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

Decentralized finite-time H ∞ 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 closedloop system is finite-time H ∞ 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 H ∞


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][4][5][6].
Due to the development of  ∞ control theory, finitetime  ∞ control problem has been well investigated in [7][8][9][10][11][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 largescale interconnected systems and the concepts of finitetime 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 finitetime  ∞ 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 largescale interconnected systems and the expanded systems.To 2 Mathematical Problems in Engineering illustrate the feasibility of the proposed method, one example is provided in Section 5.

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.
In Figure 1,  −1 is the original system composed of ( − 1) subsystems, which are described by where  = 1, 2, . . .,  − 1,   ∈    ,   ∈    ,   ∈    , and   ∈    are the state, control input, output, and external disturbance vectors of the th subsystem, respectively; V  represents the impact on the th subsystem from the other subsystems;   denotes the impact on the other subsystems by the th subsystem.  ,   ,   , Γ  ,   , and   are the constant matrices with appropriate dimensions;   represents the interconnection term from the th subsystem to the th subsystem.It is clear that   = 1 represents the fact that there is an interconnection and   = 0 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 V  () =  ,−1  −1 (), which represents the impact on the th subsystem by the original ( − 1) subsystems.After the th subsystem is added, the impact on the original ( − 1) subsystems by the th subsystem is The large-scale system with  subsystems becomes where Mathematical Problems in Engineering 3 Equation ( 5) can be written as the following compact form: ẋ () = Â x () + Bû () + Γ ω () + ĥ (, x () , ) , where ĥ (, x () , ) It is assumed that ω() 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 û() = ω() = 0 is said to be finite-time connectively stable with respect to ( 1 ,  2 , , ) for any   ,  = 1, . . ., ,  = 1, . . ., , and x(0) = x0 .
In order to obtain the main results in this paper, the following assumption and lemmas are proposed.

Decentralized Finite-Time 𝐻 ∞ 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 û() = K x() if there exist two scalars,  ≥ 0 and  > 0, matrix M, and the interconnected constraint matrix, Q, as well as symmetric positive definite matrices Ŷ and  such that where with  min () and  max () being the minimum and maximum eigenvalues of , respectively.And the control law can be determined by K = M Ŷ−1 .
Using Schur complement lemma, then (43) can be changed to with Ψ = ÂT P + P Â + KT BT P + P BK −  P + ĈT Ĉ = ÂT P + PÂ −  P + ĈT Ĉ due to Â = Â + BK .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 K 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  1 and  2 .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.

Decentralized Finite-Time 𝐻 ∞ 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.
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 q1 , q2 , and q3 are the elements of the interconnected constraint matrix Q and with  min (  ) and  max (  ) being the minimum and maximum eigenvalues of   , respectively.And the control law can be determined by   =    −1  .
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 Set Ξ = diag(,  −1  , , , , , , ) and  −1  = P .Then, the following inequality follows from pre-and postmultiplying (55) by Ξ: where 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.

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: the original system.In order to check connective stability of the system, we cut off the interconnection  12 between two subsystems.The result is shown in Figure 3, which shows that the closed-loop system is finite-time  ∞ connectively bounded.
Now the third subsystem is added to the original system with  4. It is obvious that  1 < min( 2 ) and  ∈ [0, 6], 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  23 varies from 1 to 0 and the result is shown in Figure 5.
From the figures, it can be known that both the original system and the resulting expanded system are finite-time  ∞ connectively bounded.

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 finitetime  ∞ connectively bounded.Therefore, this paper can be used as the theoretical basis for expansion of large-scale interconnected system online.

Figure 1 :
Figure 1: The basic structure of an expanded system.

Lemma 7 .
System (14) is finite-time  ∞ connectively bounded if there exist two scalars,  ≥ 0 and  > 0, and a symmetric positive definite matrix  such that

1 Figure 4 : 1 Figure 5 :
Figure 4: The simulation curve of the expanded system.