with the first-order filter was first proposed during the process of back-stepping design in [22]. The authors in [23] came up with the adaptive neural network control method based on dynamic surface control for nonlinear MIMO systems with input saturation and unknown direction control gains, wherein the influence of the explosion of complexity was removed.

However, the error caused by the first-order filter is not taken into account, which will bring a negative impact on the control performance if the dimension of the systems is enough high. Hence, the command-filtered back-stepping scheme was first proposed in [24], which utilizing the output of the designed command filter to approach the input signal of the virtual controller at each step of the back-stepping control design. Quite different from the dynamic surface control (DSC) [25], [26], the merit of the introduction of the compensating signal in command-filtered back-stepping scheme is not only to eliminate the filtering error but also to avoid the explosion of differentiation. The authors in [27] extended this method to the spectrum of the adaptive control for a class of nonlinear strict-feedback systems and remove the computation burden caused by calculating partial derivatives effectively. For a class of uncertain nonlinear systems with unmeasurable states, the state observer and command-filter-based adaptive fuzzy control was introduced in [28]. However, according to the above discussions, the influences of state and output constraints are not taken into consideration, which are the important factors to degrade the control performance. It is the first inspiration of the current research.

Owing to the restrictions of objective conditions, measuring devices and implementation structures, the state constraints phenomenon is inevitable in the realistic engineering systems, such as chemical system [30], flexible crane systems [29]. The virtual of the barrier Lyapunov function method has been widely regarded as an effective tool to handle the state constraints problem [31], [32], [33], [34]. In [35], the strategy that combining the back-stepping design method with the barrier Lyapunov functions (BLF) was first introduced for a category of nonlinear systems with output constraints. The authors in [36] proposed an adaptive fuzzy backstepping tracking control scheme for a class of strict-feedback nonlinear systems with output constraint and input delay. On the basis of [36], a class of nonlinear systems suffered from input delay as well as the full system state constraints had been considered and the neural networks-based adaptive control scheme has been designed in [46]. Note that the unknown control direction problem has been overlooked in the previous contributions. However, the impact of the unknown control direction can not be ignored, which is the second inspiration of the current research.

Similar to the state constraints, the unknown control direction also is the main factor to deteriorate the control performance of the system. If the control gains are given accurately, the strategy that combining the adaptive backstepping design with neural network (NN) or fuzzy control is feasible [37], [38]. Nevertheless, under certain conditions, the control direction is hard to detect or be decided by the physical model of the system, which adds much difficulty to get a satisfying control performance. So as to deal with the trouble, a continuous Nussbaum gain scheme was first introduced by Nussbaum in [39], and the critical point of the scheme was that utilizing Nussbaum gain function to substitute the sign of the control gains. Because of the practicability of the method, it has become a common strategy to cope with the issue [40], [41], [42], [43], [44]. However, it should be pointed out that only the signs of the actual control input coefficients need to be considered for most of the practical systems. If taking the signs of the virtual control inputs into account, the desired control system performance is difficult to obtain because of the increase of unnecessary computation burden. The author in [45] proposed the adaptive neural command-filter tracking control method for a class of nonlinear MIMO systems with unknown control direction, which the control input signals were saturated. Furthermore, it is worth noting that the adaptive tracking problem for nonlinear MIMO systems with full state constraints with unknown direction has not yet received adequate attention.

Motivated by the above observations, this paper primarily focuses on the command-filter-based adaptive fuzzy control scheme design for a class of nonlinear MIMO systems subjected to full state constraints and unknown control direction. Before proceeding further, the fuzzy logic systems are utilized to approximate the uncertain nonlinear functions of the MIMO systems. By establishing an appropriate barrier Lyapunov function, all the signals of the closed-loop can be prevented from transgressing the constraints. Besides that, the Nussbaum gain technique is used to cope with the unknown signs of the actual control input coefficients. Specifically, the main contributions of this paper can be concluded as follows: (1) Based on adaptive fuzzy control, full state constraints and unknown control direction are taken into consideration simultaneously in the process of backstepping design, which provides a referential control strategy for practical engineering; (2) The explosion of differentiation caused by traditional backstepping design is overcame with the command filter, and the compensating signals are introduced to eliminate the effect of the filtering error to gain a better tracking performance; (3) The proposed control strategy is developed to guarantee semiglobal uniform ultimate boundedness for all the signals of the closed-loop systems, and the tracking error can be adjusted around the origin with an arbitrarily small neighborhood by choosing the appropriate design parameters.

Section snippets

Problem formulation and preliminaries

Consider the following nonlinear MIMO system with time-varying external disturbance and full state constraints:ẋi,j=fi,j(x¯i,j)+gi,j(x¯i,j)xi,j+1+di,jẋi,ni=fi,ni(x¯i,ni)+gi,ni(x¯i,ni)ui+di,niyi=xi,1where i=1,2,,n and j=1,2,,ni-1,x¯i,j=[xi,1,xi,2,,xi,j]TRj denotes the state vector, ui and yi are control input and output of the system, respectively. fi,j(·) is an unknown but smooth nonlinear function. The function gi,j(·) represents the known bounded smooth nonlinear function. di,j(·)

Controller design and stability analysis

In this section, an adaptive fuzzy tracking control is designed to stabilize a class of nonlinear MIMO systems with full state constraints and unknown control direction. The main design process of the system controller is presented as follows. First of all, we should define the tracking error for the adaptive fuzzy command-filtered control asei,1=xi,1-yi,dei,j=xi,j-xi,jcfor j=2,3,,ni, where yi,d is the desired trajectory and xi,jc is the output of the command filter with the αi,j-1 as the

Simulation results

In order to demonstrate the effectiveness of the designed approach, the following MIMO nonlinear system with external disturbances will be considered:ẋ1,1=x1,2+0.5x1,1x2,1+d1,1ẋ1,2=0.8sin(x1,1x1,2)+g1u1+d1,2y1=x1,1ẋ2,1=-x2,11+x2,14+x2,2+d2,1ẋ2,2=0.7x2,2sin(x1,2)+g2u2+d2,2y2=x2,1,where xi,j are the state variables of the MIMO nonlinear system to be constrained so that the following condition need to be satisfied: x1,1<kc1,1=1.3,x1,2<kc1,2=5,x2,1<kc2,1=1.5,x2,2<kc2,2=4.8, and gi,i=1,2 are

Conclusion

In this paper, a command-filter-based adaptive fuzzy control strategy is introduced for a class of nonlinear MIMO systems with full state constraints and unknown direction. By using fuzzy logic systems, the unknown nonlinear functions can be approximated accurately in this system. The command-filter back-stepping control strategy is improved on the basis of the traditional back-stepping method, and the improved strategy can avoid ’the explosion of complexity’ effectively which doesn’t need

CRediT authorship contribution statement

Yuhao Zhou: Conceptualization, Methodology, Software, Writing – original draft. Xin Wang: 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.

Xin Wang received his Ph.D. degree in Computer Science and Technology from Chongqing University, Chongqing, China, 2015. From 2018 to 2019, He was a visiting scholar with the Humboldt University of Berlin, Berlin, Germany, and with the Potsdam Institute for Climate Impact Research, Potsdam, Germany. Since 2018, He has been an associate professor with the school of Electronic and Information Engineering, Southwest University. He has published about more than 20 journal papers. His current

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    Xin Wang received his Ph.D. degree in Computer Science and Technology from Chongqing University, Chongqing, China, 2015. From 2018 to 2019, He was a visiting scholar with the Humboldt University of Berlin, Berlin, Germany, and with the Potsdam Institute for Climate Impact Research, Potsdam, Germany. Since 2018, He has been an associate professor with the school of Electronic and Information Engineering, Southwest University. He has published about more than 20 journal papers. His current research interest covers complex networks, impulsive control, and Multi-agent systems.

    Yuhao Zhou received the B.S degree in automation from the Jiangxi University of Science and Technology, Ganzhou, China, in 2020. He is currently pursuing the M.S. degree from the College of Electronic and Information Engineering, Southwest University, Chongqing, China. His research interests include adaptive control, multiagent system, and event-triggered control.

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