Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 3959806 | https://doi.org/10.1155/2020/3959806

Min Wan, Shanshan Huang, "Adaptive Dynamic Surface Decentralized Output Feedback Control for Switched Nonstrict Feedback Large-Scale Systems with Unknown Dead Zones", Mathematical Problems in Engineering, vol. 2020, Article ID 3959806, 15 pages, 2020. https://doi.org/10.1155/2020/3959806

Adaptive Dynamic Surface Decentralized Output Feedback Control for Switched Nonstrict Feedback Large-Scale Systems with Unknown Dead Zones

Academic Editor: Jose Vicente Salcedo
Received05 Apr 2020
Revised18 Jul 2020
Accepted20 Aug 2020
Published27 Sep 2020

Abstract

This paper investigates a novel adaptive output feedback decentralized control scheme for switched nonstrict feedback large-scale systems with unknown dead zones. A decentralized linear state observer is designed to estimate the unmeasurable states of subsystems. The dead zone inverse technique is used to compensate the effect of the unknown dead zone. A variable separation approach is applied to deal with the nonstrict feedback problem. Moreover, dynamic surface control and minimal parameter learning technology are adopted to reduce the computation burden. The proof of stability and the arbitrary switching are obtained by the common Lyapunov method. Finally, simulation results are given to show the effectiveness of the proposed control scheme.

1. Introduction

With the development of science and technology, the complexity of control plant is increasing, and more and more large-scale interconnected systems appear. It is well known that the decentralized control can reduce the computational burden compared with the centralized control strategy. Therefore, as an effective control scheme, the decentralized control has been widely used for large-scale nonlinear system control. Jain and Khorrami first discussed the decentralized adaptive control of a class of large-scale interconnected nonlinear systems in [1], which did not require the unknown interconnected terms of the system to meet the so-called matching conditions. Sun et al. [2] studied the fuzzy adaptive decentralized optimal control problem of the strict feedback large-scale system. Yoo and Park[3] proposed an adaptive decentralized output feedback control method for a large-scale system with unknown time-varying delayed interactions. In [4], a decentralized adaptive output feedback controller was designed for a stochastic nonlinear interconnected system. In [5], based on backstepping technology, an adaptive decentralized neural tracking control scheme was proposed for a class of large-scale interconnected systems. In [6], to deal with the output feedback control problem of the stochastic large-scale system with unknown dead zone, an adaptive fuzzy decentralized control method based on state observer was proposed. In [7], a decentralized control scheme was discussed for a SISO large-scale system with strong interconnection. Huang and Wang [8] extended the results in [7] to control a class of large-scale MIMO systems with strong interconnection and successfully applied the control method to the automated highway systems.

The aforementioned results all focus on large-scale uncertain nonlinear systems. In the actual engineering, a lot of systems not only contain many low-dimensional subsystems but also have switched properties. That is to say, these systems are not only large-scale interconnected systems but also switched systems. For example, the robot cooperative control system, spacecraft control system, power system, large-scale traffic control system, communication system, and so on are all switched large-scale systems. Recently, the control problem of large-scale interconnected switched nonlinear systems has aroused the researchers’ strong interest, and a small number of relevant results have appeared. Wang and Tong [9] studied the adaptive output feedback control problem for a class of switched Takagi–Sugeno fuzzy large-scale systems with unmeasured states. Li and Tong [10] investigated the adaptive fuzzy output feedback control problem of a switched large-scale system with nonstrict feedback structure. In [11], by adopting switched Lyapunov function with performance, the decentralized control was designed for switched fuzzy large-scale Takagi–Sugeno systems. However, there are no results on the control problems of large-scale switched nonstrict feedback systems with input constraints.

Many actuators used for industrial control, such as the servo motor, reducer, gear, hydraulic proportional servo system, cylinder, and so on, have input constraints due to mechanical structure design, manufacturing, and friction damping [12, 13]. Unknown dead zone will seriously affect the control performance of the control system, such as reduce the control accuracy, or even completely fail or produce safety accident caused by self-excited oscillation. In recent years, in order to solve the control problem caused by the dead zone of actuators, some researchers in the field of control have carried out more in-depth research and achieved some research results. In [14], an adaptive fuzzy tracking control was designed for a class of pure feedback nonlinear systems with unknown dead zone. In [15], based on backstepping control, a fuzzy output feedback tracking control method was proposed for the strict feedback system with unknown dead zone. In [16], by using dynamic surface control, an adaptive control problem was studied for a pure feedback system with unknown dead zone. Wang and Chen [17] studied the problem of adaptive fuzzy dynamic surface control of nonstrict feedback nonlinear systems with unknown virtual control coefficients, dead zone, and full state constraints. However, none of the above results have considered the control problem of switched systems and switched large-scale systems. In [18], Tong et al. proposed an adaptive fuzzy decentralized tracking control scheme based on state observer for switched large-scale systems with unknown dead zone. But the system in [18] was strict feedback form, and there existed the problem of “explosion of complexity.”

This paper will study the control problem of large-scale switched nonlinear systems with nonstrict feedback structure and use the common Lyapunov function method to overcome the arbitrary switching problem. Moreover, adaptive parameters will be designed to identify the unknown dead zone parameters through dead zone inverse technology. Compared with the previous literature, the main advantages of the proposed control scheme are listed as follows:(1)For the first time, the control problem of large-scale switched systems with nonstrict feedback structure and unknown dead zones is studied. The proposed control scheme is quite different from the existing results. The control method in [1924] cannot be used to control switched systems. In [2528], the control problem of switched systems was studied, whereas these methods cannot control switched large-scale systems. Moreover, unknown dead zone was not considered. Though switched large-scale systems with dead zones were considered in [18, 29], they were strict feedback form, rather than nonstrict feedback system in this paper. The system in [10] was a switched large-scale nonstrict feedback system, but it was without a dead zone, and the control method had the drawback of “explosion of complexity.”(2)The control method proposed in this paper does not need order differentiable and bounded conditions of input signals and a monotonically increasing condition of unknown functions. However, these strict assumptions are common in the existing literature [30, 31]. Because of using dynamic surface control, “explosion of complexity” problem is avoided in this paper. Moreover, by applying linear state observer and minimal parameter learning technology, the proposed control strategy only contains online learning parameters for the switched large-scale system, where is the number of interconnected subsystems. Therefore, the computation burden is reduced significantly.(3)The results in this paper are developed with full consideration of the dead zone problem. By constructing a dead zone compensator, the proposed adaptive fuzzy decentralized output feedback control approach can overcome the unknown dead zone problem of uncertain switched nonlinear large-scale nonstrict feedback systems.

2. Problem Description and Preliminaries

The switched nonstrict feedback large-scale system considered in this paper has interconnected subsystems. subsystem is shown as follows [10]:where is the state vector of , and only can be measured. is the output vector of the large-scale system. and (, ) are unknown smooth functions, and represents the coupling effect between subsystems of the large-scale system. is the switching signal, which is a piecewise continuous function of time from the right. The subsystem is active when . is the actual control input of . is the output of the unknown dead zone with input , which can represented as [32]where and denote the unknown slope of the dead zone and and denote the unknown width parameters of the dead zone.

Assumption 1 (see [9, 10]). For and , there exists unknown smooth function , such thatwhere .

Assumption 2 (see [9, 10]). There exists unknown smooth function , such that , where .

Assumption 3. The reference signal and its first derivative exist and are bounded.

Assumption 4 (see [32]). The output of dead zone is unmeasurable, and the parameters , , , and are unknown, whereas the signs are known such that , , , and , respectively.

Assumption 5 (see [32]). The slopes of the dead zone are bounded by constants , , , and and satisfy and .

Remark 1. It should be mentioned that if for , the switching signal , then system (1) represents a class of nonswitched nonlinear large-scale systems in nonstrict feedback form, which is investigated commonly in the existing literature.

Remark 2. It is worth pointing out that the switched systems are different from those nonswitched systems, which brings more difficulties in the control design of switched systems than those of nonswitched systems.
In this paper, the dead zone inverse technique will be used to compensate the unknown dead zone effect. Defining as the control input from the controller to achieve the control objective for the plant without a dead zone, can be generated according to the certainty equivalence dead zone inverse [29, 32]:where , , , and are the estimations of , , , and , respectively. Moreover, satisfiesThen, we can have the error between and :where , , , and are the parameter estimate errors; is bounded, where and are as follows:Control Objective. The control objective is to design an adaptive output feedback decentralized control scheme to keep all the outputs of the subsystem tracking the desired trajectories under unknown dead zones, respectively. Moreover, the tracking errors can be kept as small as possible and all the signals of the closed system are bounded.

3. The Approximation of Fuzzy Logic Systems

A fuzzy logic system can be written as , where is the fuzzy basis function vector and is the adjustable weight parameter vector.

Lemma 1 (see [3335]). If is a continuous function defined on the compact set , then for any given small constant , there exists a fuzzy logic system such that .

4. Observer-Based Adaptive Control Scheme Design

In order to estimate the unmeasured states, we design a linear state observer for as follows [10]:where is the estimation of . are the observer design parameters. Suppose that switched subsystem is active. Define observer error vector as , where , .

Then, from (1) and (8), we can havewhere , , . We can choose appropriate parameters to ensure is a Hurwitz matrix, that is, for any given positive definite matrix , there exists a positive definite matrix , such that

Choose Lyapunov function candidate as [10]

Then, we can obtain the time derivative of :

According to Young’s inequality, we have

Based on Assumption 2, satisfieswhere is the unknown nonlinear function which can be approximated by a fuzzy logic system.

From (13) and (14), we have

Based on Assumption 1, we obtainwhere is the design parameter and .

Substituting equations (13)–(16) into (12) results in

Let . By using fuzzy system to approximate , there exists optimal parameter vectorwhere and are the compact sets of and , respectively. Then, the minimal error can be written aswhere and is the unknown positive constant.

From (18) and (19), we can obtainwhere is the minimal eigenvalue of matrix .

5. Adaptive Control Law Design

Define the tracking error , virtual error , virtual control law , and first-order filters aswhere . is the time constant of the filter, that is, by letting pass through a filter that has the time constant , we can obtain .

Step 1. Define . From (21), we haveDefine as the first filter output error; then, we can obtain and .
The time derivative of is as follows:By using fuzzy system to approximate and defining as the optimal parameter vector, we havewhere is the compact set of . Then, the minimal approximation error is as follows:where and is the unknown positive constant.
Choose Lyapunov function candidate aswhere and are design parameters, , , and and are estimations of and , respectively. Define, .
Now, we can infer that satisfiesAccording to Young’s inequality and Assumption 1, we haveSubstituting (28) into (27), we obtainBased on and Young’s inequality, we can infer the following inequalities:where , , and and are design parameters.
Substituting (30)–(32) into (29) results inwhere is a constant. Then, (33) can be rearranged aswhere and .
Design , , and as follows:where , , and are design parameters.
Substituting (35)–(37) into (34) results inwhere is the maximum absolute value of .
According to Young’s inequality,Substituting (39)–(41) into (38) results inSubstituting the inequalityinto equation (42) results in

Step 2. Define and . From (21), we haveDefine as the filter output error; then, we have and .
The time derivative of is as follows:Choose Lyapunov function candidate asThe time derivative of is as follows:where is the maximum absolute value of .
According to Young’s inequality, we haveDesign as follows:where is the design parameter.
Substituting (50) into (49) leads toStep : define and . From (21), we haveDefine the filter output error as ; then, we obtain and .
The time derivative of is as follows:Choose Lyapunov function candidate asThe time derivative of is as follows:Design as follows:where is the design parameter.
Using the same recursive method in Step 2, we can getStep : define , and we can get the time derivative of as follows:Choose Lyapunov function candidate aswhere , , , and are design parameters.
Now, we can have the time derivative of as follows: