Predictive control model method and system of nonlinear pendulum system

Abstract

The invention relates to a predictive control model method and system for a nonlinear pendulum system, and the method mainly comprises the following steps: building a kinetic model of the nonlinear pendulum system according to the working state of the nonlinear pendulum system; the control input is quantized by combining with a hysteresis quantizer, so that the limitation of network resource limitation on the collaborative performance of the nonlinear pendulum system is relieved; a learning type model prediction control algorithm capable of online learning is designed, an optimal control gain matrix is predicted online, and the influence of communication defects on system performance is effectively reduced; a Lyapunov function related to a Markov switching topology is constructed, and the Lyapunov function is used as a cost function of a learning type model prediction control algorithm. According to the method, the transmission efficiency of the nonlinear pendulum system can be further reduced, the minimization of the cost function is effectively realized through the real-time adaptive adjustment of the neural network and the learning type model prediction control algorithm, and the feedback control matrix is optimized.

Description

Technical Field

[0001] The invention belongs to the field of nonlinear pendulum system control, and in particular relates to a predictive control model method and system for a nonlinear pendulum system. Background Art

[0002] The inverted pendulum is an open-loop unstable strong nonlinear system. Its control strategy is similar to the technique of acrobats performing pole balance performances. The purpose is to keep the pendulum in a critical stable state. It is a typical experimental platform for control theory research. In the 1950s, control experts at MIT designed the first inverted pendulum experimental equipment based on the principle of rocket boosters, which started the initial related research work. As a classic nonlinear control system, the inverted pendulum system has become an ideal platform for verifying the effectiveness of various control strategies due to its own instability and high requirements for control algorithms.

[0003] The primary inverted pendulum system consists of a vertically placed pole and a cart that can move horizontally. The horizontal displacement of the cart and the angle of the pole together describe the state of the system. The control goal is to keep the inverted pendulum in an upright equilibrium state by controlling the horizontal acceleration of the cart, regardless of the initial state. Unlike the simplified model that only considers the angle, this paper considers the distance information between the cart and the equilibrium point, which is closer to the actual application scenario and also increases the complexity of control. The inverted pendulum system is a typical nonlinear, strongly coupled, multivariable and unstable system. The stable control of the inverted pendulum system is a typical problem in control theory, and it is also a classic case for learning various control theories. Summary of the invention

[0004] The following is a summary of the subject matter described in detail herein. This summary is not intended to limit the scope of the claims.

[0005] In order to solve the above technical problems, the present invention relates to a predictive control model method and system for a nonlinear pendulum system to further improve the robustness and performance of the controller.

[0006] In a first aspect, a predictive control model method for a nonlinear pendulum system is characterized by comprising the following steps:

[0007] Step S1: constructing a dynamic model of the nonlinear pendulum system according to the working state of the nonlinear pendulum system;

[0008] Step S2: quantizing the control input in combination with a hysteresis quantizer, thereby alleviating the limitation of network resources on the cooperative performance of the nonlinear pendulum system;

[0009] Step S3: According to the steps S1 and S2, a learning model predictive control algorithm capable of online learning is designed to online predict the optimal control gain matrix, thereby effectively reducing the impact of communication defects on system performance;

[0010] Step S4: According to step S3, a Lyapunov function related to the Markov switching topology is constructed and used as the cost function of the learning model predictive control algorithm.

[0011] 1. A predictive control model method for a nonlinear pendulum system according to claim 1, characterized in that it comprises the following steps:

[0012] Step S1: constructing a dynamic model of the nonlinear pendulum system according to the working state of the nonlinear pendulum system;

[0013] The dynamic model of the nonlinear pendulum system is:

[0014]

[0015] Where x and v represent the displacement and velocity of the nonlinear pendulum, respectively, and u represents the control input;

[0016] Assume i (t) = [x i (t),v i (t)] T , the continuous-time state space equation is:

[0017]

[0018] in, A and B both represent nonlinear pendulum coefficient matrices;

[0019] The above is a continuous nonlinear pendulum system. The present invention further proposes a discrete time nonlinear pendulum system model:

[0020] x i (k+1)=f(x i (k),u i (k))

[0021] 2. The predictive control model method of a nonlinear pendulum system according to claim 1, characterized in that it comprises the following steps:

[0022] Step S2: quantizing the control input in combination with a hysteresis quantizer, thereby alleviating the limitation of network resources on the cooperative performance of the nonlinear pendulum system;

[0023] Since the hysteresis quantizer can effectively reduce the oscillation phenomenon and ensure that the quantization value is more stable during the switching process, this paper considers the discretized hysteresis quantizer:

[0024]

[0025] Where ρ = (1-δ) / (1+δ), (0<ρ<1) parameter determines the quantization density, u j =ρ 1-j u0(j=1,…,n x ).

[0026] In order to optimize the network resources among multiple agents, this paper proposes a quantitative feedback controller:

[0027]

[0028] In the formula, K k is the controller gain matrix.

[0029] For the convenience of further discussion, the control input vector u(k) = [u1(k)…u N (k)] T Similarly, x(k)=[x1(k)…x N (k)] T . Combined with the above formula, the control input can be re-expressed in the following vector form:

[0030]

[0031] Let p = θ(k), then the system state vector of MASs can be described as:

[0032]

[0033] Among them, f pi (x(k)) represents the function f(x i (k),u i (k)), and Assumption 1 is satisfied.

[0034] 3. The predictive control model method of a nonlinear pendulum system according to claim 1, characterized in that it comprises the following steps:

[0035] Step S3: According to the steps S1 and S2, a learning model predictive control algorithm capable of online learning is designed to online predict the optimal control gain matrix, thereby effectively reducing the impact of communication defects on system performance;

[0036] The cost function is defined as:

[0037]

[0038] Where P p,p∈S is the positive definite matrix associated with the Markov mode.

[0039] The goal of LMPC is to find J that minimizes the cost J(e(k)). Then the optimal feedback gain matrix is The following conditions are met:

[0040]

[0041] The present invention aims to solve the optimization problem (17) using a shallow neural network. In this paper, the input layer of the shallow neural network consists of a length of l (l = n x n u ) sequence Composition, where a∈{1...n x} and b∈{1…n u}. In order to train to get the optimal gain value K k , a single hidden layer with M neurons is selected. The input layer consists of the values ​​of the cost function J(e(k)).

[0042] make The signal generated by the hidden layer is given by:

[0043] w=σ(Wz)

[0044] Where σ represents the sigmoid function. W∈R M×(l+1) Represents the weight matrix of the neuron. For a given gain K k , the cost function of the output layer J can be approximated as:

[0045]

[0046] Among them, Y∈R 1×M Represents the weight matrix of the output layer. The neural network continuously optimizes the weights W and Y during the iterative learning process to better fit the control gain.

[0047] The specific steps of the model predictive control algorithm are as follows: In the initial stage, the system is given the initial condition x(0), the initial mode θ(0) and the cost function J0. For convenience, the initial gain matrix K0 is set to a null value. Then, it is used together with J0 as the initial training data set of the neural network. Then, the neural network predicts the value of the cost function through iterative learning In each iteration, the previous K k Different combinations are provided to generate predicted values ​​of the cost function In order to improve the prediction accuracy, K k It can be adjusted by ±δ. Then, the item with the smallest predicted value is selected as the optimal control gain K k, and the control input u(k) is determined accordingly. Next, in order to estimate more accurately The algorithm introduces the cycle prediction mechanism of the LMPC protocol. The specific steps are: 1) Automatically generate the jump pattern θ(t) according to the previous moment and the probability transfer matrix Π; 2) Calculate the predicted control input and system status 3) Estimate approximate value At the end of each iteration, sum all the current Get an approximation of the cost function Finally, the predicted value And the corresponding optimal gain Stored in the data set for the next iteration. Repeat the above steps until the predicted value The preset minimum condition ξ is reached. It can be considered that the optimization problem has been solved and the training is ended.

[0048] 4. The predictive control model method of a nonlinear pendulum system according to claim 1, characterized in that it comprises the following steps:

[0049] Step S4: A Lyapunov function related to the Markov switching topology is constructed and used as the cost function of the learning model predictive control algorithm.

[0050] In the second aspect, an embodiment of the present invention further provides a predictive control model system for a nonlinear pendulum system, comprising: S001 a nonlinear inverted pendulum model, S002 a discrete-time nonlinear inverted pendulum system, S003 a sensor, S004 a hysteresis quantization strategy, S005 a controller, S006 a zero-order holder, and a computer program that can be run on a processor, wherein when the processor executes the computer program, a predictive control model method for a nonlinear pendulum system as described in the first aspect is implemented.

[0051] The present invention includes the following beneficial effects: the present invention considers the robustness and performance of the nonlinear inverted pendulum system controller, proposes a quantized output feedback control problem of a discrete nonlinear inverted pendulum system, designs a hysteresis quantized output feedback controller of a learning model predictive control model, constructs a Lyapunov function related to the Markov switching topology, and uses it as a cost function of the learning model predictive control algorithm.

[0052] Other features and advantages of the present invention will be described in the following description, and partly become apparent from the description, or understood by practicing the present invention. The purpose and other advantages of the present invention can be realized and obtained by the structures particularly pointed out in the description, claims and drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] The accompanying drawings are used to provide further understanding of the present invention and constitute a part of the specification. Together with the embodiments of the present invention, they are used to explain the technical solution of the present invention, but do not constitute a limitation to the technical solution of the present invention.

[0054] Figure 1 is a flow chart of a predictive control model method for a nonlinear pendulum system of the present invention;

[0055] Figure 2 It is a structural schematic diagram of a c-hysteresis quantized feedback control system of a nonlinear pendulum system of the present invention;

[0056] Figure 3 is a graph showing the displacement and velocity of the system changing with time provided by an embodiment of the present invention;

[0057] Figure 4 is a graph showing changes in the system controller over time provided by an embodiment of the present invention; DETAILED DESCRIPTION

[0058] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0059] It should be noted that, although the functional modules are divided in the device schematic diagram and the logical order is shown in the flowchart, in some cases, the steps shown or described may be performed in a different order than the module division in the device or the order in the flowchart. The terms "first", "second", etc. in the specification, claims or the above drawings are used to distinguish similar objects, and are not necessarily used to describe a specific order or sequence.

[0060] like Figure 1 As shown, the present invention provides a predictive control model method for a nonlinear pendulum system, the method comprising the following steps:

[0061] 2. The predictive control model method of a nonlinear pendulum system according to claim 1, characterized in that it comprises the following steps:

[0062] Step S1: constructing a dynamic model of the nonlinear pendulum system according to the working state of the nonlinear pendulum system;

[0063] The dynamic model of the nonlinear pendulum system is:

[0064]

[0065] Where x and v represent the displacement and velocity of the nonlinear pendulum, respectively, and u represents the control input;

[0066] Assume i (t) = [x i (t),v i (t)] T , the continuous-time state space equation is:

[0067]

[0068] in, A and B both represent nonlinear pendulum coefficient matrices;

[0069] The above is a continuous nonlinear pendulum system. The present invention further proposes a discrete time nonlinear pendulum system model:

[0070] x i (k+1)=f(x i (k),u i (k))

[0071] Step S2: quantizing the control input in combination with a hysteresis quantizer, thereby alleviating the limitation of network resources on the cooperative performance of the nonlinear pendulum system;

[0072] Since the hysteresis quantizer can effectively reduce the oscillation phenomenon and ensure that the quantization value is more stable during the switching process, this paper considers the discretized hysteresis quantizer:

[0073]

[0074] Where ρ = (1-δ) / (1+δ), (0<ρ<1) parameter determines the quantization density, u j =ρ 1-j u0(j=1,...,n x ).

[0075] In order to optimize the network resources among multiple agents, this paper proposes a quantitative feedback controller:

[0076]

[0077] In the formula, K k is the controller gain matrix.

[0078] For the convenience of further discussion, the control input vector u(k) = [u1(k) ... u N (k)] T Similarly, x(k)=[x1(k)…x N (k)] T . Combined with the above formula, the control input can be re-expressed in the following vector form:

[0079]

[0080] Let p = θ(k), then the system state vector of MASs can be described as:

[0081]

[0082] Among them, f pi (x(k)) represents the function f(x i (k),u i (k)), and Assumption 1 is satisfied.

[0083] Step S3: According to the steps S1 and S2, a learning model predictive control algorithm capable of online learning is designed to online predict the optimal control gain matrix, thereby effectively reducing the impact of communication defects on system performance;

[0084] The cost function is defined as:

[0085]

[0086] Where P p ,p∈S is the positive definite matrix associated with the Markov mode.

[0087] The goal of LMPC is to find J that minimizes the cost J(e(k)). Then the optimal feedback gain matrix is The following conditions are met:

[0088]

[0089] The present invention aims to solve the optimization problem (17) using a shallow neural network. In this paper, the input layer of the shallow neural network consists of a length of l (l = n x n u ) sequence Composition, where a∈{1…n x} and b∈{1…n u}. In order to train to get the optimal gain value K k , a single hidden layer with M neurons is selected. The input layer consists of the values ​​of the cost function J(e(k)).

[0090] make The signal generated by the hidden layer is given by:

[0091] w=σ(Wz)

[0092] Where σ represents the sigmoid function. W∈R M×(l+1) Represents the weight matrix of the neuron. For a given gain K k , the output layer of the cost function It can be approximated as:

[0093]

[0094] Among them, Y∈R 1×M Represents the weight matrix of the output layer. The neural network continuously optimizes the weights W and Y during the iterative learning process to better fit the control gain.

[0095] The specific steps of the model predictive control algorithm are as follows: In the initial stage, the system is given the initial condition x(0), the initial mode θ(0) and the cost function J0. For convenience, the initial gain matrix K0 is set to a null value. Then, it is used together with J0 as the initial training data set of the neural network. Then, the neural network predicts the value of the cost function through iterative learning In each iteration, the previous K k Different combinations are provided to generate predicted values ​​of the cost function In order to improve the prediction accuracy, K k It can be adjusted by ±δ. Then, the item with the smallest predicted value is selected as the optimal control gain K k , and the control input u(k) is determined accordingly. Next, in order to estimate more accurately The algorithm introduces the cycle prediction mechanism of the LMPC protocol. The specific steps are: 1) Automatically generate the jump pattern θ(t) according to the previous moment and the probability transfer matrix Π; 2) Calculate the predicted control input and system status 3) Estimate approximate value At the end of each iteration, sum all the current Get an approximation of the cost function Finally, the predicted value And the corresponding optimal gain Stored in the data set for the next iteration. Repeat the above steps until the predicted value The preset minimum condition ξ is reached. It can be considered that the optimization problem has been solved and the training is ended.

[0096] Step S4: A Lyapunov function related to the Markov switching topology is constructed and used as the cost function of the learning model predictive control algorithm.

[0097] like Figure 2As shown, the present invention also provides a recursive quantization feedback control system of a mass spring damping system, comprising: S001 a mass spring damping system, S002 a discrete time TS fuzzy Markov jump system, S003 a sensor, S004 a recursive quantization strategy, S005 a controller, S006 a zero-order holder, and a computer program that can be run on a processor, and when the processor executes the computer program, a predictive control model method for a nonlinear pendulum system as described in the first aspect is implemented.

[0098] Verify the effect of learning model predictive control method in nonlinear inverted pendulum system:

[0099] Assuming the sampling time is Ts = 10 μs, the parameters of the nonlinear inverted pendulum system are B=[0,1] T The initialization parameters of the nonlinear pendulum system are x0(0) = 0, v0(0) = 0, x1(0) = 50π / 13, v1(0) = -4.5, v2(0) = 2.2, x2(0) = 12π / 13, x3(0) = -50π / 13, x4(0) = 24π / 13v3(0) = 4.5, and v4(0) = -2.2. The coupling topology matrix of the pendulum system is: and

[0100] The above is a specific description of the preferred implementation of the present invention, but the present invention is not limited to the above-mentioned implementation mode. Technical personnel familiar with the field can also make various equivalent deformations or substitutions without violating the spirit of the present invention. These equivalent deformations or substitutions are all included in the scope defined by the claims of the present invention.

Claims

1. A predictive control model method and system for a nonlinear pendulum system, characterized in that: The following steps are involved: Step S1: constructing a dynamic model of the nonlinear pendulum system according to the working state of the nonlinear pendulum system; Step S2: quantizing the control input in combination with a hysteresis quantizer, thereby alleviating the limitation of network resources on the cooperative performance of the nonlinear pendulum system; Step S3: According to the steps S1 and S2, a learning model predictive control algorithm capable of online learning is designed to online predict the optimal control gain matrix, thereby effectively reducing the impact of communication defects on system performance; Step S4: According to step S3, a Lyapunov function related to the Markov switching topology is constructed and used as the cost function of the learning model predictive control algorithm.

2. The predictive control model method for a nonlinear pendulum system according to claim 1, characterized in that: The following steps are involved: Step S1: constructing a dynamic model of the nonlinear pendulum system according to the working state of the nonlinear pendulum system; The dynamic model of the nonlinear pendulum system is: Where x and v represent the displacement and velocity of the nonlinear pendulum, respectively, and u represents the control input; Assume i (t) = [x i (t),v i (t)] T , the continuous-time state space equation is: in, A and B both represent nonlinear pendulum coefficient matrices; The above is a continuous nonlinear pendulum system. The present invention further proposes a discrete time nonlinear pendulum system model: x i (k+1)=f(x i (k),u i (k)) 3. The predictive control model method for a nonlinear pendulum system according to claim 1, characterized in that: The following steps are involved: Step S2: quantizing the control input in combination with a hysteresis quantizer, thereby alleviating the limitation of network resources on the cooperative performance of the nonlinear pendulum system; Since the hysteresis quantizer can effectively reduce the oscillation phenomenon and ensure that the quantization value is more stable during the switching process, this paper considers the discretized hysteresis quantizer: Where ρ = (1-δ) / (1+δ), (0<ρ<1) parameter determines the quantization density, u j =ρ 1-j u0(j=1,...,n x ). In order to optimize the network resources among multiple agents, this paper proposes a quantitative feedback controller: In the formula, K k is the controller gain matrix. For the convenience of further discussion, the control input vector u(k) = [u1(k) ... u N (k)] T Similarly, x(k)=[x1(k)...x N (k)] T . Combined with the above formula, the control input can be re-expressed in the following vector form: Let p = θ(k), then the system state vector of MASs can be described as: Among them, f pi (x(k)) represents the function f(x i (k),u i (k)), and Assumption 1 is satisfied.

4. The predictive control model method for a nonlinear pendulum system according to claim 1, characterized in that: The following steps are involved: Step S3: According to the steps S1 and S2, a learning model predictive control algorithm capable of online learning is designed to online predict the optimal control gain matrix, thereby effectively reducing the impact of communication defects on system performance; The cost function is defined as: Where P p ,p∈S is the positive definite matrix associated with the Markov mode. The goal of LMPC is to find J that minimizes the cost J(e(k)). Then the optimal feedback gain matrix is The following conditions are met: The present invention aims to solve the optimization problem (17) using a shallow neural network. In this paper, the input layer of the shallow neural network consists of a length of l (l = n x n u ) sequence Composition, where a∈{1...n x } and b∈{1...n u }. In order to train to get the optimal gain value K k , a single hidden layer with M neurons is selected. The input layer consists of the values ​​of the cost function J(e(k)). make The signal generated by the hidden layer is given by: w=σ(Wz) Where σ represents the sigmoid function. W∈R M×(l+1) Represents the weight matrix of the neuron. For a given gain K k , the output layer of the cost function It can be approximated as: Among them, Y∈R 1×M Represents the weight matrix of the output layer. The neural network continuously optimizes the weights W and Y during the iterative learning process to better fit the control gain. The specific steps of the model predictive control algorithm are as follows: In the initial stage, the system is given the initial condition x(0), the initial mode θ(0) and the cost function J0. For convenience, the initial gain matrix K0 is set to a null value. Then, it is used together with J0 as the initial training data set of the neural network. Then, the neural network predicts the value of the cost function through iterative learning In each iteration, the previous K k Different combinations are provided to generate predicted values ​​of the cost function In order to improve the prediction accuracy, K k It can be adjusted by ±δ. Then, the item with the smallest predicted value is selected as the optimal control gain K k , and the control input u(k) is determined accordingly. Next, in order to estimate more accurately The algorithm introduces the cycle prediction mechanism of the LMPC protocol. The specific steps are: 1) Automatically generate the jump pattern θ(t) according to the previous moment and the probability transfer matrix Π; 2) Calculate the predicted control input and system status 3) Estimate approximate value At the end of each iteration, sum all the current Get an approximation of the cost function Finally, the predicted value And the corresponding optimal gain Stored in the data set for the next iteration. Repeat the above steps until the predicted value Reach the preset minimum condition ξ. It can be considered that the optimization problem has been solved and the training is ended.

5. The predictive control model method for a nonlinear pendulum system according to claim 1, characterized in that: The following steps are involved: Step S4: A Lyapunov function related to the Markov switching topology is constructed and used as the cost function of the learning model predictive control algorithm.

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