A dual-gain parameter design method based on output feedback controller of time-delay system

Abstract

The invention discloses a dual-gain parameter design method based on a time-delay system output feedback controller, which relates to the field of controllers. The method includes obtaining the constraint conditions of the time-delay system model based on the acquired time-delay system model; constructing a disturbance observer based on common external disturbances, the disturbance observer including a disturbance observer gain matrix to be solved; constructing an output feedback controller based on the disturbance observer, the output feedback controller including an output feedback controller gain matrix to be solved; defining a disturbance estimation error and converging it; and obtaining the value range of the disturbance observer gain matrix and the output feedback controller gain matrix according to sufficient conditions for the asymptotic stability of the closed-loop system. Based on the disturbance observer, the invention takes into account the actuator failure, makes full use of the disturbance information, reduces the conservatism of the system, and can describe the disturbances that actually exist in reality, while improving the stability and performance of the system.

Description

Technical Field

[0001] The invention relates to the technical field of control systems, in particular to a dual-gain parameter design method based on a time-delay system output feedback controller. Background Art

[0002] In practical applications, time lag is inevitable in the system, such as in neural networks, population dynamics models, ship stability analysis, and power systems, which will lead to system performance degradation or even instability. Therefore, it is necessary to solve the stability problem of time-delay systems. Existing research has designed a robust controller for a mixed time-delay and variable structure uncertainty neural network system. Considering the existence of actuator failures, this method has complex iterative learning and heavy load, and is prone to introduce uncertainty into the controller itself.

[0003] At the same time, in the process of dealing with disturbances and uncertainties, many advanced control technologies are constantly being proposed, such as stochastic control, output regulation technology, H∞ control, sliding mode control, etc. Compared with the disturbance observer method, the first three control technologies all need to solve some nonlinear differential equations or differential inequalities, and can only guarantee the stability of the system when the disturbance has an interfering effect on the system, and still have the disadvantage of being too conservative. Summary of the invention

[0004] In view of the above problems existing in the prior art, the present invention is proposed.

[0005] Therefore, the problem to be solved by the present invention is the stability problem of the time-delay system performance with uncertainty when the actuator fails and the system state is unknown.

[0006] To solve the above technical problems, in a first aspect, the present invention provides the following technical solutions: a dual-gain parameter design method based on an output feedback controller of a time-delay system, comprising: obtaining the constraint conditions of the time-delay system model based on an acquired time-delay system model with uncertainty and external disturbances; constructing a disturbance observer based on common external disturbances of the control system, wherein the disturbance observer includes a disturbance observer gain matrix to be solved; constructing an output feedback controller based on the disturbance observer, wherein the output feedback controller includes an output feedback controller gain matrix to be solved; obtaining a closed-loop system based on the time-delay system model, the disturbance observer and the output feedback controller; defining a disturbance estimation error and converging it so that the disturbance estimation error and the closed-loop system are asymptotically stable; and obtaining the value ranges of the disturbance observer gain matrix and the output feedback controller gain matrix according to sufficient conditions for the asymptotic stability of the closed-loop system.

[0007] As a preferred solution of the dual-gain parameter design method based on the time-delay system output feedback controller of the present invention, wherein: the time-delay system model is:

[0008]

[0009] Where x(t)∈R n×1 is the state variable of the time-delay system; is the first-order differential of the state variable of the time-delay system; x0(t) is the initial state of the time-delay system; φ(t) is the initial value of the system state of the time-delay system; τ(t) is the time delay time of the time-delay system; u(t)∈R m×1 is the input of the time-delay system controller; d(t)∈R m×1 is the external disturbance input to the time-delay system; y(t)∈R n×1 is the output variable of the time-delay system; A∈R n×n , B∈R n×m , A d ∈R n×n and C∈R n×n is the system matrix of the time-delay system; ΔA(t), ΔA d (t) and d(t) are the uncertainties of the time-delay system; I is the identity matrix; n and m are the dimensions of the matrix;

[0010] The constraints of the time-delay system model are:

[0011] 0≤τ(t)≤τ M(t) and

[0012] [ΔA(t)ΔA d (t)] = DF(t)[E1 E2]

[0013] rank(B)=m

[0014] Among them, τ M(t) is the upper bound of the delay time in the time-delay system; is the time-delay time change rate in the time-delay system; h is the upper bound of the time-delay time change rate in the time-delay system; E1 and E2 are structural uncertainties in the time-delay system, which are constant matrices satisfying D is the parameter adjustment gain matrix;

[0015] When considering the actuator failure, the corresponding time-delay system model is:

[0016]

[0017] Among them, ρ=diag(ρ1,ρ2,ρ3,······,ρ m ), ρk is the failure parameter of the actuator, which satisfies k=1,2,3,……m;ρ k and are the upper and lower bounds of the diagonal parameters of the matrix ρ, ρ k =diag( ρ 1, ρ 2,······, ρ m ), gather N ρ Contains up to 2 m elements.

[0018] As a preferred solution of the dual-gain parameter design method based on the time-delay system output feedback controller of the present invention, the common external disturbance is:

[0019]

[0020] Where ω(t)∈R n×1 is the time-delay system frequency; is the first-order differential of ω(t); W and V are disturbance gain matrices; W = 0, V = 1 are constant load disturbances of unknown values; V =

[10] is a harmonic disturbance of unknown amplitude and phase; d∈R m ;ω is the parameter adjustment gain;

[0021] The disturbance observer is:

[0022]

[0023] Where L∈R n×n is the disturbance observer gain matrix; is the disturbance estimate; is the frequency estimation value of the time-delay system; v(t) is the internal state of the disturbance observer, is its first-order differential; C is the output matrix and B is the input matrix.

[0024] As a preferred solution of the dual-gain parameter design method based on the time-delay system output feedback controller of the present invention, wherein: the output feedback controller is:

[0025]

[0026] Wherein, Ky(t) is the linear expression of the output of the time-delay system; is the compensation of the disturbance estimation value; K is the output feedback controller gain matrix.

[0027] As a preferred solution of the dual-gain parameter design method based on the time-delay system output feedback controller of the present invention, the disturbance estimation error is:

[0028]

[0029] Wherein, e(t) is the disturbance estimation error; ω(t)∈R n×1 is the time-delay system frequency, is the estimated value of the time-delay system.

[0030] As a preferred solution of the dual-gain parameter design method based on the time-delay system output feedback controller of the present invention, the method for converging the disturbance estimation error includes derivation of the disturbance estimation error:

[0031]

[0032] Right now:

[0033] Among them, when the output feedback controller realizes state quantity stabilization control, x(t)→0, x(t-τ(t))→0; when W+LCB(I-ρ)V<0, the exponential stability of the disturbance estimation error e(t) is achieved.

[0034] As a preferred solution of the dual-gain parameter design method based on the time-delay system output feedback controller of the present invention, the sufficient condition for the asymptotic stability of the closed-loop system is:

[0035] There exists a positive constant α i , i∈(1,2,3,4) and the positive definite matrix P1∈R n×n , P2∈R n×n , R∈R n×n And the matrix Q1∈R n×n , Q2∈R n×n , Q3∈R n×n , Q4∈R n×n So that the following first inequality holds:

[0036]

[0037] Among them, Γ 11 =He(P1(A+B(I-ρ)KC))+(α1+α2)PDD T P+(α1 -1 +α3 -1 )E1 T E1+P1-Q1-Q1 T Γ 12 =P1B(I-ρ)V+(P2LCA-P2LCBρKC)T -Q4;Γ 13 =P1A d -Q2+Q T Γ 14 =Q1 T -Q3;Γ 22 =He(P2(W+LCB(I-ρ)V))+(α3+α4)P2LCDD T C T L T P2;Γ 23 =P2LCA d +Q4 T Γ 24 =Q4 T ; Γ 34 =Q2 T +Q3;Γ 44 =Q3 T +Q3; He is the symmetrization operation; P and D are matrix parameters; A is the state matrix; A d is the delayed state matrix; B is the input matrix; C is the output matrix; and D is the transfer matrix.

[0038] As a preferred solution of the dual-gain parameter design method based on the time-delay system output feedback controller of the present invention, the first inequality is decoupled by using the Fisher lemma to transform the first inequality into the second inequality:

[0039]

[0040] Among them, Λ 11 =He(P1A+b1B(I-ρ)NC)+(α1 -1 +α3 -1 )E1 T E1+R-He(Q1); 12 =P1B(I-ρ)V+(MCA-b2BρNC) T -Q4; Λ 13 =P1A d -Q2+Q1 T ; Λ 14 =Q1 T -Q3;Λ 15 =P1B(I-ρ)V-b1B(I-ρ)Z+μ(NC) T ; Λ 16 =P1D; Λ 22 =He(P2W+MCB(I-ρ)V); 23 =MCA d +Q4 T ; Λ 24 =Q4T ; Λ 25 = b2BρZ-MCBρ; Λ 27 =MCD; Λ 34 =Q3+Q2 T ; Λ 44 =He(Q3); 55 = -μHe(Z); Λ 66 =-(α1+α2) -1 I; Λ 77 =-(α3+α4) -1 I;

[0041] For a given positive constant b1, b2, μ, α i , i∈{1,2,3,4}, if there exist positive definite matrices P1>0, P2>0, R>0 and matrices Q1, Q2, Q3, Q4 and M, N, Z such that the second inequality holds, then the closed-loop system and the disturbance observer error are both asymptotically stable;

[0042] Among them, the disturbance observer gain matrix K=Z is selected -1 N, the output feedback controller gain L = P2 -1 M.

[0043] In a second aspect, the present invention provides an electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the steps of the above-mentioned dual gain parameter design method based on the time-delay system output feedback controller when executing the computer program.

[0044] In a third aspect, the present invention provides a computer-readable storage medium storing computer-executable instructions, which, when executed by a processor, implement the steps of the above-mentioned dual-gain parameter design method based on the time-delay system output feedback controller.

[0045] The beneficial effects of the present invention are as follows: based on a disturbance observer, the present invention takes into account actuator failures, fully utilizes disturbance information, reduces the conservatism of the system, and can describe the disturbances that actually exist in reality, while improving the stability and performance of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following briefly introduces the drawings required for use in the description of the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0047] Figure 1 It is a schematic diagram of the structure of the time-delay system described in Example 1 of the present invention.

[0048] Figure 2 This is a system state quantity response curve diagram considering the existence of an actuator failure described in Example 2 of the present invention.

[0049] Figure 3 This is a simulation curve diagram of the disturbance observer error e(t) in the presence of an actuator failure as described in Example 2 of the present invention.

[0050] Figure 4 is the disturbance d(t) under the consideration of the actuator failure described in Example 2 of the present invention, Simulation curve graph.

[0051] Figure 5 This is a system state quantity response curve diagram without considering actuator failure described in Example 2 of the present invention.

[0052] Figure 6 This is a simulation curve diagram of the disturbance observer error e(t) without considering actuator failure described in Example 2 of the present invention.

[0053] Figure 7 is the disturbance d(t) described in Embodiment 2 of the present invention without considering the actuator failure, Simulation curve graph. DETAILED DESCRIPTION

[0054] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the specific embodiments of the present invention are described in detail below in conjunction with the accompanying drawings.

[0055] In the following description, many specific details are set forth to facilitate a full understanding of the present invention, but the present invention may also be implemented in other ways different from those described herein, and those skilled in the art may make similar generalizations without violating the connotation of the present invention. Therefore, the present invention is not limited to the specific embodiments disclosed below.

[0056] Secondly, the term "one embodiment" or "embodiment" as used herein refers to a specific feature, structure or characteristic that may be included in at least one implementation of the present invention. The term "in one embodiment" that appears in different places in this specification does not necessarily refer to the same embodiment, nor is it an embodiment that is mutually exclusive with other embodiments, either individually or selectively.

[0057] Example 1

[0058] Reference Figure 1, which is the first embodiment of the present invention, provides a dual-gain parameter design method based on a time-delay system output feedback controller, the method comprising:

[0059] S1: Based on the acquired time-delay system model with uncertainty and external disturbances, the constraints of the time-delay system model are obtained.

[0060] Specifically, the time-delay system model with uncertainty and external disturbance is:

[0061]

[0062] Where x(t)∈R n×1 is the state variable of the time-delay system; is the first-order differential of the state variable of the time-delay system; x0(t) is the initial state of the time-delay system; φ(t) is the initial value of the system state of the time-delay system; τ(t) is the time delay time of the time-delay system; u(t)∈R m×1 is the input of the time-delay system controller; d(t)∈R m×1 is the external disturbance input to the time-delay system; y(t)∈R n ×1 is the output variable of the time-delay system; A∈R n×n , B∈R n×m , A d ∈R n×n and C∈R n×n is the system matrix of the time-delay system; ΔA(t), ΔA d (t) and d(t) are the uncertainties of the time-delay system; I is the identity matrix; n and m are the dimensions of the matrix.

[0063] Among them, the constraints of the time-delay system model are:

[0064] Constraint 1: τ(t) satisfies the following conditions:

[0065] 0≤τ(t)≤τ M(t) and

[0066] Constraint 2: System uncertainty satisfies the following conditions:

[0067] [ΔA(t)ΔA d (t)] = DF(t)[E1 E2];

[0068] Constraint 3: Matrix B is full rank:

[0069] rank(B)=m.

[0070] Among them, τ M(t) is the upper bound of the delay time in the time-delay system; is the time-delay rate of change in the time-delay system; h is the upper bound of the time-delay rate of change in the time-delay system; E1 and E2 are structural uncertainties in the time-delay system, which are constant matrices satisfying D is the parameter adjustment gain matrix.

[0071] In addition, in order to better prove the sufficient conditions for asymptotic stability given below, three lemmas in this embodiment are given here:

[0072] Lemma 1: For any constant α>0, the following inequality holds:

[0073] X T Y+Y T X≤αX T X+α -1 Y T Y;

[0074] Lemma 2: Assume that there exists P = P T ∈R n×n and B∈R n×n , the following propositions are equivalent:

[0075] (a)v T Pv<0,v≠0,Hv=0;

[0076] (b)H T pH<0;

[0077] (c) Let P+He(SH)<0.

[0078] Lemma 3: Assume that the given symmetric matrix S 11 ∈R r×r , the following three propositions are equivalent:

[0079] (a) S<0;

[0080]

[0081] The output feedback control structure of the time-delay system based on DOB technology is shown in the figure below: Figure 1 As shown. In the prior art, most controller designs are based on state feedback. However, in real industrial systems, the system state or the estimation of the state may be unknown. The present invention proposes to handle this type of uncertain time-delay system by output feedback control and a method utilizing time-delay dependence. Assuming that the system state is unknown, a disturbance observer and controller are constructed based on the system output.

[0082] S2: Based on the common external disturbances of the control system, a disturbance observer is constructed, which includes the disturbance observer gain matrix to be solved.

[0083] Specifically, the common external disturbances of the system are:

[0084]

[0085] Where ω(t)∈R n×1 is the time-delay system frequency; is the first-order differential of ω(t); W and V are disturbance gain matrices; W = 0, V = 1 are constant load disturbances of unknown values. The following condition (3) can describe the harmonic disturbance with known frequency ω and unknown amplitude and phase:

[0086]

[0087] For the above disturbance structure, the following disturbance observer is constructed:

[0088]

[0089] Where L∈R n×n is the disturbance observer gain matrix; is the disturbance estimate; is the frequency estimate of the time-delay system; v(t) is the internal state of the disturbance observer, is its first-order differential.

[0090] By differentiating the internal state expression of the above disturbance observer (4), we can obtain:

[0091]

[0092] From equation (5), it can be concluded that the disturbance observer based on output feedback only needs to obtain the control law u(t), the system output y(t) and the derivative of the system output It has great application value for actual industrial systems whose states are unknown.

[0093] S3: Based on the disturbance observer, an output feedback controller is constructed, and the output feedback controller includes an output feedback controller gain matrix to be solved.

[0094] Specifically, based on the disturbance observer (4), assuming that the system state is unknown, the following output feedback controller (6) is designed:

[0095]

[0096] Among them, Ky(t) is the linear expression of the output of the time-delay system; is the compensation of the disturbance estimate; K is the output feedback controller gain matrix, and the control law u(t) is the linear expression of the system output y(t) and the compensation of the disturbance estimate composition.

[0097] At the same time, considering the actuator failure, the time-delay system model can be expressed as follows (7):

[0098]

[0099] Among them, the parameter ρ = diag (ρ1, ρ2, ρ3,······,ρ m ), which satisfies the following constraints:

[0100]

[0101] Among them, ρ k (k=1,2,·····,m) is the failure parameter of the actuator;

[0102] ρ k and are the upper and lower bounds of the diagonal parameters of the matrix ρ, assuming they are known constants, and define:

[0103]

[0104] The following expression can be derived:

[0105] N ρ Contains up to 2 m elements.

[0106] For a time-delay system with external disturbances, disturbance suppression can be performed by designing an H∞ controller, but the H∞ controller does not propose a specific structure for the form of disturbance, and the result of disturbance suppression is conservative. The present invention proposes a specific disturbance structure for the disturbance, and the conservatism is reduced.

[0107] S4: Define the disturbance estimation error and converge it so that the disturbance estimation error and the closed-loop system are asymptotically stable.

[0108] Specifically, define the disturbance estimation error And taking its derivative, combining equations (1), (2), (4) and (6), we can get the following expression (8):

[0109]

[0110] From the above formula (8), we can see that in order to make the disturbance estimate approach the true value, e(t) must converge. When the controller achieves state quantity stabilization control, x(t)→0, x(t-τ(t))→0. When there is a suitable L so that W+LCB(I-ρ)V<0, then formula (8) achieves exponential stability with respect to e(t).

[0111] Therefore, the disturbance observer gain matrix L plays a very important role in the performance of the disturbance observer. From the above analysis, it can be seen that solving the appropriate disturbance observer gain matrix L and output controller gain matrix K can stabilize the closed-loop system state and disturbance estimation error.

[0112] In addition, combining equations (7) and (8) we can derive the system composite structure (9):

[0113]

[0114] in,

[0115] S5: Based on the sufficient conditions for the asymptotic stability of the closed-loop system, the value ranges of the disturbance observer gain matrix and the output feedback controller gain matrix are obtained.

[0116] Specifically, the sufficient condition for the asymptotic stability of the closed-loop system is (the following expression is the derivation process):

[0117] Derivation 1:

[0118] If there exists a positive constant α i , i∈(1,2,3,4) and the positive definite matrix P1∈R n×n , P2∈R n×n , R∈R n×n And the matrix Q1∈R n×n , Q2∈R n×n , Q3∈R n×n , Q4∈R n×n So the following first inequality (10) holds true:

[0119]

[0120] in,

[0121] Γ 11 =He(P1(A+B(I-ρ)KC))+(α1+α2)PDD T P+(α1 -1 +α3 -1 )E1 T E1+P1-Q1-Q1 T ;

[0122] Γ 12 =P1B(I-ρ)V+(P2LCA-P2LCBρKC) T -Q4;

[0123] Γ 13 =P1A d -Q2+Q T ;

[0124] Γ 14 =Q1 T -Q3;

[0125] Γ 22 =He(P2(W+LCB(I-ρ)V))+(α3+α4)P2LCDD T C T L T P2;

[0126] Γ 23 =P2LCA d +Q4 T ;

[0127] Γ 24 =Q4 T ;

[0128]

[0129] Γ 34 =Q2 T +Q3;

[0130] Γ 44 =Q3 T +Q3,

[0131] Then both the system state and the disturbance controller state are asymptotically stable.

[0132] The following is the process of proving it:

[0133] The Lyapunov functional is selected as follows:

[0134]

[0135] At the same time, P should satisfy P>0 and set Among them, P1>0, P2>0.

[0136] Substituting the control law (6) into the system state (7) considering the actuator failure, the following system expression (12) can be obtained:

[0137]

[0138] Combining equations (8) and (12) and the Newton-Leibniz formula, we can derive the Lyapunov functional equation (11) to obtain equation (13):

[0139]

[0140] In order to process the and The generated nonlinear part can be scaled by Lemma 1 to obtain the following inequality:

[0141]

[0142] This inequality and equation (13) can be used to obtain equation (14):

[0143]

[0144] Formula (14) is equivalent to the following state space expression:

[0145]

[0146] In order to ensure the stability of the system, it is only necessary to satisfy v(t)<0. By rearranging the above spatial expressions, we can obtain the inequality of Γ in Derivation 1, and the proof is complete.

[0147] Furthermore, the above gives the nonlinear form of the asymptotic stability of the system, and the corresponding controller gain and disturbance observer gain cannot be directly solved. Therefore, the Fisher lemma is used to give the linear matrix inequality (LMI) that can solve the output feedback controller gain matrix K and the disturbance observer gain matrix L.

[0148] Specifically, derivation 2:

[0149] For a given positive constant b1, b2, μ, α i , i∈{1,2,3,4}, if there exists a positive definite matrix P1>0, P2>0, R>0 and matrices Q1, Q2, Q3, Q4 and M, N, Z such that the second inequality (15) holds:

[0150]

[0151] in,

[0152] Λ 11 =He(P1A+b1B(I-ρ)NC)+(α1 -1 +α3 -1 )E1 T E1+R-He(Q1);

[0153] Λ 12 =P1B(I-ρ)V+(MCA-b2BρNC) T -Q4;

[0154] Λ 13 =P1A d -Q2+Q1 T ;

[0155] Λ 14 =Q1 T -Q3;

[0156] Λ 15=P1B(I-ρ)V-b1B(I-ρ)Z+μ(NC) T ;

[0157] Λ 16 =P1D;

[0158] Λ 22 =He(P2W+MCB(I-ρ)V);

[0159] Λ 23 =MCA d +Q4 T ;

[0160] Λ 24 =Q4 T ;

[0161] Λ 25 = b2BρZ-MCBρ;

[0162] Λ 27 =MCD;

[0163]

[0164] Λ 34 =Q3+Q2 T ;

[0165] Λ 44 =He(Q3);

[0166] Λ 55 = -μHe(Z);

[0167] Λ 66 =-(α1+α2) -1 I;

[0168] Λ 77 =-(α3+α4) -1 I;

[0169] Then the closed-loop system and the disturbance observer error tend to be asymptotically stable and the disturbance observer gain matrix K = Z - 1 N, output feedback controller gain L = P2 -1 M.

[0170] The following proves it:

[0171] Derivation 1 can be transformed into the following expression (16):

[0172]

[0173] In order to decouple the nonlinear part appearing in Λ, Lemma 2 is introduced, and equation (16) can be written as follows (17):

[0174] P+He(SH)<0⒄

[0175] in,

[0176] H=(KC,0,0,0,-I)

[0177]

[0178] S1=b1B(I-ρ)Z-P1B(I-ρ)

[0179] S2=-b2BρZ+MCBρ

[0180]

[0181] Δ 11 =He(b1B(I-ρ)NC-P1B(I-ρ)KC)

[0182] Δ 12 =(-b2BρNC+MCBρ)

[0183] Δ 15 =P1B(I-ρ)-b1B(I-ρ)Z+μ(NC) T

[0184] Δ 25 =b2BρZ-MCBρ

[0185] Δ 55 =-μHe(Z)

[0186] The matrix expression of P+He(SH)<0 can be obtained as shown in the following formula (19):

[0187]

[0188] ⒆

[0189] in,

[0190] Θ 11 =He(P1A+b1B(I-ρ)NC)+(α1+α2)P1DD T P1+(α1 -1 +α3 -1 )E1 T E1+R-He(Q1);

[0191] Θ 12 =P1B(I-ρ)V+(MCA-b2BρNC) T -Q4;

[0192] Θ 13 =P1A d -Q2+Q1 T ;

[0193] Θ 14 =Q1 T -Q3;

[0194] Θ 15 =P1B(I-ρ)V-b1B(I-ρ)Z+μ(NC) T ;

[0195] Θ 22 =He(P2W+MCB(I-ρ)V)+(α3+α4)P2LCDD T C T L T P2;

[0196] Θ 23 =MCA d +Q4 T ;

[0197] Θ 24 =Q4 T ;

[0198] Θ 25 = b2BρZ-MCBρ;

[0199]

[0200] Θ 34 =Q3+Q2 T ;

[0201] Θ 44 =He(Q3);

[0202] Θ 55 = -μHe(Z);

[0203] Performing Schur expansion on equation (19) above can yield the inequality of Derivation 2, and the proof is complete.

[0204] Furthermore, when ρ = 0, Derivation 2 can be transformed into a general form, that is, considering the actuator failure.

[0205] That is, Derivation 3:

[0206] For a given positive constant b1, b2, μ, α i , i∈{1,2,3,4}, if there exist positive definite matrices P1>0, P2>0, R>0 and matrices Q1, Q2, Q3, Q4 and M, N, Z such that the following inequality (20) holds:

[0207]

[0208] Among them,

[0209] Oh 11 =He(P1A+b1BNC)+(α1 -1 +a3 -1 )E1 T E1+R-He(Q1);

[0210] Oh 12 =P1BV+(MCA) T -Q4;

[0211] Oh 13 =P1A d -Q2+Q1 T ;

[0212] Oh 14 =Q1 T -Q3;

[0213] Oh 15 =P1B-b1BZ+(b2NC) T ;

[0214] Oh 16 =P1D;

[0215] Oh 22 =He(P2W+MCBV);

[0216] Oh 23 =MCA d +Q4 T ;

[0217] Oh 24 =Q4 T ;

[0218] Oh 27 =MCD;

[0219]

[0220] Oh 34 =Q3+Q2 T ;

[0221] Oh 44 =He(Q3);

[0222] Oh 55 =He(-b2Z);

[0223] Oh 66 =-(α1+α2) -1 I;

[0224] Ω 77 =-(α3+α4) -1 I;

[0225] The closed-loop system and the disturbance observer errors tend to be asymptotically stable. Wherein, the disturbance observer gain matrix K=Z -1 N, the output feedback controller gain L = P2 -1 M.

[0226] Example 2

[0227] This embodiment is based on the previous embodiment, and this embodiment provides a simulation test verification of the dual-gain parameter design method based on the time-delay system output feedback controller proposed in Embodiment 1.

[0228] In order to verify and illustrate the technical effects adopted in the present invention, a second-order system is selected as the test object in this embodiment, and the control method proposed in the present invention is tested. The test results with and without considering the existence of actuator failure are compared by means of scientific argumentation to verify the real effect of this control method.

[0229] Compared methods: Methods that do not consider actuator failures.

[0230] The inequality without considering the actuator failure and the disturbance observer gain matrix K=Z obtained based on the inequality - 1 N and output feedback controller gain L = P2 -1 The value range of M is as described in Derivation 3 in Example 1 and will not be repeated here.

[0231] Test environment: When running a second-order system on a simulation platform, it is necessary to consider the bandwidth of the observer to cover the various disturbance frequencies that the system may suffer.

[0232] Select parameters: V = [2 0]; C = [1 0]; α1=α2=α3=α4=1; b1=10; b2=0.1; μ=1; K=22.5073; L=[0.0323 0.0012] T ; x0 = [-1, 0.5] T ;ρ=0.75.

[0233] By solving Derivation 2 described in Example 1, the output feedback controller gain matrix L and the disturbance observer gain matrix K can be obtained respectively:

[0234] L = [0.0323 0.0012] T, K = 22.5073. Assume the initial state x = [-1 0.5] T , and the fault parameter ρ = 0.75.

[0235] Reference Figure 2 , is a dual-gain parameter design method based on the output feedback controller of the time-delay system, considering the system state quantity response curve under the existence of actuator failure. Figure 2 As shown, the time-delay system has initial states x1 = -1 and x2 = 0.5, and eventually converges to the equilibrium state 0. For comparison, refer to Figure 5 , is a system state quantity response curve diagram without considering actuator failure based on the dual gain parameter design method of the output feedback controller of the time-delay system. Figure 5 As shown in Figure 2, when the running time is 25s, the state quantity of the time-delay system diverges to about 60.

[0236] Reference Figure 3 , is a dual-gain parameter design method for the output feedback controller of a time-delay system considering the disturbance observer error e(t) diagram under actuator failure. Figure 3 As shown, the disturbance observer error value converges to state 0 at the running time of 15s. Figure 6 , is the disturbance observer error e(t) diagram without considering actuator failure based on the dual gain parameter design method of the output feedback controller of the time-delay system. Figure 6 As shown, its trend is consistent with the state quantity of the time-delay system ( Figure 5 ), and ultimately did not reach a state of equilibrium convergence.

[0237] Reference Figure 4 , is a dual-gain parameter design method for the output feedback controller of a time-delay system considering the disturbance d(t) under actuator failure, Fig. Figure 4 As shown, the observer tracks the value The disturbance observer designed based on the common external disturbance structure has good tracking performance and shows high fault tolerance performance when considering the existence of actuator failure, which is further consistent with Figure 2 The error tracking performance of has a consistent characterization effect. Figure 7 , is the dual-gain parameter design method of the output feedback controller of the time-delay system without considering the disturbance d(t) under actuator fault, Fig. 2 also shows that the algorithm designed for actuator failure has poor anti-interference performance and cannot meet the stability requirements.

[0238] From the above images, we can see that: Figure 3 , Figure 4It can be seen that the reliable controller based on the disturbance observer with output feedback can track the disturbance very well, and its dynamic performance response speed is better than the running speed of the system itself, so that the disturbance estimation error converges to zero. Figure 2 It is shown that the system state also tends to be asymptotically stable, and the simulation results have achieved the expected effect of the system design strategy. Figure 5 , Figure 6 , Figure 7 It can be seen that the system state and disturbance estimation errors have not converged, and the external disturbance cannot be accurately tracked. The control strategy of the system considering actuator failure is more reliable than that without considering it, which further improves the fault tolerance performance.

[0239] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solutions of the present invention may be modified or replaced by equivalents without departing from the spirit and scope of the technical solutions of the present invention, which should all be included in the scope of the claims of the present invention.

Claims

1. A dual-gain parameter design method based on an output feedback controller for a time-delay system, characterized in that: Based on the acquired time-delay system model with uncertainty and external disturbance, obtaining the constraint conditions of the time-delay system model; Based on common external disturbances of the control system, a disturbance observer is constructed, wherein the disturbance observer includes a disturbance observer gain matrix to be solved; Based on the disturbance observer, an output feedback controller is constructed, wherein the output feedback controller includes an output feedback controller gain matrix to be solved; Based on the time-delay system model, the disturbance observer and the output feedback controller, a closed-loop system is obtained; Defining a disturbance estimation error and converging it so that the disturbance estimation error and the closed-loop system are asymptotically stable; According to the sufficient conditions for the asymptotic stability of the closed-loop system, the value ranges of the disturbance observer gain matrix and the output feedback controller gain matrix are obtained; The time-delay system model is: Where x(t)∈R n×1 is the state variable of the time-delay system; is the first-order differential of the state variable of the time-delay system; x0(t) is the initial state of the time-delay system; φ(t) is the initial value of the system state of the time-delay system; τ(t) is the time delay time of the time-delay system; u(t)∈R m×1 is the input of the time-delay system controller; d(t)∈R m×1 is the external disturbance input to the time-delay system; y(t)∈R n×1 is the output variable of the time-delay system; A∈R n×n , B∈R n×m , A d ∈R n×n and C∈R n×n is the system matrix of the time-delay system; ΔA(t), ΔA d (t) is the uncertainty of the time-delay system; I is the unit matrix; n and m are the dimensions of the matrix; The constraints of the time-delay system model are: 0≤τ(t)≤τ M(t) and [ΔA(t)ΔA d (t)]=DF(t)[E1 E2] rank(B)=m Among them, τ M(t) is the upper bound of the delay time in the time-delay system; is the time-delay time change rate in the time-delay system; h is the upper bound of the time-delay time change rate in the time-delay system; E1 and E2 are constant matrices, and are the structural uncertainties in the time-delay system. Satisfy F(t) T F(t)≤I, F(t) is the unknown real variable delay matrix; D is the parameter gain matrix; When considering the actuator failure, the corresponding time-delay system model is: Among them, ρ=diag(ρ1,ρ2,ρ3,······,ρ m );ρ k is the failure parameter of the actuator, which satisfies ρ k and are the upper and lower bounds of the diagonal parameters of the matrix ρ, ρ k =diag( ρ 1, ρ 2,······, ρ m ), ;gather N ρ Contains up to 2 m elements; The common external disturbances are: Where ω(t)∈R n×1 is the time-delay system frequency; is the first-order differential of ω(t); W and V are disturbance gain matrices; W = 0, V = 1 are constant load disturbances of unknown values; V = [10] is the harmonic disturbance of unknown amplitude and phase; ω is the tuning gain; The disturbance observer is: Where L∈R n×n is the disturbance observer gain matrix; is the disturbance estimate; is the frequency estimation value of the time-delay system; v(t) is the internal state of the disturbance observer, is its first-order differential; C is the output matrix, B is the input matrix; The output feedback controller is: Wherein, Ky(t) is the linear expression of the output of the time-delay system; K is the output feedback controller gain matrix; The disturbance estimation error is: Wherein, e(t) is the disturbance estimation error; The method for converging the disturbance estimation error includes derivatizing the disturbance estimation error: Right now: Wherein, when the output feedback controller realizes state quantity stabilization control, x(t)→0, x(t-τ(t))→0; when W+LCB(I-ρ)V<0, the exponential stability of the disturbance estimation error e(t) is achieved; The sufficient condition for the closed-loop system to be asymptotically stable is: There exists a positive constant α i , i∈(1,2,3,4) and the positive definite matrix P1∈R n×n , P2∈R n×n , R∈R n×n And the matrix Q1∈R n×n , Q2∈R n×n , Q3∈R n×n , Q4∈R n×n So that the following first inequality holds: Among them, G 11 =He(P1(A+B(I-ρ)KC))+(α1+α2)PDD T P+(α1 -1 +a3 -1 )E1 T E1+P1-Q1-Q1 T ;C 12 =P1B(I-ρ)V+(P2LCA-P2LCBρKC) T -Q4;C 13 =P1A d -Q2+Q1 T ;C 14 =Q1 T -Q3; Γ 22 =He(P2(W+LCB(I-ρ)V))+(α3+α4)P2LCDD T C T L T P2;Γ 23 =P2LCA d +Q4 T Γ 24 =Q4 T ; Γ 34 =Q2 T +Q3;Γ 44 =Q3 T +Q3; He is the symmetrization operation; P is the matrix parameter; A is the state matrix; A d is the delay state matrix; B is the input matrix; C is the output matrix; Fisher's lemma is used to decouple the first inequality and transform it into the second inequality: Among them, L 11 =He(P1A+b1B(I-ρ)NC)+(α1 -1 +a3 -1 )E1 T E1+R-He(Q1); L 12 =P1B(I-ρ)V+(MCA-b2BρNC) T -Q4;L 13 =P1A d -Q2+Q1 T ;L 14 =Q1 T -Q3; L 15 =P1B(I-ρ)V-b1B(I-ρ)Z+μ(NC) T ;L 16 =P1D;L 22 =He(P2W+MCB(I-ρ)V); L 23 =MCA d +Q4 T ;L 24 =Q4 T ;L 25 =b2BρZ-MCBρ;Λ 27 =MCD; L 34 =Q3+Q2 T ;L 44 =He(Q3);L 55 =-μHe(Z); L 66 =-(α1+α2) -1 I;L 77 =-(α3+α4) -1 I; For a given positive constant b1, b2, μ, α i , i∈{1,2,3,4}, if there exist positive definite matrices P1>0, P2>0, R>0 and matrices Q1, Q2, Q3, Q4 and M, N, Z such that the second inequality holds, then the closed-loop system and the disturbance observer error are both asymptotically stable; Among them, the disturbance observer gain matrix K=Z is selected -1 N, the output feedback controller gain matrix L = P2 -1 M.

2. An electronic device, characterized in that: The invention comprises a memory, a processor and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, the steps of the dual-gain parameter design method based on the time-delay system output feedback controller as claimed in claim 1 are implemented.

3. A computer-readable storage medium, characterized in that: It stores computer executable instructions, which, when executed by a processor, implement the steps of the dual-gain parameter design method based on the time-delay system output feedback controller as described in claim 1.

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