Robust control method of disturbance-containing nonlinear discrete interconnection system based on state observer

Abstract

The invention discloses a robust control method for obtaining a disturbance-containing nonlinear discrete interconnection system based on a state observer, relates to the technical field of robust control of interconnection systems, and aims to solve the problem of designing a control theory of a nonlinear interconnection system based on an observer. The method comprises the following steps: establishing a discrete interconnection system model containing a nonlinear term and a disturbance term; designing a state observer for the discrete interconnection system model; aiming at the interconnected system state observer, considering a nonlinear term and a disturbance term, designing a group of linear matrix inequalities, and solving an observer gain matrix of each subsystem to obtain an estimated value of the interconnected system state; designing a state feedback controller based on the state estimation value of the observer; on the basis of the designed controller, a set of linear matrix inequality is designed for the state feedback closed-loop system, a gain matrix of the controller is obtained by solving the linear matrix inequality, and therefore robust stabilization of the closed-loop system is guaranteed.

Description

Technical Field

[0001] The present invention relates to the technical field of robust control of interconnected systems, and in particular to a robust control method for a nonlinear discrete interconnected system containing disturbances based on a state observer. Background Art

[0002] Normally, state feedback control is performed under the premise that the state information of the control system is fully available. However, due to technical or economic reasons, there are often situations where the system state is unmeasurable or difficult to measure. The system state observer can achieve robust estimation of state variables based on control inputs and outputs, and solve the problem of the state not being directly measurable through reconstruction, providing a theoretical basis for the implementation of control system state feedback technology. Therefore, scholars have conducted research on robust control methods based on state observers, and applied feedback control methods based on observed states to different types of systems such as discrete systems and continuous systems.

[0003] Existing robust control methods are mostly used in normal systems, but relatively few are designed for interconnected large systems. Interconnected large systems are a type of complex large system composed of several subsystems connected by interconnected terms. The scale of the systems involved in production and life is getting larger and larger, and the complexity is getting higher and higher. It is of certain practical significance to develop robust control methods for interconnected large systems and form a control theory for nonlinear interconnected systems based on observers. Summary of the invention

[0004] The present invention proposes a robust control method for a nonlinear discrete interconnected system with disturbance based on a state observer, which comprises the following steps: S1: Establish a discrete interconnected system model with nonlinear terms and disturbance terms; S2: Designing a state observer for the discrete interconnected system model in step S1; S3: for the interconnected system state observer in step S2, a set of linear matrix inequalities is designed by considering nonlinear terms and disturbance terms, and the observer gain matrix of each subsystem is solved to obtain an estimated value of the interconnected system state; S4: designing a state feedback controller based on the state estimation value of the observer in step S3; S5: Based on the controller designed in step S4, a set of linear matrix inequalities is designed for the state feedback closed-loop system. By solving the linear matrix inequalities, the gain matrix of the controller is obtained, thereby ensuring the robust stabilization of the closed-loop system.

[0005] The present invention is further configured as follows: the discrete interconnected system model in step S1 is: Where: x i (k)∈R n 、u i (k)∈R m ,y i (t)∈R p and d i (k)∈R l are the state, control input, output and system disturbance of the ith subsystem, g[x i (k)] is the nonlinear term of the system, A i ,B i ,C i ,D i is the coefficient matrix with appropriate dimensions in the ith subsystem, H ij is the interconnection matrix between the i-th subsystem and the j-th subsystem in the interconnected system.

[0006] The present invention is further configured as follows: the discrete interconnected system model (Formula 1) satisfies the following conditions: (1) In each subsystem, the matrix [A i C i ] is observable, the matrix [A i B i ] is controllable; (2) In each subsystem, the nonlinear term g[x i (k)] satisfies the Lipschitz condition, i.e. Where L gi is the Lipschitz constant,

[0007] The present invention is further configured as follows: the state observer in step S2 is: Where: is the estimated value of the state vector and output vector of the ith subsystem, L i is the observer gain matrix of the i-th subsystem to be designed.

[0008] The present invention is further configured as follows: the linear matrix inequality in step S3 is: Where: ψ 44 =diag(-P1,-P2,…,-P N ); and X i =P i L i ,i=1,2,…,N. and Where: Ξ 11 =diag(-P1,-P2,…,-P N ); and β ii =P i A i -X i C i -o1P i ,i=1,2,…,N. β ij =P i H ij ,i≠j and i,j=1,2,…,N. Where: Ω 11 =diag(-(1-σ)P1,-(1-σ)P2,…,-(1-σ)P N );Ω 12 =ψ 12 ;Ω 22 =ψ 44 .

[0009] The present invention is further configured as follows: the specific design process of the interconnected system state observer in step S2 is as follows: S2-1: Set the disk parameters O1 and r1 to ensure that the state estimation error system is asymptotically stable; S2-2: Setting the performance index σ, shortening the convergence speed of the observation error of the interconnected system observer, and improving the performance of the observer; S2-3: Setting H ∞ Performance index γ1, which reduces the impact of system disturbance on the observer state estimate; S2-4: Based on the set index parameters o1, r1, σ and γ1, solve the matrix inequality group (Formula 3), (Formula 4) and (Formula 5) to calculate the undetermined matrix P = diag (P1, P2, ..., P N ) and X=diag(X1,X2,…,X N ), further Then the gain matrix L, R is obtained, thus completing the design of the augmented fault state observer.

[0010] The present invention is further configured as follows: in step S4, the following state feedback controller is designed:

[0011] The present invention is further configured as follows: in step S5, the linear matrix inequality designed for the state feedback closed-loop system is: Where: γ 11 =diag(-Q1,-Q2,…,-Q N ); γ 13 =diag(Q1,Q2,…,Q N ); γ 14 =diag(-B1Y1,-B2Y2,…,-B N Y N ); γ 15 =diag(Q1D1,Q2D2,…,Q N D N ); and φ ii =Q i A i +B i Y i ,i=1,2,…,N. φ ij =Q i H ij ,i≠j and i,j=1,2,…,N. Q i B i K i =B i Y i ,i=1,2,…,N. Where: ∑ 11 =diag(-Q1,-Q1,…,-Q N ); and

[0012] The present invention is further configured as follows: the state feedback controller design process in step S5 is as follows: S4-1: Set the center and radius O2 and r2 of the disk to ensure the robust stabilization of the feedback closed-loop system; S4-2: Setting H ∞ Performance index γ2, reducing the impact of system disturbance on system robust control; S4-3: Based on the set parameters O2, r2 and γ2, solve the matrix inequality group (Equation 7) and (Equation 8) to calculate the undetermined matrix Q = diag (Q1, Q2, ..., Q N ) and Y=diag(Y1,Y2,…,Y N ), further Then, the controller gain matrix is ​​obtained so that the feedback control system satisfies the performance ||x(k)||≤γ2||η(k)|| and the robust stabilization of the closed-loop system is guaranteed.

[0013] The present invention is further configured as follows: in step S5, the controller expression of the i-th system of the interconnected system based on the observer is:

[0014] The beneficial effects of the present invention are as follows: the present invention designs an observation-based robust controller in the presence of external disturbances in a nonlinear discrete interconnected system, thereby making the interconnected system robustly stabilized. In terms of the convergence of the observation estimation error, the present invention introduces a stability margin index to improve the performance of the observer; in terms of controller design, the H ∞ Performance indicators reduce the impact of disturbances on controller design; by separating the observer from the controller, it has greater freedom and is conducive to achieving robust control of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Figure 1 A flow chart of the present invention is shown.

[0016] Figure 2 The estimated error response curve of the first subsystem of the interconnected system in Example 2 is shown.

[0017] Figure 3 The estimated error response curve of the second subsystem of the interconnected system in Example 2 is shown.

[0018] Figure 4 The first system feedback control state response curve of the interconnected system in Example 2 is shown.

[0019] Figure 5The second system feedback control state response curve of the interconnected system in Example 2 is shown. DETAILED DESCRIPTION

[0020] Those skilled in the art can refer to the content of this article and appropriately improve the process parameters. It is particularly important to point out that all similar replacements and modifications are obvious to those skilled in the art, and they are all considered to be included in the present invention. The methods and applications of the present invention have been described through preferred embodiments, and relevant personnel can modify or appropriately change and combine the methods and applications described herein without departing from the content, spirit and scope of the present invention to implement and apply the technology of the present invention.

[0021] Example 1

[0022] The present invention proposes a robust control method for a nonlinear discrete interconnected system with disturbance based on a state observer, which specifically includes the following steps: S1: Establish a discrete interconnected system model with nonlinear terms and disturbance terms; S2: Design a state observer for the discrete interconnected system model in step S1; S3: For the interconnected system state observer in step S2, a set of linear matrix inequalities is designed by considering nonlinear terms and disturbance terms, and the observer gain matrix of each subsystem is solved to obtain the estimated value of the interconnected system state; S4: Design a state feedback controller based on the state estimate of the observer in step S3; S5: Based on the controller designed in step S4, a set of linear matrix inequalities is designed for the state feedback closed-loop system. By solving the linear matrix inequalities, the gain matrix of the controller is obtained, thereby ensuring the robust stabilization of the closed-loop system.

[0023] The discrete interconnected system model in step S1 is: Where: x i (k)∈R n 、u i (k)∈R m ,y i (t)∈R p and d i (k)∈R l are the state, control input, output and system disturbance of the ith subsystem, g[x i (k)] is the nonlinear term of the system, A i ,B i ,C i ,D i is the coefficient matrix with appropriate dimensions in the ith subsystem, H ijis the interconnection matrix between the i-th subsystem and the j-th subsystem in the interconnected system.

[0024] The discrete interconnected system model (Equation 1) satisfies the following conditions: (1) In each subsystem, the matrix [A i C i ] is observable, the matrix [A i B i ] is controllable; (2) In each subsystem, the nonlinear term g[x i (k)] satisfies the Lipschitz condition, i.e. Where L gi is the Lipschitz constant,

[0025] The fault estimation observer in step S2 is designed as: Where: is the estimated value of the state vector and output vector of the ith subsystem, L i is the observer gain matrix of the i-th subsystem to be designed.

[0026] The linear matrix inequality in step S3 is: Where: ψ 44 =diag(-P1,-P2,…,-P N ); and X i =P i L i ,i=1,2,…,N. and Where: Ξ 11 =diag(-P1,-P2,…,-P N ); and β ii =P i Ai -X i C i -o1P i ,i=1,2,…,N. β ij =P i H ij ,i≠j and i,j=1,2,…,N. Where: Ω 11 =diag(-(1-σ)P1,-(1-σ)P2,…,-(1-σ)P N ); Ω 12 =ψ 12 ;Ω 22 =ψ 44 .

[0027] The specific design process of the interconnected system state observer in step S2 is as follows: S2-1: Set the disk parameters o1 and r1 to ensure that the state estimation error system is asymptotically stable; S2-2: Setting the performance index σ, shortening the convergence speed of the observation error of the interconnected system observer, and improving the performance of the observer; S2-3: Setting H ∞ Performance index γ1, which reduces the impact of system disturbance on the observer state estimate; S2-4: Based on the set index parameters o1, r1, σ and γ1, solve the matrix inequality group (Formula 3), (Formula 4) and (Formula 5) to calculate the undetermined matrix P = diag (P1, P2, ..., P N ) and X=diag(X1,X2,…,X N ), further Then the gain matrix L, R is obtained, thus completing the design of the augmented fault state observer.

[0028] The state feedback controller designed in step S4 is as follows:

[0029] In step S5, the linear matrix inequality designed for the state feedback closed-loop system is: Where: γ 11 =diag(-Q1,-Q2,…,-Q N ); γ 13=diag(Q1,Q2,…,Q N ); γ 14 =diag(-B1Y1,-B2Y2,…,-B N Y N ); γ 15 =diag(Q1D1,Q2D2,…,Q N D N ); and φ ii =Q i A i +B i Y i ,i=1,2,…,N. φ ij =Q i H ij ,i≠j and i,j=1,2,…,N. Q i B i K i =B i Y i ,i=1,2,…,N. Where: ∑ 11 =diag(-Q1,-Q1,…,-Q N ); and

[0030] In step S4, the state feedback controller design process is as follows: S4-1: Set the center and radius O2 and r2 of the disk to ensure the robust stabilization of the feedback closed-loop system; S4-2: Setting H ∞ Performance index γ2, reducing the impact of system disturbance on system robust control; S4-3: Based on the set parameters O2, r2 and γ2, solve the matrix inequality group (Equation 7) and (Equation 8) to calculate the undetermined matrix Q = diag (Q1, Q2, ..., Q N ) and Y=diag(Y1,Y2,…,Y N ), further Then, the controller gain matrix is ​​obtained so that the feedback control system satisfies the performance ||x(k)||≤γ2||η(k)|| and the robust stabilization of the closed-loop system is guaranteed.

[0031] In step S5, the controller expression of the i-th system of the interconnected system based on the observer is:

[0032] Example 2

[0033] According to the control method described in Example 1, a discrete interconnected system model including two subsystems is considered.

[0034] The parameters of subsystem 1 are: d2(k)=0.01sin(k);

[0035] The parameters of subsystem 2 are: d1(k)=0.01sin(k),

[0036] For the design of the interconnected system observer, the disk area D(o1,r1)=(0.5,0.5) is selected, and the parameter L is given g =0.03, ε1=25, γ1=1.2, the undetermined matrices of each subsystem are calculated as follows:

[0037] The observer gain matrix of each subsystem is as follows:

[0038] refer to Figure 2 , 3 , respectively show the response curves of the state estimation errors of the two subsystems of the interconnected system. It can be seen from the figure that the state estimation value can quickly track the actual value of the upper state with high accuracy.

[0039] For the design of the observation state feedback controller, the disk area D(o2,r2)=(0.5,0.5) is selected, and the parameter L is given g =0.03, ε2=100, γ2=1.5, the undetermined matrices of each subsystem are as follows:

[0040] The controller gain matrix of each subsystem is as follows:

[0041] refer to Figure 4 ,5 , respectively show the state response curves of the two western systems of the interconnected system based on the state feedback closed-loop system. It can be seen from the figure that the nonlinear discrete interconnected system can quickly achieve robust stabilization.

[0042] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention. These improvements and modifications should also be within the scope of protection of the present invention.

Claims

1. A robust control method for a nonlinear discrete interconnected system with disturbance based on a state observer, characterized in that: The following steps are involved: S1: Establish a discrete interconnected system model with nonlinear terms and disturbance terms; S2: Designing a state observer for the discrete interconnected system model in step S1; S3: for the state observer in step S2, a set of linear matrix inequalities is designed by considering nonlinear terms and disturbance terms, and the observer gain matrix of each subsystem is solved to obtain an estimated value of the state of the interconnected system; S4: designing a state feedback controller based on the state estimation value of the observer in step S3; S5: Based on the controller designed in step S4, a set of linear matrix inequalities is designed for the state feedback closed-loop system. By solving the linear matrix inequalities, the gain matrix of the controller is obtained, thereby ensuring the robust stabilization of the closed-loop system.

2. The robust control method for nonlinear discrete interconnected systems with disturbances based on a state observer according to claim 1 is characterized in that: The discrete interconnected system model in step S1 is: Where: x i (k)∈R n 、u i (k)∈R m ,y i (t)∈R p and d i (k)∈R l are the state, control input, output and system disturbance of the ith subsystem, g[x i (k)] is the nonlinear term of the system, A i ,B i ,C i ,D i is the coefficient matrix with appropriate dimensions in the ith subsystem, H ij is the interconnection matrix between the i-th subsystem and the j-th subsystem in the interconnected system.

3. The robust control method for nonlinear discrete interconnected systems with disturbances based on state observer according to claim 2 is characterized in that: The discrete interconnected system model (Formula 1) satisfies the following conditions: (1) In each subsystem, the matrix [A i C i ] is observable, the matrix [A i B i ] is controllable; (2) In each subsystem, the nonlinear term g[x i (k)] satisfies the Lipschitz condition, i.e. Where L gi is the Lipschitz constant, 4. The robust control method for nonlinear discrete interconnected systems with disturbances based on a state observer according to claim 3 is characterized in that: The state observer in step S2 is: Where: is the estimated value of the state vector and output vector of the ith subsystem, L i is the observer gain matrix of the i-th subsystem to be designed.

5. The robust control method for nonlinear discrete interconnected systems with disturbances based on a state observer according to claim 1 or 4, characterized in that: The linear matrix inequality in step S3 is: Where: ψ 44 diag(-P1,-P2,...,-P N )4 and X i =P i L i ,i=1,2,…,N. and Where: Ξ 11 diag(-P1,-P2,...,-P N )4 and β ii =P i A i -X i C i -o1P i ,i=1,2,…,N. β ij =P i H ij ,i≠j and i,j=1,2,…,N. Where: Ω 11 = diag(-(1 - σ)P1, -(1 - σ)P2, …, -(1 - σ)P N ); Ω 12 = ψ 12 ; Ω 22 = ψ 44 .

6. The robust control method for nonlinear discrete interconnected systems with disturbances based on state observer according to claim 5 is characterized in that: The specific design process of the interconnected system state observer in step S2 is as follows: S2-1: Set the disk parameters o1 and r1 to ensure that the state estimation error system is asymptotically stable; S2-2: Setting the performance index σ, shortening the convergence speed of the observation error of the interconnected system observer, and improving the performance of the observer; S2-3: Setting H ∞ Performance index γ1, which reduces the impact of system disturbance on the observer state estimate; S2-4: Based on the set index parameters o1, r1, σ and γ1, solve the matrix inequality group (Formula 3), (Formula 4) and (Formula 5) to calculate the undetermined matrix P = diag (P1, P2, ..., P N ) and X=diag(X1,X2,…,X N ), further Then the gain matrix L, R is obtained, thus completing the design of the augmented fault state observer.

7. The robust control method for nonlinear discrete interconnected systems with disturbances based on state observer according to claim 1 or 6, characterized in that: The following state feedback controller is designed in step S4:

8. The robust control method for nonlinear discrete interconnected systems with disturbances based on state observer according to claim 7 is characterized in that: In step S5, the linear matrix inequality designed for the state feedback closed-loop system is: Where: γ 11 =diag(-Q1,-Q2,…,-Q N ); γ 13 =diag(Q1,Q2,…,Q N ); c 14 =diag(-B1Y1,-B2Y2,…,-B N Y N ); γ 15 days(Q1D1,Q2D2,...,Q N D N )4 and φ ii =Q i A i +B i Y i ,i=1,2,…,N. φ ij =Q i H ij ,i≠j and i,j=1,2,…,N. Q i B i K i =B i Y i ,i=1,2,…,N. Where: ∑ 11 =diag(-Q1,-Q1,…,-Q N ); and 9. The robust control method for nonlinear discrete interconnected systems with disturbances based on a state observer according to claim 8, characterized in that: The state feedback controller design process in step S4 is as follows: S4-1: Set the center and radius O2 and r2 of the disk to ensure the robust stabilization of the feedback closed-loop system; S4-2: Setting H ∞ Performance index γ2, reducing the impact of system disturbance on system robust control; S4-3: Based on the set parameters O2, r2 and γ2, solve the matrix inequality group (Equation 7) and (Equation 8) to calculate the undetermined matrix Q = diag (Q1, Q2, ..., Q N ) and Y=diag(Y1,Y2,…,Y N ), further Then, the controller gain matrix is ​​obtained so that the feedback control system satisfies the performance ||x(k)||≤γ2||η(k)|| and the robust stabilization of the closed-loop system is guaranteed.

10. The robust control method for nonlinear discrete interconnected systems with disturbances based on state observer according to claim 9, characterized in that: In step S5, the controller expression of the i-th system of the interconnected system based on the observer is:

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