A self-triggered predictive control method for continuous-time networked control systems

Abstract

This invention discloses a self-triggered predictive control method for continuous-time networked control systems. The invention includes the following steps: first, constructing a model of the continuous-time networked control system; designing and solving a finite-time optimization control problem to obtain t. k The optimal control sequence at time t and t k The optimal cost function value at time t is obtained; then, candidate feasible control sequences at time t are constructed, and a feasibility analysis is performed on the finite-time domain optimization control problem at time t using the candidate feasible control sequences to obtain the feasible cost function value at time t; then, based on the feasible cost function value at time t and t k The deviation of the optimal cost function value at any given time is used to construct a self-triggering mechanism. Finally, the update time of the optimal control input sequence and the control input applied to the system are determined based on the self-triggering mechanism. This invention can effectively reduce the number of state updates of the controlled system, alleviate the computational burden on the controller and the communication burden on the communication network, while ensuring the control performance of the system.

Description

Technical Field

[0001] This invention pertains to the non-periodic sampling control problem of networked control systems, and relates to a self-triggering predictive control method for continuous-time networked control systems. Background Technology

[0002] With the rapid development of computer technology, communication technology, and control technology, networked control systems have received widespread attention from the scientific research and industrial communities. Traditional control systems use signal lines to connect controllers, actuators, sensors, and controlled objects, updating the state information of the controlled object periodically and calculating control signals based on the latest state information. In networked control systems, controllers, actuators, sensors, and controlled objects are connected pairwise via communication networks. This connection method is easy to install and maintain, and inexpensive. However, communication networks often have limited bandwidth. Traditional time-driven methods are no longer suitable for networked control systems. Instead, event-triggered / self-triggered control mechanisms have become a focus of researchers. Event-triggered mechanisms are proactive, requiring real-time or periodic detection of event triggering conditions. Self-triggered mechanisms, on the other hand, are passive, calculating the next triggering time at the current triggering moment without requiring real-time or periodic detection of event triggering conditions.

[0003] In the real world, various controlled systems often possess physical constraints (displacement limits, voltage and power limits, etc.) or safety constraints (temperature, pressure, etc.), making constraint control a widely discussed problem. Model predictive control (MDC) has garnered significant attention due to its ability to explicitly handle complex constrained systems. Traditional MDC obtains the optimal control sequence by solving an optimization problem, applying only the first control value to the controlled object while discarding the rest. Since systems are often nonlinear, the optimization problem is frequently a nonlinear programming problem, requiring substantial computational time. Traditional time-triggered predictive control strategies incur a significant computational burden. Therefore, event-triggered / self-triggered control, a non-periodic triggering communication mechanism, can be used to reduce the computational cost of MDC algorithms. However, research on self-triggered MDC algorithms is relatively limited. Summary of the Invention

[0004] This invention addresses the shortcomings and deficiencies in current research on self-triggered model predictive control. It provides a self-triggered model predictive control method for continuous-time networked control systems, which reduces the computational and communication burden of the system by decreasing the number of system state updates.

[0005] The objective of this invention is achieved through the following technical solution:

[0006] 1) Construct a continuous-time networked control system model, and based on this model, construct and solve a finite-time domain optimization control problem to obtain t. k The optimal control sequence at time t and t k The optimal cost function value at any given time;

[0007] 2) Based on t k The optimal control sequence at time t is used to construct a candidate feasible control sequence at time t. A feasibility analysis is performed on the finite-time domain optimization control problem at time t to obtain the feasible cost function value at time t;

[0008] 3) Based on the feasible cost function value at time t and t k The deviation of the optimal cost function value at any given time is used to construct a self-triggering mechanism. Based on the self-triggering mechanism, the update time of the optimal control input sequence and the control input applied to the system are determined.

[0009] In step 1), the formula for the continuous-time networked control system model is as follows:

[0010]

[0011] in, x represents the rate of change of the system state over time. t It is the system state, u t It is a right-continuous control input, v t This represents a bounded disturbance, and f(·) is the system state change function.

[0012] In step 1), the formula for the finite-time domain optimization control problem is as follows:

[0013]

[0014]

[0015] Among them, t k Indicates the time when the event was triggered. Indicates t k The optimal control sequence at time 1. Indicates t k The first cost function value at time t. Indicates t k The actual state of the system at any given time. Given a control sequence, T p Indicates the prediction time domain, Indicates the predicted state of the system. Let Q represent the predicted state of the terminal, R be the predicted state weight matrix, and P be the control input weight matrix. ||·||P This represents the L2 norm operation of a vector with weight P, ‖·‖ Q This represents the L2 norm operation on a vector with weights R, ‖·‖ R This represents the L2 norm operation of a vector with weights R;

[0016] The formulas for the constraints are as follows:

[0017]

[0018]

[0019]

[0020]

[0021] in, For a tightly constrained set of states, It is the terminal constraint set. Indicates t k The initial predicted state of the system at time t. Let K represent the tight constraint set of the control input, and K be the optimal feedback gain matrix of the linear quadratic regulator. Let s represent the state change function of the first system under ideal conditions, and let s represent the state change function in the time domain [t]. k , t k +T p At that moment, This represents the rate of change of the predicted state over time.

[0022] Step 2) specifically refers to:

[0023] 2.1) According to t k The optimal control sequence at time t is obtained, and candidate feasible control sequences at time t are constructed. The formula is as follows:

[0024]

[0025] in, It is the predicted state at time t; Indicates t k The optimal control sequence at time T p Represents the prediction time domain, where K is the optimal feedback gain matrix of the linear quadratic regulator;

[0026] 2.2) Utilizing candidate feasible control sequences at time t A feasibility analysis is performed on the finite-time domain optimization control problem at time t, and the candidate feasible control sequence is identified. Once feasibility is confirmed, the corresponding cost function value is obtained and denoted as the feasible cost function value at time t.

[0027] Step 3) specifically refers to:

[0028] 3.1) Based on the feasible cost function value at time t and t k The upper bound of the cost function difference obtained from solving the optimal cost function value at time t is denoted as the upper bound of the cost function deviation, and the formula is as follows:

[0029]

[0030]

[0031] in, This represents the optimal state trajectory, where ρ is the first cost coefficient. Let represent the feasible cost function value at time t. Indicates t k The optimal cost function value at time ω t The system state deviation is represented by h0, which is the second cost coefficient. λ (Q) represents the largest eigenvalue of the predicted state weight matrix Q. Representation matrix The smallest eigenvalue, Representation matrix The largest eigenvalue, Let L represent the upper bound of the disturbance, ε represent the third cost coefficient, and L represent the upper bound of the disturbance. f Represents the Lipschitz coefficients of the system;

[0032] 3.2) Construct a static self-triggered mechanism based on the upper bound of the cost function deviation, as shown in the following formula:

[0033]

[0034] Among them, t k+1 The trigger time is represented by σ, where σ represents the first trigger coefficient, c represents the second trigger coefficient, and T is the trigger time. p Indicates the prediction time domain, This represents the Lipschtiz coefficient of the system obtained by using a linear quadratic regulator as the control input;

[0035] 3.3) If the self-triggering mechanism is triggered, the optimal control sequence will be... The control values ​​are sent to the intelligent actuators of the networked control system, thereby updating the control values ​​in the intelligent actuators and applying them to the controlled object; if no trigger is triggered, the control values ​​stored in the intelligent actuators are applied to the controlled object in sequence.

[0036] Alternatively, 3.2) describes a dynamic self-triggering mechanism constructed based on the upper bound of the cost function deviation, as shown in the following formula:

[0037]

[0038]

[0039] θ>0, β>0, β θ =θ(βθ+1),

[0040] Where, η t Let be the dynamic auxiliary parameter at time t. For t k The dynamic auxiliary parameters at time, where θ represents the third triggering coefficient, β represents the fourth triggering coefficient, and β... θ This is the fifth trigger coefficient. This represents the sixth trigger coefficient.

[0041] The beneficial effects of this invention are:

[0042] Compared to predictive control frameworks where the prediction time domain equals the control time domain, this invention employs a predictive control framework where the prediction time domain is larger than the control time domain, increasing the upper bound of disturbances and enhancing system robustness. The proposed static / dynamic self-triggering mechanism effectively reduces the number of system state updates while simultaneously ensuring system control performance. Compared to the proposed static event-triggered mechanism, the proposed dynamic event-triggered mechanism further increases the event triggering interval, thereby more effectively reducing system computation and communication costs. Attached Figure Description

[0043] Figure 1 This is a flowchart of the method of the present invention.

[0044] Figure 2 This is a schematic diagram of the system state trajectory.

[0045] Figure 3 This is a schematic diagram of the control input trajectory.

[0046] Figure 4 This is event triggering scenario one: 1 indicates that the event has been triggered.

[0047] Figure 5 This is event triggering scenario two: 1 indicates that the event has been triggered. Detailed Implementation

[0048] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0049] This embodiment provides a dynamic self-touch predictive control method for continuous-time networked control systems, such as... Figure 1 As shown, the present invention includes the following steps:

[0050] 1) Construct a continuous-time networked control system model, and based on this model, construct and solve a finite-time domain optimization control problem to obtain t. kThe optimal control sequence at time t and t k The optimal cost function value at any given time;

[0051] In step 1), the formula for the continuous-time networked control system model is as follows:

[0052]

[0053] in, x represents the rate of change of the system state over time. t It is the system state, u t It is a right-continuous control input, v t This indicates bounded interference, satisfying... gather These are the first to third tight constraint sets, and the upper bound of the disturbance. satisfy ‖·‖ denotes the vector 2-norm operation, and f(·) is the system state change function, which is a quadratic continuously differentiable function satisfying f(0, 0, 0) = 0, used to represent the rate of change of the system state relative to the system state x. t Control input u t Bounded interference v t The relationship.

[0054] In step 1), the formula for the finite-time domain optimization control problem is as follows:

[0055]

[0056]

[0057] Among them, t k Indicates the time when the event was triggered. Indicates t k The optimal control sequence at time 1. Indicates t k The first cost function value at time t. Indicates t k The actual state of the system at any given time. Given a control sequence, T p Indicates the prediction time domain, Indicates the predicted state of the system. Let Q represent the predicted state of the terminal, R be the predicted state weight matrix, and P be the control input weight matrix. Q, P, and R are all symmetric positive definite constant matrices. P This represents the L2 norm operation of a vector with weight P;

[0058] The formulas for the constraints are as follows:

[0059]

[0060]

[0061]

[0062]

[0063] in, For a tightly constrained set of states, It is the terminal constraint set. Indicates t k The initial predicted state of the system at time t, i.e. k The actual state of the system at any given time Let K represent the tight constraint set of the control input, and K be the optimal feedback gain matrix of the linear quadratic regulator. Let s represent the state change function of the first system under ideal conditions, and let s represent the state change function in the time domain [t]. k , t k +T p At that moment, This represents the rate of change of the predicted state over time.

[0064] 2) Based on t k The optimal control sequence at time t is used to construct a candidate feasible control sequence at time t. A feasibility analysis is performed on the finite-time domain optimization control problem at time t to obtain the feasible cost function value at time t;

[0065] Step 2) specifically involves:

[0066] 2.1) According to t k The optimal control sequence at time t is obtained, and candidate feasible control sequences at time t are constructed. The formula is as follows:

[0067]

[0068] in, It is the predicted state at time t;

[0069] 2.2) Utilizing candidate feasible control sequences at time t A feasibility analysis is performed on the finite-time domain optimization control problem at time t, and the upper bound of the disturbance is obtained. but It is a feasible solution to the finite-time optimization control problem, which is iteratively feasible. (Disturbance upper bound) Satisfy the following formula:

[0070]

[0071]

[0072]

[0073] ε≥ε f

[0074] Where δ represents the sampling interval, P represents the terminal prediction state weight matrix, and L f ε represents the Lipschitz constant of the system. f Represents the nonnegative coefficients related to the terminal constraint set. This represents the Lipschtiz coefficients of the system obtained by using a linear quadratic regulator as the control input. Let P represent the largest eigenvalue of the terminal prediction state weight matrix, and ε represent the third cost coefficient, which is a constant and satisfies the following conditions: π represents Pi, Q * Describe the intermediate gain matrix and satisfy Q * =Q+R T KR, where K is the optimal feedback gain matrix of the linear quadratic regulator, and T represents the matrix transpose.

[0075] This candidate feasible control sequence Once feasibility is confirmed, the corresponding cost function value is obtained and denoted as the feasible cost function value at time t.

[0076] 3) Based on the feasible cost function value at time t and t k The deviation of the optimal cost function value at any given time is used to construct a self-triggering mechanism. Based on the self-triggering mechanism, the update time of the optimal control input sequence and the control input applied to the system are determined.

[0077] Step 3) specifically refers to:

[0078] 3.1) Based on the feasible cost function value at time t and t k The upper bound of the cost function difference obtained from solving the optimal cost function value at time t is denoted as the upper bound of the cost function deviation, and the formula is as follows:

[0079]

[0080]

[0081] Where ρ is the first cost coefficient, Let represent the feasible cost function value at time t. Indicates t k The optimal cost function value at time ω t For the system state deviation, h0 is the second cost coefficient, which is a constant and satisfies... Φ represents the terminal constraint set. The one-step backward reachable set, f(x) t u t(0) denotes the state change function of the second system under ideal conditions, ||| denotes the vector L2 norm operation, and max denotes taking the maximum value. λ (Q) represents the largest eigenvalue of the predicted state weight matrix Q. Representation matrix The smallest eigenvalue, Representation matrix The largest eigenvalue, The square root of the terminal prediction state weight matrix P is given. Let L represent the upper bound of the disturbance, ε represent the third cost coefficient, and L represent the upper bound of the disturbance. f Represents the Lipschitz coefficients of the system;

[0082] 3.2) Construct a static self-triggered mechanism based on the upper bound of the cost function deviation, as shown in the following formula:

[0083]

[0084] Among them, t k+1 Let T represent the trigger time, σ represent the first trigger coefficient, which is a real number and satisfies 0 < σ < 1, and c represent the second trigger coefficient, which is a real number and satisfies c > 0. p Indicates the prediction time domain, This represents the Lipschtiz coefficient of the system obtained by using a linear quadratic regulator as the control input;

[0085] 3.3) If the self-triggering mechanism is triggered, the optimal control sequence will be... The control values ​​are sent to the intelligent actuators of the networked control system, thereby updating the control values ​​in the intelligent actuators and applying them to the controlled object; if no trigger is triggered, the control values ​​stored in the intelligent actuators are applied to the controlled object in sequence.

[0086] To more clearly illustrate the control principle of this invention, the following examples are provided.

[0087] Set the prediction time domain to T p =3 seconds, the event triggering mechanism detection time interval is set to 0.01 seconds, and the integration time interval is also 0.01 seconds. Then, solving the optimization problem yields the optimal control sequence containing control values ​​of 300 piecewise constants.

[0088] Step 0: At t k The second-hour event triggers the controller to solve the optimization control problem and obtain the optimal control sequence. s = 0, 1, ..., 299, which contains 300 optimal control values. These 300 optimal control values ​​are transmitted to the intelligent actuator, which then controls the input values. As a system control input, proceed to Step 1.

[0089] Step 1: at tk At +0.01 seconds, check the event triggering mechanism in 3.2). If the event is triggered, then t k+1 =t k +0.01, k = k + 1, go to Step 0; otherwise, the smart actuator will... As a system control input, proceed to Step 2.

[0090] Step 2:) In t k At +0.02 seconds, check the event triggering mechanism in 3.2). If the event is triggered, then t k+1 =t k +0.02, k = k + 1, go to Step 0; otherwise, the smart actuator will... As a system control input, proceed to Step 3.

[0091] ...

[0092] Step 299: In t k At +2.99 seconds, check the event triggering mechanism in 3.2). If the event is triggered, then t k+1 =t k +2.99, k = k + 1, go to Step 0; otherwise, the intelligent actuator will... As a system control input, proceed to Step 300.

[0093] Step 300: at t k At +3 seconds, the event is triggered, and t is set. k+1 =3, k=k+1, go to Step 0.

[0094] Alternatively, to further extend the event trigger interval, a dynamic auxiliary variable η can be introduced. t A dynamic self-triggering mechanism is constructed based on the upper bound of the cost function deviation, as shown in the following formula:

[0095]

[0096]

[0097] θ>0, β>0, β θ =θ(βθ+1),

[0098] Where, η t Let be the dynamic auxiliary parameter at time t. For t k The dynamic auxiliary parameters at time t, θ represents the third trigger coefficient, a real number greater than zero, β represents the fourth trigger coefficient, a real number greater than zero, β θ The fifth trigger coefficient satisfies β. θA real number equal to θ / (βθ+1). This represents the sixth trigger coefficient, which satisfies... The real number.

[0099] The effects of the present invention will be further described below with reference to simulation examples.

[0100] Consider the spring-cart damping system, its dynamic equations are as follows:

[0101]

[0102] Where, x 1，t The x represents the offset position of the car. 2，t u represents the speed of the car. t Indicates control input, v t This represents external bounded disturbance. The system parameters are: M = 1 kg, k0 = 0.33 N / m, h = 1.1 Ns / m. The system state constraints are χ = {x: -2 ≤ x}. 1，t ≤2, -2≤x 2，t ≤2}. The system control input constraint is The weight matrix in the cost function is chosen as Q = [0.2, 0; 0, 0.2], R = 0.2, P = [0.5512, 0.2755; 0.2755, 0.2656]. External disturbance is chosen as v. t =0.0047*(sin(3.6t)+cos(1.5t)). The parameters in the event triggering condition are chosen as β=3 and θ=0.9. σ = 0.08. The initial state of the system is chosen as x0 = [-1.9, -1.1]. T .

[0103] Using Matlab / IPOPT, the static self-triggered predictive control algorithm and the dynamic self-triggered predictive control algorithm were simulated respectively. Figure 2 and Figure 3 This represents the changes in the system state trajectory and control input trajectory under the action of two control algorithms. From... Figure 2 and Figure 3 It can be seen that the system state and control input both meet the constraints and eventually stabilize near the origin. Figure 4 and Figure 5 The event triggering scenarios under two self-triggered predictive control algorithms are presented. From Figure 4 It can be seen that the event was triggered 31 times under the static self-triggered predictive control algorithm. Figure 5It can be seen that the event was triggered 25 times under the dynamic self-triggered predictive control algorithm. This indicates that both control algorithms can effectively reduce the communication and computational burden. Compared with the static self-triggered predictive control algorithm, the dynamic self-triggered predictive control algorithm reduced the number of event triggers by 19.4%, indicating that it is more effective in reducing the communication and computational burden.

[0104] The above embodiments are used to explain and illustrate the present invention, but not to limit the present invention. Any modifications and changes made to the present invention within the spirit and scope of the claims shall fall within the protection scope of the present invention.

Claims

1. A self-triggered predictive control method for a continuous-time networked control system, characterized in that, Comprising the following steps: 1) constructing a continuous-time networked control system model, constructing a finite-time domain optimal control problem based on the continuous-time networked control system model and solving the finite-time domain optimal control problem to obtain an optimal control sequence at the time instant and an optimal cost function value at the time instant the time instant​ 2) based on optimal control sequence construction candidate feasible control sequence at time candidate feasible control sequence at time feasibility analysis of the finite horizon optimal control problem at time feasible cost function value at time​ 3) according to the deviation of the cost function value at the time instant and the optimal cost function value at the time instant, a self-triggering mechanism is constructed, and the update time instant of the optimal control input sequence and the control input acting on the system are determined according to the self-triggering mechanism. Said step 3) is specifically: 3.1) According to the feasible cost function value at time instant and the upper bound of the cost function difference value solved by the optimal cost function value at time instant is denoted as the cost function deviation upper bound, and the formula is as follows: in, This represents the optimal state trajectory. It is the first cost coefficient. express The feasible cost function value at time 1. express The optimal cost function value at time t. For system state deviation, It is the second cost coefficient. Represents the predicted state weight matrix The largest eigenvalue, Representation matrix The smallest eigenvalue, Representation matrix The largest eigenvalue, Indicates the upper bound of the disturbance. This represents the third cost coefficient. Represents the Lipschitz coefficients of the system; 3.2) Constructing a static self-triggering mechanism based on the cost function bias upper bound, the formula is as follows: wherein, denotes a triggering time instant, denotes a first triggering coefficient, denotes a second triggering coefficient, denotes a prediction horizon, denotes a system Lipschtiz coefficient resulting from a linear quadratic regulator as control input; 3.3) If the self-triggering mechanism triggers, the optimal control sequence is sent to the smart actuator of the networked control system, which in turn updates the control value in the smart actuator and acts on the controlled object; if not, the control value stored in the smart actuator is sequentially acted on the controlled object.

2. The self-triggered predictive control method for continuous-time networked control systems according to claim 1, wherein In said step 1), the formula of the continuous-time networked control system model is as follows: wherein, denotes the rate of change of the system state with respect to time, is the system state, is the right-continuous control input, denotes the bounded disturbance, is the system state change function.

3. The self-triggered predictive control method for continuous-time networked control systems according to claim 1, wherein In said step 1), the formula of the finite time domain optimal control problem is as follows: wherein, denotes the event-triggering time instant, denotes the optimal control sequence at the time instant, denotes the first cost function value at the time instant, denotes the actual state of the system at the time instant, denotes a given control sequence, denotes the prediction horizon, denotes the predicted state of the system, denotes the terminal predicted state, is the predicted state weight matrix, is the control input weight matrix, is the terminal predicted state weight matrix, denotes the vector two-norm operation with weight denotes the vector two-norm operation with weight denotes the vector two-norm operation with weight denotes the vector two-norm operation with weight denotes the vector two-norm operation with weight denotes the vector two-norm operation with weight The formula of the constraint condition is as follows: in, For a tightly constrained set of states, It is the terminal constraint set. express The initial predicted state of the system at time t. This represents the set of tight constraints for control inputs. It is the optimal feedback gain matrix of the linear quadratic regulator. This represents the state change function of the first system under ideal conditions. In the time domain At that moment, This represents the rate of change of the predicted state over time.

4. The self-triggered predictive control method for continuous-time networked control systems according to claim 1, wherein Said step 2) is specifically: 2.1) According to the optimal control sequence at the time instant, construct candidate feasible control sequences at the time instant , the formula is as follows: ； wherein is the predicted state at time instant denotes the optimal control sequence at time instant denotes the prediction horizon is the optimal feedback gain matrix of the linear quadratic regulator 2.2) Utilizing the candidate feasible control sequence at time instant the candidate feasible control sequence at time instant feasibility analysis of the finite horizon optimal control problem at time instant with feasibility, the corresponding cost function value is denoted as the feasible cost function value at time instant 5. The self-triggered predictive control method for continuous-time networked control systems according to claim 1, wherein Or said 3.2) is constructing a dynamic self-triggering mechanism based on the cost function bias upper bound, the formula is as follows: ， ， wherein is a dynamic assistance parameter at the time instant, is a dynamic assistance parameter at the time instant, denotes a third trigger coefficient, denotes a fourth trigger coefficient, is a fifth trigger coefficient, denotes a sixth trigger coefficient.

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