A New Finite-Time Bounded Control of Stochastic Itô Systems with ( x , u , V )-Dependent Noise : Different Quadratic Function Approach

and Applied Analysis 3 Lemma 8 (Gronwall Inequality). Let θ(t) be a nonnegative function such that


Introduction
It is well known that finite-time control has become one of the important robust control methods, which has been studied extensively both in theory and practical applications; see linear systems [1][2][3][4][5][6][7][8][9], nonlinear systems [10][11][12], and the inpress book [13].Recently, based on analysis on some practical problems, [14] introduced a new finite-time stability for linear stochastic Itô systems with state and control-dependent noise.Roughly speaking, a stochastic Itô system is said to be finitetime stable if, given a bound on the initial state of the system, its state trajectories do not exceed an upper bound  2 and are not less than a lower bound  1 ( 1 <  2 ) in the mean square sense during a specific time interval.
On the other hand, the model of stochastic Itô systems with state, control input, and external disturbance dependent-noise ((, , V)-dependent noise for short) is more general than stochastic Itô systems with state and control input-dependent noise ((, )-dependent noise for short).
For this class of model, some results have been obtained.
Motivated by aforementioned discussions, we extend the results in [14] to stochastic Itô systems with (, , V)dependent noise.Here, we consider finite-time stochastic boundedness and finite-time bounded control problems for such class of systems.More precisely, a system is said to be finite-time bounded if, given a bound both on the initial state of the system and the disturbance input, the state trajectories of the system do not exceed an upper bound  2 and are not less than a lower bound  1 ( 1 <  2 ) in the mean square sense during a prespecified time interval for all admissible disturbances.By stochastic analysis technology, Gronwall's inequality, and matrix transformation, a finite-time stochastic boundedness criterion and some sufficient conditions for the existence of finite-time bounded controller are derived.The contributions of this paper lie in the following two aspects: (1) a new concept of finite-time stochastic boundedness is introduced, which generalizes the finite time stochastic stability in 2 Abstract and Applied Analysis [14] to stochastic Itô systems with (, , V)-dependent noise and (2) a different quadratic function approach is introduced and its superiority to common quadratic function approach is shown.By different quadratic function approach, two new conditions for the existence of state and output feedback finite-time bounded controller are obtained.
The paper is organized as follows.In Section 2, a concept of finite-time stochastic boundedness and some preliminaries are presented.Section 3 provides a sufficient condition for finite-time stochastic boundedness.In Section 4, state and output finite-time bounded controllers are given, respectively.Section 5 employs an example to illustrate the results of the paper.Section 6 gives the conclusion.
stands for the mathematical expectation operator with respect to the given probability measure. × is  ×  identity matrix.tr() is trace of a matrix . max ()( min ()) is the maximum (minimum) eigenvalue of a real matrix .
To illustrate clearly the concept of finite-time stochastic boundedness presented below, we first introduce finite-time stochastic stability from [14].Definition 1.Given positive real scalars  1 ,  2 ,  3 ,  4 ,  with 0 <  1 <  3 <  4 <  2 , and a positive definite matrix , the following linear stochastic system is said to be finite-time stochastically stable with respect to Based on Definition 1, a new concept of finite-time stochastic boundedness for linear stochastic Itô systems is introduced.
Definition 2. Given some positive scalars  1 ,  2 ,  3 ,  4 ,  with 0 <  1 <  3 <  4 <  2 , a positive definite matrix , and a class of exogenous signals W, the following linear stochastic system is said to be finite-time stochastically bounded with respect to ( 1 ,  2 ,  3 ,  4 , W, , ), if for all V(⋅) ∈ W. Remark 3. Definition 2 is more general than Definition 1, which concerns the behavior of the state in the presence of both given initial conditions and external disturbance.
Remark 4. It is clear that finite-time stochastic boundedness implies finite-time stochastic stability, but the converse is not true.
In the next assumption, we characterize a class of signals W considered in this paper.
Assumption 5.The class W is defined as follows: where ,  1 , and  1 ≥ 0 are constant matrices and ℎ and ℎ 1 are any given positive scalars.
Remark 6.In Assumption 5, ℎ and ℎ 1 are any given positive scalars, so W actually includes a big class of signals.
Before proceeding further, we give some lemmas which will be used in the next section.

Finite-Time Stochastic Boundedness
This section is dedicated to proposing a different quadratic function approach to the finite-time stochastic boundedness problem of the system (4).The comparison on different quadratic function approach and common quadratic function approach is first given.
In [14], the key approach of obtaining main results is as follows.Let ( state ) be a positive quadratic function; then by the following inequalities L ( state ()) <  ( state ()) , L ( state ()) >  ( state ()) , the main results are derived.We call the above approach to be common quadratic function approach, because the quadratic functions in (13) and ( 14) are the same.But we find that ( state ()) satisfying ( 13) may not satisfy (14), which results in the a relatively small range of the option of ( state ()).
So the main results obtained by common quadratic function approach are of conservativeness.
The key idea of different quadratic function approach is as follows.Let  1 ( state ()) and  2 ( state ()) be a positive quadratic function; then by the following inequalities the main results of this paper will be derived.Because the quadratic functions  1 ( state ()) in ( 15) and  2 ( state ()) in ( 16) are not the same, the main results obtained by this approach are of less conservativeness than the results obtained by common quadratic function approach.

Finite-Time Stochastic Bounded Controller Design
In this section, we use different quadratic function approach to design state and output feedback finite-time bounded controller such that the closed-loop system of system ( 1) is finite-time stochastically bounded over a finite-time interval [0, ], respectively.

State Feedback Finite-Time Bounded Controller Design.
For system (1), we first consider a state feedback controller then the closed-loop system of (1) is as follows: Next, a sufficient condition of the existence for state feedback finite-time bounded controller is presented by Theorem 10.

Dynamic Output Feedback
Finite-Time Bounded Controller Design.When the system states are not completely accessible, state feedback controllers may become invalid.This motivates us to propose an output-feedback controller.Without loss of generality, we can assume the following.Assumption 13.There exists a state feedback controller () = () which has been designed using the results of Theorem 12.
We choose, as usual, a finite-dimensional observer-based controller as follows: where x() ∈ R  is the the estimate of the state of () and  is the estimator gain matrix with appropriate dimensions, which is to be determined.
Remark 15.It is easy to see that the values of  and  determine the feasibility of the above Theorems.The procedure of how to choose  and  is given in the next section.

Numerical Algorithm
This section gives an algorithm for the results of the paper.The following algorithm is used to solve the matrix inequalities in Theorem 10.Similar algorithms can be used in Theorem 12 and Theorem 14.
Step 7. Stop.If (, ) = (0, 0), then we cannot find (, ) making ( 17 Remark 17.By Algorithm 16, we can obtain a region surrounded by  and , if it exists, which is used to select  and  for appropriate conditions.

Examples
In this section, an example is given to illustrate the results we have obtained.
The evolution of E[  ()()] of the closed-loop system of (1) is illustrated by Figure 2, which shows that the closedloop system of ( 1) is finite-time stochastically bounded with respect to (1, 30, 4, 5, 0.25, , W).The corresponding control curves are illustrated by Figure 3.
Case 2 (dynamic output feedback finite-time bounded controller design).Based on state feedback controller design,   The evolution of E[  ()()] of the closed-loop system of (1) is illustrated by Figure 5, which shows that the closedloop system of ( 1) is finite-time stochastically bounded with respect to (1, 30, 4, 5, 0.25, , W).The corresponding control curves are illustrated by Figure 6. Figure 7 illustrates the evolution of E[  ()()] of the error system (54), which shows that E[  ()()] < 1.

Conclusion
In this study, finite-time stochastically bounded control linear stochastic Itô systems with (, , V)-dependent noise has been investigated.Applying different quadratic function approach, state and output feedback finite-time bounded controllers have been obtained, respectively.One example is presented to illustrate the effectiveness of the proposed results.In addition, we can also refer to [17][18][19] and extend the results of this paper to Takagi-Sugeno fuzzy systems, networked systems, linear parameter varying systems, and so on.

Figure 1 :
Figure 1: A region by  and .

Figure 4 :
Figure 4: A region by  and .