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

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

A Design Method of Memory H∞ Output Feedback Controller for Time-delay 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 design method of a time-delay LPV system with a memory H _∞ output feedback controller.

Background technique

In the machining process, the precision and surface roughness of the machined workpiece, the service life of the tool and the machine tool, and the machining cycle are all affected by the tool vibration. Therefore, the vibration control of machining has become an important issue in the machining process;

The LPV theory was first proposed by Shamma in 1988, and its main purpose is to extend the existing linear control design basis to nonlinear and time-varying systems; in the existing technology, for the milling process of milling machines, the general The vibration control system is abstracted into a time-delay LPV system, and then a corresponding controller is designed for the time-delay LPV system to reduce the impact of vibration on the tool and workpiece;

However, the existing controller design method generally adopts the robust H _∞ state feedback technology. Although this technology can ensure that the system meets the H _∞ performance index, because the state feedback needs to obtain the state information of the controlled object, in many practical problems, The state variables considered are a set of variables that describe the internal information of the system, and are often not directly measurable; sometimes even if the state of the system can be directly measured, considering factors such as the cost of implementing control and the reliability of the system, if it can be Using the output feedback of the system to meet the performance requirements of the closed-loop system is more suitable for selecting the control method of output feedback; in addition, the existing memoryless state feedback controller does not introduce the past state information of the system, so the time delay cannot affect the system. With effective control, the cutting tool cannot work accurately and smoothly; therefore, the existing controller design methods have many defects.

SUMMARY OF THE INVENTION

In view of this, the present invention provides a design method of a time-delay LPV system with a memory H _∞ output feedback controller, so that the state information that does not depend on real-time measurement can be designed, and the disturbance attenuation, robustness and stability, and closed-loop response are satisfied. The required controller makes the cutting workpiece more precise and the cutting surface smoother;

The present invention provides the following technical solutions: a design method of a memory H _∞ output feedback controller for a time-delay LPV system, the method comprising:

A. The vibration control system of the milling process of the milling machine is abstracted as a time-delay LPV model, and the standard form of the memory H _∞ output feedback controller of the time-delay LPV system is obtained through model transformation;

B. Introduce slack matrix variables and quadratic Lyapunov functionals to transform the memory H _∞ dynamic output feedback control problem satisfying the desired performance index into a finite-dimensional convex optimization problem within the framework of linear matrix inequality;

C. Choose a new convex optimization method to give a finite-dimensional parametric linear matrix inequality at the vertices of a given multicellular LPV system;

D. Solve the linear matrix inequality to obtain the corresponding positive definite parameter dependence matrices X4 and Y4; calculate and obtain the gains A _k , B _k , C _k , D _k of the memory H _∞ dynamic output feedback controller in turn, so as to determine the memory H _∞ output feedback controller K.

Preferably, the step A includes:

Considering the vibration control system in the milling process of the milling machine, the state space model of the following time-delay LPV system is abstracted:

Among them, x(t) ^∈Rn is the state variable, u(t) ^∈Rr is the control input, y(t) ^∈Rp is the measurement output, z(t) ^∈Rr is the adjusted output, w(t)∈Rr is the modulated output )∈R ^q is the disturbance input, τ>0 is the known time delay constant, φ(ρ) is the given initial condition, and the system matrix is assumed to be a function of the time-varying parameter θ(t); θ _i (where i=1, . . . , s) replaces θ(t), θ _i (t);

The above LPV system was transformed into the following multicellular LPV model:

是系统所在的有界凸多面体顶点系统的集合， (A_i,A_1i,B_1i,B_2i,C_1i,C_2i,D_i,D_1i,D_2i)表示系统的第i个顶点系统，i＝1,2,…,N，对LPV系统，考虑在反馈控制率中引入时滞项，设计鲁棒有记忆H_∞动态输出反馈控制器K：For system (1),

is the set of bounded convex polyhedron vertex systems where the system is located, (A _i ,A _1i ,B _1i ,B _2i ,C _1i ,C _2i ,D _i ,D _1i ,D _2i )represents the i-th vertex system of the system, i=1,2,...,N, for the LPV system, consider introducing a time delay term in the feedback control rate, and design a robust dynamic output feedback controller K with memory H _∞ :

Among them: A _k , B _k , C _k , D _k are the controller parameter matrix to be determined, x _k (t)∈R ⁿ is the state of the controller, and u(t) is the control input; substitute the controller K into Time-delay LPV system, the closed-loop time-delay LPV system C is obtained:

in:

C _cl (θ)=[C ₂ (θ)+D(θ)D _k (θ)C ₁ (θ) D(θ)C _k (θ)]

C _cl1 (θ)=[D(θ)D _k (θ)C ₁ (θ) 0], D _cl (θ)=D ₂ (θ)

Therefore, the problem of designing a memory H _∞ output feedback controller K in the above-mentioned LPV system can be summarized as follows: design a memory H _∞ dynamic output feedback controller (2), so that the closed-loop system (3) can control the scheduling parameter θ in all its values. Within the value range, the following indicators are met:

(1). The interior of the closed-loop system is stable;

(2).H∞ performance index: for the disturbance input signal w(t), given a performance index γ>0, make the closed-loop transfer function T _wz from the disturbance input w(t) to the controlled output z(t) ( The H _∞ norm of s) is smaller than γ, that is, it satisfies:

Preferably, consider that step B includes:

Lemma 1. For the system (1), if there is a symmetric positive definite matrix P(θ), the matrix Q satisfies the linear matrix inequality (4)

Then the closed-loop system is asymptotically stable;

in,

Lemma 2. For the system (1), if there is a symmetric positive definite matrix P(θ), a matrix Q and a given positive scalar γ such that inequality (5) holds, the system is asymptotically stable and satisfies the H∞ performance index;

in,

对不等式(5)进行全等变换，可得到不等式(6)：It can be seen from the above that the sufficient condition for the closed-loop time-delay LPV system (3) to be asymptotically stable and to satisfy the H∞ performance index γ is the existence of a symmetric positive definite matrix P(θ), the matrix Q and the given positive scalar γ satisfy the inequality (5), Utilize a symmetric block matrix of appropriate dimension

Congruent transformation of inequality (5), inequality (6) can be obtained:

in:

The matrix inequality (6) can be written as:

According to Lemma 3, for appropriate dimensional arbitrary matrices M, N and identity matrix I, the following conditions are equivalent:

(1).

(2). There is a slack matrix G of appropriate dimension such that the following formula holds

Among them, let matrix G ^T =[G ₁ (θ) 0 0 0 G ₂ (θ)], then the above formula can be transformed into inequality (7)

in:

Apply Schur's complement lemma to matrix inequality (7) to get Corollary 2;

Corollary 2 For a closed-loop time-delay LPV system (3), if there exists a continuously differentiable symmetric positive definite matrix function P(θ), a symmetric positive definite matrix Q, symmetric matrices G ₁ (θ) and G ₂ (θ) and a given positive definite matrix Scalar γ, satisfying LMI(8):

Then the system is asymptotically stable and satisfies the H _∞ performance index;

in:

和适当维矩阵Q₁,Q₂,S₁(θ),S₂(θ)，X(θ)，Y(θ)，U(θ)，

和

以及给定的正标量γ，满足LMI(9)：Theorem 1 For a closed-loop time-delay multicellular LPV system (3), if there is a continuously differentiable symmetric positive definite matrix function

and appropriate dimensional matrices Q ₁ , Q ₂ , S ₁ (θ), S ₂ (θ), X(θ), Y(θ), U(θ),

and

and a given positive scalar γ, satisfying LMI(9):

in:

Γ₁₉＝-X(θ)-X^T(θ)，

Γ ₁₉ =-X(θ) ^-XT (θ),

Γ ₂₀ ＝-IU(θ), Γ ₂₁ ＝-Y ^T (θ)-Y(θ)

满秩分解求出；Then the time-delay LPV system satisfying equation (2) has memory H _∞ dynamic output feedback controller coefficient matrix can be obtained from equation (10), where the matrices X ₄ (θ) and Y ₄ (θ) are given by

Full rank decomposition to find out;

in:

In order to obtain the above inequality (9), it is assumed that G ₁ =G ₂ =G>0, so G is invertible, denoted as W=G ^-1 , and the matrices G and W are expressed in blocks as:

definition:

According to the above definition, the following operational relationship can be easily obtained:

Multiplying inequality (8) on the left by diag{Λ ^T (θ) III Λ ^T (θ) I Λ ^T (θ)} and right multiplying the matrix diag{Λ(θ)III Λ(θ) I Λ(θ)}, we get Inequality (13):

in:

Δ ₂ =Λ ^T (θ)G ^T (θ)A _cl1 (θ),Δ ₃ =Λ ^T (θ)G ^T (θ)B _cl (θ),

Δ ₈ =-Λ ^T (θ)G(θ)Λ(θ)-Λ ^T (θ)G ^T (θ)Λ(θ)

From equations (11) and (12), the following relationship can be deduced:

in:

Δ ₁₄ =C ₂ (θ)Y(θ)+D(θ)D _k (θ)C ₁ (θ)Y(θ)+D(θ)C _k (θ)Y ₄ (θ)

Do the following variable substitutions:

The formula (14) can be written as:

According to the above inference, Equation (13) is transformed into Equation (9) in Theorem 1; according to Equation (8), it can be seen that the closed-loop time-delay LPV system is asymptotically stable under the action of the memory H∞ dynamic output feedback controller, while satisfying H∞ performance indicators.

Preferably, consider that step C includes:

In order to reduce the conservatism, a new convex optimization method is chosen to give a parameterized linear matrix inequality of finite dimension at the vertices of a given bounded multicellular LPV system;

适当维矩阵Q₁，Q₂，S_1i，S_2i，X_i，Y_i，U_i，

Δ_ij使式(16)和式(17)成立：Theorem 2 For the closed-loop time-delay multicellular LPV system (3), it is assumed that there is a given positive scalar γ and a symmetric positive definite matrix

Appropriate dimension matrices Q ₁ , Q ₂ , S _1i , S _2i , X _i , Y _i , U _i ,

_Δij makes equations (16) and (17) hold:

in:

Γ₂₀＝-I-U_i，Γ₂₁＝-Y_i ^T-Y_i

Γ ₂₀ =-IU _i , Γ ₂₁ =-Y _i ^T -Y _i

Then the time-delay LPV system satisfying equation (2) has memory H∞ dynamic output feedback controller coefficient matrix can be obtained from equation (10); where:

Preferably, consider that step D includes:

满秩分解得到矩阵X₄(θ)和Y₄(θ)；进一步求得有记忆H_∞输出反馈控制器增益矩阵：If the above inequalities (16) and (17) have feasible solutions, then

The full rank decomposition is used to obtain the matrices X ₄ (θ) and Y ₄ (θ); further, the gain matrix of the output feedback controller with memory H _∞ is obtained:

in:

As can be seen above, in the design method of the memory H∞ output feedback controller of the time-delay LPV system in the present invention, for the time-delay LPV model of the vibration control system in the milling process of the milling machine, the memory H∞ output feedback control will be solved. The problem of the controller is transformed into the problem of solving the linear matrix inequalities, and the parameter dependence matrices X4 and Y4 are obtained by solving the inequalities, 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 beneficial effect of the present invention is that the controller does not rely on real-time measured state information, and has the characteristics of interference attenuation, robust stability, closed-loop response meeting requirements, and good dynamic performance and robustness. , which makes the cutting workpiece with higher precision and smoother cutting surface.

Description of drawings

Figure 1 is a schematic flow chart of the design method of a memory H _∞ output feedback controller for a time-delay LPV system;

Figure 2 shows the position change curve and change rate curve of the two modules without disturbance under the action of the memory H∞ output feedback controller;

Figure 3 shows the position change curve and change rate curve of the two modules after disturbance under the action of the memory H∞ output feedback controller;

Fig. 4 is the position change curve and change rate curve of two modules without controller under the initial condition;

Detailed ways

In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and examples;

In the present invention, a design method of a time-delay LPV system with a memory H _∞ output feedback controller is provided, and the method is suitable for a time-delay LPV system of vibration control in the milling process of a milling machine;

1 is a schematic flowchart of a design method of a memory H _∞ output feedback controller in a time-delay LPV system in the present invention. As shown in FIG. 1 , a time-delay LPV system in an embodiment of the present invention has a memory H _∞ output feedback controller design method, including the following steps:

Step 1, abstract the vibration control system of the milling process of the milling machine into a multicellular time-delay LPV model, and obtain the standard form of the time-delay LPV system with memory H∞ output feedback controller to solve the problem through model transformation;

In the technical solution of the present invention, the system can be abstracted into a time-delay LPV model first;

For example, preferably, in a specific embodiment of the present invention, the above-mentioned step 1 includes:

Considering the vibration control system in the milling process of the milling machine, it is abstracted as a time-delay LPV system state space model:

x₁,x₂分别是刀具和机床的位置，τ>0为已知时滞常数，φ(ρ)是给定的初始条件，假定系统矩阵和时延h(θ(t))均为时变参数θ(t)的函数；为了表述方便以后以θ，θ_i(其中i＝1,···,s)代替θ(t)，θ_i(t)；Among them, the system status

x ₁ , x ₂ are the positions of the tool and the machine tool respectively, τ>0 is the known time delay constant, φ(ρ) is the given initial condition, assuming that the system matrix and the time delay h(θ(t)) are both The function of variable parameter θ(t); for the convenience of expression, θ(t), _θi (t) will be replaced by θ, θ _i (where i=1, . . . , s);

The above LPV system was transformed into the following multicellular LPV model:

是系统所在的有界凸多面体顶点系统的集合， (A_i,A_1i,B_1i,B_2i,C_1i,C_2i,D_i,D_1i,D_2i)表示系统的第i个顶点系统，i＝1,2,…,N，对 LPV系统，考虑在反馈控制率中引入时滞项，设计鲁棒有记忆H∞动态输出反馈控制器K：For system (1),

is the set of bounded convex polyhedron vertex systems where the system is located, (A _i ,A _1i ,B _1i ,B _2i ,C _1i ,C _2i ,D _i ,D _1i ,D _2i )represents the i-th vertex system of the system, i=1,2,...,N, for the LPV system, consider introducing a time delay term in the feedback control rate, and design a robust dynamic output feedback controller K with memory H∞:

Among them: A _k , B _k , C _k , D _k are the controller parameter matrix to be determined, x _k (t)∈R ⁿ is the state of the controller, u(t) is the control input; when the controller K is substituted into Hysteresis LPV system, the closed-loop time-delay LPV system C is obtained:

in:

C _cl (θ)=[C ₂ (θ)+D(θ)D _k (θ)C ₁ (θ) D(θ)C _k (θ)]

C _cl1 (θ)=[D(θ)D _k (θ)C ₁ (θ) 0], D _cl (θ)=D ₂ (θ)

Therefore, the problem of designing a memory H∞ output feedback controller K in the above-mentioned LPV system can be summarized as: designing a memory H∞ dynamic output feedback controller (2), so that the closed-loop system (3) can control the scheduling parameter θ in all its values. Within the value range, the following indicators are met:

(1). The interior of the closed-loop system is stable;

(2).H∞ performance index: for the disturbance input signal w(t), given a performance index γ>0, make the closed-loop transfer function T _wz from the disturbance input w(t) to the controlled output z(t) ( The H _∞ norm of s) is smaller than γ, that is, it satisfies:

That is, if A _k , B _k , C _k , D _k are obtained, the dynamic output feedback controller with memory H _∞ can be obtained;

Therefore, through the above model transformation, the standard form of the dynamic output feedback controller with memory H _∞ can be obtained.

Step 2: Introduce slack matrix variables and quadratic Lyapunov functionals to transform the memory H∞ robust dynamic output feedback control problem satisfying the desired performance index into a finite-dimensional convex optimization problem within the framework of linear matrix inequalities;

In the technical solution of the present invention, in order to determine whether the memory H _∞ robust dynamic output feedback controller exists and is stable, the controller solution problem of the above-mentioned time-delay LPV system can be transformed into a convex optimization problem of solving a linear matrix inequality. ;

For example, preferably, in a specific embodiment of the present invention, Lemma 1 and Lemma 2 can be introduced first:

Lemma 1. For the system (1), if there is a symmetric positive definite matrix P(θ), the matrix Q satisfies the linear matrix inequality (4)

Then the closed-loop system is asymptotically stable;

in,

Lemma 2. For the system (1), if there is a symmetric positive definite matrix P(θ), a matrix Q and a given positive scalar γ such that the linear matrix inequality (5) holds, the system is asymptotically stable and satisfies the H∞ performance index ;

in,

对不等式(5)进行全等变换，可得到不等式(6)：It can be seen from the above that the sufficient condition for the closed-loop time-delay LPV system (3) to be asymptotically stable and to satisfy the H∞ performance index γ is the existence of a symmetric positive definite matrix P(θ), the matrix Q and the given positive scalar γ satisfy the inequality (5), Utilize a symmetric block matrix of appropriate dimension

Congruent transformation of inequality (5), inequality (6) can be obtained:

in:

The matrix inequality (6) can be written as:

According to Lemma 3, for appropriate dimensional arbitrary matrices M, N and identity matrix I, the following conditions are equivalent:

(1).

(2). There is a slack matrix G of appropriate dimension such that the following formula holds

Among them, let matrix G ^T =[G ₁ (θ) 0 0 0 G ₂ (θ)], then the above formula can be transformed into inequality (7)

in:

Corollary 2 For a closed-loop time-delay LPV system (3), if there exists a continuously differentiable symmetric positive definite matrix function P(θ), a symmetric positive definite matrix Q, symmetric matrices G ₁ (θ) and G ₂ (θ) and a given positive definite matrix Scalar γ, satisfying LMI(8):

Then the system is asymptotically stable and satisfies the H _∞ performance index;

in:

和适当维矩阵Q₁,Q₂,S₁(θ),S₂(θ),X(θ),Y(θ),U(θ),

和

以及给定的正标量γ，满足LMI(9)：Theorem 1 For a closed-loop time-delay multicellular LPV system (3), if there is a continuously differentiable symmetric positive definite matrix function

and appropriate dimensional matrices Q ₁ , Q ₂ , S ₁ (θ), S ₂ (θ), X (θ), Y (θ), U (θ),

and

and a given positive scalar γ, satisfying LMI(9):

in:

Γ₁₉＝-X(θ)-X^T(θ)，

Γ ₁₉ =-X(θ) ^-XT (θ),

Γ ₂₀ =-IU(θ), Γ ₂₁ =-Y ^T (θ)-Y(θ)

满秩分解求出；Then the time-delay LPV system satisfying equation (2) has memory H∞ dynamic output feedback controller coefficient matrix can be obtained by the following equation, where the matrix X ₄ and Y ₄ are given by

Full rank decomposition to find out;

in:

In order to obtain the above inequality (9), it is assumed that G ₁ =G ₂ =G>0, so G is invertible, denoted as W=G ^-1 , and the matrices G and W are expressed in blocks as:

definition:

According to the above definition, the following operational relationship can be easily obtained:

Multiplying inequality (8) on the left by diag{Λ ^T (θ) III Λ ^T (θ) I Λ ^T (θ)} and right multiplying the matrix diag{Λ(θ)III Λ(θ) I Λ(θ)}, we get Linear matrix inequality (13):

in:

Δ ₂ =Λ ^T (θ)G ^T (θ)A _cl1 (θ),Δ ₃ =Λ ^T (θ)G ^T (θ)B _cl (θ),

Δ ₈ =-Λ ^T (θ)G(θ)Λ(θ)-Λ ^T (θ)G ^T (θ)Λ(θ)

From equations (11) and (12), the following relationship can be deduced:

Do the following variable substitutions:

in:

Δ ₁₄ =C ₂ (θ)Y(θ)+D(θ)D _k (θ)C ₁ (θ)Y(θ)+D(θ)C _k (θ)Y ₄ (θ)

The formula (14) can be written as:

in:

According to the above inference, Equation (13) is transformed into Equation (9) in Theorem 1; according to Equation (8), it can be seen that the parameters of the closed-loop time-delay LPV system are quadratic stable under the action of the memory H∞ dynamic output feedback controller, At the same time meet the H∞ performance index;

Therefore, it can be seen that the controller solution problem of the time-delay LPV system can be transformed into a convex optimization problem of solving a linear matrix inequality; when the linear matrix inequality has a solution, a memory H∞ output feedback controller exists and is stable.

Step 3, select a new convex optimization method, and at the vertex of the given multicellular LPV system, give the parameterized linear matrix inequality of finite dimension;

In the embodiment of the present invention, a variety of specific implementations can be used to implement step 3, and the following will take one of the implementations as an example to introduce the technical solution of the present invention in detail;

For example, preferably, in a specific embodiment of the present invention, the step 3 includes:

In order to reduce the conservatism, a new convex optimization method is chosen to give a parameterized linear matrix inequality of finite dimension at the vertices of a given bounded multicellular LPV system;

适当维矩阵Q₁，Q₂，S_1i，S_2i，X_i，Y_i，U_i，

Δ_ij使式(16)和式(17)成立：Theorem 2 For the closed-loop time-delay multicellular LPV system (3), it is assumed that there is a given positive scalar γ and a symmetric positive definite matrix

Appropriate dimension matrices Q ₁ , Q ₂ , S _1i , S _2i , X _i , Y _i , U _i ,

_Δij makes equations (16) and (17) hold:

in:

Γ₂₀＝-I-U_i，Γ₂₁＝-Y_i ^T-Y_i

Γ ₂₀ =-IU _i , Γ ₂₁ =-Y _i ^T -Y _i

Then the time-delay LPV system satisfying equation (2) has memory H∞ dynamic output feedback controller coefficient matrix can be obtained from equation (9);

in:

When the linear matrix inequalities in the above inequalities (16) and (17) have solutions, there is a memory H∞ dynamic output feedback controller.

Step 4: Solve the linear matrix inequality to obtain the corresponding positive definite parameter dependence matrices X4 and Y4; calculate and obtain the gains A _k , B _k , C _k , D _k of the memory H∞ dynamic output feedback controller in turn, so as to obtain the following equations: Memory H∞ output feedback controller K;

In the technical solution of the present invention, a variety of specific implementations can be used to realize the above-mentioned step 4; the following will take one of the implementations as an example to introduce the technical solution of the present invention in detail;

For example, preferably, in a specific embodiment of the present invention, the fourth step includes:

If there are feasible solutions to the above inequalities (16) and (17), the memory H∞ output feedback controller exists and is stable, and the LMI toolbox in MATLAB determines whether the controller existence conditions are established, so it can be solved by solving the linear Matrix inequality, get the corresponding positive definite parameter dependence matrix X, Y and U;

满秩分解得到矩阵X₄(θ)和Y₄(θ)；可以使用如下所述的公式确定有记忆H_∞输出反馈控制器K中的参数：right

Full-rank decomposition yields matrices X ₄ (θ) and Y ₄ (θ); the parameters in the output feedback controller K with memory H _∞ can be determined using the formulas described below:

in:

To sum up, in the design of the time-delay LPV system with memory H∞ output feedback controller in the present invention, for the time-delay LPV model of the vibration control system in the milling process of the milling machine, the memory H∞ output feedback controller will be solved. The problem of solving linear matrix inequalities is transformed into the problem of solving linear matrix inequalities, and the parameter dependence matrices X4 and Y4 are obtained by solving the inequalities, and the parameter matrix in the controller K is finally determined, so that a memory H∞ output feedback controller can be designed, which can not only ensure the system It is stable and can meet the H∞ performance index, and has good dynamic performance and robustness; in order to further illustrate the advantages of the present invention, relevant simulation data are provided: H∞ performance index γ=1.3587 and parameter-dependent memory H∞ output Feedback controller parameter matrix:

C _K1 = [3.5526 1.3313 8.2361 9.9666], D _K1 = -4.6587

C _K2 = [12.71 -49.11 1.667 30.4025], D _K2 = -4.1305

In addition, Figure 2 shows the position change curve and rate of change curve of the two modules without disturbance under the action of the H∞ output feedback controller with memory; Figure 3 shows the perturbation curve under the action of the H∞ output feedback controller with memory The position change curve and change rate curve of the latter two modules; comparing Figure 2 and Figure 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; In contrast, Fig. 4 further confirms the effectiveness of the memory output feedback controller designed by the present invention;

In addition, due to the generality of this method, a memory H∞ output feedback controller can be designed for all actual physical systems that can be abstracted as time-delay LPV models by using this method, so as to achieve a good control effect;

It will be apparent to those skilled in the art that the present invention is not limited to the details of the above-described exemplary embodiments, but that the present invention can be embodied in other specific forms without departing from the spirit or essential characteristics of the present invention; In all respects, the embodiments should be considered as exemplary and non-restrictive, the scope of the present invention is defined by the appended claims rather than the above description, and it is intended that All changes within the meaning and scope are intended to be embraced within the invention, and any reference signs in the claims shall not be construed as limiting the involved claim;

In addition, it should be understood that although this specification is described in terms of embodiments, not each embodiment only includes an independent technical solution, and this description in the specification is only for the sake of clarity, and those skilled in the art should take the specification as a whole , the technical solutions in each embodiment can also be appropriately combined to form other implementations that can be 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 controller_k,B_k,C_k,D_kThereby 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:

wherein x (t) e RⁿIs a state variable, u (t) e R^rFor control input, y (t) e R^pIs the measurement output, z (t) e R^rIs the modulated output, w (t) e R^qFor 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, theta_iInstead of theta (t), theta_i(t), wherein i ═ 1, ·, s;

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,A_1i,B_1i,B_2i,C_1i,C_2i,D_i,D_1i,D_2i) 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. the_k、B_k、C_k、D_kIs a matrix of controller parameters, x, to be determined_k(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:

wherein:

C_cl(θ)＝[C₂(θ)+D(θ)D_k(θ)C₁(θ) D(θ)C_k(θ)]

C_cl1(θ)＝[D(θ)D_k(θ)C₁(θ) 0],D_cl(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:

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)

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 used

The 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 G^T＝[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 exists

And an appropriate dimension matrix Q₁,Q₂,S₁(θ),S₂(θ),X(θ),Y(θ),U(θ),

And

and a given positive scalar γ, satisfying LMI (9):

(9)

wherein:

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

Solving by full rank decomposition;

wherein:

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:

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 Λ^T(θ) I Λ^T(theta), multiplying diag { Λ (theta) IILambda (theta) ILambda (theta) }rightwardto obtain a matrix inequality (13):

wherein:

the following relationships can be derived from equations (11) and (12):

wherein:

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.

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

Matrix Q of appropriate dimension₁，Q₂，S_1i，S_2i，X_i，Y_i，U_i，

Δ_ijEquations (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:

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

Full 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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