Aperiodic sampling deep neural network controller based on closed-loop stability guarantee

Abstract

The invention relates to the field of deep neural network control, in particular to an aperiodic sampling deep neural network controller based on closed-loop stability guarantee. The controller design comprises the following steps: step 1, firstly, dividing a discrete system into a known nominal part and an unknown uncertainty part, and constructing a virtual filter based on integral quadratic constraint to obtain an augmented system model; 2, deploying a dynamic event triggering strategy between the sampler and the DNN controller; 3, on the basis of the step 2, performing convex relaxation on the deep neural network controller, converting nonlinear mapping into linear matrix inequality constraint, and establishing a convex optimization model through a slack variable; and 4, carrying out stability analysis on the closed-loop system. Constructing a linear matrix inequality criterion and solving a corresponding convex optimization problem to obtain a closed-loop stability guarantee, and taking ellipsoid approximation of an attraction domain as stability measurement; and step 5, non-periodic sampling deep neural network control applied to the inverted pendulum.

Description

Technical Field

[0001] This invention relates to the field of deep neural network control, and more specifically to an aperiodic sampling deep neural network controller based on closed-loop stability guarantee. Background Technology

[0002] The rapid development of imitation learning has provided new ideas for the control design of cyber-physical systems (CPS). As a data-driven approach, imitation learning can directly acquire control strategies through expert demonstrations, thereby reducing the dependence on accurate system models and demonstrating advantages in complex dynamic control tasks. Deep neural networks (DNNs), with their powerful nonlinear approximation capabilities, have become the core tool of the imitation learning framework. They can characterize complex dynamic relationships in high-dimensional state spaces and replace computationally expensive methods such as model predictive control (MPC) with offline training and online inference, thus significantly improving real-time performance and scalability. At the same time, DNNs exhibit strong adaptability in dealing with uncertainties and external disturbances. Compared to the conservative design of traditional robust control, they can better extract environmental patterns through data-driven methods, reducing over-conservatism while ensuring system safety. This is particularly crucial for CPS systems with time-varying parameters, strong uncertainties, and complex interactive characteristics.

[0003] However, due to the highly nonlinear and large-scale parameterization characteristics of deep neural networks, their training process often lacks interpretability and verifiability. Therefore, providing rigorous theoretical guarantees for the learned control behavior remains a significant challenge. Especially in scenarios involving closed-loop control, the neural network and system dynamics form a complex nonlinear feedback loop, making it difficult to analyze its stability and convergence from the perspective of traditional control theory. This uncertainty not only increases the difficulty of design verification but also significantly limits the application of DNN-based controllers in practical engineering systems with extremely high requirements for safety and robustness.

[0004] On the other hand, the recent emergence of Event Triggered Control (ETC) has provided new insights into the application of neural network control in practical systems. Unlike traditional periodic sampling strategies, ETC monitors changes in system state and updates control signals only when necessary, thus achieving non-periodic sampling. This mechanism not only significantly reduces communication and computational burdens and avoids wasting resources on invalid sampling, but also improves overall energy efficiency and real-time performance while ensuring control performance and system stability. For control frameworks relying on deep neural networks, the event triggering strategy can effectively alleviate the burden of network inference computation and improve the implementability and robustness of the controller. Therefore, combining event triggering mechanisms with imitation learning and DNN control has become a key development direction in the current field of intelligent control that balances performance and resource efficiency.

[0005] Against this backdrop, based on the above analysis, it is essential to study a non-periodic sampling deep neural network controller based on closed-loop stability guarantees. Summary of the Invention

[0006] The purpose of this invention is to provide a non-periodic sampling deep neural network controller based on closed-loop stability guarantee, which can solve the problems of existing deep neural network control methods lacking stability theoretical guarantees in closed-loop systems, having large computational and communication burdens, and being unable to balance performance and resource efficiency in the presence of uncertainties and external disturbances.

[0007] The objective of this invention is achieved through the following technical solution:

[0008] A non-periodic sampling deep neural network controller based on closed-loop stability guarantee is designed by the following steps:

[0009] Step 1: First, establish a discrete-time system model, divide the system into a known nominal part and an unknown uncertainty part, and construct a virtual filter based on integral quadratic constraint (IQC) to obtain an augmented system model;

[0010] Step 2: Deploy a dynamic event triggering strategy between the sampler and the DNN controller to reduce computational and communication burden;

[0011] Step 3: Based on Step 2, perform convex relaxation on the deep neural network controller. Utilize the sector-bounded property of the activation function to transform the nonlinear mapping into linear matrix inequality constraints, and establish an overall convex optimization model through relaxation variables;

[0012] Step 4: Based on IQC, Lyapunov functions, and ring transformations, perform stability analysis on the closed-loop system. By constructing a linear matrix inequality criterion and solving the corresponding convex optimization problem, while considering system uncertainties and aperiodic sampling characteristics, the closed-loop stability guarantee is obtained. The ellipsoidal approximation of the attraction domain is used as a stability measure to optimize the triggering parameters.

[0013] Step 5: Applying aperiodic sampling deep neural network control to the inverted pendulum;

[0014] In step one, the state-space equation of the discrete system is:

[0015]

[0016] (1)

[0017] in, The sampling interval is... , , , These represent the system state, control input, and uncertainty, respectively. Input and uncertainty The output of the nominal system. Feedback with bounded random operators interconnected:

[0018]

[0019] Assumption and The interconnection is well-defined, meaning that for each input... Each has a unique response. , , This ensures the control system There exists a unique solution, and... and There are no algebraic loops in the interconnection; furthermore, all state variables... All of these can be controlled through feedback.

[0020] Characterizing uncertain operators The key to understanding the input-output relationship and combining it with IQC lies in constructing a virtual filter. The state of the filter With input uncertainty and uncertain operators Output Related, its output This results in a quadratic constraint. (Filter) It can be expressed in the following form:

[0021] (2)

[0022] in, and satisfy ,and .matrix It is a Schur matrix whose spectrum is contained within the unit open disk in the complex plane, which ensures that (2) for any There is a unique solution for each. The extended state is denoted as ,in ,and Combining equations (1) and (2), we can obtain:

[0023] (3)

[0024] in ,and

[0025]

[0026] The stable equilibrium point of the extended system (3) is denoted as .make Indicating a neural feedback loop The state response, whose initial state is Generally, it's impossible for a globally stable deep neural network (DNN) controller to exist; therefore, local stability guarantees within a specific data region are sufficient. Thus, we assume... It can converge locally to a stable equilibrium point. .

[0027] In step two, control input It can be achieved through a having This is obtained using a layered feedforward deep neural network (DNN), denoted as... Its form is:

[0028] (4)

[0029] in Indicates the first Layer activation, and . No. The perception of the layer is determined by the weight matrix. and bias Obtain the activation function. It acts element-wise, that is:

[0030]

[0031] in Represents a satisfying The scalar nonlinear function can be of the form ReLU, sigmoid, or softmax. Typically, it is assumed that all layers... They are the same, but this does not affect their generality.

[0032] Consider the system state at a specific moment Sampled and transmitted to the controller, where ,and For all Established. Available sampling states. Transmitted to and generate control input.

[0033]

[0034] Zero-order hold (ZOH) is used in this process. To improve communication efficiency, we will design an event-triggered (ETC) scheme for the neural feedback loop (5) while ensuring closed-loop stability, where the control input Represented as:

[0035] (5)

[0036] Therefore, the non-periodic sampling DNN controller (5) depends on the DNN architecture (4) and the adaptive triggering time. .

[0037] For ETC, the state trajectory occurs at discrete times. Periodic sampling, in which and constant sampling interval ,in and These represent the lower and upper bounds, respectively. Next, we will base our analysis on the sampled states. To evaluate the following triggering rules:

[0038] (6)

[0039] in These are parameters to be determined. This condition applies to Established, among which .function exist A quadratic function is defined nearby, with the following form:

[0040]

[0041] in It is the parameter that triggers the discount. It is a weight matrix. State trajectory. The corresponding state between the current time and the most recent transmission time. The error between them is defined as ,also, A dynamic parameter was constructed that satisfies the following at discrete time intervals. and

[0042] (7)

[0043] in , These are preset coefficients.

[0044] Once the event triggering logic (6) is violated, the sampling state is... This information can then be transmitted to the DNN controller to obtain new DNN control inputs, and held at zero order (ZOH) in the subsequent trigger interval. The state is maintained internally. The state from the most recent transmission is used to update the control input of the event-triggered DNN and further determine the next sampling state. Therefore, the dynamic event-triggered logic can be expressed as:

[0045]

[0046] In step three, the pre-activation of the DNN is defined as , making Therefore, by aggregating the inputs and outputs of the activation layer, we define... ,in They can be represented as and ,condition The condition holds true element-wise for repeated nonlinear activation functions. Based on (4) and combined with the principle of superposition of linear operators, the nonlinearity can be isolated, thus obtaining:

[0047] (8)

[0048] in

[0049]

[0050] in Activation functions typically satisfy the properties of sector boundedness and / or slope constraint, which can potentially be used to relax the function through quadratic constraints based on (8). .

[0051] In step four, for a series of moments If the function

[0052] The following conditions must be met: If it is called a looped function, then it is called a looped function.

[0053] For a given scalar , and satisfy of Under the event-triggered communication scheme (7), the neural feedback loop (3) asymptotically converges to a flat surface. ,in from It converges to the origin. If a matrix exists... , , , , and vectors (Its elements satisfy) As a decision variable, the following condition must be met:

[0054] (9)

[0055] for , , , ,and Established, among which Furthermore, the other items in (9) are given below. Represents the weight matrix The Okay. So, an ellipsoid ,in Indicates having appropriate dimensions The top left block, when When, it means that it is an inner approximation of the attraction domain (RoA).

[0056]

[0057]

[0058] The feasible solution of (9) characterizes an interior approximation of the robust attraction domain in (6), because the set Volume and matrix The determinant is directly proportional to the matrix, denoted as . To compute the maximum inner approximation of the robust attraction domain, we can formulate the following optimization problem:

[0059]

[0060] All decision variables defined above are convex.

[0061] In step five, the obtained controller based on aperiodic sampling deep neural network was applied to an inverted pendulum, verifying the feasibility of the controller.

[0062] The beneficial effects of this invention are as follows:

[0063] To address the problems of difficulty in guaranteeing closed-loop stability and high computational and communication resource consumption in deep neural network control in cyber-physical systems (CPS), this invention proposes an aperiodic sampling deep neural network control method based on closed-loop stability guarantee, which combines imitation learning with event-triggered control mechanism.

[0064] Compared to existing control methods, this invention introduces a closed-loop stability analysis framework into the controller design, ensuring the stability of the feedback loop after the neural network is dynamically coupled with the system. Through robust design, it effectively resists external disturbances and system uncertainties, achieving reliable control of complex CPS systems. Furthermore, unlike existing methods that require updating the control signal at every moment, this invention integrates an event-triggered control (ETC) strategy, performing data transmission and control updates only when preset conditions are met. This significantly reduces redundant computation and communication burdens, lowering the dependence on the real-time performance of the hardware platform. By leveraging the powerful nonlinear approximation capabilities of deep neural networks, combined with the efficient sampling mechanism of event triggering, it reduces excessive conservatism and resource waste while ensuring system performance and closed-loop stability, thereby improving system energy efficiency and real-time performance. Attached Figure Description

[0065] The present invention will now be described in further detail with reference to the accompanying drawings and specific implementation methods.

[0066] Figure 1 This is a schematic diagram of the non-periodic sampling deep neural network controller framework based on closed-loop stability guarantee of the present invention;

[0067] Figure 2 This is a time-varying graph showing the state response of the angular position, angular velocity, and IQC virtual filter event-triggered control and model predictive control of the present invention.

[0068] Figure 3 This is a graph showing the change of the event triggering interval over time according to the present invention;

[0069] Figure 4 This is an approximate diagram of the ellipse within the robust attraction domain of the present invention. Detailed Implementation

[0070] The present invention will now be described in further detail with reference to the accompanying drawings.

[0071] like Figures 1 to 4As shown, in order to solve the technical problem that "deep neural network control in cyber-physical systems (CPS) has difficulty in guaranteeing closed-loop stability and consumes a lot of computational and communication resources", the design steps and functions of an aperiodic sampling deep neural network controller based on closed-loop stability guarantee are described in detail below.

[0072] A non-periodic sampling deep neural network controller based on closed-loop stability guarantee is designed, and the controller design includes the following steps:

[0073] Step 1: First, establish a discrete-time system model, divide the system into a known nominal part and an unknown uncertainty part, and construct a virtual filter based on integral quadratic constraint (IQC) to obtain an augmented system model;

[0074] The state-space equations of the discrete system are:

[0075]

[0076] (1)

[0077] in, The sampling interval is... , , , These represent the system state, control input, and uncertainty, respectively. Input and uncertainty The output of the nominal system. Feedback with bounded random operators interconnected:

[0078]

[0079] Assumption and The interconnection is well-defined, meaning that for each input... Each has a unique response. , , This ensures the control system There exists a unique solution, and... and There are no algebraic loops in the interconnection; furthermore, all state variables... All of these can be controlled through feedback.

[0080] Characterizing uncertain operators The key to understanding the input-output relationship and combining it with IQC lies in constructing a virtual filter. The state of the filter With input uncertainty and uncertain operators Output Related, its output This results in a quadratic constraint. (Filter) It can be expressed in the following form:

[0081] (2)

[0082] in, and satisfy ,and .matrix It is a Schur matrix whose spectrum is contained within the unit open disk in the complex plane, which ensures that (2) for any There is a unique solution for each. The extended state is denoted as ,in ,and Combining equations (1) and (2), we can obtain:

[0083] (3)

[0084] in ,and

[0085]

[0086] The stable equilibrium point of the extended system (3) is denoted as .make Indicating a neural feedback loop The state response, whose initial state is Generally, it's impossible for a globally stable deep neural network (DNN) controller to exist; therefore, local stability guarantees within a specific data region are sufficient. Thus, we assume... It can converge locally to a stable equilibrium point. .

[0087] Step 2: Deploy a dynamic event triggering strategy between the sampler and the DNN controller to reduce computational and communication burden;

[0088] Control input It can be achieved through a having This is obtained using a layered feedforward deep neural network (DNN), denoted as... Its form is:

[0089] (4)

[0090] in Indicates the first Layer activation, and . No. The perception of the layer is determined by the weight matrix. and bias Obtain the activation function. It acts element-wise, that is:

[0091]

[0092] in Represents a satisfying The scalar nonlinear function can be of the form ReLU, sigmoid, or softmax. Typically, it is assumed that all layers... They are the same, but this does not affect their generality.

[0093] Consider the system state at a specific moment Sampled and transmitted to the controller, where ,and For all Established. Available sampling states. Transmitted to and generate control input.

[0094]

[0095] Zero-order hold (ZOH) is used in this process. To improve communication efficiency, we will design an event-triggered (ETC) scheme for the neural feedback loop (5) while ensuring closed-loop stability, where the control input Represented as:

[0096] (5)

[0097] Therefore, the non-periodic sampling DNN controller (5) depends on the DNN architecture (4) and the adaptive triggering time. .

[0098] For ETC, the state trajectory occurs at discrete times. Periodic sampling, in which and constant sampling interval ,in and These represent the lower and upper bounds, respectively. Next, we will base our analysis on the sampled states. To evaluate the following triggering rules:

[0099] (6)

[0100] in These are parameters to be determined. This condition applies to Established, among which .function exist A quadratic function is defined nearby, with the following form:

[0101]

[0102] in It is the parameter that triggers the discount. It is a weight matrix. State trajectory. The corresponding state between the current time and the most recent transmission time. The error between them is defined as ,also, A dynamic parameter was constructed that satisfies the following at discrete time intervals. and

[0103] (7)

[0104] in , These are preset coefficients.

[0105] Once the event triggering logic (6) is violated, the sampling state is... This information can then be transmitted to the DNN controller to obtain new DNN control inputs, and held at zero order (ZOH) in the subsequent trigger interval. The state is maintained internally. The state from the most recent transmission is used to update the control input of the event-triggered DNN and further determine the next sampling state. Therefore, the dynamic event-triggered logic can be expressed as:

[0106]

[0107] Step 3: Based on Step 2, perform convex relaxation on the deep neural network controller. Utilize the sector-bounded property of the activation function to transform the nonlinear mapping into linear matrix inequality constraints, and establish an overall convex optimization model through relaxation variables;

[0108] Define the pre-activation of a DNN as , making Therefore, by aggregating the inputs and outputs of the activation layer, we define... ,in They can be represented as and ,condition The condition holds true element-wise for repeated nonlinear activation functions. Based on (4) and combined with the principle of superposition of linear operators, the nonlinearity can be isolated, thus obtaining:

[0109] (8)

[0110] in

[0111]

[0112] in Activation functions typically satisfy the properties of sector boundedness and / or slope constraint, which can potentially be used to relax the function through quadratic constraints based on (8). .

[0113] Step 4: Based on IQC, Lyapunov functions, and ring transformations, perform stability analysis on the closed-loop system. By constructing a linear matrix inequality criterion and solving the corresponding convex optimization problem, closed-loop stability can be guaranteed while considering system uncertainties and aperiodic sampling characteristics. The ellipsoidal approximation of the attraction domain is used as a stability measure to optimize the triggering parameters.

[0114] For a series of moments If the function

[0115] The following conditions must be met: If it is called a looped function, then it is called a looped function.

[0116] For a given scalar , and satisfy of Under the event-triggered communication scheme (7), the neural feedback loop (3) asymptotically converges to a flat surface. ,in from It converges to the origin. If a matrix exists... , , , , and vectors (Its elements satisfy) As a decision variable, the following condition must be met:

[0117] (9)

[0118] for , , , ,and Established, among which Furthermore, the other items in (9) are given below. Represents the weight matrix The Okay. So, an ellipsoid ,in Indicates having appropriate dimensions The top left block, when When, it means that it is an inner approximation of the attraction domain (RoA).

[0119]

[0120]

[0121] The feasible solution of (9) characterizes an interior approximation of the robust attraction domain in (6), because the set Volume and matrix The determinant is directly proportional to the matrix, denoted as . To compute the maximum inner approximation of the robust attraction domain, we can formulate the following optimization problem:

[0122]

[0123] All decision variables defined above are convex.

[0124] Step 5: Applying aperiodic sampling deep neural network control to the inverted pendulum;

[0125] An inverted pendulum can be represented by the interconnection of its nominal and indeterminate parts:

[0126]

[0127] in, The sampling time interval, It is the acceleration due to gravity. The distance from the base to the center of mass of the equilibrium body. The coefficient of rotational friction is For quality. The system state is defined as follows: in and These represent angular position and angular velocity, respectively. Assume... and This holds true in practice. Nonlinear operators. This represents the uncertainty term in the inverted pendulum model, which locally satisfies the sector bounded condition. Then, they will undergo imitation learning training.

[0128] like Figure 2 As shown, the state trajectory Converging under the control of the controller, while different The trigger time and trigger interval are as follows: Figure 3 As shown. Case 1, Case 2, Case 3, and Case 4 correspond to... Figure 2 The feasibility of the controller has been verified. Figure 3Aperiodic sampling was achieved, reducing the communication and computational burden. The sampling interval was increased. This can lead to a degradation in the stability of neural feedback loops, so balancing communication efficiency with closed-loop performance is crucial for control design.

[0129] exist Figure 4 In this study, the ellipsoidal approximation of robust RoA was captured, with different bounds for the pre-activation of the first layer of the deep neural network (DNN). We stipulate in , All elements are 1. Based on IBP technology, we calculated the term and related For comparison, During simulation, the value can be chosen as 0.25, 0.35, or 0.45. We define a local upper bound. in The values ​​can be 0.97, 1.00, and 1.03. Figure 4 Reflecting different and The results under the combined conditions. Since the estimated volume of the ellipsoidal RoA varies, based on the simulation results, the optimal choice is... and This will yield the largest RoA estimate.

[0130] This neural network controller is applied to the control of the inverted pendulum, and is composed of... Figures 2 to 4 It can be seen that it has good control performance, can strictly guarantee the tracking stability of the set value, and the event triggering scheme can reduce communication costs and save communication resources, providing a theoretical and application basis for its subsequent application in complex industrial systems.

Claims

1. A non-periodic sampling deep neural network controller based on closed-loop stability guarantee, the controller design comprising the following steps: Step one: firstly, a discrete-time system model is established, the system is divided into a known nominal part and an unknown uncertainty part, and a virtual filter is constructed based on integral quadratic constraint (IQC) to obtain an augmented system model; Step two: a dynamic event-triggered strategy is deployed between the sampler and the DNN controller to reduce the computational and communication burden; Step three: on the basis of step two, the deep neural network controller is convex relaxed; the sector-bounded property of the activation function is used to convert the nonlinear mapping into a linear matrix inequality constraint, and an overall convex optimization model is established through a relaxation variable; Step four: based on IQC, Lyapunov function and loop transformation, the stability of the closed-loop system is analyzed; by constructing a linear matrix inequality criterion and solving the corresponding convex optimization problem, the closed-loop stability guarantee is obtained while considering the system uncertainty and non-periodic sampling characteristics, and the ellipsoid approximation of the attraction domain is used as the stability measure for optimizing the trigger parameters; Step five: application of the non-periodic sampling deep neural network control to an inverted pendulum.

2. The non-periodic sampling deep neural network controller based on closed-loop stability guarantee according to claim 1, wherein: In the step one, the state-space equation of the discrete system is: (1) where is the sampling interval, , , , denote the system state, the control input, the input of the uncertainty part and the output of the uncertainty part , respectively; the nominal system is interconnected in feedback with a bounded stochastic operator : Assume that the interconnection of is well posed, i.e. for each input there exists a unique response , , which guarantees that the control system has a unique solution and that there are no algebraic loops in the interconnection of and , furthermore, all state variables can be controlled by feedback; Characterizing input-output relations of uncertain operators The key to characterizing the input-output relations of uncertain operators and incorporating IQC is to construct a virtual filter whose states are related to the input uncertainty and the output of the uncertain operator in such a way that its output results in a quadratic constraint; the filter (2) where and satisfy and ; the matrix is a Schur matrix whose spectrum is contained in the open unit disc of the complex plane, which ensures that (2) has a unique solution for any ; let the extended state be denoted as where and , and (1), (2) can be obtained simultaneously: (3) wherein and The stable equilibrium points of the extended system (3) are denoted by ; let denote the state response of the neural feedback loop with initial state ; in general, there does not exist a globally stable DNN controller, so it is sufficient to guarantee local stability in a certain region of the data; thus, we assume that the system can locally converge to a stable equilibrium point .

3. The non-periodic sampling deep neural network controller based on closed-loop stability guarantee according to claim 2, wherein: Control input may be obtained by a feedforward deep neural network (DNN) with layers, denoted as in the form of: (4) wherein represents the activation of the layer, and ; the perception of the layer is obtained by a weight matrix and a bias ; the activation function is element-wise, i.e.: where represents a scalar nonlinear function satisfying ; its form can be chosen as ReLU, sigmoid, or softmax; in general, it is assumed that is the same across all layers without loss of generality; Consider the system state at a particular time is sampled and transmitted to the controller, where , and holds for all available sampled states are transmitted to and generate the control input: In this process, zero-order hold (ZOH) is adopted; in order to improve the communication efficiency, we will design an event-triggered (ETC) scheme for the neural feedback loop (5) under the premise of ensuring closed-loop stability, where the control input is represented as: (5) Thus, the aperiodic sampling DNN controller (5) relies on the DNN architecture (4) and the adaptive trigger instant ; For ETC, the state trajectory at discrete time instants Periodic sampling, where , and constant sampling interval , where and denote lower and upper bounds, respectively; next, the following trigger rule will be evaluated based on the sampled state : (6) wherein is a pending parameter, the condition holds, wherein the function defines a quadratic function in the vicinity of the form wherein is a trigger discount parameter, is a weight matrix; state trajectory the error between the state at the current time instant and the corresponding state at the last transmission time instant is defined as further, a dynamic parameter is constructed, which satisfies at discrete time instants and (7) wherein , is a preset coefficient; Once the event-triggering logic (6) is violated, the sampled state can be transmitted to the DNN controller to obtain a new DNN control input and hold through zero-order hold (ZOH) in the following triggering interval ; where the most recently transmitted state is used to update the event-triggering DNN control input and further determine the next sampled state; thus, the dynamic event-triggering logic can be expressed as: 。 4. The non-periodic sampling deep neural network controller based on closed-loop stability guarantee according to claim 3, wherein: In step three, the pre-activation of a DNN is defined as such that Thus, by aggregating the input and output of the activation layer, one defines where may be expressed as and provided that holds element-wise for the repeated non-linear activation function. Based on (4) and combined with the linear operator superposition principle, the nonlinearity can be isolated, and the following is obtained: (8) where wherein ; the activation functions typically satisfy sector-bounded and / or slope-limited properties that can potentially be used to relax the quadratic constraints based on (8) .

5. The non-periodic sampling deep neural network controller based on closed-loop stability guarantee according to claim 4, wherein: In step four, for a sequence of time instants if the function Let f(x) be a function of x, and let x0be a value of x. If the following conditions are met: , then f(x) is called a looped function. for a given scalar , and satisfying , , under the event-triggered communication scheme (7), the neural feedback loop (3) asymptotically converges to a , from to the origin; if there exists a matrix , , , , and a vector whose elements satisfy as decision variables, such that the following conditions hold: (9) For , , , , and hold, where , and the other terms in (9) are given below; furthermore, denotes the th row of the weight matrix ; then, an ellipsoid , where denotes the top-left block of with appropriate dimensions, means an inner approximation to the region of attraction (RoA) when . The feasible solutions of (9) characterize an inner approximation of the robust attractor region in (6) because the volume of the set is proportional to the determinant of the matrix , denoted by ; to compute the largest inner approximation of the robust attractor region, one can set up the following optimization problem: All the decision variables defined in the above are convex.

6. The non-periodic sampling deep neural network controller based on closed-loop stability guarantee according to claim 5, wherein: In the step five, the obtained non-periodic sampling deep neural network controller is applied to an inverted pendulum, and the feasibility of the controller is verified.

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