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State-Feedback Stabilization for a Class of Stochastic Feedforward Nonlinear Time-Delay Systems
We investigate the state-feedback stabilization problem for a class of stochastic feedforward nonlinear time-delay systems. By using the homogeneous domination approach and choosing an appropriate Lyapunov-Krasovskii functional, the delay-independent state-feedback controller is explicitly constructed such that the closed-loop system is globally asymptotically stable in probability. A simulation example is provided to demonstrate the effectiveness of the proposed design method.
In recent years, the study on stochastic lower-triangular nonlinear systems has received considerable attention from both theoretical and practical point of views see, for instance, [1–19] and the references therein. This paper will further consider the following stochastic feedforward nonlinear time-delay systems described by where and are the system state and input signal, respectively, , is the time-delayed state vector, and is the time-varying delay. is an -dimensional standard Wiener process defined on the complete probability space with being a sample space, being a -field, being a filtration, and being a probability measure. and are assumed to be locally Lipschitz with and , , .
Feedforward (also called upper-triangular) system is another important class of nonlinear systems. Firstly, from a theoretical point of view, since they are not feedback linearizable and maybe not stabilized by applying the conventional backstepping method, the stabilization problem of these systems is more difficult than that of lower-triangular systems. Secondly, many physical devices, such as the cart-pendulum system in  and the ball-beam system with a friction term in , can be described by equations with the feedforward structure. In the recent papers, the stabilization problems for feedforward nonlinear (or time-delay) systems have achieved remarkable development; see, for example, [22–29] and the references therein.
However, all these above-mentioned results are limited to deterministic systems. There are fewer results on stochastic feedforward nonlinear systems until now, due to the special characteristics of this system. To the best of the authors’ knowledge,  is the only paper to consider this kind of stochastic feedforward nonlinear systems, but the assumptions on the nonlinearities are restrictive.
The purpose of this paper is to further weaken the assumptions on the drift and diffusion terms of system (1) and solve the state-feedback stabilization problem. By using the homogeneous domination approach in  and choosing an appropriate Lyapunov-Krasovskii functional, a delay-independent state-feedback controller is explicitly constructed such that the closed-loop system is 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 are given in Sections 3 and 4, respectively, 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.
denotes the set of all nonnegative real numbers, and denotes the real -dimensional space. For a given vector or matrix , denotes its transpose, denotes its trace when is square, and is the Euclidean norm of a vector . denotes the space of continuous -value functions on endowed with the norm defined by for ; denotes the family of all -measurable bounded -valued random variables . denotes the set of all functions with continuous th partial derivatives; denotes the family of all nonnegative functions on which are in and in ; denotes the family of all functions which are in the first argument and in the second argument. denotes the set of all functions , which are continuous, strictly increasing, and vanishing at zero; denotes the set of all functions which are of class and unbounded; is the set of all functions : , which are of for each fixed and decrease to zero as for each fixed .
Consider the following stochastic time-delay system: with initial data , where is a Borel measurable function, is an -dimensional standard Wiener process defined on the complete probability space , and and are locally Lipschitz in (, ) uniformly in with and .
Definition 3 (see ). For fixed coordinates and real numbers , , one has the following.(i)The dilation is defined by for any ; are called as the weights of the coordinates. For simplicity, we define dilation weight .(ii)A function is said to be homogeneous of degree if there is a real number such that for any , .(iii)A vector field is said to be homogeneous of degree if there is a real number such that for any , , .(iv)A homogeneous -norm is defined as for any , where is a constant. For simplicity, in this paper, one chooses and writes for .
Lemma 4 (see ). For system (2), if there exist a function , two class functions , , and a class function such that then there exists a unique solution on for (2), the equilibrium is GAS in probability, and .
Lemma 5 (see ). Given a dilation weight , suppose that and are homogeneous functions of degrees and , respectively. Then is also homogeneous with respect to the same dilation weight . Moreover, the homogeneous degree of is .
Lemma 6 (see ). Suppose that is a homogeneous function of degree with respect to the dilation weight ; then (i) is homogeneous of degree with being the homogeneous weight of ; (ii) there is a constant such that . Moreover, if is positive definite, then , where is a positive constant.
Lemma 7 (see ). Let and be positive constants. For any positive number , then .
3. Design of State-Feedback Controller
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.
For system (1), we need the following assumptions.
Assumption 8. For , there exist positive constants and such that where .
Assumption 9. The time-varying delay satisfies for a constant .
Remark 10. When in diffusion term , Assumption 8 reduces to the same form as in , from which one can see that system (1) is more general than . The significance and reasonability of Assumption 8 are illustrated in that paper.
Firstly, we introduce the following coordinate transformation: where is a scalar to be designed. By (6), (1) can be expressed as where , , , .
3.2. State-Feedback Controller Design
We construct a state-feedback controller for system (7).
Step i (). In this step, we can get the following lemma.
Lemma 11. Suppose that at step , there is a set of virtual controllers defined by such that the th Lyapunov function satisfies where , , , are positive constants. Then there exists a virtual control law such that where .
Proof. See the Appendix.
At step , choosing and with the help of (3), (12), and (13), one obtains where , , , , , , are positive constants. The system (7) and (13) can be written as where , , and and are defined as in (14). Introducing the dilation weight , by (10) and , one has from which and Definition 3, we know that is homogeneous of degree 4.
4. Stability Analysis
We state the main result in this paper.
Theorem 12. If Assumptions 8 and 9 hold for the stochastic feedforward nonlinear time-delay system (1), under the state-feedback controller and (13), then(i)the closed-loop system has a unique solution on ;(ii)the equilibrium at the origin of the closed-loop system is GAS in probability.
Proof. We prove Theorem 12 by four steps.
Step 1. Since , , , are assumed to be locally Lipschitz, so the system consisting of (13) and (15) satisfies the locally Lipschitz condition.
Step 2. We consider the following entire Lyapunov function for system (15): where and are positive parameters to be determined. It is easy to verify that is on . Since is continuous, positive definite, and radially unbounded, by Lemma 4.3 in , there exist two class functions and such that
By Lemma 4.3 in  and Lemma 6, there exist positive constants and , class functions and , and a positive definite function whose homogeneous degree is such that
From and (19), it follows that where , are positive constants and is a class function. Since , . Defining , by (17), (18), and (20), one gets
Step 3. By Lemma 6 and (14), there exists a positive constant such that
By Assumption 8, (6), and , one has where is a positive constant. Using Lemmas 5–7 and (23), one gets where , , and are positive constants. Similar to (23), there is a positive constant such that from which and Lemmas 5–7, one gets where , , and are positive constants. With the help of (3), (15), (17), (22), (24), (26), and Assumption 9, one has
Since is a constant independent of , , , , and , by choosing
Equation (27) becomes , where is a positive constant. By (19), one obtains
By Steps 1–3 and Lemma 4, the system consisting of (13) and (15) has a unique solution on , is GAS in probability, and .
Step 4. Since (6) is an equivalent transformation, so the closed-loop system consisting of (1), , and (13) has the same properties as the system (13) and (15). Theorem 12 holds.
Remark 13. In this paper, the homogeneous domination idea is generalized to stochastic feedforward nonlinear time-delay systems (1). The underlying philosophy of this approach is that the state-feedback controller is first constructed for system (7) without considering the drift and diffusion terms, and then a low gain in (6) (whose the value range is (28)) is introduced to state-feedback controller to dominate the drift and diffusion terms.
Remark 14. Due to the special upper-triangular structure and the appearance of time-varying delay, there is no efficient method to solve the stabilization problem of system (1). By combining the homogeneous domination approach with stochastic nonlinear time-delay system criterion, the state-feedback stabilization of system (1) was perfectly solved in this paper.
Remark 15. One of the main obstacles in the stability analysis is how to deal with the effect of time-varying delay. In this paper, by constructing an appropriate Lyapunov-Krasovskii functional (17), this problem was effectively solved.
Remark 16. It is worth pointing out that the rigorous proof of Theorem 12 is not an easy job.
5. A Simulation Example
Design of Controller. Introducing the following coordinate transformation: system (30) becomes where . Choosing and , we obtain , where . By and , a direct calculation leads to
By Lemma 7, one has
In simulation, we choose the initial values , , and . Figure 1 demonstrates the effectiveness of the state-feedback controller.
6. A Concluding Remark
By using the homogeneous domination approach, this paper further studied the state-feedback stabilization problem for a class of stochastic feedforward nonlinear time-delay systems (1). The delay-independent state-feedback controller is explicitly constructed such that the closed-loop system is globally asymptotically stable in probability.
There still exist some problems to be investigated. One is to consider the output-feedback control of switched stochastic system (1) by using average dwell time method in . Another is to find a practical example (similar to [33–35]) for system (1). The last is to generalize the networked control systems (such as [36–41]) to stochastic feedforward networked systems.
Proof of Lemma 11. According to (3), (7), (10), and (11), one has
We concentrate on the last two terms on the right-hand side of (A.1).
Using (10) and Lemma 7, one obtains where , , , and are positive constants, .
Choosing and substituting (A.2)-(A.3) into (A.1), one gets the desired result.
Conflict of Interests
The authors declare that there is no conflict of interests.
The authors would like to express sincere gratitude to 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 the Polish-Norwegian Research Programme operated by the National Center for Research and 24 Development under the Norwegian Financial Mechanism 2009–2014 in the frame of Project Contract (no. Pol-Nor/200957/47/2013).
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