A decentralized control method for symmetric composite large-scale systems based on finite-time stability

Abstract

A decentralized control method for a large symmetric composite system based on finite-time stability belongs to the field of control engineering technology. The structural characteristics of the symmetric composite system are combined with finite-time stability to construct a simplified finite-time stability judgment condition, and the symmetric composite system is controlled for finite time stability based on state feedback using the judgment condition: the symmetric composite system is determined and assumptions are set to simplify the description of the system; the finite-time stability-related lemmas of the system are proposed to transform the finite-time stability problem into a problem of solving a group of inequalities; the simplified criterion for the finite-time stability of a high-dimensional symmetric composite system is given using the structural characteristics; and the finite-time stability of the symmetric composite system is achieved through state feedback. It is used to solve the finite-time control problem of a high-dimensional symmetric composite system, which can greatly simplify the design process of the finite-time decentralized controller and ensure that the symmetric composite system is finite-time stable while meeting given performance indicators.

Description

Technical Field

[0001] The invention belongs to the technical field of control engineering, and in particular relates to a decentralized control method for a symmetrical combined large system based on finite time stability. Background Art

[0002] With the continuous development of society, people are exposed to more and more large systems in their daily lives. The characteristics of large amount of information, complicated structure and diverse influencing factors of the system also bring many challenges to the research of control theory. In the process of large system research, symmetrical combination system is a special type of large system, which is widely distributed in the industrial field due to its symmetrical structure. For such systems, the use of centralized control will make the information exchange of the entire control system extremely complicated, which will increase the operating cost and reduce the reliability of the system when the system dimension is large. In response to these problems, decentralized control methods appeared in the 1970s. Compared with centralized control, decentralized control; first, each subsystem has a controller for its own characteristics, which improves control efficiency and control quality; second, there is no need to exchange information between subsystems, which increases the simplicity of system controller design; finally, even if a subsystem fails, the remaining subsystems can still operate normally, increasing the reliability of the system.

[0003] The stability analysis and synthesis of large systems have always been the focus of system control theory research. Among the research methods of stability problems, the most famous is the Lyapunov stability theory. In the late 1990s, many scholars have studied the stability of symmetrical combined large systems, and most of the results are based on the background of Lyapunov theory. However, Lyapunov stability theory is more inclined to study the asymptotic stability of the system, while in the process of industrial production, people are more concerned about the transient performance of the system in a short time. Therefore, the finite-time stable control method came into being, which ensures that the amplitude of the system state is always within the specified trajectory domain within a certain (short-time or transient) time range.

[0004] The emergence of the concept of finite-time stability can be traced back to the 1960s. Once proposed, it attracted the attention of many scholars and obtained many research results under the framework of finite-time stability. Due to the limitations of the computing means and operation tools at that time, even if an effective finite-time stabilization controller design method was derived, it was still difficult to implement through calculation. It was not until the 1990s that the emergence of the linear matrix inequality toolbox enabled the finite-time stability theory to be applied in engineering, which inspired a new round of research enthusiasm among scholars and further extended the finite-time stability theory to nonlinear systems, stochastic systems, adaptive systems and other fields.

[0005] So far, finite-time control theory has been combined with various systems to solve many system analysis and controller design problems. Finite-time stability theory has become an indispensable part of control theory. However, in finite-time stability theory, there are still few relevant literatures for large systems, and research on the design of finite-time decentralized controllers for large systems is still blank. Summary of the invention

[0006] The purpose of the present invention is to address the above-mentioned gaps in the prior art and provide a decentralized control method for a symmetric combined large system based on finite-time stability, which simplifies the finite-time stability analysis synthesis process by utilizing the similar structure of a large system, solves the finite-time stability analysis and finite-time stabilization problems of a symmetric combined system, and describes the finite-time stability conditions and the finite-time stability controller design strategy by utilizing linear matrix inequalities or differential linear matrix inequalities.

[0007] The present invention comprises the following steps:

[0008] Step 1, determine a symmetrical combination of large systems:

[0009]

[0010] where x i (t)∈R n ,u i (t)∈R m ,y i (t)∈R p They are the system state quantity, control input quantity and control output quantity respectively.

[0011] Step 2: Describe the symmetric combination system described in step 1 equivalently:

[0012]

[0013] in, And the matrix A∈R Nn×Nn ,B∈R Nn×Nm ,C∈R Np×Nn There are the following forms:

[0014]

[0015] B=block-diag{B 0 ,B 0 ,…,B 0}, C = block-diag {C 0 ,C 0 ,…,C 0}. Where block-diag(*) represents a block diagonal matrix with * as the main diagonal element;

[0016] The symbol Symm(X,Y,r) represents the following r×r dimensional matrix:

[0017]

[0018] Where X and Y are square matrices with the same dimensions; using this form, the system matrix A is expressed as A = Symm(A 0 ,H,N); System (2) is called a symmetric composite system, which is a composite system composed of N identical subsystems combined in a similar inline form;

[0019] Step 3: introduce a low-dimensional system containing subsystem and interconnected term information and establish assumptions; for system (2), take A p =A 0 +(N-1)H and A m =A 0 -H; Given a positive definite matrix R 0 ∈R n×n , a defined in the interval [t 0 ,t 0 +T] on the positive definite matrix-valued function Γ 0 (·)∈R Nn×Nn , which satisfies Γ 0 (t 0 )<R 0 , and let R = block-diag {R 0 ,R 0 ,…,R 0}, R∈R Nn×Nn And Γ(t)=block-diag{Γ 0 (t),Γ 0 (t),…,Γ 0 (t)},Γ∈R Nn×Nn ;

[0020] Step 4: Based on the definition of finite-time stability, apply two lemmas and propose a simplified analysis theorem for finite-time stability of symmetric composite systems.

[0021] Step 5: Based on the simplified analysis theorem described in step 4, a design method for constructing a finite-time decentralized controller based on state feedback is further proposed to achieve finite-time stable decentralized control of the symmetric composite system.

[0022] In step 4, the two lemmas are:

[0023] Lemma 1: The following two conditions are equivalent:

[0024] (1) Time-varying system x(t 0 )=x0 About (t 0 ,T,R 0 ,Γ 0 (·)) Limited time stability;

[0025] (2) The following differential linear matrix inequality with initial and terminal conditions

[0026]

[0027] P(t)>Γ 0 (t),t∈[t 0 ,t 0 +T], (6)

[0028] P(t 0 )<R 0 . (7)

[0029] At t∈[t 0 ,t 0 +T] has a piecewise continuous differentiable symmetric solution P(·).

[0030] Lemma 2: Consider a linear time-invariant system:

[0031]

[0032] Given a positive scalar T and a positive definite matrix R 0 ∈R n×n , if there exists a non-negative scalar α and a positive definite matrix P∈R n ×n Make

[0033]

[0034]

[0035] Established, of which represents the condition number of P, system (8) with respect to (T, R 0 ,ρR 0 ), 0<ρ<1 is finite time stable.

[0036] Note that when the value of the non-negative scalar α is determined, condition (9) is transformed into a traditional linear matrix inequality constraint. Furthermore, equation (10) can also be ensured to be feasible by the following linear matrix inequality:

[0037] ρe αT I<P<I. (11)

[0038] Combining equations (9) and (11), a finite-time stability criterion for linear time-invariant systems based on linear matrix inequalities is given.

[0039] Remark 1: On the one hand, if α in condition (9) is 0, compared with the Lyapunov equation of the continuous system, it is easy to obtain that system (8) is asymptotically stable, and for all t∈(0,+∞) there is x T (t)ρR 0 x(t)<1. On the other hand, if α in condition (9) is 0 and α in condition (10) is not 0, then system (8) is asymptotically stable and also has a 0 ,ρR 0 ) is stable for a limited time.

[0040] In step 4, the finite-time stability simplified analysis theorem is:

[0041] Theorem 1: If two n-dimensional systems and About (t 0 ,T,R 0 ,Γ 0 (·)) is finite-time stable, then the symmetric composite system (2) is about (t 0 ,T,R,Γ(·)) is finite-time stable.

[0042] Proof: From Lemma 1, we know that if the low-dimensional system and All about (t 0 ,T,R 0 ,Γ 0 (·)) is finite-time stable, which is equivalent to the existence of a definition in [t 0 ,t 0 +T] a piecewise continuously differentiable symmetric matrix P p (·) and P m (·) makes the following two sets of differential linear matrix inequalities with initial conditions and terminal conditions hold,

[0043]

[0044] P p (t)>Γ 0 (t),t∈[t 0 ,t 0 +T], (13)

[0045] P p (t 0 )<R 0 . (14)

[0046] and

[0047]

[0048] Pm (t)>Γ 0 (t),t∈[t 0 ,t 0 +T], (16)

[0049] P m (t 0 )<R 0 . (17)

[0050] Let P 0 (t) = [P p (t)+(N-1)P m (t)] / N,P 1 =[P p (t)-P m (t)] / N, construct a matrix-valued function P(t) of the following form,

[0051]

[0052] According to P p (t) and P m (t) is piecewise continuous and differentiable, it is easy to know that P after linear combination 0 (t) and P 1 (t) and the constructed function P(t) are also piecewise continuous symmetric and differentiable. Then take the following transformation matrix:

[0053]

[0054] Where I∈R n×n is the unit matrix, and the following results can be obtained after calculation:

[0055]

[0056] It is easy to see from formula (20) that it is an upper triangular matrix. in Refers to P m (t)-Γ 0 The eigenvalue of (t) is repeated N-1 times. Then, from equations (13) and (16), we can get 0 ,t 0 +T] has P p (t)-Γ 0 (t)>0 and P m (t)-Γ 0 (t)>0, furthermore, λ min (P p (t)-Γ 0 (t))>0 and λ min (P m (t)-Γ0 (t))>0, where λ min (*) represents the minimum eigenvalue of the matrix “*”. Combining with equation (20), we can know that λ min (Ξ(P(t)-Γ(t)Ξ -1 ))=λ min (P(t)-Γ(t))>0, that is, for P(t)>Γ(t). Similarly, from equations (14) and (17), we can infer that P(t 0 )<R.

[0057] In addition, After calculation, we can get

[0058]

[0059] in

[0060]

[0061] From equation (12) and equation (15), we can know that Both have λ max (ΞΣ(t)Ξ -1 )=λ max (Σ(t))<0, where λ max (*) represents the maximum eigenvalue of the matrix “*”, which means that Σ(t)<0. In summary, we can see that the symmetric combination system is about (t 0 ,T,R,Γ(·)) is finite time stable. The theorem is proved.

[0062] The state feedback decentralized stabilization controller design problem of the continuous symmetric composite system in step 5 is described as follows:

[0063] Problem 1: Consider system (2), given a positive definite matrix R = block-diag {R 0 ,…,R 0}, positive scalar T, initial time t 0 , defined in [t 0 ,t 0 +T] on the positive definite matrix-valued function Γ(·) = block-diag{Γ 0 (·),…,Γ 0 (·)}, satisfying Γ 0 (t 0 )<R 0 , design a state feedback controller of the following form

[0064] u i (t) = K i(t)x i (t),i=1,…,N, (22)

[0065] Make the system

[0066]

[0067] About (t 0 ,T,R,Γ(·)) is finite-time stable, where K(t)=block-diag{K 1 (t),…,K N (t)}.

[0068] The following theorem is proposed to solve problem 1 and design a finite-time state feedback decentralized controller:

[0069] Theorem 2: If there exists a piecewise continuously differentiable matrix-valued function Q(·)∈R n×n And the continuous matrix-valued function L(·)∈R m×n Satisfies the following differential linear matrix inequality system with initial and terminal conditions

[0070]

[0071]

[0072]

[0073]

[0074] Then the system finite-time state feedback decentralized stabilization controller gain is:

[0075] K i (t) = L(t)Q -1 (t): = K 0 (t),i=1,…,N. (28)

[0076] prove:

[0077] May as well and Then equation (24) and equation (25) can be transformed into the following forms respectively:

[0078]

[0079]

[0080] Take P(t) = Q -1 (t), and multiply P(t) by equations (26), (27), (29), and (30) simultaneously. We can get

[0081]

[0082]

[0083] P(t)>Γ 0 (t),t∈[t 0 ,t 0 +T], (33)

[0084] P(t)<R 0 (34)

[0085] Using the conclusion of the lemma, we can see that the system and All about (t 0 ,T,R 0 ,Γ 0 (·)) is finite time stable. Further, using the conclusion of Theorem 1, we can get that the system is stable with respect to (t 0 ,T,R,Γ(·)) is finite time stable. The proof is complete.

[0086] Considering the time-varying nature of Q(t) in Theorem 2, it is difficult to solve the differential linear matrix inequality group (24)-(27). From a practical point of view, it is more inclined to transfer the problem to the framework of linear matrix inequality for solution. The following is a comprehensive method based on linear matrix inequality, where the matrix-valued function Γ(t) is taken as a constant matrix with the same shape as the matrix R, that is, let Γ(t) = Γ = ρR, 0 < ρ < 1 and the initial time t 0 =0, under this condition, Problem 1 is restated as follows.

[0087] Problem 2: Given a positive scalar T, a positive definite matrix R = block-diag{R 0 ,…,R 0}, design a state feedback decentralized controller in the form of equation (22) so that the closed-loop system (23) is finite-time stabilized with respect to (T, R, ρR).

[0088] To solve this problem, a controller design method based on linear matrix inequality is proposed.

[0089] Theorem 3: If there exists a non-negative scalar α 1 and α 2 , positive definite matrix Q∈R n×n , the matrix M 0 ∈R m×n , which satisfies the following linear matrix inequality system:

[0090]

[0091]

[0092]

[0093] in α=max{α 1 ,α 2}. At this time, Problem 2 has a solution, and the decentralized state feedback controller gain is:

[0094]

[0095] prove:

[0096] From equation (38), we can get Substituting it into equation (35) and equation (36), we can obtain

[0097]

[0098]

[0099] Pick use By multiplying equation (39) and equation (40) on the left and right at the same time, we can get

[0100]

[0101]

[0102] Considering that Cond(P)=Cond(Q), it is easy to get i=1,2. Pick and According to the conclusion of Lemma 2, we can know that the closed-loop system and All about (T,R 0 ,ρR 0 ) is finite-time stable. Further, from the conclusion of Theorem 1, we can conclude that, in the framework of linear matrix inequalities, the closed-loop system is finite-time stable with respect to (T, R, ρR).

[0103] In summary, through state feedback, a finite-time stabilizing controller can be designed for a low-dimensional continuous system, and the obtained controller can be directly constructed into a stabilizing controller for a continuous symmetric composite system in a block diagonal form.

[0104] The main objectives of the present invention are: (1) to provide a finite-time decentralized control method for a large system to address the gaps in the prior art and enrich the theory of large systems; (2) to simplify the finite-time stability analysis synthesis process by utilizing the similar structure of large systems and solve the finite-time stability analysis and finite-time stabilization problems of symmetric combined systems; and (3) to describe the finite-time stability conditions and the finite-time stability controller design strategy by utilizing linear matrix inequalities or differential linear matrix inequalities, which is easier to solve.

[0105] The beneficial effects of the present invention are:

[0106] 1) For a class of symmetric combined large systems, the finite-time stability analysis and finite-time stabilization problems are studied to fill the gap in the theory of large systems; 2) The similar structure of large systems is used to simplify the analysis and synthesis process and reduce the computational complexity; 3) The finite-time stability conditions and finite-time stable controller design strategies are described using linear matrix inequalities or differential linear matrix inequalities. BRIEF DESCRIPTION OF THE DRAWINGS

[0107] Figure 1 Schematic diagram of the mechanical system used to verify the present invention.

[0108] Figure 2 It is a trajectory simulation curve of the system state under the control of the finite time decentralized controller in the present invention. DETAILED DESCRIPTION

[0109] The present invention is described in detail below with reference to the accompanying drawings and specific embodiments.

[0110] The present invention aims at the finite time decentralized control problem of a type of combined large system with a symmetrical structure, and obtains a simplified condition for the finite time stability of the symmetrical combined large system based on the finite time stability theory. The state feedback controller and the output feedback controller are designed according to the simplified condition, and the finite time stability condition and the finite time stability controller design strategy are described by using linear matrix inequality or differential linear matrix inequality, which is more convenient to solve.

[0111] Step 1, determine a symmetrical combination of large systems:

[0112]

[0113] where x i (t)∈R n ,u i (t)∈R m ,y i (t)∈R p They are the system state quantity, control input quantity and control output quantity respectively.

[0114] Step 2: Describe the symmetric combination system described in step 1 equivalently:

[0115]

[0116] in And the matrix A∈R Nn×Nn ,B∈R Nn×Nm ,C∈R Np×Nn There are the following forms:

[0117]

[0118] B=block-diag{B 0 ,B 0 ,…,B 0}, C = block-diag {C 0 ,C 0 ,…,C 0}. Where block-diag(*) represents a block diagonal matrix with * as the main diagonal element.

[0119] To simplify the representation, the symbol Symm(X,Y,r) is introduced to represent the following r×r dimensional matrix:

[0120]

[0121] Where X and Y are square matrices with the same dimensions. Using this form, the system matrix A can be expressed as A = Symm(A 0 ,H,N). In this paper, system (2) is called a symmetric composite system, which is a composite system composed of N identical subsystems combined in a similar inline form.

[0122] Step 3: Introduce a low-dimensional system containing subsystem and interconnected term information and establish assumptions. For system (2), take A p =A 0 +(N-1)H and A m =A 0 -H. Given a positive definite matrix R 0 ∈R n×n , a defined in the interval [t 0 ,t 0 +T] on the positive definite matrix-valued function Γ 0 (·)∈R Nn×Nn , which satisfies Γ 0 (t 0 )<R 0 , and let R = block-diag {R 0 ,R 0 ,…,R 0}, R∈R Nn×Nn And Γ(t)=block-diag{Γ 0 (t),Γ 0 (t),…,Γ 0 (t)},Γ∈R Nn×Nn .

[0123] Step 4: Based on the definition of finite-time stability, apply two lemmas and propose a simplified analysis theorem for finite-time stability of symmetric composite systems.

[0124] The two lemmas in step 4 are:

[0125] Lemma 1: The following two conditions are equivalent:

[0126] (1) Time-varying system Limited time stability;

[0127] (2) The following differential linear matrix inequality with initial and terminal conditions

[0128]

[0129] P(t)>Γ 0 (t),t∈[t 0 ,t 0 +T], (6)

[0130] P(t 0 )<R 0 . (7)

[0131] At t∈[t 0 ,t 0 +T] has a piecewise continuous differentiable symmetric solution P(·).

[0132] Lemma 2: Consider a linear time-invariant system

[0133]

[0134] Given a positive scalar T and a positive definite matrix R 0 ∈R n×n , if there exists a non-negative scalar α and a positive definite matrix P∈R n ×n Make

[0135]

[0136]

[0137] Established, of which represents the condition number of P, system (8) with respect to (T, R 0 ,ρR 0 ), 0<ρ<1 is finite time stable.

[0138] Note that when the value of the non-negative scalar α is determined, condition (9) is transformed into a traditional linear matrix inequality constraint. Furthermore, equation (10) can also be ensured to be feasible by the following linear matrix inequality:

[0139] ρe αT I<P<I. (11)

[0140] Combining equations (9) and (11), a finite-time stability criterion for linear time-invariant systems based on linear matrix inequalities is given.

[0141] Remark 1: On the one hand, if α in condition (9) is 0, compared with the Lyapunov equation of the continuous system, it is easy to obtain that system (8) is asymptotically stable, and for all t∈(0,+∞) there is x T (t)ρR 0 x(t)<1. On the other hand, if α in condition (9) is 0 and α in condition (10) is not 0, then system (8) is asymptotically stable and also has a 0 ,ρR 0 ) is stable for a limited time.

[0142] The finite time stability simplified analysis theorem in step 4 is:

[0143] Theorem 1: If two n-dimensional systems and About (t 0 ,T,R 0 ,Γ 0 (·)) is finite-time stable, then the symmetric composite system (2) is about (t 0 ,T,R,Γ(·)) is finite-time stable.

[0144] Proof: From Lemma 1, we know that if the low-dimensional system and All about (t 0 ,T,R 0 ,Γ 0 (·)) is finite-time stable, which is equivalent to the existence of a definition in [t 0 ,t 0 +T] a piecewise continuously differentiable symmetric matrix P p (·) and P m (·) makes the following two sets of differential linear matrix inequalities with initial conditions and terminal conditions hold;

[0145]

[0146] P p (t)>Γ 0 (t),t∈[t 0 ,t 0 +T], (13)

[0147] P p (t 0 )<R 0 . (14)

[0148] and

[0149]

[0150] P m (t)>Γ 0 (t),t∈[t 0 ,t 0 +T], (16)

[0151] P m (t 0 )<R 0 . (17)

[0152] Let P 0 (t) = [P p (t)+(N-1)P m (t)] / N,P 1 =[P p (t)-P m (t)] / N, construct a matrix-valued function P(t) of the following form,

[0153]

[0154] According to P p (t) and P m (t) is piecewise continuous and differentiable, it is easy to know that P after linear combination 0 (t) and P 1 (t) and the constructed function P(t) are also piecewise continuous symmetric and differentiable. Then take the following transformation matrix:

[0155]

[0156] where I∈R n×n is the unit matrix, and the following results can be obtained after calculation:

[0157]

[0158] It is easy to see from formula (20) that it is an upper triangular matrix. in Refers to P m (t)-Γ 0 The eigenvalue of (t) is repeated N-1 times. Then, from equations (13) and (16), we can get 0 ,t 0 +T] has P p (t)-Γ 0 (t)>0 and P m (t)-Γ 0 (t)>0, furthermore, λ min (P p (t)-Γ0 (t))>0 and λ min (P m (t)-Γ 0 (t))>0, where λ min (*) represents the minimum eigenvalue of the matrix “*”. Combining with equation (20), we can know that λ min (Ξ(P(t)-Γ(t)Ξ -1 ))=λ min (P(t)-Γ(t))>0, that is, for P(t)>Γ(t). Similarly, from equations (14) and (17), we can infer that P(t 0 )<R.

[0159] In addition, After calculation, we can get:

[0160]

[0161] in:

[0162]

[0163] From equations (12) and (15), we can know that Both have λ max (ΞΣ(t)Ξ -1 )=λ max (Σ(t))<0, where λ max (*) represents the maximum eigenvalue of the matrix “*”, which means that Σ(t)<0. In summary, we can see that the symmetric combination system is about (t 0 ,T,R,Γ(·)) is finite time stable. The theorem is proved.

[0164] Step 5: Based on the simplified analysis theorem described in step 4, a design method for constructing a finite-time decentralized controller based on state feedback is further proposed to achieve finite-time stable decentralized control of the symmetric composite system.

[0165] The state feedback decentralized stabilization controller design problem of the continuous symmetric composite system in step 5 is described as follows:

[0166] Problem 1: Consider system (2), given a positive definite matrix R = block-diag {R 0 ,…,R 0}, positive scalar T, initial time t 0 , defined in [t 0 ,t 0 +T] on the positive definite matrix-valued function Γ(·) = block-diag{Γ 0(·),…,Γ 0 (·)}, satisfying Γ 0 (t 0 )<R 0 , design a state feedback controller of the following form

[0167] u i (t) = K i (t)x i (t),i=1,…,N, (22)

[0168] Make the system

[0169]

[0170] About (t 0 ,T,R,Γ(·)) is finite-time stable, where K(t)=block-diag{K 1 (t),…,K N (t)}.

[0171] The following theorem is proposed to solve problem 1 and design a finite-time state feedback decentralized controller:

[0172] Theorem 2: If there exists a piecewise continuously differentiable matrix-valued function Q(·)∈R n×n And the continuous matrix-valued function L(·)∈R m×n Satisfies the following differential linear matrix inequality system with initial and terminal conditions;

[0173]

[0174]

[0175]

[0176]

[0177] Then the system finite-time state feedback decentralized stabilization controller gain is:

[0178] K i (t) = L(t)Q -1 (t): = K 0 (t),i=1,…,N. (28)

[0179] prove:

[0180] May as well and Then equation (24) and equation (25) can be transformed into the following forms respectively:

[0181]

[0182]

[0183] Take P(t) = Q -1 (t), and multiply P(t) by equations (26), (27), (29), and (30) simultaneously. We can get

[0184]

[0185]

[0186] P(t)>Γ 0 (t),t∈[t 0 ,t 0 +T], (33)

[0187] P(t)<R 0 (34)

[0188] Using the conclusion of the lemma, we can see that the system and All about (t 0 ,T,R 0 ,Γ 0 (·)) is finite time stable. Further, using the conclusion of Theorem 1, we can get that the system is stable with respect to (t 0 ,T,R,Γ(·)) is finite time stable. The proof is complete.

[0189] Considering the time-varying nature of Q(t) in Theorem 2, it is difficult to solve the differential linear matrix inequality group (24)-(27). From a practical point of view, people tend to transfer the problem to the framework of linear matrix inequalities for solution. The following is a comprehensive method based on linear matrix inequalities. The matrix-valued function Γ(t) is taken as a constant matrix with the same shape as the matrix R, that is, let Γ(t) = Γ = ρR, 0 < ρ < 1 and the initial time t 0 =0, under this condition, Problem 1 is restated as follows.

[0190] Problem 2: Given a positive scalar T, a positive definite matrix R = block-diag{R 0 ,…,R 0}, design a state feedback decentralized controller in the form of equation (22) so that the closed-loop system (23) is finite-time stabilized with respect to (T, R, ρR).

[0191] To solve this problem, a controller design method based on linear matrix inequality is proposed.

[0192] Theorem 3: If there exists a non-negative scalar α 1 and α 2 , positive definite matrix Q∈Rn×n , the matrix M 0 ∈R m×n , satisfying the following linear matrix inequality system

[0193]

[0194]

[0195]

[0196] in α=max{α 1 ,α 2}. At this time, Problem 2 has a solution, and the decentralized state feedback controller gain is:

[0197]

[0198] prove:

[0199] From equation (38), we can get Substituting it into equation (35) and equation (36), we can obtain:

[0200]

[0201]

[0202] Pick use By multiplying the left and right equations together, we can get:

[0203]

[0204]

[0205] Considering that Cond(P)=Cond(Q), it is easy to get i=1,2. Pick and According to the conclusion of Lemma 2, we can know that the closed-loop system and All about (T,R 0 ,ρR 0 ) is finite time stable. Further, from the conclusion of Theorem 1, we can conclude that, in the framework of linear matrix inequalities, the closed-loop system (23) is finite time stable with respect to (T, R, ρR). The proof is complete.

[0206] In summary, a finite-time stabilizing controller can be designed for a low-dimensional continuous system through state feedback, and the obtained controller can be directly constructed into a stabilizing controller for a continuous symmetric composite system in a block diagonal form.

[0207] The present invention is further described below with an example.

[0208] consider Figure 1 A mechanical system in which two spring-connected sliders move on a smooth surface.

[0209] Analyze system characteristics and obtain state quantities u=[u 1 ,u 2 ] T =[F 1 ,-F 2 ] T , y=[y 1 ,y 2 ] T =[s 1 ,s 2 ] T , establish the following system equation:

[0210]

[0211] y(t)=Cx(t). (44)

[0212] in Design a controller for the system (43)(44) so ​​that its state returns to the equilibrium point x ＊ =[a,0,b,0] T Place.

[0213] Take the error ξ(t) = x(t) - x ＊ , error system input Error system output The following error system can be obtained:

[0214]

[0215]

[0216] When taking M 1 =M 2 =1kg, k = 1N / m, it is noted that the original system (43) and the error system (45) are symmetrical combinations. At this time, N = 2, C 0 =[1 0].

[0217] Design a state feedback controller for system (45) that satisfies the following error constraints:

[0218] |ξ 1 (t)| / a<δ 1 , (47)

[0219] |ξ2 (t)|<δ 2 , (48)

[0220] |ξ 3 (t)| / b<δ 1 , (49)

[0221] |ξ 4 (t)|<δ 2 , (50)

[0222] where δ 1 ,δ 2 Respectively represent x 1 (t) and x 3 The relative overshoot of (t) and x 2 (t) and x 4 The absolute overshoot of (t).

[0223] To meet the control requirements of (47)-(50), c = min{|a|,|b|}, Γ=block-diag{Γ 0 ,Γ 0},R 0 =ρ -1 Γ 0 ,0<ρ<1,R=block-diag{R 0 ,R 0}, a=4, b=5, ρ=0.1, δ 1 =0.2,δ 2 =0.9.

[0224] According to the above simulation parameters, the state feedback control gain K is obtained by using the LMI toolbox in MATLAB software. 0 =[-2.42 -11.84], then the control gain of the state feedback decentralized controller of the symmetric combined system can be directly given by the following block symmetric form:

[0225]

[0226] The simulation results are as follows Figure 2 As shown in Figure 2, the controller designed based on state feedback can make the closed-loop system state tend to the given expected value x ＊ =[4,0,5,0] T And each state does not exceed the pre-set overshoot requirement. It can be known that the state feedback decentralized control obtained by the present invention can make the closed-loop system stable for a finite time, proving the effectiveness of the present invention.

[0227] The above results show that the control method of the present invention solves the design problem of finite-time stable controller for symmetrical composite systems and simplifies the computational complexity by utilizing the structural characteristics of large systems.

Claims

1. A decentralized control method for a symmetric composite large system based on finite-time stability, characterized in that The following steps are involved: Step 1, determine a symmetrical combination of large systems: y i (t)=C0x i (t),i=1,2,...,N. where x i (t)∈R n ,u i (t)∈R m ,y i (t)∈R p They are the state quantity, control input quantity and control output quantity of the system respectively; Step 2: Describe the symmetrical combination system described in step 1 equivalently: y(t)=Cx(t) in And the matrix A∈R Nn ×Nn ,B∈R Nn×Nm ,C∈R Np×Nn There are the following forms: B = block-diag{B0, B0, ..., B0}, C = block-diag{C0, C0, ..., C0}; where block-diag (*) represents a block diagonal matrix with * as the main diagonal element; The symbol Symm(X,Y,r) is introduced to represent the following r×r dimensional matrix: Where X and Y are square matrices with the same dimension. Using this form, the system matrix A can be expressed as A = Symm(A0, H, N). System (2) is called a symmetric combination system, which is a combination system composed of N identical subsystems combined in a similar inline form. Step 3: introduce a low-dimensional system containing subsystem and interconnected term information and establish assumptions; for system (2), take A p =A0+(N-1)H and A m =A0-H; given a positive definite matrix R0∈R n×n , a positive definite matrix-valued function Γ0(·)∈R defined on the interval [t0,t0+T] Nn×Nn , which satisfies Γ0(t0)<R0, and let R=block-diag{R0,R0,…,R0}, R∈R Nn×Nn And Γ(t)=block-diag{Γ0(t),Γ0(t),…,Γ0(t)}, Γ∈R Nn×Nn ; Step 4, based on the definition of finite-time stability, apply two lemmas and propose a simplified analysis theorem of finite-time stability of symmetric composite systems; The finite-time stability simplified analysis theorem is: If there exists a piecewise continuously differentiable matrix-valued function Q(·)∈R n×n And the continuous matrix-valued function L(·)∈R m×n The following differential linear matrix inequality system with initial and terminal conditions is satisfied: Then the gain of the system finite-time state feedback decentralized stabilization controller is: K i (t)=L(t)Q -1 (t):=K0(t),i=1,…,N.; Step 5: Based on the simplified analysis theorem described in step 4, a design method for constructing a finite-time decentralized controller based on state feedback is proposed, and the implementation process of the state feedback decentralized controller under time-invariant conditions is further given to realize the finite-time stable decentralized control of the symmetric composite system; The implementation process of the state feedback decentralized controller under time-invariant conditions is: If there are non-negative scalars α1 and α2, the positive definite matrix Q∈R n×n , the matrix M0∈R m×n , which satisfies the following linear matrix inequality system: in, α=max{α1,α2}; At this time, the finite-time decentralized state feedback problem has a solution, and the decentralized state feedback controller gain is:

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