Finite-time fuzzy output-feedback control for p -norm stochastic nonlinear systems with output constraints

: This paper investigates the ﬁnite-time control problem of p -norm stochastic nonlinear systems subject to output constraint. Combining a tan-type barrier Lyapunov function (BLF) with the adding a power integrator technique, a fuzzy state-feedback controller is constructed. Then, an output-feedback controller design scheme is developed by the constructed state-feedback controller and a reduce-order observer. Finally, both the rigorous analysis and the simulation results demonstrate that the designed output-feedback controller not only guarantees that the output constraint is not violated, but also ensures that the system is semi-global ﬁnite-time stable in probability (SGFSP).


Introduction
Over the past decades, a variety of control design strategies have been proposed for different nonlinear systems [1][2][3][4][5][6][7]. Especially, many approximated-based control schemes have been developed for uncertain nonlinear systems by using neural networks (NNs) or the fuzzy logic systems (FLSs) [8][9][10][11][12][13][14][15][16][17][18][19]. Among these studies, the research of stochastic systems is much more attracted (see, e.g. [16][17][18][19] and the references therein), due to their wide application. It is worth noting that the aforementioned NNs-based or FLSs-based control strategies haven't taken output constraint into account. In fact, many practical systems are usually required to satisfy an output constraint in the operation for considering the performance specifications or safety [20,21]. It is well known that, the BLF-based approaches are useful tools to settle controller design problems of output-constrained nonlinear systems, see references [22][23][24][25][26][27] for instances. In the latest research progress of constrained control, many kinds of adaptive neural or fuzzy control design methods have been presented by dx i = x p i+1 dt + φ i (x i )dt + g T i (x i )dω, i = 1, · · · , n − 1, dx n = u p dt + φ n (x)dt + g T n (x)dω, y = x 1 , where ω is a r-dimension standard Wiener process; x = (x 1 , · · · , x n ) T ∈ R n is system state vector; u ∈ R and y ∈ R are respectively control input and output; the fractional power p ∈ R ≥1 odd := {m/k|m ≥ k, m and k are positive odd integers}; for i = 1, · · · , n,x i = (x 1 , · · · , x i ) T ∈ R i ; φ i : R i → R and g i : R i → R r are unknown continuous functions satisfying φ i (0) = 0, g i (0) = 0. The system output y = x 1 is measurable and constrained in Π 1 = {y(t) ∈ R, |y(t)| < ε} with a constant ε > 0, while the other states x 2 , · · · , x n are all unmeasurable.
The objective of this paper is to design a finite-time fuzzy output-feedback controller for system (2.1) such that: 1) the output don't violate the given constrained boundary; 2) all the signals of the closed-loop system converge to a small compact of the original point in finite-time in probability in presence of unknown nonlinearities and unmeasured states x i (i = 2, · · · , n).
Firstly, some concepts and lemmas are presented for preliminaries. Consider the following where φ(x) and g(x) are continuous functions with satisfying φ(0) = g(0) = 0.
Remark 1. As stated in [17], the Eq (2.4) implies that there exists the stochastic setting time function
[8] Let p ∈ (0, ∞), for any ζ i ∈ R, i = 1, · · · , n, one has Lemma 5. [36] If ζ, η ∈ R and p > 1 is an odd number, then In this paper, the nonlinear functions φ i (·) and g i (·) are all unknown. The unknown functions will be approximated by the FLSs based on the following presented lemma.
Remark 2. In view of Lemma 7, any function F(X) which is defined and continuous on a compact set Π 0 , can be approximated by where (X) is the FLS approximation error satisfying | (X)| < δ.

State-feedback controller design
In this section, a fuzzy state-feedback controller will be explicitly designed for system (2.1) by combining a tan-type BLF and the FLSs into the adding a power integrator technique.
First of all, we introduce a coordinate transformation as follow where q 1 = 0, q j = q j−1 +1 p ( j = 2, · · · , n + 1), and H > 1 is a constant to be determined later. Based on (3.1) , system (2.1) turns into In what follows, a fuzzy state-feedback controller will be designed through n steps based on the equivalent system (3.2). Define where β i 's are the virtual signals being constructed later.
Step 1. From (3.1), we can get Choose the first Lyapunov function where b 1 > 0 is an adjustment parameter,α 1 = α 1 −α 1 is the estimate error andα 1 is the estimator of the parameter α 1 .
2ε 4 is a tan-type BLF adopted to deal with the system output constraint. Compared to the log-type BLF, V B (ξ 1 ) possesses the following characteristic: which implies that the proposed method is also applicable to the system without output constraints.
Substituting (3.11) and (3.12) into (3.10), gets In addition, it is easily obtained that Therefore, we can get Remark 5. According to Lemma 6, one getsα 1 ≥ 0, for ∀t ≥ 0. In each design step, this characteristic will be always applied.
Step 2. From (3.3) and Itô's formula, we have where Combining the definition of β 1 with the properties of f 1 (χ 1 ) and h 1 (χ 1 ), implies that β 1 is valid and continuous.
Choose the second Lyapunov function as where b 2 > 0 is an adjustment parameter,α 2 = α 2 −α 2 is the estimate error andα 2 is the estimator of the parameter α 2 . Applying (2.3), (3.14) and (3.16), it can be gotten that Besides, applying Lemma 3 renders On the other hand, we gets , and σ 12 > 0 is an adjustment parameter.
In addition, it is evident that Substituting (3.25) into (3.22), one gets where Step (3 ≤ k ≤ n). In view of above two steps, we can deduce the following similar property whose proof can be found in the Appendix. Proposition 1. For the kth Lyapunov function V k : Π k → R + as there exists a virtual controller β k and the adaptive law ofα k of the following forms

Selection of the observer gains
In this section, we will analyze the appropriate values of the gains γ i (i = 2, · · · , n) and some constant parameters in output-feedback controller.
Now, a proposition is provided for helping to determine gain constants, whose proof will be given in Appendix.

Stability analysis
To state the main result, the following theorem is presented.
ii) all the signals in the closed-loop stochastic nonlinear system (2.1) are SGFSP.
Proof. i) Let µ 0 = min{ 1 , d 1 , · · · , d n ,¯ 2 , · · ·¯ n ,¯θ 2 γ , · · ·¯θ n γ } and π 0 =Q. Then, Eq (3.51) can be expressed as (3.52) We can easily get from Eq (3.52) that For x(0) = (x 1 (t 0 ), · · · , x n (t 0 )) T satisfying x 1 (t 0 ) ∈ Π 1 , it easily obtains that the mean of V(t) is bounded, which implies that V is bounded in probability. It can be directly deduced from the definition of V that P{V B (ξ 1 ) < ∞} = 1. (3.54) Consequently, it is clear that P{|y(t)| < ε} = P{|ξ 1 (t)| < ε} = 1, which demonstrates that the output constraint of system (2.1) is not violated in the sense of probability. ii) For ∀ 0 <ς 0 < 1, it is easy to get from Lemma 3 that Further, one has Then, substituting (3.55) into (3.52) drives where χ(0) = (χ 1 (t 0 ), · · · , χ n (t 0 )) T , e(0) = (ẽ 2 (t 0 ), · · · ,ẽ n (t 0 )) T ,α(0) = (α 1 (t 0 ), · · · ,α n (t 0 )), 0 < l 0 < 1 is a constant. Then it follows from Lemma 1 that for ∀t ≥ t 0 + T * , E V 1−ς (χ,ẽ,α) ≤¯π 0 µ 0 (1−ς 0 ) , which means that all the signals in the closed-loop systems are semi-global finite-time stable in probability. Remark 6. In this paper, we construct an output-feedback controller rather than the designed statefeedback controllers in existing results about output constraints. On the other hand, it should be pointed out the considered constraint is symmetric rather than asymmetric, which leads that the proposed scheme can not be directly employed or further extended to the case of asymmetric constraints. However, a control scheme based on a new BLF can be developed for asymmetric output constraints in a similar way to this paper. In addition, another limitation is that all of the fractional powers are equal to p. If p i 's are taken different values, the proposed strategy seems not applicable. In the future, we will address the two issues.

Simulation example
The validation of the proposed strategy will be testified by the following system.
where the output y = x 1 is measurable and constrained by Π 1 = {y(t) ∈ R, |y(t)| < 1}, and the state x 2 is unmeasurable.  According to the controller design procedure, we can respectively design the finite-time outputfeedback controller, the adaptive laws and the observer as follows:     Figure 1 provides the trajectory of x 1 (t), which indicates that the system output constraint is not violated under controller (4.2). Meanwhile, the trajectories of x 2 (t) andx 2 (t) are given in Figure 2, which shows that x 2 (t) is well estimated byx 2 (t). Moreover, the trajectory of the controller u is displayed in Figure 3. Finally, Figure 4 expresses the curves of the adaptive parameter vector under the developed strategy. Also, one could evidently observe from these figures that all the signals of system (4.1) are semi-global finite-time stable in probability under controller (4.2).

Conclusion
In this paper, the output-feedback controller design problem is investigated for a class of p-norm stochastic nonlinear systems with output constraints. Through using a tan-type BLF, an adaptive fuzzy state-feedback controller is proposed by the adding a power integrator technique. Then, a finite-time fuzzy output-feedback controller is constructed by combining the proposed state-feedback controller and a reduced-order observer. Both rigorous proof and the simulation example verify that the designed controller can ensure the achievement of the system output constraint and semi-global finite-time stability of all the signals in probability. In the future, we will consider the situations of asymmetric constraints, different fractional powers, or multi-input multi-output stochastic nonlinear systems.
In addition, we have γ Hẽ