An invariant set control and guaranteed cost control method and system for a multi-equilibrium point switching generalized system

Abstract

The application provides a kind of invariant set control and guaranteed cost control method and system of multiple equilibrium point switching generalized system, belongs to the field of hybrid system control. In order to solve the problem that traditional switching generalized system is difficult to directly apply to different equilibrium points of actual engineering application, and the traditional guaranteed cost control method is invalid due to the switching characteristics of multiple equilibrium points. The application considers the more general case of multiple equilibrium points, and studies the invariant set control strategy method of multiple equilibrium point switching generalized system, so as to realize more accurate control of the behavior of complex switching system. On the other hand, for the guaranteed cost control problem in practical application, the application designs a multiple equilibrium point switching generalized guaranteed cost controller to ensure that the cost of the system can have an upper bound. This innovation provides strong technical support for the practical application of switching generalized system theory.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of hybrid system control, in particular to an invariant set control and guaranteed cost control method and system for a multi-equilibrium switching generalized system. BACKGROUND

[0002] Switched systems, as a unique class of hybrid control systems, are characterized by the integration of multiple subsystems (or modes) and a signal mechanism responsible for switching between these subsystems. In the field of physical system modeling and control involving mode transition characteristics, switched system theory plays a crucial role. This theory has permeated into multiple key application domains, such as unmanned aerial vehicle flight control, robotic arm manipulation, multi-agent cooperative system control, traffic flow management, and chemical process control, demonstrating its extensive practical value. Given the widespread application of switched systems in numerous high-tech and industrial fields, it is of great significance to explore and develop control strategies and performance guarantee techniques for switched systems to ensure invariant sets and enhance control performance.

[0003] In the field of switched system research, switched generalized systems are a special class of dynamic systems whose state equations are typically described by differential equations and algebraic equations. This class of systems not only includes the dynamic behavior of conventional switched systems but also involves the constraint characteristics of generalized systems, making them have a wider range of application scenarios. Due to the possibility of containing singular matrices and high-order dynamic constraints, the stability analysis and controller design of switched generalized systems are more challenging than those of traditional switched systems. In switched generalized systems, invariant set analysis and controller design are one of the core problems of research. Existing research has focused on the case where all subsystems share the same equilibrium point, simplifying the analysis. However, in actual engineering applications, each subsystem often has different equilibrium points, making it difficult for traditional switched generalized system methods to be directly applicable. Although some research has explored the invariant set conditions for multi-equilibrium switching systems, there is currently no invariant set analysis and controller design method suitable for generalized systems. Therefore, how to construct an invariant set analysis framework suitable for multi-equilibrium switching generalized systems and design effective control strategies based on this is still an important research problem that needs to be solved.

[0004] Guaranteed cost control is an important control strategy, which aims to ensure that the system is not only stable during switching, but also to optimize the performance index under certain constraints. Due to the existence of multiple equilibrium points, the dynamic behavior of the system is more complex, and the traditional guaranteed cost control method is difficult to be directly applied to such systems. Therefore, how to construct an effective guaranteed cost control strategy for multiple equilibrium point switching generalized systems has become a key problem in current research. The core goal of guaranteed cost control is to design a switching controller so that the performance index (usually represented by a quadratic cost function) of the system remains within a certain limit under all possible switching paths. However, due to the switching characteristics of multiple equilibrium points, the system may switch between different equilibrium points, making the traditional guaranteed cost control analysis method based on a single equilibrium point ineffective. Therefore, a new analysis framework needs to be constructed, combining with the invariant set theory to ensure that the system state can evolve within the feasible region corresponding to each equilibrium point, and on this basis, the overall performance of the system is optimized. Therefore, the study of guaranteed cost controller for multiple equilibrium point switching generalized systems has important theoretical value and practical significance for promoting the development of switching system control theory.

[0005] In summary, this patent focuses on the complex field of multiple equilibrium point switching generalized systems, aiming to innovatively propose a numerically verifiable invariant set criterion, aiming to provide an efficient method for invariant set analysis of such systems. Most importantly, this patent will first involve the invariant set control strategy and guaranteed cost control method for multiple equilibrium point switching generalized systems, which is an important supplement to the theory of switching system control. Through the method of this patent, not only a novel perspective and methodology for invariant set analysis of multiple equilibrium point switching generalized systems are provided, but also a new idea for the design of guaranteed cost controller is opened up, thereby greatly enriching the theoretical system of this field. In addition, these achievements will provide a solid theoretical foundation and practical guidance for the stable operation and efficient control of multiple equilibrium point switching generalized systems in practical engineering applications, showing a wide application prospect and far-reaching social and economic value. SUMMARY

[0006] The technical problems to be solved by the present application are:

[0007] In order to solve the problem that the traditional switching generalized system is difficult to be directly applied to different equilibrium points in practical engineering applications, and the traditional guaranteed cost control method is ineffective due to the switching characteristics of multiple equilibrium points.

[0008] The technical scheme adopted by the present application to solve the above technical problems is:

[0009] The present application provides an invariant set control and guaranteed cost control method for multiple equilibrium point switching generalized systems, which is applied to the stabilization control of air-ground cross-domain robot air mode and ground mode frequent switching scene, including the following steps:

[0010] S100, establishing a control system model of the multi-equilibrium switching generalized system, including finding two non-singular matrices of appropriate dimensions using singular value decomposition, and then using dynamic decomposition method to convert the system into an equivalent dynamic decomposition form through coordinate transformation;

[0011] S200, proposing a state jump mapping of the multi-equilibrium switching generalized system, including establishing a state jump mapping relationship of the multi-equilibrium switching generalized system at the modal switching moment based on the multi-equilibrium characteristics and the constraint relationship of algebraic equations, and then establishing a jump mapping relationship of the difference between the sub-state and the equilibrium point before and after the switching;

[0012] S300, proposing an invariant set criterion for the multi-equilibrium switching generalized system, including designing a Lyapunov function about the equilibrium point based on the multi-Lyapunov function method, and then proposing an invariant set criterion in the form of a linear matrix inequality by analyzing the descending rate of the Lyapunov function within the mode and the inclusion relationship between the sets;

[0013] S400, designing a multi-equilibrium switching generalized controller with invariant set characteristics and performance preserving characteristics, including designing a mode-dependent switching generalized performance preserving controller suitable for the multi-equilibrium switching generalized system based on the proposed invariant set criterion for the multi-equilibrium switching generalized system.

[0014] Further, in step S100, it includes,

[0015] S110, establishing a model of the multi-equilibrium switching generalized system:

[0016]

[0017] wherein, denotes the derivative of x, is a state vector, is a control input vector, σ(t) = i is a switching signal indicating that mode i is currently activated; σ(t) , A σ(t) , B σ(t) is a system matrix, respectively representing a state derivative matrix, a state transition matrix, and a control matrix; respectively represent the equilibrium state and the equilibrium input of mode i;

[0018] In the air-ground spanning robot application, the state vector x is a six-dimensional vector representing the attitude Euler angles and angular velocities in three directions of the air-ground spanning robot, the control input vector u is a three-dimensional vector representing the roll moments in three directions generated by the rotors of the air-ground spanning robot, and the equilibrium point and represent the equilibrium state and the equilibrium input in the steady flight under the air mode and the ground mode.

[0019] S120, using a dynamic decomposition technique, the multi-equilibrium switching generalized system is equivalent transformed,

[0020] Using singular value decomposition method to find non-singular matrix and matrix So that:

[0021]

[0022] Wherein, And Is the system matrix in the dynamic decomposition form; in addition,

[0023] Is a non-singular matrix;

[0024] Then introduce the state transformation relationship:

[0025]

[0026] Wherein, δ, ζ represent the system sub-state in the dynamic decomposition form, Represent the equilibrium point in the dynamic decomposition form;

[0027] In the new coordinate system, the system is equivalent to Has the following dynamic decomposition form;

[0028]

[0029] Wherein, Indicates the derivative of δ;

[0030] The definition of single-stage cost of multi-equilibrium switching generalized system is given: in the modal activation stage from t k To t, t∈[t k ,t k+1 ) and modal i is activated, the multi-equilibrium switching generalized system has single-stage cost J i :

[0031]

[0032] Wherein, Q i And R i Are positive definite weight matrix; if the single-stage cost satisfies Then It is called single-stage guaranteed cost under modal i.

[0033] Further, in step S200, it includes,

[0034] S210, considering the multi-equilibrium switching generalized system its dynamic decomposition form is At each switching instant t from mode i to mode j k , let denote the state at the post-switching instant in the dynamic decomposition form, denote the state at the pre-switching instant in the dynamic decomposition form, the state jump is mapped as

[0035]

[0036] wherein,

[0037]

[0038] At the switching instant, there is the following relationship:

[0039]

[0040] According to the state transformation relationship in the dynamic decomposition, there are and The solution of the original system is left-continuous, that is, Substitute them to obtain:

[0041]

[0042] Then, analyze the jump relationship of the distance between the state and the current equilibrium point before and after the switching, and establish the mapping relationship between and ;

[0043] S220, consider the multi-equilibrium point switching generalized system its dynamic decomposition form is At each switching instant t from mode i to mode j k , the state jump is mapped as

[0044]

[0045] wherein,

[0046]

[0047] According to the algebraic relationship obtain:

[0048]

[0049] In addition, there is a relationship and further establish the state jump relationship of the multi-equilibrium point switching generalized system at the switching instant.

[0050] Further, in step S300, comprising,

[0051] S310, considering open-loop multi-equilibrium switching generalized system Its dynamic decomposition form is Let λ i > 0, ρ i > κ i > 0 is a given constant; Lyapunov function related to equilibrium point The system energy between the sub-state δ and the equilibrium state , where L is the set of all modes of the switching system; if V i and K ∞ -type function α i , satisfy the following conditions, for :

[0052]

[0053] Where, Indicates the derivative of V i (δ) with respect to time;

[0054] Under the initial condition V σ(0) (δ(0))≤ρ σ(0) , for any switching signal with mode-dependent dwell time:

[0055]

[0056] The following invariant set properties hold:

[0057] (1) The sub-state δ of the dynamic decomposition form has a mode-dependent invariant set

[0058] (2) The state x of the original system has a mode-dependent invariant set Where,

[0059] And Indicates the pseudo-inverse of Y;

[0060] S320, based on step S310, considering open-loop multi-equilibrium switching generalized system Its dynamic decomposition form is Let the given energy reduction parameter λ i > 0, the region parameter ρ i > κ i > 0 is a given constant; assume that there exists to be solved variable ε i ≥ 0 and matrix P i > 0, so that for

[0061] ΛT i P i +P i Λ i +λ i P i ≤0 (21)

[0062]

[0063] where

[0064] Under the initial condition and the minimum dwell time switching signal of formula (20), the following properties are true:

[0065] (1) The sub-state δ has a mode-dependent invariant set Ω i :

[0066]

[0067] (2) The state x of the original system has a mode-dependent invariant set Ξ i :

[0068]

[0069] where, and

[0070] Further, the invariant set criterion of the multi-equilibrium switching generalized system in the form of a linear matrix inequality is established.

[0071] Further, in step S400, comprising,

[0072] Based on step S320, considering the multi-equilibrium switching generalized system, let λ i > 0, ρ i > κ i > 0 be a given constant; Q i and R i are positive definite weight matrices with appropriate dimensions; it is assumed that there exists a scalar ε i ≥ 0, a matrix X i > 0 and a matrix W i , such that for

[0073] Λ i X i +B i W i +(Λ i X i +B i W i ) T +λ iX i ≤0 (25)

[0074]

[0075]

[0076] wherein,

[0077] There is a multi-equilibrium switching generalized controller, so that under the initial condition and the minimum dwell time switching signal of formula (20), the closed-loop system has the following properties:

[0078] (1) The sub-state δ has a mode-dependent invariant set Ω described by formula (23) i wherein,

[0079] (2) The state x of the original system has a mode-dependent invariant set Ξ described by formula (24) i wherein,

[0080] (3) The original system has a single-stage guaranteed cost

[0081] In addition, if there is a feasible solution, the controller is given by:

[0082]

[0083] wherein,

[0084] An invariant set control and performance control system of a multi-equilibrium switching generalized system, the system has a program module corresponding to the above steps, and the steps in the above multi-equilibrium switching generalized system invariant set control and performance control method are executed when running.

[0085] A computer readable storage medium, the computer readable storage medium stores a computer program, the computer program is configured to be called by a processor to realize the steps of the multi-equilibrium switching generalized system invariant set control and performance control method.

[0086] Compared with the prior art, the beneficial effects of the present application are:

[0087] The concept of mode-dependent invariant set is proposed for a class of switching system models with different equilibrium points of each subsystem, and a numerically verifiable invariant set criterion suitable for multi-equilibrium point switching generalized systems is given.

[0088] In order to cope with the guaranteed cost control problem in actual application scenarios, the designed multi-equilibrium point switching generalized guaranteed cost controller can effectively guarantee the upper bound of the cost of the system in theory and actual effect; the multi-equilibrium point switching generalized guaranteed cost control method proposed in the application effectively reduces the error of the system and reduces the control cost, and improves the reliability and effectiveness of the control system.

[0089] In summary, the application effectively solves the problems of invariant set analysis and guaranteed cost controller design for switching generalized systems with different equilibrium points in each subsystem, and the proposed multi-equilibrium point switching generalized guaranteed cost controller has a wide range of applications, greatly improves the safety and reliability of the control system, and has high engineering application value. BRIEF DESCRIPTION OF DRAWINGS

[0090] Figure 1 The flowchart of the invariant set control and guaranteed cost control method for a multi-equilibrium point switching generalized system in an embodiment of the application is shown.

[0091] Figure 2 The state trajectory diagram of the multi-equilibrium point switching generalized system with invariant set guarantee in an embodiment of the application is shown.

[0092] Figure 3 The switching signal diagram used in the simulation experiment in an embodiment of the application is shown.

[0093] Figure 4 The state response curve diagram of the application of the multi-equilibrium point switching generalized controller in an embodiment of the application is shown.

[0094] Figure 5 The single-stage cost curve diagram of the application of the multi-equilibrium point switching generalized controller in an embodiment of the application is shown. DETAILED DESCRIPTION

[0095] In order to make the above-mentioned purposes, features and advantages of the application more obvious and easy to understand, the specific embodiments of the application will be described in detail below with reference to the accompanying drawings.

[0096] Specific implementation scheme one: combined with the drawings shown in Figure 1 and Figure 2 The application provides an invariant set control and guaranteed cost control method for a multi-equilibrium point switching generalized system, including the following steps:

[0097] S100, establishing a control system model of a multi-equilibrium switching generalized system, comprising finding two non-singular matrices of appropriate dimensions using singular value decomposition, and then using dynamic decomposition method to transform the original system into an equivalent dynamic decomposition form through coordinate transformation; specifically comprising:

[0098] S110, first considering the following multi-equilibrium switching generalized system model:

[0099]

[0100] wherein, denotes the derivative of x, is a state vector, is a control input vector, σ(t) = i is a switching signal, indicating that mode i is currently activated, which is selected from the set L = {1,...,N} of all modes, and N is the number of subsystems; σ(t) σ(t) σ(t) is a system matrix, also written as E i i i , respectively represent a state derivative matrix, a state transition matrix, and a control matrix; respectively represent the equilibrium state and the equilibrium input of mode i, written below as

[0101] S120, using dynamic decomposition technology to equivalently transform the multi-equilibrium switching generalized system, comprising,

[0102] using singular value decomposition to find a non-singular matrix and a matrix such that:

[0103]

[0104] wherein, is a system matrix in the dynamic decomposition form; in addition,

[0105] is a non-singular matrix;

[0106] then introducing a state transformation relationship:

[0107]

[0108] wherein, δ, ζ represent the system sub-state in the dynamic decomposition form, represent the equilibrium point in the dynamic decomposition form;

[0109] in the new coordinate system, the original system can be equivalently transformed into​​​​ where, denotes the derivative of the state in the dynamic decomposition form; it has the dynamic decomposition form as follows:

[0110]

[0111] where, denotes the derivative of δ; the dynamic decomposition form reveals the composite structure of the system dynamics, which consists of two parts: the r i th-order differential equation of the substate δ, and the algebraic relationship between the substate δ and ; in addition, it can be observed that the multi-equilibrium point property of the original system leads to the multi-equilibrium point property of the dynamic decomposition form;

[0112] The definition of the single-stage cost of the multi-equilibrium switching generalized system is given: from t k to t, t k and t respectively represent the switching time and the integral termination time, t∈[t k ,t k+1 ) and mode i is activated, the multi-equilibrium switching generalized system has a single-stage cost J i :

[0113]

[0114] where, Q i and R i are positive definite weight matrices with appropriate dimensions, i.e. Q i and R i have the same dimensions as x and u respectively; in addition, if the single-stage cost satisfies then is called the single-stage guaranteed cost under mode i;

[0115] S200, the state jump mapping of the multi-equilibrium switching generalized system is proposed, including the establishment of the state jump mapping relationship of the multi-equilibrium switching generalized system at the moment of mode switching based on the multi-equilibrium point property and the algebraic equation constraint relationship, and then the jump mapping relationship of the difference between the substate and the equilibrium point before and after the switching is established; specifically including,

[0116] Theorem 1: consider the multi-equilibrium switching generalized system its dynamic decomposition form is At each switching moment t k from mode i to mode j, let denote the state at the moment after switching in the dynamic decomposition form, denote the state at the moment before switching in the dynamic decomposition form, and the state jump mapping is:

[0117]

[0118] where,

[0119]

[0120] Proof: At the switching instant, there is the following relationship:

[0121]

[0122] where, The state at the switching instant is according to the state transformation relationship in dynamic decomposition, and and The solution of the original system is left continuous, that is, Substitute them into to obtain:

[0123]

[0124] Then, analyze the jump relationship of the distance between the state and the current equilibrium point before and after the switching, and establish the mapping relationship between and and

[0125] Theorem 2: Consider the multi-equilibrium switching generalized system Its dynamic decomposition form is At each switching instant t k from mode i to mode j, the state jump mapping is:

[0126]

[0127] where,

[0128]

[0129] Proof: According to the algebraic relationship

[0130]

[0131] In addition, there is a relationship Substitute these into formula (16), that is, the proof is completed;

[0132] Thus, the state jump relationship of the multi-equilibrium switching generalized system at the switching instant is established; that is, the state trajectory of the multi-equilibrium switching generalized system is discontinuous at the switching instant, and this discontinuous jump can be described by the above jump function;

[0133] ​​S300, the invariant set criterion of the multi-equilibrium switching generalized system is proposed, including the design of Lyapunov function about equilibrium point based on multi-Lyapunov function method; then, by analyzing the Lyapunov function descending rate inside the mode and the inclusion relation between sets, the linear matrix inequality form of the invariant set criterion is further proposed; including,

[0134] Theorem 3: Consider the open-loop multi-equilibrium switching generalized system The dynamic decomposition form is Let λ i > 0, ρ i > κ i > 0 be given constants; the Lyapunov function V related to the equilibrium point represents the system energy between the substate δ and the equilibrium state i and the class-K ∞ function α i , satisfy the following conditions, for

[0135]

[0136] Wherein, represents the derivative of V i (δ) with respect to time;

[0137] Then, under the initial condition V σ(0) (δ(0))≤ρ σ(0) , for any switching signal with mode-dependent dwell time:

[0138]

[0139] The following invariant set properties hold:

[0140] (1) The substate δ of the dynamic decomposition form has a mode-dependent invariant set

[0141] (2) The state x of the original system has a mode-dependent invariant set Wherein,

[0142] And represents the pseudo-inverse of Y;

[0143] Proof: Define the closed region Consider the case of ; for , on , integration can be obtained Therefore we have From the modal dependence on the residence time condition This means Thus, it is derived that This means Considering the initial condition δ(0) ∈ Ω σ(0) and δ(0 + ) = δ(0), where δ(0 + ) represents the instantaneous sub-state after the initial moment, the above process is recursively performed, and when the mode i is activated, the state is kept within Ω i ; in combination and get Therefore, the invariant set of the original system is ensured;

[0144] Using the control strategy proposed in the present application, the state trajectory of the sub-state δ in the dynamic decomposition form is as shown in Figure 2 The regions Ω i and are elliptical regions centered on Initially, mode 1 is activated and δ(0) ∈ Ω1; when mode 1 is continuously activated, the sub-state gradually enters At the moment when mode 1 switches to mode 2, the sub-state experiences an instantaneous jump, and the sub-state after the jump is located within Ω2; subsequently, the sub-state converges to the interior of and then jumps into Ω1; it can be observed that when mode 1 (or mode 2) is currently activated, the sub-state δ is kept within Ω1 (or Ω2);

[0145] Based on Theorem 3, an invariant set criterion in the form of a linear matrix inequality for a multi-equilibrium switching generalized system is designed as follows:

[0146] Theorem 4: Consider an open-loop multi-equilibrium switching generalized system Its dynamic decomposition form is Given an energy decreasing parameter λ i > 0, a region parameter ρ i > κ i > 0 is a given constant; it is assumed that there exist to-be-solved variables ε i ≥ 0 and a matrix P i > 0, such that for

[0147] Λ T i P i + P i Λ i + λ i P i ≤ 0 (21)

[0148]

[0149] where

[0150] Then, under the initial condition and the minimum dwell time switching signal (20), the following properties hold:

[0151] (1) The sub-state δ has a mode-dependent invariant set Ω i :

[0152]

[0153] (2) The state x of the original system has a mode-dependent invariant set Ξ i :

[0154]

[0155] where, and

[0156] Thus, the invariant set criterion of the multi-equilibrium switching generalized system in the form of linear matrix inequality is established;

[0157] S400, design a multi-equilibrium switching generalized controller with invariant set characteristics and performance preserving characteristics, including designing a mode-dependent switching generalized performance preserving controller suitable for a multi-equilibrium switching generalized system based on the proposed invariant set criterion of the multi-equilibrium switching generalized system; including,

[0158] Based on theorem 4, the multi-equilibrium switching generalized controller can be designed by the following method;

[0159] Theorem 5: considering a multi-equilibrium switching generalized system, let λ i > 0, ρ i > κ i > 0 be given constants; Q i and R i be positive definite weight matrices with appropriate dimensions; assume that there exists a scalar ε i ≥ 0, a matrix X i > 0 and a matrix W i such that for

[0160] Λ i X i +B i W i +(Λ i X i +B i W i ) T +λ i Xi ≤ 0 (25)

[0161]

[0162] wherein,

[0163] Then, there exists a multi-equilibrium switching generalized controller such that under the initial condition and the minimum dwell time switching signal (20), the closed-loop system has the following properties:

[0164] (1) the sub-state δ has a mode-dependent invariant set Ω described by (23) i wherein,

[0165] (2) the state x of the original system has a mode-dependent invariant set Ξ described by (24) i wherein,

[0166] (3) the original system has a single-stage guaranteed cost

[0167] In addition, if there exists a feasible solution, the controller is given by:

[0168]

[0169] wherein,

[0170] Specific implementation scheme two: the present application is a kind of invariant set control and performance guarantee control system of multi-equilibrium switching generalized system, the system has the program module corresponding to the above steps, and executes the steps in the invariant set control and performance guarantee control method of the above multi-equilibrium switching generalized system when running.

[0171] Other combinations and connection relationships of the present embodiment are the same as those of specific implementation scheme one.

[0172] Specific implementation scheme three: the present application is a kind of computer readable storage medium, the computer readable storage medium stores computer program, the computer program is configured to be called by processor to realize the steps of invariant set control and performance guarantee control method of multi-equilibrium switching generalized system.

[0173] Other combinations and connection relationships of the present embodiment are the same as those of specific implementation scheme one.

[0174] Simulation experiment

[0175] An aerial-ground transition robot is simulated, which has two moving modes, aerial mode and ground mode. The ground mode loses a degree of freedom because the passive wheels of the robot are in contact with the ground. The dynamics model of the aerial-ground transition robot is modeled as a multi-equilibrium switching generalized system by using equation (1). The state x is a six-dimensional vector, representing the Euler angles of the robot's attitude in three directions and the angular velocities in three directions. The control input u is a three-dimensional vector, representing the roll moments in three directions generated by the propeller. The equilibrium points and represent the equilibrium states and equilibrium inputs in the aerial mode and the ground mode under steady flight. A multi-equilibrium switching generalized controller is designed based on Theorem 5 in step S400. The controller considers the state jump behavior generated by the robot from the air to the ground during the design process. By optimizing the invariant set parameters and the system cost, the safety and reliability of the control system are enhanced. The design parameters are λ1=3.6, λ2=3.2, ρ1=1.8, ρ2=1.6, κ1=0.4, κ2=0.4,

[0176] Q1=Q2=diag{0.5,0.5,0.5,0.5,0.5,0.5} and R1=R2=diag{0.5,0.5,0.5}. The mode-dependent dwell time conditions are τ1≥0.4178 and τ2≥0.4332. The control gain of the multi-equilibrium switching generalized controller is:

[0177]

[0178] In the simulation results, the state response curve of the multi-equilibrium switching generalized controller is recorded. The switching signal is shown in Figure 3 , and the state response curve is shown in Figure 4 . The state experiences a sudden change when switching from mode 1 to mode 2, but no state jump occurs when switching from mode 2 to mode 1. This is because mode 1 does not involve algebraic constraints. Observations show that the state always remains within the mode-dependent invariant set Ξ i . The single-stage cost is calculated as shown in Figure 5 . The single-stage cost under mode 1 is less than 1.8, and the single-stage cost under mode 2 is less than 1.6, which indicates that the guaranteed cost of a single period is satisfied. The above simulation results verify the applicability of the proposed multi-equilibrium switching generalized controller on the aerial-ground transition UAV. The simulation results verify the effectiveness and advantages of the patent description method.

[0179] Although the present disclosure is as described above, the scope of protection of the present disclosure is not limited thereto. Those skilled in the art can make various changes and modifications without departing from the spirit and scope of the present disclosure, and these changes and modifications shall fall within the scope of protection of the present disclosure.

Claims

1. A method for invariant set control and performance-preserving control of a multi-equilibrium-point switching generalized system, characterized in that, The stabilization control applied to scenarios involving frequent switching between aerial and ground modes for air-to-ground cross-domain robots includes the following steps: S100. Establish a control system model for a generalized system with multiple equilibrium points, including finding two non-singular matrices of appropriate dimensions using singular value decomposition, and then using a dynamic decomposition method to transform the system into an equivalent dynamic decomposition form through coordinate transformation. include, S110. Establish a generalized system model with multiple equilibrium point switching: in, express The derivative of It is a state vector. It is the control input vector. It is a switching signal, indicating the mode. Currently activated; , , These are system matrices, representing the state derivative matrix, state transition matrix, and control matrix, respectively. , Representing modes The equilibrium state and the equilibrium input; In air-to-ground traversal robot applications, state vectors It is a six-dimensional vector representing the Euler angles of the air-to-ground cross-domain robot's attitude in three directions and its angular velocity in three directions, serving as the control input vector. It is a three-dimensional vector representing the rolling torque generated by the propeller of the air-to-ground cross-domain robot in three directions, and the equilibrium point. and It represents the equilibrium state and equilibrium input under stable flight conditions in both air and ground modes; S120. Employing dynamic decomposition technology, the multi-equilibrium-point switching generalized system is transformed into an equivalent system. Use singular value decomposition to find non-singular matrices. sum matrix , so that: in, and , , , It is the system matrix in the dynamic decomposition form; furthermore, A non-singular matrix; Then, the state transition relationship is introduced: in, , This represents the sub-state of the system in the dynamic decomposition form. , This represents the equilibrium point in the dynamic decomposition form. In the new coordinate system, the system is equivalent to It has the following dynamic decomposition form; （6） in, express The derivative; The definition of the single-stage cost of a multi-equilibrium-point switching generalized system is given: During the modal activation phase, from... arrive , And modality When activated, a multi-equilibrium-point switching generalized system has a single-stage cost. : （7） in, and It is a positive definite weight matrix; if the single-stage cost satisfies ,So Known as mode The cost of guaranteeing the order at each stage; S200. A state transition mapping for a multi-equilibrium-point switching generalized system is proposed, including establishing a state transition mapping relationship for the multi-equilibrium-point switching generalized system at the moment of mode switching based on the characteristics of multiple equilibrium points and the constraint relationship of algebraic equations, and then establishing a transition mapping relationship for the difference between sub-states and equilibrium points before and after switching. S300. Propose an invariant set criterion for generalized systems with multiple equilibrium points, including designing Lyapunov functions about equilibrium points based on the multi-Lyapunov function method; then, by analyzing the descent rate of Lyapunov functions within modes and the inclusion relationship between sets, propose an invariant set criterion in the form of a linear matrix inequality. S400. Design a multi-equilibrium-point switching generalized controller with invariant set properties and performance-preserving properties, including designing a mode-dependent switching generalized performance-preserving controller applicable to multi-equilibrium-point switching generalized systems based on the proposed invariant set criterion.

2. The invariant set control and performance-preserving control method for a multi-equilibrium-point switching generalized system according to claim 1, characterized in that: Step S200 includes, S210, Considering generalized systems with multiple equilibrium point switching Its dynamic decomposition form is ; in each mode Switch to mode The instant switching ,use This represents the state at the instant after the switch in the dynamic decomposition form. In the dynamic decomposition form, the state just before the transition is represented, and the state jump mapping is: （8） in, （9） （10） At the moment of switching, the following relationship exists: （11） Based on the state transition relationships in dynamic decomposition, we have , ,as well as The solution to the original system is left-continuous, i.e. Substituting them, we get: （12） Then, analyze the relationship between the distance between the current state and the current equilibrium point before and after the switch, and establish... and The mapping relationship between them; S220, Considering multi-balance-point switching generalized systems Its dynamic decomposition form is ; in each mode Switch to mode The instant switching The state jump mapping is as follows: （13） in, （14） （15） According to algebraic relations ,get: （16） In addition, there is a relationship Furthermore, we establish the state transition relationship of a multi-equilibrium-point switching generalized system at the instant of switching.

3. The invariant set control and performance-preserving control method for a multi-equilibrium-point switching generalized system according to claim 2, characterized in that: Step S300 includes, S310, Considering Open-Loop Multi-Balance-Point Switching Generalized System Its dynamic decomposition form is ;set up , Given a constant; Lyapunov function related to the equilibrium point Representing substates With equilibrium state The system energy between, of which It is the set of all modes of the switching system; if and class function , For the following conditions are met, ,but: （17） （18） （19） in, express The derivative with respect to time; Under initial conditions For any switching signal with modality-dependent residence time: （20） The following properties of invariant sets hold: (1) Substates in dynamic decomposition form Invariant sets with modality dependencies ; (2) The state of the original system Invariant sets with modality dependencies ,in, ,and , express The false reversal; S320. Based on step S310, consider the open-loop multi-balance point switching generalized system. Its dynamic decomposition form is Given a parameter for energy decrease Regional parameters Given constants; assume there exist variables to be determined. sum matrix This makes for , : （21） （22） in ; Under initial conditions And under the minimum dwell time switching signal of formula (20), the following properties hold true: (1) Substate Invariant sets with modality dependencies : （23） (2) The state of the original system Invariant sets with modality dependencies : （24） in, ,and ; Furthermore, an invariant set criterion for multi-equilibrium-switching generalized systems is established in the form of linear matrix inequalities.

4. The invariant set control and performance-preserving control method for a multi-equilibrium-point switching generalized system according to claim 3, characterized in that: In step S400, the following are included: Based on step S320, considering a multi-equilibrium-point switching generalized system, let... , Given a constant; and It is a positive definite weight matrix with appropriate dimensions; assume the existence of a scalar ,matrix sum matrix This makes for , : （25） （26） （27） in, , ; There exists a multi-equilibrium-point switching generalized controller such that, under initial conditions... Under the minimum dwell time switching signal of formula (20), the closed-loop system has the following properties: (1) Substate An invariant set having modal dependencies described by formula (23) ,in ; (2) The state of the original system An invariant set having modal dependencies described by formula (24) ,in ; (3) The original system has a single-stage guarantee cost. ; Furthermore, if a feasible solution exists, the controller is given by the following equation: （28） in, .

5. An invariant set control and performance-preserving control system for a multi-equilibrium-point switching generalized system, characterized in that: The system has a program module corresponding to the steps of any one of the claims 1-4 above, and executes the steps in the above-described method for invariant set control and performance-preserving control of multi-equilibrium-point switching generalized systems when it is run.

6. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores a computer program configured to, when invoked by a processor, implement the steps of the invariant set control and performance-preserving control method for the multi-equilibrium-point switching generalized system as described in any one of claims 1-4.

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