
This manuscript considers the state feedback stabilization problem for a class of stochastic high-order and low-order nonlinear systems with time-delay. Compared with the previous results, a distinctive feature to be studied is that the considered systems involve high-order, low-order, intricate stochastic diffusion terms and time-delay simultaneously. First, the homogeneous domination approach and suitable coordinate transformations are introduced to obtain the updating laws. Then, a state feedback controller is devised to make the closed-loop systems globally asymptotically stable in probability. Finally, a simulation example is shown to prove the proposed approach powerfully.
Citation: Yanghe Cao, Junsheng Zhao, Zongyao Sun. State feedback stabilization problem of stochastic high-order and low-order nonlinear systems with time-delay[J]. AIMS Mathematics, 2023, 8(2): 3185-3203. doi: 10.3934/math.2023163
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This manuscript considers the state feedback stabilization problem for a class of stochastic high-order and low-order nonlinear systems with time-delay. Compared with the previous results, a distinctive feature to be studied is that the considered systems involve high-order, low-order, intricate stochastic diffusion terms and time-delay simultaneously. First, the homogeneous domination approach and suitable coordinate transformations are introduced to obtain the updating laws. Then, a state feedback controller is devised to make the closed-loop systems globally asymptotically stable in probability. Finally, a simulation example is shown to prove the proposed approach powerfully.
Recently, the theory of nonlinear systems with time-delay has been a hot topic, due to its wide application in practical problems, such as physical engineering, biological systems and economic processes. Among these, the Lyapunov-Krasovskii methodology plays a crucial role in dealing with time-delay systems. Based on the above method, Pepe [1] addressed the input state stability of nonlinear systems with time-delay. Zhang [2] designed a stabilized controller for time-delay feed-forward nonlinear systems to achieve system stability. In order to address the stabilization problem of high-order nonlinear systems with time-delay, some researchers try to find new ways to design corresponding controllers. Yang and Sun [3] investigated the state feedback stabilization problem of controlled systems with high-order or/and time-delay via the homogeneous domination idea. With the help of the saturation function technique, homogeneous domination idea and Lyapunov approach, Song [4] studied the stabilization problem of high order feed-forward time-delay nonlinear systems. In addition to the above works, many results in [5,6,7,8,9,10] have established and improved the concept framework of nonlinear systems with time-delay.
Ever since the stochastic stability theory was founded and enriched by Deng and Zhu [11,12], great progress has been made on the global stabilization of stochastic nonlinear systems [13,14,15,16]. Subsequently, Florchinger [17] extended the theory of control with the Lyapunov-Krasovskii functional. With the stochastic stability theory in mind, it is still important and meaningful to address high-order stochastic nonlinear systems with time-delay. Zha [18] investigated the issue of output feedback stabilization. Liu [19] studied the output feedback stabilization problem for time-delay stochastic feed-forward systems. By using a power integrator approach, the work in [20,21,22,23] also considered the state-feedback stabilization problems. However, the state feedback stabilization problem for stochastic high-order and low-order nonlinear systems with time-delay has not been well addressed, which leads us to take the interesting problem into account.
How to deal with the state feedback stabilization problem for high-order and low-order nonlinear systems with time-delay? By using a power integrator approach, Liu & Sun [24] constructed a time-delay independent controller for the aforementioned systems to relax the growth condition and the power order limitations. However, to the best of our knowledge, research on the corresponding stochastic version is limited with scarcely a few convincing results. The main difficulties are explained from two aspects. On one hand, the Itˆo formula brings the gradient terms and the Hessian terms in the Lyapunov analysis. On the other hand, the particularity of its structure has made many traditional methods inapplicable. Therefore, we need to give a new way to consider stochastic nonlinear systems. Inspired by a large number of results in [25,26,27,28,29], stochastic high-order and low-order nonlinear systems with time-delay will be considered as follows:
{dxi(t)=xpii+1(t)dt+fi(ˉxi(t),ˉxi(t−τ),t)dt+gTi(ˉxi(t),ˉxi(t−τ),t)dω(t),dxn(t)=upn(t)dt+fn(x(t),x(t−τ),t)dt+gTn(x(t),x(t−τ),t)dω(t), | (1.1) |
where x(t)=[x1(t),…,xn(t)]T∈Rn is state, and u(t)∈R is input; the nonnegative real number τ is the time-delay of the states. ω(t)=[ω1(t),…,ωr(t)]T. The high-order can be revealed by pi∈R>1odd=:{pq|p≥q>0 andp,qareoddintegers}. The drift terms fi:Ri×Ri×R+⟶R and the diffusion terms gi:Ri×Ri×R+⟶Rr,i=1,…,n are considered as locally Lipschitz with fi(0,0,t)=0 and gi(0,0,t)=0.
The contributions are highlighted in the following:
(i) Systems considered are more general. Systems in [24] only solve the control issues for deterministic cases. It is more complex to consider the stochastic disturbance. By using the homogeneous domination idea, one can give a novel perspective to generalize the control strategy for deterministic systems to the corresponding stochastic cases.
(ii) The result extends the works [30,31,32] by relaxing the growth condition and the power order limitations. The low order of the nonlinear terms is successfully relaxed to the high-order and low-order of the nonlinear terms. Based on the above situations, we use a proper Lyapunov-Krasovskii functional to handle the stabilization problem under the weaker assumptions.
Notations: R+≜{x|x≥0,x∈R},Rn≜{xn|x≥0}. For a given vector/matrix D, DT denotes its transpose, Tr{D} is the trace when D is square, and the Euclidean norm of a vector |D|. Ci is composed of continuous and ith partial derivable functions. K is composed of continuous functions and strictly increasing; K∞ is composed of functions with K. One sometimes denotes X(t) by X to simplify the procedure.
Now, the time-delay stochastic nonlinear systems are addressed as follows:
dx(t)=f(ˉxi(t),ˉxi(t−τ),t)dt+gT(ˉxi(t),ˉxi(t−τ),t)dω(t). | (2.1) |
{x(s):−d≤s≤0}=z∈CbF0([−d,0];Rn) is an initial data, and ω(t) denotes a Brownian motion with dimension r defined on a complete probability space (Ω,F,{Ft}t≥0,P).
The following assumptions are needed:
Assumption 1. For i=1,…,n, there exist two constants a1>0 and a2>0 such that
|fi(ˉxi(t),ˉxi(t−τ),t)|≤a1i∑j=1(|xj(t)|ri+θrj+|xj(t−τ)|ri+θrj)+a1i−1∑j=1(|xj(t)|1pj…pi−1+|xj(t−τ)|1pj…pi−1)+a1(|xi(t)|+|xi(t−τ)|),‖gi(ˉxi(t),ˉxi(t−τ),t)‖≤a2i∑j=1(|xj(t)|2ri+θ2rj+|xj(t−τ)|2ri+θ2rj)+a2i−1∑j=1(|xj(t)|12pj…pi−1+|xj(t−τ)|12pj…pi−1)+a2(|xi(t)|+|xi(t−τ)|), | (2.2) |
in which θ=mn≥0, n is an odd integer, m is an even integer, and r′is have the following definitions:
r1=1,ri=ri−1+θpi−1,i=2,3,…,n+1. | (2.3) |
Remark 1. Assumption 1 encompasses and extends high-order and/or low-order results. We discuss this point from two cases.
Case I: Condition (2.2), when τ=0 it reduces to high-order growth condition with θ≥0,
|fi(ˉxi(t),t)|≤a1i∑j=1(|xj(t)|ri+θrj+|xj(t)|1pj…pi−1+a1|xi(t)|,‖gi(ˉxi(t),t)‖≤a2i∑j=1(|xj(t)|2ri+θ2rj+|xj(t)|12pj…pi−1+a2(|xi(t)|, |
and low-order growth condition with θ=0,
|fi(ˉxi(t),t)|≤a1i∑j=1|xj(t)|1pj…pi−1+a1|xi(t)|,‖gi(ˉxi(t),t)‖≤a2i∑j=1|xj(t)|12pj…pi−1+a2(|xi(t)|. |
We further discuss its significance from value ranges of both low-order and high-order. From θ∈(−1pj…pi−1,0], it is easy to see that 0<ri+θrj≤1pj…pi−1, which implies that both low-order and high-order in Assumption 1 can take any value in (0,1pj…pi−1],[1pj…pi−1,+∞), respectively.
Case II: When τ≠0, several new results [18,19,20,21,22] have been achieved on feedback stabilization of high-order nonlinear time-delay systems. The nonlinearities in [18,19,20,21,22] only have high-order terms. The nonlinearities in [24] include linear and nonlinear parts, and their nonlinear parts only allow low-order 1pj…pi−1 and high-order ri+θrj with θ≥0.
While in this paper, (2.2) not only includes time-delays but relaxes the intervals of low-order and high-order.
Remark 2. When pi=1,i=1,2,…,n−1, and τ=0, equation (1) reduces to the well-known form, for which the feedback control problem has been well developed in recent years[16,24,26].
Proposition 1. For r1,…,rn and σ=p1…pnrn+1 having the following properties:
● rk∈R≥1odd,σrk∈R≥1odd,σ∈R≥1odd,σrkpk−1…p1∈R≥1odd.
● σ≥max1≤k≤n{rk+θ}.
● There hold
4≤4−1p1…pk−1+rk+1pkrkpk−1…p1,4σ−rk+1pkrkpk−1…p1+1p1…pk−1≤4σrkpk−1…p1; |
4≤4−1p1…pk−1+rk+1pk2rkpk−1…p1,4σ−rk+1pk2rkpk−1…p1+12p1…pk−1≤4σrkpk−1…p1. |
● For i=1,…,k−1, one has
4≤4rk+1pk…p1ripi−1…p1,4σripi−1…p1≤4σripi−1…p1. |
Remark 3. It is not difficult to see that system (1.1) is a class of high-order and low-order stochastic nonlinear systems with time-delay satisfying Assumption 1. Compared with [30], it is significant to point out that system (1.1) addressed here is more general. The systems can be composed by time-delay and the coupling of the high-order and low-order terms. Moreover, if g=0, Assumption 1 will generate the same assumption as in [24]. When pi>3, the state feedback stabilization problem under constraint pi=p can give similar results as [19]. Under Assumption 1 with τ=0, we can obtain the same results with [30], if there are no low-order nonlinearities.
Remark 4. For the case of τ=0 in system (1.1), with the help of adding a power integrator, fruitful results have been achieved over the past years. However, for the case of τ≠0, some essential difficulties will inevitably be encountered in constructing the desired controller. For instance, the time-delay effect will make the common assumption on the high-order system nonlinearities infeasible, and what conditions should be placed to the nonlinearities remains unanswered. Second, due to the higher power, time-delay and assumptions on the nonlinearities, it is more complicated to find a Lyapunov-Krasovskii functional which can be behaved well in theoretical analysis.
For ease of the controller design, some helpful definitions are presented.
Definition 1. [19] Consider the stochastic system dx(t)=f(x,t)dt+g(x,t)dω. For any given C2 function V(x,t), the differential operator L is defined as follows:
LV=∂V∂t+∂V∂xf(x,t)+12Tr{gT∂2V∂2tg}, |
where 12Tr{gT∂2V∂2tg} is called the Hessian term of L.
Definition 2. [25] There exists coordinate (x1,…,xn)∈Rn,hi>0,i=1,…,n, for arbitrarily ε>0.
∙ The dilation Δε(x)=(εh1x1,…,εhnxn), and hi is referred to as the weights. And one defines dilation weight as △=(h1,…,hn).
∙ A function U∈C(Rn,R) is considered as homogeneous of degree μ, if μ∈R, then U(Δε(x))=εμU(x1,…,xn), for arbitrarily x∈Rn∖{0}.
∙ A vector field fi∈C(Rn,R) is considered as homogeneous of degree μ, if μ∈R, then fi(Δε(x))=εμ+hifi(x), for arbitrarily x∈Rn∖{0}, i=1,…,n.
∙ A homogeneous γ-norm is considered as ‖x‖△,γ=(∑ni=1|xi|γhi)1γ, for any x∈Rn, where γ≥1. We use ‖x‖△ or ‖x‖△,2 to a exhibit 2-norm.
With the above definitions, we give some lemmas which will be crucial for controller design.
Lemma 1. [13] For m∈R≥1odd, ∀a∈Rand∀b∈R, there hold
(|a|+|b|)1m≤|a|1m+|b|1m≤2m−1m(|a|+|b|)1m, |
|a−b|m≤2m−1|am−bm|. |
Lemma 2. [13] For given a,b≥0 and a given positive function f(x,y), there exists a positive function g(x,y), such that
|f(x,y)xayb|≤g(x,y)|x|a+b+ba+b(a(a+b)g(x,y))ab|f(x,y)|a+bb|y|a+b,∀x,y∈R. |
Lemma 3. [13] For a continuous function g, if it is monotone, and g(s)=0, then
|∫tsg(x)dx|≤|g(t)|⋅|t−s|. |
Lemma 4. [19] Given τi∈R,i=1,…,n satisfying 0≤τ1≤…≤τn and for given nonnegative functions ai(x,y),i=1,…,n, there holds
a1(x,y)|x|τ1+an(x,y)|x|τn≤n∑j=1aj(x,y)|x|τj≤(|x|τ1+|x|τn)n∑j=1aj(x,y),∀x,y∈R. |
Consider the stochastic high-order and low-order nonlinear systems with time-delay as follows:
{dxi=(xpii+1+fi)dt+gidω(t),i=1,…,n−1,dxn=upndt. | (3.1) |
Step 0: To begin with, introducing the complete form of the controller,
{zi(t)=xp1…pi−1i(t)−αp1…pi−1i−1(t),i=1,…,n,αi(t)=−ϱ1p1…pii(zi(t)+zri+1pirii(t))1p1…pi,i=1,…,n,u(t)=αn(t). | (3.2) |
The purpose of this work is to construct a state controller to render system (1.1) globally asymptotically stable in probability. To achieve this goal, propositions are presented as follows.
Proposition 2. For c1>0, c2>0, i=1,…,n, there hold
|fi(t,ˉxi(t),ˉxi(t−τ))|≤c1i∑j=1(|zj(t)|1p1…pi−1+|zj(t)|ri+θrjp1…pi−1)+c1i∑j=1(|zj(t−τ)|1p1…pi−1+|zj(t−τ)|ri+θrjp1…pi−1)‖gi(t,ˉxi(t),ˉxi(t−τ))‖≤c2i∑j=1(|zj(t)|12p1…pi−1+|zj(t)|2ri+θ2rjp1…pi−1)+c2i∑j=1(|zj(t−τ)|12p1…pi−1+|zj(t−τ)|2ri+θ2rjp1…pi−1). | (3.3) |
Step 1. First, we will construct a Lyapunov-candidate-function V1=∫x10s3ds+∫x10s4σ−r2p1r1ds+n∫tt−τ(z41(l)+z4σr11(l))dl. Along the solution of (3.1), one has
LV1=x31(xp12+f1)+x4σ−r2p1r11(xp12+f1)+n(z41(t)+z4σr11(t))−n(z41(t−τ)+z4σr11(t−τ))+Ψ1, |
where Ψ1=12Tr{gT1∂2V1∂x21g1}, which leads to
LV1=(z31+z4σ−r2p1r11)(xp12−αp11)+(z31+z4σ−r2p1r11)αp11+(z31+z4σ−r2p1r11)f1+n(z41(t)+z4σr11(t))−n(z41(t−τ)+z4σr11(t−τ))+Ψ1. | (3.4) |
With Proposition 2, Lemma 1 and Lemma 2 in mind, one has
(z31+z4σ−r2p1r11)f1≤c1(|z1|3+|z1|4σ−r2p1r1)(|z1|+|z1|r2p1r1+|z1(t−τ)|+|z1(t−τ)|r2p1r1)≤c1(|z1|4+|z1|4σ−r2p1r1z1+|z1|r2p1r1z31+|z1|4σr1)+c1(|z1|3|z1(t−τ)|+|z1|3|z1(t−τ)|r2p1r1+|z1|4σ−r2p1r1z1|z1(t−τ)|+|z1|4σ−r2p1r1|z1(t−τ)|r2p1r1); | (3.5) |
with the help of Lemma 4, we can see it satisfies |z1|r2p1r1z31≤|z1|r2+θ+3r1r1≤|z1|4+θ≤z41+z4σr11 when 4≤4+θ≤2r2p1σ+3≤4σr1. Similarly, one can obtain
(z31+z4σ−r2p1r11)f1≤β1(z41+z4σr11)+(z1(t−τ)4+z1(t−τ)4σr1), | (3.6) |
where β1=4c1+2c21+4σ−r2p1σ(2r2p1σ)r2p14σ−r2p1c2σ4σ−r2p11. Now, one designs the virtual controller α1 as
αp11(x1)=−(2n+β1)(z1+zr2p1r11)=−ϱ1(z1+zr2p1r11), | (3.7) |
where ϱ1>1 is a positive constant. Noticing that
−ϱ1z1+4σ−r2p1r11≤0,−ϱ1z1+r2p1r11≤0, |
and using (4.1) and (3.7) with (3.4) after complex calculations, one finally obtains
LV1≤−n(z41+z4σr11)+(z31+z4σ−r2p1r11)(xp12−αp11)−(n−1)(z41(t−τ)+z4σr11(t−τ))+Ψ1. | (3.8) |
To complete the induction, at the kth step, we now define
WLk=∫xkαk−1(sp1…pk−1−αp1…pi−1k−1)4−1p1…pi−1dsWHk=∫xkαk−1(sp1…pk−1−αp1…pi−1k−1)4σ−rk+1pkrkp1…pi−1dsWDk=(n−k+1)∫tt−τ(z4k(l)+z4σrkp1…pi−1k(l))dl. |
Lyapunov function Vk=Vk−1+WLk+WHk+WDk is C2, proper and positive definite. Moreover, for i=1,…,k−1, WLk(⋅),WHk(⋅),WDk(⋅) satisfy
∂WLk∂xk=z4−1p1…pi−1k,∂WHk∂xk=z4σ−rk+1pkrkp1…pi−1k,frac∂2WLk∂x2k=(4−1p1…pi−1)z3−1p1…pi−1k(p1…pk−1)xp1…pk−1−1 | (3.9) |
∂2WHk∂x2k=(4σ−rk+1pkrkp1…pi−1)z4σ−rk+1pkrkp1…pi−1−1k(p1…pk−1)xp1…pk−1−1∂WLk∂xi=−(4−1p1…pi−1)∫xkαk−1(sp1…pk−1−αp1…pi−1k−1)3−1p1…pi−1…pi−1ds∂αp1…pi−1k−1∂xi∂2WLk∂xkxi=−(4−1p1…pi−1)z3−1p1…pi−1k∂αp1…pi−1k−1∂xi,∂2WHk∂xkxi=−(4σ−rk+1pkrkp1…pi−1)z4σ−rk+1pkrkp1…pi−1−1k∂αp1…pi−1k−1∂xi∂2WLk∂x2i=∫xkαk−1(4−1p1…pi−1)(3−1p1…pi−1)(sp1…pk−1−αp1…pi−1k−1)2−1p1…pi−1…pi−1ds(∂αp1…pi−1k−1∂xi)2+∫xkαk−1(4−1p1…pi−1)(sp1…pk−1−αp1…pi−1k−1)3−1p1…pi−1…pi−1ds(∂2αp1…pi−1k−1∂x2i)∂WHk∂xi=−(4σ−rk+1pkrkp1…pi−1)∫xkαk−1(sp1…pk−1−αp1…pi−1k−1)4σ−rk+1pkrkp1…pi−1−1ds∂αp1…pi−1k−1∂xi∂2WHk∂x2i=∫xkαk−1(4σ−rk+1pkrkp1…pi−1)(4σ−rk+1pkrkp1…pi−1−1)(sp1…pk−1−αp1…pi−1k−1)4σ−rk+1pkrkp1…pi−1−2ds(∂αp1…pi−1k−1∂xi)2+∫xkαk−1(4σ−rk+1pkrkp1…pi−1)(sp1…pk−1−αp1…pi−1k−1)4σ−rk+1pkrkp1…pi−1−1ds(∂2αp1…pi−1k−1∂x2i). | (3.10) |
Step k (k = 2, 3, …, n): As in step k-1, there exists Lyapunov-candidate-function Vk−1, implying
LVk−1≤−(n−k+2)k−1∑i=1(z4i+z4σripi−1…p1i)−(n−k+1)k−1∑i=1(z4i(t−τ)+z4σripi−1…p1i(t−τ))+(z4−1p1…pk−2k−1+z4σ−rkpk−1rk−1pk−2…p1k−1)(xpk−1k−αpk−1k−1)+Ψk−1, | (3.11) |
where Ψk−1= 12Tr{ˉψTk−1 ∂2Vk−1∂ˉx2k−1 ˉψk−1}, ˉψk−1 = (g1, …, gk−1). Hence, one will consider Vk=Vk−1+WLk+WHk+WDk and define an appropriate virtual controller αk. Similar to step 1, one can obtain
LVk≤−(n−k+2)k−1∑i=1(z4i+z4σripi−1…p1i)−(n−k+1)k−1∑i=1(z4i(t−τ)+z4σripi−1…p1i(t−τ))+(n−k+1)(z4k+z4σrkpk−1…p1k)+(z4−1p1…pk−1k+z4σ−rkpk−1rkpk−1…p1k)(xpk−1k+1−αpkk)+(z4−1p1…pk−1k+z4σ−rkpk−1rkpk−1…p1k)αpkk+(z4−1p1…pk−1k+z4σ−rkpk−1rkpk−1…p1k)fk+(z4−1p1…pk−2k−1+z4σ−rkpk−1rk−1pk−2…p1k−1)(xpk−1k−αpk−1k−1)+k−1∑i=1(∂WLk∂xi+∂WHk∂xi)(xpii+1+fi)+Ψk, | (3.12) |
where Ψk=12Tr{ˉψTk∂2Vk∂ˉx2kˉψk}, ˉψk=(g1,…,gk). Obviously, the virtual controller αk is used to eliminate the last three terms of (3.12). In light of (3.2) and Lemma 1, it yields that
xpk−1k−αpk−1k−1≤|(xp1…pk−1k)1p1…pk−2−(αp1…pk−1k−1)1p1…pk−2|≤23−1p1…pk−2|zk|1p1…pk−2. |
In the case of 4σ−θ≤4σ, by Lemma 2, one obtains that
(z4−1p1…pk−2k−1+z4σ−rkpk−1rk−1pk−2…p1k−1)(xpk−1k−αpk−1k−1)≤21−1p1…pk−2|zk|1p1…pk−2(|zk−1|4−1p1…pk−2+|zk−1|4σ−rkpk−1rk−1pk−2…p1)≤βk1(z4k+z4σrkpk−1…p1k)+13(z4k−1+z4σrk−1pk−2…p1k−1), | (3.13) |
where βk1 denotes a positive constant. On the basis of Proposition 2 and Lemma 3, one has
(z4−1p1…pk−2k−1+z4σ−rkpk−1rk−1pk−2…p1k−1)fk≤12k−2∑i=1(z4i+z4σripi−1…p1i)+13(z4k−1+z4σrk−1pk−2…p1k−1)+12k−1∑i=1(z4i(t−τ)+zi(t−τ)4σripi−1…p1)+z4k(t−τ)+zk(t−τ)4σrkpk−1…p1+βk2(z4k+z4σrkpk−1…p1k), | (3.14) |
where βk2 denotes a positive constant. In the sequel, one estimates the last term. With the help (3.2), Lemmas 2 and 4, it is not hard to achieve
∫xkαk−1(sp1…pk−1−αp1…pk−1k−1)3−1p1…pk−1ds≤|zk|3−1p1…pk−1⋅|xk−αk−1|≤23−1p1…pk−1|zk|. | (3.15) |
Similarly, one can obtain
∫xkαk−1(sp1…pk−1−αp1…pk−1k−1)4σ−rk+1pkrkpk−1…p1−1ds≤23−1p1…pk−1|zk|4σ−θrkpk−1…p1−1. | (3.16) |
On the basis of the previous inequality, one has
(∂WLk∂xi+∂WHk∂xi)(xpii+1+fi)≤λk(|zk|+|zk|4σ−θrkpk−1…p1−1)|∂αp1…pk−1k−1∂xi|(xpii+1+fi)≤dki(z4k+z4σrkpk−1…pk−1k)+12(k−1)(k−2∑j=1(z4j+z4σrjp1…pj−1j)+13(k−1)(z4k−1+z4σrk−1p1…pk−2k−1)+12(k−1)k−1∑j=1(z4j(t−τ)+z4σrjp1…pj−1j(t−τ)), | (3.17) |
in which dki denotes a positive constant. Define βk=βk1+βk2+βk3 with βk3=∑k−1i=1dki and choose the virtual controller αk as
αp1…pkk(Xk)=−ϱk(zk+zrk+1pkrkk). | (3.18) |
By Lemma 2, one can arrive at
(z4−1p1…pk−1k+z4σ−rk+1pkrkpk−1…p1k)αpkk≤−(2(n−k+1)+βk)(z4k+z4σrkpk−1…p1k). | (3.19) |
Substituting (3.13)–(3.18) into (3.12) yields
LVk≤−(n−k+1)k∑i=1(z4i+z4σripi−1…p1i)−(n−k)k∑i=1(z4i(t−τ)+z4σripi−1…p1i(t−τ))+(z4−1p1…pk−1k+z4σ−rk+1pkrkpk−1…p1k)(xpkk+1−αpkk)+Ψk. |
It is shown that the above formula holds for k=n with virtual controllers (3.18). Similarly, we choose Vn(x)=∑ni=1(WLi(⋅)+WHi(⋅)+WDi(⋅)). There is an actual control law
u(x)=−ϱ1p1…pnn(zn+zrn+1pnrnn)1p1…pn, | (3.20) |
such that
LVk≤−n∑i=1(z4i+z4σripi−1…p1i). | (3.21) |
Until now, the recursive design has been completed. Under the new coordinates
ξ1=x1,ξi=xiLki,νp=upLkn+1, | (3.22) |
where k1=0, ki=ki−1+1pi−1, i=2,…,n and L>1 is a constructed constant, system (1.1) can be rewritten in the form
dξi=Lξpii+1dt+fi(⋅)Lkidt+gi(⋅)Lkidω(t),dξn=Lνpnndt+fn(⋅)Lkndt+gn(⋅)Lkndω(t). | (3.23) |
By (3.18) and (3.20), the system (3.1) can be integrated into the complex format
dξ=LR(ξ)dt+T(t,ξ,ξ(t−τ))dt+ψT(t,ξ,ξ(t−τ))dω(t), | (3.24) |
where ξ=(ξ1,…,ξn)T, R(ξ)=(ξp2, …,ξpn, νp)T, T(t,ξ, ξ(t−τ))=(f1, f2Lk2,…, fnLkn)T, ψ(t,ξ,ξ(t−τ))=(g1, g2Lk2,…, gnLkn). Introducing the dilation weight △=(r1,r2,…,rn), one gets
Vn(△ε(ξ))=n∑i=1∫εrixiεriαi−1(sp1…pi−1−εripi−1…p1αp1…pi−1i−1)4−1p1…pi−1ds+n∑i=1∫εrixiεriαi−1(sp1…pi−1−εripi−1…p1αp1…pi−1i−1)4σ−rk+1pkrkpk−1…p1ds+(n−k+1)∫tt−τ(z4k(s)+z4σrkpk−1…p1k(s))ds=n∑i=1∫xiαi−1(εripi−1…p1(ζp1…pi−1−αp1…pi−1i−1))4−1p1…pi−1εripi−1…p1dζ+n∑i=1∫εrixiαi−1(sp1…pi−1−εripi−1…p1αp1…pi−1i−1)4σ−rk+1pkrkpk−1…p1εripi−1…p1dζ=ε4σ−θVn(ξ), | (3.25) |
where s is defined as s=riζ. With the help of the above formula and Definition 2, it can be concluded that Vn(ξ) is homogeneous of degree 4σ−θ.
The main result of this manuscript will be stated as follows.
Theorem 1. Suppose Assumptions 1 apply to stochastic system (1), under the state feedback controller up=Lkn+1νp and (3.20), then:
(i) There exists a unique solution on [−d,∞);
(ii) The equilibrium at the origin is globally asymptotically stable in probability.
Proof. Four steps are used to verify Theorem 1.
Step 1: By the definition of ϱ>0, we know that p1…pj−1−1>1, which implies that 4−1p1…pk−1−1>2, 4σ−rk+1pkrkpk−1…p1>2. Therefore, ∂αp1…pii(t)∂xj(t) is continuous, and upn=Lkn+1νpn is C. As is known to all, the function is C. The closed-loop system satisfies the locally Lipschitz condition based on fi and gi being locally Lipschitz.
Step 2: Consider the Lyapunov-candidate-function:
V(ξ)=Vn(ξ)+n∑i=1h1+h21−γ∫tt−τ‖ξ‖4σ△dη, | (3.26) |
where h1 and h2 are positive parameters. It is straightforward to prove that V(ξ) is C2 on ξ. Since Vn(ξ) is continuous, positive definite and radially unbounded, from Lemma 1, one can have
α20(|ξ|)≤V(ξ)≤α21(|ξ|), | (3.27) |
where α20 and α21 are K∞ functions. With the help of the homogeneous theory, one finally has
ˉc0‖ξi‖4σΔ≤U(ξ)≤c_0‖ξi‖4σΔ, | (3.28) |
in which ˉc0>0, c_0>0, and U(ξ) denotes a positive definite function of the 4σ homogeneous degree. Hence, one has the formula
α20(|ξ|)≤U(ξ)≤α21(|ξ|). | (3.29) |
(3.29) leads to
h1+h21−γ∫tt−τ‖ξ‖4σ△dη≤˜c∫tt−τˉα22(|ξi|)dη≤˜c∫0−τα21(|ξi(t+s)|)d(t+s)≤csup−τ≤s≤0ˉα22(|ξi(s+t)|)≤α22(sup−τ≤s≤0|ξi(s+t)|), | (3.30) |
where η=s+t, ˜c>0, c>0 and α22 is a class K∞ function. Since
|ξ|≤(sup−τ≤s≤0|ξ(s+t)|),α21|ξ|≤α21(sup−τ≤s≤0|ξ(s+t)|). |
Defining β2=α21+α22, by (3.26)-(3.30), one gets
β1(|ξ|)≤V(|ξ|)≤β2(sup−τ≤s≤0|ξ(s+t)|). |
Step 3: With the help of Lemma 1 and (3.20), c01 is a positive constant, and one has
∂Vn(ξ)∂ξLR(ξ)≤−c01L‖ξ‖4σΔ. | (3.31) |
By Proposition 2 and L>1, one can have
|fi(t,ˉξ(t),ˉξ(t−τ))Lki|≤ˉδ1L1−γi1(i∑j=1|ξ(t)|ri+θrj+i∑j=1|ξ(t−τ)|ri+θrj)≤δ1L1−γi1(‖ξ(t)‖ri+θ△+‖ξ(t−τ)‖ri+θ△), | (3.32) |
in which ˉδ1,δ1>0. With the help of Lemmas 1, 2 and (3.32), one can obtain
|∂Vn∂ξ(t)T(t,ξ(t),ξ(t−τ))|≤˜c02L1−¯γ0(n∑i=1‖ξ(t)‖4σ−ri−θ△‖ξ(t)‖ri+θ△+i∑j=1‖ξ(t−τ)‖4σ−ri−θ△‖ξ(t−τ)‖ri+θ△)≤L1−¯γ0(ˉc02‖ξ(t)‖4σ△+ˉc02‖ξ(t)‖4σ△‖ξ(t−τ)‖4σ△), | (3.33) |
where c02, ˉc02, ˜c02 and ¯γ0=min1≤i≤nγi1 are positive constants. Similar to (3.32), we use δ2 and γi2<1/2 to show that
|gi(t,ˉξ(t),ˉξ(t−τ))Lki|≤1Lkic2i∑j=1(|zj(t)|12p1…pi−1+|zj(t)|2ri+θ2rjp1…pi−1)+1Lkic2c2i∑j=1(|zj(t−τ)|12p1…pi−1+|zj(t−τ)|2ri+θ2rjp1…pi−1)≤L12−γi2(‖ξ(t)‖+‖ξ(t−τ)‖)ri+θ2≤δ2L12−γi2(‖ξ(t)‖ri+θ2△+‖ξ(t−τ)‖ri+θ2△). |
Using Lemma 1, Lemma 3, Lemma 4 and (3.34), one obtains
12Tr{ψ(t,ξ(t),ξ(t−τ))∂2Vn∂ξ2⋅ψT(t,ξ(t),ξ(t−τ))}≤12r√rn∑i,j=1|∂2Vn∂ξ2||ψT(t,ξ(t),ξ(t−τ))||ψ(t,ξ(t),ξ(t−τ))|≤˜c03L1−¯γ0n∑i,j=1‖ξ(t)‖4σ−ri−rj−θ△×(‖ξ(t−τ)‖ri+θ2△+‖ξ(t)‖ri+θ2△)×(‖ξ(t)‖rj+θ2△+‖ξ(t−τ)‖rj+θ2△)≤˜c03L1−¯γ0(c03‖ξ(t)‖4σ△+˜c03ˉc03‖ξ(t)‖4σ△‖ξ(t−τ)‖4σ△)≤L1−¯γ0(c03‖ξ(t)‖4σ△+ˉc03‖ξ(t)‖4σ△‖ξ(t−τ)‖4σ△), | (3.34) |
in which ˜γ0=min1≤i,j≤n{γi2+γj2}>0, c03>0, ˉc03>0 and ˜c03>0 are constants. Based on L>1, we have
V(ξ)≤Vn(ξ)+h1+h21−γL1−γ0∫tt−τ‖ξ‖4σΔdη. |
By Definition 1, (3.26), (3.31), (3.33) and (3.34), one has
LV≤∂Vn∂ξLR(ξ)+∂Vn(ξ)∂ξT(t,ξ(t),ξ(t−τ)+12Tr{ψT(t,ξ(t),ξ(t−τ))∂2Vn∂ξ2ψ(t,ξ(t),ξ(t−τ))}+(h1+h2)L1−γ0⋅(11−γ‖ξ(t)‖4σ△−‖ξ(t−τ)‖4σ△)≤−L(c01−(c02+c03+h1+h21−γ)L−γ0)⋅‖ξ‖4σ△, | (3.35) |
which satisfies γ0=min{ˉγ0, ˜γ0}<1. Because c01 is a constant independent of c02,c03, we choose L>L∗=max{(c02+c03+h1+h21−γc1)1γ0,1}, and there exists a constant B=c01−(c02+c03+h1+h21−γ)L−γ0>0, such that
LV≤−LB‖ξ‖4σ△=−c0|ξ‖4σ△. |
With the help of the above formula and (3.28), one obtains
LV(ξ(t))≤−(c0˜c)α_22|(|ξ(t)|). |
Briefly, following Steps 1–3, the system has a unique solution on [−d,∞], and ξ(t)=0 is globally asymptotically stable in probability.
Step 4: Because (3.20) is an equivalent transformation, the system composed by (1) and up=Lkn+1vp is similar to the systems (3.20) and (3.22).
Remark 5. Compared with [24], we construct a state-feedback controller independent of time delays for the stochastic nonlinear system. Compared with [30], we use the methods of adding a power integrator to relax the nonlinear growth condition to cover both high-order and low-order nonlinearities. Not only does it not need to know anything information about the unknown function, but also it can reduce burdensome computations.
Remark 6. The homogeneous domination method is used for the first time to solve the stabilization problem of stochastic high-order and low-order nonlinear system (1.1) with time-delay.
Remark 7. In this paper, it is hard to adopt a Lyapunov-Krasovskii functional. In order to solve the above the problem, a suitable Lyapunov-candidate-function is designed to guarantee good system performance, and stabilization analysis is proposed to save better resources
Remark 8. The construction of the controller effectively keeps away from the zero-division problem of ∂2ξ∗μ/rii∂ξ2j. It need be noted that the non-zero-division problem and the locally Lipschitz condition (see Step 1 in the proof of Theorem 1) should to be guaranteed simultaneously, which will increase more difficulties.
Consider the following stochastic high-order and low-order nonlinear systems with time-delay:
{dx1(t)=[x32(t)+x22(t)x1(t−1)]dt+14x1(t)sinx1(t−1)dω(t),dx2(t)=[u3(t)+x2(t)cosx2(t−1)]dt. | (4.1) |
One can see that Assumption 1 is satisfied with p1=p2=3,τ=1,C=1,r1=1,θ=25. One can easily get
|f1|≤(|z1|+|z1|75+|z1(t−1)|+|z1(t−1)|75)/5, |
|g1|≤(|z1|+|z1|165+|z1(t−1)|+|z1(t−1)|165)/8, |
|f2|≤(|z1|13+|z1|1345+|z2|13+|z2|136+|z1(t−1)|13+|z1(t−1)|1345+|z2(t−1)|13+|z2(t−1)|136)/5. | (4.2) |
In this simulation, we choose V1(z1)=15z51+110z101+2∫tt−1(z41+z4σr11)dl. Several calculations lead to
LV1≤−2(z41+z5251)−(z41(t−1)+z1(t−1)525)+(z1+z91)(x32−α31), | (4.3) |
where αp1=−(2n+β1)(z31+z751). By choosing V2(¯η2)=V1(η1)+WL2+WH2+WD2, a direct calculation leads to
LV2≤−(z41+z5251)−(z42+z26072). | (4.4) |
From the previous manipulations, one obtains the following actual controller
u(t)=−ϱ19(z2+z13452)19=−306.771(z2+z13452)19. | (4.5) |
The initial condition can be given as ξ0(θ)≡[1,−1]T. Figure 1 illustrates that the globally asymptotically stable in probability has been achieved and the responses of (4.7) is given in Figure 2.
In this technical note, we investigate the state feedback stabilization problem of stochastic high-order and low-order nonlinear systems with time-delay successfully. According to the homogeneous domination method and the design of integral Lyapunov functions, the control strategy is achieved with the controller design. The above results indicate that the closed-loop system is globally asymptotically stable in probability. There still remain problems to be investigated, such as how to take into account output feedback control and how to extend our results under weaker conditions.
This work is supported by National Natural Science Foundation of China under Grant 62173208, Taishan Scholar Project of Shandong Province of China under Grant tsqn202103061, Shandong Qingchuang Science and Technology Program of Universities under Grant 2019KJN036.
The authors declare no conflicts of interest.
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