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

,is 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. ,e 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.,e 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.


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. erefore, 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 H ∞ 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.
e 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. at 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 H ∞ performance, the decentralized control was designed for switched fuzzy largescale 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." is 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. e proposed control scheme is quite different from the existing results. e control method in [19][20][21][22][23][24] cannot be used to control switched systems. In [25][26][27][28], the control problem of switched systems was studied, whereas these methods cannot control switched large-scale systems. Moreover, unknown dead zone was not considered.
ough 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. e 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) e control method proposed in this paper does not need n 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 6M online learning parameters for the switched large-scale system, where M is the number of interconnected subsystems. erefore, the computation burden is reduced significantly. (3) e 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.

Problem Description and Preliminaries
e switched nonstrict feedback large-scale system considered in this paper has M interconnected subsystems. kth subsystem k (k � 1, 2, . . . , M) is shown as follows [10]: } is the switching signal, which is a piecewise continuous function of time from the right. e sth subsystem is active when σ(t) � s. u k ∈ R is the actual control input of k . D k u k ∈ R is the output of the unknown dead zone with input u k , which can represented as [32] where m k,r and m k,l denote the unknown slope of the dead zone and d k,l and d k,r denote the unknown width parameters of the dead zone.

Assumption 3.
e reference signal y k,d (t) and its first derivative exist and are bounded.
Assumption 4 (see [32]). e output of dead zone D k u k ∈ R is unmeasurable, and the parameters m k,r , m k,l , d k,l , and d k,r are unknown, whereas the signs are known such that m k,r > 0, m k,l > 0, d k,l ≥ 0, and d k,r ≥ 0, respectively.
Assumption 5 (see [32]). e slopes of the dead zone are bounded by constants m k,r min , m k,r max , m k,l min , and m k,l max and satisfy 0 < m k,r min ≤ m k,r ≤ m k,r max and 0 < m k,l min ≤ m k,l ≤ m k,l max .

Remark 1.
It should be mentioned that if for t ∈ [0, ∞), the switching signal σ(t) � s, 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 u k,d ∈ R as the control input from the controller to achieve the control objective for the plant without a dead zone, u k ∈ R can be generated according to the certainty equivalence dead zone inverse [29,32]: where m k,r , m k,l , d k,mr , and d k,ml are the estimations of m k,r , m k,l , m k,r d k,r , and m k,l d k,l , respectively. Moreover, δ k satisfies en, we can have the error between D k (u k ) and u k,d : where is bounded, where χ k,r and χ k,l are as follows: Control Objective. e control objective is to design an adaptive output feedback decentralized control scheme to keep all the outputs y k (t) of the subsystem tracking the desired trajectories y k,d (t) 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.

The Approximation of Fuzzy Logic Systems
A fuzzy logic system can be written as where ξ(x) is the fuzzy basis function vector and θ ∈ R N is the adjustable weight parameter vector. Lemma 1 (see [33][34][35]). If f(x) is a continuous function defined on the compact set Ω, then for any given small constant ε > 0, there exists a fuzzy logic system such that

Observer-Based Adaptive Control Scheme Design
In order to estimate the unmeasured states, we design a linear state observer for kth as follows [10]: where x k,l (l � 1, 2, . . . , m k ) is the estimation of x k,l . ℓ k,1 , . . . , ℓ k,m k are the observer design parameters. Suppose that sth switched subsystem is active. Define observer error vector as Mathematical Problems in Engineering en, from (1) and (8), we can have where We can choose appropriate parameters ℓ k,1 , . . . , ℓ k,m k to ensure A k is a Hurwitz matrix, that is, for any given positive definite Choose Lyapunov function candidate as [10] en, we can obtain the time derivative of V 0 : According to Young's inequality, we have Based on Assumption 2, f s k,j (x k ) satisfies where f s k,j (x k ) is the unknown nonlinear function which can be approximated by a fuzzy logic system.
Substituting equations (13)-(16) into (12) results in where Ω k,s and U k are the compact sets of θ k,s and x k , respectively. en, the minimal error can be written as where |ε k,s (x k )| ≤ ε * k,s and ε * k,s is the unknown positive constant.

Mathematical Problems in Engineering
where λ min (Q k ) is the minimal eigenvalue of matrix Q k .

Adaptive Control Law Design
Define the tracking error z k,1 , virtual error z k,l , virtual control law α k,l− 1 , and first-order filters as where l � 2, . . . , m k . υ k,l− 1 is the time constant of the filter, that is, by letting α k,l− 1 pass through a filter that has the time constant υ k,l− 1 , we can obtain ω k,l− 1 .
Step 1. Define ω k,0 � y k,d . From (21), we have Define e k,1 as the first filter output error; then, we can obtain e k,1 � ω k,1 − α k,1 and _ ω k,1 � − (e k,1 /υ k,1 ). e time derivative of z k,1 is as follows: By using fuzzy system F k,s (x k | φ k,s ) � φ T k,s ζ k,s (x k ) to approximate f s k,1 (x k ) and defining φ * k,s as the optimal parameter vector, we have where Ω k,s is the compact set of φ k,s . en, the minimal approximation error is as follows: where |δ k,s (x k )| ≤ δ * k,s and δ * k,s is the unknown positive constant.
According to Young's inequality and Assumption 1, we have Substituting (28) into (27), we obtain

Mathematical Problems in Engineering
Based on ξ T k,s (·)ξ k,s (·) ≤ 1 and Young's inequality, we can infer the following inequalities: k,s , and τ k > 0 and λ k > 0 are design parameters.

Substituting (39)-(41) into (38) results in
Substituting the inequality into equation (42) results in Step 2. Define z k,2 � x k,2 − ω k,1 and z k, 3 Define e k,2 as the filter output error; then, we have e k,2 � ω k,2 − α k,2 and _ ω k,2 � − (e k,2 /υ k,2 ). e time derivative of z k,2 is as follows: Choose Lyapunov function candidate as e time derivative of V 2 is as follows: where ψ k,2 is the maximum absolute value of _ α k,2 . According to Young's inequality, we have (49) Design α k,2 as follows: where c k,2 > 0 is the design parameter. Substituting (50) into (49) leads to Step lth(l � 3, . . . , m k − 1): define z k,l � x k,l − ω k,l− 1 and z k,l+1 � x k,l+1 − ω k,l . From (21), we have Define the filter output error as e k,l ; then, we obtain e k,l � ω k,l − α k,l and _ ω k,l � − (e k,l /υ k,l ). e time derivative of z k,l is as follows: Choose Lyapunov function candidate as e time derivative of V l is as follows: Design α k,l as follows: where c k,l > 0 is the design parameter.
Using the same recursive method in Step 2, we can get Step m k th: define z k,m k � x k,m k − ω k,m k − 1 , and we can get the time derivative of z k,m k as follows: 8 Mathematical Problems in Engineering (58) Choose Lyapunov function candidate as where η k,1 > 0, η k,2 > 0, η k,3 > 0, and η k,4 > 0 are design parameters. Now, we can have the time derivative of V m k as follows: where |ε k,d | < ε k,d . Substituting (69) into (68) leads to

Stability Analysis
Define Lyapunov function of the closed-loop system as V ⌢ � V m k , and we can have Let the design parameters satisfy Define where i � 1, 2, . . . , m k . Define C � min k�1,...,M C k , and (70) can be rearranged as en, we have From (78), we can get that all the signals of the closed system, such as x k,i (t), x k,i (t), z k,i (t), a k,i (t), and u k (t), are semiglobally uniformly ultimately bounded (SGUUB). Moreover, the observer error satisfies ‖x k ‖ ≤ ������������������������� � (2V ⌢ (0)e − Ct + (H/C))/λ min (P k ), and the tracking error Remark 3. Note that from (78), we can only conclude that the state observer errors and tracking errors do not converge to zero. However, we can make the state observer errors and tracking errors to be small by choosing design parameters.

Remark 4.
It is worth mentioning that the influence of dead zone is compensated by inverse transformation technology, and dead zone parameters can be identified online, which is different from that in [17]. In [17], the problem of the dead zone is solved by constructing a tan-type BLF, which can compensate the influence of the dead zone but cannot get the dead zone parameters. e configuration of the output feedback control scheme proposed above for the nonstrict feedback system is shown in Figure 1.

Comparisons with Some Previous Results
To further illustrate the contributions of this method, some comparisons with previous results in [10-32, 36, 37] will be given in this section.
(1) e control methods in [13,17,[19][20][21][22][23][24] can address only nonlinear system (78) without switching signals: e control methods in [25][26][27][28]32] can only control switched systems (79), not large-scale systems: e control methods in [14-16, 18, 29, 36, 37] address the following strict feedback system: x n � f n x n + g n x n u, where x i � [x 1 , x 2 , . . . , x i ] T and f i (x i ) and g i (x i ) are the functions of partial state variables. e control method in this paper is designed for the nonlinear large-scale nonstrict feedback system (81) with arbitrary switching signals: where f σ(t) k,l (·) is the function of whole state variables and σ(t) is the arbitrary switching signal. Whether it is controller design or stability proof, switched large-scale nonstrict feedback systems are far more difficult than systems (78)-(80). We were unable to use the control methods in [13,17,[19][20][21][22][23][24] to control the switched system due to the switching problem. e control methods in [25][26][27][28]32] also cannot be used because of the interconnected influence between subsystems of the large-scale system. Moreover, when the controller of the nonstrict feedback switched system is designed by the control methods in [14-16, 18, 29, 36, 37], the virtual control signal and adaptive law of each subsystem are the functions of whole state variables. Consequently, the algebraic loop problem arises, which makes the controller design of a nonstrict feedback switched system very difficult. erefore, the controller design method of the nonstrict feedback switched largescale system (81) considered in this paper is quite different from that of the controller design methods in [13-29, 32, 36, 37].
(2) Li and Tong and Younsi et al. [10,11] proposed adaptive control methods for the nonstrict feedback switched large-scale system, but the input constraints were not considered. ough Wang et al. [12] presented the control schemes for a system with input constrained systems, all states of the system should be measurable. e strict limitation makes these control methods difficult to realize in practical applications.
erefore, control methods in [10][11][12] cannot be used to control a nonlinear system with unmeasurable state and input constraint that is discussed in this paper.
(3) is proposed adaptive control scheme does not need n-order differentiable and bounded conditions of the input signals and a monotonically increasing condition of unknown functions. However, these strict assumptions are common in the existing references [30,31]. Moreover, by adopting dynamic surface technology and first-order filters, this control scheme can avoid the problem of the "explosion of complexity." erefore, this control scheme not only conforms to engineering practice but also has a simple algorithm and requires a small number of calculations.

Remark 7.
It is worth pointing out that the selected design parameters mentioned above are the only sufficient conditions for guaranteeing the stability of the control system. Of course, in the simulation, we can select another group of design parameters based on the selection guidelines of the design parameters.
e simulation results are shown in Figures 2-7, where Figure 2 shows the trajectories of the first subsystem's output y 1 and tracking signal y 1,d ; Figure 3 shows the trajectory of the first subsystem's tracking error z 1,1 ; Figure 4 shows the trajectories of the second subsystem's output y 2 and tracking signal y 2,d ; Figure 5 shows the trajectory of the second subsystem's tracking error z 2,1 ; Figure 6 shows the trajectories of control signals u k,d and the output of dead zone D k (u k ); and Figure 7 shows the trajectory of the switched signal σ(t).
From Figures 2, 3, and 7, we can see that a good tracking performance of the first subsystem is achieved under arbitrary switched signal and the absolute value of tracking error is less than 0.014. From Figures 4, 5, and 7, we can see that a good tracking performance of the second subsystem is achieved under arbitrary switched signal and the absolute value of tracking error is less than 0.04. From Figure 6, we can see that the effect of unknown dead zone can be compensated well by the dead zone inverse technique.
From the simulation results, it can be seen that the proposed control scheme in this paper can guarantee the switched large-scale system to have a good tracking performance and can ensure the tracking errors to be kept in a small neighbourhood of zero. Moreover, all the signals of the closed-loop system are SGUUB.

Conclusion
In this paper, an adaptive fuzzy dynamic surface decentralized output feedback control scheme is proposed for large-scale interconnected uncertain nonstrict feedback switched systems. A decentralized linear state observer is designed to estimate the unmeasurable states of each subsystem. e dead zone problem is solved by using the dead zone inverse technology. e virtual control law, the real control law, and the adaptive law are designed by using the common Lyapunov function method to deal with the problem of arbitrary switching. Finally, the stability of the large-scale interconnected switched system is proved, and all the signals are guaranteed to be SGUUB. By choosing appropriate observer matrix parameters and controller design parameters, the tracking errors can be kept as small as possible. Future research directions are the extension of the results to stochastic nonlinear switched large-scale uncertain systems.

Data Availability
All the data included in this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.