Networked nonlinear system model predictive control method considering data packet loss

Abstract

The invention discloses a predictive control method based on an expected predictive model for a networked nonlinear system with data packet loss. Firstly, an expectation prediction model of a networked nonlinear system is established to compensate the possible uncertain influence of data packet loss on the system in advance. In order to improve the calculation efficiency and reduce the calculation complexity, a networked nonlinear system local linearization model is constructed in advance. And calculating expected prediction output of a future system under the condition of data packet loss by constructing an expected state containing a packet loss rate. Then, an optimal control problem is designed to obtain an optimal solution under random data packet loss. The obtained control law is applied to a networked nonlinear system, and it is ensured that the system converges to a set point.

Description

Technical Field

[0001] The invention relates to the field of networked control systems, and in particular to a predictive control method based on an expected prediction model for solving the problem of system instability caused by data packet loss in a networked nonlinear system. Background Art

[0002] With the rapid development of computer technology and network technology, networked control systems have received extensive attention. In a networked control system, sensors, actuators, and controllers exchange information and work together through the network. This architecture brings many advantages to industrial systems, including reducing equipment costs, simplifying system maintenance, and optimizing resource utilization. Networked control systems have been widely used in industrial fields, such as smart grids, drone fleets, aircraft engine systems, etc. Therefore, the research results of the present invention have broad application prospects.

[0003] However, due to the limitation of network bandwidth and network congestion, networked control systems may experience packet loss during information transmission, which can reduce the performance of the controller and even cause system instability. Therefore, it is necessary to achieve stable control of networked control systems with packet loss.

[0004] As a popular industrial process control technology, model predictive control (MPC) predicts the future dynamic response of the system based on the current state to develop the optimal control strategy. MPC is widely used to deal with packet loss problems due to its ability to handle constrained, time-varying, and multivariable systems. Most current MPC methods focus on post-packet loss compensation control, which controls the system by predicting the future behavior of the system based on observed data. These methods do not take into account the fact that accurate predictions of the occurrence of packet loss before control decisions are made may result in control measures being taken only after packet loss occurs, thus affecting the response and stability of the system. Therefore, new MPC methods are needed to meet the requirements for stable control of systems with packet loss.

[0005] The present invention designs a predictive control method based on an expected prediction model. By establishing an expected prediction model to compensate in advance for the uncertain impact that data packet loss may have on the system, a cost function is constructed based on this model, and quadratic programming is used to solve the optimal control problem to obtain the optimal control law, thereby achieving stable control of the nonlinear system with data packet loss. Summary of the invention

[0006] The present invention discloses a predictive control method based on an expected prediction model for networked nonlinear systems with packet loss. The method establishes an expected prediction model to compensate in advance for the uncertain effects that packet loss may have on the current system, and obtains the optimal control law by solving the optimal control problem, thereby achieving stable control of the system under the influence of packet loss.

[0007] The present invention adopts the following technical solutions and implementation steps:

[0008] A networked nonlinear system model predictive control method considering packet loss is provided, which realizes stable control of the networked nonlinear system with packet loss by designing an expected prediction model including the probability of packet loss and solving the optimal control problem to obtain the control law; the method is characterized by comprising the following steps:

[0009] (1) Consider the following networked nonlinear system:

[0010]

[0011] Where x(k)∈R n and x(k+1)∈R n are the system states at time k and time k+1 respectively, n is the dimension of the system state, u(k)∈R m is the system input at time k, m is the dimension of the system input, y(k)∈R z is the system output at time k, z is the dimension of the system output, f(x(k)) represents the nonlinear state equation, and h(x(k)) represents the nonlinear output equation. B and D are the input matrix and transfer matrix of the system respectively;

[0012] (2) Networked nonlinear systems suffer from packet loss, specifically:

[0013] When the networked nonlinear system sensor-to-controller channel transmits state information, there is a phenomenon of data packet loss. A random variable θ(k) is defined to characterize the communication state of the sensor-to-controller channel at time k, where θ(k)=1 means that a data packet loss occurs in the sensor-to-controller channel at time k, and the probability of data packet loss is p; θ(k)=0 means that a data packet is successfully transmitted in the sensor-to-controller channel at time k, and the probability of successful data packet transmission is 1-p.

[0014] When a data packet is lost, the controller should maintain the state of the last moment. That is,

[0015] x * (k)=(1-θ(k))x(k)+θ(k)x * (k-1) (2)

[0016] Among them, x* (k) and x * (k-1) represents the data transmitted to the controller at time k and time k-1 respectively.

[0017] (3) Design a predictive controller based on the expected prediction model, specifically:

[0018] ① Establish a linear model of the networked nonlinear system, specifically:

[0019]

[0020] in

[0021]

[0022] Among them, A x(k) is the system matrix of the linearized system obtained by taking the partial derivative of the nonlinear state equation f(x(k)) at point x(k), C x(k) is the output matrix of the linearized system obtained by taking the partial derivative of the nonlinear output equation h(x(k)) at point x(k), e x(k) and r x(k) are the state error term and output error term of the local linearization system of the nonlinear system at point x(k);

[0023] ② Establish a system output expectation prediction model based on the probability of packet loss, specifically:

[0024] Combined with formula (3), the system state at time k+i is predicted at time k, and the expression is as follows:

[0025] x(k+i|k)=A x(k) x(k+i-1|k)+Bu(k+i-1|k)+e x(k) (5)

[0026] Among them, x(k+i|k) represents the predicted value of the system state at time k for time k+i, x(k+i-1|k) represents the predicted value of the system state at time k for time k+i-1, u(k+i-1|k) represents the predicted value of the system input at time k for time k+i-1, i=1, 2,…, L is a positive integer, and L is the prediction time domain and the control time domain.

[0027] Combined with formula (3), the expected prediction value of the calculation system output is expressed as follows:

[0028]

[0029] Among them, E[y(k+i|k)] represents the expected predicted value of the system output at time k for time k+i, and the expected state prediction value E[x *(k+i|k)] is expressed as follows:

[0030]

[0031] Among them, E[x * (k+i|k)] represents the expected state prediction value at time k for time k+i, E[x * (k+i-1|k)] represents the expected state prediction value at time k for time k+i-1, E[x * (k|k)] is x * (k).

[0032] ③ Establish an auxiliary system based on the expected prediction model, specifically:

[0033]

[0034] E[y s (k)]=C x(k) E[x *s (k)]+Du s (k)+r x(k) (9)

[0035] Among them, E[x *s (k)] and E[x *s (k-1)] represent the expected state of the auxiliary system at time k and time k-1 respectively, and x s (k-1) represents the system state of the auxiliary system at time k-1, E[y s (k)] represents the expected output of the auxiliary system at time k, u s (k) and u s (k-1) represents the system input of the auxiliary system at time k and time k-1 respectively.

[0036] ④ Design the objective function, specifically:

[0037]

[0038] where u s (k), E[y s (k)] is the physical quantity to be solved at time k, [u(1|k), u(2|k), …, u(L|k)] T is the optimal control sequence of the future L steps to be solved at time k, J(k) is the cost function at time k, r is the set value of the system output, W1 is the weighted u(i+1|k) and u s (k), W2 is the weighted difference between E[y(k+i|k)] and E[y s (k)], W is the weighted E[y s(k)] and r, the initial value of the system state is x(0), E[x *s (0)]=x s (0) = x * (0) = x(0).

[0039] (4) Use quadratic programming to solve the optimization problem (10) and obtain the optimal control sequence [u(1|k), u(2|k), …, u(L|k)] for the next L steps at time k under the condition of packet loss. T , take the first item u(1|k) in the optimal control sequence as the input u(k+1) at time k+1 and act on the system.

[0040] The creativity of the present invention is mainly reflected in:

[0041] (1) The present invention aims to solve the problem that the networked nonlinear system has packet loss, which affects the controller control performance and system stability. An expected prediction model is established to compensate in advance for the uncertain impact that the packet loss may have on the system. The model is closer to the actual system and reduces the prediction error.

[0042] (2) The present invention adopts a predictive control method based on an expected prediction model to control a networked nonlinear system with data packet loss, establishes a cost function, uses quadratic programming to solve the optimal control problem to obtain an optimal control rate, and applies the optimal control rate to the system, thereby achieving stable control of the system, with the characteristics of high computational efficiency and good stability. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 is a structural block diagram of the networked nonlinear system considered in the present invention;

[0044] Figure 2a The actual output of the system converges to the set value along the equilibrium point;

[0045] Figure 2b The actual input of the system;

[0046] Figure 2c Schematic diagram of random data packet loss in the sensor-to-controller channel when p=0.2. DETAILED DESCRIPTION

[0047] The technical solution of the present invention is clearly and completely described below with reference to the accompanying drawings and embodiments.

[0048] The present invention proposes a predictive control method based on an expected prediction model for a networked nonlinear system with data packet loss; an expected prediction model is established to compensate in advance for the uncertain impact that the data packet loss may have on the system; a cost function is constructed based on the expected prediction model, and the optimal control problem is solved using quadratic programming to obtain the optimal control rate, thereby achieving stable control of the system under consideration.

[0049] In the networked nonlinear system of the present invention, there is a phenomenon of data packet loss when the state information is transmitted from the sensor to the controller channel; the random variable θ(k) is used to characterize the communication state of the sensor to the controller channel at time k, wherein θ(k)=1 indicates that a data packet loss occurs in the sensor to the controller channel at time k, and the probability of data packet loss is p; θ(k)=0 indicates that a data packet is successfully transmitted from the sensor to the controller channel at time k, and the probability of successful data packet transmission is 1-p;

[0050] The embodiment of the present invention provides a predictive control method based on an expectation model, comprising the following steps:

[0051] (1) Consider the following networked nonlinear system:

[0052]

[0053] Where x(k)∈R n and x(k+1)∈R n are the system states at time k and time k+1 respectively, n is the dimension of the system state, u(k)∈R m is the system input at time k, m is the dimension of the system input, y(k)∈R z is the system output at time k, z is the dimension of the system output, f(x(k)) represents the nonlinear state equation, and h(x(k)) represents the nonlinear output equation. B and D are the input matrix and transfer matrix of the system respectively;

[0054] (2) Design a predictive controller based on the expected prediction model, specifically:

[0055] ① Establish a linear model of the networked nonlinear system, specifically:

[0056]

[0057] in

[0058]

[0059] Among them, A x(k) is the system matrix of the linearized system obtained by taking the partial derivative of the nonlinear state equation f(x(k)) at point x(k), C x(k)is the output matrix of the linearized system obtained by taking the partial derivative of the nonlinear output equation h(x(k)) at point x(k), e x(k) and r x(k) are the state error term and output error term of the local linearization system of the nonlinear system at point x(k);

[0060] ②When a data packet is lost, the controller maintains the state of the previous moment, specifically:

[0061] x * (k)=(1-θ(k))x(k)+θ(k)x * (k-1) (4)

[0062] A system output expectation prediction model based on packet loss probability is established, specifically:

[0063] ③ Combined with formula (2), the system state at time k+i is predicted at time k, and the expression is as follows:

[0064] x(k+i|k)=A x(k) x(k+i-1|k)+Bu(k+i-1|k)+e x(k) (5)

[0065] Among them, x(k+i|k) represents the predicted value of the system state at time k for time k+i, x(k+i-1|k) represents the predicted value of the system state at time k for time k+i-1, u(k+i-1|k) represents the predicted value of the system input at time k for time k+i-1, i=1, 2,…, L is a positive integer, and L is the prediction time domain and the control time domain.

[0066] Combined with formula (2), the expected prediction value of the calculation system output is expressed as follows:

[0067]

[0068] Among them, E[y(k+i|k)] represents the expected predicted value of the system output at time k for time k+i, and the expected state prediction value E[x * (k+i|k)] is expressed as follows:

[0069]

[0070] Among them, E[x * (k+i|k)] represents the expected state prediction value at time k for time k+i, E[x * (k+i-1|k)] represents the expected state prediction value at time k for time k+i-1, E[x * (k|k)] is x * (k).

[0071] ④ Establish an auxiliary system based on the expected prediction model, specifically:

[0072]

[0073] E[y s (k)]=C x(k) E[x *s (k)]+Du s (k)+r x(k) (9)

[0074] Among them, E[x *s (k)] and E[x *s (k-1)] represent the expected state of the auxiliary system at time k and time k-1 respectively, and x s (k-1) represents the system state of the auxiliary system at time k-1, E[y s (k)] represents the expected output of the auxiliary system at time k, u s (k) and u s (k-1) represents the system input of the auxiliary system at time k and time k-1 respectively.

[0075] ⑤Design objective function, specifically:

[0076]

[0077] where u s (k), E[y s (k)] is the physical quantity to be solved at time k, [u(1|k), u(2|k), …, u(L|k)] T is the optimal control sequence of the future L steps to be solved at time k, J(k) is the cost function at time k, r is the set value of the system output, W1 is the weighted u(i+1|k) and u s (k), W2 is the weighted difference between E[y(k+i|k)] and E[y s (k)], W is the weighted E[y s (k)] and r, the initial value of the system state is x(0), E[x *s (0)]=x s (0) = x * (0) = x(0).

[0078] (3) Use quadratic programming to solve the optimization problem (10) and obtain the optimal control sequence [u(1|k), u(2|k), …, u(L|k)] for the next L steps at time k under the condition of packet loss. T , take the first item u(1|k) in the optimal control sequence as the input u(k+1) at time k+1 and act on the system.

[0079] (4) Figure 2a -c is a diagram showing the effect of a predictive control method based on an expected prediction model provided by the present invention on a continuous stirred reactor system.

[0080] The continuous stirred reactor system is discretized with a sampling period of 0.2, and the following form is obtained:

[0081]

[0082] y(k)=x2(k)

[0083] Set the initial state x(0) = [0.9492, 0.4250], set point r = 0.47, prediction domain L = 40, packet loss probability p = 0.2, W1 = I, W2 = 1, W = 500, system input lower bound u min =0, the upper limit of system input u max =1, auxiliary system input lower limit Auxiliary system input upper bound The simulation results demonstrate the effectiveness of the invented predictive control method based on the expectation model.

Claims

1. A model predictive control method for networked nonlinear systems considering packet loss, characterized in that The following steps are involved: (1) Design a networked nonlinear system, specifically: Where x(k)∈R n and x(k+1)∈R n are the system states at time k and time k+1 respectively, n is the dimension of the system state, u(k)∈R m is the system input at time k, m is the dimension of the system input, y(k)∈R z is the system output at time k, z is the dimension of the system output, f(x(k)) represents the nonlinear state equation, h(x(k)) represents the nonlinear output equation; B and D are the input matrix and transfer matrix of the system respectively; (2) Networked nonlinear systems suffer from packet loss, specifically: When the networked nonlinear system sensor-to-controller channel transmits state information, there is a packet loss phenomenon. A random variable θ(k) is defined to characterize the communication state of the sensor-to-controller channel at time k, where θ(k)=1 means that the sensor-to-controller channel has a packet loss at time k, and the probability of packet loss is p; θ(k)=0 means that the sensor-to-controller channel has a successful transmission of the packet at time k, and the probability of successful transmission of the packet is 1-p. When a packet is lost, the controller should maintain the state at the last moment; that is, x * (k)=(1-θ(k))x(k)+θ(k)x * (k-1) (2) Among them, x * (k) and x * (k-1) represents the data transmitted to the controller at time k and time k-1 respectively; (3) Design a predictive controller based on the expected prediction model, specifically: ① Establish a linear model of the networked nonlinear system, specifically: in Among them, A x(k) is the system matrix of the linearized system obtained by taking the partial derivative of the nonlinear state equation f(x(k)) at point x(k), C x(k) is the output matrix of the linearized system obtained by taking the partial derivative of the nonlinear output equation h(x(k)) at point x(k), e x(k) and r x(k) are the state error term and output error term of the local linearization system of the nonlinear system at point x(k); ② Establish a system output expectation prediction model based on the probability of packet loss, specifically: Combined with formula (3), the system state at time k+i is predicted at time k, and the expression is as follows: x(k+i|k)=A x(k) x(k+i-1|k)+Bu(k+i-1|k)+e x(k) (5) Wherein, x(k+i|k) represents the predicted value of the system state at time k for time k+i, x(k+i-1|k) represents the predicted value of the system state at time k for time k+i-1, u(k+i-1|k) represents the predicted value of the system input at time k for time k+i-1, i=1, 2, ..., L is a positive integer, and L is the prediction domain; Combined with formula (3), the expected prediction value of the calculation system output is expressed as follows: E[y(k+i|k)]=C x(k) E[x * (k+i|k)]+You(k+i|k)+r x(k) =C x(k) pE[x * (k+i-1|k)]+C x(k) (1-p)A x(k) x(k+i-1|k) +C x(k) (1-p)Bu(k+i-1|k)+Du(k+i|k)+C x(k) (1-p)e x(k) +r x(k) (6) Among them, E[y(k+i|k)] represents the expected predicted value of the system output at time k for time k+i, and the expected state prediction value E[x * (k+i|k)] is expressed as follows: Among them, E[x * (k+i|k)] represents the expected state prediction value at time k for time k+i, E[x * (k+i-1|k)] represents the expected state prediction value at time k for time k+i-1, E[x * (k|k)] is x * (k); ③ Establish an auxiliary system based on the expected prediction model, specifically: in, and denote the expected state of the auxiliary system at time k and time k-1 respectively, x s (k-1) represents the system state of the auxiliary system at time k-1, E[y s (k)] represents the expected output of the auxiliary system at time k, u s (k) and u s (k-1) represents the system input of the auxiliary system at time k and time k-1 respectively; ④ Design the objective function, specifically: where u s (k), E[y s (k)] is the physical quantity to be solved at time k, [u(1|k), u(2|k), …, u(L|k)] T is the optimal control sequence of the future L steps to be solved at time k, J(k) is the cost function at time k, r is the system output setting value, W1 is the weighted u(i+1|k) and u s (k), W2 is the weighted difference between E[y(k+i|k)] and E[y s (k)], W is the weighted E[y s (k)] and r, the initial value of the system state is x(0), E[x *s (0)]=x s (0) = x * (0) = x(0); Using quadratic programming to solve the optimization problem (10), we obtain the optimal control sequence [u(1|k), u(2|k), …, u(L|k)] for the next L steps at time k under packet loss. T , take the first item u(1|k) in the optimal control sequence as the input u(k+1) at time k+1 and act on the system.

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