Finite-Time Stabilization for p-Norm Stochastic Nonlinear Systems with Output Constraints

)is paper investigates the finite-time stability problem of p-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. )is result is demonstrated by both the rigorous analysis and the simulation example.


Introduction
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], H ∞ control [5,6], and sliding mode control [7][8][9][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][12][13][14], for examples. Especially, some works have considered p-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 [15]. Luckily, the issues have been well studied for p-norm stochastic nonlinear systems with different structures by the adding a power integrator technique in the existing literatures. For instance, Li et al. [16] have considered the adaptive state-feedback stabilization for p-norm stochastic nonlinear systems; the output-feedback control has been addressed for p-norm stochastic nonlinear systems with time-varying delays in [17]; Zhao et al. [18] have proposed a neural tracking control algorithm for p-norm switched stochastic nonlinear systems. More latest studies can be found in [19][20][21] and the references within.
However, most of the abovementioned works about p-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. [24] have first proposed the notion of the barrier Lyapunov function (BLF) and consequently have developed a control design strategy for a class of strictfeedback 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 [27], and sliding mode control for nonlinear systems with output constraints [28][29][30]. Moreover, since the finite-time control possesses some inherent advantages [31][32][33], techniques for the finitetime stabilization under output/state constraints have also been developed, respectively, for strict-feedback nonlinear systems [34], norm nonlinear systems [35][36][37], and switched nonlinear systems [38]. On the basis of these results, the constrained control schemes for some classes of stochastic nonlinear systems have also been proposed. Jin [39] has constructed an adaptive tracking controller for a class of output-constrained stochastic nonlinear systems in strictfeedback 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][43][44][45][46]. Nevertheless, the stochastic nonlinear systems with output constraints considered in most of the existing related works are in the strictfeedback form, rather than in p-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 p-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.

Problem Statement.
e following class of stochastic nonlinear systems are considered: where ω is a N-dimension standard Wiener process; x i � (x 1 , . . . , x i ) T ∈ R i , u ∈ R and y ∈ R are system state, control input, and output, respectively; ϑ i (t) is the time-varying parameter; the nonlinear functions f i : R i ⟶ R and g i : R i ⟶ R N are continuous and satisfy f i (0) � g i (0) � 0; and the fractional powers q i 's meet the requirement q i ∈ R ≥1 odd : � τ ≥ 1, τ is the positive odd integersratio . e output y is required to satisfy where b is a known positive constant.
is 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.

Assumption 2.
For i � 1, . . . , n, there are a constant μ ∈ (− [1 + n j�2 q 1 · · · q j− 1 ] − 1 , 0) and known nonnegative smooth Remark 1. Note that condition (4) is borrowed from [34]. However, the systems considered in [34] are p-norm deterministic nonlinear systems, while we consider p-norm stochastic nonlinear systems with drift terms f i 's and 2 Complexity diffusion terms g i 's in this paper. In light of μ ∈ (− [1 + n j�2 q 1 · · · q j− 1 ] − 1 , 0), the value of μ is generally taken as μ � − (m/p) for simplicity, where m and p represent even and odd integers, respectively. en, the value of each v j (j � 2, . . . , n) can be obtained by applying v 1 � 1 and v j+1 � ((v j + μ)/q j ) > 0. It can also be observed that both the denominator and numerator of each v j are odd.
Lemma 2 (see [13]). Suppose that system (3) admits a solution for each initial value. If there are κ ∞ class functions 9 1 (·) and 9 2 (·), a positive C 2 Lyapunov function V, real numbers c > 0, and 0 < c < 1, such that For all t ≥ 0, then the origin of system (3) is finite-time stable in probability.

A Tan-Type BLF.
Before carrying out the control design for system (1), we should handle the output constraint issue.
Let σ be a constant parameter satisfying σ ≥ σ 0 , where the value of σ 0 is chosen as below: en, a tan-type BLF can be constructed on Π 1 as follows: where μ is given by Assumption 2 and σ is defined as above. It is not hard to obtain from the expression of V b (x 1 ) that where .

Remark 2.
It should be noted that the BLF is modified from [34], which is constructed by fully taking the advantage of the given nonlinear growth conditions. As stated in [34], the control strategy based on V b (x 1 ) is a universal method, which can handle stochastic systems with or without output constraints.

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.  (1) is kept in a given constrained set in the sense of probability, i.e., P |y(t)| < b � 1 (ii) e origin of the closed-loop system is finite-time stable in probability.
Proof. e proof contains three parts. First of all, the design procedure of the controller is explicitly displayed. en, 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. Step 1. Let ς 1 � ⌈x 1 ⌉ σ , and choose the Lyapunov function . en, we can directly get from Definition 1 that Complexity where is a nonnegative C 2 function and ξ 2 is a virtual controller required to be design after later. en, we design Substituting (12) into (13) yields Step 2. We denote Since Using Definition 1 again, we have

Complexity
where ξ 3 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.

Complexity 5
where is a nonnegative C 2 function.

Proposition 1. Choose the ith Lyapunov function
en, V i is C 2 on Π i and there exists a virtual controller ξ i+1 such that where e proof of above Proposition 1 is provided in the Appendix.
Step 3. In light of the inductive step, when i � n and x n+1 � u, Proposition 1 holds. us, we choose the overall Lyapunov function V n as V n � V n− 1 + Ψ n with and define the virtual controller ξ n+1 as en, V n (x) is clearly a C 2 function on Π n , and it is easy to obtain that erefore, we can design which results in

Remark 3.
Comparing with the existing results in most of the literatures, the paper mainly focuses on the finite-time stabilization, instead of the boundness of tracking error. Moreover, the proposed approach can be extend to the tracking control by introducing a coordinate transformation before constructing the BLF.
us, one can select σ � σ 0 � 1. Now, let ς 1 � ⌈x 1 ⌉ and G(x 1 ) � sec 2 (π|x 1 | 48/11 /2b 48/11 ). In view of design procedure in Part I of the proof, we can design the virtual controller ξ 2 as Next, according to the design procedure, we denote ς 2 � ⌈x 2 ⌉ 11/5 − ⌈ξ 2 ⌉ 11/5 and further obtain that Finally, we suppose b � 1.5 and select some different initial states x(0)'s with each x(0) satisfying x(0) ∈ Π 2 . e simulation results of system (48) are shown in Figures 1 and  2. Figure 1 curves trajectories of x 1 (t) under different initial values, which illustrate that the output constraint is always not violated. Meanwhile, the trajectories of x 2 (t) is given in Figure 2. It can be observed from the two figures that system (48) under controller (51) is finite-time stable.

Conclusion
In this paper, the stability issue is addressed for a class of p-norm stochastic systems with output constraints and unknown time-varying parameters. Using a tan-type BLF, the finite-time control strategy is proposed by the adding a power integrator technique. On this basis, the designed controller has been proved to ensure that the origin of the closed-loop system is finite-time stable in probability and the system output is kept in a pre-given set. is conclusion has also been verified by the simulation results. It should be pointed out that the proposed approach is not applicable to the case of asymmetrical output constraints. In the future, we will try to modify the proposed method to be suitable for stochastic systems with asymmetrical output constraints or multi-input multi-output stochastic systems. en, we have In the following, we introduce some sub-propositions to simplify the proof.