Switching graph method for input and output stability of nonlinear system

Abstract

The invention discloses a switching graph method oriented to input and output stability of a nonlinear system, and relates to the technical field of switching system engineering, the method comprises the following steps: step 1, establishing a switching directed graph, in which nodes represent different subsystems, and edges represent switching among the subsystems; 2, constructing a vertex set and an edge set of the switched directed graph by analyzing the structural features of the switched directed graph; step 3, on the basis of the structure of the switched directed graph, defining an independent Lyapunov function for each subsystem, and dividing each subsystem into a stable subsystem and an unstable subsystem; 4, setting an average residence time constraint; and step 5, designing a feedback controller, suppressing pulse interference at the switching moment, and ensuring that the system keeps I < SS > in the switching process. According to the method, the conversion relation between the subsystems in the switching system is visually described through the directed graph, the system structure is clear, analysis and design are convenient, and the method is suitable for stability analysis of the complex nonlinear switching system.

Description

Technical Field

[0001] The present invention relates to the technical field of switching system engineering, and in particular to a switching diagram method for input-output stability of nonlinear systems. Background Art

[0002] A switching system consists of a set of continuous or discrete-time systems and the switching rules that govern them. The continuous (or discrete) time subsystem is typically described by a set of differential (or difference) equations. The switching rules are often referred to as switching laws. The switching signals are typically piecewise constant functions that depend on time, state, or both. Due to the presence of switching signals, the properties of a switching system are not simply the sum of the properties of its individual subsystems. Furthermore, the properties of a switching system generally do not inherit the properties of its subsystems. Because switching systems can improve their performance by applying appropriate switching rules, they have a wide range of applications in many practical engineering systems, such as furnace on / off control, automobile engine torque control systems, traffic management, chemical processes, power systems, and robotic control systems.

[0003] A Chinese patent (publication number: CN114371627A) discloses a method for determining the state stability of discrete-time nonlinear systems. This patent allows the input-output state stability (IOSS) of a system to be determined when the derivative of a Lyapunov function is positive definite within a certain interval. Furthermore, the obtained results are generalized to switching systems, and a new indefinite multiple Lyapunov function is proposed.

[0004] In the prior art, there is a problem of how to use the improved multi-Lyapunov method to make a non-autonomous discrete-time switching system an ISS, and how to achieve ISS in a pulse-switched nonlinear system with external input in the presence of an unstable subsystem. Therefore, the present invention proposes a switching diagram method for input-output stability of nonlinear systems. Summary of the Invention

[0005] The purpose of the present invention is to provide a switching diagram method for input-output stability of nonlinear systems to solve the problems mentioned in the above background technology.

[0006] The present invention can be implemented by the following technical solution: a switching diagram method for input-output stability of a nonlinear system, the method comprising the following steps:

[0007] Step 1: Create a switching directed graph, where nodes represent different subsystems and edges represent switching between subsystems;

[0008] Step 2: By analyzing the structural characteristics of the switching directed graph, the vertex set and edge set of the switching directed graph are constructed;

[0009] Step 3: Based on the structure of the switching directed graph, an independent Lyapunov function is defined for each subsystem, and each subsystem is divided into a stable subsystem and an unstable subsystem;

[0010] Step 4: Set the average dwell time constraint;

[0011] Set a minimum dwell time for each subsystem to ensure that the system maintains ISSS characteristics during switching;

[0012] Step 5: Design a feedback controller to suppress pulse interference at the switching moment and ensure that the system maintains I SS during the switching process.

[0013] A further technical improvement of the present invention is that the switching system is a pulsating switching system, and the stability of its subsystems includes:

[0014] All subsystems are stable;

[0015] All subsystems are unstable;

[0016] The Lyapunov method and state feedback control are combined to deal with the stability problem of the system. By designing an appropriate state feedback controller, the system is adjusted during the switching process to maintain the stability of the system. At the same time, the stability conditions of the switching system under the action of state feedback are derived.

[0017] Some subsystems are stable, while others are unstable;

[0018] For stable subsystems, Lyapunov method is used to prove their stability;

[0019] For unstable subsystems, state feedback controllers are designed to intervene in them at the switching moments to maintain the overall stability.

[0020] A further technical improvement of the present invention is that the design method of the state feedback controller includes:

[0021] S1. Define a Lyapunov function V(x) for the subsystem and require that the Lyapunov function V(x) decreases when the system state changes;

[0022] S2, adopt linear feedback control law u=-Kx;

[0023] Where u is the control input vector, which represents the control signal acting on the subsystem, and its purpose is to adjust the state of the system to achieve the desired stability;

[0024] K is the feedback gain matrix, which determines the magnitude and direction of the influence of the subsystem state x on the control input u;

[0025] x is the subsystem state vector, which represents the current state of the subsystem. The subsystem state vector contains the key variables of the subsystem and is used to describe the state information of the corresponding subsystem at a certain moment;

[0026] S3. Determine the feedback gain matrix K;

[0027] It adopts any one of the pole placement method, LQR (Linear Quadratic Regulation) method or H∞ control method;

[0028] S4. Check whether the state feedback controller satisfies the Lyapunov function V(x) condition;

[0029] By substituting the linear feedback control law u = -Kx into the Lyapunov function V(x), verify whether the closed-loop system satisfies the following inequality condition: V(f(x, -Kx)) ≤ ηV(x);

[0030] Where η is the shrinkage factor, 0≤η<1, which means that the Lyapunov function V(x) decreases under the action of the controller, thereby ensuring the stability of the system and ensuring that the value of the Lyapunov function V(x) is reduced to within η times of the original value after each update;

[0031] V(x) is the Lyapunov function, which is a non-negative function used to analyze the stability of the subsystem;

[0032] f(x, -Kx): The state update equation f(x, u) of the subsystem represents the evolution of the subsystem under the current state x and the control input u = -Kx;

[0033] S5, simulation verification;

[0034] After designing the state feedback controller, numerical simulation is performed to verify the effect of the state feedback controller in the actual switching system.

[0035] A further technical improvement of the present invention is that the state feedback controller adopts an adaptive design to timely adjust the feedback gain matrix K to cope with pulse interference of different strengths and timings, thereby ensuring the input state stability of the subsystem, which includes the following steps:

[0036] Q1. Detect the size and timing of pulse interference;

[0037] q1. Pulse size detection: During the state update process of the switching subsystem, the interference size is calculated through the state deviation;

[0038] By defining the interference detection threshold δ, when the subsystem state change Δx exceeds δ, that is, |Δx>δ, it is determined that a pulse interference occurs;

[0039] q2. Pulse time detection, which sets the pulse time window T and monitors the input and state changes of the subsystem within each pulse time window T. If a pulse is detected within the pulse time window T, its occurrence time t is recorded. p ;

[0040] Q2. In the absence of pulse interference or with minimal interference, standard feedback control is implemented using the basic feedback gain matrix K to maintain the input state stability of the system. At this time, the control input is: u = -Kx;

[0041] When pulse interference is detected, the control input is adjusted to u = -K p x;

[0042] Among them, K p =K+ΔK, which represents the adjusted feedback gain matrix;

[0043] ΔK is a gain compensation based on the size of the pulse interference. It is set according to the size of the pulse interference. The specific adjustment is as follows: ΔK = β × |Δx|;

[0044] Among them, β is the adjustment coefficient, which is used to control the proportional relationship between the feedback gain matrix and the disturbance size;

[0045] After the pulse interference ends and the system state returns to the allowable range, the feedback gain is reduced to the base value K. The attenuation formula is:

[0046] Where K(t) is the value of the feedback gain matrix over time t, which will gradually decay over time and gradually return to the normal gain level after the pulse interference ends;

[0047] is an exponential decay function, which is used to reduce the feedback gain matrix K(t) from K p It starts to decrease gradually, and its rate is controlled by α;

[0048] α is the attenuation coefficient, t p is the moment when the pulse interference occurs, and t is the current time;

[0049] Q3, numerical simulation verification;

[0050] The effectiveness of the dynamic adaptive design was verified through numerical simulations. Pulse interferences of different sizes and timings were simulated to observe the system's recovery and verify the input state stability of the system under the adaptive design.

[0051] And by adjusting the parameters δ, β and α, the influence of the adaptive design on the system response is observed and the control effect is optimized.

[0052] Compared with the prior art, the present invention has the following beneficial effects:

[0053] The present invention uses a directed graph to intuitively describe the conversion relationship between subsystems in a switching system, making the system structure clear and easy to analyze and design. It is particularly suitable for stability analysis of complex nonlinear switching systems.

[0054] In addition, by combining Lyapunov multi-function and average residence time methods, independent stability criteria are set for each subsystem, and global input and output state stability (ISS) is achieved in the switching system containing unstable subsystems, thereby enhancing the system's adaptability to various interferences and switching conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] To facilitate understanding by those skilled in the art, the present invention is further described below with reference to the accompanying drawings.

[0056] Figure 1 Schematic diagram of the ISS process of the pulse switching system of the present invention;

[0057] Figure 2 This is a state diagram of a switching system based on a switching directed graph of the present invention. DETAILED DESCRIPTION

[0058] In order to further illustrate the technical means and effects adopted by the present invention to achieve the predetermined purpose of the invention, the specific implementation methods, structures, features and effects of the present invention are described in detail below in conjunction with the accompanying drawings and preferred embodiments.

[0059] See also Figure 1-2 As shown, the present invention provides a switching diagram method for input-output stability of nonlinear systems, the method comprising the following steps:

[0060] Step 1: Create a switching directed graph, where nodes represent different subsystems and edges represent switching between subsystems;

[0061] Step 2: By analyzing the structural characteristics of the switching directed graph, the vertex set and edge set of the switching directed graph are constructed;

[0062] G(S, E(S)) is used to describe a directed graph, where φs represents a stable subsystem and φu represents an unstable subsystem. S is the index set of the vertex set. The self-loop at each vertex is marked by two positive integers mj,Mj,mj≤Mj,R∈{mj,mj+1,...,Mj}, where R indicates that the vertex (subsystem) is constrained when activated and must remain active in consecutive steps;

[0063] like Figure 2 As shown in the figure, there are four modes based on the Lyapunov method. The following is the mathematical expression of the method:

[0064] exist

[0065] α i ,i=1,2 belongs to the x∞ type equation, α belongs to the x type equation;

[0066] For any The following inequality holds:

[0067]

[0068] Step 3: Based on the structure of the switching directed graph, an independent Lyapunov function is defined for each subsystem, and each subsystem is divided into a stable subsystem and an unstable subsystem;

[0069] like Figure 1 As shown, the following is the mathematical expression of this method;

[0070] set up Then we have:

[0071]

[0072] Among them, R k is an integer, and

[0073] Step 4: Set the average dwell time constraint;

[0074] Set a minimum dwell time for each subsystem to ensure that the system maintains ISSS characteristics during switching;

[0075] Step 5: Design a feedback controller to suppress pulse interference at the switching moment and ensure that the system maintains I SS during the switching process.

[0076] The switching system is a pulsating switching system, and the stability of its subsystems includes:

[0077] All subsystems are stable;

[0078] All subsystems are unstable;

[0079] The Lyapunov method and state feedback control are combined to address the stability of the system. By designing an appropriate state feedback controller, the system is adjusted during the switching process to maintain system stability. At the same time, the stability conditions of the switching system under the action of state feedback are derived.

[0080] Some subsystems are stable, while others are unstable;

[0081] For stable subsystems, Lyapunov method is used to prove their stability;

[0082] For unstable subsystems, state feedback controllers are designed to intervene in them at the switching moments to maintain the overall stability.

[0083] The design methods of state feedback controller include:

[0084] S1. Define a Lyapunov function V(x) for the subsystem and require that the Lyapunov function V(x) decreases when the system state changes;

[0085] S2, adopt linear feedback control law u=-Kx;

[0086] Where u is the control input vector, which represents the control signal acting on the subsystem, and its purpose is to adjust the state of the system to achieve the desired stability;

[0087] K is the feedback gain matrix, which determines the magnitude and direction of the influence of the subsystem state x on the control input u;

[0088] x is the subsystem state vector, which represents the current state of the subsystem. The subsystem state vector contains the key variables of the subsystem and is used to describe the state information of the corresponding subsystem at a certain moment;

[0089] S3. Determine the feedback gain matrix K;

[0090] In this embodiment, the pole placement method is used to ensure system stability by placing the closed-loop eigenvalues ​​of the subsystem within the unit circle;

[0091] S4. Check whether the state feedback controller satisfies the Lyapunov function V(x) condition;

[0092] By substituting the linear feedback control law u = -Kx into the Lyapunov function V(x), verify whether the closed-loop system satisfies the following inequality condition: V(f(x, -Kx)) ≤ ηV(x);

[0093] This means that under the control input u = -Kx, the value of the Lyapunov function V(x) is reduced to η times its original value after the state update;

[0094] Where η is the shrinkage factor, 0≤η<1, which means that the Lyapunov function V(x) decreases under the action of the controller, thereby ensuring the stability of the system and ensuring that the value of the Lyapunov function V(x) is reduced to within η times of the original value after each update;

[0095] V(x) is the Lyapunov function, which is a non-negative function used to analyze the stability of the subsystem;

[0096] f(x, -Kx): The state update equation f(x, u) of the subsystem represents the evolution of the subsystem under the current state x and the control input u = -Kx;

[0097] S5, simulation verification;

[0098] After designing the state feedback controller, numerical simulation is performed to verify the effect of the state feedback controller in the actual switching system.

[0099] In this embodiment, the state feedback controller adopts an adaptive design to cope with pulse interference of different strengths and timings by timely adjusting the feedback gain matrix K, thereby ensuring the input state stability of the subsystem, which includes the following steps:

[0100] Q1. Detect the size and timing of pulse interference;

[0101] q1. Pulse size detection: During the state update process of the switching subsystem, the interference size is calculated through the state deviation;

[0102] By defining the interference detection threshold δ, when the subsystem state change Δx exceeds δ, that is, |Δx>δ, it is determined that a pulse interference occurs;

[0103] q2. Pulse time detection, which sets the pulse time window T and monitors the input and state changes of the subsystem within each pulse time window T. If a pulse is detected within the pulse time window T, its occurrence time t is recorded. p ;

[0104] Q2. In the absence of pulse interference or with minimal interference, standard feedback control is implemented using the basic feedback gain matrix K to maintain the input state stability of the system. At this time, the control input is: u = -Kx;

[0105] When pulse interference is detected, the control input is adjusted to u = -K p x;

[0106] Among them, K p =K+ΔK, which represents the adjusted feedback gain matrix;

[0107] ΔK is a gain compensation based on the size of the pulse interference. It is set according to the size of the pulse interference. The specific adjustment is as follows: ΔK = β × |Δx|;

[0108] Among them, β is the adjustment coefficient, which is used to control the proportional relationship between the feedback gain matrix and the disturbance size;

[0109] After the pulse interference ends and the system state returns to the allowable range, the feedback gain is reduced to the base value K. The attenuation formula is:

[0110] Where K(t) is the value of the feedback gain matrix over time t, which will gradually decay over time and gradually return to the normal gain level after the pulse interference ends;

[0111] is an exponential decay function, which is used to reduce the feedback gain matrix K(t) from K p It starts to decrease gradually, and its rate is controlled by α;

[0112] α is the attenuation coefficient, t p is the moment when the pulse interference occurs, and t is the current time;

[0113] Q3, numerical simulation verification;

[0114] The effectiveness of the dynamic adaptive design was verified through numerical simulations. Pulse interferences of different sizes and timings were simulated to observe the system's recovery and verify the input state stability of the system under the adaptive design.

[0115] And by adjusting the parameters δ, β and α, the influence of the adaptive design on the system response is observed and the control effect is optimized.

[0116] The above description is merely a preferred embodiment of the present invention and does not constitute any form of limitation to the present invention. Although the present invention has been disclosed as a preferred embodiment as above, it is not intended to limit the present invention. Any person skilled in the art can make some changes or modifications to equivalent embodiments using the technical contents disclosed above without departing from the scope of the technical solution of the present invention. However, any simple modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solution of the present invention are still within the scope of the technical solution of the present invention.

Claims

1. A switching diagram method for input-output stability of nonlinear systems, characterized in that: The method comprises the following steps: Step 1: Create a switching directed graph, where nodes represent different subsystems and edges represent switching between subsystems; Step 2: Construct the vertex set and edge set of the switching directed graph by analyzing the structural characteristics of the switching directed graph; Step 3: Based on the structure of the switching directed graph, an independent Lyapunov function is defined for each subsystem, and each subsystem is divided into a stable subsystem and an unstable subsystem; Step 4: Set the average residence time constraint; Set a minimum dwell time for each subsystem to ensure that the system maintains IS characteristics during switching; Step 5: Design a feedback controller to suppress pulse interference at the switching moment and ensure that the system maintains ISS during the switching process.

2. A switching diagram method for input-output stability of nonlinear systems according to claim 1, characterized in that: The switching system is a pulsating switching system, and the stability conditions of its subsystems include: all subsystems are stable, all subsystems are unstable, and some subsystems are stable, and some subsystems are unstable.

3. A switching diagram method for input-output stability of nonlinear systems according to claim 2, characterized in that: When all subsystems are unstable, the Lyapunov method and state feedback control are combined to deal with the stability problem of the system. By designing a state feedback controller, the system is adjusted during the switching process.

4. A switching diagram method for input-output stability of nonlinear systems according to claim 3, characterized in that: When some subsystems are stable and some are unstable; For stable subsystems, Lyapunov method is used to prove their stability; For unstable subsystems, state feedback controllers are designed to intervene at the switching moments.

5. A switching diagram method for input-output stability of nonlinear systems according to claim 4, characterized in that: The design method of state feedback controller includes: S1. Define a Lyapunov function V(x) for the subsystem and require that the Lyapunov function V(x) decreases when the system state changes; S2, adopting linear feedback control law u=-Kx; Among them, u is the control input vector, which represents the control signal acting on the subsystem; K is the feedback gain matrix; x is the subsystem state vector; S3, determining the feedback gain matrix K; It adopts any one of the pole placement method, LQR (Linear Quadratic Regulation) method or H∞ control method; S4, check whether the state feedback controller satisfies the Lyapunov function V(x) condition; S5, simulation verification; After designing the state feedback controller, numerical simulation is performed to verify the effect of the state feedback controller in the actual switching system.

6. A switching diagram method for input-output stability of nonlinear systems according to claim 5, characterized in that: In step S4, by replacing the linear feedback control law u=-Kx into the Lyapunov function V(x), verify whether the closed-loop system satisfies the following inequality condition: V(f(x,-Kx))≤ηV(x); Where η is the shrinkage factor, 0≤η<1; V(x) is the Lyapunov function; f(x, -Kx) is the state update equation of the subsystem, and f(x,u) represents the evolution of the subsystem under the current state x and control input u = -Kx.

7. A switching diagram method for input-output stability of nonlinear systems according to claim 5, characterized in that: The state feedback controller adopts an adaptive design and includes the following steps: Q1. Detect the size and timing of pulse interference; Q2. In the case of no pulse interference or interference less than the preset threshold, the basic feedback gain matrix K is used to implement standard feedback control to maintain the input state stability of the system. At this time, the control input is: u = -Kx; When pulse interference is detected, the control input is adjusted to u = -K p x; Among them, K p =K+ΔK, represents the adjusted feedback gain matrix; ΔK is a gain compensation based on the size of the pulse interference, which is set according to the size of the pulse interference. The specific adjustment is as follows: ΔK = β × |Δx|; Among them, β is the adjustment coefficient, which is used to control the proportional relationship between the feedback gain matrix and the disturbance size; Q3, numerical simulation verification; The effect of dynamically adjusting the adaptive design is verified through numerical simulation, pulse interferences of different sizes and times are simulated, the recovery of the system state is observed, and the input state stability of the system under the adaptive design is verified.

8. A switching diagram method for input-output stability of nonlinear systems according to claim 7, characterized in that: Q1 includes the following detection methods: q1. Pulse size detection. During the state update process of the switching subsystem, the interference size is calculated through the state deviation; By defining the interference detection threshold δ, when the subsystem state change Δx exceeds δ, that is, |Δx>δ, it is determined that pulse interference occurs; q2. Pulse time detection, which sets the pulse time window T and monitors the input and state changes of the subsystem within each pulse time window T. If a pulse is detected within the pulse time window T, its occurrence time t is recorded. p .

9. A switching diagram method for input-output stability of nonlinear systems according to claim 8, characterized in that: In step Q2, after the pulse interference ends and the system state returns to the allowable range, the feedback gain is reduced to the base value K. The attenuation formula is: Where K(t) is the value of the feedback gain matrix over time t, which decays over time; is an exponential decay function, α is the decay coefficient, t p is the moment when the pulse interference occurs, and t is the current time.

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