A method of negative virtualization with a high resonant mode flexibility system

Abstract

The application provides a negative virtualization method for a high resonance mode flexible system, first, a flexible system model is established, a dynamic feedforward compensator is introduced, and a system state space equation after feedforward compensation is given; second, a set of positive definite matrices is selected for the compensated system, and a set of matrix inequality conditions are derived, so that the compensated system has a negative virtual property, and further solving the matrix inequality conditions can obtain the parameters of the feedforward compensator; finally, an improved negative virtual controller is designed for the system after negative virtualization to ensure the stability of the closed-loop system. The method of the application ensures that the high resonance mode flexible system with non-negative virtual property can use the negative virtual theory to design the controller, and expands the application range of the negative virtual theory in the flexible system. Through the satellite model simulation experiment, it is shown that under the condition of parameter perturbation or unmodeled dynamics, the designed system can simultaneously consider the response characteristics and robustness.

Description

Technical Field

[0001] This invention relates to the field of flexible system control technology, and specifically to a negative virtualization method for a flexible system with high resonant modes. Background Technology

[0002] Flexible structures are found in many fields, such as flexible robot manipulators, payloads on space satellites (e.g., antennas, solar cell arrays), and cantilever beams in precision instruments (e.g., atomic force microscopes and optical systems). Systems that incorporate such flexible structures are generally referred to as flexible systems. However, due to parameter uncertainties and unmodeled system dynamics, the control performance and stability of flexible systems can be affected, thus requiring robust control design.

[0003] Traditional control methods, such as H∞ loop shaping, hybrid sensitivity compensation, and model matching, complicate controller design when dealing with flexible systems with uncertainties, especially when the system has weak damping characteristics. This can lead to conservative or unstable controller designs. While the latest negative virtual control theory offers significant advantages in handling flexible systems with high resonant modes, it currently only applies to flexible structures with negative virtual properties. Real-world flexible systems are far more complex than flexible structures; sensor configurations and the presence of rigid modes mean the system itself does not possess negative virtual properties, making existing negative virtual control theory unsuitable. Therefore, a negative virtualization method is urgently needed to address practical problems. This method involves negative virtualization design of flexible systems with high resonant modes (shifting the transformed Nyquist curve below the imaginary axis on the polar graph), giving the transformed system negative virtuality. This allows for a simple negative virtual controller design that guarantees system stability and robustness. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention proposes a negative virtualization method for flexible systems with high resonant modes, comprising the following steps:

[0005] Step 1: Design a dynamic feedforward compensator based on a flexible system model with high resonance modes, and establish the state-space equations of the compensated system;

[0006] Step 2: Based on the negative imaginary criterion, design a set of linear matrix inequalities (LMI) conditions to guarantee the negative imaginary property of the compensated system, and solve for the compensator parameters;

[0007] Step 3: Design an improved negative virtual controller for the negative virtualized system to ensure the performance and stability of the closed-loop system;

[0008] Step 1 includes:

[0009] Step 1.1: Establish a model representation of the flexible system in which the actuator and position sensor are located on opposite sides of the flexible structure.

[0010]

[0011] Wherein, the modal resonant frequency ω i >0, modal damping ζ i >0, modal gain c i >0, when i≠1, c i >0. When i=1, c1=-c0, G(s) represents the transfer function of the flexible system, c0 represents the rigid modal system, and s represents the complex frequency;

[0012] State-space expression of a flexible system:

[0013]

[0014] Simplifying, we obtain the state-space equation:

[0015]

[0016] in,

[0017] In the formula, A1 and A2 represent the state matrices of the flexible and rigid objects, respectively; B1 and B2 represent the input matrices of the flexible and rigid objects, respectively; C1 and C2 represent the output matrices of the flexible and rigid objects, respectively; I represents the appropriate identity matrix; B 21 B 22 C represents the input submatrix of a rigid object. 21 C 22 The output submatrix representation of a rigid object is given by G1(s), which represents the subsystem transfer function of G(s).

[0018] Step 1.2: Establish the state-space expression of the dynamic feedforward compensator:

[0019]

[0020] Among them, G K Let A represent the transfer function of the feedforward compensator. K It is the Herwitz matrix, A K B K C K For the parameters of the feedforward compensator to be solved, u k Indicates the input of the compensator, x k y represents the state variable of the compensator. k Indicates the output variable of the compensator;

[0021] Step 1.3: Further simplify the state-space equations of the system after feedforward compensation:

[0022]

[0023] in,

[0024] Step 2 includes:

[0025] Step 2.1: For the compensated system G NI (s) Choose the Lyapunov positive definite matrix Y:

[0026]

[0027] Where, matrix Y1 = Y1 T >0, Y 12 =Y2;Y1,Y 12 Y1 and Y2 represent the positive definite matrices to be solved; given the positive definite condition, matrix Y is positive definite, and the positive definiteness of Y is equivalent to:

[0028]

[0029] According to the condition Y1=Y1 T >0, Y 12 =Y2, so the positive definiteness of matrix Y is equivalent to the condition Y1-Y2>0; Step 2.2: Design feedforward compensation to make the flexible system negative virtualized, which needs to satisfy the condition

[0030]

[0031] Among them, matrix therefore Equivalent to the inequality condition:

[0032]

[0033] Secondly, the equality conditions must also be met. therefore

[0034]

[0035] in therefore Equivalent to:

[0036]

[0037]

[0038] Equivalence Multiply both sides by A on the left. K -1 Therefore, we can obtain

[0039] in So the condition Equivalent to the linear inequality condition:

[0040]

[0041]

[0042] Finally, consider the condition for the rigid modal part to satisfy negative imaginary property, when s = 0 and G is... NI When (s) has double poles, there is

[0043]

[0044] Therefore, G2 = C 21 B 22 When >0, It is the Hamiltonian matrix, and the compensated system G NI (s) has negative imaginary properties;

[0045] Step 2.3: Solve the linear matrix inequality conditions:

[0046] Y1-Y2>0

[0047]

[0048]

[0049]

[0050] Among them, matrices Y1, Y2 and matrix This is the solution to the above linear matrix inequality conditions, which can be solved using MATLAB. Due to the matrix parameters... Therefore, the feedforward compensator parameter A can be further solved. K B K C K .

[0051] Step 3 includes:

[0052] Step 3.1: Design an improved integral resonant controller:

[0053]

[0054] Where k, φ, and τ are controller parameters, φ>0, τ>0, and k>φ / τ;

[0055] Step 3.2: Establish the negative virtualized system G NI The closed-loop system stability conditions for (s) and the improved controller C(s):

[0056] G NI (0)C(0)<0

[0057] Among them, G NI (0) is G NI C(s) is the zero-frequency gain of C(s), and C(0) is the zero-frequency gain of C(s).

[0058] The beneficial effects of this invention are:

[0059] The negative virtualization design of this invention ensures that a class of systems with non-negative virtual properties can use negative virtual theory to design controllers. In addition, it considers the case where the system has rigid body modes. Compared with the initial negative virtual system which only focuses on flexible structures with resonant modes, it broadens the scope of application in practical applications and solves the robustness problems such as parameter perturbation and modal overflow in the control design of weakly damped flexible systems. Attached Figure Description

[0060] Figure 1 The flowchart shows the design process of the negative virtualization method for a flexible system with high resonant modes in this invention.

[0061] Figure 2 This is a block diagram of the negative virtualization method based on the feedforward compensator in this invention;

[0062] Figure 3 This is a block diagram of the robust scheme for a closed-loop system based on feedforward compensation in this invention.

[0063] Figure 4 The nominal system G(s) and the feedforward compensated system G in this invention NI Polar plot of (s);

[0064] Figure 5 The compensated system G in this invention NI The Nyquist curve of parameter k after being perturbed by a factor of 10 in (s);

[0065] Figure 6 These are the perturbation output curves for the two schemes under the initial condition of 0.2 rad / s in this invention. Detailed Implementation

[0066] The invention will be further described below with reference to the accompanying drawings and specific implementation examples. This invention provides a negative virtualization design method for a class of flexible systems with high resonant modes. First, a model of the flexible structure and the flexible system is established, and a feedforward dynamic compensator is introduced to further simplify the state-space equations of the system after feedforward compensation. Second, a suitable positive definite matrix is ​​selected, and a set of LMI conditions is derived to make the compensated system have negative virtual properties. The compensator parameters can be obtained by solving the LMI conditions. Finally, a negative virtual controller is designed for the negative virtualized system to ensure the stability of the closed-loop system. The negative virtualization design of this invention ensures that a class of systems with non-negative virtual properties can use negative virtuality theory to design controllers.

[0067] This invention is specifically applied to a celestial probe satellite. The system consists of two components: the satellite body and an optical instrument panel. These two components are connected by a flexible linkage, and its mathematical model is represented by a damped spring. The satellite attitude angle is the angle between the celestial sensor and the instrument panel. The parameters d and k of the flexible structure are affected by temperature.

[0068] like Figure 1 As shown, a negative virtualization method for a flexible system with high resonant modes includes:

[0069] Step 1: Design a dynamic feedforward compensator based on a flexible system model with high resonant modes; specifically:

[0070] Step 1.1: Establish a mathematical model of the flexible structure from the actuator input to the corresponding sensor measurement output:

[0071]

[0072] Where, ζ i >0,ω i >0, s is a complex variable, φ i (s) is a first-order polynomial form, φ i (s) Depending on the sensor type and the structural configuration of the actuator and sensor, when a position sensor is used for measurement and the sensor and actuator are configured on the same side, φ i (s)=c i A value greater than 0 indicates a negative imaginary system for this type of flexible structure. That is, for a single-input, single-output system, the Nyquist curve of its transfer function lies below the real axis.

[0073] Table 1 Modal Parameters of Flexible System

[0074]

[0075] Establish a model representation of a flexible system in which the actuator and position sensor are located on opposite sides of the flexible structure:

[0076]

[0077] The negative imaginary mode in the third term on the right-hand side of the above equation can be treated as an uncertainty Δ(s). According to Table 1, we can choose k = 0.091 and d = 0.0036 to design the system according to the resonant frequency ω1 = 1 and the damping ratio ζ = 0.02. Therefore, the nominal system transfer function is...

[0078]

[0079] For the state-space expression of a flexible system:

[0080]

[0081] in:

[0082] Step 1.2: Establish the state-space expression of the dynamic feedforward compensator:

[0083]

[0084] Among them, A K It is the Herwitz matrix.

[0085] Step 1.3: Further simplify the state-space equations of the system after feedforward compensation:

[0086]

[0087] in:

[0088] Figure 2 The flexible system G(s) consists of rigid modes and high-resonance modes G1(s), where G1(s) does not have negative imaginary properties, resulting in G(s) also not being negative imaginary. K (s) represents the added dynamic feedforward compensator, G K (s) acts on the flexible system G(s), which can make the compensated system G NI (s) has negative imaginary properties.

[0089] Step 2.1: Choose the Lyapunov positive definite matrix as:

[0090]

[0091] Matrix Y1 = Y1 T >0, Y 12 =Y². The matrix Y is positive definite, and the positive definiteness of Y is equivalent to... Therefore, according to the condition Y1=Y1 T >0, Y 12=Y2, so Y is positive definite if Y1-Y2>0.

[0092] Step 2.2: To design the LMI condition for feedforward compensation that enables the flexible system to exhibit negative virtualization, it is first necessary to prove...

[0093]

[0094] Among them, matrix therefore Equivalent to the inequality condition:

[0095]

[0096] Secondly, the equality conditions must also be met. therefore

[0097]

[0098] in therefore Equivalent to the condition:

[0099]

[0100]

[0101] Equivalence Multiply both sides by A on the left. K -1 Therefore, we can obtain

[0102] in So the condition Equivalent to the linear matrix inequality condition:

[0103]

[0104]

[0105] Finally, consider the condition for the rigid modal part to satisfy negative imaginary property, when s = 0 and G is... NI When (s) has double poles, there is

[0106]

[0107] Therefore, when G2 = C 21 B 22 When >0, It is the Hamiltonian matrix, and the compensated system G NI (s) has negative imaginary properties.

[0108] Step 2.3: Solve the matrix inequality conditions:

[0109] Y1-Y2>0

[0110]

[0111]

[0112]

[0113] The solution is:

[0114]

[0115] because Therefore, the feedforward compensator parameter A can be further solved. K B K C K for:

[0116]

[0117] Therefore, the system after negative virtualization can be concluded as:

[0118]

[0119] Step 3: Provide a negative virtual controller for the negative virtualized system to ensure the performance and stability of the closed-loop system; including:

[0120] Step 3.1: The improved integral resonant controller is as follows:

[0121]

[0122] Wherein, the parameters φ = 4.1028, τ = 1.4524, and k = 10.4257.

[0123] Step 3.2: Give the system G after negative virtualization NI The closed-loop system stability conditions for (s) and the improved controller C(s):

[0124] G NI (0)C(0)<0

[0125] Among them, G NI (0) is G NI The zero-frequency gain of C(s) is C(0), and C(s) is the zero-frequency gain of C(s). Since G NI (0)C(0)=-6.841, therefore the closed-loop system is stable.

[0126] Figure 3 The diagram shows a robust scheme for a closed-loop system based on feedforward compensation. The closed-loop system remains stable even when uncertainty Δ is considered. Figure 4 For the nominal system G(s) and the feedforward compensated system GNI The Nyquist curve of (s), where the nominal system G(s) does not have negative imaginary properties, while the system G after feedforward compensation... NI (s) acquires negative imaginary property, meaning the Nyquist curve shifts below the real axis. Further consideration is given to the robustness of this feedforward scheme, i.e., maintaining the negative imaginary property of the system after parameter perturbations. Figure 5 Perturb the Nyquist curve by a factor of 10 for the structural parameter k. It can be seen that the curve remains below the imaginary axis after the perturbation and has not changed. NI The negative imaginary nature of (s) means that the system G after perturbation NI (s) remains stable.

[0127] Compare the robustness and stability of our method with that of the H∞ loop forming method. From Figure 6 It can be concluded that under the initial condition of θ2 = 0.2 rad / s, the maximum perturbation peak value of the negative virtual scheme is 0.03 rad / s, and the system output tends to 0 after 0.5 s. In contrast, the maximum perturbation peak value of the loop shaping method is 0.1 rad / s, and the system output tends to 0 after 35 s. Therefore, the negative virtual scheme has a significantly faster response and smaller overshoot. Under the H∞ loop shaping scheme, the stability boundary is reached when the angular frequency perturbation is 3.5%, i.e., ω1 increases from 1 to 1.035 rad / s. However, the negative virtual scheme maintains its negative virtuality even after a 20% angular frequency perturbation, i.e., ω1 = 1.2 rad / s, thus the system is stable.

Claims

1. A method for negative virtualization of a flexible system with high resonant modes, characterized in that, include: Step 1: Design a dynamic feedforward compensator based on a flexible system model with high resonance modes, and establish the state-space equations of the compensated system; Step 2: Based on the negative imaginary criterion, design a set of linear matrix inequalities to guarantee the negative imaginary property of the compensated system, and solve for the compensator parameters; Step 2.1: For the compensated system Choose a Lyapunov positive definite matrix : Among them, matrix , , ; , , Let each represent a positive definite matrix to be solved. Given the positive definite condition, the matrix It is positive definite. The positive definiteness is equivalent to: According to the conditions , , It can be seen that the matrix Positive definiteness is equivalent to conditional ; Step 2.2: Design feedforward compensation to achieve negative virtualization of the flexible system, which must meet the following conditions. ; Among them, matrix therefore Equivalent to the inequality condition: Secondly, the equality conditions must also be met. ,therefore in ,therefore Equivalent to: Equivalence Multiply both sides by left Therefore, we can obtain ; in , so the condition Equivalent to the linear inequality condition: Finally, consider the condition for the rigid modal component to satisfy negative imaginary property, when... And for The double poles are sometimes present therefore, hour, It is the Hamiltonian matrix, and the compensated system It has negative virtual properties; Step 2.3: Solve the linear matrix inequality conditions: Among them, matrix sum matrix This is the solution to the above linear matrix inequality conditions, due to the matrix parameters. Therefore, the parameters of the feedforward compensator can be solved. ; Step 3: Design an improved negative virtual controller for the negative virtualized system to ensure the performance and stability of the closed-loop system; Step 3.1: Design an improved negative virtual controller C(s): in, , , For the parameters of the improved negative virtual controller, , , ; Step 3.2: Establish the system after negative virtualization and improved negative virtual controller Stability conditions for a closed-loop system: in, for Zero frequency gain for Zero frequency gain.

2. The negative virtualization method for a flexible system with high resonant modes according to claim 1, characterized in that, Step 1 includes: Step 1.1: Establish the state-space equations for the flexible system with the actuator and position sensor located on opposite sides of the flexible structure: in, ; In the formula, , These represent the state matrices of flexible and rigid objects, respectively. , These represent the input matrices for flexible and rigid objects, respectively. , These represent the output matrices for flexible and rigid objects, respectively. Represent the appropriate identity matrix. , Represents the input submatrix of a rigid object. Let s represent the transfer function of the flexible system, and let s represent the complex frequency. , The output submatrix representation represents a rigid object. express Subsystem transfer function; Step 1.2: Establish the state-space expression of the dynamic feedforward compensator: in, This represents the transfer function of the feedforward compensator. It is the Herwitz matrix. , , To solve for the parameters of the feedforward compensator, Indicates the input of the compensator. Represents the state variables of the compensator. Indicates the output variable of the compensator; Step 1.3: The simplified state-space equation of the system after feedforward compensation is expressed as: in, The transfer function of the compensated system, .

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