# Adaptive Decentralized Output Feedback Tracking Control for Large-Scale Interconnected Systems with Time-Varying Output Constraints

Authors: Min Wan; Yanxia Yin
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper investigates a novel adaptive output feedback decentralized control scheme for nonstrict feedback large-scale interconnected systems with time-varying constraints. A decentralized linear state observer is designed to estimate the unmeasurable states of subsystems. Time-varying barrier Lyapunov functions are designed to ensure outputs are not violating constraints. A variable separation approach is applied to deal with the nonstrict feedback problem. Moreover, dynamic surface control and minimal parameter learning technologies are adopted to reduce the computation burden, and there are only two parameters for every subsystem to be updated online. The proof of stability is obtained by the Lyapunov method. Finally, simulation results are given to show the effectiveness of the proposed control scheme.


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## Body

## 1. Introduction

No matter in practical engineering application or theoretical research field, the high-precision trajectory tracking control of the nonlinear system is a very valuable research topic. For example, the welding robot needs to carry out welding according to the given trajectory, and the accuracy of the trajectory tracking control is an important indicator of the performance of the welding robot. The drilling guide system needs to control the well trajectory continuously to ensure the drilling quality. However, it is very difficult to design the control scheme of the nonlinear system due to various uncertainties [1–3]. Because the fuzzy logic system has been proved to have universal approximation property, it has become one of the effective measures to solve the uncertainty problem [4]. In [5], an adaptive fuzzy quantized tracking control scheme was proposed for the stochastic nonlinear uncertain strict feedback system. In [6], a direct model reference adaptive fuzzy control was discussed for a class of networked SISO nonlinear uncertain systems. In [7], for the nonlinear system with parametric uncertainties and unknown modelling errors, a robust adaptive control scheme was presented. In [8], an adaptive dynamic surface asymptotic tracking control was proposed for the uncertain nonlinear system. In [9], for the nontriangular stochastic uncertain nonlinear system with unmeasured states, an adaptive robust control was studied. In [10], for the pure-feedback nonlinear system with time-varying delay and unknown dead zone, an adaptive fuzzy tracking control was investigated. However, none of the above literature has solved the output constraint problem of the uncertain nonlinear system.Output constraint is a very important engineering problem. If the system outputs exceed the given range, it will cause control performance degradation, equipment damage, and even endanger the safety of operators and environment. Therefore, in recent years, the output constraint problem has become a hot research issue, and a large number of valuable research results have emerged. When the relevant states approach the boundary, the barrier Lyapunov function tends to infinity, so as long as the function value is bounded, the correlation states can be kept in the given constraints. Because of the above property, the barrier Lyapunov function becomes an effective measure to solve the problem of constraints. In [11], by using a barrier Lyapunov function, an indirect adaptive fuzzy control was designed for the output-constrained nonlinear system. In [12], an adaptive neural network control was proposed for the nonlinear system with full-state constraints. In [13], for a class of nonlinear pure-feedback full-state constraint systems, an adaptive control based on barrier Lyapunov functions was studied. In [14], an adaptive fuzzy backstepping output constrain tracking control was proposed for the uncertain nonlinear system in strict feedback form. In [15], by adopting barrier Lyapunov functions, an adaptive neural network control was presented for nonlinear state-constrained systems with input delay. In [16, 17], adaptive full-state constraint control methods were discussed for stochastic nonlinear systems based on barrier Lyapunov functions. All of the above literature studies considered the static constraint problem, and then the time-varying constraint is more suitable for practical engineering. In [18], an adaptive fuzzy control was proposed for the nontriangular system with time-varying error constraint. In [19], for the uncertain nonlinear system with time-varying prescribed performance, a fuzzy adaptive control based on the observer was presented. In [20], a fuzzy state observer-based adaptive control for the strict feedback system with time-varying constraint was studied. However, for the time-varying output constraints of large-scale interconnected systems, there are no relevant research results.In practical engineering, many electromechanical systems consist of many subsystems, such as multirobot cooperative control system, offshore platform, drilling system, and aerospace. Because a large-scale interconnected system has the coupling effect between subsystems, the design of its controller is very difficult. Therefore, the research on the control scheme of large-scale systems has obvious practical value and theoretical significance. In [21, 22], decentralized H∞ performance control methods were studied for switched and nonswitched large-scale systems, respectively. In [23, 24], based on the observer and backstepping technology, adaptive decentralized control methods were presented for uncertain large-scale systems with unmeasured states. In [25], for the switched uncertain large-scale system with dead zones, an adaptive output decentralized tracking control scheme was proposed. However, the output constraints of large-scale systems are not considered in the above literature. Although the control problem of large-scale systems with output constraints has been studied in [26], the system is in strict feedback form, and there are no corresponding research results about the control scheme of nonstrict feedback large-scale systems with time-varying output constraints.Based on the above results, this paper studies an adaptive output feedback decentralized control for the nonstrict feedback large-scale system with unmeasurable states and time-varying output constraints. Compared with the existed works, there are two contributions in this paper: (1) for the first time, the output feedback control problem of nonstrict feedback large-scale systems with time-varying constraints is studied. The proposed control scheme is quite different from the existed results. The proposed control scheme can not only solve the output feedback tracking control problem for a class of uncertain large-scale systems in nonstrict feedback form but also ensure all the outputs are not violating the time-varying constraints. Moreover, time-varying constraint relaxes the initial conditions of the system. (2) The control method which is proposed in this paper does not needn-order differentiable and bounded conditions of input signals and the monotonically increasing condition of unknown functions, which are common in the existing literature [27, 28]. Because of using the dynamic surface control, the “complexity explosion” problem is avoided. Moreover, each subsystem has only two adaptive parameters, and the number of parameters does not increase with the increase of system’s order.


## 2. Problem Description and Preliminaries

The nonstrict feedback large-scale system considered in this paper hasM interconnected subsystems. kth k=1,2,…M subsystem is shown as follows:(1)x˙k,1=xk,2+fk,1x¯k+hk,1y¯,x˙k,2=xk,3+fk,2x¯k+hk,2y¯,⋮x˙k,nk−1=xk,nk+fk,nk−1x¯k+hk,nk−1y¯,x˙k,nk=uk+fk,nkx¯k+hk,nky¯,yk=xk,1,where x¯k=xk,1,xk,2,…,xk,nkT∈Rnk is the state vector, and only xk,1 can be measured. y¯=y1,…,yM∈RM is the output vector of the large-scale system. fk,lx¯k and hk,ly¯ 1≤k≤M,1≤l≤nk are unknown smooth functions, and hk,ly¯ represents the coupling effect between M subsystems of the large-scale system. uk∈R is the actual control input of the kth subsystem. In practical engineering, the mathematical models of the two-stage chemical reactor, air traffic control, spring connected two-stage inverted pendulum, and other large-scale systems can be expressed as (1) [22, 29–31].Assumption 1.
(see [23, 24]). For 1≤k≤M and 1≤l≤nk, there exists unknown smooth function ℏk,l,iyi such that(2)hk,ly¯2≤∑i=1Myiℏk,l,iyi2.Assumption 2.
(see [23, 24]). There exists unknown smooth function f¯k,lx¯k such that fk,lx¯k=xk,1f¯k,lx¯k, where l=1,2,…,nk.Assumption 3.
(see [32, 33]). The reference signal yk,dt and its first derivative are bounded.
Control objective: the control objective is to design an adaptive output feedback decentralized control scheme to keep all the outputsykt of the subsystems tracking desired trajectories yk,dt, respectively. Moreover, the tracking errors can be kept as small as possible, and all the signals of the closed system are bounded.


## 3. Fuzzy Logic System and Observer Design

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 [34, 35]). If fx 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.
In order to estimate the unmeasured states, we design a linear state observer for thekth subsystem as follows [23]:(3)x^˙k,1=x^k,2−ℓk,1x^k,1−yk,x^˙k,2=x^k,3−ℓk,2x^k,1−yk,⋮x^˙k,nk−1=x^k,nk−ℓk,nk−1x^k,1−yk,x^˙k,nk=uk−ℓk,nkx^k,1−yk,where x^k,ll=1,2,…,nk is the estimation of xk,l. ℓk,1,…,ℓk,nk are the observer design parameters. Define observer error vector as x˜¯k=x¯k−x^¯k, where x¯k=xk,1,xk,2,…,xk,nkT and x^¯k=x^k,1,x^k,2,…,x^k,nkT.
Then, from (1) and (3), we can have(4)x˜˙¯k=Akx˜¯k+Fk+Hk,where Ak=−ℓk,1⋮Ink−1×nk−1−ℓk,nk00, Fk=fk,1x¯k,…,fk,nkx¯kT, and Hk=hk,1y¯,…,hk,nky¯T. We can choose appropriate parameters ℓk,1,…,ℓk,nk 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(5)AkTPk+PkAk=−Qk.
Choose the Lyapunov function candidate as [23](6)V0=∑k=1Mx˜¯kTPkx˜¯k.
Then, we can obtain the time derivative ofV0:(7)V˙0=−∑k=1Mx˜¯kTQkx˜¯k+∑k=1M2x˜¯kTPkFk+Hk.
According to Yang’s inequalities, we have(8)2x˜¯kTPkFk≤x˜¯k2+Pk2∑j=1nkfk,jx¯k2.
Based on Assumption2, fk,jx¯k satisfies(9)fk,jx¯k=ykf¯k,jx¯k,where f¯k,jx¯k is an unknown nonlinear function which can be approximated by a fuzzy logic system.
From (8) and (9), we have(10)Pk2∑j=1nkfk,jx¯k2=Pk2∑j=1nkyk2f¯k,jx¯k2=ykPk2∑j=1nkykf¯k,jx¯k2.
Based on Assumption1, we obtain(11)∑k=1M2x˜¯kTPkHk≤∑k=1M1λx˜¯k2+∑k=1MλPk2Hk2≤1λ∑k=1Mx˜¯k2+λ¯∑k=1M∑l=1nk∑i=1Myiℏk,l,iyi2≤1λ∑k=1Mx˜¯k2+λ¯∑i=1M∑l=1nk∑k=1Myiℏk,l,iyi2≤1λ∑k=1Mx˜¯k2+λ¯∑k=1M∑i=1M∑l=1nkyk2yiℏk,l,iyi2≤1λ∑k=1Mx˜¯k2+∑k=1Myk2∑i=1M∑l=1nkλ¯ℏk,l,iyi2,where λ>0 is the design parameter and λ¯=λmax1≤k≤MPk2.
Substituting equations (8)–(11) into (7) results in(12)V˙0≤∑k=1M−x˜¯kTQkx˜¯k+1+1λx˜¯k2+yk2∑i=1M∑l=1nkλ¯ℏi,l,kyk2+ykPk2∑j=1nkykf¯k,jx¯k2≤∑k=1M−x˜¯kTQkx˜¯k+1+1λx˜¯k2+ykyk∑i=1M∑l=1nkλ¯ℏi,l,ksyk2+Pk2∑j=1nkykf¯k,jx¯k2+∑i=1Mykℏi,1,kyk2−∑i=1Myk2ℏi,1,kyk2.
LetΦkx¯k=yk∑i=1M∑l=1nkλ¯ℏi,l,kyk2+Pk2∑j=1nkykf¯k,jx¯k2+∑i=1Mykℏi,1,kyk2. By using fuzzy system f^kx¯kθk=θkTξkx¯k to approximate Φkx¯k, there exists optimal parameter vector(13)θk∗=arg minθk∈Ωksupx¯k∈Ukf^kx¯kθk−Φkx¯k,where Ωk and Uk are the compact set of θk and x¯k, respectively. Then, the minimal error can be written as(14)εkx¯k=Φkx¯k−f^kx¯kθk∗,where εkx¯k≤εk∗ and εk∗ is an unknown positive constant.
From (13) and (14), we can obtain(15)V˙0≤∑k=1M−λminQk−1−1λx˜¯k2+ykθk∗Tξkx¯k+εkx¯k−∑i=1Myk2ℏi,1,kyk2,where λminQk is the minimal eigenvalue of matrix Qk.


## 4. Adaptive Control Law Design

Define the tracking errorzk,1, virtual error zk,l, virtual control law αk,l−1, and first-order filters as(16)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,…,nk. 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 (16), we have(17)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:(18)z˙k,1=xk,2+fk,1+hk,1−ω˙k,0=zk,2+ek,1+αk,1+x˜k,2+fk,1x¯k+hk,1y¯−y˙k,d.
By using fuzzy systemF^kx¯kφk=φkTζkx¯k to approximate fk,1x¯k and defining φk∗ as the optimal parameter vector, we have(19)φk∗=argminφk∈Ω¯ksupx¯k∈UkF^kx¯kφk−fk,1x¯k,where Ω¯k is the compact set of φk. Then, the minimal approximation error is as follows:(20)δkx¯k=fk,1x¯k−F^kx¯kφk∗,where δkx¯k≤δk∗, and δk∗ is an unknown positive constant.
Define(21)kk,at=yk,dt−k¯k,c1t,kk,bt=k¯k,c1t−yk,dt,qkzk,1=1,zk,1>0,0,zk,1≤0.
Choose the Lyapunov function candidate as(22)V1=V0+∑k=1Mqkzk,12logkk,b2tkk,b2t−zk,12+1−qkzk,12logkk,a2tkk,a2t−zk,12+12ek,12+12γk,1ϖ˜k,12+12γk,2ϖ˜k,22,where γk,1>0 and γk,2>0 are design parameters, ϖk,1∗=θk∗Tθk∗, ϖk,2∗=φk∗Tφk∗, and ϖk,1 and ϖk,2 are estimations of ϖk,1∗ and ϖk,2∗, respectively. Define ϖ˜k,1=ϖk,1∗−ϖk,1 and ϖ˜k,2=ϖk,2∗−ϖk,2.
Letςk,a=zk,1t/kk,at, ςk,b=zk,1t/kk,bt, and ςk=qkςk,b+1−qkςk,a; then, we can have(23)V1=V0+∑k=1M12log11−ςk2+12ek,12+12γk,1ϖ˜k,12+12γk,2ϖ˜k,22.
Now, we can infer thatV˙1 satisfies(24)V˙1=V˙0+∑k=1Mςkς˙k1−ςk2+ek,1e˙k,1−1γk,1ϖ˜k,1ϖ˙k,1−1γk,2ϖ˜k,2ϖ˙k,2.
Because we have(25)ςkς˙k1−ςk2=qkςk,b+1−qkςk,a1−ςk2qkς˙k,b+1−qkς˙k,a,ς˙k,b=z˙k,1kk,bt−zk,1k˙k,bttkk,b2t,ς˙k,a=z˙k,1kk.at−zk,1k˙k,atkk,a2t,V˙1=V˙0+∑k=1Mqkςk,b+1−qkςk,a1−ςk2qkς˙k,b+1−qkς˙k,a+ek,1e˙k,1−1γk,1ϖ˜k,1ϖ˙k,1−1γk,2ϖ˜k,2ϖ˙k,2=V˙0+∑k=1Mqkςk,b1−ςk2ς˙k,b+1−qkςk,a1−ςk2ς˙k,a+ek,1e˙k,1−1γk,1ϖ˜k,1ϖ˙k,1−1γk,2ϖ˜k,2ϖ˙k,2=V˙0+∑k=1Mqkςk,bkk,b1−ςk,b2z˙k,1−zk,1k˙k,bkk,b+1−qkςk,akk,a1−ςk,a2z˙k,1−zk,1k˙k,akk,a+ek,1e˙k,1−1γk,1ϖ˜k,1ϖ˙k,1−1γk,2ϖ˜k,2ϖ˙k,2.
Letμk=qk/kk,b2−zk,12+1−qk/kk,a2−zk,12, and we can have(26)V˙1=V˙0+∑k=1Mμkzk,1z˙k,1−qkzk,1k˙k,bkk,b−1−qkzk,1k˙k,akk,a+ek,1e˙k,1−1γk,1ϖ˜k,1ϖ˙k,1−1γk,2ϖ˜k,2ϖ˙k,2=V˙0+∑k=1Mμkzk,1zk,2+ek,1+αk,1+x˜k,2+fk,1x¯k+hk,1y¯−y˙k,d−qkzk,1k˙k,bkk,b−1−qkzk,1k˙k,akk,a+ek,1e˙k,1−1γk,1ϖ˜k,1ϖ˙k,1−1γk,2ϖ˜k,2ϖ˙k,2.
According to Yang’s inequalities and Assumption1, we have(27)∑k=1Mμkzk,1hk,1y¯≤∑k=1Mμk2zk,124+∑k=1Mhk,1y¯2≤∑k=1Mμk2zk,124+∑k=1M∑i=1Myk2ℏk,1,iyk2.
Substituting (15) and (27) into (26), we obtain(28)V˙1≤∑k=1M−λminQk−1−1λx˜¯k2+ykθk∗Tξkx¯k+εkx¯k+μk2zk,124+∑k=1Mμkzk,1zk,2+ek,1+αk,1+x˜k,2+φk∗Tζkx¯k+δkx¯k−y˙k,d−qkzk,1k˙k,bkk,b−1−qkzk,1k˙k,akk,a+ek,1e˙k,1−1γk,1ϖ˜k,1ϖ˙k,1−1γk,2ϖ˜k,2ϖ˙k,2.
Based onξkT⋅ξk⋅≤1 and Yang’s inequalities, we can infer inequalities (29)–(31):(29)ykθk∗Tξkx¯k+εkx¯k≤zk,1θk∗Tξkx¯k+εkx¯k+yk,dθk∗Tξkx¯k+εkx¯k≤zk,12ϖk,1∗4τk+zk,124+yk,d22+ϖk,1∗+2εk∗2+τk,(30)μkzk,1φk∗Tζkx¯k+δkx¯k≤μk2zk,12ϖk,2∗4τk+μk2zk,124+δk∗2+τk,(31)μkzk,1x˜k,2≤1λkx˜¯k2+λkμk2zk,124,where τk>0 and λk>0 are design parameters.
Substituting (29)–(31) into (28) results in(32)V˙1≤∑k=1M−λminQk−1−1λ−1λkx˜¯k2+∑k=1Mμkzk,1zk,2+ek,1+αk,1+zk,1ϖk,1∗4τkμk+μkzk,1ϖk,2∗4τk+1+2μk+μkλkzk,14−y˙k,d−qkzk,1k˙k,bkk,b−1−qkzk,1k˙k,akk,a+ek,1e˙k,1−1γk,1ϖ˜k,1ϖ˙k,1−1γk,2ϖ˜k,2ϖ˙k,2+y¯k,d22+ϖk,1∗+2εk∗2+δk∗2+2τk,where y¯k,d=maxt∈0,+∞yk,dt is a constant. Then, (32) can be rearranged as(33)V˙1≤∑k=1M−πk,1x˜¯k2+∑k=1Mμkzk,1zk,2+ek,1+αk,1+zk,1ϖk,14τkμk+μkzk,1ϖk,24τk+1+2μk+μkλkzk,14−y˙k,d−qkzk,1k˙k,bkk,b−1−qkzk,1k˙k,akk,a+1γk,1ϖ˜k,1γk,1zk,124τk−ϖ˙k,1+1γk,2ϖ˜k,2γk,2μk2zk,124τk−ϖ˙k,2+ek,1e˙k,1+dk,1,where πk,1=λminQk−1−1/λ−1/λk and dk,1=y¯k,d2/2+ϖk,1∗+2εk∗2+δk∗2+2τk.
Now, we can chooseαk,1, ϖk,1, and ϖk,2 as follows:(34)αk,1=−ck,1zk,1+c⌢k,1zk,1−zk,1ϖk,14τkμk−μkzk,1ϖk,24τk−1+2μk+μkλkzk,14+y˙k,d,(35)ϖ˙k,1=γk,1zk,124τk−2σk,1ϖk,1,(36)ϖ˙k,2=γk,2μk2zk,124τk−2σk,2ϖk,2,where ck,1>0, c⌢k,1=k˙k,b/kk,b2+k˙k,a/kk,a2+βk, and σk,1>0 and σk,2>0 are design parameters. βk is a positive design constant.
Substituting (34)–(36) into (33) results in(37)V˙1≤∑k=1M−πk,1x˜¯k2+∑k=1M−ck,1μkzk,12+μkzk,1zk,2+μkzk,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−πk,1x˜¯k2+∑k=1M−ck,1μkzk,12+μkzk,1zk,2+μkzk,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 Yang’s inequalities, there exist (38)–(40):(38)2σk,1γk,1ϖ˜k,1ϖk,1≤−σk,1γk,1ϖk,12+σk,1γk,1ϖk,1∗2,(39)2σk,2γk,2ϖ˜k,2ϖk,2≤−σk,2γk,2ϖk,22+σk,2γk,2ϖk,2∗2,(40)μkzk,1ek,1≤μkzk,12+μk4ek,12.
Substituting (38)–(40) into (37) results in(41)V˙1≤∑k=1M−πk,1x˜¯k2+∑k=1M−ck,1−1μkzk,12+μkzk,1zk,2+1+μk4−1υk,1ek,12−∑i=12σk,iγk,iϖk,i2+∑i=12σk,iγk,iϖk,i∗2+14ψk,12+dk,1.
Substituting the following inequality(42)−σk,i2γk,iϖ˜k,i2≥−σk,iγk,iϖk,i2−σk,iγk,iϖk,i∗2,into equation (41) results in(43)V˙1≤∑k=1M−πk,1x˜¯k2+∑k=1M−ck,1−1μkzk,12+μkzk,1zk,2+1+μk4−1υk,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 (16), we have(44)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:(45)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 the Lyapunov function candidate as(46)V2=V1+∑k=1M12zk,22+12ek,22.
The time derivative ofV2 is as follows:(47)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 Yang’s inequalities, we have(48)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−πk,1x˜¯k2+∑k=1M−ck,1−1μkzk,12+μkzk,1zk,2+1+μk4−1υk,1ek,12−∑i=12σk,i2γk,iϖ˜k,i2+∑i=122σk,iγk,iϖk,i∗2+14ψk,12+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:(49)αk,2=−ck,2zk,2−μkzk,1+ℓk,2x^k,1−yk−ωk,1−αk,1υk,1,where ck,2>0 is the design parameter.
Substituting (49) into (48) leads to(50)V˙2≤∑k=1M−πk,1x˜¯k2+∑k=1M−ck,1−1μkzk,12+μkzk,1zk,2+1+μk4−1υk,1ek,12−∑i=12σk,i2γk,iϖ˜k,i2+∑i=122σk,iγk,iϖk,i∗2+14ψk,12+dk,1+∑k=1Mzk,2zk,3−μkzk,2zk,1−ck,2−1zk,22+54−1υk,2ek,22+14ψk,22≤∑k=1M−πk,1x˜¯k2+∑k=1M−ck,1−1μkzk,12−ck,2−1zk,22+zk,2zk,3+1+μk4−1υk,1ek,12+54−1υk,2ek,22−∑i=12σk,i2γk,iϖ˜k,i2+∑i=122σk,iγk,iϖk,i∗2+∑i=1214ψk,i2+dk,1.
Steplth l=3,…,nk−1: define zk,l=x^k,l−ωk,l−1 and zk,l+1=x^k,l+1−ωk,l. We have(51)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:(52)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 the Lyapunov function candidate as(53)Vl=Vl−1+∑k=1M12zk,l2+12ek,l2.
The time derivative ofVl is as follows:(54)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:(55)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(56)V˙l≤∑k=1M−πk,1x˜¯k2+∑k=1M−ck,1−1μkzk,12−∑i=2lck,i−1zk,i2+zk,lzk,l+1+1+μk4−1υk,1ek,12+∑i=2l54−1υk,iek,i2−∑i=12σk,i2γk,iϖ˜k,i2+∑i=122σk,iγk,iϖk,i∗2+∑i=1214ψk,i2+dk,1.
Stepnkth: define zk,nk=x^k,nk−ωk,nk−1, and we can get the time derivative of zk,nk as follows:(57)z˙k,nk=uk−ℓk,nkx^k,1−yk−ω˙k,nk−1.
Choose the Lyapunov function candidate as(58)Vnk=Vnk−1+∑k=1M12zk,nk2.
Now, we can have the time derivative ofVnk as follows:(59)V˙nk=V˙nk−1+∑k=1Mzk,nkz˙k,nk=V˙nk−1+∑k=1Mzk,nkuk−ℓk,nkx^k,1−yk−ω˙k,nk−1.
Choose the final control law as(60)uk=−ck,nkzk,nk−zk,nk−1+ℓk,nkx^k,1−yk−ωk,nk−1−αk,nk−1υk,nk−1,where ck,nk>0 is the design parameter.
Substituting (60) into (59) results in(61)V˙nk≤∑k=1M−πk,1x˜¯k2+∑k=1M−ck,1−1μkzk,12−∑i=2nkck,i−1zk,i2+1+μk4−1υk,1ek,12+∑i=2nk54−1υk,iek,i2−∑i=12σk,i2γk,iϖ˜k,i2+∑i=122σk,iγk,iϖk,i∗2+∑i=1nk−114ψk,i2+dk,1.
Define(62)H=∑i=12σk,iγk,iϖk,i∗2+∑i=1nk−114ψk,i2+dk,1.
Substituting (62) into (61) leads to(63)V˙nk≤∑k=1M−πk,1x˜¯k2+∑k=1M−ck,1−1μkzk,12−∑i=2nkck,i−1zk,i2+1+μk4−1υk,1ek,12+∑i=2nk54−1υk,iek,i2−∑i=12σk,i2γk,iϖ˜k,i2+H.


## 5. Stability Analysis

Define Lyapunov function of the closed-loop system asV⌢=Vnk, and we can have V⌢˙=V˙nk. Let the design parameters satisfy(64)ck,i−1>0,i=1,2,…,nk,1+μk4−1υk,1<0,54−1υk,i<0,i=2,…,nk−1.Define(65)Ck=min2πk,1λmaxPk,2ck,i−1,2σk,1,2σk,2,21υk,1−1−μk4,21υk,i−54,where i=2,…,nk.DefineC=mink=1,…,MCk, and (75) can be rearranged as(66)V⌢≤−CV⌢+H.Then, we have(67)eCtV⌢˙≤−CV⌢+HeCt,(68)ddtV⌢eCt≤HeCt,(69)V⌢eCt−V⌢0≤HCeCt−1,(70)0≤V⌢t≤V⌢0e−Ct+HC1−e−Ct≤V⌢0+HC.From (70), 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.


## 6. Comparisons with Some Previous Results

Comparisons with previous results will be given in this section.(1)
The control methods in [11–20, 34] can deal with only nonlinear system (71) without interconnected subsystems:(71)x˙i=xi+1+fix,x˙n=ut+fnx,y=x1,wherex=x1,x2,…,xnT and i=1,2,…,n−1. y is the only output of the system and is only affected by x, fix, and ut. Therefore, there is no coupling effect hk,ly¯ between subsystems.The control methods in [22, 25, 26] deal with large-scale systems in strict feedback form while not in nonstrict feedback structure (72):(72)x˙k,1=xk,2+fk,1xk,1+hk,1y¯,x˙k,2=xk,3+fk,2xk,1,xk,2+hk,2y¯,⋮x˙k,nk−1=xk,nk+fk,nk−1xk,1,…,xk,nk−1+hk,nk−1y¯,x˙k,nk=uk+fk,nkxk,1,…,xk,nk+hk,nky¯,yk=xk,1,wherefk,ixk,1,…,xk,ii=1,2,…,n are the functions of partial state variables, and large-scale system (72) has a strict lower triangle structure.However, the control method in this paper is designed for nonstrict feedback large-scale systems (73) that are more complex compared with (71) and (72):(73)x˙k,1=xk,2+fk,1x¯k+hfk,1y¯,x˙k,2=xk,3+fk,2x¯k+hfk,2y¯,⋮x˙k,nk−1=xk,nk+fk,nk−1x¯k+hk,nk−1y¯,x˙k,nk=uk+fk,nkx¯k+hk,nky¯,yk=xk,1,wherex¯k=xk,1,xk,2,…,xk,nkT∈Rnk. Then, fk,ixk,1,…,xk,i is a function of full-state variables.It is well known that an interconnected large-scale system comprises some subsystems with obvious interconnections, which lead to the increasing difficulty of controller design and stability proof for the large-scale system. We unable to use the control methods in [11–20, 34] to control the large-scale system due to the coupling effect between subsystems. Moreover, when the controller of the nonstrict feedback large-scale system is designed by the control methods in [22, 25, 26], the virtual control signal and adaptive law of each subsystem are the functions of full-state variables. Consequently, the algebraic loop problem arises, which makes the controller design of a nonstrict feedback large-scale system very difficult. Therefore, the controller design method of nonstrict feedback large-scale system (73) considered in this paper is quite different from that of the controller design methods in [11–20, 22, 25, 26, 34].(2)
[21–25, 29–31] proposed adaptive control methods for the large-scale system, but output constraints were not considered. Though [32, 33] presented the control schemes for constrained systems, the system considered in [32, 33] is not a large-scale system, and all states should be measurable. The strict limitation makes these control methods difficult to realize in practical applications. Therefore, control methods in [21–25, 32, 33] cannot be used to control a large-scale system with unmeasurable state and output constraints that is discussed in this paper. To the best of our knowledge, by far, no results have been reported on the adaptive control for the nonstrict feedback large-scale nonlinear system with output constraints and unmeasured states.(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 [27, 28]. Moreover, this control scheme has only 2M adaptive parameters, and the number of parameters does not increase with the increase of system’s order nk. Therefore, this control scheme not only conforms to engineering practice but also has a simple algorithm and requires a small number of calculations.


## 7. Simulations

Consider the following nonstrict feedback large-scale system [23]:(74)x˙1,1=x1,2+x1,121−cosx1,1x1,2+x1,2−0.2x1,1sinx1,1x1,2+y1y2,x˙1,2=u1+x1,1x1,2+x1,1+y12+y22,y1=x1,1,x˙2,1=x2,2+x2,1sinx2,1x2,2−0.5x2,1sinx2,1x2,2+y13y23,x˙2,2=u2+x2,1x2,2+y1−2y2,y2=x2,1.The given tracking signals arey1,d=sin0.5t and y2,d=0.5sint. The constraints are given as k¯1,c1=−0.2×2−t−0.05+y1,d, k¯1,c1=0.2×2−t+0.05+y1,d, k¯2,c1=−0.2×2−t−0.05+y2,d, and k¯2,c1=0.2×2−t+0.05+y2,d.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, and υ1,1=υ1,2=υ2,1=υ2,2=0.5.To show the superiority and validity of the method proposed in this paper, we compare it with a scheme proposed for large-scale systems without considering the output constraints [23]. The design parameters and initial conditions of the two methods are the same. In the simulation, system 1 is controlled by the method in this paper, whereas system 2 is controlled by the method without output constraints [23].Figures1–5 show the simulation results. From Figures 1–4, we can see that the output y1,d,y2,d and the tracking error z1,1,z2,1 of system 1 can both be kept within the constraints, whereas the output and the tracking error of system 2 violate the constraints. Figure 5 shows control input signals uk of the two systems, respectively. From the simulation results, it can be seen that the proposed adaptive control approach in this paper not only guarantees the boundedness of all the signals and not violating of output constraints but also achieves better control performance than that of the control method without considering output constraints [23].Figure 1
Trajectories ofy1,d, y1, and the output constraints.Figure 2
The tracking errorz1,1 and the error constraints.Figure 3
Trajectories ofy2,d, y2, and the output constraints.Figure 4
The tracking errorz2,1 and the error constraints.Figure 5
Control input signalsuk.


## 8. Conclusion

This paper proposes an adaptive fuzzy dynamic surface decentralized output feedback control scheme for a class of large-scale interconnected uncertain nonstrict feedback systems with time-varying output constraints. By using a decentralized linear state observer, the unmeasurable states can be estimated. Based on fuzzy logic systems, the uncertain nonlinear functions and interconnected influence between the subsystems can be compensated. The problems of nonstrict feedback and time-varying constraints are solved by variable separation technology and time-varying barrier Lyapunov function, respectively. The proposed scheme can not only achieve good tracking performance but also keep the output trajectories within the given ranges. Finally, the stability of the large-scale interconnected system is proved by using the Lyapunov direct method.



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*Source: 6760521-2020-04-21.xml*