Output-Feedback Control for a Class of Stochastic High-Order Feedforward Nonlinear Systems with Delay

The problem of global output-feedback stabilization for a class of stochastic high-order time-delay feedforward nonlinear systems with different power orders is investigated. By combining the adding one power integrator technique with the homogeneous domination approach, an output-feedback controller design is proposed, which ensures the global asymptotical stability in probability of the closed-loop system.


Introduction
In the last decades, stochastic systems have received much more attention since stochastic modeling has come to play an important role in many branches of science and engineering applications.For this type of systems, the authors in [1][2][3][4][5][6][7][8][9][10] presented the basic stability theory of the stochastic control systems.
Global stabilization of triangular structural stochastic nonlinear systems has been a major issue in control theory over the last decades.For lower-triangular systems (namely, feedback systems), the stochastic asymptotic stabilization has been studied by using backstepping approach; see [11][12][13] and the references therein.Upper-triangular systems, which are also called feedforward systems, have been fully used to model many physical devices, such as the ball-beam with a friction term [14] and the cart-pendulum system [15].By using a homogeneous domination approach, the problem of global output-feedback stabilization was addressed in [16] for a class of upper-triangular systems with higher-order nonlinearities.Then, the case of lower-order nonlinearities was considered in [17].With the help of a generalized definition of homogeneity, the problem of using small controls to globally stabilize a class of upper-triangular systems was investigated in [18].Inspired by the works [19,20], the problem of using a sampled data controller to globally stabilize a class of uncertain upper-triangular systems was considered in [21].When parameter uncertainties appeared in a system model, an adaptive stabilizer for feedforward nonlinear systems with general dynamic order in [22] was extended to [23].
On the other hand, time delays may arise naturally, which are usually the key factors that influence the stability of nonlinear systems.Many researchers have paid more attention to studying the stochastic nonlinear time-delay systems over the last decades and various results concerning stochastic lower-triangular nonlinear systems with time delays have been reported in [24][25][26][27].For the case where the delays are of unknown length, an adaptive control design was presented in [28] for a system in feedforward form.An approach for compensating input delay of arbitrary length was presented [29] for forward complete and strict-feedforward nonlinear systems.When the nonlinear functions were higher order in states, the problem of global stabilization by state feedback was addressed in [30,31].It should be pointed out that these results were obtained for the case of state measurement.Naturally, one may ask a challenging and interesting question: if not all the states are measurable, how do we design an outputfeedback controller for stochastic high-order feedforward systems with time delay?To the best of our knowledge, no output-feedback stabilization control scheme has so far been proposed for stochastic high-order feedforward systems with time delay.
Inspired by the aforementioned discussion, we deal with the problem of global output-feedback stabilization for a class of stochastic high-order feedforward nonlinear systems with time-varying delay and different power orders.Firstly, we design a state-feedback controller for the nominal system using the adding one power integrator technique.Then, by designing a homogeneous observer for the nominal system, we explicitly construct a homogeneous output-feedback stabilizer under the certainty equivalence principle.At last, we propose a scaled controller which guarantees global asymptotic stability in probability of the closed-loop system.
The outline of this paper is as follows.Sections 2 and 3 offer some preliminary results and problem formulation, respectively.The output-feedback controller is designed and analyzed in Section 4. This paper is concluded in Section 5.

Notations
For the sake of simplicity, sometimes function () is denoted by ;   represents the ith element of vector  and   = ( 1 , . . .,   )  .  denotes the set of all functions with continuous th partial derivative.K denotes the set of all functions:  + →  + which are continuous, strictly increasing, and vanishing at zero; K ∞ denotes the set of all functions which are of class K and unbounded.
In the following, we borrow some definitions and lemmas which play an important role in stabilizing and analyzing the stochastic time-delay systems for later development in this paper.
(i) /  is homogeneous of degree  −   with   being the homogeneous weight of   .
For any positive number  > 0, the following inequality holds

Problem Formulation
In this paper, we will consider the following stochastic highorder feedforward nonlinear system with time-varying delay in the form where  = [ where , and  0 = max =1,..., {  }.
In this paper, we aim to constructively design a homogeneous output-feedback controller for system (9) under Assumptions 10-11, such that the closedloop system is GAS in probability.

Controller Design and Stability Analysis
In this section, an output-feedback controller will be explicitly constructed for the nonlinear system (9).We will combine the adding one power integrator technique with the homogeneous domination approach for output-feedback stabilization.The design procedure can be divided into three steps: (i) we first design a state-feedback controller for the nominal system without the drift and diffusion functions using the adding one power integrator technique; (ii) then, by designing a homogeneous observer for the nominal system, we explicitly construct a homogeneous output-feedback stabilizer under the certainty equivalence principle; (iii) in the end, we propose a scaled controller which guarantees global asymptotic stability in probability of the closed-loop system.
For simplicity, we assume  = / in this paper with  being an even integer and  being an odd integer.Based on this assumption, it is obvious that   is a ratio of odd integers.

Homogeneous State-Feedback Control of Nominal Nonlinear System.
In this subsection, we firstly introduce a key lemma, which avoids the zero-division problem and serves as a basis in the following design procedure.The proof is similar to that in [32].
Now, we design a homogeneous state-feedback controller for the nominal system Lemma 13.For system (13), there are positive definite, proper, and  2 Lyapunov function   , a state-feedback controller  * , and two positive constants, such that where   ,  * are defined in the following form: Remark 14.For the deterministic work, it is well known that there only needs a positive definite, proper, and  1 Lyapunov function for the controller design and analysis.However, for the stochastic system, the Lyapunov function must be positive definite, proper, and at least  2 because of the appearance of the Hessian term.
Next, we construct a scaled homogeneous observer where  2 > 0, . . .,   > 0 are the gains selected in (28) and the controller  is designed with the same construction of (17).
Theorem 20.Under Assumptions 10 and 11, there is an outputfeedback controller V   =       +1 such that the closed-loop system consisting of (9), (16), and (17) has a global unique solution and the equilibrium  = 0 is GAS in probability.
Proof.We prove Theorem 20 as follows.
Step 2. Construct a Lyapunov-Krasovskii functional which is positive definite, proper, and  2 on Φ, where  0 ,  0 ,  0 are positive parameters to be determined later.
According to Lemma 4.3 in [33], there are two K ∞ functions  1 and  21 such that Then, with the notation in [26], it is easy to verify that where with  2 =  21 +  22 .
Using Assumption 10 and the new coordinates in (32) and the power of  in (42) is Owing to  1 = 0,   = ( −1 + 1)/ −1 , one has For  ≥  + 2, (44) gives In summary, it can be concluded that there exists a positive constant where  0 is a positive constant.
From Steps 1 to 3 and Lemma 4, it is obtained that the closed-loop system (33), (34), and (17) has a global unique solution and the equilibrium Φ = 0 is GAS in probability.
Step 4. Because of the equivalence in (32), then there is a global unique solution for the closed-loop system consisting of ( 9), (16), and ( 17) and V   =       +1 and the equilibrium  = 0 is GAS in probability.
Remark 21.Using the adding one power integrator method and the homogeneous domination technique, [34] has designed a state-feedback stabilizer for a class of stochastic high-order feedforward nonlinear systems.However, to deal with the unmeasurable states and time-delay terms in this paper, we choose a reduced-order homogeneous observer and an appropriate Lyapunov-Krasovskii functional, which is not an easy work.

Conclusions
For a class of stochastic high-order feedforward nonlinear systems with different power orders and time-varying delay, an output-feedback stabilizer has been designed by virtue of the adding one power integrator technique and homogeneous domination approach.It globally stabilizes the origin of the closed-loop system.The proof of stability we have adopted depends on the construction of a Lyapunov-Krasovskii functional.