Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 132517, 11 pages
http://dx.doi.org/10.1155/2015/132517

Research Article

Decentralized Finite-Time Connective Control for a Class of Large-Scale Systems with Different Structural Forms

Xiaohua Li,¹ Yang Liu,¹ and Xiaoping Liu1,2

¹University of Science and Technology Liaoning, Anshan, Liaoning 114051, China
²Department of Electrical Engineering, Lakehead University, Thunder Bay, ON, Canada P7G 1A4

Received 16 February 2015; Revised 4 June 2015; Accepted 8 June 2015

Academic Editor: Rafael Morales

Copyright © 2015 Xiaohua Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Decentralized finite-time connective control problem for a class of large-scale interconnected systems is studied in this paper. The research aims at two structural forms, namely, the interconnected structure and the one with expanding construction. A new method is proposed to design a decentralized state feedback control law for a large-scale interconnected system so that the closed-loop system is finite-time connectively bounded. The sufficient conditions for the existence of such a decentralized control law are deduced by using LMI method. Another method is presented for a large-scale interconnected system with expanding construction which can be used without changing the decentralized state feedback control law of the original system to design a controller for the newly added subsystem so that both the new subsystem and the resulting expanded system are finite-time connectively bounded. The feasibility and effectiveness of the proposed method are verified by some simulation results.

1. Introduction

In many practical applications, the operating time of a given dynamic system is finite. It is natural to study finite-time stability rather than the classical Lyapunov asymptotic stability. The concept of finite-time stability was first presented in [1], which was extended to finite-time boundedness in [2]. Nowadays, finite-time control problem has been paid close attention by many scholars and a large number of results have been reported in the literature [3–6].

Due to the development of control theory, finite-time control problem has been well investigated in [7–12]. Therein, the finite-time control problems of a variety of systems have been studied, such as linear continuous systems, linear stochastic systems, uncertain discrete-time singular systems, switched systems with time-varying delay, linear time-invariant and time-varying systems, and a class of nonlinear time-delay Hamiltonian systems. But all the above results are based on the single system. In [13], decentralized finite-time control method is applied to interconnected linear systems. However, so far, there is no research report on decentralized finite-time connective control problem of large-scale interconnected systems and the one with expanding construction.

In this paper, connective boundedness and finite-time performance index are applied to a class of large-scale interconnected systems and the concepts of finite-time connective stability, finite-time connective boundedness, and finite-time connective boundedness are given. A decentralized state feedback control law is designed for a large-scale interconnected system so that the corresponding closed-loop system is finite-time connectively bounded. Furthermore, without changing the decentralized state feedback control law of the original system, a control law is constructed for the newly added subsystem so that both the new subsystem and the resulting expanded system are finite-time connectively bounded.

The paper is organized as follows. In Section 2, some definitions and preliminary results are provided and the mathematical models of large-scale interconnected systems and the one with expanding construction are precisely stated. Sections 3 and 4 present the sufficient conditions for the existence of decentralized state feedback control laws for finite-time connective boundedness of large-scale interconnected systems and the expanded systems. To illustrate the feasibility of the proposed method, one example is provided in Section 5.

2. Mathematical Description and Preliminary Results

Consider a large-scale interconnected system and the one with expanding construction as in [14]. The basic structure of these systems is shown in Figure 1.

Figure 1: The basic structure of an expanded system.

In Figure 1, is the original system composed of subsystems, which are described by

where , , , , and are the state, control input, output, and external disturbance vectors of the th subsystem, respectively; represents the impact on the th subsystem from the other subsystems; denotes the impact on the other subsystems by the th subsystem. , , , , , and are the constant matrices with appropriate dimensions; represents the interconnection term from the th subsystem to the th subsystem. It is clear that represents the fact that there is an interconnection and means that there is no interconnection. For simplification, system (1) can be rewritten as

where

In Figure 1, is the th subsystem newly added to the original system, which is modeled by

where , which represents the impact on the th subsystem by the original subsystems. After the th subsystem is added, the impact on the original subsystems by the th subsystem is . and . The large-scale system with subsystems becomes

where

Equation (5) can be written as the following compact form:

where

It is assumed that satisfies

The following definitions for the large-scale systems with expanding construction are given.

Definition 1 (finite-time connective stability). The large-scale system described by (7) with is said to be finite-time connectively stable with respect to with and if ; then

for any , , , and .

Definition 2 (finite-time connective boundedness). The large-scale system described by (7) with is said to be finite-time connectively bounded with respect to with and if ; then

for any , , , and any nonzero satisfying (9).

Definition 3 (finite-time connective boundedness). The large-scale system described by (7) with is said to be finite-time connectively bounded if

(1)equation (7) with is finite-time connectively bounded, and(2)with zero initial condition

for any , , , and any nonzero satisfying (9), where is a positive constant and .

The decentralized state feedback control law of the form

with is introduced to control system (7). Then the closed-loop system becomes

where

Definition 4. The decentralized finite-time connective control problem is said to be solvable if there exists a decentralized state feedback control law (13) so that the closed-loop system (14) is finite-time connectively bounded.

In order to obtain the main results in this paper, the following assumption and lemmas are proposed.

Assumption 5. The interconnection item is uncertain and satisfies the following inequality:

where is bounding parameter and is the interconnected constraint matrix.

Lemma 6. System (7) with is finite-time connectively bounded if there exist two scalars, and , and a symmetric positive definite matrix such that

with and and being the minimum and maximum eigenvalues of , respectively.

Proof. Let ; then

From (17) and (19), we have

Multiplying (20) by , we can obtain

which is equivalent to

Integrating (22) from to gives

Noting , then

Due to , it follows from (24) that

Now, we have

Putting (25) and (26) together, we have

Condition (18) implies that, for all , . Because (16) holds for any , then system (7) with is connectively bounded. The proof is completed.

Lemma 7. System (14) is finite-time connectively bounded if there exist two scalars, and , and a symmetric positive definite matrix such that

with .

Proof. Note that

Therefore, condition (28) implies that

From Lemma 6, conditions (29) and (31) guarantee that system (14) is finite-time connectively bounded. Now, we need to proof that (12) holds.
Let ; then

It follows from (28) and (32) that

which implies that

that is,

Taking into consideration, integrating (35) from to produces

which is equivalent to

Due to for and for , it follows that

Equation (12) can be obtained from (37)-(38). Because (16) holds for any , then system (14) with is connectively bounded. The proof is completed.

3. Decentralized Finite-Time Connective Control

The main result is given in Theorem 8 which is based on Lemmas 6 and 7 in Section 2.

Theorem 8. The decentralized finite-time connective control problem for (14) including N subsystems is solvable with feedback control laws if there exist two scalars, and , matrix , and the interconnected constraint matrix, , as well as symmetric positive definite matrices and such that

where

with and being the minimum and maximum eigenvalues of , respectively. And the control law can be determined by

Proof. According to Lemmas 6 and 7, let . Then (40) is equivalent to the following inequality:

with .
Using Schur complement lemma, then (43) can be changed to

with .
Set and . Then, pre- and postmultiplying (44) by yields

with due to .
Using Schur complement lemma again, it can be proved that (45) is equivalent to (28), which completes the proof. When varies from 1 to 0 or from 0 to 1, the change of interconnection is still within limited range. So the control law can make the system stable and robustly connectively stable.

It is obvious that (41) is not an LMI. However, it is guaranteed by imposing the conditions

for two positive numbers and . Using Schur complements, inequality (47) can be converted to the following LMI:

Therefore, (41) holds if LMIs (46) and (48) are true. From a computational point of view, it is important to notice that, for a given , the feasibility of the conditions stated in Theorem 8 can be turned into the following feasibility problem of LMIs.

LMI Feasibility Problem 1 (from Theorem 8). For system (14), given , and , let ; find a symmetric positive definite matrix and two positive scalars, and , satisfying LMIs (39), (40), (46), and (48). If the problem is feasible, the decentralized finite-time connective control problem can be solved by .

4. Decentralized Finite-Time Connective Control for Large-Scale Systems with Expanding Construction

Section 3 has introduced a method for constructing a decentralized state feedback control law to solve the finite-time connective control problem for large-scale systems. This section will propose a control design method for the finite-time connective control problem of large-scale systems with expanding construction. The technical difficulty of dealing with this kind of systems is that a control law is constructed for the newly added subsystem so that both the new subsystem and the resulting expanded system are finite-time connectively bounded without changing the decentralized state feedback control law of the original system. To this end, the following assumption is made.

Assumption 9. The decentralized finite-time connective control problem for the original system (1) can be solved by the control law .

The main result is given in Theorem 10.

Theorem 10. The decentralized finite-time connective control problem for the th subsystem in (14) is solvable with feedback control law , if there are positive constants, and , as well as symmetrical positive definite matrices, and , and matrix , so that the following inequalities are feasible:

where , , and are the elements of the interconnected constraint matrix and

with and being the minimum and maximum eigenvalues of , respectively. And the control law can be determined by

Proof. The result of Theorem 10 is based on Lemmas 6 and 7 and Theorem 8. Let . Then, (50) is equivalent to

where

Using Schur complement lemma results in

with .
Set and . Then, the following inequality follows from pre- and postmultiplying (55) by :

where

Set . Due to

then (56) is equivalent to

Applying Schur complement lemma to (59) shows that (28) is true. It follows from Lemma 7 that Theorem 10 holds. Similar to Theorem 8, the control law can make the system stable and robustly connectively stable and the following feasibility problem of LMIs is given.

LMIs Feasibility Problem 2 (from Theorem 10). For system (14), given , and , let ; find a symmetric positive definite matrix, , matrix , and two positive scalars, and , satisfying LMIs (46), (48), (49), and (50). If the problem is feasible, the decentralized finite-time connective control problem for the expanded subsystem can be solved by .

5. Simulation Example

Consider a class of multiarea interconnected power systems, in which each area includes a hydroelectric power unit and a thermal power unit. The mathematical model, state variables, and output variables can be found in [15, 16]. This is a deviation model of automatic generation control (AGC). The th area-subsystem model can be described as

where , and are the state, control input, output, and uncertain disturbance input of subsystems, respectively. And the matrices in (60) are, respectively,

The specific parameters of (61) can be found in [15, 16]. And other parameters are given as follows:

Suppose that the original system consists of two subsystems. The initial values are . The connective relations are . According to Theorem 8, the corresponding control gain matrices , and are obtained as

and . is plotted as shown in Figure 2. It can be easily seen from Figure 2 that and , which means that the decentralized finite-time connective control problem is solvable for the original system. In order to check connective stability of the system, we cut off the interconnection between two subsystems. The result is shown in Figure 3, which shows that the closed-loop system is finite-time connectively bounded.

Figure 2: The simulation curve of the original system.

Figure 3: The simulation curve of the original system when varies from 1 to 0.

Now the third subsystem is added to the original system with

The connective relations are given by

According to Theorem 10, the control gain matrix is obtained as

and . is shown in Figure 4. It is obvious that and , which implies that the decentralized finite-time connective control problem is solvable for the expanded system. In order to check the connective stability of the expanded system, the connection varies from 1 to 0 and the result is shown in Figure 5.

Figure 4: The simulation curve of the expanded system.

Figure 5: The simulation curve of the expanded system when varies from 1 to 0.

From the figures, it can be known that both the original system and the resulting expanded system are finite-time connectively bounded.

6. Conclusion

Decentralized finite-time connective control problem for a class of large-scale systems is studied in this paper. The large-scale systems include the original structural interconnected system and the systems with expanding construction. Based on state feedback, the sufficient conditions of decentralized finite-time connective boundedness for large-scale systems are deduced by using LMI method. The design methods for the decentralized finite-time connective control problem are given. The simulation examples verify the feasibility and effectiveness of the proposed method. In particular, the paper proposes a method for the structure expansion of large-scale systems. A controller can be designed for the newly added subsystem on the basis of keeping the decentralized state feedback control laws of the original construction systems unchanged so that both the new subsystem and the resulting expanded system are finite-time connectively bounded. Therefore, this paper can be used as the theoretical basis for expansion of large-scale interconnected system online.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (no. 61273011).

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