CN109991849A - A kind of time lag LPV system has memory H ∞ output feedback controller design method - Google Patents
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Abstract
The invention discloses a kind of time lag LPV systems memory H∞Output feedback controller design method, comprising the following steps: the vibration control system of milling machine milling process is abstracted as time lag LPV model first, memory H has been obtained by model conversion∞The canonical form of output feedback controller Solve problems；Secondly, introduce relaxation matrix variable and secondary Lyapunov functional, there is memory H for meet expected performance index∞Output feedback ontrol problem is converted into the problem of convex optimization based on linear matrix inequality；Then, a kind of method for selecting new convex optimization provides the parametrization linear matrix inequality of finite dimension in the apex of given more born of the same parents LPV systems；Finally, having obtained memory H by the linear matrix inequality∞Output feedback controller K.By using method provided by the present invention, can design has memory H with interference attenuation, robust stability∞Output feedback controller makes cutter have good dynamic property always during the cutting process.
Description
Technical Field
The invention relates to the field of vibration control in the milling process of a milling machine, in particular to a time-lag LPV system with memory H∞An output feedback controller design method.
Background
In the machining process, the precision and the surface roughness of a machined workpiece, the service life of a cutter and a machine tool, the machining period and the like are all influenced by the vibration of the cutter, so the vibration control of the machining becomes an important problem in the machining process;
the LPV theory was first proposed by Shamma in 1988, and its main objective was to extend the existing linear control design basis to non-linear and time-varying systems; in the prior art, a vibration control system in a milling process of a milling machine is generally abstracted into a time-delay LPV system, and then a corresponding controller is designed for the time-delay LPV system so as to reduce the influence of vibration on a cutter and a workpiece;
however, existing controller design methods typically employ robust H∞State feedback technique capable of securing systemSatisfy H∞Performance indexes, but because state feedback requires acquiring state information of a controlled object, in many practical problems, considered state variables are a group of variables describing system internal information and often cannot be directly measured; even though the state of the system can be directly measured, considering the cost for implementing control, the reliability of the system and other factors, if the performance requirement of the closed-loop system can be met by the output feedback of the system, the control mode of the output feedback is more suitable to be selected; in addition, the existing memoryless state feedback controller cannot effectively control the influence of time lag on the system because the past state information of the system is not introduced, so that the cutting tool cannot accurately and stably work; therefore, the existing design method of the controller has many defects.
Disclosure of Invention
In view of the above, the present invention provides a time-lapse LPV system with memory H∞The design method of the output feedback controller can design the controller which does not depend on the state information measured in real time, has the advantages of interference attenuation, stable robustness and closed loop response meeting the requirements, so that the precision of the cut workpiece is higher, and the cut surface is smoother;
the invention provides the following technical scheme: time-lag LPV system with memory H∞An output feedback controller design method, the method comprising:
A. abstracting a vibration control system of a milling process of a milling machine into a time-lag LPV model, and obtaining a memory H of the time-lag LPV system through model conversion∞Outputting a standard form of the feedback controller solution problem;
B. the relaxation matrix variable and the secondary Lyapunov functional are introduced, so that the memory H meeting the expected performance index is realized∞The dynamic output feedback control problem is converted into a finite dimension convex optimization problem in a linear matrix inequality framework;
C. selecting a new convex optimization method, and giving a finite-dimension parameterized linear matrix inequality at the vertex of a given multi-cell LPV system;
D. solving the linear matrix inequality to obtain corresponding positive definite parameter dependent matrixes X4 and Y4; sequentially calculating to obtain memory H∞Gain A of dynamic output feedback controllerk,Bk,Ck,DkThereby determining that there is a memory H∞And outputting the feedback controller K.
Preferably, the step a includes:
considering a vibration control system in the milling process of the milling machine, abstracting the vibration control system into a state space model of a time-lag LPV system as follows:
wherein x (t) e RnIs a state variable, u (t) e RrFor control input, y (t) e RpIs the measurement output, z (t) e RrIs the modulated output, w (t) e RqFor disturbance input, τ>0 is a known time-lag constant, phi (p) is a given initial condition, assuming the system matrix is a function of the time-varying parameter theta (t); for convenience of description, the following terms theta, thetai(where i ═ 1,. cndot., s) is substituted for θ (t), θi(t)；
The LPV system described above was converted into a multicellular LPV model as follows:
with regard to the system (1),is the set of bounded convex polyhedral vertex systems in which the system is located, (A)i,A1i,B1i,B2i,C1i,C2i,Di,D1i,D2i) Watch (A)The ith vertex system of the system is shown, i is 1,2, …, N, and for the LPV system, a robust memorial H is designed by considering the introduction of a time lag term in a feedback control rate∞Dynamic output feedback controller K:
wherein: a. thek、Bk、Ck、DkIs a matrix of controller parameters, x, to be determinedk(t)∈RnIs the state of the controller, u (t) is the control input; substituting the controller K into the time-delay LPV system to obtain a closed-loop time-delay LPV system C:
wherein:
Ccl(θ)＝[C2(θ)+D(θ)Dk(θ)C1(θ) D(θ)Ck(θ)]
Ccl1(θ)＝[D(θ)Dk(θ)C1(θ) 0]，Dcl(θ)＝D2(θ)
thus, the LPV system design described above has a memory H∞The problem of the output feedback controller K can be summarized as: design of memory H∞And dynamically outputting the feedback controller (2) to ensure that the closed-loop system (3) meets the following indexes in all value ranges of the scheduling parameter theta:
(1) the closed loop system is internally stable;
(2) h ∞ performance index: for disturbance input signal w (t), a performance index gamma is given>0, closed loop transfer function T from disturbance input w (T) to controlled output z (T)wzH of(s)∞The norm ratio gamma is small, namely that:
preferably, consider that step B comprises:
lemma 1. for system (1), if there is a symmetric positive definite matrix P (θ), matrix Q satisfies the linear matrix inequality (4)
The closed loop system asymptotically stabilizes;
wherein,
2, for the system (1), if a symmetric positive definite matrix P (theta), a matrix Q and a given positive scalar gamma are existed, so that an inequality (5) is established, the system is asymptotically stable and meets the H-infinity performance index;
wherein,
from the above, it is understood that sufficient conditions for the closed-loop time-lag LPV system (3) to be asymptotically stable and satisfy the H ∞ performance index γ are that a symmetric positive definite matrix P (θ) exists, that the matrix Q and a given positive scalar γ satisfy the inequality (5), and that a symmetric block matrix of an appropriate dimension is usedThe inequality (5) is subjected to congruent transformation to obtain an inequality (6):
wherein:
the matrix inequality (6) can be written as:
according to lemma 3, for the arbitrary matrices M, N and the identity matrix I of the appropriate dimension, the following conditions are equivalent:
(1).
(2) there is a relaxation matrix G of appropriate dimensions such that the following holds
Therein, let matrix GT＝[G1(θ) 0 0 0 G2(θ)]Then the above formula can be converted into inequality (7)
Wherein:
applying Schur complement theorem to the matrix inequality (7) to obtain inference 2;
inference 2 for a closed-loop time-lapse LPV system (3), if there is a continuously differentiable symmetric positive definite matrix function P (theta), a symmetric positive definite matrix Q, a symmetric matrix G1(theta) and G2(θ) and a given positive scalar γ, satisfy LMI (8):
the system asymptotically stabilizes and satisfies H∞Performance index;
wherein:
theorem 1 for closed-loop time-lag multi-cell LPV system (3), if continuous differentiable symmetrical positive definite matrix function existsAnd an appropriate dimension matrix Q1,Q2,S1(θ),S2(θ)，X(θ)，Y(θ)，U(θ)，Andand a given positive scalar γ, satisfying LMI (9):
wherein:
Γ19＝-X(θ)-XT(θ)，
Γ20＝-I-U(θ),Γ21＝-YT(θ)-Y(θ)
the time-lag LPV system satisfying the formula (2) has a memory H∞The dynamic output feedback controller coefficient matrix can be obtained by equation (10), where matrix X4(theta) and Y4(theta) ofSolving by full rank decomposition;
wherein:
to obtain the above inequality (9), assume G1＝G2G is reversible and is denoted as W ═ G > 0-1And the matrices G and W are partitioned as:
defining:the following operational relationships can be readily obtained from the above definitions:
left-hand diag { Lambda over inequality (8)T(θ) I I I ΛT(θ) I ΛT(θ) }, right-multiplying the matrix diag { Λ (θ) iiΛ (θ) I Λ (θ) }, to obtain the inequality (13):
wherein:
Δ2＝ΛT(θ)GT(θ)Acl1(θ),Δ3＝ΛT(θ)GT(θ)Bcl(θ),
Δ8＝-ΛT(θ)G(θ)Λ(θ)-ΛT(θ)GT(θ)Λ(θ)
the following relationships can be derived from equations (11) and (12):
wherein:
Δ14＝C2(θ)Y(θ)+D(θ)Dk(θ)C1(θ)Y(θ)+D(θ)Ck(θ)Y4(θ)
the following variable substitutions are made:
equation (14) can be written as:
according to the above inference, formula (13) is converted to formula (9) in theorem 1; according to the formula (8), the closed-loop time-lag LPV system is asymptotically stable under the action of the memorized H-infinity dynamic output feedback controller, and simultaneously meets the H-infinity performance index.
Preferably, consider that step C comprises:
in order to reduce conservatism, a new convex optimization method is selected, and parameterized linear matrix inequalities with finite dimensions are given at the vertex of a given bounded multi-cell LPV system;
theorem 2 for a closed-loop time-lapse multi-cell LPV system (3), it is assumed that there is a given positive scalar γ and a symmetric positive matrixMatrix Q of appropriate dimension1，Q2，S1i，S2i，Xi，Yi，Ui， ΔijEquations (16) and (17) are satisfied:
wherein:
Γ20＝-I-Ui，Γ21＝-Yi T-Yi
the time-lag LPV system satisfying the formula (2) has a memory H ∞ dynamic output feedback controller coefficient matrix which can be obtained by the formula (10); wherein:
preferably, step D is considered to comprise:
if the above inequalities (16) and (17) have feasible solutions, the pairFull rank decomposition to obtain matrix X4(theta) and Y4(θ); further obtain the memory H∞Output feedback controller gain matrix:
wherein:
as seen from the above, in the design method of the time-lag LPV system with the memory H ∞ output feedback controller in the invention, for the time-lag LPV model of the vibration control system in the milling process of the milling machine, the problem of solving the memory H ∞ output feedback controller is converted into the problem of solving the linear matrix inequality, the parameter dependent matrices X4 and Y4 are obtained by solving the inequality, and the parameter matrix in the controller K is finally determined, so that the memory H ∞ output feedback controller can be designed;
compared with the prior art, the controller has the advantages that the controller does not depend on the state information measured in real time, has the characteristics of interference attenuation, stable robustness and closed-loop response meeting the requirements, and has good dynamic performance and robustness, so that the precision of a cut workpiece is higher, and the cut surface is smoother.
Drawings
FIG. 1 shows a memory H of a time-lapse LPV system∞Outputting a flow schematic diagram of a design method of a feedback controller;
FIG. 2 is a graph of the position change curves and rate of change curves of two modules without disturbance under the action of a memorized H ∞ output feedback controller;
FIG. 3 is a graph of the position change curves and the change rate curves of two modules after disturbance is added under the action of a memorized H ∞ output feedback controller;
FIG. 4 is a graph of position change and rate of change for two modules without a controller under initial conditions;
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail below with reference to the accompanying drawings and examples;
in the invention, a time-lag LPV system with memory H is provided∞The design method of the output feedback controller is suitable for a time-lag LPV system for vibration control in the milling process of a milling machine;
FIG. 1 shows a time-lapse LPV system with memory H according to the present invention∞FIG. 1 shows a flow chart of a design method of an output feedback controller, in which a time-lag LPV system in an embodiment of the present invention has a memory of H∞The design method of the output feedback controller comprises the following steps:
step one, abstracting a vibration control system in the milling process of a milling machine into a multi-cell time-delay LPV model, and obtaining a standard form of a problem solved by an H-infinity output feedback controller with memory of a time-delay LPV system through model conversion;
in the technical scheme of the invention, the system can be abstracted into a time-lag LPV model;
for example, in a preferred embodiment of the present invention, the step one includes:
considering a vibration control system in the milling process of a milling machine, abstracting to a state space model of a time-delay LPV system:
wherein the system statex1,x2The position of the tool and the machine tool, τ, respectively>0 is a known time lag constant, phi (rho) is given initial conditions, and the system matrix and the time delay h (theta (t)) are assumed to be functions of a time-varying parameter theta (t); for convenience of description, the following terms theta, thetai(where i ═ 1,. cndot., s) is substituted for θ (t), θi(t)；
The LPV system described above was converted into a multicellular LPV model as follows:
with regard to the system (1),is the set of bounded convex polyhedral vertex systems in which the system is located, (A)i,A1i,B1i,B2i,C1i,C2i,Di,D1i,D2i) The ith vertex system, i ═ 1,2, …, N, representing the system, for the LPV system, a robust memorised H ∞ dynamic output feedback controller K was designed, considering the introduction of a lag term in the feedback control rate:
wherein: a. thek、Bk、Ck、DkIs a matrix of controller parameters, x, to be determinedk(t)∈RnIs the state of the controller, u (t) is the control input; substituting the controller K into the time-delay LPV system to obtain a closed-loop time-delay LPV system C:
wherein:
Ccl(θ)＝[C2(θ)+D(θ)Dk(θ)C1(θ) D(θ)Ck(θ)]
Ccl1(θ)＝[D(θ)Dk(θ)C1(θ) 0]，Dcl(θ)＝D2(θ)
therefore, the problem of designing the H ∞ output feedback controller K with memory as described above for LPV systems can be summarized as: a memory H-infinity dynamic output feedback controller (2) is designed, so that the closed-loop system (3) meets the following indexes in all value ranges of a scheduling parameter theta:
(1) the closed loop system is internally stable;
(2) h ∞ performance index: for disturbance input signal w (t), a performance index gamma is given>0, closed loop transfer function T from disturbance input w (T) to controlled output z (T)wzH of(s)∞The norm ratio gamma is small, namely that:
that is, if A is obtainedk,Bk,Ck,DkCan obtain the memory H∞A dynamic output feedback controller;
therefore, through the model conversion, the solved memory H can be obtained∞A standard form of dynamic output feedback controller.
Step two, introducing a relaxation matrix variable and a quadratic Lyapunov functional, and converting the problem of memorized H infinity robust dynamic output feedback control meeting the expected performance index into a problem of finite dimensional convex optimization in a linear matrix inequality framework;
in the technical scheme of the invention, in order to determine the existence of the memory H∞If the robust dynamic output feedback controller exists and is stable, the problem solved by the controller of the time-delay LPV system can be converted into a convex optimization problem for solving a linear matrix inequality;
for example, preferably, in the embodiment of the present invention, lemma 1 and lemma 2 can be introduced first:
lemma 1. for system (1), if there is a symmetric positive definite matrix P (θ), matrix Q satisfies the linear matrix inequality (4)
The closed loop system asymptotically stabilizes;
wherein,
2, for the system (1), if a symmetric positive definite matrix P (theta), a matrix Q and a given positive scalar gamma are existed, so that the linear matrix inequality (5) is established, the system is asymptotically stable and meets the H infinity performance index;
wherein,
from the above, it is understood that sufficient conditions for the closed-loop time-lag LPV system (3) to be asymptotically stable and satisfy the H ∞ performance index γ are that a symmetric positive definite matrix P (θ) exists, that the matrix Q and a given positive scalar γ satisfy the inequality (5), and that a symmetric block matrix of an appropriate dimension is usedThe inequality (5) is subjected to congruent transformation to obtain an inequality (6):
wherein:
the matrix inequality (6) can be written as:
according to lemma 3, for the arbitrary matrices M, N and the identity matrix I of the appropriate dimension, the following conditions are equivalent:
(1).
(2) there is a relaxation matrix G of appropriate dimensions such that the following holds
Therein, let matrix GT＝[G1(θ) 0 0 0 G2(θ)]Then the above formula can be converted into inequality (7)
Wherein:
inference 2 for a closed-loop time-lapse LPV system (3), if there is a continuously differentiable symmetric positive definite matrix function P (theta), a symmetric positive definite matrix Q, a symmetric matrix G1(theta) and G2(θ) and a given positive scalar γ, satisfy LMI (8):
the system asymptotically stabilizes and satisfies H∞Performance index;
wherein:
theorem 1 for closed-loop time-lag multi-cell LPV system (3), if continuous differentiable symmetrical positive definite matrix function existsAnd an appropriate dimension matrix Q1,Q2,S1(θ),S2(θ),X(θ),Y(θ),U(θ), Andand a given positive scalar γ, satisfying LMI (9):
wherein:
Γ19＝-X(θ)-XT(θ)，
Γ20＝-I-U(θ)，Γ21＝-YT(θ)-Y(θ)
the time-lag LPV system satisfying the formula (2) has a memory H ∞ dynamic output feedback controller coefficient matrix which can be obtained by the following formula, wherein the matrix X4And Y4FromSolving by full rank decomposition;
wherein:
to obtain the above inequality (9), assume G1＝G2G is reversible and is denoted as W ═ G > 0-1And the matrices G and W are partitioned as:
defining:
the following operational relationships can be readily obtained from the above definitions:
left-hand diag { Lambda over inequality (8)T(θ) I I I ΛT(θ) I ΛT(theta), right-multiplying the matrix diag { Λ (theta) IILambda (theta) ILambda (theta) }, to obtain a linear matrix inequality (13):
wherein:
Δ2＝ΛT(θ)GT(θ)Acl1(θ),Δ3＝ΛT(θ)GT(θ)Bcl(θ),
Δ8＝-ΛT(θ)G(θ)Λ(θ)-ΛT(θ)GT(θ)Λ(θ)
the following relationships can be derived from equations (11) and (12):
the following variable substitutions are made:
wherein:
Δ14＝C2(θ)Y(θ)+D(θ)Dk(θ)C1(θ)Y(θ)+D(θ)Ck(θ)Y4(θ)
equation (14) can be written as:
wherein:
according to the above inference, formula (13) is converted to formula (9) in theorem 1; according to the formula (8), the closed-loop time-lag LPV system has the advantages that the parameters are secondarily stabilized under the action of the memorized H-infinity dynamic output feedback controller, and the H-infinity performance index is simultaneously met;
therefore, the problem solved by the controller of the time-lag LPV system can be converted into a convex optimization problem for solving a linear matrix inequality; when the linear matrix inequality has a solution, a memory H ∞ output feedback controller exists and is stable.
Selecting a new convex optimization method, and giving a parameterized linear matrix inequality with finite dimension at the vertex of a given multi-cell LPV system;
in the embodiment of the present invention, the third step may be implemented by using various specific implementations, and one implementation of the third step will be taken as an example to describe in detail the technical solution of the present invention;
for example, in a preferred embodiment of the present invention, the third step includes:
in order to reduce conservatism, a new convex optimization method is selected, and parameterized linear matrix inequalities with finite dimensions are given at the vertex of a given bounded multi-cell LPV system;
theorem 2 for a closed-loop time-lapse multi-cell LPV system (3), it is assumed that there is a given positive scalar γ and a symmetric positive matrixMatrix Q of appropriate dimension1，Q2，S1i，S2i，Xi，Yi，Ui， ΔijEquations (16) and (17) are satisfied:
wherein:
Γ20＝-I-Ui，Γ21＝-Yi T-Yi
the time-lag LPV system satisfying the formula (2) has a memory H ∞ dynamic output feedback controller coefficient matrix which can be obtained by the formula (9);
wherein:
when the linear matrix inequality in the inequality (16) and the inequality (17) has a solution, a memory H infinity dynamic output feedback controller exists.
Solving the linear matrix inequality to obtain corresponding positive definite parameter dependent matrixes X4 and Y4; sequentially calculating gain A of feedback controller with memory H infinity dynamic outputk,Bk,Ck,DkThereby obtaining a feedback controller K with memory H infinity output;
in the technical solution of the present invention, the step four can be implemented by using various specific embodiments; the technical scheme of the invention will be described in detail below by taking one implementation manner thereof as an example;
for example, in a preferred embodiment of the present invention, the fourth step includes:
if the inequalities (16) and (17) are feasible, the memory H-infinity output feedback controller exists and is stable, and the LMI tool kit in the MATLAB judges whether the existence condition of the controller is satisfied, so that the corresponding positive definite parameter dependent matrixes X, Y and U can be obtained by solving the linear matrix inequalities;
to pairFull rank decomposition to obtain matrix X4(theta) and Y4(θ); memory H can be determined using the formula described below∞Parameters in the output feedback controller K:
wherein:
in summary, in the design of the time-lag LPV system with the memory H-infinity output feedback controller, aiming at a time-lag LPV model of a vibration control system in the milling process of a milling machine, the problem of solving the memory H-infinity output feedback controller is converted into the problem of solving a linear matrix inequality, parameter dependent matrixes X4 and Y4 are obtained through solving the inequality, and a parameter matrix in a controller K is finally determined, so that the memory H-infinity output feedback controller can be designed, the stability of the system can be ensured, the H-infinity performance index can be met, and the dynamic performance and the robustness are good; to further illustrate the superiority of the present invention, relevant simulation data are provided: h-infinity performance index γ 1.3587 and a parameter dependent memorized H-infinity output feedback controller parameter matrix:
CK1＝[3.5526 1.3313 8.2361 9.9666],DK1＝-4.6587
CK2＝[12.71 -49.11 1.667 30.4025],DK2＝-4.1305
in addition, FIG. 2 is a graph of the position change curves and the change rate curves of two modules without disturbance under the action of the memorized H ∞ output feedback controller; FIG. 3 is a graph of the position change curves and the change rate curves of two modules after disturbance is added under the action of a memorized H ∞ output feedback controller; comparing fig. 2 and fig. 3, it is found that the H ∞ output feedback controller with memory can effectively reduce the influence of disturbance on the system, and improve the cutting accuracy and surface smoothness of the workpiece; by way of comparison, FIG. 4 further demonstrates the effectiveness of the feedback controller with memory output of the present design;
in addition, because the method has certain universality, a memory H-infinity output feedback controller can be designed for all practical physical systems which can be abstracted into a time-lag LPV model by using the method so as to achieve a good control effect;
it will be evident to those skilled in the art that the invention is not limited to the details of the foregoing illustrative embodiments, and that the present invention may be embodied in other specific forms without departing from the spirit or essential attributes thereof; the present embodiments are therefore to be considered in all respects as illustrative and not restrictive, the scope of the invention being indicated by the appended claims rather than by the foregoing description, and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein, and any reference signs in the claims are not intended to be construed as limiting the claim concerned;
furthermore, it should be understood that although the present description refers to embodiments, not every embodiment may contain only a single embodiment, and such description is for clarity only, and those skilled in the art should integrate the description, and the embodiments may be combined as appropriate to form other embodiments understood by those skilled in the art.
Claims (5)
1. Time-lag LPV system with memory H∞A method of designing an output feedback controller, the method comprising:
A. abstracting a vibration control system of a milling process of a milling machine into a time-lag multi-cell LPV model, and obtaining the memory H of the time-lag LPV system through model conversion∞Outputting a standard form of the feedback controller solution problem;
B. the relaxation matrix variable and the secondary Lyapunov functional are introduced, so that the memory H meeting the expected performance index is realized∞Conversion of dynamic output feedback control problem into linear matrixThe problem of finite dimension convex optimization within the framework of equations;
C. selecting a new convex optimization method, and giving a finite-dimension parameterized linear matrix inequality at the vertex of a given multi-cell LPV system;
D. solving the linear matrix inequality to obtain corresponding positive definite parameter dependent matrixes X4 and Y4; sequentially calculating to obtain memory H∞Gain A of dynamic output feedback controllerk,Bk,Ck,DkThereby determining that there is a memory H∞And outputting the feedback controller K.
2. The design method of time-lag LPV system with memory H ∞ output feedback controller of claim 1, wherein said step A includes:
considering a vibration control system in the milling process of a milling machine, abstracting to a state space model of a time-delay LPV system:
wherein x (t) e RnIs a state variable, u (t) e RrFor control input, y (t) e RpIs the measurement output, z (t) e RrIs the modulated output, w (t) e RqFor disturbance input, τ>0 is a known time lag constant, phi (rho) is given initial conditions, and the system matrix and the time delay h (theta (t)) are assumed to be functions of a time-varying parameter theta (t); for convenience of description, the following terms theta, thetai(where i ═ 1,. cndot., s) is substituted for θ (t), θi(t)；
The LPV system described above was converted into a multicellular LPV model as follows:
with regard to the system (1),is the set of bounded convex polyhedral vertex systems in which the system is located, (A)i,A1i,B1i,B2i,C1i,C2i,Di,D1i,D2i) An ith vertex system of the system is shown, i is 1,2, …, N, and for an LPV system, a robust memorial H is designed by considering the introduction of a time lag term in a feedback control rate∞Dynamic output feedback controller K:
wherein: a. thek、Bk、Ck、DkIs a matrix of controller parameters, x, to be determinedk(t)∈RnIs the state of the controller, u (t) is the control input; substituting the controller K into the time-delay LPV system to obtain a closed-loop time-delay LPV system:
wherein:
Ccl(θ)＝[C2(θ)+D(θ)Dk(θ)C1(θ) D(θ)Ck(θ)]
Ccl1(θ)＝[D(θ)Dk(θ)C1(θ) 0]，Dcl(θ)＝D2(θ)
thus, the LPV system design described above has a memory H∞The problem of the output feedback controller K can be summarized as: design of memory H∞And dynamically outputting the feedback controller (2) to ensure that the closed-loop system (3) meets the following indexes in all value ranges of the scheduling parameter theta:
(1) the closed loop system is internally stable;
(2) h ∞ performance index: for disturbance input signal w (t), a performance index gamma is given>0, closed loop transfer function T from disturbance input w (T) to controlled output z (T)wzH of(s)∞The norm ratio gamma is small, namely that:
3. the design method of time-lag LPV system with memory H ∞ output feedback controller of claim 1, wherein said step B comprises:
lemma 1. for system (1), if there is a symmetric positive definite matrix P (θ), matrix Q satisfies the linear matrix inequality (4)
The closed loop system asymptotically stabilizes;
wherein,
2, for the system (1), if a symmetric positive definite matrix P (theta), a matrix Q and a given positive scalar gamma are existed, so that an inequality (5) is established, the system is asymptotically stable and meets the H-infinity performance index;
wherein,
from the above, it is understood that sufficient conditions for the closed-loop time-lag LPV system (3) to be asymptotically stable and satisfy the H ∞ performance index γ are that a symmetric positive definite matrix P (θ) exists, that the matrix Q and a given positive scalar γ satisfy the inequality (5), and that a symmetric block matrix of an appropriate dimension is usedThe inequality (5) is subjected to congruent transformation to obtain an inequality (6):
wherein:
the matrix inequality (6) can be written as:
according to lemma 3, for the arbitrary matrices M, N and the identity matrix I of the appropriate dimension, the following conditions are equivalent:
(1).
(2) there is a relaxation matrix G of appropriate dimensions such that the following holdsTherein, let matrix GT＝[G1(θ) 0 0 0 G2(θ)]Then the above formula can be converted into inequality (7)
Wherein:
inference 2 for a closed-loop time-lapse LPV system (3), if there is a continuously differentiable symmetric positive definite matrix function P (theta), a symmetric positive definite matrix Q, a symmetric matrix G1(theta) and G2(θ) and a given positive scalar γ, satisfy LMI (8):
the system asymptotically stabilizes and satisfies H∞Performance index;
wherein:
theorem 1 for closed-loop time-lag multi-cell LPV system (3), if continuous differentiable symmetrical positive definite matrix function existsAnd an appropriate dimension matrix Q1,Q2,S1(θ),S2(θ),X(θ),Y(θ),U(θ), Andand a given positive scalar γ, satisfying LMI (9):
wherein:
Γ20＝-I-U(θ),Γ21＝-YT(θ)-Y(θ)
then the formula (A) is satisfied2) The time-lag LPV system with memory H infinity dynamic output feedback controller coefficient matrix can be obtained by the following formula, wherein the matrix X4And Y4FromSolving by full rank decomposition;
wherein:
wherein, to obtain the above inequality (9), assume G1＝G2G is reversible and is denoted as W ═ G > 0-1，
And the matrices G and W are denoted in blocks as:
defining:
the following operational relationships can be readily obtained from the above definitions:
left-hand diag { Lambda over inequality (8)T(θ) I I I ΛT(θ) I ΛT(theta), multiplying diag { Λ (theta) IILambda (theta) ILambda (theta) }rightwardto obtain a matrix inequality (13):
wherein:
Δ2＝ΛT(θ)GT(θ)Acl1(θ),Δ3＝ΛT(θ)GT(θ)Bcl(θ),
Δ8＝-ΛT(θ)G(θ)Λ(θ)-ΛT(θ)GT(θ)Λ(θ)
the following relationships can be derived from equations (11) and (12):
wherein:
Δ14＝C2(θ)Y(θ)+D(θ)Dk(θ)C1(θ)Y(θ)+D(θ)Ck(θ)Y4(θ)
the following variable substitutions are made:
equation (14) can be written as:
according to the above inference, formula (13) is converted to formula (9) in theorem 1; according to the formula (8), the closed-loop time-lag LPV system is asymptotically stable under the action of the memorized H-infinity dynamic output feedback controller, and simultaneously meets the H-infinity performance index.
4. The time-lag LPV system with memory H ∞ output feedback controller design method of claim 1, wherein said step C comprises:
in order to reduce conservatism, a new convex optimization method is selected, and parameterized linear matrix inequalities with finite dimensions are given at the vertex of a given bounded multi-cell LPV system;
theorem 2 for a closed-loop time-lapse multi-cell LPV system (3), it is assumed that there is a given positive scalar γ and a symmetric positive matrixMatrix Q of appropriate dimension1，Q2，S1i，S2i，Xi，Yi，Ui， ΔijEquations (16) and (17) are satisfied:
wherein:
the time-lag LPV system satisfying the formula (2) has a memory H ∞ dynamic output feedback controller coefficient matrix which can be obtained by the formula (10);
wherein:
5. the lag LPV system with memory H ∞ output feedback controller design method of claim 1, wherein said step D comprises:
if the above inequalities (16) and (17) have feasible solutions, the pairFull rank decomposition to obtain matrix X4(theta) and Y4(θ); further obtain the memory H∞Output feedback controller gain matrix:
wherein:
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