A Distributed Control Method for Unknown Nonlinear Systems Based on State-Event Triggering

Abstract

The present invention relates to a distributed control method for an unknown nonlinear system based on state-event triggering, comprising the following steps: constructing a multi-agent system composed of multiple agents, presetting an expected trajectory of a control signal and constructing an intermittent state feedback distributed control model; calculating an operation tracking path of the input simulated control signal through the model; calculating a tracking error between the operation tracking path and the expected trajectory, and performing online training; stopping the training when the training reaches that the tracking error value decays to the residual range, so as to obtain a trained intermittent state feedback distributed control model. The method described in the present invention can achieve stable tracking control of the simulated control signal of the multi-intelligent system.

Description

Technical Field

[0001] The present invention relates to the field of automatic control, and particularly relates to a distributed control method for an unknown nonlinear system based on state event triggering. Background Art

[0002] Nowadays, control systems are usually implemented through networks. If network resources are shared in sensor / actuator and control input channels, onboard communication bandwidth and stored energy can be saved, but both of these resources are limited in an autonomously operating network system. Therefore, maintaining stability under certain communication and energy constraints is crucial for networked control systems. The commonly used data transmission / communication method in traditional digital control technology is fixed-time scheduling sampling, which however may lead to unnecessary overload of the communication network.

[0003] To address the above problems of resource-limited microprocessors and bandwidth-limited networks, event-triggering technology has been introduced. However, existing event-triggering control is mainly applied to linear systems, and although it can be applied to some single nonlinear systems, there are certain limitations. However, a large number of actual networked engineering systems are not within the above application scope.

[0004] On the other hand, most networked multi-agent systems usually lack communication and energy resources. Especially when the agents themselves or their internal devices are powered by batteries, the communication bandwidth and channels between subsystems are limited, which has promoted the research on distributed event triggering. Existing distributed state triggering control either has some strict condition limitations or has a problem of explosive growth in complexity due to the repeated / recursive derivation of virtual controllers in the backstepping technique. In addition, under the event-triggering control framework, the research results for networked nonlinear strict-feedback systems with mismatched uncertainties are still limited, and related problems have not been well solved. Summary of the Invention

[0005] Aiming at the above problems existing in the prior art, the technical problem to be solved by the present invention is that the current technical methods cannot perform stable tracking control on the analog control signals of multi-intelligent systems.

[0006] To solve the above technical problems, the present invention adopts the following technical solutions:

[0007] A distributed control method for an unknown nonlinear system based on state event triggering, comprising the following steps:

[0008] S100: Construct a multi-agent system composed of multiple agents:

[0009]

[0010]

[0011] y i = x i，1 ;(1)

[0012] where \(i\) represents the \(i\)-th agent, \(i = 1, 2, \ldots, N\); \(x\) i，k : \(\mathbb{R}\) + \(\to \mathbb{R}\), \(u\) i : \(\mathbb{R}\) + \(\to \mathbb{R}\), \(y\) i : \(\mathbb{R}\) + \(\to \mathbb{R}\) represent the state, control input, and control output of the \(i\)-th agent respectively, \(f\) i，k : \(\mathbb{R}\) k \(\to \mathbb{R}\), \(k = 1, \ldots, n\) represent unknown smooth nonlinear functions, represents the first derivative of the state of the \(i\)-th agent at time \(k\), \(x\) i，k+1 represents the state of the \(i\)-th agent at time \(k + 1\), \(x\) i，1 represents the initial state of the \(i\)-th agent, \(x\) i，k represents the state of the \(i\)-th agent at time \(k\), represents the first derivative of the state of the \(i\)-th agent at the last \(n\) time steps, \(f\) i，n represents the unknown smooth nonlinear function corresponding to the last \(n\) time steps of the \(i\)-th agent, \(x\) i，n represents the state of the \(i\)-th agent at the last \(n\) time steps;

[0013] where

[0014]

[0015] where \(x\) i，k \(\in \mathbb{R}\) l is the neural network input vector, \(W\) i，k \(\in \mathbb{R}\) p is the ideal weight matrix, is the transpose of \(W\) i，k \(\varphi\) i，k (x i，k ) = [\varphi i，k1 (x i，k ), \ldots, \varphi i，kp (x i，k )] T \(\in \mathbb{R}\) p is the basis function vector; \(\varepsilon\) i，k (x i，k ) \(\in \mathbb{R}\) is the approximation error, satisfying \(\|\varphi i，k (x i，k )\| \leq \varphi m , |\varepsilon i，k (x i，k )| \leq \varepsilon m , where \(\varphi m and \(\varepsilon\)m is an unknown positive constant, φ i，kh (x i，k ) represents the Gaussian function, C = [C1, …, C l ) T is the center of the acceptance region, b h is the width of the Gaussian function;

[0016] S200: Preset the desired trajectory y0 of the control signal and construct the intermittent state feedback distributed control model of the multi-agent system. The specific steps are as follows:

[0017] S210: Construct the event-triggering mechanism of the intermittent state feedback, and the expression is as follows:

[0018]

[0019]

[0020]

[0021]

[0022] where and are the triggering thresholds, is the subsystem i, when i = 0, it represents itself, the first moment to complete formula (3), is the first triggering time of the adjacent subsystem j, l = 0, 1, 2, …, k = 1, ..., n.; and respectively represent the time used by agent i and its adjacent agent j to publish their respective state information in the l-th event, indicating that the states of agent i and its adjacent agent j remain unchanged, that is and

[0023] S220: Define the tracking error of the intermittent state feedback, and the expression is as follows:

[0024]

[0025]

[0026]

[0027] where

[0028]

[0029] where represents the triggering threshold, is The first trigger time after completion;

[0030] S230: Construct an intermittent state feedback distributed control model according to the event trigger mechanism and tracking error of the intermittent state feedback. The expression is as follows:

[0031]

[0032]

[0033]

[0034] and The expression of is as follows:

[0035]

[0036]

[0037] Where, represents the estimation of W i,k ; Γ i,k is a positive definite matrix, represents the intermittent state feedback tracking error,;

[0038] S300: Take any simulated control signal of the multi-agent system as the input of the intermittent state feedback distributed control model to obtain the running tracking path y of the signal i ;

[0039] S400: Calculate The tracking error ε = y i - y0;

[0040] S500: If the tracking error ε decays to within the residual set, obtain the trained intermittent state feedback distributed control model. At this time, it is considered that the signal can perform stable tracking control;

[0041] If the tracking error ε does not decay to within the residual set, then update using the least squares method and return to S300; The expression of the residual set is as follows:

[0042]

[0043] Where, ‖ε(t)‖ [0,T] represents the integral value of the output tracking error in [0, T], T is a certain moment greater than 0, κ1 is a positive design parameter, Q is a positive definite symmetric matrix, λ min (Q) represents the minimum eigenvalue of the matrix Q, V n(0) represents the initial value of the Lyapunov function V n , and Δ n is a constant.

[0044] Preferably, the limiting conditions for the event-triggering mechanism in S210 are expressed as follows:

[0045]

[0046]

[0047] where Δz i,k , ρ i,kr and τ i,k are positive constants, z i，k represents z i,k = x i,k - α i,kf , k = 2, …, n - 1, which is the continuous state tracking error, represents the triggering threshold at k = 1, and Δz i，1 represents the absolute value of the difference between the two at k = 1, and α i，k represents the first-order filter in the continuous state where k = 2, …, n, u i,k > 0 is a time constant; α i,k-1 is the virtual control and serves as the input of the first-order filter, while α i,kf represents the output of α i,k-1 , α i,k , γ yi0 and σ yi0 are positive design parameters, is a positive design parameter in the intermittent state, and Δα i，k represents the absolute value of the difference between the design parameters in the continuous and intermittent states, where Δz i,k , ρ i,kr and τ i,k are positive constants, depending on the triggering threshold Δy0, the topological parameter d i , μ i , the design parameters κ1, μ i,k , c i,k and b h , l = 2, …, n, h = 1, …, p.

[0048] Since the proposed distributed state-triggering control is constructed by replacing the states with the states at which they are triggered, it is important that this replacement not only saves communication and energy resources but also maintains consensus stability. By setting the triggering conditions, the application of bandwidth can be effectively reduced, saving system resources.

[0049] Compared with the prior art, the present invention has at least the following advantages:

[0050] 1. Based on the construction of an event trigger mechanism, this method constructs an intermittent state feedback distributed control on the basis of the existing continuous state feedback distributed control. The signal is only allowed to continue to be transmitted backward when it meets the trigger mechanism, so as to save the system communication resources and maintain the consensus stability. In addition, by adding the trigger mechanism and improving the control conditions, the tracking trajectory of the signal can basically coincide with the desired trajectory, realizing stable tracking control of the multi-agent system.

[0051] 2. The application conditions of this method are more relaxed than those of the existing distributed state trigger control, and it has better universality. In addition, this control method also solves the problem of explosive growth of complexity caused by repeated / recursive derivation of virtual controllers in the backstepping technique.

[0052] 3. The existing distributed state trigger control is either restricted by some harsh conditions or has the problem of explosive growth of complexity due to repeated / recursive derivation of virtual controllers in the backstepping technique. The control scheme of this invention avoids the problem of explosive growth of complexity while relaxing the application object conditions, and can perform stable tracking control on the multi-agent system. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 It is a control strategy block diagram of agent i based on state trigger setting.

[0054] Figure 2 It is a communication topology structure diagram of the multi-agent system.

[0055] Figure 3 It is the simulation result of the experiment using this method. DETAILED DESCRIPTION OF THE INVENTION

[0056] The present invention will be further described in detail below.

[0057] Refer to Figure 1-2 , a distributed control method for an unknown nonlinear system based on state event triggering, including the following steps:

[0058] S100: Construct a multi-agent system composed of multiple agents:

[0059]

[0060]

[0061] y i = x i，1 (1)

[0062] where \(i\) represents the \(i\)-th agent, \(i = 1, 2, \cdots, N\); \(x\) i，k : \(\mathbb{R}\) + \(\to\mathbb{R}\), \(u\) i : \(\mathbb{R}\) + \(\to\mathbb{R}\), \(y\) i : \(\mathbb{R}\) + \(\to\mathbb{R}\) represent the state, control input, and control output of the \(i\)-th agent respectively, \(f\) i，k : \(\mathbb{R}\) k \(\to\mathbb{R}\), \(k = 1, \cdots, n\) represent unknown smooth nonlinear functions, represents the first derivative of the state of the \(i\)-th agent at time \(k\), \(x\) i，k+1 represents the state of the \(i\)-th agent at time \(k + 1\), \(x\) i，1 represents the initial state of the \(i\)-th agent, \(x\) i，k represents the state of the \(i\)-th agent at time \(k\), represents the first derivative of the state of the \(i\)-th agent at the last \(n\) time steps, \(f\) i，n represents the unknown smooth nonlinear function corresponding to the last \(n\) time steps of the \(i\)-th agent, \(x\) i，n represents the state of the \(i\)-th agent at the last \(n\) time steps;

[0063] where,

[0064]

[0065] where \(x\) i,k \(\in\mathbb{R}\) l is the neural network input vector, \(W\) i,k \(\in\mathbb{R}\) p is the ideal weight matrix, is the transpose of \(W\) i,k , \(\varphi\) i,k (x i,k ) = [\varphi i,k1 (x i,k ), \cdots, \varphi i,kp (x i,k )] T \(\in\mathbb{R}\) p is the basis function vector; \(\varepsilon\) i,k (x i,k ) \(\in\mathbb{R}\) is the approximation error, satisfying \(\|\varphi i,k (x i,k )\| \leq \varphi m , |\varepsilon i,k (x i,k )| \leq \varepsilon m , where \(\varphi m and \(\varepsilon m are unknown positive constants, \(\varphi i,kh (x i,k ) represents the Gaussian function, \(C = [C_1, \cdots, Cl T is the center of the acceptance region, and b h is the width of the Gaussian function;

[0066] S200: Set the desired trajectory y0 of the control signal, and construct an intermittent state feedback distributed control model for the multi-agent system. The specific steps are as follows:

[0067] S210: Construct an event-triggering mechanism for intermittent state feedback, and the expression is as follows:

[0068]

[0069]

[0070]

[0071]

[0072] Among them, and are the triggering thresholds, is the subsystem i, when i = 0, it represents itself, and it is the first instant to complete formula (3). is the first triggering time of the adjacent subsystem j, l = 0, 1, 2,..., k = 1,..., n.; and respectively represent the time when agent i and its adjacent agent j release their respective state information in the l-th event, indicating that the states of agent i and its adjacent agent j remain unchanged, that is and

[0073] The limiting conditions for the event-triggering mechanism in S210 are expressed as follows:

[0074]

[0075]

[0076] Among them, Δz i,k , ρ i,kr and τ i,k are positive constants, z i，k represents z i,k = x i,k - α i,kf , k = 2,..., n - 1, is the continuous state tracking error, represents the triggering threshold at k = 1, and Δz i，1

[0077] represents the absolute value of the difference between the two at k = 1, and α​i，k Represents a first-order filter in the continuous state where k = 2, …, n, u i,k > 0 is a time constant; α i,k-1 is the virtual control and serves as the input of the first-order filter, while α i,kf represents α i,k-1 's output, α i,k , γ yi0 and σ yi0 are positive design parameters, is a positive design parameter in the intermittent state, Δα i，k represents the absolute value of the difference between the design parameters in the continuous and intermittent states, where Δz i,k , ρ i,kr and τ i,k are positive constants, depending on the triggering threshold Δy0, topological parameter d i , μ i , design parameters κ1, μ i,k , c i,k and b h , l = 2, …, n, h = 1, …, p.

[0078] S220: Define the tracking error of the intermittent state feedback, and the expression is as follows:

[0079]

[0080]

[0081]

[0082] where,

[0083]

[0084] where, represents the triggering threshold, is the first triggering time after completion;

[0085] S230: Construct the intermittent state feedback distributed control model according to the event-triggered mechanism of the intermittent state feedback and the tracking error, and the expression is as follows:

[0086]

[0087]

[0088]

[0089] and The expression is as follows:

[0090]

[0091]

[0092] Among them, represents the estimate of W i,k ; Γ i,k is a positive definite matrix, represents the intermittent state feedback tracking error;

[0093] S300: Take any multi-agent system simulation control signal as the input of the intermittent state feedback distributed control model to obtain the running tracking path y of the signal i ;

[0094] S400: Calculate The tracking error ε = y i - y0;

[0095] S500: When the tracking error ε decays to within the residual set, stop updating At this time, it is considered that the signal can perform stable tracking control; if it does not decay to within the residual set, then use the least squares method to update

[0096] Return to step S300 until the tracking error ε decays to within the residual set and stop training to obtain the trained intermittent state feedback distributed control model. The expression of the residual set is as follows:

[0097]

[0098] Among them, ‖ε(t)‖ [0,T] represents the integral value of the output tracking error in [0, T], T is a certain moment greater than 0, κ1 is a positive design parameter, Q is a positive definite symmetric matrix, λ min (Q) represents the minimum eigenvalue of the matrix Q, V n (0) represents the initial value of the Lyapunov function V n ; Δ n is a constant.

[0099] Here, the adaptive online training method is adopted for training the intermittent state feedback distributed control model, which belongs to the prior art. The adaptive online training method means that the simulated control signal is input into the intermittent state feedback distributed control model. When the simulated control signal of the multi-agent system meets the limit conditions of the event-triggering mechanism, the signal will be transmitted to the intermittent state feedback distributed control model for adaptive online training according to the triggering conditions, and it will be continuously updated This can make the update be able to adapt to the disturbances of factors such as system external noise and the changes of the intermittent state feedback distributed control model until the output tracking error decays to within the residual set range.

[0100] Simulation verification

[0101] Consider a system consisting of 4 nonlinear subsystems with the following dynamic model:

[0102]

[0103]

[0104] y i =x i,1

[0105] where i = 1,..., 4. Figure 2 shows the interaction topology of the multi-agent system. In the simulation, the desired trajectory y0 = 0.5sin(0.1t) + 0.5sin(0.05t) is set, the initial state is x i,1 (0) = 1.0, x i,2 (0) = 0, the triggering threshold is Δy0 = 0.005, the parameter settings are: k1 = 0.5, c1 = c2 = 5.0, γ yi0 = 1.5, σ yi0 = 0.001, σ yi1 = 0.2. The RBFNN contains 25 nodes, the centers are distributed in the space [-5, 5], b h = 2. The results are as Figure 3 shown: Figure 3 (a) shows the output trajectories of all subsystems. From Figure 3 (b), it can be determined that the output tracking error converges to a compact set near the origin. Figure 3 (c) gives the distributed protocol u i , Figure 3 (d) shows the state x i,2 triggering time.

[0106] In addition, in order to test the influence of the triggering threshold on the system tracking performance, select and using the same set of other design parameters, the results are as shown in Figure 3 (e)-3(f): showing the number of trigger events for x at two different trigger thresholds, which indicates that the larger the trigger threshold used, the less trigger time is required. However, the output tracking error variance is slightly larger. i,1 x i,2 Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that the technical solutions of the present invention can be modified or equivalently replaced without departing from the spirit and scope of the technical solutions of the present invention, and they should all be covered within the scope of the claims of the present invention.

[0107] ​

Claims

1. A distributed control method for an unknown nonlinear system based on state-event triggering, characterized in that: It includes the following steps: S100: Construct a multi-agent system composed of multiple agents: y i = x i，1 ; (1) where \(i\) represents the \(i\)-th agent, \(i = 1, 2, \ldots, N\); \(x\) i，k R + \(\to R\), \(u\) i : \(R\) + \(\to R\), \(y\) i : \(R\) + \(\to R\) represent the state, control input, and control output of the \(i\)-th agent respectively, \(f\) i，k : \(R\) k \(\to R\), \(k = 1, \ldots, n\) represent unknown smooth nonlinear functions, represents the first derivative of the state of the \(i\)-th agent at time \(k\), \(x\) i，k+1 represents the state of the \(i\)-th agent at time \(k + 1\), \(x\) i，1 represents the initial state of the \(i\)-th agent, \(x\) i，k represents the state of the \(i\)-th agent at time \(k\), represents the first derivative of the state of the \(i\)-th agent at the last \(n\) time steps, \(f\) i，n represents the unknown smooth nonlinear function corresponding to the last \(n\) time steps of the \(i\)-th agent, \(x\) i，n represents the state of the \(i\)-th agent at the last \(n\) time steps; Among them, where x i,k ∈R l is the neural network input vector, W i,k ∈R p is the ideal weight matrix, is the transpose of W i,k , φ i,k (x i,k ) = [φ i,k1 (x i,k ), …, φ i,kp (x i,k )] T ∈R p is the basis function vector; ε i,k (x i,k ) ∈ R is the approximation error, satisfying ||φ i,k (x i,k )|| ≤ φ m , |ε i,k (x i,k )| ≤ ε m , where φ m and ε m are unknown positive constants, φ i,kh (x i,k ) represents the Gaussian function, C = [C1, …, C l ) T is the center of the receptive field, b h is the width of the Gaussian function; S200: Preset the desired trajectory y0 of the control signal, and construct an intermittent state feedback distributed control model for the multi-agent system. The specific steps are as follows: S210: Construct an event-triggering mechanism for intermittent state feedback, and the expression is as follows: Among them, and are trigger thresholds, is subsystem i, when i = 0, it represents itself, and it is the first instant to complete formula (3). is the first trigger time of the adjacent subsystem j, l = 0, 1, 2,..., k = 1,..., n.; and respectively represent the time used by agent i and its adjacent agent j to publish their respective state information in the l-th event, indicating that the states of agent i and its adjacent agent j remain unchanged, that is and S220: Define the tracking error of intermittent state feedback, and the expression is as follows: Among them, Among them, represents the trigger threshold, is the first trigger time after completion; S230: Construct an intermittent state feedback distributed control model according to the event-triggering mechanism and tracking error of intermittent state feedback, and the expression is as follows: and The expressions are as follows: wherein, denotes the estimation of W i,k , Γ i,k is a positive definite matrix, denotes the intermittent state feedback tracking error; S300: Take any multi-agent system simulation control signal as the input intermittent state feedback distributed control model to obtain the running tracking path y of the signal i ; S400: Calculate The tracking error ε of i is ε = y - y0; S500: When the tracking error ε decays to within the residual set range, obtain the trained intermittent state feedback distributed control model. At this time, it is considered that the signal can be stably tracked and controlled; If the tracking error ε does not decay within the residual set, the least squares method is used for updating and S300 is returned; the residual set expression is as follows: where, ||ε(t)|| [0,T] denotes the integral value of the output tracking error over [0, T], where T is a certain moment greater than 0, κ1 is a positive design parameter, Q is a positive definite symmetric matrix, and λ min (Q) represents the minimum eigenvalue of the matrix Q, and V n (0) represents the initial value of the Lyapunov function V n , and Δ n is a constant.

2. The distributed control method for an unknown nonlinear system based on state-event triggering according to claim 1, characterized in that: The limitation conditions for the event-triggering mechanism in S210 are expressed as follows: where, Δz i,k , ρ i,kr and τ i,k are positive constants, i = 1, …, N, k = 1, …, n,, z i,k denotes z i,k = x i,k - α i,kf , k = 2, …, n - 1, is the continuous state tracking error, Δz i,1 denotes the absolute value of the difference between the two at k = 1, α i，k denotes the first-order filter in the continuous state α i,kf (0) = α i,k-1 (0), where k = 2, …, n, u i,k > 0 is a time constant; α i,k-1 is the virtual control and serves as the input of the first-order filter, while α i,kf denotes the output of α i,k-1 , α i,k , γ yi0 and σ yi0 are positive design parameters, is the positive design parameter in the intermittent state, Δα i,k denotes the absolute value of the difference between the design parameters in the continuous and intermittent states, where Δz i,k , ρ i,kr and τ i,k are positive constants, dependent on the triggering threshold Δy0, topological parameter d i , μ i , design parameters κ1, μ i,k , c i,k as well as b h , l = 2, …, n, h = 1, …, p.

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