# Finite-Time Stabilization forp-Norm Stochastic Nonlinear Systems with Output Constraints

Authors: Shijun Guan; Liandi Fang
Journal: Complexity
 (2020)

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## Abstract

This paper investigates the finite-time stability problem ofp-norm stochastic nonlinear systems subject to output constraint. To cope with the constraint on system output, a tan-type barrier Lyapunov function (BLF) is constructed. By using the constructed BLF and the backstepping technique, a new control algorithm is proposed with a continuous state-feedback controller being designed, which guarantees not only that the requirement of output constraint is always achieved but also that the origin of the system is finite-time stable. This result is demonstrated by both the rigorous analysis and the simulation example.


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## Body

## 1. Introduction

During the past decades, the control problem of nonlinear systems has long been a hot topic, and many control design approaches have been proposed for various kinds of nonlinear systems, such as adaptive fuzzy control [1, 2], output tracking control [3, 4], H∞ control [5, 6], and sliding mode control [7–10]. Due to their important roles in many science and industry applications, the stochastic nonlinear systems have attracted much interest in recent years. With the development of stochastic theory, various control design strategies have been developed for types of stochastic nonlinear systems by the backstepping technique, see [11–14], for examples. Especially, some works have considered p-norm stochastic nonlinear systems, which are inherently nonlinear due to the fractional powers of such systems being not identically equal to one. It should be noted that the inherent nonlinearities cause the stability and control design problems, which are not very easy to be solved [15]. Luckily, the issues have been well studied for p-norm stochastic nonlinear systems with different structures by the adding a power integrator technique in the existing literatures. For instance, Li et al. [16] have considered the adaptive state-feedback stabilization for p-norm stochastic nonlinear systems; the output-feedback control has been addressed for p-norm stochastic nonlinear systems with time-varying delays in [17]; Zhao et al. [18] have proposed a neural tracking control algorithm for p-norm switched stochastic nonlinear systems. More latest studies can be found in [19–21] and the references within.However, most of the abovementioned works aboutp-norm stochastic nonlinear systems did not take the output constraint into consideration. As it is well known, many actual systems are subject to output constraint due to the consideration of the system performance and operation safety [22, 23]. For this reason, the constrained control issue of nonlinear systems has drawn attention from many scholars. Tee et al. [24] have first proposed the notion of the barrier Lyapunov function (BLF) and consequently have developed a control design strategy for a class of strict-feedback deterministic nonlinear systems with output constraints. After then, with the aid of BLFs, control design schemes have been presented for many deterministic nonlinear systems with different types of constraints, including stability control for nonlinear systems with time-varying or asymmetric output constraints [25, 26], adaptive control for nonlinear systems with full-state constrains [27], and sliding mode control for nonlinear systems with output constraints [28–30]. Moreover, since the finite-time control possesses some inherent advantages [31–33], techniques for the finite-time stabilization under output/state constraints have also been developed, respectively, for strict-feedback nonlinear systems [34], norm nonlinear systems [35–37], and switched nonlinear systems [38]. On the basis of these results, the constrained control schemes for some classes of stochastic nonlinear systems have also been proposed. Jin [39] has constructed an adaptive tracking controller for a class of output-constrained stochastic nonlinear systems in strict-feedback form. Later, the adaptive control problem and the finite-time control problem have been, respectively, addressed for stochastic nonlinear systems with full-state constraints in [40, 41]. Furthermore, the adaptive neural network or fuzzy constrained control problems have attracted some attention [42–46]. Nevertheless, the stochastic nonlinear systems with output constraints considered in most of the existing related works are in the strict-feedback form, rather than in p-normal form. On the contrary, the existing research has mainly focused on the adaptive control problem but did not take the finite-time stabilization into account.Motivated by the above discussions, we will investigate the problem of the finite-time stabilization for a class ofp-norm stochastic nonlinear systems with output constraints and unknown time-varying parameters. First of all, a BLF-based control strategy will be developed by the backstepping approach. Secondly, applying stochastic Lyapunov theorems and ItÔ’s formula, the constructed state-feedback controller is rigorously proved to be able to ensure the achievement of the output constraint and the finite-time stability of the considered systems simultaneously. Finally, the main result of this paper will be further demonstrated by a simulation example.


## 2. Problem and Preliminaries

### 2.1. Problem Statement

The following class of stochastic nonlinear systems are considered:(1)dxi=ϑitxi+1qidt+fix¯idt+giTx¯idω,i=1,…,n−1,dxn=ϑntuqndt+fnx¯ndt+gnTx¯ndω,y=x1,where ω is a N-dimension standard Wiener process; x¯i=x1,…,xiT∈Ri,u∈Randy∈R are system state, control input, and output, respectively; ϑit is the time-varying parameter; the nonlinear functions fi:Ri⟶R and gi:Ri⟶RN are continuous and satisfy fi0=gi0=0; and the fractional powers qi’s meet the requirement qi∈Rodd≥1:=τ≥1,τis the positive odd integersratio. The output y is required to satisfy(2)y∈∏1=yt∈R,yt<b=x1∈R,x1t<b,where b is a known positive constant.This paper aims to design a continuous state-feedback controller for system (1), which can ensure that the origin of the closed-loop system is finite-time stable in probability and the requirement of the output constraint is achieved.


### 2.2. Preliminaries

Notations 1.
Fork=1,…,n, let gk−Tx¯k=g1Tx1,…,giTx¯k and Πk=x¯k∈Rk,x1t<b. For any ς∈R and θ>0, denote φς=ςθ:=ςθsgnς.
Consider the following stochastic system:(3)dx=fxdt+gxdω,where fx and gx are continuous satisfying f0=0 and g0=0.Definition 1 (see [13]).
For any givenVx∈C2Rn, associated with system (1), the second-order differential operator is defined as follows:(4)ℓV=∂V∂xfx+12trgTx∂2V∂x2gx.Definition 2 (see [24]).
Suppose thatΠ is an open set containing the origin and V:Π⟶R is positive definite and continuously differentiable. Then, for system x˙=gx,Vxt is called a BLF if for each solution xt starting from xt0∈Π,Vxt⟶∞, as xt⟶∂Π, Vxt≤τ for all t≥t0 and for some τ∈R+.Assumption 1 (see [36]).
For∀i=1,…,n, there exist known positive constants ϑ¯i and ϑ¯i such that ϑ¯i≤ϑit≤ϑ¯i.Assumption 2.
Fori=1,…,n, there are a constant μ∈−1+∑j=2nq1⋯qj−1−1,0 and known nonnegative smooth functions ψix¯i, ηix¯i such that(5)fix¯i≤ψix¯i∑j=1ixjvi+μ/vj,gix¯i≤ηix¯i∑j=1ixj2vi+μ/2vj,for all t≥0, where v1=1,vj+1=vj+μ/qj>0,j=1,…,n.Remark 1.
Note that condition (4) is borrowed from [34]. However, the systems considered in [34] are p-norm deterministic nonlinear systems, while we consider p-norm stochastic nonlinear systems with drift terms fi’s and diffusion terms gi’s in this paper. In light of μ∈−1+∑j=2nq1⋯qj−1−1,0, the value of μ is generally taken as μ=−m/p for simplicity, where m and p represent even and odd integers, respectively. Then, the value of each vjj=2,…,n can be obtained by applying v1=1 and vj+1=vj+μ/qj>0. It can also be observed that both the denominator and numerator of each vj are odd.Lemma 1 (see [13]).
Suppose that there exists a positive Lyapunov functionV∈C2Rn, which satisfies limx⟶∞Vx=∞. If ℓV is with respect to (3) and satisfies ℓV≤0,∀x∈Rn, then system (3) has a solution for any initial value.Lemma 2 (see [13]).
Suppose that system (3) admits a solution for each initial value. If there are κ∞ class functions ϱ1⋅ and ϱ2⋅, a positive C2 Lyapunov function V, real numbers c>0, and 0<γ<1, such that(6)ϱ1x≤Vx≤ϱ2x,x∈Rn,ℓVx≤−cVγx,x∈Rn\0.For allt≥0, then the origin of system (3) is finite-time stable in probability.Lemma 3 (see [37]).
Leta,b,ρ,andς be positive real numbers. For any z1,z2∈R, we have(7)ρz1az2b≤ςaa+bz1a+b+ba+bρa+b/bς−a/bz2a+b.Lemma 4 (see [8]).
Letp∈0,∞; for any ςi∈R,i=1,…,n, one has(8)ς1+⋯+ςnp≤dς1p+⋯+ςnp,where d=np−1 if p≥1 and d=1 if 0<p<1.Lemma 5 (see [10]).
Leta,d∈R+witha≥1. For any ς1,ς2∈R, we have(i)
ς1a−ς2a≤a2a−2+2ς1−ς2ς1−ς2a−1+ς2a−1(ii)
ς1d/a−ς2d/a≤21−1/aς1d−ς2d1/a(iii)
ς1+ς21/a≤ς11/a+ς21/a≤21−1/aς1+ς21/a


## 2.1. Problem Statement

The following class of stochastic nonlinear systems are considered:(1)dxi=ϑitxi+1qidt+fix¯idt+giTx¯idω,i=1,…,n−1,dxn=ϑntuqndt+fnx¯ndt+gnTx¯ndω,y=x1,where ω is a N-dimension standard Wiener process; x¯i=x1,…,xiT∈Ri,u∈Randy∈R are system state, control input, and output, respectively; ϑit is the time-varying parameter; the nonlinear functions fi:Ri⟶R and gi:Ri⟶RN are continuous and satisfy fi0=gi0=0; and the fractional powers qi’s meet the requirement qi∈Rodd≥1:=τ≥1,τis the positive odd integersratio. The output y is required to satisfy(2)y∈∏1=yt∈R,yt<b=x1∈R,x1t<b,where b is a known positive constant.This paper aims to design a continuous state-feedback controller for system (1), which can ensure that the origin of the closed-loop system is finite-time stable in probability and the requirement of the output constraint is achieved.


## 2.2. Preliminaries

Notations 1.
Fork=1,…,n, let gk−Tx¯k=g1Tx1,…,giTx¯k and Πk=x¯k∈Rk,x1t<b. For any ς∈R and θ>0, denote φς=ςθ:=ςθsgnς.
Consider the following stochastic system:(3)dx=fxdt+gxdω,where fx and gx are continuous satisfying f0=0 and g0=0.Definition 1 (see [13]).
For any givenVx∈C2Rn, associated with system (1), the second-order differential operator is defined as follows:(4)ℓV=∂V∂xfx+12trgTx∂2V∂x2gx.Definition 2 (see [24]).
Suppose thatΠ is an open set containing the origin and V:Π⟶R is positive definite and continuously differentiable. Then, for system x˙=gx,Vxt is called a BLF if for each solution xt starting from xt0∈Π,Vxt⟶∞, as xt⟶∂Π, Vxt≤τ for all t≥t0 and for some τ∈R+.Assumption 1 (see [36]).
For∀i=1,…,n, there exist known positive constants ϑ¯i and ϑ¯i such that ϑ¯i≤ϑit≤ϑ¯i.Assumption 2.
Fori=1,…,n, there are a constant μ∈−1+∑j=2nq1⋯qj−1−1,0 and known nonnegative smooth functions ψix¯i, ηix¯i such that(5)fix¯i≤ψix¯i∑j=1ixjvi+μ/vj,gix¯i≤ηix¯i∑j=1ixj2vi+μ/2vj,for all t≥0, where v1=1,vj+1=vj+μ/qj>0,j=1,…,n.Remark 1.
Note that condition (4) is borrowed from [34]. However, the systems considered in [34] are p-norm deterministic nonlinear systems, while we consider p-norm stochastic nonlinear systems with drift terms fi’s and diffusion terms gi’s in this paper. In light of μ∈−1+∑j=2nq1⋯qj−1−1,0, the value of μ is generally taken as μ=−m/p for simplicity, where m and p represent even and odd integers, respectively. Then, the value of each vjj=2,…,n can be obtained by applying v1=1 and vj+1=vj+μ/qj>0. It can also be observed that both the denominator and numerator of each vj are odd.Lemma 1 (see [13]).
Suppose that there exists a positive Lyapunov functionV∈C2Rn, which satisfies limx⟶∞Vx=∞. If ℓV is with respect to (3) and satisfies ℓV≤0,∀x∈Rn, then system (3) has a solution for any initial value.Lemma 2 (see [13]).
Suppose that system (3) admits a solution for each initial value. If there are κ∞ class functions ϱ1⋅ and ϱ2⋅, a positive C2 Lyapunov function V, real numbers c>0, and 0<γ<1, such that(6)ϱ1x≤Vx≤ϱ2x,x∈Rn,ℓVx≤−cVγx,x∈Rn\0.For allt≥0, then the origin of system (3) is finite-time stable in probability.Lemma 3 (see [37]).
Leta,b,ρ,andς be positive real numbers. For any z1,z2∈R, we have(7)ρz1az2b≤ςaa+bz1a+b+ba+bρa+b/bς−a/bz2a+b.Lemma 4 (see [8]).
Letp∈0,∞; for any ςi∈R,i=1,…,n, one has(8)ς1+⋯+ςnp≤dς1p+⋯+ςnp,where d=np−1 if p≥1 and d=1 if 0<p<1.Lemma 5 (see [10]).
Leta,d∈R+witha≥1. For any ς1,ς2∈R, we have(i)
ς1a−ς2a≤a2a−2+2ς1−ς2ς1−ς2a−1+ς2a−1(ii)
ς1d/a−ς2d/a≤21−1/aς1d−ς2d1/a(iii)
ς1+ς21/a≤ς11/a+ς21/a≤21−1/aς1+ς21/a


## 3. Main Results

### 3.1. A Tan-Type BLF

Before carrying out the control design for system (1), we should handle the output constraint issue.Firstly, we denotev0=max1≤i≤nvi and ci=v0/vii=1,…,n. Let σ be a constant parameter satisfying σ≥σ0, where the value of σ0 is chosen as below:(i)
If for all2≤i≤n, 1≤ci≤2, then σ0=2v0(ii)
If for all2≤i≤n, ci≥2, then σ0=v0Consequently, it is clear thatσ/vi≥2.Then, a tan-type BLF can be constructed onΠ1 as follows:(9)Vbx1=2b4σ−μ4σ−μπtanπx14σ−μ2b4σ−μ,where μ is given by Assumption 2 and σ is defined as above.It is not hard to obtain from the expression ofVbx1 that(10)∂Vb∂x1=sec2πx14σ−μ2b4σ−μx14σ−μ−1=Gx1x14σ−μ−1,∂2Vb∂x12=4σ−μ−1Gx1x14σ−μ−2+π4σ−μb4σ−μGx1tanπx14σ−μ2b4σ−μx124σ−μ−1,where Gx1=sec2πx14σ−μ/2b4σ−μ.Remark 2.
It should be noted that the BLF is modified from [34], which is constructed by fully taking the advantage of the given nonlinear growth conditions. As stated in [34], the control strategy based on Vbx1 is a universal method, which can handle stochastic systems with or without output constraints.


### 3.2. Controller Design and Stability Analysis

In what follows, a continuous state-feedback controller will be constructed, and the stability of system (1) under the designed controller will be rigorously analysed. To this end, a theorem is presented to describe the main result.Theorem 1.
Suppose Assumptions 1-2 hold for system (1). For any constant b>0, there is a continuous state-feedback controller such that(i)
The output of system (1) is kept in a given constrained set in the sense of probability, i.e., Pyt<b=1(ii)
The origin of the closed-loop system is finite-time stable in probability.Proof.
The proof contains three parts. First of all, the design procedure of the controller is explicitly displayed. Then, the system output is proved to be kept in the given constrained set with probability one. In the last part, the finite-time stability of system (1) is rigorously analysed.


#### 3.2.1. Part I: Design Procedure

Step 1.
Letς1=x1σ, and choose the Lyapunov function V1x1=Vbx1. Then, we can directly get from Definition 1 that(11)ℓV1≤ϑ1tGx1x14σ−μ−1x2q1+4σ−μ−12Gx1x14σ−μ−2η12x12+μ+ϑ¯1Gx1x14σ−μ−1ψ1x11+μ+π4σ−μ2b4σ−μx124σ−μ−1Gx1tanπx14σ−μ2b4σ−μη2x12+μ≤ϑ1tGx1ς14σ−μ−1/σξ2q1+ϑ1tGx1ς14σ−μ−1/σx2q1−ξ2q1+H1x1Gx1ς14,where H1x1≥ϑ¯1ψ1+1/2η124σ−μ−1+π4σ−μη12/2b4σ−μtanπx14σ−μ/2b4σ−μx14σ−μ is a nonnegative C2 function and ξ2 is a virtual controller required to be design after later.
Then, we design(12)ξ2=−λ1x1ς1v2/σwithλ1x1=n+H1x1ϑ¯11/q1>0.
Substituting (12) into (13) yields(13)ℓV1x1≤−nGx1ς14+ϑ1tGx1ς14σ−μ−1/σx2q1−ξ2q1=−12Gx1ς14−n−12Gx1ς14+ϑ1tGx1ς14σ−μ−1/σx2q1−ξ2q1≤−12Gx1ς14−n−12ς14+ϑ1tGx1ς14σ−μ−1/σx2q1−ξ2q1.Step 2.
We denoteς2=x2σ/v2−ξ2σ/v2 and define the positive Lyapunov function V2 on Π2 as V2=V1+Ψ2 with(14)Ψ2=∫ξ2x2rσ/v2−ξ2σ/v24σ−μ−1/σdr.
Since∂ξ2σ/v2/∂x1=−∂λ1σ/v2x1/x1σ−σλ1σ/v2x1x1σ−1 is valid, one can obtain(15)∂Ψ2∂x2=ς24σ−μ−1/σ,∂Ψ2∂x1=−4σ−μ−v2σ∂ξ2σ/v2∂x1∫ξ2x2rσ/v2−ξ2σ/v23σ−μ−v2/σdr,∂2Ψ2∂x1∂x2=4σ−μ−v2σς23σ−μ−v2/σ∂ξ2σ/v2∂x1,∂2Ψ2∂x22=4σ−μ−v2v2x2σ−v2/v2ς23σ−μ−v2/σ,∂2Ψ2∂x12=−4σ−μ−v2σ∂2ξ2σ/v2∂x12∫ξ2x2rσ/v2−ξ2σ/v23σ−μ−v2/σdr+4σ−μ−v2σ×3σ−μ−v2σ∂ξ2σ/v2∂x12∫ξ2x2rσ/v2−ξ2σ/v22σ−μ−v2/σdr.
Using Definition1 again, we have(16)ℓV2≤−12Gx1ζ14−n−12ζ14+ϑ1tGx1x14σ−μ−1x2q1−ξ2q1+∂Ψ2∂x1ϑ1tx2q1+f1x1+∂Ψ2∂x1f2x¯2+12trg¯2T∂2Ψ2∂x¯22g¯2+∂Ψ2∂x2ϑ2tξ3q2+∂Ψ2∂x2ϑ2tx3q2−ξ3q2,where ξ3 is the virtual controller required to be designed later.
In the following, each term in the right hand of (16) will be estimated by its upper bound.
Firstly, applying0<v2/σ≤1,0<1+μ/σ≤1 and Lemma 5, it is easily obtained that(17)x2−ξ2≤x2σ/υ2−ξ2σ/υ2υ2/σ×21−υ2/σ≤2ζ2υ2/σ,(18)x2q1≤ζ2+ξ2σ/υ21+μ/σ≤ζ21+μ/σ+λ11+μ/υ2ζ21+μ/σ,(19)x2q1−ξ2q1≤21−1+μ/σx2σ/υ2−ξ2σ/υ21+μ/σ≤2ζ21+μ/σ.
It can be deduced from (19) and Lemma 3 that(20)ϑ1tGx1x14σ−μ−1x2q1−ξ2q1≤2ϑ¯1Gx1ζ14σ−μ−1/σζ2μ+1/σ≤18ζ14+H21x¯2ζ24,where H21x¯2≥μ+1/4σ2ϑ¯1Gx14σ/μ+14σ/84σ−μ−1−4σ−μ−1/1+μ≥0 is a C2 function.
Secondly, one can obtain from Assumption1 and Lemma 4 that(21)f2x¯2≤ψ˜2x¯2x1υ2+μ/υ1+x2υ2+μ/υ2≤ψ˜2x¯2ζ1υ2+μ/σ+λ1υ2+μ/υ2ζ1υ2+μ/σ+ζ2υ2+μ/σ≤ψ˜2˜x¯2ζ1υ2+μ/σ+ζ2υ2+μ/σ,(22)g2x¯2≤η2x¯2x12υ2+μ/2υ1+x22υ2+μ/2υ2≤η2x¯2ζ12υ2+μ/2σ+λ12υ2+μ/2υ2ζ12υ2+μ/2σ+ζ22υ2+μ/2σ≤η˜2x¯2ζ12υ2+μ/2σ+ζ22υ2+μ/2σ,where ψ˜2x¯2≥ψ2x¯21+λ1v2+μ/v2 and η˜2x¯2≥η2x¯21+λ12v2+μ/2v2 are nonnegative smooth functions.
Additionally, note thatσ/v2≥2. Then, as stated in [36], there exist C2 functions M21x¯2≥0,K21x¯2≥0, and M˜21x¯2≥0, such that(23)∂ξ2σ/υ2∂x12≤∂λ1σ/υ2∂x1x1σ+σλ1σ/υ2x1σ−1≤M21x¯2ζ11−1/σ,(24)∂2ξ2σ/υ2∂x12≤∂2λ1σ/υ2∂x12x1σ+2σ∂λ1σ/υ2∂x1x1σ−1+σσ−1λ1σ/υ2x1σ−2≤Κ21x¯2ζ11−2/σ,(25)∂Ψ2∂x1≤4σ−μ−υ2σ∂ξ2σ/υ2∂x1x2σ/υ2−ξ2σ/υ23σ−μ−υ2/σx2−ξ2≤4σ−μ−υ2σ×M21x¯2ζ11−1/σζ23σ−μ−υ2/σ×2ζ2υ2/σ≤M˜21x¯2ζ11−1/σζ23σ−μ/σ,(26)12∂2Ψ2∂x12≤4σ−μ−υ22σ∂2ξ2σ/υ2∂x12x2σ/υ2−ξ2σ/υ23σ−μ−υ2/σx2−ξ2+4σ−μ−υ23σ−μ−υ22σ2∂ξ2σ/υ2∂x1x2σ/υ2−ξ2σ/υ22σ−μ−υ2/σx2−ξ2≤4σ−μ−υ2σK21x¯2ζ11−2/σζ23σ−μ/σ+4σ−μ−υ23σ−μ−υ2σ2M212x¯2ζ12−2/σζ22σ−μ/σ.
Then, using (18), (21), (25), Assumptions 1-2, and Lemma 3, we can infer(27)∂Ψ2∂x1ϑ1tx2q1+f1x1≤M˜21x¯2ϑ¯1ζ11−1/σζ23σ−μ/σζ21−μ/σ+λ11+μ/σ+M˜21x¯2ζ11−1/σζ23σ−μ/σψ˜1ζ11−1/σ≤18ζ14+Η22x¯2ζ24,where(28)H22x¯2≥3σ+14σϑ¯1M˜214σ/3σ+14σ32σ−1−σ−1/3σ+1+3σ−μ4σϑ¯1M˜21λ11+σ/v24σ/3σ−μ4σ32σ+μ−σ+μ/3σ−μ+3σ−μ4σψ1M˜214σ/3σ−μ4σ16σ+μ−σ+μ/3σ−μ,is a nonnegative C2 function.
Moreover, from (19), Assumption 2, and Lemma 3, it can be deduced that(29)∂Ψ2∂x2f2x¯2≤ζ24σ−μ−υ2/σψ˜2ζ1υ2+μ/σ+ζ2υ2+μ/σ≤18ζ14+H23x¯2ζ24,where H23x¯2≥ψ˜2+4σ−μ−v2/4σψ˜24σ/4σ−μ−v24σ/8μ+v2−μ+v2/4σ−μ−v2≥0 is a C2 function.
On the contrary, it is noted that(30)12trg¯2T∂2Ψ2∂x¯22g¯2=12∂2Ψ2∂x12g12+g1T∂2Ψ2∂x1∂x2g2+12∂2Ψ2∂x22g22.
Then, applying (22), (26), Assumption 2, and Lemma 3, there clearly exist nonnegative C2 functions H241x¯2,H242x¯2, and H243x¯2 such that(31)12∂2Ψ2∂x12g12≤4σ−μ−υ2σK21ζ11−2/σζ23σ−μ/ση12ζ12+μ/σ+4σ−μ−υ23σ−μ−υ22σ2M212ζ12−2/σζ22σ−μ/ση12ζ12+μ/σ≤124ζ14+H241x¯2ζ24,(32)g1T∂2ψ2∂x1∂x2g2≤η1ζ1μ+2/2σ4σ−μ−v2σζ23σ−μ−v2/σ×M21ζ1σ−1/ση˜2ζ12v2+μ/2σ+ζ22v2+μ/2σ≤124ζ14+H242x¯2ζ24,(33)12∂2ψ2∂x22g22≤4σ−μ−v22v2ζ23σ−μ−v2/σζ2σ−v2/σ+λ1σ−v2/v2ζ1σ−v2/σ×2η˜22ζ12v2+μ/σ+ζ22v2+μ/σ≤124ζ14+H243x¯2ζ24.
Substituting equations (31)–(33) into (30), one obtains(34)12trg¯2T∂2Ψ2∂x¯22g¯2≤18ζ14+H24x¯2ζ24,where H24x¯2=H241x¯2+H242x¯2+H243x¯2≥0 is a C2 function.
LetH2x¯2=H21x¯2+H22x¯2+H23x¯2+H24x¯2≥0. Design the virtual controller ξ3 as(35)ξ3=−λ2x¯2ζ2v3/σwithλ2x¯2=n−1+H2x¯2ϑ¯21/q2>0.
Substituting (20), (27), (29), (34), and (35) into (16), one can obtain(36)ℓV2≤−12Gx1ζ14−n−1ζ14+ζ24+ϑ2tζ24σ−μ−v2/σx3q2−ξ3q2.
Inductive Step. Suppose at step i−1, there exist a C2 Lyapunov function Vi−1:Πi−1⟶R+, and a range of continuous virtual controllers ξ1,ξ2,…,ξi defined as(37)ξ1=0,ζ1=x1σ/v1−ξ1σ/v1,ξ2=−λx1ζ1v2/σ,ζ2=x2σ/v2−ξ2σ/v2,ξ3=−λ2x¯2ζ2v3/σ,ζ3=x3σ/v3−ξ3σ/v3,⋮ξi=−λi−1x¯i−1ζi−1vi/σ,ζi=xiσ/vi−ξiσ/vi,with λkx¯k>0,fork=1,…,i−1, such that(38)ℓVi−1≤−12Gx1ζ14−n+2−i∑l=1i−1ζi4+ϑi−1tζi−14σ−μ−vi/σxiqi−1−ξiqi−1.
Then, the following property can be inferred.Proposition 1.
Choose theith Lyapunov function Vi:Πi⟶R+ as Vi=Vi−1+Ψi with(39)Ψi=∫ξixirσ/vi−ξiσ/vi4σ−μ−vi/σdr.Then,Vi is C2 on Πi and there exists a virtual controller ξi+1 such that(40)ℓVi≤−12Gx1ζ14−n+1−i∑l=1iζl4+ϑi−1tζi4σ−μ−vi/σxi+1qi−ξi+1qi,where(41)ξi+1=−λix¯iζivi+1/σwithλix¯i=n−i+1+Hix¯iϑ¯i1/qi>0.The proof of above Proposition1 is provided in the Appendix.Step 3.
In light of the inductive step, wheni=n and xn+1=u, Proposition 1 holds. Thus, we choose the overall Lyapunov function Vn as Vn=Vn−1+Ψn with(42)Ψn=∫ξnxnrσ/vn−ξnσ/vn4σ−μ−vn/σdr,and define the virtual controller ξn+1 as(43)ξn+1=−λnxςnvn+1/σwithλnx=1+Hnxϑ¯n1/qn>0.
Then,Vnx is clearly a C2 function on Πn, and it is easy to obtain that(44)ℓVn≤−12Gx1ζl4−∑l=1nζl4+ϑntζn4σ−μ−vn/σuqn−ξn+1qn.
Therefore, we can design(45)u=ξn+1x=−λnxxnσ/vn+λn−1σ/vnxn−1σ/vn−1+λn−1σ/vnλn−2σ/vn−2xn−2σ/vn−2+⋯+λn−1σ/vnλn−2σ/vn−1⋯λ1σ/v2x1σ/v1vn+μ/σqn,which results in(46)ℓVn≤−12Gx1ζ14−∑l=1nζl4≤0.


#### 3.2.2. Part II: Verification of Keeping the Output Constraint

For anyx0=x10T∈Πn, by Ito’s formula and (46), we can deduce(47)0≤EVnxt=Vnx0+E∫0tℓVnxτdr≤Vnx0<∞.FromVnx=V1x1+∑l=2nΨl>0 and (47), it can be further verified that(48)0≤EV1x1t≤Vnx0<∞,which indicates(49)PV1x1t<∞=1.Hence,Pyt<b=Px1t<b=1. Then, Part I of Theorem 1 is proved.


#### 3.2.3. Part III: Stability Analysis

Based on the definition ofVn, we easily obtain the fact that Vn is radially unbounded. Combining the fact with (40) and Lemma 1 directly infers that system (1) has a solution xt,x0 for any x0∈Πn.On the contrary, by a simple calculation, one obtains(50)Vn=V1+∑l=2nΨl=V1+∑l=2n∫ξlxlrσvl−ξlσvl4σ−μ−vl/σdr≤2b4σ−μ4σ−μπtanπx14σ−μ2b4σ−μ+2∑l=2nζl4σ−μ/σ.In addition, since4σ−μ>1, it is easy to get that 0≤πx14σ−μ/2b4σ−μ<π/2 for all x1∈Π1. Then, applying the characteristics of tangent functions, it is not difficult to infer that(51)tanπx14σ−μ2b4σ−μ≤π2b4σ−μx14σ−μGx1≤π4σ−μ2b4σ−μx14σ−μGx1.Now, letc=2μ−8σ/4σ−μ4>0 and 0<γ=4σ/4σ−μ<1. According to (44), (45), and Lemma 4, we obtain(52)cVnγ≤2μ−8σ/4σ−μ2b4σ−μ4σ−μπtanπx14σ−μ2b4σ−μ+2∑l=1nς14σ−μσ4σ/4σ−μ≤2μ−8σ/4σ−μGx14σ/4σ−μς14+12∑l=1nζ14≤12Gx1ζ14+12∑l=1nς14.So, one further obtains(53)ℓVn+cVnγ=ℓVn+2μ−8σ/4σ−μVn4σ/4σ−μ≤−12Gx1ζ14−∑l=1nς14+12Gx1ζ14+12∑l=1nζ14≤−12∑l=1nζ14≤0.In other words,ℓVn≤−cVnγ. Consequently, it is directly deduced from Lemma 2 that the origin of system (1) under controller (39) is finite-time stable in probability.Remark 3.
Comparing with the existing results in most of the literatures, the paper mainly focuses on the finite-time stabilization, instead of the boundness of tracking error. Moreover, the proposed approach can be extend to the tracking control by introducing a coordinate transformation before constructing the BLF.


## 3.1. A Tan-Type BLF

Before carrying out the control design for system (1), we should handle the output constraint issue.Firstly, we denotev0=max1≤i≤nvi and ci=v0/vii=1,…,n. Let σ be a constant parameter satisfying σ≥σ0, where the value of σ0 is chosen as below:(i)
If for all2≤i≤n, 1≤ci≤2, then σ0=2v0(ii)
If for all2≤i≤n, ci≥2, then σ0=v0Consequently, it is clear thatσ/vi≥2.Then, a tan-type BLF can be constructed onΠ1 as follows:(9)Vbx1=2b4σ−μ4σ−μπtanπx14σ−μ2b4σ−μ,where μ is given by Assumption 2 and σ is defined as above.It is not hard to obtain from the expression ofVbx1 that(10)∂Vb∂x1=sec2πx14σ−μ2b4σ−μx14σ−μ−1=Gx1x14σ−μ−1,∂2Vb∂x12=4σ−μ−1Gx1x14σ−μ−2+π4σ−μb4σ−μGx1tanπx14σ−μ2b4σ−μx124σ−μ−1,where Gx1=sec2πx14σ−μ/2b4σ−μ.Remark 2.
It should be noted that the BLF is modified from [34], which is constructed by fully taking the advantage of the given nonlinear growth conditions. As stated in [34], the control strategy based on Vbx1 is a universal method, which can handle stochastic systems with or without output constraints.


## 3.2. Controller Design and Stability Analysis

In what follows, a continuous state-feedback controller will be constructed, and the stability of system (1) under the designed controller will be rigorously analysed. To this end, a theorem is presented to describe the main result.Theorem 1.
Suppose Assumptions 1-2 hold for system (1). For any constant b>0, there is a continuous state-feedback controller such that(i)
The output of system (1) is kept in a given constrained set in the sense of probability, i.e., Pyt<b=1(ii)
The origin of the closed-loop system is finite-time stable in probability.Proof.
The proof contains three parts. First of all, the design procedure of the controller is explicitly displayed. Then, the system output is proved to be kept in the given constrained set with probability one. In the last part, the finite-time stability of system (1) is rigorously analysed.


### 3.2.1. Part I: Design Procedure

Step 1.
Letς1=x1σ, and choose the Lyapunov function V1x1=Vbx1. Then, we can directly get from Definition 1 that(11)ℓV1≤ϑ1tGx1x14σ−μ−1x2q1+4σ−μ−12Gx1x14σ−μ−2η12x12+μ+ϑ¯1Gx1x14σ−μ−1ψ1x11+μ+π4σ−μ2b4σ−μx124σ−μ−1Gx1tanπx14σ−μ2b4σ−μη2x12+μ≤ϑ1tGx1ς14σ−μ−1/σξ2q1+ϑ1tGx1ς14σ−μ−1/σx2q1−ξ2q1+H1x1Gx1ς14,where H1x1≥ϑ¯1ψ1+1/2η124σ−μ−1+π4σ−μη12/2b4σ−μtanπx14σ−μ/2b4σ−μx14σ−μ is a nonnegative C2 function and ξ2 is a virtual controller required to be design after later.
Then, we design(12)ξ2=−λ1x1ς1v2/σwithλ1x1=n+H1x1ϑ¯11/q1>0.
Substituting (12) into (13) yields(13)ℓV1x1≤−nGx1ς14+ϑ1tGx1ς14σ−μ−1/σx2q1−ξ2q1=−12Gx1ς14−n−12Gx1ς14+ϑ1tGx1ς14σ−μ−1/σx2q1−ξ2q1≤−12Gx1ς14−n−12ς14+ϑ1tGx1ς14σ−μ−1/σx2q1−ξ2q1.Step 2.
We denoteς2=x2σ/v2−ξ2σ/v2 and define the positive Lyapunov function V2 on Π2 as V2=V1+Ψ2 with(14)Ψ2=∫ξ2x2rσ/v2−ξ2σ/v24σ−μ−1/σdr.
Since∂ξ2σ/v2/∂x1=−∂λ1σ/v2x1/x1σ−σλ1σ/v2x1x1σ−1 is valid, one can obtain(15)∂Ψ2∂x2=ς24σ−μ−1/σ,∂Ψ2∂x1=−4σ−μ−v2σ∂ξ2σ/v2∂x1∫ξ2x2rσ/v2−ξ2σ/v23σ−μ−v2/σdr,∂2Ψ2∂x1∂x2=4σ−μ−v2σς23σ−μ−v2/σ∂ξ2σ/v2∂x1,∂2Ψ2∂x22=4σ−μ−v2v2x2σ−v2/v2ς23σ−μ−v2/σ,∂2Ψ2∂x12=−4σ−μ−v2σ∂2ξ2σ/v2∂x12∫ξ2x2rσ/v2−ξ2σ/v23σ−μ−v2/σdr+4σ−μ−v2σ×3σ−μ−v2σ∂ξ2σ/v2∂x12∫ξ2x2rσ/v2−ξ2σ/v22σ−μ−v2/σdr.
Using Definition1 again, we have(16)ℓV2≤−12Gx1ζ14−n−12ζ14+ϑ1tGx1x14σ−μ−1x2q1−ξ2q1+∂Ψ2∂x1ϑ1tx2q1+f1x1+∂Ψ2∂x1f2x¯2+12trg¯2T∂2Ψ2∂x¯22g¯2+∂Ψ2∂x2ϑ2tξ3q2+∂Ψ2∂x2ϑ2tx3q2−ξ3q2,where ξ3 is the virtual controller required to be designed later.
In the following, each term in the right hand of (16) will be estimated by its upper bound.
Firstly, applying0<v2/σ≤1,0<1+μ/σ≤1 and Lemma 5, it is easily obtained that(17)x2−ξ2≤x2σ/υ2−ξ2σ/υ2υ2/σ×21−υ2/σ≤2ζ2υ2/σ,(18)x2q1≤ζ2+ξ2σ/υ21+μ/σ≤ζ21+μ/σ+λ11+μ/υ2ζ21+μ/σ,(19)x2q1−ξ2q1≤21−1+μ/σx2σ/υ2−ξ2σ/υ21+μ/σ≤2ζ21+μ/σ.
It can be deduced from (19) and Lemma 3 that(20)ϑ1tGx1x14σ−μ−1x2q1−ξ2q1≤2ϑ¯1Gx1ζ14σ−μ−1/σζ2μ+1/σ≤18ζ14+H21x¯2ζ24,where H21x¯2≥μ+1/4σ2ϑ¯1Gx14σ/μ+14σ/84σ−μ−1−4σ−μ−1/1+μ≥0 is a C2 function.
Secondly, one can obtain from Assumption1 and Lemma 4 that(21)f2x¯2≤ψ˜2x¯2x1υ2+μ/υ1+x2υ2+μ/υ2≤ψ˜2x¯2ζ1υ2+μ/σ+λ1υ2+μ/υ2ζ1υ2+μ/σ+ζ2υ2+μ/σ≤ψ˜2˜x¯2ζ1υ2+μ/σ+ζ2υ2+μ/σ,(22)g2x¯2≤η2x¯2x12υ2+μ/2υ1+x22υ2+μ/2υ2≤η2x¯2ζ12υ2+μ/2σ+λ12υ2+μ/2υ2ζ12υ2+μ/2σ+ζ22υ2+μ/2σ≤η˜2x¯2ζ12υ2+μ/2σ+ζ22υ2+μ/2σ,where ψ˜2x¯2≥ψ2x¯21+λ1v2+μ/v2 and η˜2x¯2≥η2x¯21+λ12v2+μ/2v2 are nonnegative smooth functions.
Additionally, note thatσ/v2≥2. Then, as stated in [36], there exist C2 functions M21x¯2≥0,K21x¯2≥0, and M˜21x¯2≥0, such that(23)∂ξ2σ/υ2∂x12≤∂λ1σ/υ2∂x1x1σ+σλ1σ/υ2x1σ−1≤M21x¯2ζ11−1/σ,(24)∂2ξ2σ/υ2∂x12≤∂2λ1σ/υ2∂x12x1σ+2σ∂λ1σ/υ2∂x1x1σ−1+σσ−1λ1σ/υ2x1σ−2≤Κ21x¯2ζ11−2/σ,(25)∂Ψ2∂x1≤4σ−μ−υ2σ∂ξ2σ/υ2∂x1x2σ/υ2−ξ2σ/υ23σ−μ−υ2/σx2−ξ2≤4σ−μ−υ2σ×M21x¯2ζ11−1/σζ23σ−μ−υ2/σ×2ζ2υ2/σ≤M˜21x¯2ζ11−1/σζ23σ−μ/σ,(26)12∂2Ψ2∂x12≤4σ−μ−υ22σ∂2ξ2σ/υ2∂x12x2σ/υ2−ξ2σ/υ23σ−μ−υ2/σx2−ξ2+4σ−μ−υ23σ−μ−υ22σ2∂ξ2σ/υ2∂x1x2σ/υ2−ξ2σ/υ22σ−μ−υ2/σx2−ξ2≤4σ−μ−υ2σK21x¯2ζ11−2/σζ23σ−μ/σ+4σ−μ−υ23σ−μ−υ2σ2M212x¯2ζ12−2/σζ22σ−μ/σ.
Then, using (18), (21), (25), Assumptions 1-2, and Lemma 3, we can infer(27)∂Ψ2∂x1ϑ1tx2q1+f1x1≤M˜21x¯2ϑ¯1ζ11−1/σζ23σ−μ/σζ21−μ/σ+λ11+μ/σ+M˜21x¯2ζ11−1/σζ23σ−μ/σψ˜1ζ11−1/σ≤18ζ14+Η22x¯2ζ24,where(28)H22x¯2≥3σ+14σϑ¯1M˜214σ/3σ+14σ32σ−1−σ−1/3σ+1+3σ−μ4σϑ¯1M˜21λ11+σ/v24σ/3σ−μ4σ32σ+μ−σ+μ/3σ−μ+3σ−μ4σψ1M˜214σ/3σ−μ4σ16σ+μ−σ+μ/3σ−μ,is a nonnegative C2 function.
Moreover, from (19), Assumption 2, and Lemma 3, it can be deduced that(29)∂Ψ2∂x2f2x¯2≤ζ24σ−μ−υ2/σψ˜2ζ1υ2+μ/σ+ζ2υ2+μ/σ≤18ζ14+H23x¯2ζ24,where H23x¯2≥ψ˜2+4σ−μ−v2/4σψ˜24σ/4σ−μ−v24σ/8μ+v2−μ+v2/4σ−μ−v2≥0 is a C2 function.
On the contrary, it is noted that(30)12trg¯2T∂2Ψ2∂x¯22g¯2=12∂2Ψ2∂x12g12+g1T∂2Ψ2∂x1∂x2g2+12∂2Ψ2∂x22g22.
Then, applying (22), (26), Assumption 2, and Lemma 3, there clearly exist nonnegative C2 functions H241x¯2,H242x¯2, and H243x¯2 such that(31)12∂2Ψ2∂x12g12≤4σ−μ−υ2σK21ζ11−2/σζ23σ−μ/ση12ζ12+μ/σ+4σ−μ−υ23σ−μ−υ22σ2M212ζ12−2/σζ22σ−μ/ση12ζ12+μ/σ≤124ζ14+H241x¯2ζ24,(32)g1T∂2ψ2∂x1∂x2g2≤η1ζ1μ+2/2σ4σ−μ−v2σζ23σ−μ−v2/σ×M21ζ1σ−1/ση˜2ζ12v2+μ/2σ+ζ22v2+μ/2σ≤124ζ14+H242x¯2ζ24,(33)12∂2ψ2∂x22g22≤4σ−μ−v22v2ζ23σ−μ−v2/σζ2σ−v2/σ+λ1σ−v2/v2ζ1σ−v2/σ×2η˜22ζ12v2+μ/σ+ζ22v2+μ/σ≤124ζ14+H243x¯2ζ24.
Substituting equations (31)–(33) into (30), one obtains(34)12trg¯2T∂2Ψ2∂x¯22g¯2≤18ζ14+H24x¯2ζ24,where H24x¯2=H241x¯2+H242x¯2+H243x¯2≥0 is a C2 function.
LetH2x¯2=H21x¯2+H22x¯2+H23x¯2+H24x¯2≥0. Design the virtual controller ξ3 as(35)ξ3=−λ2x¯2ζ2v3/σwithλ2x¯2=n−1+H2x¯2ϑ¯21/q2>0.
Substituting (20), (27), (29), (34), and (35) into (16), one can obtain(36)ℓV2≤−12Gx1ζ14−n−1ζ14+ζ24+ϑ2tζ24σ−μ−v2/σx3q2−ξ3q2.
Inductive Step. Suppose at step i−1, there exist a C2 Lyapunov function Vi−1:Πi−1⟶R+, and a range of continuous virtual controllers ξ1,ξ2,…,ξi defined as(37)ξ1=0,ζ1=x1σ/v1−ξ1σ/v1,ξ2=−λx1ζ1v2/σ,ζ2=x2σ/v2−ξ2σ/v2,ξ3=−λ2x¯2ζ2v3/σ,ζ3=x3σ/v3−ξ3σ/v3,⋮ξi=−λi−1x¯i−1ζi−1vi/σ,ζi=xiσ/vi−ξiσ/vi,with λkx¯k>0,fork=1,…,i−1, such that(38)ℓVi−1≤−12Gx1ζ14−n+2−i∑l=1i−1ζi4+ϑi−1tζi−14σ−μ−vi/σxiqi−1−ξiqi−1.
Then, the following property can be inferred.Proposition 1.
Choose theith Lyapunov function Vi:Πi⟶R+ as Vi=Vi−1+Ψi with(39)Ψi=∫ξixirσ/vi−ξiσ/vi4σ−μ−vi/σdr.Then,Vi is C2 on Πi and there exists a virtual controller ξi+1 such that(40)ℓVi≤−12Gx1ζ14−n+1−i∑l=1iζl4+ϑi−1tζi4σ−μ−vi/σxi+1qi−ξi+1qi,where(41)ξi+1=−λix¯iζivi+1/σwithλix¯i=n−i+1+Hix¯iϑ¯i1/qi>0.The proof of above Proposition1 is provided in the Appendix.Step 3.
In light of the inductive step, wheni=n and xn+1=u, Proposition 1 holds. Thus, we choose the overall Lyapunov function Vn as Vn=Vn−1+Ψn with(42)Ψn=∫ξnxnrσ/vn−ξnσ/vn4σ−μ−vn/σdr,and define the virtual controller ξn+1 as(43)ξn+1=−λnxςnvn+1/σwithλnx=1+Hnxϑ¯n1/qn>0.
Then,Vnx is clearly a C2 function on Πn, and it is easy to obtain that(44)ℓVn≤−12Gx1ζl4−∑l=1nζl4+ϑntζn4σ−μ−vn/σuqn−ξn+1qn.
Therefore, we can design(45)u=ξn+1x=−λnxxnσ/vn+λn−1σ/vnxn−1σ/vn−1+λn−1σ/vnλn−2σ/vn−2xn−2σ/vn−2+⋯+λn−1σ/vnλn−2σ/vn−1⋯λ1σ/v2x1σ/v1vn+μ/σqn,which results in(46)ℓVn≤−12Gx1ζ14−∑l=1nζl4≤0.


### 3.2.2. Part II: Verification of Keeping the Output Constraint

For anyx0=x10T∈Πn, by Ito’s formula and (46), we can deduce(47)0≤EVnxt=Vnx0+E∫0tℓVnxτdr≤Vnx0<∞.FromVnx=V1x1+∑l=2nΨl>0 and (47), it can be further verified that(48)0≤EV1x1t≤Vnx0<∞,which indicates(49)PV1x1t<∞=1.Hence,Pyt<b=Px1t<b=1. Then, Part I of Theorem 1 is proved.


### 3.2.3. Part III: Stability Analysis

Based on the definition ofVn, we easily obtain the fact that Vn is radially unbounded. Combining the fact with (40) and Lemma 1 directly infers that system (1) has a solution xt,x0 for any x0∈Πn.On the contrary, by a simple calculation, one obtains(50)Vn=V1+∑l=2nΨl=V1+∑l=2n∫ξlxlrσvl−ξlσvl4σ−μ−vl/σdr≤2b4σ−μ4σ−μπtanπx14σ−μ2b4σ−μ+2∑l=2nζl4σ−μ/σ.In addition, since4σ−μ>1, it is easy to get that 0≤πx14σ−μ/2b4σ−μ<π/2 for all x1∈Π1. Then, applying the characteristics of tangent functions, it is not difficult to infer that(51)tanπx14σ−μ2b4σ−μ≤π2b4σ−μx14σ−μGx1≤π4σ−μ2b4σ−μx14σ−μGx1.Now, letc=2μ−8σ/4σ−μ4>0 and 0<γ=4σ/4σ−μ<1. According to (44), (45), and Lemma 4, we obtain(52)cVnγ≤2μ−8σ/4σ−μ2b4σ−μ4σ−μπtanπx14σ−μ2b4σ−μ+2∑l=1nς14σ−μσ4σ/4σ−μ≤2μ−8σ/4σ−μGx14σ/4σ−μς14+12∑l=1nζ14≤12Gx1ζ14+12∑l=1nς14.So, one further obtains(53)ℓVn+cVnγ=ℓVn+2μ−8σ/4σ−μVn4σ/4σ−μ≤−12Gx1ζ14−∑l=1nς14+12Gx1ζ14+12∑l=1nζ14≤−12∑l=1nζ14≤0.In other words,ℓVn≤−cVnγ. Consequently, it is directly deduced from Lemma 2 that the origin of system (1) under controller (39) is finite-time stable in probability.Remark 3.
Comparing with the existing results in most of the literatures, the paper mainly focuses on the finite-time stabilization, instead of the boundness of tracking error. Moreover, the proposed approach can be extend to the tracking control by introducing a coordinate transformation before constructing the BLF.


## 3.2.1. Part I: Design Procedure

Step 1.
Letς1=x1σ, and choose the Lyapunov function V1x1=Vbx1. Then, we can directly get from Definition 1 that(11)ℓV1≤ϑ1tGx1x14σ−μ−1x2q1+4σ−μ−12Gx1x14σ−μ−2η12x12+μ+ϑ¯1Gx1x14σ−μ−1ψ1x11+μ+π4σ−μ2b4σ−μx124σ−μ−1Gx1tanπx14σ−μ2b4σ−μη2x12+μ≤ϑ1tGx1ς14σ−μ−1/σξ2q1+ϑ1tGx1ς14σ−μ−1/σx2q1−ξ2q1+H1x1Gx1ς14,where H1x1≥ϑ¯1ψ1+1/2η124σ−μ−1+π4σ−μη12/2b4σ−μtanπx14σ−μ/2b4σ−μx14σ−μ is a nonnegative C2 function and ξ2 is a virtual controller required to be design after later.
Then, we design(12)ξ2=−λ1x1ς1v2/σwithλ1x1=n+H1x1ϑ¯11/q1>0.
Substituting (12) into (13) yields(13)ℓV1x1≤−nGx1ς14+ϑ1tGx1ς14σ−μ−1/σx2q1−ξ2q1=−12Gx1ς14−n−12Gx1ς14+ϑ1tGx1ς14σ−μ−1/σx2q1−ξ2q1≤−12Gx1ς14−n−12ς14+ϑ1tGx1ς14σ−μ−1/σx2q1−ξ2q1.Step 2.
We denoteς2=x2σ/v2−ξ2σ/v2 and define the positive Lyapunov function V2 on Π2 as V2=V1+Ψ2 with(14)Ψ2=∫ξ2x2rσ/v2−ξ2σ/v24σ−μ−1/σdr.
Since∂ξ2σ/v2/∂x1=−∂λ1σ/v2x1/x1σ−σλ1σ/v2x1x1σ−1 is valid, one can obtain(15)∂Ψ2∂x2=ς24σ−μ−1/σ,∂Ψ2∂x1=−4σ−μ−v2σ∂ξ2σ/v2∂x1∫ξ2x2rσ/v2−ξ2σ/v23σ−μ−v2/σdr,∂2Ψ2∂x1∂x2=4σ−μ−v2σς23σ−μ−v2/σ∂ξ2σ/v2∂x1,∂2Ψ2∂x22=4σ−μ−v2v2x2σ−v2/v2ς23σ−μ−v2/σ,∂2Ψ2∂x12=−4σ−μ−v2σ∂2ξ2σ/v2∂x12∫ξ2x2rσ/v2−ξ2σ/v23σ−μ−v2/σdr+4σ−μ−v2σ×3σ−μ−v2σ∂ξ2σ/v2∂x12∫ξ2x2rσ/v2−ξ2σ/v22σ−μ−v2/σdr.
Using Definition1 again, we have(16)ℓV2≤−12Gx1ζ14−n−12ζ14+ϑ1tGx1x14σ−μ−1x2q1−ξ2q1+∂Ψ2∂x1ϑ1tx2q1+f1x1+∂Ψ2∂x1f2x¯2+12trg¯2T∂2Ψ2∂x¯22g¯2+∂Ψ2∂x2ϑ2tξ3q2+∂Ψ2∂x2ϑ2tx3q2−ξ3q2,where ξ3 is the virtual controller required to be designed later.
In the following, each term in the right hand of (16) will be estimated by its upper bound.
Firstly, applying0<v2/σ≤1,0<1+μ/σ≤1 and Lemma 5, it is easily obtained that(17)x2−ξ2≤x2σ/υ2−ξ2σ/υ2υ2/σ×21−υ2/σ≤2ζ2υ2/σ,(18)x2q1≤ζ2+ξ2σ/υ21+μ/σ≤ζ21+μ/σ+λ11+μ/υ2ζ21+μ/σ,(19)x2q1−ξ2q1≤21−1+μ/σx2σ/υ2−ξ2σ/υ21+μ/σ≤2ζ21+μ/σ.
It can be deduced from (19) and Lemma 3 that(20)ϑ1tGx1x14σ−μ−1x2q1−ξ2q1≤2ϑ¯1Gx1ζ14σ−μ−1/σζ2μ+1/σ≤18ζ14+H21x¯2ζ24,where H21x¯2≥μ+1/4σ2ϑ¯1Gx14σ/μ+14σ/84σ−μ−1−4σ−μ−1/1+μ≥0 is a C2 function.
Secondly, one can obtain from Assumption1 and Lemma 4 that(21)f2x¯2≤ψ˜2x¯2x1υ2+μ/υ1+x2υ2+μ/υ2≤ψ˜2x¯2ζ1υ2+μ/σ+λ1υ2+μ/υ2ζ1υ2+μ/σ+ζ2υ2+μ/σ≤ψ˜2˜x¯2ζ1υ2+μ/σ+ζ2υ2+μ/σ,(22)g2x¯2≤η2x¯2x12υ2+μ/2υ1+x22υ2+μ/2υ2≤η2x¯2ζ12υ2+μ/2σ+λ12υ2+μ/2υ2ζ12υ2+μ/2σ+ζ22υ2+μ/2σ≤η˜2x¯2ζ12υ2+μ/2σ+ζ22υ2+μ/2σ,where ψ˜2x¯2≥ψ2x¯21+λ1v2+μ/v2 and η˜2x¯2≥η2x¯21+λ12v2+μ/2v2 are nonnegative smooth functions.
Additionally, note thatσ/v2≥2. Then, as stated in [36], there exist C2 functions M21x¯2≥0,K21x¯2≥0, and M˜21x¯2≥0, such that(23)∂ξ2σ/υ2∂x12≤∂λ1σ/υ2∂x1x1σ+σλ1σ/υ2x1σ−1≤M21x¯2ζ11−1/σ,(24)∂2ξ2σ/υ2∂x12≤∂2λ1σ/υ2∂x12x1σ+2σ∂λ1σ/υ2∂x1x1σ−1+σσ−1λ1σ/υ2x1σ−2≤Κ21x¯2ζ11−2/σ,(25)∂Ψ2∂x1≤4σ−μ−υ2σ∂ξ2σ/υ2∂x1x2σ/υ2−ξ2σ/υ23σ−μ−υ2/σx2−ξ2≤4σ−μ−υ2σ×M21x¯2ζ11−1/σζ23σ−μ−υ2/σ×2ζ2υ2/σ≤M˜21x¯2ζ11−1/σζ23σ−μ/σ,(26)12∂2Ψ2∂x12≤4σ−μ−υ22σ∂2ξ2σ/υ2∂x12x2σ/υ2−ξ2σ/υ23σ−μ−υ2/σx2−ξ2+4σ−μ−υ23σ−μ−υ22σ2∂ξ2σ/υ2∂x1x2σ/υ2−ξ2σ/υ22σ−μ−υ2/σx2−ξ2≤4σ−μ−υ2σK21x¯2ζ11−2/σζ23σ−μ/σ+4σ−μ−υ23σ−μ−υ2σ2M212x¯2ζ12−2/σζ22σ−μ/σ.
Then, using (18), (21), (25), Assumptions 1-2, and Lemma 3, we can infer(27)∂Ψ2∂x1ϑ1tx2q1+f1x1≤M˜21x¯2ϑ¯1ζ11−1/σζ23σ−μ/σζ21−μ/σ+λ11+μ/σ+M˜21x¯2ζ11−1/σζ23σ−μ/σψ˜1ζ11−1/σ≤18ζ14+Η22x¯2ζ24,where(28)H22x¯2≥3σ+14σϑ¯1M˜214σ/3σ+14σ32σ−1−σ−1/3σ+1+3σ−μ4σϑ¯1M˜21λ11+σ/v24σ/3σ−μ4σ32σ+μ−σ+μ/3σ−μ+3σ−μ4σψ1M˜214σ/3σ−μ4σ16σ+μ−σ+μ/3σ−μ,is a nonnegative C2 function.
Moreover, from (19), Assumption 2, and Lemma 3, it can be deduced that(29)∂Ψ2∂x2f2x¯2≤ζ24σ−μ−υ2/σψ˜2ζ1υ2+μ/σ+ζ2υ2+μ/σ≤18ζ14+H23x¯2ζ24,where H23x¯2≥ψ˜2+4σ−μ−v2/4σψ˜24σ/4σ−μ−v24σ/8μ+v2−μ+v2/4σ−μ−v2≥0 is a C2 function.
On the contrary, it is noted that(30)12trg¯2T∂2Ψ2∂x¯22g¯2=12∂2Ψ2∂x12g12+g1T∂2Ψ2∂x1∂x2g2+12∂2Ψ2∂x22g22.
Then, applying (22), (26), Assumption 2, and Lemma 3, there clearly exist nonnegative C2 functions H241x¯2,H242x¯2, and H243x¯2 such that(31)12∂2Ψ2∂x12g12≤4σ−μ−υ2σK21ζ11−2/σζ23σ−μ/ση12ζ12+μ/σ+4σ−μ−υ23σ−μ−υ22σ2M212ζ12−2/σζ22σ−μ/ση12ζ12+μ/σ≤124ζ14+H241x¯2ζ24,(32)g1T∂2ψ2∂x1∂x2g2≤η1ζ1μ+2/2σ4σ−μ−v2σζ23σ−μ−v2/σ×M21ζ1σ−1/ση˜2ζ12v2+μ/2σ+ζ22v2+μ/2σ≤124ζ14+H242x¯2ζ24,(33)12∂2ψ2∂x22g22≤4σ−μ−v22v2ζ23σ−μ−v2/σζ2σ−v2/σ+λ1σ−v2/v2ζ1σ−v2/σ×2η˜22ζ12v2+μ/σ+ζ22v2+μ/σ≤124ζ14+H243x¯2ζ24.
Substituting equations (31)–(33) into (30), one obtains(34)12trg¯2T∂2Ψ2∂x¯22g¯2≤18ζ14+H24x¯2ζ24,where H24x¯2=H241x¯2+H242x¯2+H243x¯2≥0 is a C2 function.
LetH2x¯2=H21x¯2+H22x¯2+H23x¯2+H24x¯2≥0. Design the virtual controller ξ3 as(35)ξ3=−λ2x¯2ζ2v3/σwithλ2x¯2=n−1+H2x¯2ϑ¯21/q2>0.
Substituting (20), (27), (29), (34), and (35) into (16), one can obtain(36)ℓV2≤−12Gx1ζ14−n−1ζ14+ζ24+ϑ2tζ24σ−μ−v2/σx3q2−ξ3q2.
Inductive Step. Suppose at step i−1, there exist a C2 Lyapunov function Vi−1:Πi−1⟶R+, and a range of continuous virtual controllers ξ1,ξ2,…,ξi defined as(37)ξ1=0,ζ1=x1σ/v1−ξ1σ/v1,ξ2=−λx1ζ1v2/σ,ζ2=x2σ/v2−ξ2σ/v2,ξ3=−λ2x¯2ζ2v3/σ,ζ3=x3σ/v3−ξ3σ/v3,⋮ξi=−λi−1x¯i−1ζi−1vi/σ,ζi=xiσ/vi−ξiσ/vi,with λkx¯k>0,fork=1,…,i−1, such that(38)ℓVi−1≤−12Gx1ζ14−n+2−i∑l=1i−1ζi4+ϑi−1tζi−14σ−μ−vi/σxiqi−1−ξiqi−1.
Then, the following property can be inferred.Proposition 1.
Choose theith Lyapunov function Vi:Πi⟶R+ as Vi=Vi−1+Ψi with(39)Ψi=∫ξixirσ/vi−ξiσ/vi4σ−μ−vi/σdr.Then,Vi is C2 on Πi and there exists a virtual controller ξi+1 such that(40)ℓVi≤−12Gx1ζ14−n+1−i∑l=1iζl4+ϑi−1tζi4σ−μ−vi/σxi+1qi−ξi+1qi,where(41)ξi+1=−λix¯iζivi+1/σwithλix¯i=n−i+1+Hix¯iϑ¯i1/qi>0.The proof of above Proposition1 is provided in the Appendix.Step 3.
In light of the inductive step, wheni=n and xn+1=u, Proposition 1 holds. Thus, we choose the overall Lyapunov function Vn as Vn=Vn−1+Ψn with(42)Ψn=∫ξnxnrσ/vn−ξnσ/vn4σ−μ−vn/σdr,and define the virtual controller ξn+1 as(43)ξn+1=−λnxςnvn+1/σwithλnx=1+Hnxϑ¯n1/qn>0.
Then,Vnx is clearly a C2 function on Πn, and it is easy to obtain that(44)ℓVn≤−12Gx1ζl4−∑l=1nζl4+ϑntζn4σ−μ−vn/σuqn−ξn+1qn.
Therefore, we can design(45)u=ξn+1x=−λnxxnσ/vn+λn−1σ/vnxn−1σ/vn−1+λn−1σ/vnλn−2σ/vn−2xn−2σ/vn−2+⋯+λn−1σ/vnλn−2σ/vn−1⋯λ1σ/v2x1σ/v1vn+μ/σqn,which results in(46)ℓVn≤−12Gx1ζ14−∑l=1nζl4≤0.


## 3.2.2. Part II: Verification of Keeping the Output Constraint

For anyx0=x10T∈Πn, by Ito’s formula and (46), we can deduce(47)0≤EVnxt=Vnx0+E∫0tℓVnxτdr≤Vnx0<∞.FromVnx=V1x1+∑l=2nΨl>0 and (47), it can be further verified that(48)0≤EV1x1t≤Vnx0<∞,which indicates(49)PV1x1t<∞=1.Hence,Pyt<b=Px1t<b=1. Then, Part I of Theorem 1 is proved.


## 3.2.3. Part III: Stability Analysis

Based on the definition ofVn, we easily obtain the fact that Vn is radially unbounded. Combining the fact with (40) and Lemma 1 directly infers that system (1) has a solution xt,x0 for any x0∈Πn.On the contrary, by a simple calculation, one obtains(50)Vn=V1+∑l=2nΨl=V1+∑l=2n∫ξlxlrσvl−ξlσvl4σ−μ−vl/σdr≤2b4σ−μ4σ−μπtanπx14σ−μ2b4σ−μ+2∑l=2nζl4σ−μ/σ.In addition, since4σ−μ>1, it is easy to get that 0≤πx14σ−μ/2b4σ−μ<π/2 for all x1∈Π1. Then, applying the characteristics of tangent functions, it is not difficult to infer that(51)tanπx14σ−μ2b4σ−μ≤π2b4σ−μx14σ−μGx1≤π4σ−μ2b4σ−μx14σ−μGx1.Now, letc=2μ−8σ/4σ−μ4>0 and 0<γ=4σ/4σ−μ<1. According to (44), (45), and Lemma 4, we obtain(52)cVnγ≤2μ−8σ/4σ−μ2b4σ−μ4σ−μπtanπx14σ−μ2b4σ−μ+2∑l=1nς14σ−μσ4σ/4σ−μ≤2μ−8σ/4σ−μGx14σ/4σ−μς14+12∑l=1nζ14≤12Gx1ζ14+12∑l=1nς14.So, one further obtains(53)ℓVn+cVnγ=ℓVn+2μ−8σ/4σ−μVn4σ/4σ−μ≤−12Gx1ζ14−∑l=1nς14+12Gx1ζ14+12∑l=1nζ14≤−12∑l=1nζ14≤0.In other words,ℓVn≤−cVnγ. Consequently, it is directly deduced from Lemma 2 that the origin of system (1) under controller (39) is finite-time stable in probability.Remark 3.
Comparing with the existing results in most of the literatures, the paper mainly focuses on the finite-time stabilization, instead of the boundness of tracking error. Moreover, the proposed approach can be extend to the tracking control by introducing a coordinate transformation before constructing the BLF.


## 4. Simulation

In this section, we will provide the simulation results of the following example to illustrate the validity of the proposed strategy:(54)dx1=x27/5dt,dx2=udt+12x12x21/5dt+18sinx22x23/5dω,y=x1,where q1=7/5, q2=1, and μ=−4/11∈−5/12,0. Then, v1=1, v2=5/11, and v3=1/11. Since ϑ1t=ϑ2t=1, Assumption 1 is satisfied. And Assumption 2 is also satisfied with ψ1=η1=0, ψ2=1/2x12, and η2=1/8sinx22. Furthermore, note that c1=1,c2=11/5>2,andσ0=1. Thus, one can select σ=σ0=1.Now, letς1=x1 and Gx1=sec2πx148/11/2b48/11. In view of design procedure in Part I of the proof, we can design the virtual controller ξ2 as(55)ξ2=−25/7ζ15/11≔−λ1ζ15/11.Next, according to the design procedure, we denoteς2=x211/5−ξ211/5 and further obtain that(56)H21=7442Gx144/744296−37/7,H22=8611λ111/5+37448611λ118/544/374456−7/37,H23=12x12λ11/7+434412x12λ11/744/43448−1/43,H24=37320sinx24+324437320sinx24λ112/544/324496−12/32.LetH2=H21+H22+H23+H24 and λ2=1+H2; then, the controller u can be designed as(57)u=−λ2ζ21/11=−λ2x211/5−ξ211/51/11.Finally, we supposeb=1.5 and select some different initial states x0’s with each x0 satisfying x0∈Π2. The simulation results of system (48) are shown in Figures 1 and 2. Figure 1 curves trajectories of x1t under different initial values, which illustrate that the output constraint is always not violated. Meanwhile, the trajectories of x2t is given in Figure 2. It can be observed from the two figures that system (48) under controller (51) is finite-time stable.Figure 1
Trajectories ofx1t.Figure 2
Trajectories ofx2t.


## 5. Conclusion

In this paper, the stability issue is addressed for a class ofp-norm stochastic systems with output constraints and unknown time-varying parameters. Using a tan-type BLF, the finite-time control strategy is proposed by the adding a power integrator technique. On this basis, the designed controller has been proved to ensure that the origin of the closed-loop system is finite-time stable in probability and the system output is kept in a pre-given set. This conclusion has also been verified by the simulation results. It should be pointed out that the proposed approach is not applicable to the case of asymmetrical output constraints. In the future, we will try to modify the proposed method to be suitable for stochastic systems with asymmetrical output constraints or multi-input multi-output stochastic systems.



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*Source: 2486210-2020-08-12.xml*