CN109991849B - Design method of feedback controller with memory H-infinity output of time-lag LPV system - Google Patents

Design method of feedback controller with memory H-infinity output of time-lag LPV system Download PDF

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CN109991849B
CN109991849B CN201910269377.XA CN201910269377A CN109991849B CN 109991849 B CN109991849 B CN 109991849B CN 201910269377 A CN201910269377 A CN 201910269377A CN 109991849 B CN109991849 B CN 109991849B
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黄金杰
潘晓真
郝现志
何瑾洁
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Harbin University of Science and Technology
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Abstract

The invention discloses a time-lag LPV system with memory H∞The design method of the output feedback controller comprises the following steps: firstly, abstracting a vibration control system of a milling process of a milling machine into a time-lag LPV model, and obtaining a memory H through model conversion∞Outputting a standard form of the feedback controller solution problem; secondly, a relaxation matrix variable and a secondary Lyapunov functional are introduced, so that the memorized H meeting the expected performance index∞The output feedback control problem is converted into a convex optimization problem based on a linear matrix inequality; then, 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; finally, obtaining the memorized H through the linear matrix inequality∞And outputting the feedback controller K. By using the method provided by the invention, the memory H with interference attenuation and stable robustness can be designed∞And a feedback controller is output, so that the cutter has good dynamic performance all the time in the cutting process.

Description

Design method of feedback controller with memory H-infinity output of time-lag LPV system
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, the design method of the existing controller generally adopts robust H∞State feedback technique that can guarantee the system to satisfy 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; sometimes even if the state of the system is directly measurable, it is more appropriate to choose the output feedback of the system to meet the performance requirements of the closed loop system, considering the cost of implementing the control and the reliability of the systemOutputting a feedback control mode; 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∞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 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 a state space model of the following time delay LPV system:
Figure BDA0002016653660000021
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:
Figure BDA0002016653660000022
with regard to the system (1),
Figure BDA0002016653660000023
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:
Figure BDA0002016653660000031
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:
Figure BDA0002016653660000032
wherein:
Figure BDA0002016653660000033
Figure BDA0002016653660000034
Figure BDA0002016653660000035
Figure BDA0002016653660000036
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:
Figure BDA0002016653660000037
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)
Figure BDA0002016653660000041
The closed loop system asymptotically stabilizes;
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002016653660000042
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;
Figure BDA0002016653660000043
wherein the content of the first and second substances,
Figure BDA0002016653660000044
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 used
Figure BDA0002016653660000045
The inequality (5) is subjected to congruent transformation to obtain an inequality (6):
Figure BDA0002016653660000046
wherein:
Figure BDA0002016653660000047
the matrix inequality (6) can be written as:
Figure BDA0002016653660000048
Figure BDA0002016653660000051
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).
Figure BDA0002016653660000052
(2) there is a relaxation matrix G of appropriate dimensions such that the following holds
Figure BDA0002016653660000053
Therein, let matrix GT＝[G1(θ) 0 0 0 G2(θ)]Then the above formula can be converted into inequality (7)
Figure BDA0002016653660000054
Wherein:
Figure BDA0002016653660000055
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):
Figure BDA0002016653660000056
the system becomes asymptotically stable and fullFoot H∞Performance index;
wherein:
Figure BDA0002016653660000057
theorem 1 for closed-loop time-lag multi-cell LPV system (3), if continuous differentiable symmetrical positive definite matrix function exists
Figure BDA0002016653660000061
And an appropriate dimension matrix Q1,Q2,S1(θ),S2(θ)，X(θ)，Y(θ)，U(θ)，
Figure BDA0002016653660000062
And
Figure BDA0002016653660000063
and a given positive scalar γ, satisfying LMI (9):
Figure BDA0002016653660000064
wherein:
Figure BDA0002016653660000065
Figure BDA0002016653660000066
Figure BDA0002016653660000067
Figure BDA0002016653660000068
Figure BDA0002016653660000069
Figure BDA00020166536600000610
Figure BDA00020166536600000611
Figure BDA00020166536600000612
Figure BDA00020166536600000613
Figure BDA00020166536600000614
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) of
Figure BDA00020166536600000615
Solving by full rank decomposition;
Figure BDA0002016653660000071
Figure BDA0002016653660000072
wherein:
Figure BDA0002016653660000073
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:
Figure BDA0002016653660000074
defining:
Figure BDA0002016653660000075
the following operational relationships can be readily obtained from the above definitions:
Figure BDA0002016653660000076
left-hand diag { Lambda over inequality (8)T(θ) I I I ΛT(θ) I ΛT(θ) } right-multiplying the matrix diag { Λ (θ) ii Λ (θ) I Λ (θ) } to obtain an inequality (13):
Figure BDA0002016653660000077
wherein:
Figure BDA0002016653660000078
Δ2＝ΛT(θ)GT(θ)Acl1(θ),Δ3＝ΛT(θ)GT(θ)Bcl(θ),
Figure BDA0002016653660000079
Figure BDA00020166536600000710
Δ8＝-ΛT(θ)G(θ)Λ(θ)-ΛT(θ)GT(θ)Λ(θ)
the following relationships can be derived from equations (11) and (12):
Figure BDA00020166536600000711
Figure BDA0002016653660000081
wherein:
Figure BDA0002016653660000082
Figure BDA0002016653660000083
Figure BDA0002016653660000084
Figure BDA0002016653660000085
Figure BDA00020166536600000815
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:
Figure BDA0002016653660000086
Figure BDA0002016653660000087
Figure BDA0002016653660000088
Figure BDA0002016653660000089
Figure BDA00020166536600000810
equation (14) can be written as:
Figure BDA00020166536600000811
Figure BDA00020166536600000812
Figure BDA00020166536600000813
Figure BDA00020166536600000814
Figure BDA0002016653660000091
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 memory 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 matrix
Figure BDA0002016653660000092
Matrix Q of appropriate dimension1，Q2，S1i，S2i，Xi，Yi，Ui，
Figure BDA0002016653660000093
Figure BDA0002016653660000094
ΔijEquations (16) and (17) are satisfied:
Figure BDA0002016653660000095
Figure BDA0002016653660000096
Figure BDA0002016653660000097
wherein:
Figure BDA0002016653660000098
Figure BDA0002016653660000099
Figure BDA00020166536600000910
Figure BDA0002016653660000101
Figure BDA0002016653660000102
Figure BDA0002016653660000103
Figure BDA0002016653660000104
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:
Figure BDA0002016653660000105
Figure BDA0002016653660000106
preferably, step D is considered to comprise:
if the above inequalities (16) and (17) have feasible solutions, the pair
Figure BDA0002016653660000107
Full rank decomposition to obtain matrix X4(theta) and Y4(θ); further obtain the memory H∞Output feedback controller gain matrix:
Figure BDA0002016653660000108
Figure BDA0002016653660000109
wherein:
Figure BDA00020166536600001010
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 present invention, aThe time-lag LPV system has memory H∞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:
Figure BDA0002016653660000111
wherein the system state
Figure BDA0002016653660000112
x1,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:
Figure BDA0002016653660000113
Figure BDA0002016653660000121
with regard to the system (1),
Figure BDA0002016653660000122
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:
Figure BDA0002016653660000123
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:
Figure BDA0002016653660000124
wherein:
Figure BDA0002016653660000125
Figure BDA0002016653660000126
Figure BDA0002016653660000127
Figure BDA0002016653660000128
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:
Figure BDA0002016653660000131
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-lag LPV system can be converted into a convex optimization problem for solving a linear matrix inequalityTitle to be obtained;
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)
Figure BDA0002016653660000132
The closed loop system asymptotically stabilizes;
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002016653660000133
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;
Figure BDA0002016653660000134
wherein the content of the first and second substances,
Figure BDA0002016653660000135
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 used
Figure BDA0002016653660000141
The inequality (5) is subjected to congruent transformation to obtain an inequality (6):
Figure BDA0002016653660000142
wherein:
Figure BDA0002016653660000143
the matrix inequality (6) can be written as:
Figure BDA0002016653660000144
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).
Figure BDA0002016653660000145
(2) there is a relaxation matrix G of appropriate dimensions such that the following holds
Figure BDA0002016653660000146
Therein, let matrix GT＝[G1(θ) 0 0 0 G2(θ)]Then the above formula can be converted into inequality (7)
Figure BDA0002016653660000147
Wherein:
Figure BDA0002016653660000151
inference 2 for closed-loop skew LPV systems (3), symmetric positive definite matrix function P (θ), symmetric positive definite matrix Q, symmetric matrix G if there is a continuous differentiable1(theta) and G2(θ) and a given positive scalar γ, satisfy LMI (8):
Figure BDA0002016653660000152
the system asymptotically stabilizes and satisfies H∞Performance index;
wherein:
Figure BDA0002016653660000153
theorem 1 for closed-loop time-lag multi-cell LPV system (3), if continuous differentiable symmetrical positive definite matrix function exists
Figure BDA0002016653660000154
And an appropriate dimension matrix Q1,Q2,S1(θ),S2(θ),X(θ),Y(θ),U(θ),
Figure BDA0002016653660000155
Figure BDA0002016653660000156
And
Figure BDA0002016653660000157
and a given positive scalar γ, satisfying LMI (9):
Figure BDA0002016653660000158
wherein:
Figure BDA0002016653660000161
Figure BDA0002016653660000162
Figure BDA0002016653660000163
Figure BDA0002016653660000164
Figure BDA0002016653660000165
Figure BDA0002016653660000166
Figure BDA0002016653660000167
Figure BDA0002016653660000168
Figure BDA0002016653660000169
Figure BDA00020166536600001610
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 Y4From
Figure BDA00020166536600001612
Solving by full rank decomposition;
Figure BDA00020166536600001613
Figure BDA00020166536600001614
wherein:
Figure BDA00020166536600001615
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:
Figure BDA00020166536600001616
defining:
Figure BDA00020166536600001617
the following operational relationships can be readily obtained from the above definitions:
Figure BDA00020166536600001618
left-hand diag { Lambda over inequality (8)T(θ) I I I ΛT(θ) I ΛT(theta), right-multiplying the matrix diag { Λ (theta) IIILambda (theta) ILambda (theta) }, to obtain a linear matrix inequality (13):
Figure BDA0002016653660000171
wherein:
Figure BDA0002016653660000172
Δ2＝ΛT(θ)GT(θ)Acl1(θ),Δ3＝ΛT(θ)GT(θ)Bcl(θ),
Figure BDA0002016653660000173
Figure BDA0002016653660000174
Δ8＝-ΛT(θ)G(θ)Λ(θ)-ΛT(θ)GT(θ)Λ(θ)
the following relationships can be derived from equations (11) and (12):
Figure BDA0002016653660000175
the following variable substitutions are made:
Figure BDA0002016653660000176
Figure BDA0002016653660000177
Figure BDA0002016653660000178
Figure BDA0002016653660000179
wherein:
Figure BDA0002016653660000181
Figure BDA0002016653660000182
Figure BDA0002016653660000183
Figure BDA0002016653660000184
Figure BDA0002016653660000185
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:
Figure BDA0002016653660000187
wherein:
Figure BDA0002016653660000188
Figure BDA0002016653660000189
Figure BDA00020166536600001810
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 parameters are secondarily stabilized under the action of a memory 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 step three 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 matrix
Figure BDA0002016653660000191
Matrix Q of appropriate dimension1，Q2，S1i，S2i，Xi，Yi，Ui，
Figure BDA0002016653660000192
Figure BDA0002016653660000193
ΔijEquations (16) and (17) are satisfied:
Figure BDA0002016653660000194
Figure BDA0002016653660000195
Figure BDA0002016653660000196
wherein:
Figure BDA0002016653660000197
Figure BDA0002016653660000198
Figure BDA0002016653660000199
Figure BDA00020166536600001910
Figure BDA00020166536600001911
Figure BDA00020166536600001912
Figure BDA00020166536600001913
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:
Figure BDA0002016653660000201
Figure BDA0002016653660000202
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 the 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 H-infinity output feedback controller exists and is stable, and the LMI toolbox in the MATLAB judges whether the existence condition of the controller is established, so that the corresponding positive definite parameter dependent matrixes X, Y and U can be obtained by solving the linear matrix inequalities;
to pair
Figure BDA0002016653660000203
Full 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:
Figure BDA0002016653660000204
Figure BDA0002016653660000205
wherein:
Figure BDA0002016653660000206
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:
Figure BDA0002016653660000211
CK1＝[3.5526 1.3313 8.2361 9.9666],DK1＝-4.6587
Figure BDA0002016653660000212
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 (3)

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∞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∞Outputting a feedback controller K;
the step A comprises the following steps:
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:
Figure FDA0003527791290000011
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, thetaiInstead of theta (t), thetai(t), wherein i ═ 1, ·, s;
the LPV system described above was converted into a multicellular LPV model as follows:
Figure FDA0003527791290000012
Figure FDA0003527791290000021
with regard to the system (1),
Figure FDA0003527791290000022
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:
Figure FDA0003527791290000023
wherein: a. thek、Bk、Ck、DkIs a matrix of controller parameters, x, to be determinedk(t)∈RnIs to controlState 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:
Figure FDA0003527791290000024
wherein:
Figure FDA0003527791290000025
Figure FDA0003527791290000026
Figure FDA0003527791290000027
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 with 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:
Figure FDA0003527791290000031
the step B comprises the following steps:
lemma 1. for system (1), if there is a symmetric positive definite matrix P (θ), matrix Q satisfies the linear matrix inequality (4)
Figure FDA0003527791290000032
The closed loop system asymptotically stabilizes;
wherein the content of the first and second substances,
Figure FDA0003527791290000033
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;
Figure FDA0003527791290000034
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003527791290000035
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 used
Figure FDA0003527791290000036
The inequality (5) is subjected to congruent transformation to obtain an inequality (6):
Figure FDA0003527791290000037
wherein:
Figure FDA0003527791290000038
the matrix inequality (6) can be written as:
Figure FDA0003527791290000041
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).
Figure FDA0003527791290000042
(2) there is a relaxation matrix G of appropriate dimensions such that the following holds
Figure FDA0003527791290000043
Therein, let matrix GT＝[G1(θ) 0 0 0 G2(θ)]Then the above formula can be converted into inequality (7)
Figure FDA0003527791290000044
Wherein:
Figure FDA0003527791290000045
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):
Figure FDA0003527791290000046
Figure FDA0003527791290000051
the system asymptotically stabilizes and satisfies H∞Performance index;
wherein:
Figure FDA0003527791290000052
theorem 1 for closed-loop time-lag multi-cell LPV system (3), if continuous differentiable symmetrical positive definite matrix function exists
Figure FDA0003527791290000053
And an appropriate dimension matrix Q1,Q2,S1(θ),S2(θ),X(θ),Y(θ),U(θ),
Figure FDA0003527791290000054
And
Figure FDA0003527791290000055
and a given positive scalar γ, satisfying LMI (9):
Figure FDA0003527791290000056
(9)
wherein:
Figure FDA0003527791290000057
the time-lag LPV system having a memory H ∞ dynamic output feedback controller coefficient matrix satisfying equation (2) can be obtained by the following equation, wherein the matrix X4And Y4From
Figure FDA0003527791290000061
Solving by full rank decomposition;
Figure FDA0003527791290000062
wherein:
Figure FDA0003527791290000063
wherein, to obtain the above inequality (9), assume G1＝G2G > 0, so G is reversible and is denoted as W-G-1And the matrices G and W are partitioned as:
Figure FDA0003527791290000064
defining:
Figure FDA0003527791290000065
the following operational relationships can be readily obtained from the above definitions:
Figure FDA0003527791290000066
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):
Figure FDA0003527791290000067
wherein:
Figure FDA0003527791290000068
the following relationships can be derived from equations (11) and (12):
Figure FDA0003527791290000071
wherein:
Figure FDA0003527791290000072
the following variable substitutions are made:
Figure FDA0003527791290000073
Figure FDA0003527791290000074
Figure FDA0003527791290000075
Figure FDA0003527791290000076
equation (14) can be written as:
Figure FDA0003527791290000077
Figure FDA0003527791290000078
Figure FDA0003527791290000079
Figure FDA0003527791290000081
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.
2. 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 matrix
Figure FDA0003527791290000082
Matrix Q of appropriate dimension1，Q2，S1i，S2i，Xi，Yi，Ui，
Figure FDA0003527791290000083
ΔijEquations (16) and (17) are satisfied:
Figure FDA0003527791290000084
Figure FDA0003527791290000085
Figure FDA0003527791290000086
wherein:
Figure FDA0003527791290000091
Figure FDA0003527791290000092
Figure FDA0003527791290000093
Figure FDA0003527791290000094
Figure FDA0003527791290000095
Figure FDA0003527791290000096
Figure FDA0003527791290000097
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:
Figure FDA0003527791290000098
Figure FDA0003527791290000099
3. 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 pair
Figure FDA00035277912900000910
Full rank decomposition to obtain matrix X4(theta) and Y4(θ); further obtain the memory H∞Output feedback controller gain matrix:
Figure FDA00035277912900000911
wherein:
Figure FDA00035277912900000912
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