A switching graph method for input-output stability of nonlinear systems

Abstract

This invention discloses a switching graph method for input-output stability analysis of nonlinear systems, relating to the field of switching system engineering technology. The method includes the following steps: Step 1: Establishing a switching directed graph, where nodes represent different subsystems and edges represent the switching between subsystems; Step 2: Constructing the vertex set and edge set of the switching directed graph by analyzing its structural characteristics; Step 3: Based on the structure of the switching directed graph, defining an independent Lyapunov function for each subsystem and dividing each subsystem into stable and unstable subsystems; Step 4: Setting average dwell time constraints; Step 5: Designing a feedback controller to suppress impulse interference at the switching time, ensuring that the system maintains ISS (Inter-Standard Situation) during the switching process. This invention intuitively describes the transition relationships between subsystems in a switching system using a directed graph, resulting in a clear system structure that facilitates analysis and design, and is suitable for stability analysis of complex nonlinear switching systems.

Description

Technical Field

[0001] This invention relates to the field of switching system engineering technology, specifically a switching graph method for input-output stability of nonlinear systems. Background Technology

[0002] A switched system consists of a set of continuous or discrete-time subsystems and a switching rule acting upon them. The continuous (or discrete)-time subsystems are typically described by a set of differential (difference) equations. The switching rule, also known as the switching law, is a piecewise constant function that depends on time, state, or both. Due to the presence of the switching signal, the properties of a switched system are not simply a superposition of the properties of its subsystems. Furthermore, the properties of a switched system generally do not inherit those of its subsystems. Because the performance of a switched system can be improved by applying appropriate switching rules, switched systems have wide applications in many practical engineering systems, such as furnace switching control, automobile engine torque control systems, traffic management, chemical processes, power systems, and robot control systems.

[0003] A search revealed a Chinese patent (publication number: CN114371627A) that discloses a method for determining the state stability of a discrete-time nonlinear system. This patent allows for determining the input-output state stability (IOSS) of a system when the derivative of the Lyapunov function is positive definite within certain intervals. Furthermore, the results are extended to switched systems, proposing a new indeterminate multiple Lyapunov function.

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

[0005] The purpose of this invention is to provide a switching graph method for input-output stability of nonlinear systems, so as to solve the problems mentioned in the background art.

[0006] This invention can be achieved through the following technical solution: a switching graph method for input-output stability of nonlinear systems, the method comprising the following steps:

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

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

[0009] Step 3: Based on the structure of the switching directed graph, define an independent Lyapunov function for each subsystem, and divide each subsystem into stable subsystems and unstable subsystems;

[0010] Step 4: Set average dwell time constraints;

[0011] To set a minimum dwell time for each subsystem to ensure that the system maintains its IS characteristics during handover;

[0012] Step 5: Design a feedback controller to suppress impulse interference at the switching time and ensure that the system maintains IS during the switching process.

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

[0014] All subsystems are stable;

[0015] All subsystems are unstable;

[0016] This paper combines the Lyapunov method and state feedback control to address the system stability problem. By designing an appropriate state feedback controller, the paper attempts to adjust the system during switching to maintain its stability. Furthermore, the stability condition of the switching system under state feedback is derived.

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

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

[0019] For unstable subsystems, design a state feedback controller to intervene at switching times to maintain overall stability.

[0020] A further technical improvement of the present invention lies in the fact 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 as the system state changes;

[0022] S2. A linear feedback control law u = -Kx is adopted;

[0023] Where u is the control input vector, representing the control signal acting on the subsystem, the purpose of which 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, representing 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 employs any one of the following methods: pole placement, LQR (linear quadratic control), or H∞ control.

[0028] S4. Check the state feedback controller's satisfaction with the Lyapunov function V(x) conditions;

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

[0030] Where η is the contraction 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 the original value after each update;

[0031] V(x) is the Lyapunov function, a non-negative function used to analyze the stability of a 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 simulations were performed to verify its effectiveness in a real switching system.

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

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

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

[0038] By defining an interference detection threshold δ, when the subsystem state change Δx exceeds δ, i.e. |Δx>δ, it is determined that an impulse interference has occurred;

[0039] q2. Pulse timing detection: This method monitors the input and state changes of the subsystem within a set 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 stability of the system's input state. In this case, 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, representing the adjusted feedback gain matrix;

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

[0044] Where β is the adjustment coefficient, used to control the proportional relationship between the feedback gain matrix and the magnitude of the disturbance;

[0045] After the pulse interference ends and the system state returns to within acceptable limits, reduce the feedback gain to the base value K, with the attenuation formula as follows:

[0046] Wherein, K(t) is the value of the feedback gain matrix with time t. This value will gradually decay over time, and thus gradually recover to the normal gain level after the pulse interference ends.

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

[0048] α is the attenuation coefficient, t p The moment the pulse interference occurs is t, where t is the current time.

[0049] Q3. Numerical simulation verification;

[0050] The effect of dynamic adjustment adaptive design is verified by numerical simulation. Impulse interference of different magnitudes and times is simulated, the recovery of system state is observed, and the stability of system input state under adaptive design is verified.

[0051] Furthermore, by adjusting the parameters δ, β, and α, the impact of adaptive design on the system response was observed, and the control effect was optimized.

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

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

[0054] Furthermore, by combining Lyapunov multifunction and average dwell time methods, an independent stability criterion is set for each subsystem, achieving global input and output state stability (ISS) in a switched system containing unstable subsystems, thereby enhancing the system's adaptability to various disturbances and switching conditions. Attached Figure Description

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

[0056] Figure 1 This is a schematic diagram of the ISS process of the pulse switching system of the present invention;

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

[0058] To further illustrate the technical means and effects of the present invention in achieving its intended purpose, the following detailed description of the specific implementation methods, structures, features, and effects of the present invention, in conjunction with the accompanying drawings and preferred embodiments, is provided.

[0059] Please see Figure 1-2 As shown, this invention provides a switching graph method for input-output stability of nonlinear systems, which includes the following steps:

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

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

[0062] The directed graph is described by G(S, E(S)), where φs represents a stable subsystem and φu represents an unstable subsystem. S is the index set of the vertex set. A self-loop at each vertex is labeled with two positive integers mj, Mj, mj ≤ Mj, R ∈ {mj, mj+1, ..., Mj}, where R indicates that the vertex (subsystem) is constrained at activation and must remain active in successive steps.

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

[0064] exist

[0065] α i ,i=1,2 belong to the x∞ class of equations, and α belongs to the x class of equations;

[0066] For any The following inequalities hold:

[0067]

[0068] Step 3: Based on the structure of the switching directed graph, define an independent Lyapunov function for each subsystem, and divide each subsystem into stable subsystems and unstable subsystems;

[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 It is an integer, and

[0073] Step 4: Set average dwell time constraints;

[0074] To set a minimum dwell time for each subsystem to ensure that the system maintains its IS characteristics during handover;

[0075] Step 5: Design a feedback controller to suppress impulse interference at the switching time and ensure that the system maintains IS during the switching process.

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

[0077] All subsystems are stable;

[0078] All subsystems are unstable;

[0079] This paper combines the Lyapunov method and state feedback control to address the system's stability problem. By designing an appropriate state feedback controller, it attempts to regulate the system during switching processes to maintain stability. Furthermore, the stability condition of the switching system under state feedback is derived.

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

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

[0082] For unstable subsystems, design a state feedback controller to intervene at switching times to maintain overall stability.

[0083] The design method of state feedback controllers includes:

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

[0085] S2. A linear feedback control law u = -Kx is adopted;

[0086] Where u is the control input vector, representing the control signal acting on the subsystem, the purpose of which 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, representing 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 the state feedback controller's satisfaction with the Lyapunov function V(x) conditions;

[0092] By substituting the linear feedback control law u=-Kx into the Lyapunov function V(x), we can 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 contraction 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 the original value after each update;

[0095] V(x) is the Lyapunov function, a non-negative function used to analyze the stability of a 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 simulations were performed to verify its effectiveness in a real switching system.

[0099] Furthermore, in this embodiment, the state feedback controller adopts an adaptive design, adjusting the feedback gain matrix K in a timely manner to cope with pulse interference of different intensities and timings, thereby ensuring the stability of the subsystem's input state. This includes the following steps:

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

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

[0102] By defining an interference detection threshold δ, when the subsystem state change Δx exceeds δ, i.e. |Δx>δ, it is determined that an impulse interference has occurred;

[0103] q2. Pulse timing detection: This method monitors the input and state changes of the subsystem within a set 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 stability of the system's input state. In this case, 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, representing the adjusted feedback gain matrix;

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

[0108] Where β is the adjustment coefficient, used to control the proportional relationship between the feedback gain matrix and the magnitude of the disturbance;

[0109] After the pulse interference ends and the system state returns to within acceptable limits, reduce the feedback gain to the base value K, with the attenuation formula as follows:

[0110] Wherein, K(t) is the value of the feedback gain matrix with time t. This value will gradually decay over time, and thus gradually recover to the normal gain level after the pulse interference ends.

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

[0112] α is the attenuation coefficient, t p The moment the pulse interference occurs is t, where t is the current time.

[0113] Q3. Numerical simulation verification;

[0114] The effect of dynamic adjustment adaptive design is verified by numerical simulation. Impulse interference of different magnitudes and times is simulated, the recovery of system state is observed, and the stability of system input state under adaptive design is verified.

[0115] Furthermore, by adjusting the parameters δ, β, and α, the impact of adaptive design on the system response was observed, and the control effect was optimized.

[0116] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make some modifications or alterations to the above-disclosed technical content to create equivalent embodiments without departing from the scope of the present invention. Any simple modifications, equivalent changes and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the scope of the present invention.

Claims

1. A switching graph method for input-output stability of nonlinear systems, characterized in that, The method includes the following steps: Step 1: Establish a switching directed graph, where nodes represent different subsystems and edges represent the switching between subsystems; Step 2: By analyzing the structural features of the switching directed graph, construct the vertex set and edge set of the switching directed graph; Step 3: Based on the structure of the switching directed graph, define an independent Lyapunov function for each subsystem, and divide each subsystem into stable subsystems and unstable subsystems; Step 4: Set average dwell time constraints; To set a minimum dwell time for each subsystem to ensure that the system maintains ISS characteristics during switching; Step 5: Design a feedback controller to suppress impulse interference at the switching time and ensure that the system maintains ISS during the switching process; The state feedback controller employs an adaptive design, including the following steps: Q1. Detect the magnitude and timing of pulse interference; Q2. Under conditions of no pulse interference or interference less than a preset threshold, the basic feedback gain matrix is ​​used. To achieve standard feedback control and maintain the stability of the system's input state, the control input is: ; When pulse interference is detected, the control input is adjusted to... ; in, , representing the adjusted feedback gain matrix; This is gain compensation based on the magnitude of the pulse interference. It is set according to the magnitude of the pulse interference, and the specific adjustment is as follows: ; in, This is an adjustment coefficient used to control the proportional relationship between the feedback gain matrix and the magnitude of the disturbance; Q3. Numerical simulation verification; The effect of dynamic adjustment adaptive design is verified by numerical simulation. Impulse interference of different magnitudes and times is simulated, the recovery of system state is observed, and the stability of system input state under adaptive design is verified.

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

3. The switching graph method for input-output stability of nonlinear systems according to claim 2, characterized in that, When all subsystems are unstable, the stability problem of the system is addressed by combining the Lyapunov method and state feedback control. By designing a state feedback controller, the system is adjusted during the switching process.

4. The switching graph method for input-output stability of nonlinear systems according to claim 3, characterized in that, When some subsystems are stable and others are unstable; For stable subsystems, the Lyapunov method is used to prove their stability. For unstable subsystems, design a state feedback controller to intervene at switching times.

5. The switching graph method for input-output stability of a nonlinear system according to claim 4, characterized in that, The design method of state feedback controllers includes: S1. Define a Lyapunov function for the subsystem. It also requires that the Lyapunov function responds to changes in the system state. Decreasing; S2. Using a linear feedback control law ; in, The control input vector represents the control signal acting on the subsystem; This is the feedback gain matrix; This represents the subsystem state vector; S3. Determine the feedback gain matrix ; It employs any one of the following methods: pole placement, LQR method, or H∞ control method; S4. Check the state feedback controller's response to the Lyapunov function. Satisfaction of conditions; S5. Simulation verification; After designing the state feedback controller, numerical simulations were performed to verify its effectiveness in a real switching system.

6. The switching graph method for input-output stability of a nonlinear system according to claim 5, characterized in that, In step S4, the linear feedback control law is replaced. To Lyapunov functions In the middle, verify whether the closed-loop system satisfies the following inequality conditions: ; in, For contraction factor, 0 ≤ <1; For Lyapunov functions; State update equations for the subsystem Indicates the current state of the subsystem. and control input The evolution of the following.

7. The switching graph method for input-output stability of nonlinear systems according to claim 1, 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 by the state deviation. By defining an interference detection threshold When the state of the subsystem changes Exceed ,Right now At that time, it is determined that pulse interference has occurred; q2. Pulse timing detection: This method monitors the input and state changes of the subsystem within a set pulse time window T. If a pulse is detected within the pulse time window T, its occurrence time is recorded. .

8. A switching graph method for input-output stability of a nonlinear system according to claim 7, characterized in that, In step Q2, after the pulse interference ends and the system state returns to within acceptable limits, the feedback gain is reduced to the base value. Its attenuation formula is: ; in, This is the value of the feedback gain matrix over time t, which decays over time. It is an exponential decay function, where α is the decay coefficient. t represents the moment when the pulse interference occurs, and t represents the current time.

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