Finite-Time H 2 / H ∞ Control for Linear Itô Stochastic Systems with x , u , v-Dependent Noise

This paper deals with the problem of the H2/H∞ control based on finite-time boundedness for linear stochastic systems. The motivation for investigating this problem comes from one observation that the H2/H∞ control does not involve systems’ transient performance. To express this problem clearly, a concept called finite-time H2/H∞ control is introduced. Moreover, state feedback and observer-based finite-time H2/H∞ controllers are designed, which guarantee finite-time boundedness, H2 performance index, and H∞ performance index of the closed-loop systems. Furthermore, an optimization algorithm on the finite-time H2/H∞ control is presented to obtain the minimum values of the H2 index and H∞ index. Finally, we use an example to show the validity of the obtained results.


Introduction
Recently, stochastic systems have been receiving more and more attention, which is due to their applications in many practical fields, such as finance systems [1] and power systems [2].A lot of results on stochastic systems have been obtained.For example, [3] gave some Razumikhin-type theorems on the pth moment input-to-state stability for impulsive stochastic delayed systems by utilizing the Razumikhin technique and average dwell-time approach.By using the interval matrix transformation method, the problems of finite-time dissipative control for stochastic interval systems are investigated in [4].The literature [5] investigated the state prediction problem for nonlinear stochastic differential systems by utilizing the Carleman embedding technique.In addition, many other nice results on stochastic systems have also been obtained; see, e.g., Lyapunov stability conditions [6], reliable output feedback control [7] and distributed containment control [8], and the references therein.
On the other hand, one of the important control methods in dealing with robust control problems is the H 2 /H ∞ control method.Its concrete implication is to seek a controller which not only minimizes the H 2 cost function but also restrains the effect of external disturbance.The H 2 /H ∞ control method has been applied to many fields, such as communication systems [9] and synthetic gene network design [10].Meanwhile, H 2 /H ∞ control problems have been extended to many control system models, and many nice results have been obtained, such as stochastic systems [11], descriptor systems [12,13], Markovian jump systems [14], and the recent monograph [15].Nevertheless, most of the results on H 2 /H ∞ control problems in existing literature are based on Lyapunov's asymptotic stability, which do not reflect the transient performance of the systems.In some cases, large transient performance has a negative effect on the practical systems.For example, in power systems, a large transient current cannot be permitted, because it can damage the system [16].In order to describe the phenomenon precisely, finite-time (FT) stability was introduced in the literature [17][18][19].In the sequel, considering external disturbance, [20] extended FT stability to FT boundedness.Currently, the issues on FT stability and FT boundedness have been extensively studied for many kinds of system models; see, e.g.[21][22][23][24][25][26][27][28][29][30] and the references therein.Taking into account the advantages of FT stability and H 2 /H ∞ control, the finite-time H 2 /H ∞ control problem is proposed in this paper, which not only guarantees FTB but also satisfies the H 2 /H ∞ performance indices.Up to date, there is almost no literature to consider the FT H 2 /H ∞ control of stochastic systems.
This paper will study the problems of FT H 2 /H ∞ control for linear Itô stochastic systems.Because of the complexity of the problems, the appropriate controller design will be more difficult.By using the method of stochastic analysis, state feedback and observer-based FT H 2 /H ∞ controllers are designed.The main contributions of this paper are as follows: (i) The definition of the FT H 2 /H ∞ control for linear stochastic systems with x, u, v -dependent noise (which is called state-, control-, and disturbance-dependent noise) is first given, which simultaneously presents FT boundedness, H 2 performance index, and H ∞ performance index, respectively, of the closed-loop system.(ii) The two new sufficient conditions for the existence of the state feedback controller and observer-based controller are obtained in the form of linear matrix inequalities (LMIs).(iii) A parameter optimization algorithm is given to obtain the minimum values of the H 2 performance index and H ∞ performance index, simultaneously.
The organization of this paper is summarized as follows: In Section 2, we give some preliminaries.In Section 3, the state feedback FT H 2 /H ∞ controller is designed.In Section 4, the observer-based FT H 2 /H ∞ controller is designed.In Section 5, an optimization algorithm is provided for obtaining the minimum values of the H 2 performance index and H ∞ performance index, simultaneously.In Section 6, a numerical example is discussed, and some remarks are concluded in Section 7.
Notation.Ω, F , F t t≥0 , ℙ : a complete probability space with a filtration F t t≥0 .M ′ : transpose of a matrix M. M > 0: M is a positive definite symmetric matrix.I n×n : n × n identity matrix.L 2 F R + , R l : the space of nonanticipative stochastic process ξ t with respect to filtration F t satisfying and λ min M : the maximum and minimum eigenvalue of matrix M. E • is the mathematical expectation of the stochastic process.The asterisk " * " in a matrix stands for the symmetry term.diag ⋯ represents a diagonal matrix.The "wrt" denotes "with respect to."

Preliminaries
We consider the linear stochastic system as follows: where and z t ∈ R l are called state, control input, external disturbance, measurement output, and controlled output, respectively.x 0 is the initial state.w t is a one-dimensional Wiener process, which is defined as Ω, F , F t t≥0 , ℙ .v t is the disturbance and satisfies the following set: Next, the definition of FT stochastic boundedness for system (1) is introduced, which is the generalization of FT boundedness in [20,[31][32][33].For the subsequent analysis, the following lemmas are useful for the FT H 2 /H ∞ controller design.

Complexity
then we obtain ρ t ≤ f exp gt 5 Lemma 2 [34].Let V t, x ∈ C 1,2 R + , R + × R n be a scalar function and V t, x > 0. For the following stochastic system dx t = a x dt + b x dw t , 6 the Itô formula of V t, x is given as follows: where This section gives a problem formulation of state-feedback finite-time (SFFT) H 2 /H ∞ control for system (1).And also, the two sufficient conditions are given for the existence of a SF controller.
Considering the following linear state-feedback (SF) controller where K is the feedback gain matrix.Substituting the controller (9) into system (1) leads to the closed-loop system where Associated with system (1), the following cost function is provided: where P 1 > 0 and P 2 > 0 are the given positive scalars or given weighting matrices.
Also, by substituting ( 9) into (11), the corresponding performance function (11) becomes For a given γ > 0, under the condition of zero initial value, the discretional nonzero disturbance v t and the control output z t satisfy the following form: Based on the above preparations, we are in a position to give the definition of the state-feedback finite-time H 2 /H ∞ control.Definition 2. For some given positive scalars γ, d 1 , d 2 , T, R, and h, considering system (1), cost function (11), and the inequality (13), if there exist a positive scalar J * s and a SF controller ( 9) such that (i) system ( 10) is FT stochastically bounded wrt d 1 , (iii) for any nonzero disturbance v t , under the zero initial condition, the inequality ( 13) is satisfied then ( 9) is a state-feedback finite-time H 2 /H ∞ controller of the stochastic system (1).
Remark 1. Definition 2 considers three aspects of actual systems: FT boundedness of systems' states, minimum performance cost, and ability to suppress interference of the closed-loop systems, which is more complex than only considering the problems of the H 2 performance index or H ∞ performance index.In addition, these three aspects involved by the FT H 2 /H ∞ control are often required by the actual systems.For example, in the power systems, a large transient current cannot be permitted, electric energy consumption is expected to be minimum, and the system is expected to have better ability to suppress interference [16].
A sufficient condition for the existence of the SFFT H 2 / H ∞ controller will be given by the following theorem.
Theorem 1.For the given positive scalars γ, d 1 , d 2 , T, R, and h, if there exist a matrix M > 0, a matrix K, and a scalar α ≥ 0 such that hold, where and the upper bound of the H 2 index can be given as Proof.The proof will be divided into three steps.
Step 1. Prove that the condition (i) is satisfied.
It is noticed that Therefore, the condition ( 14) implies that 10) is defined as follows: where Pre-and postmultiplying the inequality ( 18) by diag M −1 , I, M −1 , it leads to the following inequality: According to the Schur complement, ( 20) is equivalent to Considering the conditions ( 19) and ( 21), it follows that Integrating ( 22) from 0 to t and then taking the expectation, it follows that By Lemma 1, we get Ev ′ s v s ds 24 According to known conditions, it yields From ( 24), ( 25), (26), and (27), it is easy to obtain According to condition (16), it follows that (28) leads to E x ′ t Rx t < d 2 for all t ∈ 0, T .So, 4 Complexity the closed-loop system (10) is FT stochastically bounded wrt d 1 , d 2 , T, R, h .
Step 2. Prove that the condition (ii) is satisfied.
Under the condition of v t = 0, the infinitesimal operator L 2 V x t of system ( 10) is as follows: 29 By Schur's complement, it can be seen that ( 15) is equivalent to Premultiplying and postmultiplying (30) by M −1 , we obtain According to (29) and (31), we get Integrating (32) from 0 to t, t ∈ 0, T , and taking the expectation, it is obtained that From (33), we get From (34), by Lemma 1, we obtain According to (35) and (36), it is obtained that Step 3. Prove that the condition (iii) is satisfied.
Premultiplying and postmultiplying ( 14) by diag M −1 , I, M −1 , and according to the Schur complement, we have where According to (19) and (38), we have Premultiplying and postmultiplying (39) by e −αt , we have By applying Lemma 2, we can obtain that According to (40) and (41), we get Under the condition of zero initial value, integrating both sides of (42) from 0 to t, t ∈ 0, T , taking the expectation, and after some calculations, we have Because e −αT EV x t > 0, we can get This completes the proof.
It should be noted that ( 14), (15), and ( 16) are not linear matrix inequalities and difficult to solve.To overcome the above difficulties, the following theorem is obtained.
Theorem 2. For the given positive scalars γ, d 1 , d 2 , T, R, and h, if there exist a matrix M > 0, a matrix Y, a scalar λ > 0, and a scalar α ≥ 0 such that hold, where

Observer-Based Finite-Time H 2 /H ∞ Controller Design
In the above Section 3, the SFFT H 2 /H ∞ control problem of system (1) has been discussed.However, in many practical cases, it is difficult to measure the total states.Therefore, considering the problem of the observer-based finite-time (OBFT) H 2 /H ∞ controller design is necessary.Typically, an observer-based (OB) controller is provided as follows: x 0 = 0, 49 where x t ∈ R n is the estimation of x t and L is the estimator gain.Substituting (49) into system (1), we will obtain the closed-loop system and the corresponding closed-loop cost function where

53
For a given γ > 0, under the condition of zero initial value, the discretional nonzero disturbance v t and the control output z t satisfy the following form: Then, the definition of the observer-based finite-time H 2 /H ∞ control is given as follows.
Definition 3.For the given positive scalars γ, d 1 , d 2 , T, R, and h, considering system (1), cost function (11) 49) is an OBFT H 2 /H ∞ controller and the upper bound of the H 2 index can be given as Proof.The proof will be divided into three steps.
Step 1. Prove that the condition (i) is satisfied.
Let V x t = x ′ t Qx t with Q > 0 be solutions to (55), (56), and (57).The infinitesimal operator L 3 V x t of system (50) is defined as follows: Therefore, the inequality (55) implies that According to the Schur complement, (60) can be transformed into According to (58) and ( 61), we have Integrating (62) from 0 to t and then taking the expectation, it follows that

Ev′ s v s ds 64
According to known conditions, it yields that Step 2. Prove that the condition (ii) is satisfied.
Under the condition of v t = 0, the infinitesimal operator L 4 V x t of system (50) is defined as follows: According to (56), we get Integrating (70) from 0 to t and taking the expectation, the following results are obtained: From (71), we have From (72), by Lemma 1, we obtain From ( 73) and (74), we have Step 3. Prove that the condition (iii) is satisfied.
According to the Schur complement, it can be seen that ( 55) is equivalent to According to (58) and (76), we get Using the similar procedure to prove Step 3 in Theorem 3, we can obtain

s ds 78
This completes the proof.
hold, where , and 49) is an OBFT H 2 /H ∞ controller and the upper bound of the H 2 index can be given as J * o = ηd 1 e βT .In this case, an appropriate estimator gain matrix is given by 52) and ( 53) into (55), and letting N = Q 22 L, (79) can be obtained from (55).It is obvious that (80) can be obtained from (56).And then, (81) and ( 82) imply (57).The proof is completed.

Algorithm
In this section, an optimization algorithm on the FT H 2 /H ∞ control is presented.The following algorithm is given for Theorem 2. Similarly, the algorithm can be utilized in Theorem 4.
By running Algorithm 1, we get the following results: (1) If U i = empty, F i = empty, and H i = empty, then it shows that the above optimization problem (OP) is not solvable.
(2) If U i ≠ empty, F i ≠ empty, and H i ≠ empty, then it shows that the above optimization problem is solvable.Moreover, we can respectively plot the curves of U i , F i , U i , H i , and F i , H i .The three curves can help us choose a suitable finite-time H 2 /H ∞ controller.
(3) This algorithm can find the most suitable α to decrease the conservativeness of the conditions of Theorem 2.

Examples
This section provides an example to show the effectiveness of the obtained results.System (10) with parameters is as follows: 6.1.SFFT H 2 /H ∞ Controller Design.By running Algorithm 1, the relations of α and ξ (Figure 1), α and γ (Figure 2), and γ and ξ (Figure 3) are obtained.From Figure 1, it can be seen that the bigger the number of α, the bigger ξ is.Also, the minimum value of the H 2 performance index is ξ = 1 when α = 0, and the maximum value of the H 2 performance index is ξ = 4 4817 when α = 1 5.The range of α is 0, 1 5 , and the range of ξ is 1, 4 4817 .
From Figure 2, it can be seen that the bigger the number of α, the smaller γ is.Also, the maximum value of γ is 9 Complexity 0.6757 when α = 0, and the minimum value of γ is 0.6734 when α = 1 5.The range of α is 0, 1 5 and the range of γ is 1, 4 4817 .
In fact, Figures 1 and 2 imply Figure 3.To show the relations between γ and ξ clearly, Figure 3 is also drawn.From Figure 3, it can be seen that the bigger the number of ξ, the smaller γ is.Figures 1, 2, and 3 can provide us a method for how to select the suitable SFFT H 2 /H ∞ controller.If we are mainly concerned with the H 2 performance, a smaller α can be selected.If we are mainly concerned with the H ∞ performance, a bigger α can be selected.

OBFT H
2 /H ∞ Controller Design.On the basis of the SF case, Algorithm 1 can also be used to solve the similar OP for the OBFT H 2 /H ∞ control.The relations of β and ξ (Figure 5), β and γ (Figure 6), and γ and ξ (Figure 7) are obtained.From Figure 5, it can be seen that the bigger the number of β, the bigger ξ is.Also, the minimum value of the H 2 performance index is ξ = 1 when β = 0, and the maximum value of the H 2 performance index is ξ = 4 0552 when β = 1 4. The range of α is 0, 1 4 , and the range of ξ is 1, 4 0552 .
From Figure 6, it can be seen that the bigger the number of β, the smaller γ is.Also, the maximum value of γ is 0.8656 when β = 0, and the minimum value of γ is 0 8413 when β 1 4. The range of β is 0, 1 4 , and the range of γ is 0 8413, 0 8656 .
In fact, Figures 5 and 6 also imply Figure 7.To show the relations between γ and ξ clearly, Figure 7 is also drawn.From Figure 7, it can be seen that the bigger the number of ξ, the smaller γ is.Figures 5-7 can provide us a method for how to select the suitable OBFT H 2 /H ∞ controller.If we are mainly concerned with the H 2 performance, a smaller β can be selected.If we are mainly concerned with the H ∞ performance, a bigger β can be selected.

Conclusions
In this study, state-feedback and observer-based finite-time H 2 /H ∞ control problems have been investigated.The problems have been transformed into optimization problems with the constraint of matrix inequalities, and a detailed optimization algorithm has been given to obtain the minimum values of the H 2 index and H ∞ index.An example has been given to demonstrate the results obtained.

1 d 2 = 5 Figure 4 :
Figure 4: The response for E x′ t Rx t .
(15)the upper bound of the H 2 index can be given as J * state = λ −1 d 1 e αT Proof.Letting Y = KM, (45) and (46) can lead to inequalities (14) and(15), respectively, and it is easy to check that (16) in Theorem 1 can be guaranteed by inequalities (47) and (48).The proof is completed.
From Definition 3, it can be seen that the existence of the observer-based finite-time H 2 /H ∞ controller (49) implies the existence of the SFFT H 2 /H ∞ controller.By the results in Theorem 2, we have designed a statefeedback finite-time H 2 /H ∞ controller u t = Kx t .Next, a design condition of the observer-based finite-time H 2 /H ∞ controller for the stochastic system (1) will be given.For the given positive scalars γ, d 1 , d 2 , T, R, and h, if there exist a positive matrix Q and a scalar β ≥ 0 such that 3: If α i makes the following optimization problem min solvable, then store α i into U i , ξ min into F i , and γ min into H i and let α i+1 = α i + d α , loop.
of x ′ t Rx t (30 curves) and E x ′ t Rx t of system