# State-Feedback Stabilization of Stochastic Nonholonomic Systems with an Unknown Time-Varying Delay

Authors: Qinghui Du; Qianqian Cui; Dongkai Zhang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper investigates the state-feedback stabilization of stochastic nonholonomic systems with an unknown time-varying delay. Without imposing any assumptions on the time-varying delay, a state-feedback controller is skillfully designed by using input-state-scaling technique and backstepping control approach. The switching strategy is proposed to eliminate the phenomenon of uncontrollability and to guarantee that the closed-loop system has an almost surely unique solution for any initial state and the equilibrium of interest is globally asymptotically stable in probability. A simulation example demonstrates the effectiveness of the proposed scheme.


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## Body

## 1. Introduction

Nonholonomic systems represent an important class of control systems, which arises in many mechanical systems such as mobile robots, car-like vehicles, underactuated satellites, and the knife-edge. It is well known that the control of nonholonomic systems is extremely challenging, largely due to the limitation imposed by Brockett’s stability condition [1]. This class of nonlinear systems cannot be stabilized to a point by smooth or even continuous pure state feedback, which makes the well-developed smooth and continuous control methods cannot be directly applied to these systems. This motivates researchers to seek for effective control strategies such as discontinuous feedback, time-varying feedback, and hybrid control laws.Since the existences of stochastic disturbances in many practical systems may change deterministic systems into stochastic systems, the control and design of stochastic systems has become an important role in the field of engineering. In [2], the backstepping design was firstly introduced, which makes the stochastic nonlinear control to obtain a breakthrough. Then, based on the backstepping technique and quartic Lyapunov functions, many results have been obtained for stochastic nonlinear systems with different structures [3–8]. Specially, by using the stochastic Lyapunov-like theorem and backstepping design technique, the state or output-feedback stabilization for stochastic nonholonomic systems was obtained in [9–14]. Zhang et al. [11] studied the problem of adaptive stabilization of stochastic nonholonomic systems with nonhomogeneous uncertainties. Zhao et al. [12] designed a state-feedback controller to stabilize a class of more general high-order stochastic nonholonomic systems. Du et al. [13, 14] studied the design of controllers for a class of stochastic high-order nonholonomic systems, which were considered to cancel the power order restriction in [12]. However, the aforementioned contributions have not taken into account the effect of time delay on the systems.The research of time delay systems has been received widespread attentions [15–25]. One reason is that the time-delay phenomenon frequently arises and is inevitable in many practical systems, such as chemical engineering systems, communication systems, and mechanical systems. Another reason is that time delay is a primary source of instability and performance and makes practical control systems hard to control. Naturally, the research of this kind of systems plays an important role in control theory and practical applications. Compared with the previous results without time delay, the control problem of a time-delay system is more challenging. One main obstacle that exists in the stabilization procedure is the restrictive growth conditions imposed on delay-dependent nonlinear functions, for example, Liu et al. [16], Chen et al. [17], and Liu and Wu [18]. Wu and Wu [19] proposed a robust state-feedback switching controller to stabilize time-delay nonholonomic systems with strongly nonlinear uncertainties by using discontinuous transformation and dynamic feedback approach. Subsequently, Wu and Liu [20] addressed the output-feedback stabilization problem for time-delay nonholonomic systems whose time-delay exists in polynomial nonlinear growing conditions, and they considered the output-feedback control problem for high-order nonholonomic time-delay systems [21]. However, all the above references considered the time-delay nonholonomic systems in the deterministic case, and up to now, fewer researchers studied this kind of time-delay systems with stochastic disturbance. Compared with the deterministic systems, obviously, stochastic time-delay systems contain more uncertainties in drift and diffusion terms than those in the related papers. Qin and Min [22] studied the adaptive stabilization problem for a class of stochastic nonholonomic systems with time delays. This work gave the adaptive state-feedback control at the expense of delay-dependent nonlinear diffusion terms and time-delay assumption, i.e., di≤δ for a known positive constant δ. Wu et al. [23] studied state-feedback stabilization of stochastic nonholonomic systems under arbitrary switchings with time-varying delays. But the time-delay terms of stochastic nonholonomic systems is required to satisfy d˙t≤η<1. Hence, it is necessary to introduce stochastic nonholonomic systems with an unknown time-varying delay and explore how to remove the restrictions assumed on time-varying delay and to weaken the growth assumptions imposed on fi and gi.Recently, Min et al. [24] have studied the problem of globally adaptive control for stochastic nonlinear time-delay systems with perturbations and its application. Without imposing any assumptions on the time-varying delay, an adaptive state-feedback controller was skillfully designed by using adaptive backstepping control technique, and it was proven that the constructed controller can guarantee the closed-loop system to be globally asymptotically stable in probability. So, how to extend this method to stochastic nonholonomic systems with an unknown time-varying delay is very interesting and significant.In the following, we will make efforts to solve this issue. The main contributions are summarized as follows:(i)
Stochastic nonholonomic systems with an unknown time-varying delay are globally stabilized for the first time. To compensate for the unknown time-varying delay, the Lyapunov–Razumikhin lemma is employed.(ii)
We use subtly input-state-scaling technique and Itô formula to change the stochastic nonholonomic systems into stochastic nonlinear systems that is a more complicated system in the study of Min et al. [24], which is very important. Since Itô stochastic differentiation involves not only the gradient but also the Hessian term in the Lyapunov design procedure of stochastic systems, it will produce much more nonlinear terms than those in the stochastic nonlinear systems in the study of Min et al. [24]. How to deal with them skillfully is one of the main technical obstacles in our paper.(iii)
Compared with the related references, a distinctive feature is that the restrictions assumed on time-varying delay are removed and the growth assumptions imposed onfi and gi are somewhat weakened. Then, a state-feedback controller is skillfully designed which renders the closed-loop system globally asymptotically stable (GAS) in probability.The paper is organized as follows:Section 2 begins with the preliminary results. Section 3 shows the input-state-scaling technique and the backstepping design procedure, while Section 4 provides the switching control strategy and Section 5 provides the main result. Section 6 gives a simulation example to illustrate the theoretical finding of this paper. Finally, Section 7 concludes the paper.


## 2. Preliminary Results

Consider the following stochastic nonlinear time-delay system:(1)dx=fx,xt−dtdt+gx,xt−dtdω,∀x0∈Rn,where x∈Rn is the state, dt:R+⟶0,d is a time-varying delay which is the Borel measurable function, the initial value is defined as xτ:−d≤τ≤0=ξ∈Cℱ0b−d,0;Rn, and ω∈Rr is an r-dimensional independent standard Wiener process defined on the complete probability space Ω,ℱ,P. The Borel measurable functions fandg are locally the Lipschitz functions with f0,0=0 and g0,0=0.For any givenVx∈C2, the differential operator ℒ along (1) is defined as(2)ℒV=∂V∂xf+12TrgT∂2V∂x2g,where 1/2TrgT∂2V/∂x2g is said to be the Hessian term of ℒ.The following definitions and lemmas will be used throughout the paper:Definition 1 [2].
The equilibriumxt=0 of system (1) with f0,0=0 and g0,0=0 is(i)
Globally stable in probability if for anyε>0, there exists a class K function γ⋅ such that Pxt<γx0≥1−ε,∀t≥0,x0∈Rn\0(ii)
Globally asymptotically stable in probability if it is globally stable in probability and(3)Plimt⟶∞xt=0=1,∀x0∈Rn.Lemma 1 [2].
Consider the stochastic system (1); if there exist a C2 function Vx, class Κ∞ functions α1 and α2, constants c1>0,c2≥0, and a nonnegative function Wx such that(4)α1x≤Vx≤α2x,ℒV≤−c1Wx+c2,then(1)
For (1), there exists an almost surely unique solution on 0,∞ for each x0 belongs to Rn(2)
Whenc2=0,f0,0=0,g0,0=0, and Wx is continuous, then the equilibrium x=0 is globally stable in probability and Plimt⟶∞Wxt=0=1Lemma 2 [24].
For system (1), let α1,α2,andr be positive constants and q>1. Assume that there exists a continuous, positive definite Lyapunov function Vx∈C2 and W such that α1xr≤Vx≤α2xr, for all x∈Rn, and moreover, EℒV≤−EW provided ϕ=ϕτ:−d≤τ≤0∈ℒℱtr−d,0;Rn, satisfying EVϕτ<qEVϕ0,−d≤τ≤0. Then, for all initial values ξ∈Cℱ0b−d,0;Rn, the solution x=0 of system (1) is GAS in probability.Lemma 3 [26].
Letx1,…,xn,p>0 be real numbers, then(5)x1+⋯+xnp≤maxnp−1,1x1p+⋯+xnp.Lemma 4 [27].
Letxandy be real variables. For any positive real numbers a and b, the following inequality holds:(6)axmyn≤bxm+n+nm+nm+nm−m/nam+n/nb−m/nym+n.


## 3. State-Feedback Controller Design

Let us consider the following stochastic nonholonomic systems with an unknown time-varying delay:(7a)dx0=u0dt+g0Tx0dω,(7b)dxi=u0xi+1dt+fix0,x¯i,x¯it−dtdt+giTx0,x¯i,x¯it−dtdω,i=1,…,n−1,dxn=udt+fnx0,x,xt−dtdt+gnTx0,x,xt−dtdω,where x0∈R and x=x1,…,xnT∈Rn are the system states, u0 and u are the control inputs; x¯i=x1,…,xiT, x¯it−dt=x1t−dt,…,xit−dtT,x¯i=x1,…,xiT and x¯n=x; and di≥0 is an unknown time-varying delay. ω is an r-dimensional standard Wiener process defined on a probability space Ω,ℱ,P, with Ω being a sample space, ℱ being a filtration, and P being a probability measure; the drift terms fix0,x¯i,x¯it−dt:Ri+1×Ri⟶R and the diffusion terms gix0,x¯i,x¯it−dt:Ri+1×Ri⟶ Rr, g0x0:R⟶Rr are unknown locally Lipschitz functions with fi0,0,0=0,gi0,0,0=0, g00=0, i=1,…,n.The following assumptions are made on systems (7a) and (7b). For simplicity, sometimes we write xit−dt as xid and fit−dt as fid.Assumption 1.
For smooth functiong0⋅, there is a known constant vector m1∈Rr such that(8)g0x0=m1x0.Assumption 2.
For the drift terms and diffusion termsfid and gid in system (7a) and (7b), there exist smooth nonnegative functions γ¯ijx0,x¯j and ξijx0,x¯j such that(9)fix0,x¯i,x¯it−dt≤∑j=1iγ¯ijx0,x¯jxj+xjt−dt,gix0,x¯i,x¯it−dt≤∑j=1iξijx0,x¯jxj+xjt−dt.Remark 1.
There are some differences between the assumptions here and the existing ones. Wheng0x0=0,gix0,x¯i,x¯it−dt=0,i=1,2,…,n, systems (7a) and (7b) become the time-delay nonholonomic systems in the deterministic case, which has been considered in the literature [19–21]. In [20], the constant time-delay was studied for the deterministic system and the time-delay terms are required to satisfy d˙t≤η<1, which were studied in [18, 21]. Compared with the deterministic systems, obviously, systems (7a) and (7b) contain more uncertainties in drift and diffusion terms than those in the related papers. In detail, γ¯ijx0,x¯j,ξijx0,x¯j of Assumption 2 was assumed being positive constants in [18]. Compared with the related references, a distinctive feature is that the restrictions assumed on time-varying delay are removed and the growth assumptions imposed on f and g are somewhat weakened by allowing γ¯ij and ξij being functions related to system states. These differences show the assumptions of this paper are more general.


## 4. State-Feedback Control Design

In this section, we present the design procedure. For clarity, the case thatx0t0≠0 is considered first. Then the case that the initial x0t0=0 is dealt later. The inherent structure of system (7a) suggests that we should design the control inputs u0 and u in two separate stages.Let us consider the subsystem (7a) in stochastic nonholonomic time-delay systems (7a) and (7b). In order to guarantee that x0 converges to zero, one can take u0 as follows:(10)u0=−η0x0,η0=λ+32m1m1TF,where λ is a positive constant.If we take a Lyapunov function of the form(11)Vx0=14x04,from (2), Assumption 1, one can obtain(12)ℒV0=x03u0+32x02Trg0x0g0Tx0=−η0x04+32m1m1TFx04=−λx04≤0.Theorem 1.
If Assumption 1 holds for the stochastic nonholonomic time-delay subsystem (7a), then under the smooth controller (10), one has the following:(1)
The closed-loop system has an almost surely unique solution ont0,∞ for ∀x00(2)
The equilibriumx0=0 of the closed-loop system is globally asymptotically stable in probabilityRemark 2.
Proof of Theorem 1 is similar to Theorem 1 in [13]. Due to the space limitation, we omit the proof details.
Consider the second subsystem of stochastic nonholonomic time-delay system (7b). In order to design a state-feedback controller, the following state-input scaling transformation is needed:(13)zit=xitu0n−i,t≥t0;xit,t0−δ≤t<t0.
Under the newz coordinates, with choice u0 as in (10), the x subsystem is transformed into(14)dzi=zi+1dt+ϕix0,z¯i,z¯iddt+ψiTx0,z¯i,z¯iddω,i=1,…,n−1,dzn=udt+ϕnx0,zi,ziddt+ψnTx0,zi,ziddω,where(15)ϕix0,z¯i,z¯id=fidu0n−i+η0n−izi+12η02n−in−i+1zig0Tg0u02+η0n−igidTg0u0n−i+1,ψiTx0,z¯i,z¯id=gidTu0n−i+η0n−izig0Tu0,g0≔g0x0,fid≔fix0,x¯i,x¯id,gid≔gix0,x¯i,x¯id,1≤i≤n.
For using backstepping technique, we define the error variable:(16)ε1=z1,εi=zi−zi∗x0,z¯i−1,2≤i≤n,where zi∗2≤i≤n are the virtual control laws to be designed.
Then, byIto^’s rule in [28], (14)–(16), one has(17)dεi=zi+1dt+Fix0,z¯i,z¯iddt+GjTx0,z¯i,z¯iddω,i=1,…,n−1,dεn=udt+Fnx0,z¯n,z¯nddt+GnTx0,z¯n,z¯nddω,where z¯i=z1,z2,…,zi,z¯id=z1d,z2d,…,zid,1≤i≤n.(18)Fix0,z¯i,z¯id=fidu0n−i+η0n−izi+12η02n−in−i+1zig0Tg0u02−∂zi∗∂x0u0+η0n−igidTg0u0n−i+1−12∂2zi∗∂x02g0Tg0−∑k=1i−1∂zi∗∂zkzk+1+fkdu0n−k+η0n−kzk+12η02n−kn−k+1zkg0Tg0u02+η0n−kgkdTg0u0n−k+1−12∑k=1i−1∂2zi∗∂x0∂zkgkdTu0n−k+η0n−kzkg0Tu0g0−12∑j,k=1i−1∂2zi∗∂zj∂zkgjdTu0n−j+η0n−jzjg0Tu0gkdTu0n−k+η0n−kzkg0Tu0T,GiTx0,z¯i,z¯id=gidTu0n−i+η0n−izig0Tu0−∂zi∗∂x0g0T−∑k=1i−1∂zi∗∂zkgkdTu0n−k+η0n−kzkg0Tu0.Lemma 5.
For1≤i≤n, there exist positive smooth functions γijkx0,x¯i,i=1,…,n,j=1,…,i,k=1,2,3,4,5 such that(19)fidx0,x¯i,x¯idu0n−i≤∑j=1iγij1x0,ε¯jεj+εjt−dt,gidx0,x¯i,x¯idu0n−i≤∑j=1iγij2x0,ε¯jεj+εjt−dt,gidTu0n−i+η0n−izig0Tu0−∂zi∗∂x0g0T−∑k=1i−1∂zi∗∂zkgkdTu0n−k+η0n−kzkg0Tu0≤∑j=1iγij3x0,ε¯jεj+εjt−dt,−12∑k=1i∂2zi∗∂x0∂zkgkdTu0n−k+η0n−kzkg0Tu0g0≤∑j=1iγij4x0,ε¯jεj+εjt−dt,−12∑j,k=1i∂2zi∗∂zj∂zkgjdTu0n−j+η0n−jzjg0Tu0gkdTu0n−k+η0n−kzkg0Tu0T≤∑j=1iγij5x0,ε¯jεj+εjt−dt.Remark 3.
Proof of the above inequalities is based on (13), (15), and Assumption 2. Due to the space limitation, we omit the proof details.
By the following the standard procedures, a state-feedback controller can be obtained:Step 1.
Consider the Lyapunov function(20)V1x0,z1=14x04+14ε14,from Lemma 5, there exist nonnegative smooth functions γij1x0,x¯i,γij2x0,x¯i,andγij3x0,x¯i, such that(21)f1du0n−1≤γ111x0,ε1ε1+ε1t−dt,g1dTu0n−1≤γ112x0,ε1ε1+ε1t−dt,g1dTu0n−1+η0n−1z1g0Tu0≤γ113x0,ε1ε1+ε1t−dt.
Thus, we have(22)ℒV1≤−λx04+z2ε13+ε12γ111ε1+ε1d+η0n−1+12nn−1m12ε14+m1n−1γ112ε13ε1+ε1d+32ε12γ1132ε1+ε1d2≤−λx04+z2ε13+γ111+m1n−1γ112ε13ε1d+3ε12γ1132ε1d2+γ111+m1n−1γ112+3γ1132+η0n−1+12nn−1m12ε14≤−λx04+ε13z2−z2∗+ε13z2∗+γ111+344b111−1/3γ1114/3+η0n−1+12nn−1m12+n−1m1γ112+344b112−1/3n−1m1γ1124/3+3γ1132+122b113−13γ11322ε14+b111+b112+b113ε1d4≤−λx04+ε13z2−z2∗+ε13z2∗+b111+b112+b113ε1d4+ρ1,1+ρ1,2+ρ1,3+ρ1,4+ρ1,5ε14,where b111,b112,andb113 are positive design constants:(23)ρ1,1=γ111+344b111−1/3γ1114/3,ρ1,2=η0n−1,ρ1,3=12nn−1m12,ρ1,4=n−1m1γ112+344b112−1/3n−1m1γ1124/3,ρ1,5=3γ1132+122b113−13γ11322.
Constructingz2∗x0,z1=−α1x0,z1ε1, α1⋅=c¯¯1+ρ1,1+ρ1,2+ρ1,3+ρ1,4+ρ1,5. Then, adding and subtracting the term c¯¯1ε14 on the right-hand side of (22), one gets(24)ℒV1≤−λx04−c¯¯1ε14+ε13z2−z2∗+b11ε1d4,where c¯¯1>0 is a parameter to be designed, b11=b111+b112+b113.
Inductive step2≤i≤n: assume that, at step i−1, there are C2, proper and positive definite Lyapunov function Vi−1, and the virtual controllers z1∗,…,zi∗ defined by(25)z1∗=0z2∗=−ε1α1x0,z1⋮zi∗=−εi−1αi−1x0,z1,…,zi−1ε1=z1−z1∗ε2=z2−z2∗⋮εi=zi−zi∗,with α1x0,z1>0,…,αi−1x0,z1,…,zi−1>0 being smooth such that(26)ℒVi−1≤−λ−∑j=2i−1βjx04−∑j=1i−1c¯¯j−∑k=j+1i−1c¯¯kjεj4+εi−13zi−zi∗+∑j=1i−1∑l=ji−1bljzjd4,where c¯¯i>0,i=1,…,k−1, are the design parameters. Then, for the Lyapunov function(27)Vi=Vi−1+14εi4,there exists a virtual control law(28)zi+1∗x0,z¯i=−αix0,z¯iεi,αix0,z¯i=c=i+Hi2,such that(29)ℒVi≤−λ−∑j=2iβjx04−∑j=1ic¯¯j−∑k=j+1ic¯¯kjεj4+εi3zi+1−zi+1∗+∑j=1i∑l=jibljzjd4,where αix0,z¯i>0 is the known continuous function and c¯¯k>0 and blj>0 for j=1,…,i,l=j,…,i are the design constants.Proof.
Firstly, according to Definition 1, (26) and (27), one obtains(30)ℒVi≤−λ−∑j=2i−1βjx04−∑j=1i−1c¯¯j−∑k=j+1i−1c¯¯kjεj4+εi−13zi−zi∗+∑j=1i−1∑l=ji−1bljzjd4+εi3zi+1∗+εi3zi+1−zi+1∗+εi3fidu0n−i+η0n−izi+12η02n−in−i+1zig0Tg0u02+η0n−igidTg0u0n−i+1−∂zi∗∂x0u0−12∂2zi∗∂x02g0Tg0−∑k=1i−1∂zi∗∂zkzk+1+fkdu0n−k+η0n−kzk+12η02n−kn−k+1zkg0Tg0u02+η0n−kgkdTg0u0n−k+1−12∑j,k=1i−1∂2zi∗∂zj∂zkgjdTu0n−j+η0n−jzjg0Tu0gkdTu0n−k+η0n−kzkg0Tu0T+32εi2gidTu0n−i+η0n−izig0Tu0−∂zi∗∂x0g0T−∑k=1i−1∂zi∗∂zkgkdTu0n−k+η0n−kzkg0Tu02−12∑k=1i−1∂2zi∗∂x0∂zkgkdTu0n−k+η0n−kzkg0Tu0g0.
To proceed further, we estimate each term on the right-hand side of (30). Using Assumption 2 and Lemma 4, we have(31)εi−13zi−zi∗≤34εi,i−1,1εi−14+14εi,i−1,1−3εi4≤34εi,i−1,1εi−14+ρi,1εi4,(32)εi3fidu0n−i≤14∑k=1i−1εi,k,2εk4+34∑k=1i−1εi,k,2−1/3γi114/3+34∑j=1i−14bij1−1/3γij14/3+γii1εi4+∑j=1ibij1εjd4≤14∑k=1i−1εi,k,2εk4+ρi,2εi4+∑j=1ibij1εjd4,(33)εi3η0n−izi≤14η0n−iεi,i−1,3εi−14+34η0n−i1+εi,i−1,3−1/3αi−14/3εi4≤14η0n−iεi,i−1,3εi−14+ρi,3εi4,(34)12η02n−in−i+1εi3zig0Tg0u02≤18n−in−i+1εi,i−1,4εi−14+12n−in−i+1m12+34εi,i−1,4−1/3m12αi−14/3εi4≤18n−in−i+1εi,i−1,4εi−14+ρi,4εi4,(35)η0n−iεi3gidTg0u0n−i+1≤14n−i∑k=1i−1εi,k,5εk4+n−iγii2m1+34∑k=1i−1εi,k,5−1/3γi2m14/3+34∑k=1i−14bik2−1/3γik2m14/3εi4+∑j=1ibij2εjd4≤14n−i∑k=1i−1εi,k,5εk4+ρi,5εi4+∑j=1ibij2εjd4,(36)−εi3∂zi∗∂x0u0≤14εi,0,6x04+34εi,0,6−1/31+η0∂zi∗∂x024/3εi4≤14εi,0,6x04+ρi,6εi4,(37)−12εi3∂2zi∗∂x02g0Tg0≤18εi,0,7x04+38εi,0,7−1/31+x0m12∂2zi∗∂x0224/3εi4≤18εi,0,7x04+ρi,7εi4,(38)−εi3∑k=1i−1∂zi∗∂zkzk+1≤∑k=1i−214εi,k+1,8εk+14+∑k=1i−114εi,k,9εk4+∑k=1i−234εi,k+1,8−1/31+∂zi∗∂zk24/3+1+∂zi∗∂zi−12+∑k=1i−134εi,k,9−1/31+αk∂zi∗∂zk24/3εi4≤∑k=1i−214εi,k+1,8εk+14+∑k=1i−114εi,k,9εk4+ρi,8+ρi,9εi4,(39)−εi3∑k=1i−1∂zi∗∂zkfkdu0n−k≤∑k=1i−1∑j=1k14εi,j,10εj4+∑k=1i−1∑j=1k34εi,j,10−1/3+4bij3−1/31+γik1∂zi∗∂zk24/3εi4+∑k=1i−1∑j=1kbij3′εjd4≤∑k=1i−1∑j=1k14εi,j,10εj4+ρi,10εi4+∑j=1i−1bij3εjd4,(40)−εi3∑k=1i−1∂zi∗∂zkη0n−kzk≤14∑k=1i−1n−kεi,k−1,12εk−14+εi,k,11εk4+∑k=1i−1n−k34εi,k,11−1/31+η0∂zi∗∂zk24/3+34εi,k−1,12−1/31+η0αk−1∂zi∗∂zk24/3εi4≤14∑k=1i−1n−kεi,k−1,12εk−14+εi,k,11εk4+ρi,11+ρi,12εi4,(41)−12η02∑k=1i−1n−kn−k+1εi3∂zi∗∂zkzkg0Tg0u02≤18∑k=1i−1n−kn−k+1εi,k,13εk4+18∑k=1i−1n−kn−k+1εi,k−1,14εk−14+38∑k=1i−1n−kn−k+1εi,k,13−1/31+m12∂zi∗∂zk24/3εi4+38∑k=1i−1n−kn−k+1εi,k−1,14−1/31+αk−1m12∂zi∗∂zk24/3εi4≤18∑k=1i−1n−kn−k+1εi,k,13εk4+18∑k=1i−1n−kn−k+1εi,k−1,14εk−14+ρi,13+ρi,14εi4,(42)−η0εi3∑k=1i−1n−k∂zi∗∂zkgkdTg0u0n−k+1≤14∑k=1i−1n−k∑j=1kεi,j,15εj4+34∑k=1i−1n−k∑j=1kεi,j,15−1/3+4bij4−1/31+γik2m1∂zi∗∂zk24/3εi4+∑k=1i−1∑j=1kbij4′εjd4≤14∑k=1i−1n−k∑j=1kεi,j,15εj4+ρi,15εi4+∑j=1i−1bij4εjd4,(43)−12εi3∑k=1i−1∂2zi∗∂x0∂zkgkdTu0n−k+η0n−kzkg0Tu0g0≤14∑k=1i−1εi,k,16εk4+34∑k=1i−1εi,k,16−1/3γik44/3+4bik5−1/3γik44/3εi4+∑j=1i−1bij5εjd4≤14∑k=1i−1εi,k,16εk4+ρi,16εi4+∑j=1i−1bij5εjd4,(44)−12εi3∑j,k=1i−1∂2zi∗∂zj∂zkgjdTu0n−j+η0n−jzjg0Tu0gkdTu0n−k+η0n−kzkg0Tu0T≤14∑k=1i−1εi,k,17εk4+34∑k=1i−1εi,k,17−1/3γik54/3+4bik6−1/3γik54/3εi4+∑j=1i−1bij6εjd4≤14∑k=1i−1εi,k,17εk4+ρi,17εi4+∑j=1i−1bij6εjd4,(45)32εi2gidTu0n−i+η0n−izig0Tu0−∂zi∗∂x0g0T−∑k=1i−1∂zi∗∂zkgkdTu0n−k+η0n−kzkg0Tu02≤34i∑k=1i−1εi,k,18εk4+34i∑k=1i−1εi,k,18−1γik34+32γik32+12i4bik7−13γik322εi4+∑j=1ibij7εjd4≤34i∑k=1i−1εi,k,18εk4+34i∑k=1i−1εi,k,18−1γik34+32γik32+12i4bik7−13γik322εi4+∑j=1ibij7εjd4≤34i∑k=1i−1εi,k,18εk4+ρi,18εi4+∑j=1ibij7εjd4,where εi,k,qk=0,…,i−1,q=1,…,18 are the arbitrary position constants; ρi,l,l=1,…,18, are the nonnegative continuous functions; and bijk,k=1,…,7 are the positive design constants, and letting bij=∑k=17bijk. Thus, by substituting (31)–(45) into (30) and adding and subtracting the term c¯¯iεi4 on the right-hand side of (30), one gets(46)ℒVi≤−λ−∑j=2i−1βjx04−∑j=1i−1c¯¯j−∑k=j+1i−1c¯¯kjεj4−c¯¯iεi4+εi3zi+1−zi+1∗+εi3zi+1∗+βix04+Hi2′εi4+14εi,1,2+14n−iεi,1,5+14εi,1,9+14i−1εi,1,10+14n−1εi,1,11+14n−1εi,1,12+14εi,1,16+18n−1n−2εi,1,13+18n−2n−1εi,1,14+18i−12n−iεi,1,15+14εi,1,17+34iεi,1,18ε14+34εi,i−1,1+14εi,i−1,2+14η0n−iεi,i−1,3+14n−iεi,i−1,5+18n−in−i+1εi,i−1,4+14εi,i−1,8+14εi,i−1,9+14εi,i−1,10+14n−i+1εi,i−1,11+18n−i+1n−i+2εi,i−1,13+14n−i+1εi,i−1,15+14εi,i−1,16+14εi,i−1,17+34iεi,i−1,18εi−14+∑k=2i−214εi,k,2+14n−iεi,k,5+14εi,k,8+14εi,k,9+14i−kεi,1,10+14n−kεi,k,11+14n−k−1εi,k,12+18n−kn−k+1εi,k,13+18n−k−1n−kεi,k,14+18i−k2n−i+1−kεi,k,15+14εi,k,16+14εi,k,17+34iεi,k,18εk4+∑j=1i∑l=jibljzjd4=−λ−∑j=2iβjx04−∑j=1i−1c¯¯j−∑k=j+1i−1c¯¯kjεj4+εi3zi+1∗+εi3zi+1−zi+1∗−c¯¯iεi4+c¯¯i1ε14+∑k=2i−2c¯¯ikεk4+c¯¯i,i−1εi−14+Hi2εi4+∑j=1i∑l=jibljzjd4,where βi=1/4εi,0,6+1/8εi,0,7,(47)c¯¯i1=14εi,1,2+14n−iεi,1,5+14εi,1,9+14i−1εi,1,10+14n−1εi,1,11+14n−1εi,1,12+14εi,1,16+18n−1n−2εi,1,13+18n−2n−1εi,1,14+18i−12n−iεi,1,15+14εi,1,17+34iεi,1,18,c¯¯ik=14εi,k,2+14n−iεi,k,5+14εi,k,8+14εi,k,9+14i−kεi,1,10+14n−kεi,k,11+14n−k−1εi,k,12+18n−kn−k+1εi,k,13+18n−k−1n−kεi,k,14+18i−k2n−i+1−kεi,k,15+14εi,k,16+14εi,k,17+34iεi,k,18,k=1,…,i−2,c¯¯i,i−1=34εi,i−1,1+14εi,i−1,2+14η0n−iεi,i−1,3+18n−in−i+1εi,i−1,4+14n−iεi,i−1,5+14εi,i−1,8+14εi,i−1,9+14εi,i−1,10+14n−i+1εi,i−1,11+18n−i+1n−i+2εi,i−1,13+14n−i+1εi,i−1,15+14εi,i−1,16+14εi,i−1,17+34iεi,i−1,18,Hi2=∑k=118ρi,k,
Thus, according to (46), one obtains(48)ℒVi≤−λ−∑j=2iβjx04−∑j=1i−1c¯¯j−∑k=j+1i−1c¯¯kjεj4+εi3zi+1∗+εi3zi+1−zi+1∗−c¯¯iεi4++c¯¯i1ε14+∑k=2i−2c¯¯ikεk4+c¯¯i,i−1εi−14+Hi2εi4+∑j=1i∑l=jibljzjd4≤−λ−∑j=2iβjx04−∑j=1ic¯¯j−∑k=j+1ic¯¯kjεj4+εi3zi+1−zi+1∗+εi3zi+1∗+Hi2εi4+∑j=1i∑l=jibljzjd4.
Constructing a virtual control law(49)zi+1∗x0,z¯i=−αix0,z¯iεi,αix0,z¯i=c=n+Hi2,and substituting it into (49) yield(50)ℒVi≤−λ−∑j=2iβjx04−∑j=1ic¯¯j−∑k=j+1ic¯¯kjεj4+εi3zi+1−zi+1∗+∑j=1i∑l=jibljzjd4.Stepn.
By exactly following the design procedure at stepi, one can get the following state-feedback controller:(51)ux0,z¯n=−αnx0,z¯nεn,αnx0,z¯n=c=n+Hn2,which can make the nth Lyapunov function,(52)Vnx0,ε¯n=14x04+∑i=1n14εi4,to satisfy(53)ℒVn≤−λ−∑k=2nβkx04−∑k=1nc¯¯k−∑i=k+1nc¯¯ikεk4+∑j=1n∑l=jnbljzjd4.where αn≥0 and Hn2 are the smooth functions; c¯¯n>0 is a design constant; bni,i=1,…,n, are the positive parameters; and Hn2 is the nonnegative continuous function.Theorem 2.
If Assumptions 1-2 and Lemma 2 hold for systems (7a) and (7b), then under the control law (10) and state-feedback control law (51), one has the following:(1)
The closed-loop system has an almost surely unique solution on−d,0(2)
The solution of the closed-loop system is GAS in probabilityProof.
On the one hand, in terms of (25) and (51), one can obtain that u is C1. This further gives that the local Lipschitz condition in x of the controller u holds, which together with the fact that fid and gid are locally Lipschitz functions. From stochastic stability theory in [28], one can obtain the closed‐loop system has an almost surely solution on −d,0.
On the other hand, choosing the design parametersλ>∑k=2nβj,c¯¯k>∑i=k+1nc¯¯ik, one can obtain ℒVn in (53), which becomes the same form as (35) in [24]. Using (53) and Lemma 2 and following the same procedure as in the proof of Theorem 1 in [24], one can easily prove Theorem 2. This completes the proof.


## 5. Switching Control Stability

In Section4, we have considered the case of x0t0≠0. The controllers (10) and (51) for systems (7a) and (7b) are given. Now, we turn to the case of x0t0=0. If the initial is zero, one can choose an open-loop control u0=−u0∗≠0 to drive the state x0 away from zero. So there exists ts∗>0 such that x0ts∗≠0. After that, controllers u0 and u given in (10) and (51) can be used.Based on the above analysis, now we give the main results of this paper.Theorem 3.
Suppose that Assumptions 1-2 hold. If the following switching control procedure is applied to systems (7a) and (7b):(i)
When the initial state belongs tox0t0,…,xnt0︸n+1∈Rn+1x0t0≠0,we design control inputsu0 and u in the forms (10) and (51), respectively(ii)
When the initial state belongs to(54)x0t0,…,xnt0︸n+1∈Rn+1∣x0t0≠0,if t∈t0,ts∗, one can choose the control law u0=−u0∗ and u=0; if t∈ts∗,∞ at the time t=ts∗, we switch the control inputs u0 and u into (10) and (51), respectively.Then, for any initial conditions in the state space, systems (7a) and (7b) will be almost asymptotically stabilized in probability at the equilibrium and, specifically, the states are globally asymptotically regulated to zero in probability.Proof.
Firstly, we consider the case that the initial state belongs tox0t0,…,xnt0︸n+1∈Rn+1∣x0t0≠0. From Theorem 1 and Theorem 2, one can obtain that signals x0,ε1,…,εn are bounded in probability and x0,ε1,…,εn are globally stable, especially, x0 will not reach zero. From (13) and (25), such that zk∗x0,ε1,…,εk−1,k=2,…,n, is a smooth function of x0,ε1,…,εk−1 and ε1=z1. One can obtain z1 and z2∗,…,zn∗ bounded and globally stable in probability. By zk=εk+zk∗, it is easy to see that zk is bounded in probability and converges to zero in probability. As a result of (13), x0,x1,…xn are all bounded in probability and converge to zero in probability. Therefore, the stochastic nonholonomic systems with unknown time-varying delays (7a) and (7b) are globally asymptotically stable.
Secondly, when the initial state belongs tox0t0,…,xnt0︸n+1∈Rn+1∣x0t0=0, we use the constant control u0=u0∗≠0 in order to drive the state x0 away from origin. When x0ts∗≠0,, the switching control strategy is applied to systems (7a) and (7b) at the time instant ts∗>0. This completes the proof.


## 6. Simulation Example

Consider the following system:(55)dx0=u0dt+12x0dω,dx1=u0x2dt+x1dt+13x1dω,dx2=udt+x2+x1t−0.11+sint+x2t−0.11+sintdt+x2dω,and it is obvious that (55) is in the form of (7a) and (7b). It can be verified that Assumption 1 is satisfied with γ¯11=1,γ¯21=1,γ¯22=1,ξ11=1/3,ξ21=1,andξ22=1.By exactly following the design procedure in Section3, one finally constructs the following state-feedback controller:(56)m1=0.5,η0=3.8,u0=−η0x0,ε1=z1,ε2=z2−z2∗x0,z1,z2∗x0,z1=−α1x0,z1ε1,∂z2∗∂z1=−α1,α1⋅=c=1+ρ1,1+ρ1,2+ρ1,3+ρ1,4+ρ1,5,u=−α2x0,z1,z2ε2,α2⋅=c=2+∑j=115ρ2,j,b11=∑k=13b11k,b21=∑k=17b21k,b22=∑k=17b22k,c=2,1=34ε2,1,1+14ε2,1,2+14ε2,1,4+14ε2,1,8+14ε2,1,9+14ε2,1,10+14ε2,1,11+14ε2,1,13+14ε2,1,15+14ε2,1,16+14ε2,1,17+32ε2,1,18,c=2,2=14ε2,2,2+14ε2,2,8+14ε2,2,9+14ε2,2,12+14ε2,2,16+14ε2,2,17+32ε2,2,18,ρ1,1=γ111+344b111−1/3γ1114/3,ρ1,2=η0,ρ1,3=m12,ρ1,4=m1γ112+344b112−1/3m1γ1124/3,ρ1,5=3γ1132+122b113−13γ11322;ρ2,1=14ε2,1,1−3,ρ2,2=34ε2,1,2−1/3γ2114/3+344b211−1/3γ2114/3+γ221,ρ2,3=0,ρ2,4=0,ρ2,5=0,ρ2,6=34ε2,0,6−1/31+η0∂z2∗∂x024/3,ρ2,7=38ε2,0,7−1/31+x0m12∂2z2∗∂x0224/3,ρ2,8+ρ2,9=34ε2,1,8−1/31+∂z2∗∂z124/3+1+∂z2∗∂z12+34ε2,1,9−1/31+α1∂z2∗∂z124/3,ρ2,10=34ε2,1,10−1/3+4b213−1/31+γ211∂z2∗∂z124/3,ρ2,11+ρ2,12=34ε2,1,11−1/31+η0∂z2∗∂z124/3,ρ2,13+ρ2,14=34ε2,1,13−1/31+m12∂z2∗∂z124/3,ρ2,15=34ε2,1,15−1/3+4b214−1/31+γ212m1∂z2∗∂z124/3,ρ2,16=34ε2,1,16−1/3γ2144/3+4b215−1/3γ2144/3,ρ2,17=34ε2,1,17−1/3γ2154/3+4b216−1/3γ2154/3,ρ2,18=32ε2,1,18−1γ2134+32γ2132+124b217−13γ21322.In the practical simulation, one choosesε2,k,q=1,k=0,1,q=1,…,18; γ21k=1,k=1,…,5; b21k=1/7,b22k=1/7, k=1,…,7; and c¯¯1=4.8, c¯¯2=2.1 and the initial values x0τ,x1τ,x2τT=xτ=3,−1,−10T for τ∈−0.2,0. Figure 1 gives the responses of states x0,x1,andx2 and controllers u0 and u with respect to time.Figure 1
The responses of states (a)x0,x1,andx2 and controllers (b) u0 and (c) u with respect to time.
(a)
(b)
(c)


## 7. Conclusion

For the stochastic nonholonomic systems with unknown time-varying delays, this paper investigates the problem of state-feedback stabilization. By further weakening the commonly used growth assumptions, a state-feedback controller has been proposed by using the Lyapunov–Razumikhin lemma and backstepping technique. The result extends the work [24] to nonholonomic systems. As a consequence, based on the switching control strategy, global asymptotic regulation to zero in probability of the closed-loop system states is achieved.



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