Abstract
This paper deals with the problem of the control based on finite-time boundedness for linear stochastic systems. The motivation for investigating this problem comes from one observation that the control does not involve systems’ transient performance. To express this problem clearly, a concept called finite-time control is introduced. Moreover, state feedback and observer-based finite-time controllers are designed, which guarantee finite-time boundedness, performance index, and performance index of the closed-loop systems. Furthermore, an optimization algorithm on the finite-time control is presented to obtain the minimum values of the index and index. Finally, we use an example to show the validity of the obtained results.
1. 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 control method. Its concrete implication is to seek a controller which not only minimizes the cost function but also restrains the effect of external disturbance. The control method has been applied to many fields, such as communication systems [9] and synthetic gene network design [10]. Meanwhile, 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 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 [17–19]. 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. [21–30] and the references therein. Taking into account the advantages of FT stability and control, the finite-time control problem is proposed in this paper, which not only guarantees FTB but also satisfies the performance indices. Up to date, there is almost no literature to consider the FT control of stochastic systems.
This paper will study the problems of FT 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 controllers are designed. The main contributions of this paper are as follows: (i) The definition of the FT control for linear stochastic systems with -dependent noise (which is called state-, control-, and disturbance-dependent noise) is first given, which simultaneously presents FT boundedness, performance index, and 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 performance index and 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 controller is designed. In Section 4, the observer-based FT controller is designed. In Section 5, an optimization algorithm is provided for obtaining the minimum values of the performance index and performance index, simultaneously. In Section 6, a numerical example is discussed, and some remarks are concluded in Section 7.
Notation. : a complete probability space with a filtration . : transpose of a matrix . : is a positive definite symmetric matrix. : identity matrix. : the space of nonanticipative stochastic process with respect to filtration satisfying . and : the maximum and minimum eigenvalue of matrix . is the mathematical expectation of the stochastic process. The asterisk “” in a matrix stands for the symmetry term. represents a diagonal matrix. The “wrt” denotes “with respect to.”
2. Preliminaries
We consider the linear stochastic system as follows: where , , , , , , , , , and are constant matrices with appropriate dimensions. , , , , and are called state, control input, external disturbance, measurement output, and controlled output, respectively. is the initial state. is a one-dimensional Wiener process, which is defined as . 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, 31–33].
Definition 1. System (1) with is FT stochastically bounded wrt , if where , , and .
For the subsequent analysis, the following lemmas are useful for the FT controller design.
Lemma 1 [27]. If , , and satisfy then we obtain
Lemma 2 [34]. Let be a scalar function and . For the following stochastic system the Itô formula of is given as follows: where
3. State Feedback Finite-Time Controller Design
This section gives a problem formulation of state-feedback finite-time (SFFT) 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 is the feedback gain matrix.
Substituting the controller (9) into system (1) leads to the closed-loop system where , , and .
Associated with system (1), the following cost function is provided: where and are the given positive scalars or given weighting matrices.
Also, by substituting (9) into (11), the corresponding performance function (11) becomes
For a given , under the condition of zero initial value, the discretional nonzero disturbance and the control output satisfy the following form:
Based on the above preparations, we are in a position to give the definition of the state-feedback finite-time control.
Definition 2. For some given positive scalars , , , , , and , considering system (1), cost function (11), and the inequality (13), if there exist a positive scalar and a SF controller (9) such that (i)system (10) is FT stochastically bounded wrt (ii) cost function (12) satisfies under the condition of (iii)for any nonzero disturbance , under the zero initial condition, the inequality (13) is satisfiedthen (9) is a state-feedback finite-time 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 performance index or performance index. In addition, these three aspects involved by the FT 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 controller will be given by the following theorem.
Theorem 1. For the given positive scalars , , , , , and , if there exist a matrix , a matrix , and a scalar such that hold, where , , and , then is a SFFT controller of system (1) and the upper bound of the 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 Let , the infinitesimal operator of system (10) is defined as follows: where . Pre- and postmultiplying the inequality (18) by , 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 and then taking the expectation, it follows that By Lemma 1, we get 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 for all . So, the closed-loop system (10) is FT stochastically bounded wrt .Step 2.Prove that the condition (ii) is satisfied. Under the condition of , the infinitesimal operator of system (10) is as follows: By Schur’s complement, it can be seen that (15) is equivalent to Premultiplying and postmultiplying (30) by , we obtain According to (29) and (31), we get Integrating (32) from 0 to , , 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 , and according to the Schur complement, we have where . According to (19) and (38), we have Premultiplying and postmultiplying (39) by , 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 , , taking the expectation, and after some calculations, we have Because , 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 , , , , , and , if there exist a matrix , a matrix , a scalar , and a scalar such that hold, where , then is a SFFT controller and the upper bound of the index can be given as
Proof. Letting , (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.
Remark 2. It is noted that the inequalities in Theorem 2 are not LMIs, but if is fixed, (45), (46), (47), and (48) can be regarded as LMIs.
4. Observer-Based Finite-Time Controller Design
In the above Section 3, the SFFT 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) controller design is necessary. Typically, an observer-based (OB) controller is provided as follows: where is the estimation of and is the estimator gain.
Substituting (49) into system (1), we will obtain the closed-loop system and the corresponding closed-loop cost function where
For a given , under the condition of zero initial value, the discretional nonzero disturbance and the control output satisfy the following form:
Then, the definition of the observer-based finite-time control is given as follows.
Definition 3. For the given positive scalars , , , , , and , considering system (1), cost function (11), and the inequality (13), if there exist an OB controller (49) and a positive scalar such that (i)system (50) is FT stochastically bounded wrt (ii) cost function (51) satisfies under the condition of (iii)for any nonzero disturbance, under the zero initial condition, the inequality (54) is satisfiedthen (49) is said to be an observer-based finite-time controller for the stochastic system (1).
Remark 3. From Definition 3, it can be seen that the existence of the observer-based finite-time controller (49) implies the existence of the SFFT controller.
By the results in Theorem 2, we have designed a state-feedback finite-time controller . Next, a design condition of the observer-based finite-time controller for the stochastic system (1) will be given.
Theorem 3. For the given positive scalars , , , , , and , if there exist a positive matrix and a scalar such that hold, where and , then (49) is an OBFT controller and the upper bound of the index can be given as .
Proof. The proof will be divided into three steps. Step 1.Prove that the condition (i) is satisfied. Let with be solutions to (55), (56), and (57). The infinitesimal operator of system (50) is defined as follows: Note that 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 and then taking the expectation, it follows that According to Lemma 1, it is obtained that According to known conditions, it yields that From (64), (65), (66), and (67), it is easy to obtain that According to condition (57), it follows that (68) leads to for all . So, the closed-loop system (50) is FT stochastically bounded wrt .Step 2.Prove that the condition (ii) is satisfied. Under the condition of , the infinitesimal operator of system (50) is defined as follows: According to (56), we get Integrating (70) from 0 to 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 This completes the proof.
We find that inequalities (55), (56), and (57) in Theorem 3 are not easy to solve, so the following theorem will transform (55), (56), and (57) into LMIs.
Theorem 4. For the given positive scalars , , , , , and , if there exist a scalar , two matrices and , a matrix , and a scalar such that hold, where , , , , , , , , and , then (49) is an OBFT controller and the upper bound of the index can be given as . In this case, an appropriate estimator gain matrix is given by .
Proof. Setting , substituting (52) and (53) into (55), and letting , (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.
5. Algorithm
In this section, an optimization algorithm on the FT control is presented. The following algorithm is given for Theorem 2. Similarly, the algorithm can be utilized in Theorem 4.
Analysis. By Theorem 2, let , and then minimize , subject to (45), (46), (47), and (48), and where , , , and .
By analyzing (45), (46), (47), and (48) in Theorem 2, we find the following: if (45), (46), (47), and (48) have no feasible solutions at the case of , then (45), (46), (47), and (48) will have no feasible solutions for all . Therefore, we seek that makes (45), (46), (47), and (48) have feasible solutions from by linear searching. The detailed algorithm will be given as follows. One can use this algorithm to obtain the minimum value of and .
| ||||||||||||||||
By running Algorithm 1, we get the following results: (1)If , , and , then it shows that the above optimization problem (OP) is not solvable.(2)If , , and , then it shows that the above optimization problem is solvable. Moreover, we can respectively plot the curves of , , and . The three curves can help us choose a suitable finite-time controller.(3)This algorithm can find the most suitable to decrease the conservativeness of the conditions of Theorem 2.
6. Examples
This section provides an example to show the effectiveness of the obtained results. System (10) with parameters is as follows: and , ,