# State Feedback Control for Stochastic Feedforward Nonlinear Systems

Authors: Liang Liu; Ming Gao
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper considers the state feedback stabilization problem for a class of stochastic feedforward
nonlinear systems. By using the homogeneous domination approach, a state feedback controller is constructed to render the closed-loop system globally asymptotically stable in probability. A simulation example is provided to show the effectiveness of the designed controller.


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## Body

## 1. Introduction

Consider the following stochastic feedforward nonlinear systems described by(1)dx1=x2dt+f1(x-3,u)dt+g1T(x-2,u)dω,⋮dxn-2=xn-1dt+fn-2(x-n,u)dt+gn-2T(x-n-1,u)dω,dxn-1=xndt+fn-1(u)dt+gn-1T(x-n,u)dω,dxn=udt,
where x=(x1,…,xn)T∈Rn and u∈R are the system state and input, respectively. x-i=(xi,…,xn), i=2,…,n. ω is an m-dimensional standard Wiener process defined on the complete probability space (Ω,ℱ,{ℱt}t≥0,P) with Ω being a sample space, ℱ being a σ-field, {ℱt}t≥0 being a filtration, and P being a probability measure. fi:Rn-i-1×R→R and gi:Rn-i×R→Rm are assumed to be locally Lipschitz with fi(0,0)=0 and gi(0,0)=0, i=1,…,n-1.Since the stochastic stability theory was established, the stabilization problems for stochastic lower-triangular nonlinear systems have made a great number of achievements in recent years; see, for example, [1–16] and the other references.Feedforward system is an another important class of nonlinear systems. From the theoretical viewpoint, they are not feedback linearizable and cannot be stabilized by the conventional backstepping method; to some extent, the control problem of these systems is more difficult than feedback systems. On the other hand, some simple physical models, for example, the cart-pendulum system in [17] and the ball-beam with a friction term in [18], can be described by equations with the feedforward form. In recent papers on feedforward systems, [19] studied delay-adaptive feedback for linear systems. The input delay compensation for forward complete and strict-feedforward nonlinear systems was solved by [20]. Reference [21] considered the adaptive stabilization problem for feedforward nonlinear systems with time delays by taking a nested saturation feedback. The global output feedback stabilization problem for system (1) without stochastic noise was addressed by [22]. Reference [23] investigated the state and output feedback control for a class of feedforward nonlinear time-delay systems. For high-order nonlinear feedforward systems, [24] considered global stabilization problem by using the generalized adding a power integrator method and a series of nested saturation functions, [25, 26] respectively dealt with the state feedback control for this kind of systems with time delay, but all these results are limited to deterministic systems. Due to the special form of this system, there are few results on stochastic feedforward systems at present.The purpose of this paper is to solve the state feedback stabilization problem of system (1) by using the homogeneous domination approach in [22]. The underlying idea of this approach is that the homogeneous controller is first developed without considering the drift and diffusion terms, and then a low gain is introduced to the state feedback controller to dominate the drift and diffusion terms. By adopting this method, a state feedback controller is explicitly constructed to render the closed-loop system globally asymptotically stable in probability.The paper is organized as follows. Section2 provides some preliminary results. The design and analysis of state feedback controller is given in Sections 3 and 4, following a simulation example in Section 5. Section 6 concludes this paper.


## 2. Preliminary Results

The following notations, definitions, and lemmas are to be used throughout the paper.R
+ denotes the set of all nonnegative real numbers and Rn denotes the real n-dimensional space. 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,x(0)=x0∈Rn,
where x∈Rn is the system state and ω is an m-dimensional standard Wiener process defined on the complete probability space (Ω,ℱ,{ℱt}t≥0,P). The Borel measurable functions f:Rn→Rn and g:Rn→Rm×n are locally Lipschitz with f(0)=0 and g(0)=0.Definition 1 (see [1]).
For any givenV(x)∈𝒞2 associated with stochastic system (2), the differential operator ℒ is defined as
(3)ℒV=∂V∂xf(x)+12Tr{g(x)∂2V∂x2gT(x)}.Definition 2 (see [1]).
For system (2), the equilibrium x=0 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}.Definition 3 (see [22]).
For fixed coordinates(x1,…,xn)∈Rn and real numbers ri>0, i=1,…,n.(i)
The dilationΔε(x) is defined by Δε(x)=(εr1x1,…,εrnxn) for any ε>0 and r1,…,rn are called the weights of the coordinates. For simplicity, we define dilation weight as Δ=(r1,…,rn).(ii)
A functionV∈𝒞(Rn,R) is said to be homogeneous of degree τ if there is a real number τ∈R such that V(Δε(x))=ετV(x1,…,xn) for any x∈Rn∖{0}, ε>0.(iii)
A vector fieldh∈𝒞(Rn,Rn) is said to be homogeneous of degree τ if there is a real number τ∈R such that hi(Δε(x))=ετ+rihi(x) for any x∈Rn∖{0}, ε>0, i=1,…,n.(iv)
A homogeneousp-norm is defined as ∥x∥Δ,p=(∑i=1n‍|xi|p/ri)1/p for any x∈Rn, where p≥1 is a constant. For simplicity, in this paper, we choose p=2 and write ∥x∥Δ for ∥x∥Δ,2.Lemma 4 (see [1]).
Consider system (2) and suppose that there exist a 𝒞2 function V(x), class 𝒦∞ functions α1 and α2, and a class 𝒦 function α3 such that
(4)α1(|x|)≤V(x)≤α2(|x|),ℒV≤-α3(|x|).
Then there exists an almost surely unique solution on [0,∞), the equilibrium x=0 is GAS in probability, and P{limt→∞|x(t)|=0}=1.Lemma 5 (see [22]).
Given a dilation weightΔ=(r1,…,rn), suppose that V1(x) and V2(x) are homogeneous functions of degrees τ1 and τ2, respectively. Then V1(x)V2(x) is also homogeneous with respect to the same dilation weight Δ. Moreover, the homogeneous degree of V1·V2 is τ1+τ2.Lemma 6 (see [22]).
Suppose thatV:Rn→R is a homogeneous function of degree τ with respect to the dilation weight Δ; then(i)
∂V/∂xi is homogeneous of degree τ-ri with ri being the homogeneous weight of xi;(ii)
there is a constantc such that V(x)≤c∥x∥Δτ. Moreover, if V(x) is positive definite, then V(x)≥c_∥x∥Δτ, where c_ is a positive constant.Lemma 7 (see [4]).
Letc and d be positive constants. For any positive number γ-, then |x|c|y|d≤(c/(c+d))γ-|x|c+d+(d/(c+d))γ-(-c/d)|y|c+d.


## 3. Design of State Feedback Controller

### 3.1. Assumption

For system (1), we need the following Assumption.Assumption 8.
Fori=1,…,n-1, there exist positive constants a1 and a2 such that
(5)|fi(x-i+2,u)|≤a1(|xi+2|+⋯+|xn|+|u|),|gi(x-i+1,u)|≤a2(|xi+1|+⋯+|xn|+|u|),
where xn+1=0.Remark 9.
Obviously, system (1) satisfying Assumption 8 is a stochastic feedforward nonlinear system. As discussed in the deterministic feedforward references such as [21–26], that is, gi(·)≡0 for i=1,…,n-1, and stochastic feedforward reference [27], Assumption 8 is a general and frequently used condition.
Due to the special form of stochastic feedforward system, almost all the existing methods fail to be applicable to solve the stabilization problem of system (1). Based on this reason, the objective of this paper is to design a state feedback controller for system (1) such that the equilibrium of the closed-loop system is globally asymptotically stable in probability.
To achieve this objective, we first introduce the following coordinate transformation:(6)ηi=xiκi-1,v=uκn,i=1,…,n,
where 0<κ<1 is a designed constant. With the help of (6), (1) can be rewritten as
(7)dη1=κη2dt+f-1(η-3,v)dt+g-1T(η-2,v)dω,⋮dηn-2=κηn-1dt+f-n-2(η-n,v)dt+g-n-2T(η-n-1,v)dω,dηn-1=κηndt+f-n-1(v)dt+g-n-1T(η-n,v)dω,dηn=κvdt,
where f-i=fi/κi-1 and g-i=gi/κi-1, i=1,…,n-1.


### 3.2. State Feedback Control of Nominal Nonlinear System

We construct a state feedback controller for the following nominal nonlinear system of (7):
(8)dηi=κηi+1dt,i=1,…,n-1,dηn=κvdt.Step 1.
Introducingξ1=η1 and choosing V1(η1)=(1/4)ξ14, by (3) and (8), it can be verified that ℒV1=κξ13η2. The first virtual controller
(9)η2*=-c11ξ1=:-α1ξ1,c11>0
leads to ℒV1≤-κc11ξ14+κξ13(η2-η2*).
Step
i  (i=2,…,n). In this step, one can obtain the similar property for the ith subsystem, which is presented by the following lemma.Lemma 10.
Suppose that at Stepi-1 there are a set of virtual controllers η1*,…,ηi* defined by
(10)η1*=0,ξ1=η1-η1*=η1,ηk*=-αk-1ξk-1,ξk=ηk-ηk*,k=2,…,i,
such that the (i-1)th Lyapunov function Vi-1(η-i-1)=(1/4)∑j=1i-1‍ξj4 satisfies
(11)ℒVi-1≤-κ∑j=1i-1‍ci-1,jξj4+κξi-13(ηi-ηi*),
where αj, ci-1,j, j=1,…,i-1, are positive constants. Then there exists a virtual control law ηi+1*=-αiξi such that
(12)ℒVi≤-κ∑j=1i‍cijξj4+κξi3(ηi+1-ηi+1*),
where Vi(η-i)=(1/4)∑j=1i‍ξj4.Proof.
From (3), (8), (10), and (11), it follows that
(13)ℒVi≤-κ∑j=1i-1‍ci-1,jξj4+κξi3ηi+1+κξi-13ξi-κξi3∑k=1i-1‍∂ηi*∂ηkηk+1.
We concentrate on the last two terms on the right-hand side of (13).
Using (10) and Lemma 7, one obtains
(14)ξi-13ξi≤li,i-1,1ξi-14+ρi1ξi4,-ξi3∑k=1i-1‍∂ηi*∂ηkηk+1≤|ξi|3|∑k=1i-1‍αi-1⋯αk(ξk+1-αkξk)|≤|ξi|3(∑k=1i-1‍(αi-1⋯αk-1hhhhhhhh+αi-1⋯αk+1αk2)|ξk|+αi-1|ξi|∑k=1i-1‍)≤∑k=1i-1‍lik2ξk4+ρi2ξi4,
where li,i-1,1, lik2  (k=1,…,i-1), ρi1, and ρi2 are positive constants, α0=0.
Choosing(15)cij={ci-1,j-lij2>0,j=1,…,i-2,ci-1,i-1-li,i-1,1-li,i-1,2>0,j=i-1,ηi+1*=-(cii+ρi1+ρi2)ξi=:-αiξi,cii>0,
and substituting (14)-(15) into (13), one gets the desired result.At Stepn, choosing Vn(η-n)=(1/4)∑i=1n‍ξi4 and
(16)v=ηn+1*=-αnξn=-(α-nηn+α-n-1ηn-1+⋯+α-1η1),
by (3), (12), and (16), one gets
(17)ℒVn≤-κ∑i=1n‍cniξi4+κξn3(v-ηn+1*)=-κ∑i=1n‍cniξi4,
where ξn=ηn-ηn*, α-i=αn⋯αi, cni, and i=1,…,n, are positive constants. The system (7) and (16) can be written as
(18)dη=κE(η)dt+F(η,v)dt+GT(η,v)dω,
where η=η-n=(η1,…,ηn)T, E(η)=(η2,…,ηn,v)T, F(η,v)=(f-1,…,f-n-1,0)T, and G(η,v)=(g-1,…,g-n-1,0). Introducing the dilation weight Δ=(1,1,…,1︸forη1,…,ηn), by (10) and Vn(η)=(1/4)∑i=1n‍ξi4, one obtains
(19)Vn(Δε(η))=14∑i=1n‍(εηi+αi-1εηi-1+⋯+αi-1⋯α1εη1)4=ε4Vn(η),
from which and Definition 3, we know that Vn(η) is homogeneous of degree 4.


## 3.1. Assumption

For system (1), we need the following Assumption.Assumption 8.
Fori=1,…,n-1, there exist positive constants a1 and a2 such that
(5)|fi(x-i+2,u)|≤a1(|xi+2|+⋯+|xn|+|u|),|gi(x-i+1,u)|≤a2(|xi+1|+⋯+|xn|+|u|),
where xn+1=0.Remark 9.
Obviously, system (1) satisfying Assumption 8 is a stochastic feedforward nonlinear system. As discussed in the deterministic feedforward references such as [21–26], that is, gi(·)≡0 for i=1,…,n-1, and stochastic feedforward reference [27], Assumption 8 is a general and frequently used condition.
Due to the special form of stochastic feedforward system, almost all the existing methods fail to be applicable to solve the stabilization problem of system (1). Based on this reason, the objective of this paper is to design a state feedback controller for system (1) such that the equilibrium of the closed-loop system is globally asymptotically stable in probability.
To achieve this objective, we first introduce the following coordinate transformation:(6)ηi=xiκi-1,v=uκn,i=1,…,n,
where 0<κ<1 is a designed constant. With the help of (6), (1) can be rewritten as
(7)dη1=κη2dt+f-1(η-3,v)dt+g-1T(η-2,v)dω,⋮dηn-2=κηn-1dt+f-n-2(η-n,v)dt+g-n-2T(η-n-1,v)dω,dηn-1=κηndt+f-n-1(v)dt+g-n-1T(η-n,v)dω,dηn=κvdt,
where f-i=fi/κi-1 and g-i=gi/κi-1, i=1,…,n-1.


## 3.2. State Feedback Control of Nominal Nonlinear System

We construct a state feedback controller for the following nominal nonlinear system of (7):
(8)dηi=κηi+1dt,i=1,…,n-1,dηn=κvdt.Step 1.
Introducingξ1=η1 and choosing V1(η1)=(1/4)ξ14, by (3) and (8), it can be verified that ℒV1=κξ13η2. The first virtual controller
(9)η2*=-c11ξ1=:-α1ξ1,c11>0
leads to ℒV1≤-κc11ξ14+κξ13(η2-η2*).
Step
i  (i=2,…,n). In this step, one can obtain the similar property for the ith subsystem, which is presented by the following lemma.Lemma 10.
Suppose that at Stepi-1 there are a set of virtual controllers η1*,…,ηi* defined by
(10)η1*=0,ξ1=η1-η1*=η1,ηk*=-αk-1ξk-1,ξk=ηk-ηk*,k=2,…,i,
such that the (i-1)th Lyapunov function Vi-1(η-i-1)=(1/4)∑j=1i-1‍ξj4 satisfies
(11)ℒVi-1≤-κ∑j=1i-1‍ci-1,jξj4+κξi-13(ηi-ηi*),
where αj, ci-1,j, j=1,…,i-1, are positive constants. Then there exists a virtual control law ηi+1*=-αiξi such that
(12)ℒVi≤-κ∑j=1i‍cijξj4+κξi3(ηi+1-ηi+1*),
where Vi(η-i)=(1/4)∑j=1i‍ξj4.Proof.
From (3), (8), (10), and (11), it follows that
(13)ℒVi≤-κ∑j=1i-1‍ci-1,jξj4+κξi3ηi+1+κξi-13ξi-κξi3∑k=1i-1‍∂ηi*∂ηkηk+1.
We concentrate on the last two terms on the right-hand side of (13).
Using (10) and Lemma 7, one obtains
(14)ξi-13ξi≤li,i-1,1ξi-14+ρi1ξi4,-ξi3∑k=1i-1‍∂ηi*∂ηkηk+1≤|ξi|3|∑k=1i-1‍αi-1⋯αk(ξk+1-αkξk)|≤|ξi|3(∑k=1i-1‍(αi-1⋯αk-1hhhhhhhh+αi-1⋯αk+1αk2)|ξk|+αi-1|ξi|∑k=1i-1‍)≤∑k=1i-1‍lik2ξk4+ρi2ξi4,
where li,i-1,1, lik2  (k=1,…,i-1), ρi1, and ρi2 are positive constants, α0=0.
Choosing(15)cij={ci-1,j-lij2>0,j=1,…,i-2,ci-1,i-1-li,i-1,1-li,i-1,2>0,j=i-1,ηi+1*=-(cii+ρi1+ρi2)ξi=:-αiξi,cii>0,
and substituting (14)-(15) into (13), one gets the desired result.At Stepn, choosing Vn(η-n)=(1/4)∑i=1n‍ξi4 and
(16)v=ηn+1*=-αnξn=-(α-nηn+α-n-1ηn-1+⋯+α-1η1),
by (3), (12), and (16), one gets
(17)ℒVn≤-κ∑i=1n‍cniξi4+κξn3(v-ηn+1*)=-κ∑i=1n‍cniξi4,
where ξn=ηn-ηn*, α-i=αn⋯αi, cni, and i=1,…,n, are positive constants. The system (7) and (16) can be written as
(18)dη=κE(η)dt+F(η,v)dt+GT(η,v)dω,
where η=η-n=(η1,…,ηn)T, E(η)=(η2,…,ηn,v)T, F(η,v)=(f-1,…,f-n-1,0)T, and G(η,v)=(g-1,…,g-n-1,0). Introducing the dilation weight Δ=(1,1,…,1︸forη1,…,ηn), by (10) and Vn(η)=(1/4)∑i=1n‍ξi4, one obtains
(19)Vn(Δε(η))=14∑i=1n‍(εηi+αi-1εηi-1+⋯+αi-1⋯α1εη1)4=ε4Vn(η),
from which and Definition 3, we know that Vn(η) is homogeneous of degree 4.


## 4. Stability Analysis

We state the main result in this paper.Theorem 11.
If Assumption8 holds for the stochastic feedforward nonlinear system (1), under the state feedback controller u=κnv and (16), then(i)
the closed-loop system has an almost surely unique solution on[0,∞);(ii)
the equilibrium at the origin of the closed-loop system is GAS in probability.Proof.
We prove Theorem11 by three steps.
Step  1. Since fi and gi are assumed to be locally Lipschitz, so the system consisting of (7) and (16) satisfies the locally Lipschitz condition.
Step  2. By Lemma 6 and (17), there exists a positive constant c01 such that
(20)∂Vn∂ηκE(η)≤-c01κ∥η∥Δ4.
By Assumption 8, (6), (16), and 0<κ<1, one has
(21)|f-i(η-i+2,v)|≤a1(κi+1|ηi+2|+⋯+κn-1|ηn|+κn|v|)κi-1≤a1κ2(|ηi+2|+⋯+|ηn|+|v|)≤λ1κ2∥η∥Δ,
where λ1 is a positive constant. According to Lemmas 5–6 and (21), one obtains
(22)∂Vn∂ηF(η,v)=∑i=1n-1‍∂Vn∂ηif-i(η-i+2,v)≤c02κ2∥η∥Δ4,
where c02 is a positive constant. Similar to (21), there is a positive constant λ2 such that
(23)|g-i(η-i+1,v)|≤λ2κ∥η∥Δ,
from which and Lemmas 5-6, one leads to
(24)12Tr{G(η,v)∂2Vn∂η2GT(η,v)}≤12mm∑i,j=1n-1‍|∂2Vn∂ηi∂ηj||g-i(η-i+1,v)||g-j(η-j+1,v)|≤c03κ2∥η∥Δ4,
where c03 is a positive constant. By (3), (18), (20), (22), and (24), one has
(25)ℒVn=∂Vn∂ηκE(η)+∂Vn∂ηF(η,v)+12Tr{G(η,v)∂2Vn∂η2GT(η,v)}≤-c01κ∥η∥Δ4+(c02+c03)κ2∥η∥Δ4=-κ(c01-(c02+c03)κ)∥η∥Δ4.
Since c01 is a constant independent of c02 and c03, by choosing
(26)0<κ<κ*=:min{1,c01c02+c03},
(25) becomes ℒVn≤-c0∥η∥Δ4, where c0 is a positive constant.
By Steps 1-2 and Lemma4, the system consisting of (7) and (16) has an almost surely unique solution on [0,∞), η=0 is GAS in probability, and P{limt→∞|η|=0}=1.
Step  3. Since (6) is an equivalent transformation, the closed-loop system consisting of (1), u=κnv, and (16) has the same properties as the system (7) and (16). Theorem 11 holds.Remark 12.
This paper extends the homogeneous domination idea from deterministic systems to stochastic system (1) and explicitly constructs a state feedback controller. It should be emphasized that the rigorous proof of Theorem 11 is not an easy work.


## 5. A Simulation Example

Consider the following stochastic nonlinear system:(27)dx1=x2dt+130udt+15sinx2dω,dx2=udt.
It is easy to verify that Assumption 8 is satisfied with a1=1/30 and a2=1/5.Now, we give the controller design of system (27). Introducing the coordinate transformation
(28)η1=x1,η2=x2κ,v=uκ2,
system (27) becomes
(29)dη1=κη2dt+κ230vdt+15sinκη2dω,dη2=κvdt.
Choosing ξ1=η1 and V1(η1)=(1/4)ξ14, we obtain ℒV1≤-2κξ14+κξ13(η2-η2*), where η2*=-2η1=:-α1ξ1. By ξ2=η2-η2* and V2(η-2)=V1(η1)+(1/4)ξ24, a direct calculation leads to
(30)ℒV2≤-2κξ14+κξ13ξ2+κξ23v+κα1ξ23η2.
By Lemma 7, one has
(31)ξ13ξ2≤12ξ14+2732ξ24,α1ξ23η2≤12ξ14+(323+2)ξ24.
Choosing
(32)v=-(3+323+2732)ξ2=:-α2ξ2,
and substituting (31) into (30), it leads to
(33)ℒV2≤-κ(ξ14+ξ24).
By (28) and (32), one obtains the actual controller
(34)u=-α2(κx2+α1κ2x1).
Defining ∥η∥Δ=(η12+η22)1/2, one gets ℒV2≤-κ∥η∥Δ4. From (22) and (24), it follows that
(35)∂V2∂η1·κ230v≤46.7133κ2∥η∥Δ4,12∂2V2∂η12·125sin2κη2≤2κ2∥η∥Δ4.
By (29), one has ℒV2≤-κ(1-48.7133κ)∥η∥Δ4, from which one obtains κ*=0.0205.In simulation, we choose the initial valuesx1(0)=2, x2(0)=-3, and κ=0.02. Figure 1 demonstrates the effectiveness of the state feedback controller.The responses of the closed-loop system (27) and (34).
(a)
(b)


## 6. A Concluding Remark

In this paper, the homogeneous domination approach is introduced to solve the state feedback stabilization problem for the stochastic feedforward nonlinear system (1). There still exist some problems to be investigated. One is to consider the more general switched stochastic feedforward nonlinear systems by adopting average dwell time method in [28]. Another is to consider stochastic feedforward networked or fuzzy systems (similar to [29–33]).



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