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

Authors: Min Wan; Shanshan Huang
Journal: Mathematical Problems in Engineering
 (2020)

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## 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.


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## Body

## 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 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.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 H∞ 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 [19–24] cannot be used to control switched systems. In [25–28], the control problem of switched systems was studied, whereas these methods cannot control switched large-scale systems. Moreover, unknown dead zone was not considered. 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 needn 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. 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 hasM interconnected subsystems. kth subsystem ∑kk=1,2,…,M is shown as follows [10]:(1)x˙k,1=xk,2+fk,1σtx¯k+hk,1σty¯,x˙k,2=xk,3+fk,2σtx¯k+hk,2σty¯,⋮x˙k,mk−1=xk,mk+fk,mk−1σtx¯k+hk,mk−1σty¯,x˙k,mk=Dkuk+fk,mkσtx¯k+hk,mkσty¯,yk=xk,1,where x¯k=xk,1,xk,2,…,xk,mkT∈Rmk is the state vector of ∑k, and only xk,1 can be measured. y¯=y1,…,yM∈RM is the output vector of the large-scale system. fk,lσtx¯k and hk,lσty¯ (1≤k≤M, 1≤l≤mk) are unknown smooth functions, and hk,lσty¯ represents the coupling effect between M subsystems of the large-scale system. σt:0,∞⟶N=1,2,…,n is the switching signal, which is a piecewise continuous function of time from the right. The sth subsystem is active when σt=s. uk∈R is the actual control input of ∑k. Dkuk∈R is the output of the unknown dead zone with input uk, which can represented as [32](2)Dkuk=mk,ruk−dk,r,uk≥dk,r,0,−dk,l<uk<dk,r,mk,luk+dk,l,uk≤−dk,l,where mk,r and mk,l denote the unknown slope of the dead zone and dk,l and dk,r denote the unknown width parameters of the dead zone.Assumption 1 (see [9, 10]).
For1≤k≤M and 1≤l≤mk, there exists unknown smooth function ℏk,l,isyi, such that(3)hk,lsy¯2≤∑i=1Myiℏk,l,isyi2,where s∈N.Assumption 2 (see [9, 10]).
There exists unknown smooth functionf¯k,lsx¯k, such that fk,lsx¯k=xk,1f¯k,lsx¯k, where l=1,2,…,mk.Assumption 3.
The reference signalyk,dt and its first derivative exist and are bounded.Assumption 4 (see [32]).
The output of dead zoneDkuk∈R is unmeasurable, and the parameters mk,r, mk,l, dk,l, and dk,r are unknown, whereas the signs are known such that mk,r>0, mk,l>0, dk,l≥0, and dk,r≥0, respectively.Assumption 5 (see [32]).
The slopes of the dead zone are bounded by constantsmk,rmin, mk,rmax, mk,lmin, and mk,lmax and satisfy 0<mk,rmin≤mk,r≤mk,rmax and 0<mk,lmin≤mk,l≤mk,lmax.Remark 1.
It should be mentioned that if fort∈0,∞, the switching signal σt=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. Defininguk,d∈R as the control input from the controller to achieve the control objective for the plant without a dead zone, uk∈R can be generated according to the certainty equivalence dead zone inverse [29, 32]:(4)uk=Dk−1uk,d=uk,d+d^k,mrm^k,rδk+uk,d−d^k,mlm^k,l1−δk,where m^k,r, m^k,l, d^k,mr, and d^k,ml are the estimations of mk,r, mk,l, mk,rdk,r, and mk,ldk,l, respectively. Moreover, δk satisfies(5)δk=1,uk,d≥0,0,uk,d<0.
Then, we can have the error betweenDkuk and uk,d:(6)Dkuk−uk,d=d˜k,mr−uk,d+d^k,mrm^k,rm˜k,rδk+−d˜k,ml−uk,d−d^k,mlm^k,lm˜k,l1−δk+εk,d,where m˜k,r=m^k,r−mk,r, m˜k,l=m^k,l−mk,l, d˜k,mr=d^k,mr−dk,mr, and d˜k,ml=d^k,ml−dk,ml are the parameter estimate errors; εk,d=−mk,rχk,ruk−dk,r−mk,lχk,luk+dk,l is bounded, where χk,r and χk,l are as follows:(7)χk,r=1,0≤uk<dk,r,0,uk<0,uk≥dk,r,χk,l=1,−dk,l≤uk<0,0,uk<−dk,l,uk≥0.
Control Objective. The control objective is to design an adaptive output feedback decentralized control scheme to keep all the outputs ykt of the subsystem tracking the desired trajectories yk,dt 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 asf^xθ=θTξx, where ξx is the fuzzy basis function vector and θ∈RN is the adjustable weight parameter vector.Lemma 1 (see [33–35]).
Iffx 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 supx∈Ωfx−θTξx≤ε.


## 4. Observer-Based Adaptive Control Scheme Design

In order to estimate the unmeasured states, we design a linear state observer forkth as follows [10]:(8)x^˙k,1=x^k,2−ℓk,1x^k,1−yk,x^˙k,2=x^k,3−ℓk,2x^k,1−yk,⋮x^˙k,mk−1=x^k,mk−ℓk,mk−1x^k,1−yk,x^˙k,mk=Dkuk−ℓk,mkx^k,1−yk,where x^k,ll=1,2,…,mk is the estimation of xk,l. ℓk,1,…,ℓk,mk are the observer design parameters. Suppose that sth switched subsystem is active. Define observer error vector as x˜¯k=x¯k−x^¯k, where x¯k=xk,1,xk,2,…,xk,mkT, x^¯k=x^k,1,x^k,2,…,x^k,mkT.Then, from (1) and (8), we can have(9)x˜˙¯k=Akx˜¯k+Fks+Hks,where Ak=−ℓk,1⋮Imk−1×mk−1−ℓk,mk00, Fks=fk,1sx¯k,…,fk,mksx¯kT, Hks=hk,1sy¯,…,hk,mksy¯T. We can choose appropriate parameters ℓk,1,…,ℓk,mk to ensure Ak is a Hurwitz matrix, that is, for any given positive definite matrix Qk=QkT>0, there exists a positive definite matrix Pk=PkT>0, such that(10)AkTPk+PkAk=−Qk.Choose Lyapunov function candidate as [10](11)V0=∑k=1Mx˜¯kTPkx˜¯k.Then, we can obtain the time derivative ofV0:(12)V˙0=−∑k=1Mx˜¯kTQkx˜¯k+∑k=1M2x˜¯kTPkFks+Hks.According to Young’s inequality, we have(13)2x˜¯kTPkFks≤x˜¯k2+Pk2∑j=1mkfk,jsx¯k2.Based on Assumption2, fk,jsx¯k satisfies(14)fk,jsx¯k=ykf¯k,jsx¯k,where f¯k,jsx¯k is the unknown nonlinear function which can be approximated by a fuzzy logic system.From (13) and (14), we have(15)Pk2∑j=1mkfk,jsx¯k2=Pk2∑j=1mkyk2f¯k,jsx¯k2=ykPk2∑j=1mkykf¯k,jsx¯k2.Based on Assumption1, we obtain(16)∑k=1M2x˜¯kTPkHks≤∑k=1M1λx˜¯k2+∑k=1MλPk2Hks2≤1λ∑k=1Mx˜¯k2+λ¯∑k=1M∑l=1mk∑i=1Myiℏk,l,isyk2≤1λ∑k=1Mx˜¯k2+λ¯∑i=1M∑l=1mk∑k=1Myiℏi,l,ksyk2≤1λ∑k=1Mx˜¯k2+λ¯∑k=1M∑i=1M∑l=1mkyk2ℏi,l,ksyk2≤1λ∑k=1Mx˜¯k2+∑k=1Myk2∑i=1M∑l=1mkλ¯ℏi,l,ksyk2,where λ>0 is the design parameter and λ¯=λmax1≤k≤MPk2.Substituting equations (13)–(16) into (12) results in(17)V˙0≤∑k=1M−x˜¯kTQkx˜¯k+1+1λx˜¯k2+yk2∑i=1M∑l=1mkλ¯ℏi,l,ksyk2+ykPk2∑j=1mkykf¯k,jsx¯k2≤∑k=1M−x˜¯kTQkx˜¯k+1+1λx˜¯k2+ykyk∑i=1M∑l=1mkλ¯ℏi,l,ksyk2+Pk2∑j=1mkykf¯k,jsx¯k2+∑i=1Mykℏi,1,ksyk2−∑i=1Myk2ℏi,1,ksyk2.LetΦk,sx¯k=yk∑i=1M∑l=1mkλ¯ℏi,l,ksyk2+Pk2∑j=1mkykf¯k,jsx¯k2+∑i=1Mykℏi,1,ksyk2. By using fuzzy system f^k,sx¯kθk,s=θk,sTξk,sx¯k to approximate Φk,sx¯k, there exists optimal parameter vector(18)θk,s∗=argminθk,s∈Ωk,ssupx¯k∈Ukf^k,sx¯kθk,s−Φk,sx¯k,where Ωk,s and Uk are the compact sets of θk,s and x¯k, respectively. Then, the minimal error can be written as(19)εk,sx¯k=Φk,sx¯k−f^k,sx¯kθk,s∗,where εk,sx¯k≤εk,s∗ and εk,s∗ is the unknown positive constant.From (18) and (19), we can obtain(20)V˙0≤∑k=1M−λminQk−1−1λx˜¯k2+ykθk,s∗Tξk,sx¯k+εk,sx¯k−∑i=1Myk2ℏi,1,ksyk2,where λminQk is the minimal eigenvalue of matrix Qk.


## 5. Adaptive Control Law Design

Define the tracking errorzk,1, virtual error zk,l, virtual control law αk,l−1, and first-order filters as(21)zk,1=yk−yk,d,zk,l=x^k,l−ωk,l−1,υk,l−1ω˙k,l−1+ωk,l−1=αk,l−1,ωk,l−10=αk,l−10,where l=2,…,mk. υk,l−1 is the time constant of the filter, that is, by letting αk,l−1 pass through a filter that has the time constant υk,l−1, we can obtain ωk,l−1.Step 1.
Defineωk,0=yk,d. From (21), we have(22)υk,1ω˙k,1+ωk,1=αk,1,ωk,10=αk,10.
Defineek,1 as the first filter output error; then, we can obtain ek,1=ωk,1−αk,1 and ω˙k,1=−ek,1/υk,1.
The time derivative ofzk,1 is as follows:(23)z˙k,1=xk,2+fk,1s+hk,1s−ω˙k,0=zk,2+ek,1+αk,1+x˜k,2+fk,1sx¯k+hk,1sy¯−y˙k,d.
By using fuzzy systemF^k,sx¯kφk,s=φk,sTζk,sx¯k to approximate fk,1sx¯k and defining φk,s∗ as the optimal parameter vector, we have(24)φk,s∗=argminφk,s∈Ω¯k,ssupx¯k∈UkF^k,sx¯kφk,s−fk,1sx¯k,where Ω¯k,s is the compact set of φk,s. Then, the minimal approximation error is as follows:(25)δk,sx¯k=fk,1sx¯k−F^k,sx¯kφk,s∗,where δk,sx¯k≤δk,s∗ and δk,s∗ is the unknown positive constant.
Choose Lyapunov function candidate as(26)V1=V0+∑k=1M12zk,12+12ek,12+12γk,1ϖ˜k,12+12γk,2ϖ˜k,22,where γk,1>0 and γk,2>0 are design parameters, ϖk,1∗=maxs∈Nθk,s∗Tθk,s∗, ϖk,2∗=maxs∈Nφk,s∗Tφk,s∗, and ϖk,1 and ϖk,2 are estimations of ϖk,1∗ and ϖk,2∗, respectively. Defineϖ˜k,1=ϖk,1∗−ϖk,1, ϖ˜k,2=ϖk,2∗−ϖk,2.
Now, we can infer thatV˙1 satisfies(27)V˙1=V˙0+∑k=1Mzk,1z˙k,1+ek,1e˙k,1−1γk,1ϖ˜k,1ϖ˙k,1−1γk,2ϖ˜k,2ϖ˙k,2=V˙0+∑k=1Mzk,1zk,2+ek,1+αk,1+x˜k,2+fk,1sx¯k+hk,1sy¯−y˙k,d+ek,1e˙k,1−1γk,1ϖ˜k,1ϖ˙k,1−1γk,2ϖ˜k,2ϖ˙k,2.
According to Young’s inequality and Assumption1, we have(28)∑k=1Mzk,1hk,1sy¯≤∑k=1Mzk,124+∑k=1Mhk,1sy¯2≤∑k=1Mzk,124+∑k=1M∑i=1Myk2ℏk,1,isyk2.
Substituting (28) into (27), we obtain(29)V˙1=V˙0+∑k=1Mzk,1z˙k,1+ek,1e˙k,1−1γk,1ϖ˜k,1ϖ˙k,1−1γk,2ϖ˜k,2ϖ˙k,2≤∑k=1M−λminQk−1−1λx˜¯k2+ykθk,sT∗ξk,sx¯k+εk,sx¯k+zk,124+∑k=1Mzk,1zk,2+ek,1+αk,1+x˜k,2+φk,s∗Tξk,sx¯k+δk,sx¯k−y˙k,d+ek,1e˙k,1−1γk,1ϖ˜k,1ϖ˙k,1−1γk,2ϖ˜k,2ϖ˙k,2.
Based onξk,sT⋅ξk,s⋅≤1 and Young’s inequality, we can infer the following inequalities:(30)ykθk,s∗Tξk,sx¯k+εk,sx¯k≤zk,1θk,s∗Tξk,sx¯k+εk,sx¯k+yk,dθk,s∗Tξk,sx¯k+εk,sx¯k≤zk,12ϖk,1∗4τk+zk,124+yk,d22+ϖk,1∗+2εk∗2+τk,(31)zk,1φk,s∗Tξk,sx¯k+δk,sx¯k≤zk,12ϖk,2∗4τk+zk,124+δk∗2+τk,(32)zk,1x˜k,2≤1λkx˜¯k2+λkzk,124,where εk∗=maxs∈Nεk,s∗2, δk∗=maxs∈Nδk,s∗2, and τk>0 and λk>0 are design parameters.
Substituting (30)–(32) into (29) results in(33)V˙1≤∑k=1M−λminQk−1−1λ−1λkx˜¯k2+∑k=1Mzk,1zk,2+ek,1+αk,1+zk,1ϖk,1∗4τk+zk,1ϖk,2∗4τk+3+λkzk,14−y˙k,d+ek,1e˙k,1−1γk,1ϖ˜k,1ϖ˙k,1−1γk,2ϖ˜k,2ϖ˙k,2y¯k,d22+ϖk,1∗+2εk∗2+δk∗2+2τk,where y¯k,d=maxt∈0,+∞yk,dt is a constant. Then, (33) can be rearranged as(34)V˙1≤∑k=1M−qk,1x˜¯k2+∑k=1Mzk,1zk,2+ek,1+αk,1+zk,1ϖk,14τk+zk,1ϖk,24τk+3+λkzk,14−y˙k,d+1γk,1ϖ˜k,1γk,1zk,124τk−ϖ˙k,1+1γk,2ϖ˜k,2γk,2zk,124τk−ϖ˙k,2+ek,1e˙k,1+dk,1,where qk,1=λminQk−1−1/λ−1/λk and dk,1=y¯k,d2/2+ϖk,1∗+2εk∗2+δk∗2+2τk.
Designαk,1, ϖk,1, and ϖk,2 as follows:(35)αk,1=−ck,1zk,1−zk,1ϖk,14τk−zk,1ϖk,24τk−3+λkzk,14+y˙k,d,(36)ϖ˙k,1=γk,1zk,124τk−2σk,1ϖk,1,(37)ϖ˙k,2=γk,2zk,124τk−2σk,2ϖk,2,where ck,1>0, σk,1>0, and σk,2>0 are design parameters.
Substituting (35)–(37) into (34) results in(38)V˙1≤∑k=1M−qk,1x˜¯k2+∑k=1M−ck,1zk,12+zk,1zk,2+zk,1ek,1+2σk,1γk,1ϖ˜k,1ϖk,1+2σk,2γk,2ϖ˜k,2ϖk,2+ek,1−ek,1υk,1−α˙k,1+dk,1≤∑k=1M−qk,1x˜¯k2+∑k=1M−ck,1zk,12+zk,1zk,2+zk,1ek,1+2σk,1γk,1ϖ˜k,1ϖk,1+2σk,2γk,2ϖ˜k,2ϖk,2−ek,12υk,1+ek,12+14ψk,12+dk,1,where ψk,1 is the maximum absolute value of α˙k,1.
According to Young’s inequality,(39)2σk,1γk,1ϖ˜k,1ϖk,1≤−σk,1γk,1ϖk,12+σk,1γk,1ϖk,1∗2,(40)2σk,2γk,2ϖ˜k,2ϖk,2≤−σk,2γk,2ϖk,22+σk,2γk,2ϖk,2∗2,(41)zk,1ek,1≤zk,12+14ek,12.
Substituting (39)–(41) into (38) results in(42)V˙1≤∑k=1M−qk,1x˜¯k2+∑k=1M−ck,1−1zk,12+zk,1zk,2+54−1υk,1ek,12−∑i=12σk,iγk,iϖk,i2+∑i=12σk,iγk,iϖk,i∗2+14ψk,12+dk,1.
Substituting the inequality(43)−σk,i2γk,iϖ˜k,i2≥−σk,iγk,iϖk,i2−σk,iγk,iϖk,i∗2,into equation (42) results in(44)V˙1≤∑k=1M−qk,1x˜¯k2+∑k=1M−ck,1−1zk,12+zk,1zk,2+54−1υk,1ek,12−∑i=12σk,i2γk,iϖ˜k,i2+∑i=122σk,iγk,iϖk,i∗2+14ψk,12+dk,1.Step 2.
Definezk,2=x^k,2−ωk,1 and zk,3=x^k,3−ωk,2. From (21), we have(45)υk,2ω˙k,2+ωk,2=αk,2,ωk,20=αk,20.
Defineek,2 as the filter output error; then, we have ek,2=ωk,2−αk,2 and ω˙k,2=−ek,2/υk,2.
The time derivative ofzk,2 is as follows:(46)z˙k,2=x^k,3−ℓk,2x^k,1−yk−ω˙k,1=zk,3+αk,2+ek,2−ℓk,2x^k,1−yk−ω˙k,1.
Choose Lyapunov function candidate as(47)V2=V1+∑k=1M12zk,22+12ek,22.
The time derivative ofV2 is as follows:(48)V˙2=V˙1+∑k=1Mzk,2z˙k,2+ek,2e˙k,2=V˙1+∑k=1Mzk,2zk,3+αk,2+ek,2−ℓk,2x^k,1−yk−ω˙k,1+ek,2e˙k,2≤V˙1+∑k=1Mzk,2zk,3+αk,2+ek,2−ℓk,2x^k,1−yk−ω˙k,1−ek,22υk,2+ek,22+14ψk,22,where ψk,2 is the maximum absolute value of α˙k,2.
According to Young’s inequality, we have(49)V˙2≤V˙1+∑k=1Mzk,2zk,3+αk,2−ℓk,2x^k,1−yk−ω˙k,1+zk,22+14ek,22−ek,22υk,2+ek,22+14ψk,22≤∑k=1M−qk,1x˜¯k2+∑k=1M−ck,1−1zk,12+zk,1zk,2+54−1υk,1ek,12−∑i=12σk,i2γk,iϖ˜k,i2+∑i=122σk,iγk,iϖk,i∗2+14ψk,22+dk,1+∑k=1Mzk,2zk,3+αk,2−ℓk,2x^k,1−yk−ω˙k,1+zk,22+54−1υk,2ek,22+14ψk,22.
Designαk,2 as follows:(50)αk,2=−ck,2zk,2−zk,1+ℓk,2x^k,1−yk−ωk,1−αk,1υk,1,where ck,2>0 is the design parameter.
Substituting (50) into (49) leads to(51)V˙2≤∑k=1M−qk,1x˜¯k2+∑k=1M−ck,1−1zk,12+zk,1zk,2+54−1υk,1ek,12−∑i=12σk,i2γk,iϖ˜k,i2+∑i=122σk,iγk,iϖk,i∗2+14ψk,12+dk,1+∑k=1Mzk,2zk,3−zk,2zk,1−ck,2−1zk,22+54−1υk,2ek,22+14ψk,22≤∑k=1M−qk,1x˜¯k2+∑k=1M−∑i=12ck,i−1zk,i2+zk,2zk,3+∑i=1254−1υk,iek,i2−∑i=12σk,i2γk,iϖ˜k,i2+∑i=122σk,iγk,iϖk,i∗2+∑i=1214ψk,i2+dk,1.
Steplthl=3,…,mk−1: define zk,l=x^k,l−ωk,l−1 and zk,l+1=x^k,l+1−ωk,l. From (21), we have(52)υk,lω˙k,l+ωk,l=αk,l,ωk,l0=αk,l0.
Define the filter output error asek,l; then, we obtain ek,l=ωk,l−αk,l and ω˙k,l=−ek,l/υk,l.
The time derivative ofzk,l is as follows:(53)z˙k,l=x^k,l+1−ℓk,lx^k,1−yk−ω˙k,l−1=zk,l+1+αk,l+ek,l−ℓk,lx^k,1−yk−ω˙k,l−1.
Choose Lyapunov function candidate as(54)Vl=Vl−1+∑k=1M12zk,l2+12ek,l2.
The time derivative ofVl is as follows:(55)V˙l=V˙l−1+∑k=1Mzk,lz˙k,l+ek,le˙k,l=V˙l−1+∑k=1Mzk,lzk,l+1+αk,l+ek,l−ℓk,lx^k,1−yk−ω˙k,l−1+ek,le˙k,l.
Designαk,l as follows:(56)αk,l=−ck,lzk,l−zk,l−1+ℓk,lx^k,1−yk−ωk,l−1−αk,l−1υk,l−1,where ck,l>0 is the design parameter.
Using the same recursive method in Step2, we can get(57)V˙l≤∑k=1M−qk,1x˜¯k2+∑k=1M−∑i=1lck,i−1zk,i2+zk,lzk,l+1+∑i=1l54−1υk,iek,i2−∑i=12σk,i2γk,iϖ˜k,i2+∑i=122σk,iγk,iϖk,i∗2+∑i=1l14ψk,i2+dk,1.
Stepmkth: define zk,mk=x^k,mk−ωk,mk−1, and we can get the time derivative of zk,mk as follows:(58)z˙k,mk=Dkuk−ℓk,mkx^k,1−yk−ω˙k,mk−1=uk,d+d˜k,mr−uk,d+d^k,mrm^k,rm˜k,rδk−d˜k,ml+uk,d−d^k,mlm^k,lm˜k,l1−δk+εk,d−ℓk,mkx^k,1−yk−ω˙k,mk−1.
Choose Lyapunov function candidate as(59)Vmk=Vmk−1+∑k=1M12zk,mk2+12ηk,1m˜k,r2+12ηk,2m˜k,l2+12ηk,3d˜k,mr2+12ηk,4d˜k,ml2,where ηk,1>0, ηk,2>0, ηk,3>0, and ηk,4>0 are design parameters.
Now, we can have the time derivative ofVmk as follows:(60)V˙mk=V˙mk−1+∑k=1Mzk,mkz˙k,mk+1ηk,1m˜k,rm˜˙k,r+1ηk,2m˜k,lm˜˙k,l+1ηk,3d˜k,mrd˜˙k,mr+1ηk,4d˜k,mld˜˙k,ml=V˙mk−1+∑k−1Mzk,mkuk,d+d˜k,mr−uk,d+d^k,mrm^k,rm˜k,rδk−d˜k,ml+uk,d−d^k,mlm^k,lm˜k,l1−δk+εk,d−ℓk,mkx^k,1−yk−ω˙k,mk−1+1ηk,1m˜k,rm˜˙k,r+1ηk,2m˜k,lm˜˙k,l+1ηk,3d˜k,mrd˜˙k,mr+1ηk,4d˜k,mld˜˙k,ml=V˙mk−1+∑k−1Mzk,mkuk,d+εk,d−ℓk,mkx^k,1−yk−ω˙k,mk−1+1ηk,1m˜k,rm˜˙k,r−uk,d+d^k,mrm^k,rzk,mkηk,1δk+1ηk,2m˜k,lm˜˙k,l−uk,d−d^k,mlm^k,lzk,mkηk,21−δk+1ηk,3d˜k,mrd˜˙k,mr+zk,mkηk,3δk+1ηk,4d˜k,mld˜˙k,ml−zk,mkηk,41−δk.
Because(61)zk,mkεk,d≤zk,mk2+14εk,d2,we can obtain(62)V˙mk≤V˙mk−1+∑k=1Mzk,mkuk,d+zk,mk−ℓk,mkx^k,1−yk−ω˙k,mk−1+1ηk,1m˜k,rm˜˙k,r−uk,d+d^k,mrm^k,rzk,mkηk,1δk+1ηk,2m˜k,lm˜˙k,l−uk,d−d^k,mlm^k,lzk,mkηk,21−δk+1ηk,3d˜k,mrd˜˙k,mr+zk,mkηk,3δk+1ηk,4d˜k,mld˜˙k,ml−zk,mkηk,41−δk+14εk,d2.
Designuk,d, m^k,r, m^k,l, d^k,mr, and d^k,ml as follows:(63)uk,d=−ck,mkzk,mk−zk,mk−1+ℓk,mkx^k,1−yk−ωk,mk−1−αk,mk−1υk,mk−1,(64)m^˙k,r=uk,d+d^k,mrm^k,rzk,mkηk,1δk+2μk,1m^k,r,(65)m^˙k,l=uk,d−d^k,mlm^k,lzk,mkηk,21−δk+2μk,2m^k,l,(66)d^˙k,mr=−zk,mkηk,3δk+2μk,3d^k,mr,(67)d^˙k,ml=zk,mkηk,41−δk+2μk,4d^k,ml,where ck,mk>0, μk,1>0, μk,2>0, μk,3>0, and μk,4>0 are design parameters.
Substituting (63)–(67) into (62) results in(68)V˙mk≤∑k=1M−qk,1x˜¯k2+∑k=1M−∑i=1mkck,i−1zk,i2+∑i=1mk−154−1υk,iek,i2−∑i=12σk,i2γk,iϖ˜k,i2+∑i=122σk,iγk,iϖk,i∗2+∑i=1mk−114ψk,i2+dk,1+14εk,d2+2μk,1ηk,1m˜k,rm^k,r+2μk,2ηk,2m˜k,lm^k,l+2μk,3ηk,3d˜k,mrd^k,mr+2μk,4ηk,4d˜k,mld^k,ml.
Define(69)H=∑i=12σk,iγk,iϖk,i∗2+μk,1ηk,1mk,r2+μk,2ηk,2mk,l2+μk,3ηk,3dk,mr2+μk,4ηk,4dk,ml2+∑i=1mk−114ψk,i2+dk,1+14ε¯k,d2,where εk,d<ε¯k,d.
Substituting (69) into (68) leads to(70)V˙mk≤∑k=1M−qk,1x˜¯k2+∑k=1M−∑i=1mkck,i−1zk,i2−∑i=1mk−11υk,i−54ek,i2−∑i=12σk,i2γk,iϖ˜k,i2−μk,1ηk,1m˜k,r2−μk,2ηk,2m˜k,l2−μk,3ηk,3d˜k,mr2−μk,4ηk,4d˜k,ml2+H.


## 6. Stability Analysis

Define Lyapunov function of the closed-loop system asV⌢=Vmk, and we can have V⌢˙=V˙mk. Let the design parameters satisfy(71)ck,i−1>0,i=1,2,…,mk,1υk,i−54>0,i=1,2,…,mk−1.Define(72)Ck=min2qk,1λmaxPk,2ck,i−1,2σk,1,2σk,2,2μk,1,2μk,2,2μk,3,2μk,4,21υk,i−54,where i=1,2,…,mk.DefineC=mink=1,…,MCk, and (70) can be rearranged as(73)V⌢≤−CV⌢+H.Then, we have(74)eCtV⌢˙≤−CV⌢+HeCt,(75)ddtV⌢eCt≤HeCt,(76)V⌢eCt−V⌢0≤HCeCt−1,(77)0≤V⌢t≤V⌢0e−Ct+HC1−e−Ct≤V⌢0+HC.From (78), we can get that all the signals of the closed system, such as xk,it, x^k,it, zk,it, ak,it, and ukt, are semiglobally uniformly ultimately bounded (SGUUB). Moreover, the observer error satisfies x˜¯k≤2V⌢0e−Ct+H/C/λminPk, and the tracking error satisfies zk,1≤2V⌢0e−Ct+H/C.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.The configuration of the output feedback control scheme proposed above for the nonstrict feedback system is shown in Figure1.Figure 1
Block diagram of the control scheme.


## 7. 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)
The control methods in [13, 17, 19–24] can address only nonlinear system (78) without switching signals:(78)x˙i=xi+1+fix1,x2,…,xn,i=1,2,…,n−1,x˙n=ut+fnx1,x2,…,xn,y=x1.The control methods in [25–28, 32] can only control switched systems (79), not large-scale systems:(79)x˙1=x2+f1,σtx1,x2,…,xn,⋮x˙i=xi+1+fi,σtx1,x2,…,xn,⋮x˙n=uσt+fn,σtx1,x2,…,xn,y=x1.The control methods in [14–16, 18, 29, 36, 37] address the following strict feedback system:(80)x˙i=fix¯n+gix¯nxi+1,i=1,2,…,n−1,x˙n=fnx¯n+gnx¯nu,y=x1,wherex¯i=x1,x2,…,xiT and fix¯i and gix¯i are the functions of partial state variables.The control method in this paper is designed for the nonlinear large-scale nonstrict feedback system (81) with arbitrary switching signals:(81)x˙k,1=xk,2+fk,1σtx¯k+hk,1σty¯,x˙k,2=xk,3+fk,2σtx¯k+hk,2σty¯,⋮x˙k,mk−1=xk,mk+fk,mk−1σtx¯k+hk,mk−1σty¯,x˙k,mk=Dkuk+fk,mkσtx¯k+hk,mkσty¯,yk=xk,1,wherefk,lσt⋅ is the function of whole state variables and σt 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, 19–24] to control the switched system due to the switching problem. The control methods in [25–28, 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. Therefore, the controller design method of the nonstrict feedback switched large-scale 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. Though Wang et al. [12] presented the control schemes for a system with input constrained systems, all states of the system should be measurable. The strict limitation makes these control methods difficult to realize in practical applications. Therefore, control methods in [10–12] cannot be used to control a nonlinear system with unmeasurable state and input constraint that is discussed in this paper.(3)
This proposed adaptive control scheme does not needn-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.” Therefore, this control scheme not only conforms to engineering practice but also has a simple algorithm and requires a small number of calculations.


## 8. Simulations

Consider the following switched nonstrict feedback large-scale system [10]:(82)x˙1,1=x1,2+f1,1σtx1,1,x1,2+h1,1σty1,y2,x˙1,2=D1u1+f1,2σtx1,1,x1,2+h1,2σty1,y2,y1=x1,1,x˙2,1=x2,2+f2,1σtx2,1,x2,2+h2,1σty1,y2,x˙2,2=D2u2+f2,2σtx2,1,x2,2+h2,2σty1,y2,y2=x2,1,where σt=1,2, f1,11x1,1,x1,2=x1,121−cosx1,1x1,2+x1,2−0.2x1,1sinx1,1x1,2, h1,11y1,y2=y1y2, f1,21x1,1,x1,2=x1,1x1,2+x1,1, h1,21y1,y2=y12+y22, f2,11x2,1,x2,2=x2,1sinx2,1x2,2−0.5x2,1sinx2,1x2,2, h2,11y1,y2=y13y23, f2,21x2,1,x2,2=x2,1x2,2, h2,21y1,y2=y1−2y2, f1,12x1,1,x1,2=x1,121−cosx1,1x1,2+x1,2−0.3x1,1sinx1,1x1,2, h1,12y1,y2=4y1y2, f1,22x1,1,x1,2=3x1,1, h1,22y1,y2=y12+2y22, f2,12x2,1,x2,2=cosx2,1x2,2, h2,12y1,y2=3y12y22, f2,22x2,1,x2,2=0.8x2,1x2,2, and h2,22y1,y2=y1−3y2. The given tracking signals are y1,d=y2,d=0.25sint+0.25sin0.5t.Remark 5.
Note that if system (82) does not contain the switching signal σt, it turns into a nonlinear large-scale system, which was extensively studied in [17–22].Remark 6.
Note that functionsfk,lσtx¯k in (82) do not satisfy the assumptions in [28, 29], i.e., we cannot find increasing smooth functions ϕk,lσtx¯k with ϕk,lσt0=0, such that fk,lσtx¯k≤ϕk,lσtx¯k. In addition, (82) is not a switched strict feedback nonlinear system. Therefore, the control methods in [13–16, 27, 34, 35] cannot be used to control the system (82).
Choose the parameters asℓ1,1=ℓ1,2=ℓ2,1=ℓ2,2=10, c1,1=40, c1,2=20, c2,1=20, c2,2=20, λ1=λ2=1, τ1=τ2=1, γ1,1=γ1,2=1, γ2,1=γ2,2=2, σ1,1=σ1,2=0.05, σ2,1=σ2,2=0.1, υ1,1=υ1,2=υ2,1=υ2,2=0.01, m1,r=2, m1,l=1.3, d1,r=0.4, d1,l=0.6, m2,r=1.1, m2,l=2, d2,r=0.2, and d2,l=0.5.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.
The simulation results are shown in Figures2–7, where Figure 2 shows the trajectories of the first subsystem’s output y1 and tracking signal y1,d; Figure 3 shows the trajectory of the first subsystem’s tracking error z1,1; Figure 4 shows the trajectories of the second subsystem’s output y2 and tracking signal y2,d; Figure 5 shows the trajectory of the second subsystem’s tracking error z2,1; Figure 6 shows the trajectories of control signals uk,d and the output of dead zone Dkuk; and Figure 7 shows the trajectory of the switched signal σt.
From Figures2, 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.Figure 2
Trajectories ofy1 and y1,d.Figure 3
The tracking errorz1,1.Figure 4
Trajectories ofy2 and y2,d.Figure 5
The tracking errorz2,1.Figure 6
Control inputsuk,d and Dkuk.
(a)(b)Figure 7
The switched signalσt.


## 9. 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. The dead zone problem is solved by using the dead zone inverse technology. The 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.



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