Self-adaptive optimization control method of direct current motor system with state constraint
By establishing a mechanistic model of a DC motor system and introducing a nonlinear mapping function and a backstepping recursive framework, combined with neural networks and reinforcement learning strategies, an optimal feedback controller was designed, solving the problem of accurate tracking control of the DC motor system under complex operating conditions, and achieving stable system operation and performance improvement.
Patent Information
- Application Number
- CN202511309153.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-09-15
- Publication Date
- 2025-12-12
AI Technical Summary
DC motor systems are difficult to control precisely under complex operating conditions, especially in dynamic operation and performance improvement without violating state constraints.
By establishing a mechanistic model and introducing a nonlinear mapping function, the system is transformed into an unconstrained system. A feedforward virtual controller is designed under the backstepping recursive framework. Combining the approximation performance of neural networks and reinforcement learning strategies, an optimal feedback controller is designed to achieve adaptive optimization control of the system.
Without violating state constraints, precise tracking control and dynamic balance of energy consumption of DC motor system are achieved, improving system operation stability and performance.
Smart Images
Figure CN121124630A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of automatic control design and application, and specifically relates to an adaptive optimization control method for a DC motor system with state constraints. Background Technology
[0002] A DC motor control system is an electronic system used to control the speed, torque, and position of a DC motor. Its control performance directly affects industrial production efficiency and equipment operation safety. With the intelligent development of industrial automation, robotics, and electric vehicles, DC motor systems face increasingly complex operating environments. On the one hand, they typically exhibit significant nonlinear and uncertain characteristics in actual operation, making accurate dynamic modeling of the system difficult. On the other hand, the angular velocity of the DC motor system is also constrained by boundary conditions. Therefore, designing adaptive optimization control strategies for DC motor systems without violating state constraints is of significant engineering value for improving the safe operating boundaries of motor systems under high dynamic conditions and enhancing system performance. Summary of the Invention
[0003] To address the aforementioned shortcomings of existing technologies, this invention provides an adaptive optimization control method for a DC motor system with state constraints, comprising the following steps:
[0004] A mechanistic model of a DC motor system is established, and a nonlinear mapping function is introduced to convert the system with state constraints into an unconstrained system.
[0005] A feedforward virtual controller is designed within the backstepping recursive framework to ensure the convergence of system tracking error and stable operation.
[0006] The optimal feedback controller is designed based on a reinforcement learning strategy. The HJB equations related to the optimal performance function are solved by combining the approximate approximation performance of the neural network, thereby obtaining the optimal control strategy for the DC motor system.
[0007] Furthermore, the mechanistic model of the DC motor system is as follows:
[0008]
[0009] Where α1 represents the angular position of the motor, α2 represents the angular velocity of the motor; J represents the known moment of inertia; f represents the unmeasurable viscous friction force; T f It is an unmeasurable linear frictional force; d(t) represents an unknown disturbance, but it is bounded and satisfies ||d(t)||≤d M ; y∈R represents the output of the DC motor system; u represents the motor torque, and the constraint condition satisfied by the output y of the DC motor system is: and The upper and lower bounds of the constraints representing the motor's angular position α1, respectively, can also be written as: The range of values for the motor angular velocity α2 is: and These represent the upper and lower bounds of the constraint on the motor's angular velocity α2, respectively. (The constraint is satisfied.) and All are positive numbers, where i = 1, 2.
[0010] Furthermore, a nonlinear mapping function is introduced to transform a system with state constraints into an unconstrained system, specifically:
[0011] The introduced nonlinear mapping function is a state-dependent transformation function used to handle the state constraint problem of a DC motor system. The specific form of the state-dependent transformation function is as follows:
[0012]
[0013] Next, by differentiating the state-dependent transition function, we obtain:
[0014]
[0015] Among them, y d The trajectory representing the desired angular position of the motor.
[0016]
[0017] ψ i and ψ d It is a computable and continuous function.
[0018] The state-constrained DC motor system (1) is transformed into a novel unconstrained system equation:
[0019]
[0020] Where F1=ψ1α2-ξ2, J -1 It represents the reciprocal of the known inertia J.
[0021] Furthermore, the specific steps for designing the feedforward virtual controller within the backstepping recursive framework are as follows:
[0022] Within the framework of backstepping recursion, the coordinate transformation equation is defined as follows:
[0023]
[0024] Where σ1 represents the virtual feedforward controller, z1 represents the conversion error between the motor angular position and the desired motor angular position, and z2 represents the conversion error between the motor angular velocity and the virtual controller.
[0025] Differentiating the transformation error z1 in equation (5) and combining it with the differentiated state-dependent transformation function (3), we obtain:
[0026]
[0027] Construct a Lyapunov function of the following form:
[0028]
[0029] Differentiating the Lyapunov function V1 and combining it with equation (6), we can obtain:
[0030]
[0031] The virtual feedforward controller σ1 is designed as follows:
[0032]
[0033] Where k1 represents a positive constant.
[0034] Furthermore, the approximate approximation of the neural network refers to the approximation of an unknown nonlinear function based on the neural network;
[0035] Construct three neural networks R(X), J * (Z) and F * (z);
[0036] set up
[0037] Where X represents the input vector of the neural network R(X),
[0038] W R Let W represent the optimal weight vector of the neural network R(X), and satisfy ||W R ||≤υ R , where υ R It represents the upper bound of the norm of the optimal weight vector;
[0039] S R (X) represents the Gaussian function vector of the neural network R(X), and satisfies This represents the upper bound of the norm of a Gaussian function vector;
[0040] Approximation error δ R satisfy The approximation error δR The upper boundary.
[0041] definition W represents the optimal weight vector R The estimate, This represents the estimation error of the optimal weight vector;
[0042] In addition, definition Represents θ R The estimate, and the definition This represents the estimation error;
[0043]
[0044] Where Z represents neural network J * The input vector of (Z), where
[0045] W p Representing neural network J * The optimal weight vector of (Z) that satisfies ||W p ||≤υ p , where υ R It represents the upper bound of the norm of the optimal weight vector;
[0046] S(Z) represents the neural network J * (Z) is a Gaussian function vector that satisfies This represents the upper bound of the norm of a Gaussian function vector;
[0047] Approximation error δ p satisfy The approximation error δ p Upper boundary;
[0048] definition W represents the optimal weight vector p The estimate, This represents the estimation error of the optimal weight vector.
[0049]
[0050] Where z represents the input vector of the neural network F(z), and z = z2, this input vector is only related to z2;
[0051] Furthermore, let the unknown nonlinear function
[0052] Using a neural network R(X) to approximate an unknown nonlinear function R' (X), combining equations (12) and (16), we can obtain:
[0053]
[0054] Combining Young's inequality with bounding, we obtain:
[0055]
[0056] Where k1, d and d M All represent positive constants; J -2 This represents the reciprocal of the square of the known moment of inertia, where r1 and r2 represent positive constants.
[0057] Furthermore, based on the reinforcement learning strategy, the optimal performance function J'(Z) related to the transition error z2 and the optimal feedback controller u′ is defined as follows:
[0058]
[0059] Where β>0 represents the discount factor, and τ represents the time variable, specifically the time parameter within the integration interval [t,∞) starting from the current time t.
[0060] According to Leibniz's integral law, differentiating both sides of equation (24) yields:
[0061]
[0062] Construct HJB equations in the following form:
[0063]
[0064] in, Represents the costate equation, and
[0065] Solve The optimal feedback controller u′ is obtained as follows:
[0066]
[0067] Furthermore, the conversion error z2 is expressed as:
[0068]
[0069] Where F'(z2) represents an unknown nonlinear function;
[0070] The unknown nonlinear function F'(z2) is approximated by a neural network F*(z2), and the unknown nonlinear function J'(z) is approximated by a neural network J*(Z), which can be written as follows:
[0071]
[0072]
[0073] Furthermore, the optimal feedback controller u′ is transformed into:
[0074]
[0075] Substituting equation (31) into the HJB equation yields:
[0076]
[0077] in, Represents the residual term;
[0078]
[0079] residuals It is bounded and satisfies Representing residual terms The upper boundary.
[0080] Since the residual term includes the gradient term The optimal weight vector W can be omitted. p It is unknown, therefore, in combination Obtain the approximate optimal feedback controller for:
[0081]
[0082] Furthermore, approximation Convert to:
[0083]
[0084] Design Feedback Update Law and
[0085]
[0086] in, And b; c1, c2, c3; d1, d2, d3 represent positive constants.
[0087] Beneficial Effects: The adaptive optimization control method for a state-constrained DC motor system described in this invention is based on a DC motor system model and designs an adaptive optimization control method. By introducing a nonlinear mapping, the state-constrained DC motor system is transformed into an unconstrained model. Under the backstepping recursive framework, a reinforcement learning strategy is combined with a neural network to design a feedback optimal controller, thereby achieving a dynamic balance between control accuracy and energy consumption without violating state constraints. Specifically applied to DC motor systems, this invention mainly achieves accurate tracking of motor angular velocity under complex operating conditions, which has significant engineering implications for improving the efficient and stable operation of industrial equipment. Attached Figure Description
[0088] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are merely some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without any creative effort.
[0089] Figure 1 This is a flowchart of an adaptive optimization control method for a DC motor system with state constraints according to the present invention.
[0090] Figure 2 The relationship between the motor angular position α1 and the desired trajectory y in the DC motor system of this invention is... d Tracking performance graph;
[0091] Figure 3 This is a trajectory diagram of the motor angular velocity α2 in the DC motor system of the present invention;
[0092] Figure 4 This is a trajectory diagram of the conversion error z1 between the motor angular position and the desired trajectory in the DC motor system of the present invention;
[0093] Figure 5 This is the trajectory diagram of the optimal controller u in the DC motor system of the present invention;
[0094] Figure 6 The feedforward adaptive law in the DC motor system of this invention The change curve;
[0095] Figure 7 The feedback update law in the DC motor system of this invention and The change curve. Detailed Implementation
[0096] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.
[0097] The present invention will be further described below with reference to embodiments.
[0098] like Figure 1 As shown, an adaptive optimization control method for a constrained DC motor system includes the following steps:
[0099] Step 1: Establish a mechanistic model of the DC motor system, introduce nonlinear mapping, and transform the DC motor system with state constraints into an unconstrained motor system;
[0100] The established mechanistic model of the DC motor system is as follows:
[0101]
[0102] Where α1 represents the angular position of the motor, α2 represents the angular velocity of the motor; J represents the known moment of inertia; f represents the unmeasurable viscous friction force; T f It is an unmeasurable linear frictional force; d(t) represents an unknown disturbance, but it is bounded and satisfies ||d(t)||≤d M ; y∈R represents the output of the DC motor system; u represents the motor torque, and the constraint condition satisfied by the output y of the DC motor system is: and The upper and lower bounds of the constraints representing the motor's angular position α1, respectively, can also be written as: The range of values for the motor angular velocity α2 is: and These represent the upper and lower bounds of the constraint on the motor's angular velocity α2, respectively. (The constraint is satisfied.) and All are positive numbers, where i = 1, 2.
[0103] A state-dependent transition function of the following form is constructed to directly handle the state constraint problem of a DC motor system:
[0104]
[0105] Next, by differentiating the state-dependent transition function, we obtain:
[0106]
[0107] Among them, y d The trajectory representing the desired angular position of the motor.
[0108]
[0109] ψ i and ψ d It is a computable and continuous function.
[0110] The state-constrained DC motor system (1) is transformed into a novel unconstrained system equation:
[0111]
[0112] Where F1=ψ1α2-ξ2, J -1 It represents the reciprocal of the known moment of inertia.
[0113] Step 2: Based on Step 1, an adaptive optimization control method for the DC motor system is designed. Within the framework of backstepping recursion, a Lyapunov function is constructed. By designing a feedforward virtual controller, each subsystem of the DC motor system is gradually stabilized, ensuring the convergence of the conversion error between the motor angular position and the desired motor angular position, and achieving stable operation of the DC motor system. Specifically, this includes the following:
[0114] Step 2.1: Within the framework of the backstep recursion technique, define a new coordinate transformation equation:
[0115]
[0116] Where σ1 represents the virtual feedforward controller, z1 represents the conversion error between the motor angular position and the desired motor angular position, and z2 represents the conversion error between the motor angular velocity and the virtual controller.
[0117] Furthermore, by differentiating the transformation error z1 in equation (5) and combining it with the differentiated state-dependent transformation function (3), we can obtain:
[0118]
[0119] Step 2.2: Construct a Lyapunov function of the following form:
[0120]
[0121] Step 2.3: Differentiating the Lyapunov function V1 and combining it with equation (6) yields:
[0122]
[0123] Step 2.4: Design the virtual feedforward controller σ1 as follows:
[0124]
[0125] Where k1 represents a positive constant.
[0126] Substituting equation (9) into equation (8) yields The formula introduces coupling terms z1z2, so by using Young's inequality to scale the coupling terms z1z2, we can obtain:
[0127]
[0128] The Lyapunov function V1 can be retrieved again as follows:
[0129]
[0130] Step 3: Combining reinforcement learning strategies with the approximation capabilities of neural networks, design a feedback-optimal controller to ensure that the motor angular velocity of the DC motor system does not exceed the constraint range and that the optimal performance index function is minimized without violating the constraint boundaries. Specifically, this includes the following:
[0131] Step 3.1: To achieve optimal control of the DC motor system and improve its performance, reliability, and economy, we first introduce the powerful ability of neural networks to learn and approximate complex nonlinear dynamics. This invention uses three neural networks sequentially to approximate different unknown nonlinear functions. By introducing three neural networks R(X), J... * (Z) and F * (z), the specific details of the three unknown nonlinear functions will be presented in steps 3.3 and 3.6. To simplify the overall process, this step comprehensively summarizes the three neural networks. The specific form is as follows:
[0132]
[0133] Where X represents the input vector of the neural network, The input vector contains the derivative of the motor angular position with respect to the virtual feedforward controller. This neural network will be incorporated into the design of the optimal feedforward controller in step 3.3.
[0134] W R Let W represent the optimal weight vector of the neural network R(X), and satisfy ||W R ||≤υ R , where υ R It represents the upper bound of the norm of the optimal weight vector;
[0135] S R(X) represents the Gaussian function vector of the neural network R(X), and satisfies This represents the upper bound of the norm of a Gaussian function vector;
[0136] Approximation error δ R satisfy The approximation error δ R The upper boundary.
[0137] definition W represents the optimal weight vector R The estimate, This represents the estimation error of the optimal weight vector. Furthermore, it defines... Represents θ R The estimate, and the definition This represents the estimation error.
[0138]
[0139] Where Z represents neural network J * The input vector of (Z), where Includes motor angular position and virtual feedforward controller, and the virtual controller is related to α1, z1, y d , All are correlated; the neural network J * (Z) will be used in step 3.6 to approximate the optimal performance function;
[0140] W p Representing neural network J * The optimal weight vector of (Z) that satisfies ||W p ||≤υ p , where υ R It represents the upper bound of the norm of the optimal weight vector;
[0141] S(Z) represents the neural network J * (Z) is a Gaussian function vector that satisfies This represents the upper bound of the norm of a Gaussian function vector;
[0142] Approximation error δ p satisfy The approximation error δ p Upper boundary;
[0143] definition W represents the optimal weight vector p The estimate, This represents the estimation error of the optimal weight vector.
[0144]
[0145] Here, z represents the input vector of the neural network F(z), and z = z², meaning this input vector is only related to z². This neural network will be used in step 3.6 to approximate the unknown nonlinear function related to the transformation error z².
[0146] W F Let W represent the optimal weight vector of the neural network F(z), and satisfy ||W|| F ||≤υ F , where υ R It represents the upper bound of the norm of the optimal weight vector;
[0147] S(Y) represents the Gaussian function vector of the neural network F(z), and satisfies The upper bound of the norm of the Gaussian function vector; the approximation error δ F satisfy The approximation error δ F Upper boundary;
[0148] definition Represents W F The estimate, This represents the estimation error.
[0149] Step 3.2: Construct a Lyapunov function V2 of the following form:
[0150]
[0151] Differentiating the Lyapunov function V² and combining it with (3), (4) and (11), we can obtain:
[0152]
[0153] in, Indicates y d The first derivative;
[0154] In addition, let u=u+u′, u * is the optimal feedforward controller, and u′ is the optimal feedback controller.
[0155] Step 3.3: Let Due to the unmeasurable linear friction force T f Since both the unmeasurable viscous friction force f and the unknown viscous friction force are unknown, R'(X) is an unknown nonlinear function.
[0156] By using a neural network R(X) to approximate the unknown nonlinear function R'(X), and combining equations (12) and (16), we can obtain:
[0157]
[0158] Combining Young's inequality with bounding, we obtain:
[0159]
[0160] Where k1, d and d M All represent positive constants; J -2 This represents the reciprocal of the square of the known moment of inertia, where r1 and r2 represent positive constants.
[0161] Step 3.4: Design the optimal feedforward controller u * With feedforward adaptive law as follows:
[0162]
[0163] Where k2 and μ represent positive constants, substituting (21) and (22) into (17) yields:
[0164]
[0165] Step 3.5: Based on the reinforcement learning strategy, define the optimal performance function J related to the transition error z2 and the optimal feedback controller u′. * (Z):
[0166]
[0167] Where β>0 represents the discount factor, and τ represents the time variable, specifically the time parameter within the integration interval [t,∞) starting from the current time t.
[0168] According to Leibniz's integral law, differentiating both sides of equation (24) yields:
[0169]
[0170] Construct the HJB (Hamilton-Jacobi-Bellman) equation in the following form:
[0171]
[0172] in, Represents the costate equation, and
[0173] Solve The optimal feedback controller u′ is obtained as follows:
[0174]
[0175] Step 3.6: Consider the nonlinear system related to the conversion error z2:
[0176]
[0177] Here, F'(z2) represents the unknown nonlinear function. Since both the unknown nonlinear function F'(z2) and the optimal performance function J'(Z) are unknown functions, and the DC motor system exhibits strong nonlinearity and uncertainty, the optimal control strategy cannot be directly achieved by solving the HJB equations. Therefore, considering its optimal tracking control problem is of paramount importance. Combining the ability of neural networks to approximate the performance of continuous nonlinear functions, the unknown nonlinear function F'(z2) and the optimal performance function J'(Z) can be approximated as follows:
[0178]
[0179] Furthermore, the optimal feedback controller u′ can be transformed into:
[0180]
[0181] Substituting (31) into the HJB equation yields:
[0182]
[0183] in, Represents the residual term;
[0184]
[0185] residuals It is bounded and satisfies Representing residual terms The upper boundary.
[0186] Since the residual term includes the gradient term This can be omitted. And the optimal weight vector W p It is unknown, therefore, in combination Obtain the approximate optimal feedback controller for:
[0187]
[0188] Furthermore, approximation It can be converted to:
[0189]
[0190] Step 3.7: It is worth noting that the constructed HJB equations are difficult to solve directly to achieve optimal tracking control. Therefore, a reinforcement learning algorithm based on neural networks is used to overcome this challenge. Feedback update law. and The design must simultaneously achieve minimization And to ensure the stability of the motor system. The specific design is as follows:
[0191]
[0192] in, And b; c1, c2, c3; d1, d2, d3 represent positive constants.
[0193] Combining Young's inequality with bounding, we can obtain:
[0194]
[0195] Where υ p Represents a positive constant.
[0196] Substituting (35)-(40) into (16) yields:
[0197]
[0198] in,
[0199] and
[0200] β, o, υ p υ F All represent positive numbers;
[0201] Choose appropriate parameters b, c2, d2, and d3 to satisfy the following conditions:
[0202]
[0203] Combining (41) and (42), It can be transformed into:
[0204]
[0205] in,
[0206] Finally, we can obtain:
[0207]
[0208] in,
[0209] In this embodiment of the invention, the basic physical parameters of the DC motor system are J = 0.018 kg·m. 2 f = sint + 1.82, T f =0.987, d=sint. Expected trajectory y d =2sin(0.6πt)+0.6. The initial values of the motor angular position α1 = 0.6, the motor angular velocity α2 = 0.2, and the initial values of the feedforward adaptive parameters and the feedback update law are respectively selected as... as well as Furthermore, the discount factor β = 4 in the optimal cost function, and the specific design parameters for the virtual controller, optimal controller, feedforward adaptive law, and feedback update law are: k1 = 70, k2 = 25, r1 = r2 = 2, b = 0.2, c1 = c2 = c3 = 4, d1 = 0.1, d2 = 0.31, d3 = 0.21, μ = 6. The constraint boundary conditions must be satisfied. k c1 =-2, k c2 =-6.
[0210] Under these initial conditions, simulation verification was performed, and the simulation results are shown in the figure below. Figures 2-7 As shown. Figure 2 This shows the motor angular position σ1 and the desired trajectory y in the DC motor system. d Tracking performance graph; Figure 3 The trajectory diagram of the motor angular velocity σ2 in the DC motor system is shown; Figure 4 The diagram shows the conversion error z1 trajectory between the motor angular position and the desired trajectory of the DC motor system. Figure 5 The trajectory diagram of the optimal controller u for the DC motor system is shown; Figure 6 This demonstrates the feedforward adaptive law for DC motor systems. The change curve; Figure 7 The feedback update law of the DC motor system is shown. and The trajectory diagram.
[0211] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions will not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.
Claims
1. An adaptive optimization control method for a DC motor system, characterized in that, Includes the following steps: A mechanistic model of a DC motor system is established, and a nonlinear mapping function is introduced to convert the system with state constraints into an unconstrained system. A feedforward virtual controller is designed within the backstepping recursive framework to ensure the convergence of system tracking error and stable operation. The optimal feedback controller is designed based on a reinforcement learning strategy. The HJB equations related to the optimal performance function are solved by combining the approximate approximation performance of the neural network, thereby obtaining the optimal control strategy for the DC motor system.
2. The adaptive optimization control method for a DC motor system according to claim 1, characterized in that, The mechanism model of the DC motor system is as follows: Where α1 represents the angular position of the motor, α2 represents the angular velocity of the motor; J represents the known moment of inertia; f represents the unmeasurable viscous friction force; T f It is an unmeasurable linear frictional force; d(t) represents an unknown disturbance, but it is bounded and satisfies ||d(t)||≤d M ; y∈R represents the output of the DC motor system; u represents the motor torque, and the constraint condition satisfied by the output y of the DC motor system is: and The upper and lower bounds of the constraints representing the motor's angular position α1, respectively, can also be written as: The range of values for the motor angular velocity α2 is: and These represent the upper and lower bounds of the constraints for the motor's angular velocity α2, respectively; satisfying... and All are positive numbers, where i = 1, 2.
3. The adaptive optimization control method for a DC motor system according to claim 2, characterized in that, By introducing a nonlinear mapping function, a system with state constraints is transformed into an unconstrained system, specifically as follows: The introduced nonlinear mapping function is a state-dependent transformation function used to handle the state constraint problem of a DC motor system. The specific form of the state-dependent transformation function is as follows: Next, by differentiating the state-dependent transition function, we obtain: Among them, y d The trajectory representing the desired angular position of the motor. ψ i and ψ d It is a computable and continuous function; The state-constrained DC motor system (1) is transformed into a novel unconstrained system equation: Where F1=ψ1α2-ξ2, J -1 It represents the reciprocal of the known inertia J.
4. The adaptive optimization control method for a DC motor system according to claim 3, characterized in that, The specific steps for designing a feedforward virtual controller within the backstepping recursive framework are as follows: Within the framework of backstepping recursion, the coordinate transformation equation is defined as follows: Where σ1 represents the virtual feedforward controller, z1 represents the conversion error between the motor angular position and the desired motor angular position, and z2 represents the conversion error between the motor angular velocity and the virtual controller. Differentiating the transformation error z1 in equation (5) and combining it with the differentiated state-dependent transformation function (3), we obtain: Construct a Lyapunov function of the following form: Differentiating the Lyapunov function V1 and combining it with equation (6), we can obtain: The virtual feedforward controller σ1 is designed as follows: Where k1 represents a positive constant.
5. The adaptive optimization control method for a DC motor system according to claim 4, characterized in that, The approximation of the neural network refers to the approximation of an unknown nonlinear function based on the neural network; Construct three neural networks R(X), J * (Z) and F * (z); set up Where X represents the input vector of the neural network R(X), W R Let W represent the optimal weight vector of the neural network R(X), and satisfy ||W R ||≤υ R , where υ R It represents the upper bound of the norm of the optimal weight vector; S R (X) represents the Gaussian function vector of the neural network R(X), and satisfies This represents the upper bound of the norm of a Gaussian function vector; Approximation error δ R satisfy The approximation error δ R The upper bound; definition W represents the optimal weight vector R The estimate, This represents the estimation error of the optimal weight vector; furthermore, it defines... Represents θ R The estimate, and the definition This represents the estimation error; Where Z represents neural network J * The input vector of (Z), where W p Representing neural network J * The optimal weight vector of (Z) that satisfies ||W p ||≤υ p , where υ R It represents the upper bound of the norm of the optimal weight vector; S(Z) represents the neural network J * (Z) is a Gaussian function vector that satisfies This represents the upper bound of the norm of a Gaussian function vector; Approximation error δ p satisfy The approximation error δ p Upper boundary; definition W represents the optimal weight vector p The estimate, This represents the estimation error of the optimal weight vector; Here, z represents the input vector of the neural network F(z), and z = z2. This input vector is only related to z2.
6. The adaptive optimization control method for a DC motor system according to claim 5, characterized in that, Let the unknown nonlinear function By approximating the unknown nonlinear function R'(X) using a neural network R(X), and combining equations (12) and (16), we can obtain: Combining Young's inequality with bounding, we obtain: Where k1, d and d M All represent positive constants; J -2 This represents the reciprocal of the square of the known moment of inertia, where r1 and r2 represent positive constants.
7. The adaptive optimization control method for a DC motor system according to claim 6, characterized in that, Based on the reinforcement learning strategy, the optimal performance function J'(Z) related to the transformation error z2 and the optimal feedback controller u′ is defined as follows: Where β>0 represents the discount factor, and τ represents the time variable, specifically the time parameter within the integration interval [t,∞) starting from the current time t. According to Leibniz's integral law, differentiating both sides of equation (24) yields: Construct HJB equations in the following form: in, Represents the costate equation, and Solve The optimal feedback controller u′ is obtained as follows:
8. The adaptive optimization control method for a DC motor system according to claim 7, characterized in that, The conversion error z2 is expressed as: Where F'(z2) represents an unknown nonlinear function; The unknown nonlinear function F'(z2) is approximated by a neural network F*(z2), and the unknown nonlinear function J'(z) is approximated by a neural network J*(Z), which can be written as follows: Furthermore, the optimal feedback controller u′ is transformed into: Substituting equation (31) into the HJB equation, we get: in, Represents the residual term; residuals It is bounded and satisfies Representing residual terms The upper bound; Since the residual term includes the gradient term The optimal weight vector W can be omitted. p It is unknown, therefore, in combination Obtain the approximate optimal feedback controller for:
9. The adaptive optimization control method for a DC motor system according to claim 8, characterized in that, approximate Convert to: Design Feedback Update Law and in, And b; c1, c2, c3; d1, d2, d3 represent positive constants.