Finite-Time H 2 / H ‘ Control Design for Stochastic Poisson Systems with Applications to Clothing Hanging Device

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Introduction
It is known to all that stochastic systems have been studied extensively and applied to biological network [1], power systems [2], financial systems [3,4], and other fields. ere are also many other applications of stochastic systems (see, e.g., [5][6][7]). In the past few decades, stochastic systems driven by Wiener noise have been widely investigated. For example, Shaikin [8] solved the optimization problem for multiplicative stochastic systems with several external disturbances and vector Wiener processes. Xiang et al. [9] introduced the finite-time properties and state feedback H ∞ control problem for switched stochastic systems with Wiener noise. Yan et al. [10] were concerned with finite-time H 2 control of the Markovian stochastic systems with Wiener noise. However, in the real world, an actual physical system is inevitably affected by Wiener noise and Poisson jump noise. At present, some achievements have been made in the research of stochastic Poisson systems (see, e.g., [11][12][13][14]).
On the other hand, H 2 /H ∞ optimization control is one of the most important problems in the controlled system. by Wiener noise, but also by Poisson noise. So far, there are few literature studies to investigate this problem of stochastic Poisson systems affected by both Wiener and Poisson noises.
Motivated by aforementioned discussions, the problems of finite-time H 2 /H ∞ control for stochastic Poisson systems with both Wiener noise and Poisson noise are considered in this paper. e main work of this paper consists of the following three aspects: (i) Unlike the model considered in [37], this paper studies the model of stochastic Poisson systems with Wiener and Poisson noises. e former considers only Wiener noise, and the latter considers both Wiener and Poisson noises. Moreover, in the former model, the measurement output y(t) is composed of only the state, but the measurement output considered in the latter model is composed of both the state and external interference. e latter model is more general than the former model in [37], which is used to model many real systems.
(ii) e two theorems ( eorems 2 and 4) are obtained to guarantee the existence of state feedback finitetime (SFFT) and observer-based finite-time (OBFT) H 2 /H ∞ controllers, respectively. e two theorems ( eorems 2 and 4) contain the parameters both α and Poisson jump intensity λ, which are complex than the corresponding conditions in [37]. By adjusting the two parameters, the most satisfying finite-time H 2 /H ∞ controllers will be designed. (iii) A new optimization algorithm constrained by matrix inequality is proposed to demonstrate the relationships among α, λ, and optimal H 2 /H ∞ index, which is more complex than that in [37].
Notations: the notations presented in this work are standard. For specific contents, one can refer to [37].

Preliminaries
Consider a continuous-time stochastic Poisson system where A 11 , A 12 , A 13 , B 11 , B 12 , B 13 , C 11 , C 12 , D 11 , D 12 , F 1 , F 2 , and F 3 are known constant matrices. x(t) ∈ R l , y(t) ∈ R q , z(t) ∈ R s , and v(t) ∈ R n are the state vector, measurement output, control output, and control input, respectively. x 0 is the initial condition of the system. W(t) presents one-dimensional standard Wiener process and N(t) is the marked Poisson process with Poisson jump intensity λ. r(t) ∈ R p is the disturbance input which satisfies the following equation: Next, the definition of mean-square FTB of system (1) is introduced. Definition 1. Given some scalars b 2 > b 1 > 0 and T > 0 and a matrix R > 0, the above stochastic system (1) Remark 1. From Definition 1, we can know that the concept of FTB describes the specific behavior of the stochastic system (1) in a prescribed time interval.
Lemma 1 (see [38]). Let V(t, x) ∈ C 2 (R 1 , R n ) and V(t, x) > 0. Consider the following system its stochastic differential of V(t, x) is given by where 2 Complexity

Design of SFFT H 2 /H ' Controller
In this section, a SFFT H 2 /H ∞ controller for system (1) is designed. Consider a linear SF controller where K is the required SF gain matrix. Substituting (7) into (1), the following closed-loop system is obtained: where Next, we choose the following H 2 cost function: where G 1 > 0 and G 2 > 0 are known weighting scalars or positive matrices. Similarly, substituting the SF controller (7) into (9), the following formula is obtained: Given c > 0 and assuming zero initial condition, the control output z(t) and the disturbance input r(t) satisfy the following equation: Based on the above preparations, the definition of the SFFT H 2 /H ∞ controller is introduced.
Definition 2. Given positive scalars b 1 , b 2 , T, and f and a matrix R > 0. If a positive scalar J * 1 exists, a SF controller (7) can be designed to make the following conditions hold: Assuming that the initial state is zero and the nonzero disturbance input and the control output satisfy inequality (11); then (7) is the SFFT H 2 /H ∞ controller for system (1) Remark 2. Definition 3 implies that a SFFT H 2 /H ∞ controller not only makes the closed-loop system FTB, but also gets minimum performance cost and better interference suppression capability. In actual systems, these three aspects really need to be considered. For example, in industrial steel rolling heating furnace, excessive instantaneous furnace temperature cannot be permitted. Moreover, it is hoped that the fuel consumption is less and the anti-interference ability is stronger in the rolling furnace. Next, the following theorem is given for obtaining the SFFT H 2 /H ∞ controller.
Proof. Here are three steps to prove eorem 1.
Step 1: prove that system (3) is mean-square FTB. Obviously, erefore, condition (12) means where , and applying Lemma 1 (8) is given by where Pre-and postmultiplying (16) we can get the following inequality: where By utilizing Schur complement, (18) is equivalent to Taking conditions (17) and (19) into consideration, it follows Integrating from 0 to t on both sides of (20), then taking mathematical expectation, one has Utilizing Gronwall inequality in [26], it follows Er ′ (s)r(s)ds.

(22)
On the basis of above conditions, we have From (22) to (25), the following inequality is obtained: According to condition (14), we get that (26) leads to Step 2: prove that the H 2 cost function (10) satisfies When r(t) � 0, we get that the L 2 V(x(t)) of system (8) is given by By Schur complement, the equivalent condition of (13) is given by Pre-and postmultiplying (28) by N − 1 , it yields According to (27) and (29), we get Integrating from 0 to t on both sides of (30), then taking mathematical expectation, the following inequality is obtained: From (31), we get From (32), by Gronwall inequality, one has Combining (33) and (34), it is obtained that Step 3: prove that the nonzero disturbance and the control output satisfy inequality (11).
Pre-and postmultiplying (12) respectively by diag N − 1 , I, N − 1 , N − 1 , and then using Schur complement, we have where Combining (17), (36), we get Pre-and postmultiplying (37) by e − αt , one has By applying Lemma 1, we obtain (39) □ According to (38) and (39), it yields Because e − αt is between 0 and 1, for (40), we have Integrating from 0 to t on both sides of (41), then taking mathematical expectation, the following inequality can be obtained under zero initial condition: We know that e − αt EV(x(t)) > 0, so it yields is completes the proof. It is obvious that conditions (12)- (14) are not linear matrix inequalities. In order to simplify the solving process, the following theorem is given.

Design of OBFT H 2 /H ' Controller
In some practical cases, not all states can be measured directly. erefore, the design of OBFT H 2 /H ∞ controller is necessary. Typically, an OB dynamic controller is given by where x(t) ∈ R n is the estimation of x(t) and L is the desired estimator gain. Substituting the OB controller (48) into system (1), we will obtain the following closed-loop system: and then we get the closed-loop cost function where Assuming that the initial state is zero, the control output z(t) and the arbitrary nonzero disturbance input r(t) satisfy the following equation: en, we give the definition of OBFT H 2 /H ∞ control.
Definition 3. Given positive scalars b 1 , b 2 , T, and f and a matrix R > 0. If a positive scalar J * 2 exists, an OBFT controller (48) can be designed to make the following conditions hold: Assuming that the initial state is zero, the nonzero disturbance input and the control output satisfy inequality (53); then (48) is an OBFT H 2 /H ∞ controller for system (1) Next, the following theorem is given for obtaining the OBFT H 2 /H ∞ controller for system (1).
Proof. Here are three steps to prove the theorem.
From (72) and (73), we have (74) Step 3: prove that the nonzero disturbance and the control output satisfy the inequality (53).
By using Schur complement, we can obtain the following equivalent conditions of (54): According to (57) and (75), we get Repeating the proof process of Step 3 in eorem 1, it yields is completes the proof. Because the nonlinear problem of inequalities (54)-(56) in eorem 3 is difficult to solve, we transform the inequalities (54)-(56) into LMIs.

Algorithm
In this section, we propose an algorithm to optimize H 2 index and H ∞ index.

Examples
In this section, system (8) can be used to simulate a clothing hanging device and the parameters are as follows: and R � I, f � 0.4, and λ � 2.5.
6.1. Design of SFFT H 2 /H ∞ Controller. By using the above algorithm in Section 5, the relationships of α and ξ (Figure 1), α and c (Figure 2), and ξ and c ( Figure 3) are derived, respectively. It can be seen from Figure 1 that the value of ξ increases with the increase of α. Besides, it is obvious that 8 Complexity ξ � 1 when α � 0 and ξ � 3.0344 when α � 1.11, that is, the minimum and maximum values of H 2 performance index are 1 and 3.0344, respectively. Also, the range of α is [0, 1.11].
As you can see from Figure 2, the value of c decreases first and then increases with the increase of α. When α � 0.85, c can take the minimum value of 0.8688; at this point, we can get the optimal value of H ∞ performance index. When α � 1.11, c can take the minimum value of 0.8822. Also, α can be taken within [0, 1.11].
In fact, Figure 3 reflects the relation between ξ and c. As shown in Figure 3, with the increase of ξ, the value of c decreases first, and at the point of ξ � 2.3397, the value of c begins to increase. From Figures 1 to 3, we can see how to choose the right state feedback finite-time H 2 /H ∞ controller. If the cost problem is mainly considered, a smaller α can be selected. If the ability to suppress interference is mainly considered, we need to refer to Figure 2 to select the appropriate α.
Next, substituting α � 0 into eorem 2, we get en, we get the controller gain matrix as follows: Because the state x(t) in this example is two-dimensional, we make Figure 4 describes the trajectories of x 1 (t), x 2 (t), and E[x ′ (t)Rx(t)] with stochastic fluctuation driven by both Wiener and Poisson noises in Figures 5 and 6 versus the dimensionless time λt. From Figure 4, we can see that the trajectory of E[x ′ (t)Rx(t)] does not exceed b 2 � 4 in the time interval λT � 2.5. Obviously, when the time interval is T � 1, the trajectory does not exceed the given range, so we conclude that system (8)  e relationships of β and ξ (Figure 7), β and c (Figure 8), and ξ and c ( Figure 9) are derived, respectively. It can be seen from Figure 7 that the value of ξ increases with the increase of β. Besides, it is obvious that ξ � 1 when β � 0 and ξ � 2.5857 when β � 0.95, that is, the minimum and maximum values of H 2 performance index are 1 and 2.5857, respectively. Also, the range of β is [0, 0.95].
As you can see from Figure 8, the value of c decreases first and then increases with the increase of β. When β � 0.72, c can take the minimum value of 1.1456, and at this point, we can get the optimal value of H ∞ performance index. Besides, the maximum value of H ∞ performance index is 1.1650 when β � 0.95. Also, the range of β is [0, 0.95].
In fact, Figure 9 reflects the relation between ξ and c. As shown in Figure 9, with the increase of ξ, the value of c decreases first, and at the point of ξ � 2.0544, the value of c begins to increase. From Figures 7 to 9, we can see how to Step 1: given b 1 , b 2 , R, T, f, and λ.
Step 2: take an appropriate step size d α for α, and then the values of α are expressed as α i .
Step 4: if α i makes the following problems min s.t. Complexity choose the right OBFT H 2 /H ∞ controller. If the cost problem is mainly considered, select the smaller β with reference to Figure 7. If the ability to suppress interference is mainly considered, we need to refer to Figure 8 to select the appropriate β.
en, we obtain the following observer gain matrix: Because the state x in this example is two-dimensional, we make x � [x 1 x 2 ] ′ . Figure 10 describes the trajectories of x 1 , x 2 , and E[x ′ (t)Rx(t)] with stochastic fluctuation driven by both Wiener and Poisson noises in Figures 5 and 6 versus the dimensionless time λt. From Figure 10, it is obvious that the trajectory of E[x ′ (t)Rx(t)] does not exceed b 2 � 4 in the time interval λT � 2.5. Obviously, when the time interval is T � 1, the trajectory does not exceed the given range, so we draw a conclusion that system (8) is mean-square FTB w.r.t. (1, 4, 1, I, 0.4). Among them, we assume that r(t) � sin t( 1 0 sin 2 tdt < f � 0.4).

Conclusions
In this paper, state feedback and observer-based finitetime H 2 /H ∞ controllers for stochastic Poisson systems have been designed, respectively. Two sufficient conditions for guaranteeing the existence of controllers have been proposed and converted to matrix inequality constrained optimization problems, and an algorithm for all eorems has been provided to derive the optimal H 2 index and H ∞ index under the condition of the finite-time boundedness.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.