Abstract
In this paper, the design of finite-time controller for linear Itô stochastic Poisson systems is considered. First, the definition of finite-time control is proposed, which considers the transient performance, index, and index simultaneously in a predetermined finite-time interval. Then, the state feedback and observer-based finite-time controllers are presented and some new sufficient conditions are obtained. Moreover, an algorithm is given to optimize and index, simultaneously. Finally, a simulation example indicates the effectiveness of the results.
1. 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. There are also many other applications of stochastic systems (see, e.g., [5–7]). 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 control problem for switched stochastic systems with Wiener noise. Yan et al. [10] were concerned with finite-time 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., [11–14]).
On the other hand, optimization control is one of the most important problems in the controlled system. The optimal control system has good system performance, while the control theory can deal with the system robustness problem well. In view of this, Bernstein and Haddad [15] proposed the mixed control problem, which can solve both the problems of system performance and robustness. Since then, the control has been developed and used extensively (see, e.g., [16–19]). Besides, in some engineering research, such as communication system [20–23], robotic operating system [24], and industrial production system [25], more attention should be paid to the system transient performance. In order to describe system transient performance clearly, the concepts of finite-time stability (FTS) and finite-time boundedness (FTB) are proposed, which reflect the specific system behavior in a relatively short time interval. Nowadays, the problems of FTS and FTB have been deeply investigated (see, e.g., [26–36]). In consideration of the merits of FTB and control, the finite-time control for stochastic systems with Wiener noise is first presented in [37], which satisfies both FTB and performance index. However, in many practical systems, it is not only disturbed 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 control for stochastic Poisson systems with both Wiener noise and Poisson noise are considered in this paper. The 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. The former considers only Wiener noise, and the latter considers both Wiener and Poisson noises. Moreover, in the former model, the measurement output is composed of only the state, but the measurement output considered in the latter model is composed of both the state and external interference. The latter model is more general than the former model in [37], which is used to model many real systems.(ii)The two theorems (Theorems 2 and 4) are obtained to guarantee the existence of state feedback finite-time (SFFT) and observer-based finite-time (OBFT) controllers, respectively. The two theorems (Theorems 2 and 4) contain the parameters both and Poisson jump intensity , which are complex than the corresponding conditions in [37]. By adjusting the two parameters, the most satisfying finite-time controllers will be designed.(iii)A new optimization algorithm constrained by matrix inequality is proposed to demonstrate the relationships among , , and optimal 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].
2. Preliminaries
Consider a continuous-time stochastic Poisson systemwhere , , , , , , , , , , , , and are known constant matrices. , , , and are the state vector, measurement output, control output, and control input, respectively. is the initial condition of the system. presents one-dimensional standard Wiener process and is the marked Poisson process with Poisson jump intensity . 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 and and a matrix , the above stochastic system (1) with is mean-square FTB w.r.t. , if
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 and . Consider the following systemits stochastic differential of is given bywhere
3. Design of SFFT Controller
In this section, a SFFT controller for system (1) is designed. Consider a linear SF controllerwhere is the required SF gain matrix.
Substituting (7) into (1), the following closed-loop system is obtained:where , , , and .
Next, we choose the following cost function:where and are known weighting scalars or positive matrices.
Similarly, substituting the SF controller (7) into (9), the following formula is obtained:
Given and assuming zero initial condition, the control output and the disturbance input satisfy the following equation:
Based on the above preparations, the definition of the SFFT controller is introduced.
Definition 2. Given positive scalars , , , and and a matrix . If a positive scalar exists, a SF controller (7) can be designed to make the following conditions hold:(i)The closed-loop system (8) is mean-square FTB w.r.t. (ii)The cost function (10) meets under condition(iii)Assuming that the initial state is zero and the nonzero disturbance input and the control output satisfy inequality (11); then (7) is the SFFT controller for system (1)
Remark 2. Definition 3 implies that a SFFT 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 controller.
Theorem 1. Given positive scalars , , , and and a matrix , if there exist a nonnegative scalar and two matrices and such thathold, where and , then is said to be a SFFT controller and we can get the upper bound of index, that is, .
Proof. Here are three steps to prove Theorem 1. Step 1: prove that system (3) is mean-square FTB. Obviously, Therefore, condition (12) means where . Let , and applying Lemma 1 for , the of system (8) is given by where , , and . Pre- and postmultiplying (16) by , we can get the following inequality: where and . By utilizing Schur complement, (18) is equivalent to Taking conditions (17) and (19) into consideration, it follows Integrating from 0 to on both sides of (20), then taking mathematical expectation, one has Utilizing Gronwall inequality in [26], it follows 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 for all . So, system (8) is mean-square FTB w.r.t. . Step 2: prove that the cost function (10) satisfies under condition. When , we get that the of system (8) is given by By Schur complement, the equivalent condition of (13) is given by Pre- and postmultiplying (28) by , it yields According to (27) and (29), we get Integrating from 0 to 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 , and then using Schur complement, we have where . Combining (17), (36), we get Pre- and postmultiplying (37) by , one has By applying Lemma 1, we obtain
According to (38) and (39), it yields
Because is between 0 and 1, for (40), we have
Integrating from 0 to on both sides of (41), then taking mathematical expectation, the following inequality can be obtained under zero initial condition:
We know that , so it yields
This 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.
Theorem 2. Given positive scalars , , , and and a matrix , if there exist two scalars and and two matrices and such thathold, where , , , and , then is said to be a SFFT controller and we can get the upper bound of index, that is, .
Proof. Let , inequalities (12) and (13) can be obtained from (44) and (45), respectively, and (14) in Theorem 1 can be obtained from (46) and (47) easily. This ends the proof.
Remark 3. In Theorem 2, when is fixed, (44)–(47) can be treated as LMIs which are easy to solve.
4. Design of OBFT Controller
In some practical cases, not all states can be measured directly. Therefore, the design of OBFT controller is necessary. Typically, an OB dynamic controller is given bywhere is the estimation of and 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 functionwhere
Assuming that the initial state is zero, the control output and the arbitrary nonzero disturbance input satisfy the following equation:
Then, we give the definition of OBFT control.
Definition 3. Given positive scalars , , , and and a matrix . If a positive scalar exists, an OBFT controller (48) can be designed to make the following conditions hold:(i)System (49) is mean-square FTB w.r.t. , that is, , where , and (ii)The cost function (50) meets under condition(iii)Assuming that the initial state is zero, the nonzero disturbance input and the control output satisfy inequality (53); then (48) is an OBFT controller for system (1)Next, the following theorem is given for obtaining the OBFT controller for system (1).
Theorem 3. Given positive scalars , , , and and a matrix , if there exist a nonnegative scalar and a positive matrix such thathold, where , then (48) is said to be an OBFT controller and we can get the upper bound of index, that is, .
Proof. Here are three steps to prove the theorem. Step 1: prove that system (49) is mean-square FTB. Let where . Applying generalized Itô formula for , the of system (49) is given by where , , and . Note that Therefore, inequality (54) means where . By utilizing Schur complement, (59) can be converted into Combining (57) and (60), we get Integrating from 0 to on both sides of (61), then taking mathematical expectation, the following inequality is obtained: According to Gronwall inequality, it yields According to known conditions, it yields From (63) to –(66), we obtain According to (56) and (67), we get for all . So, system (49) is FTB w.r.t. . Step 2: prove that the cost function (50) satisfies under condition. When , we get that the of system (49) is given by According to (55), we have Integrating from 0 to on both sides of (69), then taking mathematical expectation, it yields From (70), we have Using Gronwall inequality for (71), one has From (72) and (73), we have 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 getRepeating the proof process of Step 3 in Theorem 1, it yieldsThis completes the proof.
Because the nonlinear problem of inequalities (54)–(56) in Theorem 3 is difficult to solve, we transform the inequalities (54)–(56) into LMIs.
Theorem 4. Given positive scalars , , , and , if there exist two positive scalars and and three matrices , , and such thathold, where , then (48) is said to be an OBFT controller and we can get the upper bound of index, that is, . Furthermore, the estimator gain matrix is obtained.
Proof. Let and , by substituting (51) and (52) into (54) and (55), (78) and (79) can be easily derived, respectively. From (80) and (81), we can deduce that (49) holds. This ends the proof.
5. Algorithm
In this section, we propose an algorithm to optimize index and index.
Analysis: in Theorem 2, let , the following inequality is derived:where , , , and .
The main purpose of the algorithm is to check whether inequalities (44)–(47) in Theorem 2 have feasible solutions by changing the value of . If there exist feasible solutions, then and are optimized to get the minimum values. The detailed algorithm will be given as follows (Algorithm 1).
| ||||||||||
6. Examples
In this section, system (8) can be used to simulate a clothing hanging device and the parameters are as follows:and , , , , , , , , , and .
6.1. Design of SFFT Controller
By using the above algorithm in Section 5, the relationships of and (Figure 1), and (Figure 2), and and (Figure 3) are derived, respectively. It can be seen from Figure 1 that the value of increases with the increase of . Besides, it is obvious that when and when , that is, the minimum and maximum values of performance index are 1 and 3.0344, respectively. Also, the range of is [0, 1.11].
As you can see from Figure 2, the value of decreases first and then increases with the increase of . When , can take the minimum value of 0.8688; at this point, we can get the optimal value of performance index. When , can take the minimum value of 0.8822. Also, can be taken within [0, 1.11].
In fact, Figure 3 reflects the relation between and . As shown in Figure 3, with the increase of , the value of decreases first, and at the point of , the value of begins to increase. From Figures 1 to 3, we can see how to choose the right state feedback finite-time controller. If the cost problem is mainly considered, a smaller can be selected. If the ability to suppress interference is mainly considered, we need to refer to Figure 2 to select the appropriate .
Next, substituting into Theorem 2, we get
Then, we get the controller gain matrix as follows:
Because the state in this example is two-dimensional, we make . Figure 4 describes the trajectories of , , and with stochastic fluctuation driven by both Wiener and Poisson noises in Figures 5 and 6 versus the dimensionless time . From Figure 4, we can see that the trajectory of does not exceed in the time interval . Obviously, when the time interval is , the trajectory does not exceed the given range, so we conclude that system (8) is mean-square FTB w.r.t. (1, 4, 1, I, 0.4). Among them, we assume that .
6.2. Design of OBFT Controller
As in the case of state feedback, similar results can be obtained in the case of observer-based finite-time control. The relationships of and (Figure 7), and (Figure 8), and and (Figure 9) are derived, respectively. It can be seen from Figure 7 that the value of increases with the increase of . Besides, it is obvious that when and when , that is, the minimum and maximum values of performance index are 1 and 2.5857, respectively. Also, the range of is [0, 0.95].
As you can see from Figure 8, the value of decreases first and then increases with the increase of . When , can take the minimum value of 1.1456, and at this point, we can get the optimal value of performance index. Besides, the maximum value of performance index is 1.1650 when . Also, the range of is [0, 0.95].
In fact, Figure 9 reflects the relation between and . As shown in Figure 9, with the increase of , the value of decreases first, and at the point of , the value of begins to increase. From Figures 7 to 9, we can see how to choose the right OBFT controller. If the cost problem is mainly considered, select the smaller 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 .
Substituting into Theorem 4, we have
Then, we obtain the following observer gain matrix:
Because the state in this example is two-dimensional, we make . Figure 10 describes the trajectories of , , and with stochastic fluctuation driven by both Wiener and Poisson noises in Figures 5 and 6 versus the dimensionless time . From Figure 10, it is obvious that the trajectory of does not exceed in the time interval . Obviously, when the time interval is , 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 .
7. Conclusions
In this paper, state feedback and observer-based finite-time 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 Theorems has been provided to derive the optimal index and index under the condition of the finite-time boundedness.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant nos. 61877062 and 61977043), China Postdoctoral Science Foundation (Grant no. 2017M610425), and Open Foundation of Key Laboratory of Pulp and Paper Science and Technology of Ministry of Education of China (Grant no. KF201419).