State Feedback Control for Stochastic Feedforward Nonlinear Systems

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.

1. Introduction

Consider the following stochastic feedforward nonlinear systems described by 𝑑𝑥₁=𝑥₂𝑑𝑡+𝑓₁𝑥₃,𝑢𝑑𝑡+𝑔^𝑇₁𝑥₂⋮,𝑢𝑑𝜔,𝑑𝑥_𝑛−2=𝑥_𝑛−1𝑑𝑡+𝑓_𝑛−2𝑥_𝑛,𝑢𝑑𝑡+𝑔^𝑇_𝑛−2𝑥_𝑛−1,𝑢𝑑𝜔,𝑑𝑥_𝑛−1=𝑥_𝑛𝑑𝑡+𝑓_𝑛−1(𝑢)𝑑𝑡+𝑔^𝑇_𝑛−1𝑥_𝑛,𝑢𝑑𝜔,𝑑𝑥_𝑛=𝑢𝑑𝑡, (1) where 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇∈𝑅^𝑛 and 𝑢∈𝑅 are the system state and input, respectively. 𝑥_𝑖=(𝑥_𝑖,…,𝑥_𝑛), 𝑖=2,…,𝑛. 𝜔 is an 𝑚-dimensional standard Wiener process defined on the complete probability space (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃) with Ω being a sample space, ℱ being a 𝜎-field, {ℱ_𝑡}_𝑡≥0 being a filtration, and 𝑃 being a probability measure. 𝑓_𝑖∶𝑅^{𝑛−𝑖−1}×𝑅→𝑅 and 𝑔_𝑖∶𝑅^𝑛−𝑖×𝑅→𝑅^𝑚 are assumed to be locally Lipschitz with 𝑓_𝑖(0,0)=0 and 𝑔_𝑖(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. Section 2 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 𝑅^𝑛 denotes the real 𝑛-dimensional space. For a given vector or matrix 𝑋, 𝑋^𝑇 denotes its transpose, Tr{𝑋} denotes its trace when 𝑋 is square, and |𝑋| is the Euclidean norm of a vector 𝑋. 𝒞^𝑖 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: 𝑑𝑥=𝑓(𝑥)𝑑𝑡+𝑔^𝑇(𝑥)𝑑𝜔,𝑥(0)=𝑥₀∈𝑅^𝑛, (2) where 𝑥∈𝑅^𝑛 is the system state and 𝜔 is an 𝑚-dimensional standard Wiener process defined on the complete probability space (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃). The Borel measurable functions 𝑓:𝑅^𝑛→𝑅^𝑛 and 𝑔:𝑅^𝑛→𝑅^𝑚×𝑛 are locally Lipschitz with 𝑓(0)=0 and 𝑔(0)=0.

Definition 1 (see [1]). For any given 𝑉(𝑥)∈𝒞² associated with stochastic system (2), the differential operator ℒ is defined as ℒ𝑉=𝜕𝑉𝑓(𝑥)+1𝜕𝑥2𝑔(𝑥)𝜕Tr²𝑉𝜕𝑥²𝑔^𝑇(𝑥). (3)

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∈𝑅^𝑛⧵{0}.

Definition 3 (see [22]). For fixed coordinates (𝑥₁,…,𝑥_𝑛)∈𝑅^𝑛 and real numbers 𝑟_𝑖>0, 𝑖=1,…,𝑛.(i)The dilation Δ_𝜀(𝑥) is defined by Δ_𝜀(𝑥)=(𝜀^𝑟1𝑥₁,…,𝜀^𝑟𝑛𝑥_𝑛) for any 𝜀>0 and 𝑟₁,…,𝑟_𝑛 are called the weights of the coordinates. For simplicity, we define dilation weight as Δ=(𝑟₁,…,𝑟_𝑛).(ii)A function 𝑉∈𝒞(𝑅^𝑛,𝑅) is said to be homogeneous of degree 𝜏 if there is a real number 𝜏∈𝑅 such that 𝑉(Δ_𝜀(𝑥))=𝜀^𝜏𝑉(𝑥₁,…,𝑥_𝑛) for any 𝑥∈𝑅^𝑛⧵{0}, 𝜀>0.(iii)A vector field ℎ∈𝒞(𝑅^𝑛,𝑅^𝑛) is said to be homogeneous of degree 𝜏 if there is a real number 𝜏∈𝑅 such that ℎ_𝑖(Δ_𝜀(𝑥))=𝜀^{𝜏+𝑟𝑖}ℎ_𝑖(𝑥) for any 𝑥∈𝑅^𝑛⧵{0}, 𝜀>0, 𝑖=1,…,𝑛.(iv)A homogeneous 𝑝-norm is defined as ‖𝑥‖_Δ,𝑝=(∑^𝑛_𝑖=1|𝑥_𝑖|^{𝑝/𝑟𝑖})^1/𝑝 for any 𝑥∈𝑅^𝑛, 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 𝛼₁(|𝑥|)≤𝑉(𝑥)≤𝛼₂(|𝑥|),ℒ𝑉≤−𝛼₃(|𝑥|). (4) Then there exists an almost surely unique solution on [0,∞), the equilibrium 𝑥=0 is GAS in probability, and 𝑃{lim_𝑡→∞|𝑥(𝑡)|=0}=1.

Lemma 5 (see [22]). Given a dilation weight Δ=(𝑟₁,…,𝑟_𝑛), 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 that 𝑉∶𝑅^𝑛→𝑅 is a homogeneous function of degree 𝜏 with respect to the dilation weight Δ; then(i)𝜕𝑉/𝜕𝑥_𝑖 is homogeneous of degree 𝜏−𝑟_𝑖 with 𝑟_𝑖 being the homogeneous weight of 𝑥_𝑖;(ii)there is a constant 𝑐 such that 𝑉(𝑥)≤𝑐‖𝑥‖^𝜏_Δ. Moreover, if 𝑉(𝑥) is positive definite, then 𝑉(𝑥)≥𝑐‖𝑥‖^𝜏_Δ, where 𝑐 is a positive constant.

Lemma 7 (see [4]). Let 𝑐 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. For 𝑖=1,…,𝑛−1, there exist positive constants 𝑎₁ and 𝑎₂ such that 𝑓_𝑖𝑥_𝑖+2,𝑢≤𝑎₁𝑥_𝑖+2+⋯+𝑥_𝑛+|𝑢|,𝑔_𝑖𝑥_𝑖+1,𝑢≤𝑎₂𝑥_𝑖+1+⋯+𝑥_𝑛+|𝑢|, (5) where 𝑥_𝑛+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, 𝑔_𝑖(⋅)≡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: 𝜂_𝑖=𝑥_𝑖𝜅^𝑖−1𝑢,𝑣=𝜅^𝑛,𝑖=1,…,𝑛, (6) where 0<𝜅<1 is a designed constant. With the help of (6), (1) can be rewritten as 𝑑𝜂₁=𝜅𝜂₂𝑑𝑡+𝑓₁𝜂₃,𝑣𝑑𝑡+𝑔^𝑇₁𝜂₂⋮,𝑣𝑑𝜔,𝑑𝜂_𝑛−2=𝜅𝜂_𝑛−1𝑑𝑡+𝑓_𝑛−2𝜂_𝑛,𝑣𝑑𝑡+𝑔^𝑇_𝑛−2𝜂_𝑛−1,𝑣𝑑𝜔,𝑑𝜂_𝑛−1=𝜅𝜂_𝑛𝑑𝑡+𝑓_𝑛−1(𝑣)𝑑𝑡+𝑔^𝑇_𝑛−1𝜂_𝑛,𝑣𝑑𝜔,𝑑𝜂_𝑛=𝜅𝑣𝑑𝑡, (7) where 𝑓_𝑖=𝑓_𝑖/𝜅^𝑖−1 and 𝑔_𝑖=𝑔_𝑖/𝜅^𝑖−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): 𝑑𝜂_𝑖=𝜅𝜂_𝑖+1𝑑𝑡,𝑖=1,…,𝑛−1,𝑑𝜂_𝑛=𝜅𝑣𝑑𝑡. (8)

Step 1. Introducing 𝜉₁=𝜂₁ and choosing 𝑉₁(𝜂₁)=(1/4)𝜉⁴₁, by (3) and (8), it can be verified that ℒ𝑉₁=𝜅𝜉³₁𝜂₂. The first virtual controller 𝜂^∗₂=−𝑐₁₁𝜉₁=∶−𝛼₁𝜉₁,𝑐₁₁>0 (9) leads to ℒ𝑉₁≤−𝜅𝑐₁₁𝜉⁴₁+𝜅𝜉³₁(𝜂₂−𝜂^∗₂).
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 Step 𝑖−1 there are a set of virtual controllers 𝜂^∗₁,…,𝜂^∗_𝑖 defined by 𝜂^∗₁=0,𝜉₁=𝜂₁−𝜂^∗₁=𝜂₁,𝜂^∗_𝑘=−𝛼_𝑘−1𝜉_𝑘−1,𝜉_𝑘=𝜂_𝑘−𝜂^∗_𝑘,𝑘=2,…,𝑖, (10) such that the (𝑖−1)th Lyapunov function 𝑉_𝑖−1(𝜂_𝑖−1)=(1/4)∑^𝑖−1_𝑗=1𝜉⁴_𝑗 satisfies ℒ𝑉_𝑖−1≤−𝜅^𝑖−1_𝑗=1𝑐_{𝑖−1,𝑗}𝜉⁴_𝑗+𝜅𝜉³_𝑖−1𝜂_𝑖−𝜂^∗_𝑖, (11) where 𝛼_𝑗, 𝑐_{𝑖−1,𝑗}, 𝑗=1,…,𝑖−1, are positive constants. Then there exists a virtual control law 𝜂^∗_𝑖+1=−𝛼_𝑖𝜉_𝑖 such that ℒ𝑉_𝑖≤−𝜅^𝑖_𝑗=1𝑐_𝑖𝑗𝜉⁴_𝑗+𝜅𝜉³_𝑖𝜂_𝑖+1−𝜂^∗_𝑖+1, (12) where 𝑉_𝑖(𝜂_𝑖)=(1/4)∑^𝑖_𝑗=1𝜉⁴_𝑗.

Proof. From (3), (8), (10), and (11), it follows that ℒ𝑉_𝑖≤−𝜅^𝑖−1_𝑗=1𝑐_{𝑖−1,𝑗}𝜉⁴_𝑗+𝜅𝜉³_𝑖𝜂_𝑖+1+𝜅𝜉³_𝑖−1𝜉_𝑖−𝜅𝜉³_𝑖^𝑖−1_𝑘=1𝜕𝜂^∗_𝑖𝜕𝜂_𝑘𝜂_𝑘+1. (13) We concentrate on the last two terms on the right-hand side of (13).
Using (10) and Lemma 7, one obtains 𝜉³_𝑖−1𝜉_𝑖≤𝑙_{𝑖,𝑖−1,1}𝜉⁴_𝑖−1+𝜌_𝑖1𝜉⁴_𝑖,−𝜉³_𝑖^𝑖−1_𝑘=1𝜕𝜂^∗_𝑖𝜕𝜂_𝑘𝜂_𝑘+1≤𝜉_𝑖^3𝑖−1_𝑘=1𝛼_𝑖−1⋯𝛼_𝑘𝜉_𝑘+1−𝛼_𝑘𝜉_𝑘≤𝜉_𝑖^3𝑖−1_𝑘=1𝛼_𝑖−1⋯𝛼_𝑘−1+𝛼_𝑖−1⋯𝛼_𝑘+1𝛼²_𝑘𝜉_𝑘+𝛼_𝑖−1𝜉_𝑖≤^𝑖−1_𝑘=1𝑙_𝑖𝑘2𝜉⁴_𝑘+𝜌_𝑖2𝜉⁴_𝑖, (14) where 𝑙_{𝑖,𝑖−1,1}, 𝑙_𝑖𝑘2  (𝑘=1,…,𝑖−1), 𝜌_𝑖1, and 𝜌_𝑖2 are positive constants, 𝛼₀=0.
Choosing 𝑐_𝑖𝑗=𝑐_{𝑖−1,𝑗}−𝑙_𝑖𝑗2𝑐>0,𝑗=1,…,𝑖−2,_{𝑖−1,𝑖−1}−𝑙_{𝑖,𝑖−1,1}−𝑙_{𝑖,𝑖−1,2}𝜂>0,𝑗=𝑖−1,_∗𝑖+1=−𝑐_𝑖𝑖+𝜌_𝑖1+𝜌_𝑖2𝜉_𝑖=∶−𝛼_𝑖𝜉_𝑖,𝑐_𝑖𝑖>0, (15) and substituting (14)-(15) into (13), one gets the desired result.

At Step 𝑛, choosing 𝑉_𝑛(𝜂_𝑛)=(1/4)∑^𝑛_𝑖=1𝜉⁴_𝑖 and 𝑣=𝜂^∗_𝑛+1=−𝛼_𝑛𝜉_𝑛=−𝛼_𝑛𝜂_𝑛+𝛼_𝑛−1𝜂_𝑛−1+⋯+𝛼₁𝜂₁, (16) by (3), (12), and (16), one gets ℒ𝑉_𝑛≤−𝜅^𝑛_𝑖=1𝑐_𝑛𝑖𝜉⁴_𝑖+𝜅𝜉³_𝑛𝑣−𝜂^∗_𝑛+1=−𝜅^𝑛_𝑖=1𝑐_𝑛𝑖𝜉⁴_𝑖, (17) where 𝜉_𝑛=𝜂_𝑛−𝜂^∗_𝑛, 𝛼_𝑖=𝛼_𝑛⋯𝛼_𝑖, 𝑐_𝑛𝑖, and 𝑖=1,…,𝑛, are positive constants. The system (7) and (16) can be written as 𝑑𝜂=𝜅𝐸𝜂𝑑𝑡+𝐹𝜂,𝑣𝑑𝑡+𝐺^𝑇𝜂,𝑣𝑑𝜔, (18) where 𝜂=𝜂_𝑛=(𝜂₁,…,𝜂_𝑛)^𝑇, 𝐸(𝜂)=(𝜂₂,…,𝜂_𝑛,𝑣)^𝑇, 𝐹(𝜂,𝑣)=(𝑓₁,…,𝑓_𝑛−1,0)^𝑇, and 𝐺(𝜂,𝑣)=(𝑔₁,…,𝑔_𝑛−1,0). Introducing the dilation weight Δ=(1,1,…,1_{for𝜂1,…,𝜂𝑛}), by (10) and 𝑉_𝑛(𝜂)=(1/4)∑^𝑛_𝑖=1𝜉⁴_𝑖, one obtains 𝑉_𝑛Δ_𝜀𝜂=14^𝑛_𝑖=1𝜀𝜂_𝑖+𝛼_𝑖−1𝜀𝜂_𝑖−1+⋯+𝛼_𝑖−1⋯𝛼₁𝜀𝜂₁⁴=𝜀⁴𝑉_𝑛𝜂, (19) from which and Definition 3, we know that 𝑉_𝑛(𝜂) is homogeneous of degree 4.

4. Stability Analysis

We state the main result in this paper.

Theorem 11. If Assumption 8 holds for the stochastic feedforward nonlinear system (1), under the state feedback controller 𝑢=𝜅^𝑛𝑣 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 Theorem 11 by three steps.
Step  1. Since 𝑓_𝑖 and 𝑔_𝑖 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 𝜕𝑉_𝑛𝜂𝜕𝜂𝜅𝐸≤−𝑐₀₁𝜅𝜂⁴_Δ. (20) By Assumption 8, (6), (16), and 0<𝜅<1, one has 𝑓_𝑖𝜂_𝑖+2≤𝑎,𝑣¹𝜅^𝑖+1𝜂_𝑖+2+⋯+𝜅^𝑛−1𝜂_𝑛+𝜅^𝑛|𝑣|𝜅^𝑖−1≤𝑎₁𝜅²𝜂_𝑖+2+⋯+𝜂_𝑛+|𝑣|≤𝜆₁𝜅²𝜂_Δ, (21) where 𝜆₁ is a positive constant. According to Lemmas 5–6 and (21), one obtains 𝜕𝑉_𝑛𝐹=𝜕𝜂𝜂,𝑣^𝑛−1_𝑖=1𝜕𝑉_𝑛𝜕𝜂_𝑖𝑓_𝑖𝜂_𝑖+2,𝑣≤𝑐₀₂𝜅²𝜂⁴_Δ, (22) where 𝑐₀₂ is a positive constant. Similar to (21), there is a positive constant 𝜆₂ such that 𝑔_𝑖𝜂_𝑖+1,𝑣≤𝜆₂𝜅𝜂_Δ, (23) from which and Lemmas 5-6, one leads to 12𝐺𝜕Tr𝜂,𝑣²𝑉_𝑛𝜕𝜂²𝐺^𝑇≤1𝜂,𝑣2𝑚√𝑚^𝑛−1_𝑖,𝑗=1𝜕²𝑉_𝑛𝜕𝜂_𝑖𝜕𝜂_𝑗𝑔_𝑖𝜂_𝑖+1,𝑣𝑔_𝑗𝜂_𝑗+1,𝑣≤𝑐₀₃𝜅²𝜂⁴_Δ, (24) where 𝑐₀₃ is a positive constant. By (3), (18), (20), (22), and (24), one has ℒ𝑉_𝑛=𝜕𝑉_𝑛𝜂+𝜕𝜂𝜅𝐸𝜕𝑉_𝑛𝐹+1𝜕𝜂𝜂,𝑣2𝐺𝜕Tr𝜂,𝑣²𝑉_𝑛𝜕𝜂²𝐺^𝑇𝜂,𝑣≤−𝑐₀₁𝜅𝜂⁴_Δ+𝑐₀₂+𝑐₀₃𝜅²𝜂⁴_Δ=−𝜅𝑐₀₁−𝑐₀₂+𝑐₀₃𝜅𝜂⁴_Δ. (25) Since 𝑐₀₁ is a constant independent of 𝑐₀₂ and 𝑐₀₃, by choosing 0<𝜅<𝜅^∗𝑐=∶min1,⁰¹𝑐₀₂+𝑐₀₃, (26) (25) becomes ℒ𝑉_𝑛≤−𝑐₀‖𝜂‖⁴_Δ, where 𝑐₀ is a positive constant.
By Steps 1-2 and Lemma 4, the system consisting of (7) and (16) has an almost surely unique solution on [0,∞), 𝜂=0 is GAS in probability, and 𝑃{lim_𝑡→∞|𝜂|=0}=1.
Step  3. Since (6) is an equivalent transformation, the closed-loop system consisting of (1), 𝑢=𝜅^𝑛𝑣, 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: 𝑑𝑥₁=𝑥₂1𝑑𝑡+130𝑢𝑑𝑡+5sin𝑥₂𝑑𝜔,𝑑𝑥₂=𝑢𝑑𝑡. (27) 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 𝜂₁=𝑥₁,𝜂₂=𝑥₂𝜅𝑢,𝑣=𝜅², (28) system (27) becomes 𝑑𝜂₁=𝜅𝜂₂𝜅𝑑𝑡+²130𝑣𝑑𝑡+5sin𝜅𝜂₂𝑑𝜔,𝑑𝜂₂=𝜅𝑣𝑑𝑡. (29) Choosing 𝜉₁=𝜂₁ and 𝑉₁(𝜂₁)=(1/4)𝜉⁴₁, we obtain ℒ𝑉₁≤−2𝜅𝜉⁴₁+𝜅𝜉³₁(𝜂₂−𝜂^∗₂), where 𝜂^∗₂=−2𝜂₁=∶−𝛼₁𝜉₁. By 𝜉₂=𝜂₂−𝜂^∗₂ and 𝑉₂(𝜂₂)=𝑉₁(𝜂₁)+(1/4)𝜉⁴₂, a direct calculation leads to ℒ𝑉₂≤−2𝜅𝜉⁴₁+𝜅𝜉³₁𝜉₂+𝜅𝜉³₂𝑣+𝜅𝛼₁𝜉³₂𝜂₂. (30) By Lemma 7, one has 𝜉³₁𝜉₂≤12𝜉⁴₁+27𝜉32⁴²,𝛼₁𝜉³₂𝜂₂≤12𝜉⁴₁+3√³2+2𝜉⁴₂. (31) Choosing 𝑣=−3+3√³2+27𝜉32²=∶−𝛼₂𝜉₂, (32) and substituting (31) into (30), it leads to ℒ𝑉₂≤−𝜅𝜉⁴₁+𝜉⁴₂. (33) By (28) and (32), one obtains the actual controller 𝑢=−𝛼₂𝜅𝑥₂+𝛼₁𝜅²𝑥₁. (34) Defining ‖𝜂‖_Δ=(𝜂²₁+𝜂²₂)^1/2, one gets ℒ𝑉₂≤−𝜅‖𝜂‖⁴_Δ. From (22) and (24), it follows that 𝜕𝑉₂𝜕𝜂₁⋅𝜅²30𝑣≤46.7133𝜅²𝜂⁴_Δ,12𝜕²𝑉₂𝜕𝜂²₁⋅125sin²𝜅𝜂₂≤2𝜅²𝜂⁴_Δ. (35) By (29), one has ℒ𝑉₂≤−𝜅(1−48.7133𝜅)‖𝜂‖⁴_Δ, from which one obtains 𝜅^∗=0.0205.

In simulation, we choose the initial values 𝑥₁(0)=2, 𝑥₂(0)=−3, and 𝜅=0.02. Figure 1 demonstrates the effectiveness of the state feedback controller.

(a)

(b)

Figure 1 

The responses of the closed-loop system (27) and (34).

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]).

Acknowledgments

The authors would like to express their sincere gratitude to the editor and reviewers for their helpful suggestions in improving the quality of this paper. This work was partially supported by the National Natural Science Foundation of China (nos. 61304002, 61304003, 61203123, and 61304054), the Fundamental Research Funds for the Central Universities of China (no. 11CX04044A), the Shandong Provincial Natural Science Foundation of China (no. ZR2012FQ019), and Doctoral Start-up Fund of Bohai University.