# Adaptive Neural Network Control Scheme of Switched Systems with Input Saturation

Authors: Xiaoli Jiang; Mingyue Liu; Siqi Liu; Jing Xu; Lina Liu
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

This paper investigates a scheme of adaptive neural network control for a stochastic switched system with input saturation. The unknown smooth nonlinear functions are approximated directly by neural networks. A modified approach is proposed to deal with unknown functions with nonstrict feedback form in the design process. Furthermore, by combining the auxiliary design signal and the adaptive backstepping design, a valid adaptive neural tracking controller design algorithm is presented such that all the signals of the switched closed-loop system are in probability semiglobally, uniformly, and ultimately bounded, and the tracking error eventually converges to a small neighborhood of the origin in probability. In the end, the effectiveness of the proposed method is verified by a simulation example.


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## Body

## 1. Introduction

Switched systems as a class of hybrid systems is made up of a race of continuous or discrete subsystems and switching rules [1–4]. In the last decades, switched systems have attracted more and more attention due to their significance in the modeling of many engineering applications, such as chemical processes, robot manipulators, and power systems. So far, many remarkable achievements about the stability analysis and synthesis have been made in the field of switched systems. In general, there are two ways of dealing with it: one is to find a common Lyapunov function to ensure stability of the switched systems under arbitrary switching laws; the other is to use a multiple Lyapunov functional technique to stabilize the switched systems under some designed switching laws, see, e.g., [5–10] and the references therein.Almost all feedback controls are involved in saturation, due to the limitation of maximum and minimum values, and the sensors of actuators are saturated. If the controller is designed without considering input saturation, it can degrade the performance of the control system and even lead to instability. In recent years, the problem of input saturation limitation has attracted wide attention, and many methods for designing controllers have been proposed. Among them, PID controller was an earlier antisaturation method [11]. Workman designed a time optimal controller to reduce input saturation. Later, scholars proposed a saturation compensator, for example, the CNF controller was designed by Hu et al. [12], Grimm et al. [13], and Mulder et al. Mulder et al. [14] gave antisaturation compensation controllers by using linear matrix inequalities. However, there are still many control questions about switched systems with input saturation to be studied in depth.According to recent research, the switching strategy plays an important role in the switching process. Due to the switching effect, the originally linear characteristics will become nonlinear. In addition, the features of multiple subsystems vary greatly and interact with each other and further increase the complexity and make the whole system show complex dynamic behavior. Therefore, how to choose the switching strategy and ensure the stability of the whole system is a hot topic at present. The adaptive backstepping technology can effectively solve the stabilization and tracking control [15–33]. Such as in [26], an output feedback scheme based on discrete adaptive approximation was proposed for uncertain switched stochastic interconnected systems, and the unknown nonlinear function was modeled by combining a fuzzy logic system. For stochastic systems with output constraints, an adaptive neural controller based on approximation was constructed [27]. Niu et al. [28] investigate the global adaptive control problem for a class of switched uncertain nonlinear systems. By improving mode-dependent average dwell time (MDADT) method, a new adaptive control scheme is established which ensures the global bounded. It should be noted that an adaptive tracking controller based on neural networks was considered for the single input switching system of lower-triangular structure in [29, 34]. As for the switching systems, because it contains multiple subsystems, its motion pattern is richer and more complex than the previous single model. In all dynamic behaviors, there may be good performance, bad performance, or even unstable dynamics. Therefore, on the premise of ensuring the stability of the system, this work attempts to design an appropriate control strategy to make the system achieve better switching effect and optimized performance index.Moreover, as it is known to all, neural networks (NNs) have strong approximation ability, which can be used to deal with the unknown nonlinear functions, and it can be discovered in many literatures [35–42]. Recently, the authors in [36] gave an adaptive scheme using multilayer neural networks. State feedback strategy was firstly investigated for nonaffine systems through neural network approximation [39]. Neural network output feedback controller has been designed in [42] for a large-scale stochastic systems. For nonstrict feedback systems, a dynamic surface method based on observer was established in [43]. However, the above control results were obtained under a conservative condition, which the considered system structures are always in strict feedback form. The drift functions may be a mixture of all state variables, which makes the systems be in the nonstrict feedback. Therefore, we aim to propose an adaptive neural networks scheme for a switched system with nonstrict feedback form. Different from the relevant literatures, we construct a new scheme of stochastic systems with input saturation and design switching strategies to resist the saturation.Based on the above research progress, this work tries to establish a neural network framework for a stochastic switched system with input saturation. Compared with the existing literatures, the innovation points of this paper are stated as follows:(i)
Applying neural networks technique, a modified backstepping algorithm for a stochastic switched system in lower-triangular is established. A novel controller is constructed to track the unknown states.(ii)
It needs to adopt the Ito^ formula for stochastic system, which makes it difficult to structure Lyapunov function corresponding to gradient term and high-order Hessen term. To solve this problem, we successfully developed neural networks to stochastic switching situation, which reduces the conservativeness caused by feedback form when the input saturation happens.The other parts of this article are organized as follows. In Section2, we introduce some assumptions and preliminaries and describe problem statements. Furthermore, we give the neural control process and stability analysis in this section. The simulation results are shown in Section 3, and the conclusions are analyzed in Section 4.


## 2. Preliminaries and Controller Design

In this section, the switched system and its descriptions are formulated in Section2.1, and some primary lemmas and definitions including stability theorem and Young’s inequality are also described. In Section 2.2, we construct the adaptive neural controller and state the stability analysis.


### 2.1. Problem Descriptions

Now, let us first present some lemmas, definitions, and assumptions, and they will be used in the subsequent developments. Consider a stochastic switched system with nonstrict feedback form:(1)dξi=Fσt,iξ¯i,ξi+1,ξdt+ψσt,iTξdw,1≤i≤n−1,dξn=Fσt,nξ¯n,u,ξdt+ψσt,nTξdw,η=ξ1,where ξ¯i=ξ1,ξ2,…,ξiT∈R,i=1,2,…,N, and ξ=ξ1,ξ2,…,ξnT∈Rn and η are the state and the output of the system, respectively. Let random function ψσt,iξ be smooth, and the switching signal is denoted by σt:0,+∞⟶Iq=1,2,…,q. For ∀i=1,2,…,n,p∈Iq, assume that fp,i⋅ and hp,i are assumed to be unknown and locally Lipschitz functions. The unknown system functions gσt,iξ¯i,fσt,iξ, and hσt,iξ are contained in the function Fσt,i and satisfy gσt,iξ¯iξi+1+fσt,iξ+hσt,iξ=Fσt,iξ¯i,ξi+1,ξ and uv+fσt,nξ+hσt,nξ=Fσt,nξ¯i,u,ξ, which are smooth and fp,i0=0. The variate ω is an independent r-dimensional standard Wiener process. It is important to note that there is a regular convention for switching systems, solutions are continuous everywhere, and the states do not jump at the switching moment. Let uv be the controller with input saturation, which can be written as(2)uvt=satvt=signvtε0,vt≥ε0,vt,vt<ε0.Lemma 1 (see [43]).
For each pairx,y∈R2, Young’s inequality holds(3)xy≤εaaxa+1bεbyb,where ε>0,a>1,b>1,a−1b−1=1.Assumption 1.
The constantshi0 satisfy hiξ≤hi0, i=1,2,…,n. Consider the following stochastic system:(4)dξt=fξtdt+gξtdw,where ξ∈Rn is the system state, ω is a standard Wiener process, and f:Rn⟶Rn and g:Rn⟶Rn are locally Lipschitz functions and satisfy f0=g0=0.Definition 1 (see [44]).
For any givenVξ∈C2, which is associated with the stochastic system (4), the infinitesimal generator ℒ is defined as follows:(5)ℒVξ=∂V∂xfξ+12TrgξT∂2V∂x2gξ,where TrA is the trace of a matrix A.Definition 2.
The trajectoryξt of system (4) is said to be semiglobally uniformly ultimately bounded in pth moment if, for some compact set Ω∈Rn and any initial state ξ0=ξt0, there exist constant ε>0 and time constant T=Tε,ξ0, such that Eξtp<ε for all t>t0+T, p∈Iq=1,2,…,q. In particular, when p=2, it is usually called semiglobally, uniformly, and ultimately bounded in mean square.Lemma 2 (see [45]).
Consider the stochastic system (4). If there exist functions Vξ∈C2,σ¯1, and σ¯2∈K∞ and constants m0>0 and n0>0, such that(6)σ¯1ξ≤Vξ≤σ¯2ξ,LVξ≤−m0Vξ+n0,then, there is a unique solution of system (4) for each ξ0∈Rn, and it holds(7)EVξ≤Vξ0e−m0t+m0n0,∀t>t0.In this paper, approximation-based NNs will be used to approximate the unknown nonlinear functions. For any continuous unknown smooth nonlinear functionfZ over a compact set ΩZ⊂Rq, there exists NNs W∗TSZ, such that, for a desired level of accuracy ε,(8)fZ=W∗TSZ+δZ,δZ≤ε,where W∗ is the ideal constant weight vector and defined by(9)W∗=argminW∈RNsupZ∈ΩZfZ−WTSZ,δZ is the approximation error, W=w1,…,wNT is the weight vector, and SZ=s1Z,…,sNZT is the radial basis function vector with N being the number of the NNs nodes and N>1:(10)siZ=exp−Z−uiTZ−uiηi2,i=1,2,…,N,where ui=ui1,ui2,…,uinT is the center of the receptive field and ηi∈R is the width of the Gaussian function.


### 2.2. Adaptive Controller Design and Stability Analysis

A neural controller will be constructed for the stochastic switched system (1). At the same time, the adaptive laws will be given in this section.At first, let us define some essential functions:(11)α1=−k1z1−94z1−z13Θ14τψ1Tξ1,η˙dψ1ξ1,η˙d,(12)Θ˙1=z16γ14τψ1Tξ1,η˙dψ1ξ1,η˙d−σ1Θ1,where α1 is the first virtual control signal and Θ˙1 is an adaptive law with k1,τ,γ1, and σ1 being positive design parameters.Similarly, fori=2,…,n−1, we design the virtual signals and adaptive laws as follows:(13)αi=−kizi−134zi−zi3Θi4τψiTξ¯i,miψiξ¯i,mi,(14)Θ˙i=zi6γi4τψiTξ¯i,miψiξ¯i,mi−σiΘi,where mi=ηd,η˙d,…,ηdi−1,Θ^1,…,Θ^i−1,αi is the ith virtual control signal and Θ˙i is the adaptive law with ki,τ,γi, and σn being positive design parameters. The actual control input v and the adaptive law Θ˙n are defined as follows:(15)v=−knzn−52zn−zn3Θn4τψnTξ,mnψnξ,mn−h˜,(16)Θ˙n=zn6γn4τψnTξ,mnψnξ,mn−σnΘn,where mn=ηd,η˙d,…,ηdi−1,Θ^1,…,Θ^i−1, kn,τ,γn, and σn being positive design parameters. h˜ is an auxiliary design signal, which will be defined later. The coordinate transformations are as follows:(17)zi=ξi−αi−1,i=1,2,…,n−1,(18)zn=ξn−αn−1−h˜,where α0=ηdt. Combining (1) with coordinate transformations (17) and (18), the following system can be obtained:(19)dzi=gi,pξ¯iξi+1+fi,pξ+hi,pξ−lαi−1dt+ψi,pξ−∑j=1i−1∂αi−1∂xjψj,pTξTdw,dzn=uv+fn,pξ+hn,pξ−lαn−1−h˜˙dt+ψn,pξ−∑j=1n−1∂αn−1∂xjψj,pTξTdw,where(20)lαi−1=∑j=1i−1∂αi−1∂Θ^jΘ^˙j+∑j=1i−1∂αi−1∂ξjgj,pξj¯ξj+1+fj,pξ+hj,pξ+∑j=1i−1∂αi−1∂ηdjηdj+1+∑m,n=1,1i−112∂2αi−1∂ξm∂ξnψm,pTξψn,pξ,with lα0=η˙d for i=2,…,n.Theorem 1.
Let Assumption1 hold, the closed-loop structure be consisted of the virtual control signals in (11) and (13), the adaptive laws (12) and (14) together with (16) and the actual control signal (15), which are designed from the stochastic switched system (1). All signals are bounded with some suitable parameters and the tracking error enters inside the area Ω1 for all t>T1:(21)Ω1=ηt∈REη−ηd4≤8Dnd0,∀t>T1,where d0=min4k1g¯1,…,4kn−1g¯n−1,4kn,σi,1≤i≤n,(22)Dn=∑i=1nai22+εi∗44+σiΘi∗22γi+nτ+∑i=1nhi044g¯i4+hn044,and T1=max0,1/d0lnd0v0/Dn, gi¯=infgσt,iξ¯i, and gi¯>0.Proof.
Step 1: choose the following Lyapunov function:(23)V1=14g¯1z14+12γ12Θ˜12,where g¯1>0andγ1>0 are design parameters, and Θ˜1=Θ1∗−Θ1 with Θ1 being the estimation of Θ1∗. In the light of the above equations in (4), (5), and (19), it yields(24)LV1=1g¯1z13g1,pξ1ξ2+f1,pξ+h1,pξ−η˙d+32g¯1z12ψ1,pTξ−1γΘ˜1Θ˙1.
Employing Young’s inequality (3), we can obtain(25)32g¯1z12ψ1,pTξψ1,pξ≤32g¯1z12ψ1,pξ2≤1a1g¯13/2z13ψ1,pξ3+a122,(26)1g¯1z13h1,pξ≤34z14+14g¯14h104,where a1 is a positive constant. The NNs ω1∗,ϕpT1ξ,η˙d can estimate the following unknown function:(27)1g¯1f1ξ−1g¯1η˙d+1a1g¯13/2ψ1,pξ3=ω1∗,ϕpT1ξ,η˙d+δ1ξ,η˙d.
According to (17), (25), and (27), formula (24) can be rewritten as(28)LV1≤z13z2+34z1+α1+ω1∗,ϕpT1ξ,η˙d+δ1ξ,η˙d+a122+14g¯14h104−1γ1Θ˜1Θ˙1.
Therefore, the following result can be obtained:(29)z13ω1∗,ϕpT1ξ,η˙d≤z16ω1∗,ϕpT1Tξ,η˙dϕ1ξ,η˙d4τ+τ≤z16w1,p∗2ϕ1Tξ,η˙dϕ1ξ,η˙d4τϕ1Tξ,η˙dϕ1ξ,η˙d+τ≤z16Θ1∗4τϕ1Tξ,η˙dϕ1ξ,η˙d+τ,(30)z13δ1ξ,η˙d≤34z14+14δ1∗4,(31)z13z2≤34z14+14z24,where Θ1∗=maxω1,p∗2:p∈Ip, δ1ξ,η˙d≤δ1∗, δ1∗ and τ are positive constants, and(32)0<ϕ1Tξ,η˙dϕ1ξ,η˙d≤1.
Substituting (29)–(31) into (28), we have(33)LV1≤z1394z1+α1+z13Θ14τϕ1Tξ,η˙dϕ1ξ,η˙d+z16Θ˜14τϕ1Tξ,η˙dϕ1ξ,η˙d+a122+14g¯14h104+τ+14z24+14δ1∗4−1γ1Θ˜1Θ˙1.
Substituting the virtual controllerα1 in (11) and the adaptive law in (12) into (33) yields(34)LV1≤−k1z14+σ1γ1Θ˜1Θ˙1+14z24+a122+14g¯14h104+τ+14δ1∗4.
We can get the following result by Young’s inequality:(35)σ1γ1Θ˜1Θ1=σ1γ1Θ˜1Θ1∗−Θ˜1≤−σ12γ1Θ˜12+σ12γ1Θ˜1∗2.
Then, one has(36)LV1≤−k1z14−σ12γ1Θ˜12+14z24+D1,where D1=a12/2+τ+δ1∗4/4+h104/4g¯14+σ1Θ1∗2/2γ1.
Stepii=2,3,…,n−1: similar to Step 1, choose the following Lyapunov function:(37)Vi=Vi−1+14g¯izi4+12γiΘ˜i2,where g¯iandγi are positive constants and Θi is the estimation of Θi∗,Θ˜i=Θi∗−Θi. Then, we can obtain the following formula:(38)LVi=LVi−1+1g¯izi3gi,pξ¯iξi+1+fi,pξ+hi,pξ−∑j=1i−1∂αi−1∂ξjgj,pξ¯jξj+1+fj,pξ+hj,pξ−∑j=0i−1∂αi−1∂ηdjηdj+1−∑j=1i−1∂αi−1∂Θ^jΘ^˙j−1γ1Θ˜iΘ˙i−12∑m,n=1i−1∂2αi−1∂ξp∂ξqψm,pTξψn,pξ+32g¯izi2ψi,pξ−∑j=1i−1∂αi−1∂ξjψj,pξTψi,pξ−∑j=1i−1∂αi−1∂ξjψj,pξ,where(39)32g¯izi2ψi,pξ−∑j=1i−1∂αi−1∂ξjψj,pξTψi,pξ−∑j=1i−1∂αi−1∂ξjψj,pξ≤32g¯iz22ψi,pξ−∑j=1i−1∂αi−1∂ξjψj,pξ2≤zi3aig¯i3/2ψi,pξ−∑j=1i−1∂αi−1∂ξjψj,pξ3+ai22,where ai is a positive constant. By Young’s inequality, one has(40)1g¯izi3hi,pξ≤34zi4+14g¯i4hi04,(41)−1g¯izi3∑j=1i−1∂αi−1∂ξjhj,pξ≤14g¯i4∑j=1i−1∂αi−1∂ξj4hj04+34zi4.
Then, we apply NNsωi∗,ϕpTiξ,mi to replace the unknown function:(42)wi∗,ϕpTiξ,mi+δiξ,mi=1g¯jfi,pξ−∑j=1i−1∂αi−1∂ξjgj,pξ¯jξj+1+fi,pξ−∑j=0i−1∂αi−1∂ηdjηdj+1−∑j=1i−1∂αi−1∂Θ^jΘ^˙j−12∑m,n=1i−1∂2αi−1∂ξm∂ξnψm,pTξψn,pξ+1aig¯i3/2ψi,pξ−∑j=1i−1∂αi−1∂ξjψj,pξ3+1aig¯4∑j=1i−1∂αi−1∂ξj4hj04.
Due to (17) and (39)–(42), we have(43)LVi≤LVi−1+zi3zi+1+32zi+wi∗,ϕpTiξ,mi+δiξ,miαi+ai22+14g¯i4h104−1γiΘ˜iΘ˙i.
In line with Young’s inequality (3), one can obtain(44)zi3wi∗,ϕpTiξ,mi≤zi6wi∗,wpTi,p∗ϕiTξ,miϕiξ,mi4τ+τ≤zi6wi,p∗2ϕiTξ,miϕiξ,mi4τϕiTξ¯i,miϕiξ,mi+τ≤zi6Θi∗4τϕiTξ¯i,miϕiξ¯i,mi+τ,(45)zi3δiξ,mi≤34zi4+14δi∗4,(46)zi3z2≤34zi4+14zi+14,where Θi∗=maxωi,p∗2:p∈Ip,δiξ,mi≤δi∗,(47)0<ϕiTξ,miϕiξ,mi≤1,δi∗ and τ are positive constants. Substituting (44)–(46) into (43), we can obtain(48)LVi≤zi33zi+αi+zi3Θi∗4τϕiTξ¯i,miϕiξ¯i,mi+LVi−1+14zi+14+14δi∗4+ai22+τ−1γiΘ˜iΘ˙i+14g¯i4h104≤∑m=1i−1kmzm4−∑m=1i−1σm2γmΘ˜m2+14zi+14+zi3134zi+αi+zi3Θi4τϕiTξ¯i,miϕiξ¯i,mi+zi6Θ˜i4τϕiTξ¯i,miϕiξ¯i,mi+αi22+τ+14g¯i4h104+14δi∗4−1γiΘ˜iΘ˙i+Di−1,where Di−1=∑j=1i−1aj2/2+1/4g¯j4hj04+1/4δj∗4+σjΘj∗2/2γj+i−1τ. Based on the designed virtual control signal αi in (13) and the adaptive law Θ˙i in (14), one can obtain(49)LVi≤−∑m=1ikmzm4−∑m=1i−1σm2γmΘ˜m2+σiγiΘ˜iΘi+ai22+14g¯i4hi04+τ+14zi+14+14δi∗4+Di−1.
Similar to (23), we can derive the following conclusion:(50)σiγiΘ˜iΘi=σiγiΘ˜iΘi∗−Θ˜i≤−σi2γiΘ˜i2+σi2γiΘ˜i∗2.
Moreover, substituting (50) into (49) implies(51)LVi≤−∑m=1ikmzm4−∑m=1iσm2γmΘ˜m2+14zi+14+Di,where Di=∑j=1iaj2/2+1/4g¯j4hj04+1/4δj∗4+σjΘj∗2/2γj+iτ.
Stepn: the Lyapunov function is chosen as follows:(52)Vn=Vn−1+14g¯nzn4+12γnΘ˜n2,where g¯nandγn are positive constants and Θn is the estimation of Θn∗,Θ˜n=Θn∗−Θn. From Ito^ differential formula, we can obtain the following result:(53)LVn=LVn−1+zn3uv+fn,pξ+hn,pξ−∑j=1n−1∂αn−1∂ξjgj,pξ¯jξj+1+fj,pξ+hj,pξ−∑j=0n−1∂αn−1∂θ^jθ^.j−∑j=0n−1∂αn−1∂ηdjηdj+1−h˜˙−12∑m,n=1n−1∂2αn−1∂ξm∂ξnψm,pTξψn,pξ−1γnΘ˜nΘ˙n+3zn22ψn,pξ−∑j=1n−1∂αn−1∂ξjψj,pξTψn,pξ−∑j=1n−1∂αn−1∂ξjψj,pξ.
We introduce the following dynamic system in order to simplify the controller design:(54)h˜˙=−h˜+uv−v.
Based on (5), one has(55)zn3hn,pξ≤34zn4+hn04,(56)−zn3∑j=1n−1∂αn−1∂ξjhj,pξ≤14∑j=1n−1∂αn−1∂ξj4hj04+34zn4.
Similar to (39), the following inequality can be written as(57)32zn2ψn,pξ−∑j=1n−1∂αn−1∂ξjψj,pξTψn,pξ−∑j=1n−1∂αn−1∂ξjψj,pξ≤32zn2ψn,pξ−∑j=1n−1∂αn−1∂ξjψj,pξ2≤zn3anψn,pξ−∑j=1n−1∂αn−1∂ξjψj,pξ3+an22,where an>0. Substituting (54)–(57) into (53), we obtain(58)LVn=LVn−1+zn332zn+fn,pξ−∑j=1n−1∂αn−1∂ξjgj,pξ¯jξj+1+fj,pξ−∑j=0n−1∂αn−1∂ηdjηdj+1−∑j=1n−1∂αn−1∂Θ^jΘ^˙j−12∑m,n=1n−1∂2αn−1∂ξp∂ξqψm,pTξψn,pξ+h˜+v+1anψn,pξ−∑j=1n−1∂αn−1∂ξjψj,pξ3+14∑j=1n−1∂αn−1∂ξj4hj04+14hn04−1γnΘ˜nΘ˙n+an22.
The neural networks can estimate the unknown nonlinear function:(59)wn∗,ϕpTnξ,mn+δnξ,mn=fi,pξ−∑j=1n−1∂αi−1∂ξjgj,pξ¯jξj+1+fi,pξ−∑j=0n−1∂αn−1∂ηdjηdj+1−∑j=1n−1∂αn−1∂Θ^jΘ^˙j−∑j=0n−1∂αn−1∂ηdjηdj+1−∑j=1n−1∂αn−1∂Θ^jΘ^˙j−12∑m,n=1n−1∂2αn−1∂ξp∂ξqψm,pTξψn,pξ+14∑j=1n−1∂αn−1∂ξj4hj04+1anψn,pξ−∑j=1n−1∂αn−1∂ξjψj,pξ3.
Similarly, it is noted that the following formula holds:(60)zn3wn∗,ϕpTnξ,mn≤zn6wn∗,ϕpTnTξ,mnϕnξ,mn4τ+τ≤zn6wn,p∗2ϕnTξ,mnϕnξ,mn4τϕnTξ,mnϕnξ,mn+τ≤z16Θ1∗4τϕnTξ,mnϕnξ,mn+τ,(61)zn3δnξ,mn≤34zn4+14δn∗4,where δn∗ and τ are positive constants, δnξ,mn≤δn∗,(62)0<ϕnTξ,mnϕnξ,mn≤1,Θn∗=maxωn,p∗2:p∈Ip.
Based on (59)–(61), the equality of (58) can be rewritten as(63)LVn≤−∑i=1n−1kizi4−∑i=1n−1σi2γiΘ˜i2+zn3h˜+v+52zn+zn3Θn4τϕnTξ,mnϕnξ,mn+an22+τ+14δn∗4+Dn−1+14hn04+Θ˜nγnzn6γn4τϕnTξ,mnϕnξ,mn−Θ˙n.
Taking formulas (15) and (16) into (63), it yields(64)LVn≤−∑i=1nkizi4−∑i=1n−1σi2γiΘ˜i2+σnγnΘ˜nΘn+an22+τ+14δn∗4+Dn−1+14hn04,where(65)σnγnΘ˜nΘn=σnγnΘ˜nΘn∗−Θ˜n≤−σn2γnΘ˜n2+σn2γnΘn∗2.
In addition, substituting (65) into (64), we have(66)LVn≤−∑i=1nkizi4−∑i=1nσi2γiΘ˜i2+σn2γnΘn∗2+an22+τ+14εn∗4+Dn−1≤−∑i=1nkizi4−∑i=1nσi2γiΘ˜i2+Dn.
Then, (66) can be rewritten as(67)LVn≤−d0Vn+Dn,where Dn=∑i=1nai2/2+1/4εn∗4+σi/2γiΘ˜i2+nτ+∑i=1nhi04/4g¯i4+hi04/4,(68)d0=min4k1g¯1,…,4kn−1g¯n−1,4kn,σ1,…,σn.
Based on (63), we can easily obtain the following inequality:(69)dEVtdt≤−d0EVt+Dn.
Then, integrating inequality (69), the following formula can be derived:(70)0≤EVt≤V0−Dnd0e−a0t+Dnd0.
Thus, the inequality,(71)Ez14≤EVt≤8Dnd0,holds for ∀t>T1 with(72)T1=max0,1d01nd0v0Dn.
The proof is completed.Remark 1.
The controller in (54) contains neural networks structure, which will cause tedious calculation in practical application. Nevertheless, the inequality 0<SiTZiSiZi≤1 should also be emphasized, and it can be obtained from the theories in [46].
From Remark1, the base functions in controller (54) can be conveniently deleted to further improve the design. Then, we give the following corollary.Corollary 1.
For1≤i≤n,k∈L, we assume that the functions fk,iξi¯ are, instead of by neural networks system, within bounded error εi. Consider the switching strategy among observer (11), the following are the intermediate virtual and the actual controllers:(73)α1=−x1z1−34z1−Θ^12a12z13−3n−14z1φ1φ1T2,αi=−xizi−34zi−Θ^i2ai2zi3,2≤i≤n−1,uk=−xnzn−34zn−Θ^n2an2zn3.The adaptive laws denote as follows:(74)Θ^˙i=r2ai2zi6−ϱiΘ^i,1≤i≤n.Then, the semiglobal boundedness of all signals in the closed-loop structure can be established under bounded initial values, and an arbitrary switching rate satisfiesτa>logμ/a0. Meanwhile, we can obtain the arbitrarily small output with some suitable parameters.Proof.
This proof is rather similar to Theorem1; therefore, we omit the procedure here.


## 2.1. Problem Descriptions

Now, let us first present some lemmas, definitions, and assumptions, and they will be used in the subsequent developments. Consider a stochastic switched system with nonstrict feedback form:(1)dξi=Fσt,iξ¯i,ξi+1,ξdt+ψσt,iTξdw,1≤i≤n−1,dξn=Fσt,nξ¯n,u,ξdt+ψσt,nTξdw,η=ξ1,where ξ¯i=ξ1,ξ2,…,ξiT∈R,i=1,2,…,N, and ξ=ξ1,ξ2,…,ξnT∈Rn and η are the state and the output of the system, respectively. Let random function ψσt,iξ be smooth, and the switching signal is denoted by σt:0,+∞⟶Iq=1,2,…,q. For ∀i=1,2,…,n,p∈Iq, assume that fp,i⋅ and hp,i are assumed to be unknown and locally Lipschitz functions. The unknown system functions gσt,iξ¯i,fσt,iξ, and hσt,iξ are contained in the function Fσt,i and satisfy gσt,iξ¯iξi+1+fσt,iξ+hσt,iξ=Fσt,iξ¯i,ξi+1,ξ and uv+fσt,nξ+hσt,nξ=Fσt,nξ¯i,u,ξ, which are smooth and fp,i0=0. The variate ω is an independent r-dimensional standard Wiener process. It is important to note that there is a regular convention for switching systems, solutions are continuous everywhere, and the states do not jump at the switching moment. Let uv be the controller with input saturation, which can be written as(2)uvt=satvt=signvtε0,vt≥ε0,vt,vt<ε0.Lemma 1 (see [43]).
For each pairx,y∈R2, Young’s inequality holds(3)xy≤εaaxa+1bεbyb,where ε>0,a>1,b>1,a−1b−1=1.Assumption 1.
The constantshi0 satisfy hiξ≤hi0, i=1,2,…,n. Consider the following stochastic system:(4)dξt=fξtdt+gξtdw,where ξ∈Rn is the system state, ω is a standard Wiener process, and f:Rn⟶Rn and g:Rn⟶Rn are locally Lipschitz functions and satisfy f0=g0=0.Definition 1 (see [44]).
For any givenVξ∈C2, which is associated with the stochastic system (4), the infinitesimal generator ℒ is defined as follows:(5)ℒVξ=∂V∂xfξ+12TrgξT∂2V∂x2gξ,where TrA is the trace of a matrix A.Definition 2.
The trajectoryξt of system (4) is said to be semiglobally uniformly ultimately bounded in pth moment if, for some compact set Ω∈Rn and any initial state ξ0=ξt0, there exist constant ε>0 and time constant T=Tε,ξ0, such that Eξtp<ε for all t>t0+T, p∈Iq=1,2,…,q. In particular, when p=2, it is usually called semiglobally, uniformly, and ultimately bounded in mean square.Lemma 2 (see [45]).
Consider the stochastic system (4). If there exist functions Vξ∈C2,σ¯1, and σ¯2∈K∞ and constants m0>0 and n0>0, such that(6)σ¯1ξ≤Vξ≤σ¯2ξ,LVξ≤−m0Vξ+n0,then, there is a unique solution of system (4) for each ξ0∈Rn, and it holds(7)EVξ≤Vξ0e−m0t+m0n0,∀t>t0.In this paper, approximation-based NNs will be used to approximate the unknown nonlinear functions. For any continuous unknown smooth nonlinear functionfZ over a compact set ΩZ⊂Rq, there exists NNs W∗TSZ, such that, for a desired level of accuracy ε,(8)fZ=W∗TSZ+δZ,δZ≤ε,where W∗ is the ideal constant weight vector and defined by(9)W∗=argminW∈RNsupZ∈ΩZfZ−WTSZ,δZ is the approximation error, W=w1,…,wNT is the weight vector, and SZ=s1Z,…,sNZT is the radial basis function vector with N being the number of the NNs nodes and N>1:(10)siZ=exp−Z−uiTZ−uiηi2,i=1,2,…,N,where ui=ui1,ui2,…,uinT is the center of the receptive field and ηi∈R is the width of the Gaussian function.


## 2.2. Adaptive Controller Design and Stability Analysis

A neural controller will be constructed for the stochastic switched system (1). At the same time, the adaptive laws will be given in this section.At first, let us define some essential functions:(11)α1=−k1z1−94z1−z13Θ14τψ1Tξ1,η˙dψ1ξ1,η˙d,(12)Θ˙1=z16γ14τψ1Tξ1,η˙dψ1ξ1,η˙d−σ1Θ1,where α1 is the first virtual control signal and Θ˙1 is an adaptive law with k1,τ,γ1, and σ1 being positive design parameters.Similarly, fori=2,…,n−1, we design the virtual signals and adaptive laws as follows:(13)αi=−kizi−134zi−zi3Θi4τψiTξ¯i,miψiξ¯i,mi,(14)Θ˙i=zi6γi4τψiTξ¯i,miψiξ¯i,mi−σiΘi,where mi=ηd,η˙d,…,ηdi−1,Θ^1,…,Θ^i−1,αi is the ith virtual control signal and Θ˙i is the adaptive law with ki,τ,γi, and σn being positive design parameters. The actual control input v and the adaptive law Θ˙n are defined as follows:(15)v=−knzn−52zn−zn3Θn4τψnTξ,mnψnξ,mn−h˜,(16)Θ˙n=zn6γn4τψnTξ,mnψnξ,mn−σnΘn,where mn=ηd,η˙d,…,ηdi−1,Θ^1,…,Θ^i−1, kn,τ,γn, and σn being positive design parameters. h˜ is an auxiliary design signal, which will be defined later. The coordinate transformations are as follows:(17)zi=ξi−αi−1,i=1,2,…,n−1,(18)zn=ξn−αn−1−h˜,where α0=ηdt. Combining (1) with coordinate transformations (17) and (18), the following system can be obtained:(19)dzi=gi,pξ¯iξi+1+fi,pξ+hi,pξ−lαi−1dt+ψi,pξ−∑j=1i−1∂αi−1∂xjψj,pTξTdw,dzn=uv+fn,pξ+hn,pξ−lαn−1−h˜˙dt+ψn,pξ−∑j=1n−1∂αn−1∂xjψj,pTξTdw,where(20)lαi−1=∑j=1i−1∂αi−1∂Θ^jΘ^˙j+∑j=1i−1∂αi−1∂ξjgj,pξj¯ξj+1+fj,pξ+hj,pξ+∑j=1i−1∂αi−1∂ηdjηdj+1+∑m,n=1,1i−112∂2αi−1∂ξm∂ξnψm,pTξψn,pξ,with lα0=η˙d for i=2,…,n.Theorem 1.
Let Assumption1 hold, the closed-loop structure be consisted of the virtual control signals in (11) and (13), the adaptive laws (12) and (14) together with (16) and the actual control signal (15), which are designed from the stochastic switched system (1). All signals are bounded with some suitable parameters and the tracking error enters inside the area Ω1 for all t>T1:(21)Ω1=ηt∈REη−ηd4≤8Dnd0,∀t>T1,where d0=min4k1g¯1,…,4kn−1g¯n−1,4kn,σi,1≤i≤n,(22)Dn=∑i=1nai22+εi∗44+σiΘi∗22γi+nτ+∑i=1nhi044g¯i4+hn044,and T1=max0,1/d0lnd0v0/Dn, gi¯=infgσt,iξ¯i, and gi¯>0.Proof.
Step 1: choose the following Lyapunov function:(23)V1=14g¯1z14+12γ12Θ˜12,where g¯1>0andγ1>0 are design parameters, and Θ˜1=Θ1∗−Θ1 with Θ1 being the estimation of Θ1∗. In the light of the above equations in (4), (5), and (19), it yields(24)LV1=1g¯1z13g1,pξ1ξ2+f1,pξ+h1,pξ−η˙d+32g¯1z12ψ1,pTξ−1γΘ˜1Θ˙1.
Employing Young’s inequality (3), we can obtain(25)32g¯1z12ψ1,pTξψ1,pξ≤32g¯1z12ψ1,pξ2≤1a1g¯13/2z13ψ1,pξ3+a122,(26)1g¯1z13h1,pξ≤34z14+14g¯14h104,where a1 is a positive constant. The NNs ω1∗,ϕpT1ξ,η˙d can estimate the following unknown function:(27)1g¯1f1ξ−1g¯1η˙d+1a1g¯13/2ψ1,pξ3=ω1∗,ϕpT1ξ,η˙d+δ1ξ,η˙d.
According to (17), (25), and (27), formula (24) can be rewritten as(28)LV1≤z13z2+34z1+α1+ω1∗,ϕpT1ξ,η˙d+δ1ξ,η˙d+a122+14g¯14h104−1γ1Θ˜1Θ˙1.
Therefore, the following result can be obtained:(29)z13ω1∗,ϕpT1ξ,η˙d≤z16ω1∗,ϕpT1Tξ,η˙dϕ1ξ,η˙d4τ+τ≤z16w1,p∗2ϕ1Tξ,η˙dϕ1ξ,η˙d4τϕ1Tξ,η˙dϕ1ξ,η˙d+τ≤z16Θ1∗4τϕ1Tξ,η˙dϕ1ξ,η˙d+τ,(30)z13δ1ξ,η˙d≤34z14+14δ1∗4,(31)z13z2≤34z14+14z24,where Θ1∗=maxω1,p∗2:p∈Ip, δ1ξ,η˙d≤δ1∗, δ1∗ and τ are positive constants, and(32)0<ϕ1Tξ,η˙dϕ1ξ,η˙d≤1.
Substituting (29)–(31) into (28), we have(33)LV1≤z1394z1+α1+z13Θ14τϕ1Tξ,η˙dϕ1ξ,η˙d+z16Θ˜14τϕ1Tξ,η˙dϕ1ξ,η˙d+a122+14g¯14h104+τ+14z24+14δ1∗4−1γ1Θ˜1Θ˙1.
Substituting the virtual controllerα1 in (11) and the adaptive law in (12) into (33) yields(34)LV1≤−k1z14+σ1γ1Θ˜1Θ˙1+14z24+a122+14g¯14h104+τ+14δ1∗4.
We can get the following result by Young’s inequality:(35)σ1γ1Θ˜1Θ1=σ1γ1Θ˜1Θ1∗−Θ˜1≤−σ12γ1Θ˜12+σ12γ1Θ˜1∗2.
Then, one has(36)LV1≤−k1z14−σ12γ1Θ˜12+14z24+D1,where D1=a12/2+τ+δ1∗4/4+h104/4g¯14+σ1Θ1∗2/2γ1.
Stepii=2,3,…,n−1: similar to Step 1, choose the following Lyapunov function:(37)Vi=Vi−1+14g¯izi4+12γiΘ˜i2,where g¯iandγi are positive constants and Θi is the estimation of Θi∗,Θ˜i=Θi∗−Θi. Then, we can obtain the following formula:(38)LVi=LVi−1+1g¯izi3gi,pξ¯iξi+1+fi,pξ+hi,pξ−∑j=1i−1∂αi−1∂ξjgj,pξ¯jξj+1+fj,pξ+hj,pξ−∑j=0i−1∂αi−1∂ηdjηdj+1−∑j=1i−1∂αi−1∂Θ^jΘ^˙j−1γ1Θ˜iΘ˙i−12∑m,n=1i−1∂2αi−1∂ξp∂ξqψm,pTξψn,pξ+32g¯izi2ψi,pξ−∑j=1i−1∂αi−1∂ξjψj,pξTψi,pξ−∑j=1i−1∂αi−1∂ξjψj,pξ,where(39)32g¯izi2ψi,pξ−∑j=1i−1∂αi−1∂ξjψj,pξTψi,pξ−∑j=1i−1∂αi−1∂ξjψj,pξ≤32g¯iz22ψi,pξ−∑j=1i−1∂αi−1∂ξjψj,pξ2≤zi3aig¯i3/2ψi,pξ−∑j=1i−1∂αi−1∂ξjψj,pξ3+ai22,where ai is a positive constant. By Young’s inequality, one has(40)1g¯izi3hi,pξ≤34zi4+14g¯i4hi04,(41)−1g¯izi3∑j=1i−1∂αi−1∂ξjhj,pξ≤14g¯i4∑j=1i−1∂αi−1∂ξj4hj04+34zi4.
Then, we apply NNsωi∗,ϕpTiξ,mi to replace the unknown function:(42)wi∗,ϕpTiξ,mi+δiξ,mi=1g¯jfi,pξ−∑j=1i−1∂αi−1∂ξjgj,pξ¯jξj+1+fi,pξ−∑j=0i−1∂αi−1∂ηdjηdj+1−∑j=1i−1∂αi−1∂Θ^jΘ^˙j−12∑m,n=1i−1∂2αi−1∂ξm∂ξnψm,pTξψn,pξ+1aig¯i3/2ψi,pξ−∑j=1i−1∂αi−1∂ξjψj,pξ3+1aig¯4∑j=1i−1∂αi−1∂ξj4hj04.
Due to (17) and (39)–(42), we have(43)LVi≤LVi−1+zi3zi+1+32zi+wi∗,ϕpTiξ,mi+δiξ,miαi+ai22+14g¯i4h104−1γiΘ˜iΘ˙i.
In line with Young’s inequality (3), one can obtain(44)zi3wi∗,ϕpTiξ,mi≤zi6wi∗,wpTi,p∗ϕiTξ,miϕiξ,mi4τ+τ≤zi6wi,p∗2ϕiTξ,miϕiξ,mi4τϕiTξ¯i,miϕiξ,mi+τ≤zi6Θi∗4τϕiTξ¯i,miϕiξ¯i,mi+τ,(45)zi3δiξ,mi≤34zi4+14δi∗4,(46)zi3z2≤34zi4+14zi+14,where Θi∗=maxωi,p∗2:p∈Ip,δiξ,mi≤δi∗,(47)0<ϕiTξ,miϕiξ,mi≤1,δi∗ and τ are positive constants. Substituting (44)–(46) into (43), we can obtain(48)LVi≤zi33zi+αi+zi3Θi∗4τϕiTξ¯i,miϕiξ¯i,mi+LVi−1+14zi+14+14δi∗4+ai22+τ−1γiΘ˜iΘ˙i+14g¯i4h104≤∑m=1i−1kmzm4−∑m=1i−1σm2γmΘ˜m2+14zi+14+zi3134zi+αi+zi3Θi4τϕiTξ¯i,miϕiξ¯i,mi+zi6Θ˜i4τϕiTξ¯i,miϕiξ¯i,mi+αi22+τ+14g¯i4h104+14δi∗4−1γiΘ˜iΘ˙i+Di−1,where Di−1=∑j=1i−1aj2/2+1/4g¯j4hj04+1/4δj∗4+σjΘj∗2/2γj+i−1τ. Based on the designed virtual control signal αi in (13) and the adaptive law Θ˙i in (14), one can obtain(49)LVi≤−∑m=1ikmzm4−∑m=1i−1σm2γmΘ˜m2+σiγiΘ˜iΘi+ai22+14g¯i4hi04+τ+14zi+14+14δi∗4+Di−1.
Similar to (23), we can derive the following conclusion:(50)σiγiΘ˜iΘi=σiγiΘ˜iΘi∗−Θ˜i≤−σi2γiΘ˜i2+σi2γiΘ˜i∗2.
Moreover, substituting (50) into (49) implies(51)LVi≤−∑m=1ikmzm4−∑m=1iσm2γmΘ˜m2+14zi+14+Di,where Di=∑j=1iaj2/2+1/4g¯j4hj04+1/4δj∗4+σjΘj∗2/2γj+iτ.
Stepn: the Lyapunov function is chosen as follows:(52)Vn=Vn−1+14g¯nzn4+12γnΘ˜n2,where g¯nandγn are positive constants and Θn is the estimation of Θn∗,Θ˜n=Θn∗−Θn. From Ito^ differential formula, we can obtain the following result:(53)LVn=LVn−1+zn3uv+fn,pξ+hn,pξ−∑j=1n−1∂αn−1∂ξjgj,pξ¯jξj+1+fj,pξ+hj,pξ−∑j=0n−1∂αn−1∂θ^jθ^.j−∑j=0n−1∂αn−1∂ηdjηdj+1−h˜˙−12∑m,n=1n−1∂2αn−1∂ξm∂ξnψm,pTξψn,pξ−1γnΘ˜nΘ˙n+3zn22ψn,pξ−∑j=1n−1∂αn−1∂ξjψj,pξTψn,pξ−∑j=1n−1∂αn−1∂ξjψj,pξ.
We introduce the following dynamic system in order to simplify the controller design:(54)h˜˙=−h˜+uv−v.
Based on (5), one has(55)zn3hn,pξ≤34zn4+hn04,(56)−zn3∑j=1n−1∂αn−1∂ξjhj,pξ≤14∑j=1n−1∂αn−1∂ξj4hj04+34zn4.
Similar to (39), the following inequality can be written as(57)32zn2ψn,pξ−∑j=1n−1∂αn−1∂ξjψj,pξTψn,pξ−∑j=1n−1∂αn−1∂ξjψj,pξ≤32zn2ψn,pξ−∑j=1n−1∂αn−1∂ξjψj,pξ2≤zn3anψn,pξ−∑j=1n−1∂αn−1∂ξjψj,pξ3+an22,where an>0. Substituting (54)–(57) into (53), we obtain(58)LVn=LVn−1+zn332zn+fn,pξ−∑j=1n−1∂αn−1∂ξjgj,pξ¯jξj+1+fj,pξ−∑j=0n−1∂αn−1∂ηdjηdj+1−∑j=1n−1∂αn−1∂Θ^jΘ^˙j−12∑m,n=1n−1∂2αn−1∂ξp∂ξqψm,pTξψn,pξ+h˜+v+1anψn,pξ−∑j=1n−1∂αn−1∂ξjψj,pξ3+14∑j=1n−1∂αn−1∂ξj4hj04+14hn04−1γnΘ˜nΘ˙n+an22.
The neural networks can estimate the unknown nonlinear function:(59)wn∗,ϕpTnξ,mn+δnξ,mn=fi,pξ−∑j=1n−1∂αi−1∂ξjgj,pξ¯jξj+1+fi,pξ−∑j=0n−1∂αn−1∂ηdjηdj+1−∑j=1n−1∂αn−1∂Θ^jΘ^˙j−∑j=0n−1∂αn−1∂ηdjηdj+1−∑j=1n−1∂αn−1∂Θ^jΘ^˙j−12∑m,n=1n−1∂2αn−1∂ξp∂ξqψm,pTξψn,pξ+14∑j=1n−1∂αn−1∂ξj4hj04+1anψn,pξ−∑j=1n−1∂αn−1∂ξjψj,pξ3.
Similarly, it is noted that the following formula holds:(60)zn3wn∗,ϕpTnξ,mn≤zn6wn∗,ϕpTnTξ,mnϕnξ,mn4τ+τ≤zn6wn,p∗2ϕnTξ,mnϕnξ,mn4τϕnTξ,mnϕnξ,mn+τ≤z16Θ1∗4τϕnTξ,mnϕnξ,mn+τ,(61)zn3δnξ,mn≤34zn4+14δn∗4,where δn∗ and τ are positive constants, δnξ,mn≤δn∗,(62)0<ϕnTξ,mnϕnξ,mn≤1,Θn∗=maxωn,p∗2:p∈Ip.
Based on (59)–(61), the equality of (58) can be rewritten as(63)LVn≤−∑i=1n−1kizi4−∑i=1n−1σi2γiΘ˜i2+zn3h˜+v+52zn+zn3Θn4τϕnTξ,mnϕnξ,mn+an22+τ+14δn∗4+Dn−1+14hn04+Θ˜nγnzn6γn4τϕnTξ,mnϕnξ,mn−Θ˙n.
Taking formulas (15) and (16) into (63), it yields(64)LVn≤−∑i=1nkizi4−∑i=1n−1σi2γiΘ˜i2+σnγnΘ˜nΘn+an22+τ+14δn∗4+Dn−1+14hn04,where(65)σnγnΘ˜nΘn=σnγnΘ˜nΘn∗−Θ˜n≤−σn2γnΘ˜n2+σn2γnΘn∗2.
In addition, substituting (65) into (64), we have(66)LVn≤−∑i=1nkizi4−∑i=1nσi2γiΘ˜i2+σn2γnΘn∗2+an22+τ+14εn∗4+Dn−1≤−∑i=1nkizi4−∑i=1nσi2γiΘ˜i2+Dn.
Then, (66) can be rewritten as(67)LVn≤−d0Vn+Dn,where Dn=∑i=1nai2/2+1/4εn∗4+σi/2γiΘ˜i2+nτ+∑i=1nhi04/4g¯i4+hi04/4,(68)d0=min4k1g¯1,…,4kn−1g¯n−1,4kn,σ1,…,σn.
Based on (63), we can easily obtain the following inequality:(69)dEVtdt≤−d0EVt+Dn.
Then, integrating inequality (69), the following formula can be derived:(70)0≤EVt≤V0−Dnd0e−a0t+Dnd0.
Thus, the inequality,(71)Ez14≤EVt≤8Dnd0,holds for ∀t>T1 with(72)T1=max0,1d01nd0v0Dn.
The proof is completed.Remark 1.
The controller in (54) contains neural networks structure, which will cause tedious calculation in practical application. Nevertheless, the inequality 0<SiTZiSiZi≤1 should also be emphasized, and it can be obtained from the theories in [46].
From Remark1, the base functions in controller (54) can be conveniently deleted to further improve the design. Then, we give the following corollary.Corollary 1.
For1≤i≤n,k∈L, we assume that the functions fk,iξi¯ are, instead of by neural networks system, within bounded error εi. Consider the switching strategy among observer (11), the following are the intermediate virtual and the actual controllers:(73)α1=−x1z1−34z1−Θ^12a12z13−3n−14z1φ1φ1T2,αi=−xizi−34zi−Θ^i2ai2zi3,2≤i≤n−1,uk=−xnzn−34zn−Θ^n2an2zn3.The adaptive laws denote as follows:(74)Θ^˙i=r2ai2zi6−ϱiΘ^i,1≤i≤n.Then, the semiglobal boundedness of all signals in the closed-loop structure can be established under bounded initial values, and an arbitrary switching rate satisfiesτa>logμ/a0. Meanwhile, we can obtain the arbitrarily small output with some suitable parameters.Proof.
This proof is rather similar to Theorem1; therefore, we omit the procedure here.


## 3. Simulation Example

We will construct a numerical simulation example to verify the effectiveness of the proposed controllers. The following two stochastic switched second-order systems are listed as follows.System (1) is defined as(75)dx1=g1,1x¯1x2+f1,1x+h1,1xdt+ψ1,1Txdω,dx2=uv+f2,1x+h2,1xdt+ψ2,1Txdω,y=x1,where the switched signal p=1, and the nonlinear functions are(76)g1,1x¯1=0.5sinx1,f1,1x=1−cosx1x3+x2+x3,h1,1x=−x3tsin0.5x1tx3t,ψ1,1x=0.5x2tcosx1t,f2,1x=1+cosx2x3+x3,h2,1ξ=0.5ξ1tξ3t−ξ3t,ψ2,1ξ=ξ2tsinξ1t.System (2) is defined as(77)dx1=g1,2x¯1x2+f1,2x+h1,2xdt+ψ1,2Txdω,dx2=uv+f2,2x+h2,2xdt+ψ2,2Txdω,y=x1,where the switched signal p=2, and the nonlinear functions are(78)g1,2x¯1=sinx1x2,f1,2x=1−x12x3+x3,h1,2x=x3tcosx1tx3t,ψ1,2x=x2tsinx1t,f2,2x=x22x3+x1,h2,2x=x3tsinx1t,ψ2,2x=x2tx1tcosx2t.The input saturation is defined as(79)uvt=satvt=3signvtε0,vt≥3,vt,vt<3.Furthermore, the reference signal is given asyd=0.4sint+sin0.5t, and the designed parameters are chosen to be τ=0.2,k1=k2=1.3,γ1=γ2=1.4,andσ1=σ2=0.2. The initial conditions are x0=0.010.030.15T and Θ^0=0.020.030.01T.The simulation results are illustrated in Figures1–5, respectively. Figure 1 demonstrates control output and target. Figure 2 gives the trajectories of adaptive parameter θ1 and θ2. Figure 3 shows the control input signal u. The input signal u is asymptotically stable with the change of time. Figure 4 shows the response curves of state variable v, and it can be drawn that v converge to zero after a few seconds. Figure 5 shows the response curves of the giving switching signal σt. From the above five figures, it can be seen that all signals in the closed-loop structure are bounded and the states can track the given reference signals. Based on the simulation results, we can conclude that the proposed control scheme is effective for the switched system.Figure 1
The response of the variablesy and yd.Figure 2
The response of the adaptive lawsθ1,θ2.Figure 3
The response of the control input signalu.Figure 4
The response curves of state variablev.Figure 5
The response of the giving switching signalσt.


## 4. Conclusion

An actual operating system is complex. When it is dominated by multiple switched subsystems with input saturation, external interference, nonstrict feedback, nonlinearity, and randomness, it is a difficult problem how to control it stably. This work comprehensively considers the influence of all the above factors, attempts to design a new switching controller and an adaptive neural scheme, combining with neural network structure, variable separation technology, reverse step method, Ito^ formula, and the auxiliary signal, gives the control design conditions of antisaturation and avoiding signal attenuation, and achieves the asymptotic stability performance of the closed-loop system. There are two difficulties in this paper, one is that the switching functions are unknown and nonstrict feedback, and the neural network structure is an effective function approximator at present. Therefore, we chose it to estimate these functions. The other is that how to design the switching strategy and controller to resist the saturation when the subsystem switches; to deal with this problem, we introduced a set of auxiliary system. Furthermore, we introduce a neural controller into the control scheme to track the variation of the error. The controller is designed based on the improved backstepping technique. It is proved that all the signals are bounded with some suitable parameters, and the trajectory error converges to zero.



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*Source: 7259613-2020-09-10.xml*