Dynamic event-triggered control method and system for unknown nonlinear systems based on adp and ism control

Abstract

This invention provides a dynamic event-triggered control method and system for unknown nonlinear systems based on ADP and ISM control, belonging to the field of nonlinear cyber-physical system control and optimization. It addresses the problem that existing adaptive dynamic programming methods use imprecise mathematical models during the modeling process, leading to system instability or security vulnerabilities. This invention uses a recurrent neural network (RNN) to reconstruct the system dynamics. Considering the vulnerability of the system's network layers to spoofed data injection attacks, it reconstructs the system again. Utilizing integral sliding mode control (ISMC) and adaptive dynamic programming (ADP), it designs a hybrid event-triggered control scheme with two different triggering components. This includes designing event-triggered sliding mode control to keep the system state on the sliding surface, eliminating the impact of spoofed data injection attacks and mismatch uncertainties on the system. For equivalent sliding mode dynamics, it designs optimal event-triggered control based on adaptive dynamic programming (ADP), using dynamic event triggering to save communication resources and extend actuator lifespan.

Description

Technical Field

[0001] This invention relates to the field of control and optimization technology for nonlinear cyber-physical systems, and more specifically, to a dynamic event-triggered control method and system for unknown nonlinear systems based on ADP and ISM control. Background Technology

[0002] In recent years, adaptive dynamic programming has emerged as a rapidly developing optimal control method. By constructing a function structure to estimate the cost function in dynamic programming, it avoids solving the Hamilton-Jacobi-Bellman (HJB) equations, significantly reducing the computational burden in solving optimal control problems for high-dimensional complex nonlinear systems. This allows the method to effectively address the optimal control problems of complex nonlinear systems. However, in real-world systems, due to the high complexity of their internal structure and external operating environment, models built using mechanistic modeling or system identification methods are only approximate models, still deviating from the real system. In other words, unmodeled dynamics and other uncertainties inevitably arise during the modeling process. If such an imprecise mathematical model is used when building the controller, it will certainly cause various unexpected obstacles in actual operation, and may even lead to system instability or safety hazards. Summary of the Invention

[0003] The technical problem to be solved by this invention is:

[0004] To address the problem that existing adaptive dynamic programming methods use imprecise mathematical models during the modeling process, which can even lead to system instability or security risks.

[0005] The technical solution adopted by the present invention to solve the above-mentioned technical problems is as follows:

[0006] This invention provides a dynamic event-triggered control method for unknown nonlinear systems based on ADP and ISM control, comprising the following steps:

[0007] S100, Constructing an uncertain nonlinear system under a fake data injection attack;

[0008] S200: Dynamic reconstruction of unknown nonlinear systems based on recurrent neural networks (RNNs);

[0009] S300. Design an event-triggered integral sliding mode control (ISMC) based on the information reconstructed from a recurrent neural network (RNN).

[0010] S400. Based on the information reconstructed by the recurrent neural network (RNN), design an adaptive dynamic programming (ADP) control scheme based on dynamic event triggering, including a dynamic event triggering mechanism and neural network approximation, to ensure that the system is asymptotically stable.

[0011] Further, in step S100, the following are included:

[0012] Consider the following continuous-time input affine nonlinear system:

[0013]

[0014] in, Indicates the system status. Indicates control input, and It is an unknown dynamic function, and m is the control input order, and n is the system state order.

[0015] Assumption In a compact set containing the origin The above is a continuation of Lipschitz; making Assuming the initial state of the system, let the solution of system (1) be... For each and They are all unique;

[0016] Attacked control input signals It is given by the following formula:

[0017]

[0018] in, This indicates that the channel from the controller to the actuator is vulnerable to spoofed data injection attacks. ,in These are known constants;

[0019] Assumption For a column to have full rank and for any Bounded, system (1) during injection The system is controllable; therefore, the uncertain nonlinear system under a false data injection attack is:

[0020]

[0021] in, This indicates uncertainty in an unknown system. ,in It is a known constant.

[0022] Further, in step S200, the following are included:

[0023] Consider the following system formula (1) expressed by a recurrent neural network (RNN):

[0024]

[0025] in, The weighted matrix representing the system state. and They represent and The unknown ideal weight matrix, , This represents the activation function vector, which is the sigmoidal function. This represents the RNN reconstruction estimation error;

[0026] System (1) is estimated as:

[0027]

[0028] in, This represents the state of an identifier recurrent neural network (RNN). , and These represent the effects of using recurrent neural networks (RNNs) on... and The estimated weight matrix, The feedback item is defined as follows: ,in Indicate design parameters and Indicates recognition error;

[0029] Using formulas (4) and (5), the following dynamic identification error is obtained. :

[0030]

[0031] in, , , , and ;

[0032] Assumption 1: The ideal RNN weighting matrix A, and satisfy:

[0033]

[0034] in, It is a known positive definite constant matrix;

[0035] Assumption 2: There exists a constant greater than 0. , making ;

[0036] If the estimated RNN weight matrix given in formula (5) passes the design matrix:

[0037]

[0038] in, It is a design matrix;

[0039] Then the RNN recognizer formula (5) maintains the error. Asymptotic stability; By using formula (8), we can obtain the result when... hour, , , All are 0, that is All are convergent; therefore, the unknown nonlinear system (1) is reconstructed as:

[0040]

[0041] in, and , and They are and The reconstructed unknown dynamic function.

[0042] Further, in step S300, the following are included:

[0043] For the system, i.e., formula (2), design an ADP-based integral sliding mode controller:

[0044]

[0045] in, It is an integral sliding mode compensator. It is an optimal controller based on ADP;

[0046] First, the design scheme of the integral sliding mode compensator is given:

[0047] Consider the integral sliding mode function S(x) as follows:

[0048]

[0049] in, This represents the initial value of the integral sliding mode function. , and It is obtained from formula (8);

[0050] Assumption 3: For any There exists a satisfying Full column rank function ; yes The left pseudo-reversal, (x) is The right pseudo-reversal, ; and It is a known positive constant;

[0051] Assumption 4: The boundary is ,and , It is reversible. It is a known positive constant;

[0052] Differentiating formula (11) yields:

[0053]

[0054] According to hypothesis 4, Then the equivalent control input of the uncertain nonlinear system (3) is... for:

[0055]

[0056] The equivalent sliding mode dynamics expression is:

[0057]

[0058] in, ;

[0059] To reduce the multiplication matrix The impact and minimization of mismatch interference ,choose As the optimal function, and can be obtained ;

[0060] By satisfying stable sliding mode , Designed as:

[0061]

[0062] in, For sliding mode gain;

[0063] satisfy:

[0064]

[0065] For cases where ISM events are triggered, discontinuous control actions occur at specific times. Updated to ,in For the triggering time, and in two consecutive triggering times The internal remains unchanged:

[0066]

[0067] Triggering error Written as:

[0068]

[0069] If an error is triggered:

[0070]

[0071] in, , ;

[0072] When an event is triggered, the control input is updated, and the event trigger error is checked. Reset to zero;

[0073] Theorem 1: For an integral sliding mode compensator For the uncertain nonlinear system under equation (3), there exists a stable sliding mode from the initial time. ;

[0074] Theorem 2: Considering the event triggering condition, i.e., formula (18), to avoid Zeno's behavior, the event triggering interval condition... Designed as follows:

[0075]

[0076] Further, in step S400, the following is included:

[0077] S410. Based on assumption 4, define... The equivalent sliding mode dynamic representation is:

[0078]

[0079] Value function for:

[0080]

[0081] in, , Q and R represent positive definite constant matrices. This indicates a predetermined level of uncertainty attenuation.

[0082] Assumption It is continuously differentiable, so we get:

[0083]

[0084] in, express right The gradient;

[0085] Hamiltonian of formula (22) for:

[0086]

[0087] According to Bellman's principle of optimality, the following holds:

[0088]

[0089] in, Describes the optimal value function. ;

[0090] ADP-based control law satisfy:

[0091]

[0092] get for:

[0093]

[0094] Similarly, satisfy:

[0095]

[0096] get for:

[0097]

[0098] Substituting equations (26) and (28) into equation (23), we obtain the HJI equation as follows:

[0099]

[0100] S420. Based on the above, design an ADP-based dynamic event-triggered control scheme to ensure the optimal performance of the unknown nonlinear system, including:

[0101] S421, Dynamic event triggering mechanism

[0102] Assuming at the triggering time Control Law Recorded as Until the next triggering time Update; The system state corresponding to time is represented as ;for Formula (20) can be written as:

[0103]

[0104] Triggering error for:

[0105]

[0106] The Hamiltonian of formula (30) is:

[0107]

[0108] At the trigger time Event-triggered control law based on ADP for:

[0109]

[0110] in, ;

[0111] According to formula (33), the Hamiltonian triggered by the event is:

[0112]

[0113] Assumption 5: Control input based on ADP It is a Lipschitz continuous function, and the Lipschitz constant is... satisfy:

[0114]

[0115] in, It relies on system parameter information reconstructed by RNN;

[0116] Define an internal dynamic variable It evolves according to the following differential equation:

[0117]

[0118] in, This indicates the internal dynamic variable adjustment parameter, and , This indicates the design parameters related to interference suppression;

[0119] Will Designed to be Related filtered values ​​should be avoided under the following conditions. Always non-negative:

[0120]

[0121] in, ;

[0122] Lemma 1: Using the event triggering rule (37), the dynamic variables defined by (36) Always non-negative;

[0123] Theorem 3: Optimal Value Function The equivalent sliding mode dynamics (20) are as follows: when (37) is true, the event is triggered and the control law is updated; at the same time, the trigger error is also triggered. Reset to zero, the equivalent sliding mode dynamics (20) are asymptotically stable;

[0124] In addition, the dynamic triggering rule, i.e. formula (27), avoids Zeno behavior;

[0125] Theorem 4: For the event triggering condition in formula (37), in order to avoid Zeno behavior, the minimum time interval is... for:

[0126]

[0127] In the formula, L f and L g It is the Lipschitz constant;

[0128] S422, Neural Network Approximation

[0129] Optimal value function Approximated by an existing critical neural network structure, there exists:

[0130]

[0131] in, and These represent the ideal weights and basis function vectors of the evaluation neural network, respectively. Indicates the approximation error;

[0132] definition approximate ,have:

[0133]

[0134] Based on (26) and (28), the event-triggered optimal control law Input Law Approximately:

[0135]

[0136] The Hamiltonian is approximated as:

[0137]

[0138] To minimize the following equation:

[0139]

[0140] in, This represents performing least squares on formula (42);

[0141] make,

[0142]

[0143] in, The gradient descent method is expressed as:

[0144]

[0145] in, The learning rate; This represents a portion of the weight estimation for a neural network;

[0146] Introducing parallel learning to obtain the update pattern of neural network weights: There exists a pattern defined as... The stored dataset, whose error set is defined as ,in To store the number of samples; at any given time ,exist satisfy:

[0147]

[0148] To minimize the following errors:

[0149]

[0150] in, This represents performing least squares on formula (45);

[0151] Based on formulas (31) and (33), the gradient descent method for parallel learning is proposed as follows:

[0152]

[0153] in, ,and This represents a portion of the weight estimation for a neural network. ;

[0154] To meet the requirements of initial admissible control, Lyapunov functions are introduced. ,have:

[0155]

[0156] To ensure system stability, there exists ,get:

[0157]

[0158] in, , Indicates the learning rate. This represents a portion of the weight estimation for the neural network; therefore, the improved gradient descent method is:

[0159]

[0160] Weight estimation error Defined as:

[0161]

[0162] get:

[0163]

[0164] in, , ;

[0165] Assumption 6: Assumption , , All are known positive numbers; and they exist. ,in It is a known constant;

[0166] Assumption 7: Assumption and It is Lipschitz continuous and exists. , ,in and All are normal numbers. Represents the Lipsitz constant;

[0167] Theorem 5: For a system (30) with a weight update law (50), if the following conditions are satisfied:

[0168]

[0169] in, , , , It is a positive definite matrix, B0 is an auxiliary constant, B is a constant, B1 is a scalar coefficient composed of the upper bound of the interference channel, the upper bound of the gradient of the neural network basis function, and the anti-interference parameter; B2 is a combination coefficient generated by the approximation error of the neural network. It is a positive definite matrix. , This is the normalized regression vector corresponding to the data at the current moment; The first in historical stored data The normalized regression vector of each sample;

[0170] The system is asymptotically stable, and It's UUB.

[0171] The system is a dynamic event-triggered control system for unknown nonlinear systems based on ADP and ISM control. The system has program modules corresponding to the above steps and executes the steps in the dynamic event-triggered control method for unknown nonlinear systems based on ADP and ISM control during runtime.

[0172] A computer-readable storage medium storing a computer program configured to, when invoked by a processor, implement steps of a dynamic event-triggered control method for an unknown nonlinear system based on ADP and ISM control.

[0173] Compared with the prior art, the beneficial effects of the present invention are:

[0174] This invention addresses continuously-time nonlinear systems with unknown dynamics by employing recurrent neural networks (RNNs) to reconstruct the system dynamics. Considering the vulnerability of the network layer to spoofed data injection attacks, a hybrid event-triggered control scheme with two distinct triggering components is designed based on the reconstructed system dynamics, utilizing integral sliding mode control (ISM) and adaptive dynamic programming (ADP) methods. This includes designing an event-triggered sliding mode control component that ensures the system state lies on the sliding surface. This controller can eliminate the impact of spoofed data injection attacks and mismatch uncertainties on the system. For equivalent sliding mode dynamics, this invention designs an optimal event-triggered control component based on adaptive dynamic programming (ADP), utilizing dynamic event triggering to save communication resources and extend actuator lifespan. Attached Figure Description

[0175] Figure 1 System state in an embodiment of the present invention and The trajectory diagram;

[0176] Figure 2 System state in an embodiment of the present invention and The trajectory diagram;

[0177] Figure 3 This is a diagram showing the trajectory of neural network weight changes in an embodiment of the present invention.

[0178] Figure 4 This is a trajectory diagram of an event-triggered controller based on ISM in an embodiment of the present invention;

[0179] Figure 5 This is a trajectory diagram of the event-triggered controller based on ADP in an embodiment of the present invention;

[0180] Figure 6 This is a diagram showing the event triggering time interval in an embodiment of the present invention;

[0181] Figure 7 This is a sliding surface trajectory diagram in an embodiment of the present invention;

[0182] Figure 8 This is a dynamic event triggering dynamic variable change trajectory diagram in an embodiment of the present invention. Detailed Implementation

[0183] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0184] Specific Implementation Scheme 1: This invention provides a dynamic event-triggered control method for unknown nonlinear systems based on ADP and ISM control, comprising the following steps:

[0185] S100. Consider the following continuous-time input affine nonlinear system:

[0186]

[0187] in, Indicates the system status. Indicates control input, and It is an unknown dynamic function, and m is the control input order, and n is the system state order.

[0188] Assumption In a compact set containing the origin The above is a continuation of Lipschitz; making Assuming the initial state of the system, let the solution of system (1) be... For each and They are all unique;

[0189] Because the controller-to-actuator channel is vulnerable to spoofed data injection attacks, which can exploit network vulnerabilities to inject false data into the control input signal, leading to system instability; the attacked control input signal It can be given by the following formula:

[0190]

[0191] in, This indicates that the controller-to-actuator channel is vulnerable to spoofed data injection attacks. ,in These are known constants;

[0192] Assumption For a column to have full rank and for any Bounded, system (1) during injection The system is controllable; therefore, an uncertain nonlinear system under a fake data injection attack can be written as:

[0193]

[0194] in, This indicates uncertainty in an unknown system. ,in These are known constants;

[0195] S200, RNN-based dynamic system reconfiguration

[0196] Consider the following system formula (1), which can be represented by an RNN:

[0197]

[0198] in, The weighted matrix representing the system state. and They represent and The unknown ideal weight matrix, , This represents the activation function vector, which is the sigmoidal function. This represents the RNN reconstruction estimation error;

[0199] System (1) can be estimated as:

[0200]

[0201] in, This represents the state of an identifier recurrent neural network (RNN). , and These represent the effects of using recurrent neural networks (RNNs) on... and The estimated weight matrix, The feedback item is defined as follows: ,in Indicate design parameters and Indicates recognition error;

[0202] Using formulas (4) and (5), the following dynamic identification error is obtained. :

[0203]

[0204] in, , , , and ;

[0205] Assumption 1: The ideal RNN weighting matrix A, and satisfy:

[0206]

[0207] in, It is a known positive definite constant matrix;

[0208] Assumption 2: There exists a constant greater than 0. , making ;

[0209] If the estimated RNN weight matrix given in formula (5) passes the design matrix:

[0210]

[0211] in, It is a design matrix;

[0212] Therefore, the RNN recognizer formula (5) can maintain the error. Asymptotic stability; Using formula (8), we can obtain the result when... hour, , , All are 0, that is All are convergent; therefore, the unknown nonlinear system (1) can be reconstructed as:

[0213]

[0214] in, and , and They are and The reconstructed unknown dynamic function;

[0215] S300, event-triggered ISM control design,

[0216] For the system, i.e., formula (2), design an integral sliding mode controller based on ADP, which is considered as follows:

[0217]

[0218] in, It is an integral sliding mode compensator. It is an optimal controller based on ADP;

[0219] First, the design scheme of the integral sliding mode compensator is given:

[0220] Consider the integral sliding mode function S(x) as follows:

[0221]

[0222] in, This represents the initial value of the integral sliding mode function. , and It can be obtained from formula (8);

[0223] Assumption 3: For any There exists a satisfying Full column rank function ; yes The left pseudo-reversal, (x) is The right pseudo-reversal, ; and It is a known positive constant;

[0224] Assumption 4: The boundary is ,and , It is reversible. It is a known positive constant;

[0225] Differentiating formula (11) yields:

[0226]

[0227] According to hypothesis 4, Then the equivalent control input of the uncertain nonlinear system (3) is... for:

[0228]

[0229] The equivalent sliding mode dynamics expression is:

[0230]

[0231] in, ;

[0232] To reduce the multiplication matrix The impact and minimization of mismatch interference ,choose As the optimal function, and can be obtained ;

[0233] By satisfying the stability of the slipper , It can be designed as:

[0234]

[0235] in, For sliding mode gain;

[0236] satisfy:

[0237]

[0238] For cases where ISM events are triggered, discontinuous control actions occur at specific times. Updated to ,in For the triggering time, and in two consecutive triggering times The inner remains unchanged, as follows:

[0239]

[0240] Triggering error Written as:

[0241]

[0242] If an error is triggered:

[0243]

[0244] in, , ;

[0245] When an event is triggered, the control input is updated, and the event trigger error is checked. Reset to zero;

[0246] Theorem 1: For an integral sliding mode compensator For the uncertain nonlinear system under equation (3), there exists a stable sliding mode from the initial time. ;

[0247] Theorem 2: Considering the event triggering condition, i.e., formula (18), to avoid Zeno's behavior, the event triggering interval condition... Designed as follows:

[0248]

[0249] S400, ADP control design based on dynamic event triggering.

[0250] Based on assumption 4, define The equivalent sliding mode dynamic representation is:

[0251]

[0252] Consider value functions for:

[0253]

[0254] in, , Q and R represent positive definite constant matrices. This indicates a predetermined level of uncertainty attenuation. Represents the integral over t;

[0255] Assumption It is continuously differentiable, so we can obtain:

[0256]

[0257] in, express right The gradient;

[0258] Furthermore, the Hamiltonian of formula (22) for:

[0259]

[0260] According to Bellman's principle of optimality, the following holds:

[0261]

[0262] in, Describes the optimal value function. ;

[0263] ADP-based control law satisfy:

[0264]

[0265] Therefore, we can obtain for:

[0266]

[0267] Similarly, satisfy:

[0268]

[0269] Therefore, we can obtain for:

[0270]

[0271] Substituting formulas (26) and (28) into formula (23), we obtain HJI (Hamilton-Jacobi-Isaacs). equation) The equation is as follows:

[0272]

[0273] Next, based on the existing research, we will design a dynamic event-triggered control scheme based on ADP to ensure the optimal performance of the unknown nonlinear system:

[0274] (1) Dynamic event triggering mechanism

[0275] Assuming at the triggering time Control Law Recorded as Until the next triggering time Update; The system state corresponding to time is represented as ;for Formula (20) can be written as:

[0276]

[0277] Triggering error for:

[0278]

[0279] The Hamiltonian of formula (30) can be expressed as:

[0280]

[0281] At the trigger time Event-triggered control law based on ADP It can be represented as:

[0282]

[0283] in, ;

[0284] According to formula (33), the Hamiltonian for event triggering can be obtained as:

[0285]

[0286] Assumption 5: Control input based on ADP It is a continuous Lipschitz function, and the Lipschitz constant is... satisfy:

[0287]

[0288] in, It relies on system parameter information reconstructed by RNN;

[0289] Define an internal dynamic variable It evolves according to the following differential equation:

[0290]

[0291] in, This indicates the internal dynamic variable adjustment parameter, and , This indicates the design parameters related to interference suppression;

[0292] Will Designed to be The relevant filter values, if the following conditions are used, can be avoided by this new dynamic ET technique. Always non-negative:

[0293]

[0294] in, ;

[0295] Lemma 1: Using the event triggering rule (37), the dynamic variables defined by (36) Always non-negative;

[0296] Theorem 3: Optimal Value Function The equivalent sliding mode dynamics (20) are given. When formula (37) holds, the event is triggered and the control law is updated; at the same time, the triggering error... Reset to zero, the equivalent sliding mode dynamics (20) are asymptotically stable;

[0297] Next, we will prove that the dynamic triggering rule, i.e., formula (27), avoids Zeno's behavior;

[0298] Theorem 4: For the event triggering condition in formula (37), in order to avoid Zeno behavior, the minimum time interval is... for:

[0299]

[0300] In the formula, L f and L g It is the Lipschitz constant;

[0301] (2) Neural Network Approximation

[0302] Optimal value function Approximated by an existing critical neural network structure, there exists:

[0303]

[0304] in, and These represent the ideal weights and basis function vectors of the evaluation neural network, respectively. Indicates the approximation error;

[0305] definition approximate ,have:

[0306]

[0307] Based on (26) and (28), the event-triggered optimal control law and input law are approximately as follows:

[0308]

[0309] The Hamiltonian is approximated as:

[0310]

[0311] To minimize the following equation:

[0312]

[0313] in, This represents performing least squares on formula (42);

[0314] make,

[0315]

[0316] in, The gradient descent method is expressed as:

[0317]

[0318] in, The learning rate; This represents a portion of the weight estimation for a neural network;

[0319] Introducing parallel learning to obtain the update pattern of neural network weights: There exists a pattern defined as... The stored dataset, whose error set is defined as ,in To store the number of samples; at any given time ,exist satisfy:

[0320]

[0321] To minimize the following errors:

[0322]

[0323] in, This represents performing least squares on formula (45);

[0324] Based on formulas (31) and (33), the gradient descent method for parallel learning is proposed as follows:

[0325]

[0326] in, ,and This represents a portion of the weight estimation for a neural network. ;

[0327] To meet the requirements of initial admissible control, the Lyapunov function is introduced. ,have:

[0328]

[0329] To ensure system stability, there exists ,get:

[0330]

[0331] in, , Indicates the learning rate. This represents a portion of the weight estimation for the neural network; therefore, the improved gradient descent method:

[0332]

[0333] Weight estimation error Defined as:

[0334]

[0335] We can obtain:

[0336]

[0337] in, , ;

[0338] Assumption 6: Assumption , , All are known positive numbers; and they exist. ,in It is a known constant;

[0339] Assumption 7: Assumption and It is Lipschitz continuous and exists. , ,in and All are normal numbers. Represents the Lipsitz constant;

[0340] Theorem 5: For a system (30) with a weight update law (50), if the following conditions are satisfied:

[0341]

[0342] in, , , B0 is an auxiliary constant, B is a constant, B1 is a scalar coefficient composed of the upper bound of the interference channel, the upper bound of the gradient of the neural network basis function, and the anti-interference parameter; B2 is a combination coefficient generated by the approximation error of the neural network. It is a positive definite matrix. , This is the normalized regression vector corresponding to the data at the current moment; The first in historical stored data The normalized regression vector of each sample; The system is asymptotically stable, and It is UUB (Ultimately Uniformly Bounded);

[0343] Specific Implementation Scheme 2: The present invention provides a dynamic event triggering control system for an unknown nonlinear system based on ADP and ISM control. The system has a program module corresponding to the above steps, and executes the steps in the above-mentioned dynamic event triggering control method for an unknown nonlinear system based on ADP and ISM control when running.

[0344] The other combinations and connections in this implementation scheme are the same as in Specific Implementation Scheme 1.

[0345] Specific Implementation Scheme 3: The present invention provides a computer-readable storage medium storing a computer program configured to implement, when called by a processor, the steps of a dynamic event-triggered control method for an unknown nonlinear system based on ADP and ISM control.

[0346] The other combinations and connections in this implementation scheme are the same as in Specific Implementation Scheme 1.

[0347] Example

[0348] To illustrate the effectiveness of the proposed method, the following example is used. A nonlinear single-link robotic arm has the following dynamic characteristics:

[0349]

[0350] in It refers to the angular position of the robotic arm. It is the mass of the payload. It is the moment of inertia. It is gravitational acceleration. It is the length of the arm. It is viscous friction.

[0351] make Then we have:

[0352]

[0353] in, This refers to the system state. Assume the initial system state is... , , , , . For an ISM-based event-triggered controller, It is an ADP-based dynamic event-triggered controller.

[0354] Assuming a fake data injection attack occurs in 5 seconds... The system uncertainty is .

[0355] For the integral sliding mode function, assume The design gain K is 2. In the simulation, the function is used. to replace ,in It is a constant, let it be equal to 0.009. For the equivalent kinetic assumption... , , .design , , Design the initial weight matrix. Activation function .

[0356] System state trajectory as follows Figure 1 and Figure 2 As shown in the figure, the blue line represents the actual system state, and the red dashed line represents the system state reconstructed by the RNN. It can be seen from the figure that the RNN recognizer can ensure asymptotic recognition of the unknown affine nonlinear system (1). Furthermore, based on the information obtained from the recognition, the designed control scheme eliminates the impact of attacks on the system.

[0357] Figure 3This represents the trajectory of weight changes in a neural network. Assume that at the initial moment... ,from Figure 3 We can see the weight update law Finally converges to . Figure 4 For ISM-based control inputs under event-triggered conditions, Figure 5 The diagram shows the ADP-based control input under event-triggered conditions. As can be seen from the figure, the control signal only changes at the trigger moment. Figure 6 This is the time interval for the event to be triggered. Figure 7 The figure shows the motion trajectory of the sliding surface. The sliding surface eventually converges to 0. Figure 8 This refers to the trajectory of dynamic variables in the dynamic event triggering conditions.

[0358] In summary, the control scheme designed in this invention can effectively identify unknown dynamics while enabling the system to achieve optimal control performance. Even if there are spurious data injection attack signals, the system can still compensate for the impact of the attack signals and achieve reliable control. The event-triggered control scheme can save communication resources, reduce the actuator update frequency, and extend the actuator's service life.

[0359] While the present invention has been disclosed above, its scope of protection is not limited thereto. Those skilled in the art can make various changes and modifications without departing from the spirit and scope of the present invention, and all such changes and modifications will fall within the scope of protection of the present invention.

Claims

1. A dynamic event-triggered control method for an unknown nonlinear system based on ADP and ISM control, characterized in that, Includes the following steps: S100, Constructing an uncertain nonlinear system under a fake data injection attack; S200: Dynamic reconstruction of unknown nonlinear systems based on recurrent neural networks (RNNs); S300. Design an event-triggered integral sliding mode control (ISMC) based on the information reconstructed from a recurrent neural network (RNN). S400. Based on the information reconstructed by the recurrent neural network (RNN), design an adaptive dynamic programming (ADP) control scheme based on dynamic event triggering, including a dynamic event triggering mechanism and neural network approximation, to ensure that the system is asymptotically stable.

2. The dynamic event-triggered control method for unknown nonlinear systems based on ADP and ISM control according to claim 1, characterized in that, In step S100, the following are included: Consider the following continuous-time input affine nonlinear system: in, Indicates the system status. Indicates control input, and It is an unknown dynamic function, and m is the control input order, and n is the system state order. Assumption In a compact set containing the origin The above is a continuation of Lipschitz; making Assuming the initial state of the system, let the solution of system (1) be... For each and They are all unique; Attacked control input signals It is given by the following formula: in, This indicates that the channel from the controller to the actuator is vulnerable to spoofed data injection attacks. ,in These are known constants; Assumption For a column to have full rank and for any Bounded, system (1) during injection The system is controllable; therefore, the uncertain nonlinear system under a false data injection attack is: in, This indicates uncertainty in an unknown system. ,in It is a known constant.

3. The dynamic event-triggered control method for unknown nonlinear systems based on ADP and ISM control according to claim 2, characterized in that, Step S200 includes, Consider the following system formula (1) expressed by a recurrent neural network (RNN): in, The weighted matrix representing the system state. and They represent and The unknown ideal weight matrix, , This represents the activation function vector, which is the sigmoidal function. This represents the RNN reconstruction estimation error; System (1) is estimated as: in, This represents the state of an identifier recurrent neural network (RNN). , and These represent the effects of using recurrent neural networks (RNNs) on... and The estimated weight matrix, The feedback item is defined as follows: ,in Indicate design parameters and Indicates recognition error; Using formulas (4) and (5), the following dynamic identification error is obtained. : in, , , , and ; Assumption 1: The ideal RNN weighting matrix A, and satisfy: in, It is a known positive definite constant matrix; Assumption 2: There exists a constant greater than 0. , making ; If the estimated RNN weight matrix given in formula (5) passes the design matrix: in, It is a design matrix; Then the RNN recognizer formula (5) maintains the error. Asymptotic stability; By using formula (8), we can obtain the result when... hour, , , All are 0, that is All are convergent; therefore, the unknown nonlinear system (1) is reconstructed as: in, and , and They are and The reconstructed unknown dynamic function.

4. The dynamic event-triggered control method for unknown nonlinear systems based on ADP and ISM control according to claim 3, characterized in that, Step S300 includes, For the system, i.e., formula (2), design an ADP-based integral sliding mode controller: in, It is an integral sliding mode compensator. It is an optimal controller based on ADP; First, the design scheme of the integral sliding mode compensator is given: Consider the integral sliding mode function S(x) as follows: in, This represents the initial value of the integral sliding mode function. , and It is obtained from formula (8); Assumption 3: For any There exists a satisfying Full column rank function ; yes The left pseudo-reversal, (x) is The right pseudo-reversal, ; and It is a known positive constant; Assumption 4: The boundary is ,and , It is reversible. It is a known positive constant; Differentiating formula (11) yields: According to hypothesis 4, Then the equivalent control input of the uncertain nonlinear system (3) is... for: The equivalent sliding mode dynamics expression is: in, ; To reduce the multiplication matrix The impact and minimization of mismatch interference ,choose As the optimal function, and can be obtained ; By satisfying stable sliding mode , Designed as: in, For sliding mode gain; satisfy: For cases where ISM events are triggered, discontinuous control actions occur at specific times. Updated to ,in For the triggering time, and in two consecutive triggering times The internal remains unchanged: Triggering error Written as: If an error is triggered: in, , ; When an event is triggered, the control input is updated, and the event trigger error is checked. Reset to zero; Theorem 1: For an integral sliding mode compensator For the uncertain nonlinear system under equation (3), there exists a stable sliding mode from the initial time. ; Theorem 2: Considering the event triggering condition, i.e., formula (18), to avoid Zeno's behavior, the event triggering interval condition... Designed as follows:

5. The dynamic event-triggered control method for unknown nonlinear systems based on ADP and ISM control according to claim 4, characterized in that, In step S400, the following are included: S410. Based on assumption 4, define... The equivalent sliding mode dynamic representation is: Value function for: in, , Q and R represent positive definite constant matrices. This indicates a predetermined level of uncertainty attenuation. Assumption It is continuously differentiable, so we get: in, express right The gradient; Hamiltonian of formula (22) for: According to Bellman's principle of optimality, the following holds: in, Describes the optimal value function. ; ADP-based control law satisfy: get for: Similarly, satisfy: get for: Substituting equations (26) and (28) into equation (23), we obtain the HJI equation as follows: S420. Based on the above, design an ADP-based dynamic event-triggered control scheme to ensure the optimal performance of the unknown nonlinear system, including: S421, Dynamic event triggering mechanism Assuming at the triggering time Control Law Recorded as Until the next triggering time Update; The system state corresponding to time is represented as ;for Formula (20) can be written as: Triggering error for: The Hamiltonian of formula (30) is: At the trigger time Event-triggered control law based on ADP for: in, ; According to formula (33), the Hamiltonian triggered by the event is: Assumption 5: Control input based on ADP It is a Lipschitz continuous function, and the Lipschitz constant is... satisfy: in, It relies on system parameter information reconstructed by RNN; Define an internal dynamic variable It evolves according to the following differential equation: in, This indicates the internal dynamic variable adjustment parameter, and , This indicates the design parameters related to interference suppression; Will Designed to be Related filtered values ​​should be avoided under the following conditions. Always non-negative: in, ; Lemma 1: Using the event triggering rule (37), the dynamic variables defined by (36) Always non-negative; Theorem 3: Optimal Value Function The equivalent sliding mode dynamics (20) are as follows: when (37) is true, the event is triggered and the control law is updated; at the same time, the trigger error is also triggered. Reset to zero, the equivalent sliding mode dynamics (20) are asymptotically stable; In addition, the dynamic triggering rule, i.e. formula (27), avoids Zeno behavior; Theorem 4: For the event triggering condition in formula (37), in order to avoid Zeno behavior, the minimum time interval is... for: In the formula, L f and L g It is the Lipschitz constant; S422, Neural Network Approximation Optimal value function Approximated by an existing critical neural network structure, there exists: in, and These represent the ideal weights and basis function vectors of the evaluation neural network, respectively. Indicates the approximation error; definition approximate ,have: Based on (26) and (28), the event-triggered optimal control law Input Law Approximately: The Hamiltonian is approximated as: To minimize the following equation: in, This represents performing least squares on formula (42); make, in, The gradient descent method is expressed as: in, The learning rate; This represents a portion of the weight estimation for a neural network; Introducing parallel learning to obtain the update pattern of neural network weights: There exists a pattern defined as... The stored dataset, whose error set is defined as ,in To store the number of samples; at any given time ,exist satisfy: To minimize the following errors: in, This represents performing least squares on formula (45); Based on formulas (31) and (33), the gradient descent method for parallel learning is proposed as follows: in, ,and This represents a portion of the weight estimation for a neural network. ; To meet the requirements of initial admissible control, Lyapunov functions are introduced. ,have: To ensure system stability, there exists ,get: in, , Indicates the learning rate. This represents a portion of the weight estimation for the neural network; therefore, the improved gradient descent method is: Weight estimation error Defined as: get: in, , ; Assumption 6: Assumption , , All are known positive numbers; and they exist. ,in It is a known constant; Assumption 7: Assumption and It is Lipschitz continuous and exists. , ,in and All are normal numbers. Represents the Lipsitz constant; Theorem 5: For a system (30) with a weight update law (50), if the following conditions are satisfied: in, , , , It is a positive definite matrix, B0 is an auxiliary constant, B is a constant, B1 is a scalar coefficient composed of the upper bound of the interference channel, the upper bound of the gradient of the neural network basis function, and the anti-interference parameter; B2 is a combination coefficient generated by the approximation error of the neural network. It is a positive definite matrix. , This is the normalized regression vector corresponding to the data at the current moment; The first in historical stored data The normalized regression vector of each sample; The system is asymptotically stable, and It's UUB.

6. A dynamic event-triggered control system for an unknown nonlinear system based on ADP and ISM control, characterized in that: The system has a program module corresponding to the steps of any one of the claims 1-5 above, and executes the steps in the above-described dynamic event triggering control method for unknown nonlinear systems based on ADP and ISM control when it runs.

7. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores a computer program configured to, when invoked by a processor, implement the steps of the dynamic event-triggered control method for an unknown nonlinear system based on ADP and ISM control as described in any one of claims 1-6.

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