H-infinity control method of high-dimensional distributed parameter system

Abstract

The invention provides an H-infinity control method based on a modal decomposition high-dimensional distributed parameter system. The H-infinity control method based on the modal decomposition high-dimensional distributed parameter system comprises the following steps: establishing a distributed parameter system model based on a two-dimensional Kuramoto-Savashinsky equation; according to a modal decomposition method, the system is projected to a feature space of a two-dimensional SURM-Liouvile operator; designing a finite dimension controller according to the current state; establishing a closed-loop system, and deducing sufficiency conditions enabling the closed-loop system to be stable according to a direct Lyapunov method; introducing an interference term, and designing a finite dimension controller for a system with interference based on a modal decomposition method; and a closed-loop system is established, and the H-infinity control problem of the closed-loop system is solved according to a given performance index. The method has the advantages that the engineering implementation is simple, a large number of sensors and actuators are not needed, and the method has suppression and robustness on uncertain factors such as interference.

Description

Technical Field

[0001] The present invention relates to the field of distributed parameter system control technology, and more specifically to a high-dimensional distributed parameter system H based on modal decomposition. ∞ Control method. Background Art

[0002] Distributed parameter systems are widely used in physical fields, chemical reaction processes, etc. In recent decades, the control problem of distributed parameter systems has become one of the difficulties in engineering. Among them, the control of the Kuramoto-Sivashinsky equation is a hot issue. The Kuramoto-Sivashinsky equation is a parabolic partial differential equation, which is often used to study chaotic phenomena, strange attractors and soliton waves in partial differential equations. It can describe many important phenomena, including magnetized plasma, flame front propagation, viscous flow and chemical reaction-diffusion processes, and has many applications in practical engineering.

[0003] As the spatial dimension of the distributed parameter system increases, the controller design will become more complicated. For the high-dimensional Kuramoto-Sivashinsky equation, the spatial decomposition method can be used to design a controller to stabilize the system, but this method may require the deployment of many sensors and actuators, and the actuators need to cover almost the entire spatial domain. In contrast, the main idea of ​​the modal decomposition method is to project the state of the partial differential equation onto a finite-dimensional subspace, and then design a control strategy for the resulting reduced-order model to stabilize the system. Therefore, this problem can be avoided, the execution efficiency can be improved, and unnecessary waste of resources can be avoided. In addition, there may be some uncertain interference in actual engineering, which will have a certain impact on the stability of the system, and the corresponding controller design will become more complicated, and H ∞ The control method has the characteristics of suppression and robustness to the uncertain factors such as disturbance. Therefore, the H ∞ Control methods have great advantages in practical problems.

[0004] In summary, the controller designed by the modal decomposition method is finite-dimensional, easy to implement in engineering, and can avoid unnecessary waste of resources. ∞ The control method can cope with external disturbances well and improve the robustness of the system. ∞ Control methods are necessary to stabilize high-dimensional distributed parameter systems. Summary of the invention

[0005] The present invention relates to a high-dimensional distributed parameter system based on modal decomposition. ∞ Control methods are used to solve interference problems and system stabilization problems in practical engineering.

[0006] The technical solution of the present invention is:

[0007] (1) Establishment of distributed parameter system model based on the two-dimensional Kuramoto-Sivashinsky equation.

[0008] Specifically, a two-dimensional nonlinear Kuramoto-Sivashinsky equation model is adopted, and distributed control is adopted, wherein a distributed parameter system model is used to describe the movement of a viscous liquid on an inclined plane under the action of an external electric field.

[0009] (2) High-dimensional distributed parameter system H based on modal decomposition ∞ Control method.

[0010] In order to avoid the use of a large number of sensors and actuators and overcome the influence of external interference on the system, the present invention provides a high-dimensional distributed parameter system H based on modal decomposition. ∞ Control methods. Specifically include:

[0011] Step 1. According to the modal decomposition method, the system is projected into the eigenspace of the two-dimensional Sturm-Liouvile operator;

[0012] Step 2. Design a finite-dimensional controller based on the current state;

[0013] Step 3. Establish a closed-loop system and derive sufficient conditions for regional stability of the closed-loop system based on the direct Lyapunov method;

[0014] Step 4. Introduce interference terms and design a finite-dimensional controller for the system with interference based on the modal decomposition method;

[0015] Step 5. Establish a closed-loop system and solve the H of the closed-loop system according to the given performance indicators. ∞ Control issues.

[0016] Compared with the prior art, the present invention has the following beneficial effects:

[0017] 1) The controller is designed using the modal decomposition method. The system state is projected and expanded into the form of Fourier series, and divided into two parts: an unstable finite-dimensional part and a stable infinite-dimensional part. Based on the unstable mode, a finite-dimensional controller is designed. Therefore, a large number of sensors and actuators are not required, and the designed controller is finite-dimensional and easy to implement in engineering.

[0018] 2) By giving performance indicators, using H ∞ The control method can better cope with external disturbances and improve the robustness of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1.H of the two-dimensional Kuramoto-Sivashinsky equation based on modal decomposition ∞ Control method block diagram;

[0020] Figure 2 . The parameters are set to M = N0 = 1, κ = -20, σ = 1, K0 = 74.575, and the image of the closed-loop system state z(x1, x2, t) at different times t∈{0, 10, 100} without interference;

[0021] Figure 3 .Image of the performance indicator J changing over time when the parameter is set to γ=10. DETAILED DESCRIPTION

[0022] In order to make the purpose, technical solutions and advantages of the present invention more clear, the present invention will be described in detail below in conjunction with specific embodiments and drawings.

[0023] 1. Establishment of distributed parameter system model based on two-dimensional Kuramoto-Sivashinsky equation

[0024] Consider the two-dimensional Kuramoto-Sivashinsky equation on the region Ω = [0,1] × [0,1]:

[0025]

[0026] Where x = (x1, x2) ∈ Ω, time t> 0, κ represents the angle of the substrate relative to the horizontal direction, and the system state Control input u(t)=col[u1(t),…,u M (t)], characteristic function

[0027] 2. Two-dimensional Kuramoto-Sivashinsky equation H based on modal decomposition ∞ control

[0028] The present invention adopts H ∞ Control method, the ultimate goal of control is that for external disturbances and a given γ, the closed-loop system is internally stable and satisfies the performance index J<0, where

[0029] The specific implementation steps are as follows:

[0030] Step 1. According to the modal decomposition method, the system is projected into the eigenspace of the two-dimensional Sturm-Liouvile operator.

[0031] Consider the two-dimensional Sturm-Liouvile problem

[0032]

[0033] The boundary conditions are Note m =m 2 π 2 ,λ n =n 2 π 2 , then the eigenvalue and eigenfunction corresponding to the Sturm-Liouvile problem are λ m +λ n =(m 2 +n 2 )π 2 , And the characteristic functions constitute L 2 (Ω) is a complete set of orthonormal bases.

[0034] Then the solution of the two-dimensional Kuramoto-Sivashinsky equation can be expressed as where z mn (t)= <z(·,t),φ mn >.

[0035] Deriving both sides of (1) under the integral sign and integrating by parts, we obtain

[0036]

[0037] where z mn (0)= <z(·,0),φ mn >, b mn =[ <b1,φ mn >,…, M ,φ mn >], because then exists For the expected decay rate δ>0, we get -(λ m +λ n ) 2 -κ(λ m +λ m )+λ m <-δ,m,n>N0.

[0038] here is the controller dimension.

[0039] Step 2. Design a finite-dimensional controller based on the current state.

[0040] Define the following matrix:

[0041] ​

[0042] Design the following Dimension Controller:

[0043]

[0044] Where K0 is the controller gain.

[0045] Step 3. Establish a closed-loop system and derive sufficient conditions for regional stability of the closed-loop system based on the direct Lyapunov method.

[0046] make Building a closed-loop system

[0047]

[0048] The following derivation makes the closed-loop system regionally stable. According to the Lyapunov function

[0049]

[0050] in So

[0051]

[0052] From Young's inequality, we know that there exists α>0 such that

[0053]

[0054]

[0055] in

[0056]

[0057] Let σ>0, assuming According to Young's inequality, we get

[0058]

[0059] Where α1>0.

[0060] According to the boundary conditions of (1), we get

[0061]

[0062] Using Parseval's equation, we can get

[0063]

[0064] in make Can get

[0065]

[0066] if and

[0067]

[0068] Established, of which according to The monotonicity of μ mn <0, m, n> N0 holds if and only if Established. According to Schur's complement lemma, μ mn <0 holds true if and only if

[0069]

[0070] Let Q = P -1 , Multiply both sides of Ψ by diag{P -1 ,I}, according to Schur's complement lemma, Ψ<0 holds if and only if

[0071]

[0072] Next, we can prove that if the initial conditions are satisfied

[0073]

[0074] in Can get

[0075]

[0076] Assume that there exists t1>0 such that and According to the function and The continuity of t * ∈(0,t1] such that and According to the comparison principle, we can get V(t)≤e -2δt V(0),t∈[0,t * )and

[0077]

[0078] Thus we get

[0079]

[0080] This contradicts the known answer, so V(t)≤e -2δt V(0),t∈[0,∞).

[0081] And the H of V(t) and z(x,t) 1 (Ω) norm is equivalent, so

[0082]

[0083] Where M0≥1.

[0084] Step 4. Introduce disturbance terms and design a finite-dimensional controller for the system with disturbance based on the modal decomposition method.

[0085] Consider the two-dimensional Kuramoto-Sivashinsky equation with interference on the region Ω = [0,1] × [0,1]:

[0086]

[0087] where ω(x, t) is the external disturbance and z(x, t0) = 0, and the remaining parameters and variables are the same as in (1).

[0088] Projecting the system state into the characteristic space of the two-dimensional Sturm-Liouvile operator, we get

[0089]

[0090] where ω mn (t)=<ω,φ mn >.

[0091] Design the following Dimension Controller:

[0092]

[0093] Where K0 is the controller gain.

[0094] Step 5. Establish a closed-loop system and solve the H of the closed-loop system according to the given performance indicators. ∞ Control issues.

[0095] make Building a closed-loop system

[0096]

[0097] Next, consider the closed-loop system H ∞ Control issues.

[0098] Given γ>0, the performance index is Note that if After integrating it with respect to t, we can get J<0.

[0099]

[0100]

[0101] make Similarly, according to the proof of step 3, we get the following result:

[0102] If exists and scalar 0<α,α1, satisfying the linear matrix inequality

[0103] Ψ1<0,Ψ2<0

[0104] in

[0105]

[0106] Then the system has an L2 gain less than γ.

[0107] The following is an explanation of the H based on modal decomposition provided by the present invention. ∞ The effectiveness of the control method is verified.

[0108] Set the following parameters: M=N0=1,δ=0.01,σ=1,κ=-20, verify that the linear matrix inequality in step 3 is feasible, and the feasible solution is α=71.933,α1=11.572,Q=0.676,Y0=50.412,ρ=0.274, Figure 2 Describes the parameters The initial condition is z(x1,x2,0)=0.236sin(πx1)sin(πx2), (x1,x2)∈Ω, the state z(x1,x2,t) of the closed-loop system (2) at different times t∈{0,10,100}, Figure 3 Depicts the parameters γ=10,ω(x,t)=(1+x1+x2)cost·e -0.1t The performance index J changes with time under the condition. From the simulation results, it can be seen that the closed-loop system gradually tends to be stable, so it can be concluded that the designed controller is effective and can maintain good robustness against external disturbances.

Claims

1. H of a high-dimensional distributed parameter system ∞ A control method, the method comprising: The distributed parameter system model is established based on the two-dimensional Kuramoto-Sivashinsky equation; According to the modal decomposition method, the system is projected into the eigenspace of the two-dimensional Sturm-Liouvile operator; Design a finite-dimensional controller based on the current state; Establish a closed-loop system and derive sufficient conditions for regional stability of the closed-loop system based on the direct Lyapunov method; Introduce disturbance terms and design finite-dimensional controllers for systems with disturbances based on modal decomposition methods; Establish a closed-loop system and solve the H of the closed-loop system according to the given performance indicators. ∞ Control issues.

2. H of the high-dimensional distributed parameter system according to claim 1 ∞ The control method is characterized in that A distributed parameter system model is established based on the two-dimensional Kuramoto-Sivashinsky equation, including: Dirichlet boundary conditions The model parameters include: bounded region Ω = [0,1] × [0,1], the angle k of the substrate relative to the horizontal direction; the model variables include: position variable x = (x1,x2)∈Ω, time variable t>0, system state Control input u(t)=col[u1(t),…,u M (t)], characteristic function 3. The modal decomposition method according to claim 1, characterized in that: According to the eigenvalues ​​and eigenfunctions of the two-dimensional Sturm-Liouvile operator l m +λ n =(m 2 +n 2 )p 2 , Projecting the two-dimensional Kuramoto-Sivashinsky equation into the eigenspace of the Sturm-Liouvile operator, we obtain where z mn (0)= <z(·,0),φ mn >, b mn =[ <b1,φ mn >,…, M ,φ mn >], ​ 4. The controller design method according to claim 1, characterized in that: Using the formula A finite-dimensional controller is established, where K0 is the controller gain.

5. The closed-loop system stabilization strategy according to claim 1, characterized in that: Establishing a closed-loop system according to the finite-dimensional controller of claim 4; According to the Lyapunov method, using the Lyapunov function The linear matrix inequality conditions that make the closed-loop system regionally stable are derived.

6. The control strategy for the two-dimensional Kuramoto-Sivashinsky system with disturbance according to claim 1, characterized in that: The model of the perturbed two-dimensional Kuramoto-Sivashinsky equation includes: Dirichlet boundary conditions and z(x,t0)=0; wherein ω(x,t) is an external disturbance, and the remaining parameters and variables are the same as those in claim 2; Using the formula A finite-dimensional controller is established, where K0 is the controller gain.

7. The perturbed two-dimensional Kuramoto-Sivashinsky system H according to claim 1 ∞ The control problem is characterized by Establishing a closed-loop system according to the finite-dimensional controller of claim 6; According to performance indicators Given a system that is internally stable and satisfies H ∞ Performance conditions, where γ>0 is a given constant.

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