This paper investigates the problem of state-feedback stabilization for a class of upper-triangular stochastic nonlinear systems with time-varying control coefficients. By introducing effective coordinates, the original system is transformed into an equivalent one with tunable gain. After that, by using
the low gain homogeneous domination technique and choosing the low gain parameter skillfully, the closed-loop system can be proved to be globally asymptotically stable in probability. The efficiency of the state-feedback controller is demonstrated by a simulation example.
1. Introduction
Consider a class of upper-triangular stochastic nonlinear systems with time-varying control coefficients described by
(1)dx1=(d1(t)x2+f1(x~3))dt+g1T(x~3)dω,dx2=(d2(t)x3+f2(x~4))dt+g2T(x~4)dω,⋮dxn-2=(dn-2(t)xn-1+fn-2(x~n))dt+gn-2T(x~n)dω,dxn-1=dn-1(t)xndt,dxn=dn(t)udt,
where x=(x1,…,xn)T∈ℝn, u∈ℝ are the measurable state and the input of system, respectively. x~i=(xi,…,xn)T. 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 functions fi:ℝn-i-1→ℝ and gi:ℝn-i-1→ℝr, i=1,…,n-2, are assumed to be 𝒞1 with their arguments and fi(0)=0, gi(0)=0. di:R+→R, i=1,…,n, are unknown time-varying control coefficients with known sign.
In recent years, the global controller design for stochastic nonlinear systems has been attracting more and more attention. According to the difference of selected Lyapunov functions, the existing literature on controller design can be mainly divided into two types. One type is to derive the backstepping controller design by using quadratic Lyapunov function and a risk-sensitive cost criterion [1–3]. Another essential improvement belongs to Krstić and Deng. By introducing the quartic Lyapunov function, [4–12] present asymptotical stabilization control under the assumption that the nonlinearities equal zero at the equilibrium point of the open-loop system. Subsequently, for several classes of stochastic high-order nonlinear systems, by combining Krstić and Deng’s method with stochastic analysis, [13, 14] study the problem of state-feedback stabilization and the output-feedback stabilization problem is considered in [15, 16].
The study of stabilization control for upper-triangular nonlinear systems has long been recognized as difficult due to the inherent nonlinearity. In the existing literature, most results are established using the nested-saturation method [17, 18] and forwarding technique [19]. When no a priori information of the system nonlinearities is known, the work [20] proposes a universal stabilizer for feedforward nonlinear systems by employing a switching controller. Note that the listed results above do not consider the stochastic noise. However, from both practical and theoretical points of view, it is more important to study the control of upper-triangular stochastic nonlinear systems with time-varying control coefficients. Therefore, in this paper, under some appropriate assumptions, we consider the stabilization for system (1). To the best of the authors' knowledge, there are not any results about this topic.
In this paper, based on the low gain homogeneous domination technique, for system (1), we design a stabilization state-feedback controller, under which the closed-loop systems can be proved to be globally asymptotically stable in probability.
The contributions of this paper are highlighted as follows.
This paper is the first result about state-feedback stabilization of upper-triangular stochastic nonlinear systems with time-varying control coefficients.
Due to the complex of upper-triangular system structure, how to deal with stochastic noise and time-varying control coefficients in the controller design is a nontrivial work.
The remainder of this paper is organized as follows. Section 2 offers some preliminary results. The state-feedback controller is designed and analyzed in Section 3. After that, in Section 4, a simulation example is presented to show the effectiveness of the state-feedback controller. Finally, the paper is concluded in Section 5.
2. Preliminary Results
The following notation will be used throughout the paper. ℝ+ denotes the set of all nonnegative real numbers. For a given vector or matrix X, XT denotes its transpose, Tr{X} denotes its trace when X is square, and |X| is the Euclidean norm of a vector X. 𝒞i denotes the set of all functions with continuous ith partial derivatives. 𝒦 denotes the set of all functions: ℝ+→ℝ+, which are continuous, strictly increasing, and vanishing at zero; 𝒦∞ denotes the set of all functions which are of class 𝒦 and unbounded; 𝒦ℒ denotes the set of all functions β(s,t): ℝ+×ℝ+→ℝ+, which are of 𝒦 for each fixed t and decrease to zero as t→∞ for each fixed s.
Consider the following stochastic nonlinear system:
(2)dx=f(x)dt+gT(x)dω,
where x∈ℝn is the state of the system and ω is an r-dimensional standard Wiener process defined on the probability space (Ω,ℱ,P). The Borel measurable functions f:ℝn→ℝn and gT:ℝn→ℝn×r are local Lipschitz in x∈ℝn.
The following definitions and lemma will be used throughout the paper.
Definition 1 (see [5]).
For any given V(x)∈𝒞2 associated with stochastic system (2), the differential operator ℒ is defined as
(3)ℒV(x)≜∂V(x)∂xf(x)+12Tr{g(x)∂2V(x)∂x2gT(x)}.
Definition 2 (see [5]).
For the stochastic system (2) with f(0)=0,g(0)=0, the equilibrium x(t)=0 of (2) is globally asymptotically stable (GAS) in probability if, for any ε>0, there exists a class 𝒦ℒ function β(·,·) such that P{|x(t)|<β(|x0|,t)}≥1-ε for any t≥0 and x0∈ℝn∖{0}.
Lemma 3 (see [5]).
Consider the stochastic system (2); if there exist a 𝒞2 function V(x), class 𝒦∞ functions α1 and α2, constants c1>0 and c2≥0, and a nonnegative function W(x) such that
(4)α1(|x|)≤V(x)≤α2(|x|),ℒV≤-c1W(x)+c2,
then
for (2), there exists an almost surely unique solution on [0,∞);
when c2=0, f(0)=0, g(0)=0, and W(x)=α3(|x|), where α3(·) is a class 𝒦 function, then the equilibrium x=0 is GAS in probability and P{limt→∞|x(t)|=0}=1.
3. Controller Design and Stability Analysis
The following assumptions are made on system (1).
Assumption 1.
For i=1,…,n, there exists a constant b>0 such that
(5)|fi(x~i+2)|≤b(|xi+2|+⋯+|xn|),|gi(x~i+2)|≤b(|xi+2|+⋯+|xn|).
Assumption 2.
Without loss of generality, the sign of di(t) is assumed to be positive, and there exist known positive constants λi and μi such that, for any t∈ℝ+ and i=1,…,n,
(6)0<λi≤di(t)≤μi.
Remark 4.
From Assumption 1, the system investigated has an upper-triangular form. Due to the complex of upper-triangular system structure and the effect of stochastic noise, the stabilization of such systems is usually very difficult. In this paper, by using the low gain homogeneous domination approach, the state-feedback stabilization problem is investigated for the first time.
Remark 5.
By Assumption 2, we know that di(t)s are time-varying control coefficients; how to effectively deal with them in the design process is nontrivial work.
Firstly, introduce the following coordinate transformation:
(7)zi=xiεi-1,υ=uεn,i=1,…,n,
where 0<ε<1 is a parameter to be designed. System (1) can be rewritten as
(8)dz1=(εd1(t)z2+f-1(z~3))dt+g¯1T(z~3)dω,dz2=(εd2(t)z3+f¯2(z~4))dt+g¯2T(z~4)dω,⋮dzn-2=(εdn-2(t)zn-1+f¯n-2(z~n))dt+g¯n-2T(z~n)dω,dzn-1=εdn-1(t)zndt,dzn=εdn(t)υdt,
where f-i(z~i+2)=fi(x~i+2)/εi-1, g-i(z~i+2)=gi(x~i+2)/εi-1.
The nominal system for (8) is
(9)dz1=d1(t)z2,dz2=d2(t)z3,⋮dzn-2=dn-2(t)zn-1,dzn-1=dn-1(t)zndt,dzn=dn(t)υdt.
Theorem 6.
For nominal system (9), with Assumption 2, one can design a stabilizing state-feedback controller to guarantee that
the closed-loop system has an almost surely unique solution on [0,∞);
the equilibrium of the closed-loop system is GAS in probability.
Proof.
The controller design process proceeds step by step.
Step 1. Defining ξ1=z1 and choosing V1=(1/4)z14, from (9), it follows that
(10)ℒV1≤d1(t)z13z2.
Suppose that z2*=-z1α1=-ξ1α1, where α1≥0 is a constant to be chosen. Thus, by Assumption 2, we have
(11)d1(t)z13z2*≤λ1z13z2*≤0.
By (10) and (11), one gets
(12)ℒV1≤λ1z13z2*+d1(t)z13(z2-z2*).
Choosing the virtual smooth control z2* as
(13)z2*=-nλ1ξ1≜-ξ1α1,
which substitutes into (12), yields
(14)ℒV1≤-nξ14+d1(t)z13(z2-z2*).
Deductive Step. Assume that, at step k-1, there are a 𝒞2, proper and positive definite Lyapunov function Vk-1, and the virtual controllers zj* defined by
(15)z1*=0,ξ1=z1-z1*,z2*=-ξ1α1,ξ2=z2-z2*,⋮zk*=-ξk-1αk-1,ξk=zk-zk*,
where αi≥0, 1≤i≤k-1, are positive constants, such that
(16)ℒVk-1(z-k-1)≤-(n-k+2)∑i=1k-1ξi4+dk-1(t)ξk-13(zk-zk*),
where z-k-1=(z1,…,zk-1)T. To complete the induction, at the kth step, one can choose the following Lyapunov function:
(17)Vk(z-k)=Vk-1(z-k-1)+14ξk4,
where z-k=(z1,…,zk)T.
By (15)–(17), one has
(18)ℒVk(z-k)≤-(n-k+2)∑i=1k-1ξi4+dk-1(t)ξk-13ξk+ξk3(dk(t)zk+1-∑i=1k-1∂zk*∂zidi(t)zi+1).
By using Young’s inequality and Assumption 2, one has
(19)dk-1(t)ξk-13ξk≤12ξk-14+ckξk4,-ξk3∑i=1k-1∂zk*∂zidi(t)zi+1≤ck1|ξk|3∑i=1k|zi|≤12∑i=1k-1ξi4+c^kξk4,
where ck>0, ck1>0, and c^k>0 are constants. Suppose that
(20)zk+1*=-ξkαk,
where αk≥0 is a constant to be chosen. Then, by Assumption 2, one has
(21)dk(t)ξk3zk+1*≤λkξk3zk+1*.
Substituting (19) and (21) into (18) yields
(22)ℒVk(z-k)≤-(n-k+1)∑i=1k-1ξi4+dk(t)ξk3(zk+1-zk+1*)+λkξk3zk+1*+(ck+c^k)ξk4.
Choosing the virtual smooth control
(23)zk+1*=-1λk(n-k+1+ck+c^k)ξk≜-ξkαk,
which substitutes into (22), yields
(24)ℒVk(z-k)≤-(n-k+1)∑i=1kξi4+dk(t)ξk3(zk+1-zk+1*).
Step n. By choosing the actual control law
(25)υ=-ξnαn,
where αn≥0 is a constant and ξn=xn-xn*, one gets
(26)ℒVn(z-n)≤-∑i=1nξi4,
where
(27)Vn(z-n)=Vn-1(z-n-1)+14ξn4.
Finally, based on (26) and (27), by Lemma 3, one immediately gets the conclusion.
Now, we are in a position to get the main results of this paper.
Theorem 7.
If Assumptions 1 and 2 hold for the upper-triangular stochastic nonlinear systems (1), with the coordinate transformation (7), by appropriately choosing the parameter 0<ε<1, then, under the state-feedback controller (25), one has the following:
the closed-loop system has an almost surely unique solution on [0,∞);
the equilibrium of the closed-loop system is GAS in probability.
Proof.
For system (8), with the state-feedback controller (25) and Lyapunov function (27), one has
(28)ℒVn(z-n)≤-ε∑i=1nξi4+∑i=1n∂Vn∂zif-i(z~i+2)+12Tr{G∂2Vn∂z2GT},
where z=(z1,…,zn)T, G=(g-1,…,g-n-2,0,0). From (15) and (27), one has
(29)Vn(z-n)=14∑i=1nξi4=14∑i=1n(zi+ci,i-1zi-1+⋯+ci,1z1)4,
where ci,j, j=1,…,i-1, are constants. By (7), (15) and Assumption 1, one can get
(30)|f¯i(z~i+2)|=|fi(x~i+2)εi-1|≤bε2∑j=i+2n|zj|≤bε2∑j=i+2n(|ξj|+αj-1|ξj-1|).
By Young’s inequality, using (29) and (30), one has
(31)∑i=1n∂Vn∂zif-i(z~i+2)≤bfε2∑i=1nξi4.
Similarly, one can prove that
(32)12Tr{G∂2Vn∂z2GT}≤bgε2∑i=1nξi4.
Substituting (31) and (32) into (28), one has
(33)ℒVn(z-n)|(8)≤-ε∑i=1nξi4+(bf+bg)ε2∑i=1nξi4=-ε(1-(bf+bg)ε)∑i=1nξi4.
By choosing 0<ε<1 appropriately, (33) can be written as
(34)ℒVn(z-n)|(8)≤-c0∑i=1nξi4,
where c0>0 is a constant.
By (34) and the coordinate transformation (7), using Lemma 3, the conclusions hold.
Remark 8.
Theorems 6 and 7 provide us a new perspective to deal with the state-feedback control problem for upper-triangular stochastic nonlinear systems with time-varying coefficients. The main technical obstacle in the Lyapunov design for stochastic upper-triangular systems is that Itô stochastic differentiation involves not only the gradient but also the higher order Hessian term. The traditional design methods are invalid to deal with these terms. However, with the design methodology provided in Theorems 6 and 7, there is no need to estimate the bounds of drift and diffusion terms step by step. Based on this technique, a homogeneous nonlinear controller for the nominal nonlinear system is firstly constructed. Then we will design a scaled controller which can effectively dominate the drift and diffusion terms by taking advantage of the homogenous structure of the controller.
4. A Simulation Example
Consider the following system:
(35)dx1=((2-sin2t)x2+x3sinx3)dt+x3cosx3dω,dx2=(2-cost)x3dt,dx3=(1+sin2t)udt.
Obviously, Assumptions 1 and 2 hold.
Introduce the following coordinate transformation:
(36)z1=x1,z2=x2ε,z3=x3ε2,υ=uε3,
where 0<ε<1 is a design parameter. Then (35) can be written as
(37)dz1=(ε(2-sin2t)z2+ε2z3sin2(ε2z3))dt+ε2z3cos(ε2z3)dω,dz2=ε(2-cost)z3dt,dz3=ε(1+sin2t)υdt.
By following the design procedure in Section 3, one gets
(38)υ(z1,z2,z3)=-1310(372z1+124z2+z3).
By choosing ε=0.001, with the initial values z1(0)=3, z2(0)=2, and z3(0)=5, Figure 1 gives the system response of the closed-loop system consisting of (35)–(38), from which the efficiency of the tracking controller is demonstrated.
The response of closed-loop system (35)–(38).
5. Concluding Remarks
For a class of upper-triangular stochastic nonlinear systems with time-varying control coefficients, this paper investigates the state-feedback stabilization problem. The designed controller can guarantee that the closed-loop system has a unique solution and the closed-loop system can be proved to be GAS in probability.
There are many related problems to be investigated, for example, how to generalize the result in this paper to more general stochastic upper-triangular nonlinear systems.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by National Natural Science Foundation (NNSF) of China under Grant nos. 61104128, 61174097; China Postdoctoral Science Foundation under Grant nos. 2012M520418, 2013T60185; Promotive Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province under Grant no. BS2013DX001; School Research Fund of Ludong University under Grant no. LY2012014.
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