Finite-time Control of Complex Systems and Their ApplicationsView this Special Issue
Finite-Time Stabilization for -Norm Stochastic Nonlinear Systems with Output Constraints
This paper investigates the finite-time stability problem of -norm stochastic nonlinear systems subject to output constraint. To cope with the constraint on system output, a tan-type barrier Lyapunov function (BLF) is constructed. By using the constructed BLF and the backstepping technique, a new control algorithm is proposed with a continuous state-feedback controller being designed, which guarantees not only that the requirement of output constraint is always achieved but also that the origin of the system is finite-time stable. This result is demonstrated by both the rigorous analysis and the simulation example.
During the past decades, the control problem of nonlinear systems has long been a hot topic, and many control design approaches have been proposed for various kinds of nonlinear systems, such as adaptive fuzzy control [1, 2], output tracking control [3, 4], control [5, 6], and sliding mode control [7–10]. Due to their important roles in many science and industry applications, the stochastic nonlinear systems have attracted much interest in recent years. With the development of stochastic theory, various control design strategies have been developed for types of stochastic nonlinear systems by the backstepping technique, see [11–14], for examples. Especially, some works have considered -norm stochastic nonlinear systems, which are inherently nonlinear due to the fractional powers of such systems being not identically equal to one. It should be noted that the inherent nonlinearities cause the stability and control design problems, which are not very easy to be solved . Luckily, the issues have been well studied for -norm stochastic nonlinear systems with different structures by the adding a power integrator technique in the existing literatures. For instance, Li et al.  have considered the adaptive state-feedback stabilization for -norm stochastic nonlinear systems; the output-feedback control has been addressed for -norm stochastic nonlinear systems with time-varying delays in ; Zhao et al.  have proposed a neural tracking control algorithm for -norm switched stochastic nonlinear systems. More latest studies can be found in [19–21] and the references within.
However, most of the abovementioned works about -norm stochastic nonlinear systems did not take the output constraint into consideration. As it is well known, many actual systems are subject to output constraint due to the consideration of the system performance and operation safety [22, 23]. For this reason, the constrained control issue of nonlinear systems has drawn attention from many scholars. Tee et al.  have first proposed the notion of the barrier Lyapunov function (BLF) and consequently have developed a control design strategy for a class of strict-feedback deterministic nonlinear systems with output constraints. After then, with the aid of BLFs, control design schemes have been presented for many deterministic nonlinear systems with different types of constraints, including stability control for nonlinear systems with time-varying or asymmetric output constraints [25, 26], adaptive control for nonlinear systems with full-state constrains , and sliding mode control for nonlinear systems with output constraints [28–30]. Moreover, since the finite-time control possesses some inherent advantages [31–33], techniques for the finite-time stabilization under output/state constraints have also been developed, respectively, for strict-feedback nonlinear systems , norm nonlinear systems [35–37], and switched nonlinear systems . On the basis of these results, the constrained control schemes for some classes of stochastic nonlinear systems have also been proposed. Jin  has constructed an adaptive tracking controller for a class of output-constrained stochastic nonlinear systems in strict-feedback form. Later, the adaptive control problem and the finite-time control problem have been, respectively, addressed for stochastic nonlinear systems with full-state constraints in [40, 41]. Furthermore, the adaptive neural network or fuzzy constrained control problems have attracted some attention [42–46]. Nevertheless, the stochastic nonlinear systems with output constraints considered in most of the existing related works are in the strict-feedback form, rather than in -normal form. On the contrary, the existing research has mainly focused on the adaptive control problem but did not take the finite-time stabilization into account.
Motivated by the above discussions, we will investigate the problem of the finite-time stabilization for a class of -norm stochastic nonlinear systems with output constraints and unknown time-varying parameters. First of all, a BLF-based control strategy will be developed by the backstepping approach. Secondly, applying stochastic Lyapunov theorems and ItÔ’s formula, the constructed state-feedback controller is rigorously proved to be able to ensure the achievement of the output constraint and the finite-time stability of the considered systems simultaneously. Finally, the main result of this paper will be further demonstrated by a simulation example.
2. Problem and Preliminaries
2.1. Problem Statement
The following class of stochastic nonlinear systems are considered:where is a -dimension standard Wiener process; are system state, control input, and output, respectively; is the time-varying parameter; the nonlinear functions and are continuous and satisfy ; and the fractional powers ’s meet the requirement . The output is required to satisfywhere is a known positive constant.
This paper aims to design a continuous state-feedback controller for system (1), which can ensure that the origin of the closed-loop system is finite-time stable in probability and the requirement of the output constraint is achieved.
Notations 1. For , let and . For any and , denote .
Consider the following stochastic system:where and are continuous satisfying and .
Definition 2 (see ). Suppose that is an open set containing the origin and is positive definite and continuously differentiable. Then, for system is called a BLF if for each solution starting from , as , for all and for some .
Assumption 1 (see ). For , there exist known positive constants and such that .
Assumption 2. For , there are a constant and known nonnegative smooth functions , such thatfor all , where .
Remark 1. Note that condition (4) is borrowed from . However, the systems considered in  are -norm deterministic nonlinear systems, while we consider -norm stochastic nonlinear systems with drift terms ’s and diffusion terms ’s in this paper. In light of , the value of is generally taken as for simplicity, where and represent even and odd integers, respectively. Then, the value of each can be obtained by applying and . It can also be observed that both the denominator and numerator of each are odd.
For all , then the origin of system (3) is finite-time stable in probability.
Lemma 3 (see ). Let be positive real numbers. For any , we have
Lemma 4 (see ). Let ; for any , one haswhere if and if .
Lemma 5 (see ). Let. For any , we have(i)(ii)(iii)
3. Main Results
3.1. A Tan-Type BLF
Before carrying out the control design for system (1), we should handle the output constraint issue.
Firstly, we denote and . Let be a constant parameter satisfying , where the value of is chosen as below:(i)If for all , , then (ii)If for all , , then
Consequently, it is clear that .
Then, a tan-type BLF can be constructed on as follows:where is given by Assumption 2 and is defined as above.
It is not hard to obtain from the expression of thatwhere .
Remark 2. It should be noted that the BLF is modified from , which is constructed by fully taking the advantage of the given nonlinear growth conditions. As stated in , the control strategy based on is a universal method, which can handle stochastic systems with or without output constraints.
3.2. Controller Design and Stability Analysis
In what follows, a continuous state-feedback controller will be constructed, and the stability of system (1) under the designed controller will be rigorously analysed. To this end, a theorem is presented to describe the main result.
Theorem 1. Suppose Assumptions 1-2 hold for system (1). For any constant , there is a continuous state-feedback controller such that(i)The output of system (1) is kept in a given constrained set in the sense of probability, i.e., (ii)The origin of the closed-loop system is finite-time stable in probability.
Proof. The proof contains three parts. First of all, the design procedure of the controller is explicitly displayed. Then, the system output is proved to be kept in the given constrained set with probability one. In the last part, the finite-time stability of system (1) is rigorously analysed.
3.2.1. Part I: Design Procedure
Step 1. Let , and choose the Lyapunov function . Then, we can directly get from Definition 1 thatwhere is a nonnegative function and is a virtual controller required to be design after later.
Then, we designSubstituting (12) into (13) yields
Step 2. We denote and define the positive Lyapunov function on as withSince is valid, one can obtainUsing Definition 1 again, we havewhere is the virtual controller required to be designed later.
In the following, each term in the right hand of (16) will be estimated by its upper bound.
Firstly, applying and Lemma 5, it is easily obtained thatIt can be deduced from (19) and Lemma 3 thatwhere is a function.
Secondly, one can obtain from Assumption 1 and Lemma 4 thatwhere and are nonnegative smooth functions.
Additionally, note that . Then, as stated in , there exist functions , and , such thatThen, using (18), (21), (25), Assumptions 1-2, and Lemma 3, we can inferwhereis a nonnegative function.
Moreover, from (19), Assumption 2, and Lemma 3, it can be deduced thatwhere is a function.
On the contrary, it is noted thatThen, applying (22), (26), Assumption 2, and Lemma 3, there clearly exist nonnegative functions , and such thatSubstituting equations (31)–(33) into (30), one obtainswhere is a function.
Let . Design the virtual controller asSubstituting (20), (27), (29), (34), and (35) into (16), one can obtainInductive Step. Suppose at step , there exist a Lyapunov function , and a range of continuous virtual controllers defined aswith , such thatThen, the following property can be inferred.
Proposition 1. Choose the th Lyapunov function as with
Then, is on and there exists a virtual controller such thatwhere
The proof of above Proposition 1 is provided in the Appendix.
Step 3. In light of the inductive step, when and , Proposition 1 holds. Thus, we choose the overall Lyapunov function as withand define the virtual controller asThen, is clearly a function on , and it is easy to obtain thatTherefore, we can designwhich results in
3.2.2. Part II: Verification of Keeping the Output Constraint
For any , by Ito’s formula and (46), we can deduce
From and (47), it can be further verified thatwhich indicates
Hence, . Then, Part I of Theorem 1 is proved.
3.2.3. Part III: Stability Analysis
On the contrary, by a simple calculation, one obtains
In addition, since , it is easy to get that for all . Then, applying the characteristics of tangent functions, it is not difficult to infer that
So, one further obtains