A decentralized event-triggered adaptive backstepping control method

Abstract

The present invention discloses a decentralized event-triggered adaptive backstepping control method, which defines a nonlinear system and sets three conditions that the system must satisfy; secondly, coordinate transformation is carried out to design a decentralized adaptive backstepping controller under continuous state feedback, and finally, a decentralized adaptive backstepping controller under event-triggered conditions is constructed on the basis of the decentralized adaptive backstepping controller under continuous state feedback. By combining the principle of the modified aggregation method based on variables with the special treatment of non-triangular structure uncertainties, this method avoids the derivation of time-varying parameters, eliminates the limitation of the triangular structure condition, thereby broadening the application scope; directly replaces the continuous state with the triggered state, completely eliminating the non-differentiability problem of the virtual controller caused by intermittent state feedback, enabling each subsystem of the interconnected system to exchange information only with adjacent subsystems and only using intermittent states and inputs, ensuring the balance between communication resource usage and control performance.

Description

Technical Field

[0001] The present invention relates to a decentralized intermittent feedback adaptive control method, and particularly to a decentralized intermittent feedback adaptive control method for a non-linear time-varying system with non-triangular structured uncertainties. Background Art

[0002] In large-scale non-linear control systems, networked control systems have advantages such as low cost, convenient maintenance, and high reliability. Among them, the communication network is a necessary means for signal transmission. However, in this framework, there is a gap between decentralized control and network control because of limited communication bandwidth and channels, and sensors cannot transmit or update data in real time, which will reduce the control performance of large-scale non-linear systems.

[0003] To maintain the balance between communication resource usage and control performance, an event-triggered control method that communicates only when certain predefined conditions are triggered is adopted to save energy and communication resources. Early achievements on event-triggered control mainly targeted linear systems, and subsequent extended work on non-linear systems was carried out, but the system models considered needed to be completely known. To address the uncertainties of non-linear systems, an event-triggered adaptive control scheme based on the backstepping design method was proposed. However, these design schemes only allow intermittent transmission of control signals in the network, while the system states are still transmitted based on continuous feedback. Therefore, they can only save communication resources in the controller-to-actuator channel and are not applicable to saving communication resources in the sensor-to-controller channel.

[0004] In the past few years, control design through intermittent state feedback has attracted increasing attention. In this direction, there are mainly two control schemes. The first is state-triggered control that only uses intermittent outputs. In this scheme, only the output is triggered, and the reduction of the communication burden is still limited. The second is state-triggered control through intermittent full-state feedback. However, most of the research objects are low-order or standard-form models, and the research on large-scale interconnected systems with mismatched and non-parametric uncertainties is also limited to non-linear systems with triangular structures, and it is required that the parameters of the controlled object be constant. However, in most applications, the parameters of the controlled object change rapidly with time. For example, in the control of highway traffic systems, the free-flow speed is a time-varying parameter, and changes in weather, air pressure, and wind speed will have a great impact on it. Summary of the Invention

[0005] In view of the above problems existing in the prior art, in the present invention, we relax the overly strong restrictions, expand the applicable scope of the state-triggered stability theory based on the backstepping method, and develop a decentralized event-triggered adaptive backstepping control method for nonlinear interconnected systems with non-triangular structured uncertainties and unknown time-varying parameters, enabling the system to well balance the use of communication resources and control performance.

[0006] To solve the above technical problems, the present invention adopts the technical solutions described in the specific implementation manners.

[0007] Compared with the prior art, the present invention has at least the following advantages:

[0008] 1. Based on the backstepping technique, this method proposes a globally decentralized adaptive control scheme. By combining the principle of the modified aggregation method based on variables with the special treatment of non-triangular structured uncertainties, the derivation of time-varying parameters is avoided, and the limitation of the triangular structure condition is eliminated, thus broadening the application scope; directly replacing the continuous state with the triggered state completely eliminates the non-differentiability problem of the virtual controller caused by intermittent state feedback, enabling each subsystem of the interconnected system to only exchange information with adjacent subsystems and only utilize the intermittent state and input, ensuring the balance between the use of communication resources and control performance.

[0009] 2. This method relaxes the overly strong restrictive conditions, expands the applicable scope of the state-triggered stability theory based on the backstepping method, and makes this method more universal.

[0010] 3. The event-triggered mechanism proposed by this method has a strict positive lower limit on the transmission time, thus avoiding the Zeno phenomenon, and using the idea of dynamic filtering technology and projection operator, extending the results of this invention to a more general system with multiple mismatched time-varying parameters. Description of the Drawings

[0011] Figure 1 For the case of x 1,k (k = 1, 2) when selecting the group with a larger triggering threshold.

[0012] Figure 2 For the case of x 2,k (k = 1, 2) when selecting the group with a larger triggering threshold.

[0013] Figure 3 For the triggering times of x i,k (i, k = 1, 2) under different triggering thresholds.

[0014] Figure 4 For the triggering times of x i,k (i, k = 1, 2) under different triggering thresholds. Specific Implementation Manners

[0015] The present invention will be further described in detail below.

[0016] The present invention studies the decentralized stabilization problem of a class of interconnected systems in the presence of non-triangular structural uncertainties and time-varying parameters, where each subsystem exchanges information only with its neighbors and uses only intermittent (instead of continuous) states and inputs. We expand the applicable scope of the state-triggered stability theory based on backstepping and propose a global decentralized intermittent feedback adaptive control scheme.

[0017] A decentralized event-triggered adaptive backstepping control method includes the following steps:

[0018] S1: Define the following nonlinear system, which consists of N interconnected subsystems, and the i-th subsystem is modeled as:

[0019]

[0020] where i = 1,..., N, the system state where are the control input and output respectively, and are known functions. Among them, is an unknown parameter vector, represents the nonlinear coupling effect from the j-th subsystem when j ≠ i, and represents the modeling error of the i-th subsystem when j = i.

[0021] The objective of the present invention is to develop a global decentralized adaptive backstepping controller for system (1), using only local intermittent feedback signals, such that:

[0022] i) The global uniform boundedness of the closed-loop signals is guaranteed, and the outputs of all subsystems are guided to an assignable residual set near zero;

[0023] ii) Zeno behavior is excluded.

[0024] Assumption 1: The unknown nonlinear function f ij,k (x j , u j , t) satisfies the following linear growth condition:

[0025]

[0026] For i, j = 1,..., N, where is an unknown coupling gain, representing the magnitude / intensity of the modeling error and the nonlinear coupling interaction, ∈ ij,k ≥0 is an unknown constant.

[0027] Assumption 2: When t ≥ 0, the parameter θ i(t) is piecewise continuous, and θ i (t) ∈ Ω i0 , where Ω i0 is an unknown compact set. The "radius" of Ω i0 , denoted by , is assumed to be bounded but not necessarily known.

[0028] Assumption 3: The functions and ψ i (x i ), i = 1, …, N satisfy the global Lipschitz continuity condition, such that

[0029]

[0030] where and are unknown bounded constants.

[0031] S2: A decentralized adaptive backstepping controller is designed using local continuous state signals. This scheme can serve as the basis for an intermittent state feedback controller. To this end, we perform the following coordinate transformations:

[0032] z i,1 = x i,1 ⑷

[0033] z i,k = x i,k - α i,k-1 , k = 2, …, n i ⑸

[0034] where z i,1 represents the coordinate transformation of the system state x i,1 in the continuous state; z i,k represents the coordinate transformation of the state signal x i,k in the continuous state; α i,k-1 represents the virtual controller in the continuous state;

[0035] S3: The decentralized adaptive backstepping controller under continuous state feedback is designed as:

[0036]

[0037]

[0038] where, α i,1 , α i,k and are the virtual controllers in the continuous state; c i,1 , c i,k , and are positive design parameters; is α i,k-1 The partial derivative with respect to x i,l is a constant that depends on c i,k , and ; x i,l+1 represents the continuous state information of the i-th subsystem at time l + 1; z i,k-1 represents the state information x i,k-1 under continuous state in a coordinate transformation; The update law of

[0039]

[0040] is designed as: i where σ is a positive design parameter, i is the estimate of θ Γ i is a positive definite design matrix, represents the state information after coordinate transformation at the end (n i ) time and in continuous state.

[0041] The following lemma is introduced here

[0042] Lemma 1: The state vector x i and its transformation vector z i obey the following relationship,

[0043]

[0044] where A i and B i are constant matrices.

[0045] For the interconnected non - linear non - triangular system (1) that satisfies Assumptions 1 and 2, if the decentralized adaptive controller (8) is adopted with the adaptive law (9), then it is considered that: i) The global uniform boundedness of the closed - loop signals is guaranteed; ii) The outputs of the subsystems are all controlled within the residual sets near zero, and the stability performance of the system can be improved by reasonably selecting the design parameters.

[0046] Proof: The proof process includes two parts: stability analysis and performance analysis.

[0047] 1) Stability analysis: This part consists of the following n i steps.

[0048] Step 1: Define the Lyapunov function According to

[0049]

[0050] Using (6) and (11), Denoted as

[0051]

[0052] Step k (k = 2, …, n i - 1): Define From Assumption 1, it can be deduced that

[0053]

[0054] Combining (1), (5), (7) and (12) - (14), we can obtain

[0055]

[0056] Step n: Define the Lyapunov function where k θ,i is an unknown bounded constant vector. From equations (1), (5) and (8), we get

[0057]

[0058] Substituting (9) into (16), we get

[0059]

[0060] where According to Assumption 1, Assumption 2 and Lemma 1, we can obtain

[0061]

[0062] Due to the participation of ETM, the parameter - induced perturbation term is processed in a non - compensatory manner. Using (18), it can be further scaled as

[0063]

[0064] Let From Lemma 1, it can be further obtained that

[0065]

[0066] where and Choose a sufficiently large c j and σ i , where c j = min{c j,1 , …, c j,ni}. It can be obtained that Among them

[0067] It can be easily obtained from the above analysis that This ensures that z i,k ∈L ∞ and θ i ∈L ∞ . From equations (4), (5), (6), and (7), it can be seen that x i,k is bounded, where k = 1, …, n i -1. Combining with equation (8) gives the boundedness of u i . Therefore, all signals in the closed-loop system are globally uniformly bounded.

[0068] 2) Performance analysis:

[0069] According to the definition of V, we can obtain This means that z i,1 is guaranteed to decay to a residual set near zero. In addition, increasing the design parameters c i,k and Γ i , or decreasing and where k = 1, …, n i -1, can reduce the upper bound of |z i,1 |.

[0070] S4: Based on S3, a decentralized adaptive backstepping controller under an event-triggered condition is constructed:

[0071] S41: Event-triggering mechanism

[0072] We denote and u i , i, j = 1, …, N (j ≠ i), k = 1, …, n i as local state information, other subsystem state information, and drive signal information respectively, and transmit their information according to the designed event-triggering mechanism. Since and represent the l-th event time when system i, other subsystem j, and the drive signal transmit their information respectively, thus, we can obtain and u i remain unchanged as and where l = 0, 1, 2, ….

[0073] Now we propose the following triggering condition that only depends on locally available information:

[0074]

[0075] where Δxi,k , Δx j,k and Δu i are positive triggering thresholds, and respectively represent the first instant when (21)-(23) are satisfied.

[0076] S42: Controller Design

[0077] Since in the case of state triggering, the control system only has local intermittent state signals available, we modify the coordinate transformation defined by Eqs. (4)-(5) as:

[0078]

[0079] where and represent the coordinate transformation of the system state in the intermittent state; represents the system state information in the intermittent state; represents the virtual controller in the intermittent state;

[0080] Based on intermittent state feedback, a decentralized event-triggered adaptive backstepping controller is constructed as:

[0081]

[0082] where, c i,1 , c i,k , and are positive design parameters, and represent the virtual controller in the intermittent state; v i represents the designed controller in the intermittent state, The update law of

[0083]

[0084] where σ i is a positive design parameter, Γ i is a positive definite design matrix, represents the system state after coordinate transformation at the end (n i ) moment and in the intermittent state.

[0085] To ensure that all closed-loop signals are globally uniformly bounded, we introduce the following lemma.

[0086] Lemma 2: The influence of event triggering is as follows:

[0087]

[0088] where Δzi,k and Δα i,k are positive constants that depend on the triggering threshold Δx i,k , Δx j,k and Δu i , as well as the design parameter c i,k , and

[0089] For the interconnected nonlinear non-triangular system (1) satisfying Assumptions 1 to 3, if the decentralized adaptive controller (28) with the adaptive law (29) and the triggering conditions (21), (23) is adopted, it is considered that: i) the global uniform boundedness of the closed-loop signals is guaranteed; ii) the subsystem outputs are all controlled within the residual sets near zero, and the stability performance of the system can be improved by reasonably selecting the design parameters; and iii) the Zeno phenomenon is excluded.

[0090] Proof: The proof process consists of three parts: stability analysis, performance analysis, and the exclusion of Zeno behavior.

[0091] 1) Stability analysis: This part consists of the following n i steps.

[0092] Step 1: Define the Lyapunov function The derivative of V obtained from Eqs. (1), (4), (5), and (6) is: i,1

[0093]

[0094] Step k (k = 2, …, n i - 1): Define From (1), (5), (7), and (32), it can be deduced that

[0095]

[0096] Step n: Define the Lyapunov function where the definition of k θ,i is the same as before. Rewrite the control law v in (28) as i

[0097]

[0098] From (1), (5), (33), and (34), can be expressed as

[0099]

[0100] Substitute (29) into (35) to get

[0101] ​​​

[0102] wherein and According to Hypothesis 3, it can be known that

[0103]

[0104] It can be deduced from Lemma 1 and (37)(38) that

[0105]

[0106] wherein Combining (18), (36) and (39), we get

[0107]

[0108] wherein Let It can be further obtained from (40) that

[0109]

[0110] wherein and Select respectively c j and σ i to be large enough, wherein c j = min{c j,1 , …, c j,ni}. It can be obtained from (41) that wherein

[0111] Similar to the proof of Theorem 1, it can be analyzed that V(t) ∈ L ∞ , which ensures that z i,k ∈ L ∞ and θ i ∈ L ∞ . From equations (24), (25), (26), (27), it can be known that x i,k is bounded, k = 1, …, n i -1. Combining with equation (28), the boundedness of v i is obtained. Therefore, all signals in the closed-loop system are globally uniformly bounded.

[0112] 2) Performance analysis:

[0113] Similar to the proof of Theorem 1, according to the definition of V, we can obtain that |z i,1 | is less than a positive constant related to the design parameters, which means that z i,1The set of residuals that is guaranteed to decay to near zero. Additionally, increasing the design parameter c i,k and Γ i , or decreasing and where k = 1, …, n i -1, can reduce the upper bound of |z i,1 |.

[0114] 3) Excluding Zeno behavior:

[0115] Finally, we prove that result iii) is guaranteed. Define Then When is the case, remains unchanged, then we can obtain and Combining the boundedness of x i,k , u i , ψ i (x i ) and f ij,k , k = 1, …, n i , we get This means where is a positive constant. Similarly, and where T0, T1, and T2 are positive constants. Thus, Zeno behavior is excluded.

[0116] Simulation verification

[0117] Consider the following numerical example:

[0118]

[0119] In the simulation, we set the initial state x 1,1 (0) = 0.2, x 1,2 (0) = 0.2, x 2,1 (0) = 0.1, x 2,2 (0) = 0.1, the design parameters c 1,1 = 0.5, c 1,2 = 0.3, c 2,1 = 1.8, c 2,2 = 1.5, σ i = 0.001, Γ i = 0.5, the time-varying parameters θ1(t) = 0.1 + 0.1sin(0.2t), θ2(t) = 0.1 + 0.1cos(0.2t), and the function In all of the above functions, i, j, k, l = 1, 2. To examine the impact of the triggering threshold on the system performance, with other design parameters remaining unchanged, we selected two different triggering thresholds for comparison, namely: 1) Δx 1,1 = 0.001, Δx 1,2 = 0.002, Δx 2,1 = 0.002, Δx 2,2 = 0.002, Δu1 = 0.01, Δu2 = 0.01; and 2) Δx′ 1,1 = 0.005, Δx′ 1,2 = 0.005, Δx′ 2,1 = 0.003, Δx′ 2,2 = 0.003, Δu′1 = 0.03, Δu′2 = 0.03.

[0120] The simulation results are as Figures 1 to 4 shown. Figure 1 and Figure 2 respectively show the variations of the states x 1,k , x 2,k (k = 1, 2) under the group with a larger triggering threshold. The triggering times of x i,k (i, k = 1, 2) and u i are respectively as Figure 3 and Figure 4 shown. It can be observed that the larger the triggering threshold, the smaller the triggering times, but the system performance will decline to a certain extent.

Claims

1. A decentralized event-triggered adaptive backstepping control method, characterized in that, It includes the following steps: S1: Define the following nonlinear system, which consists of N interconnected subsystems. The i-th subsystem is modeled as: where \(i = 1,\ldots,N\), the system state where the state vector are the control input and output respectively, \(x\) i,k+1 represents the state of the system at time \(k + 1\); and are known functions; where is an unknown parameter vector, represents the non - linear coupling effect from the \(j\) - th subsystem when \(j\neq i\), and represents the modeling error of the \(i\) - th subsystem when \(j = i\), \(f\) ij,ni (x j ,u j ,t) represents the non - linear coupling effect from the \(j\) - th subsystem when \(j\neq i\) corresponding to the last \(n\) i time instant, and represents the modeling error of the \(i\) - th subsystem when \(j = i\); Hypothesis 1: f ij,k (x j , u j , t) satisfies the following linear growth condition: For \(i, j = 1, \ldots, N\), where is the unknown coupling gain, \(\in\) ij,k \(\geq 0\) is an unknown constant; Hypothesis 2: When t ≥ 0, the parameter θ i (t) is piecewise continuous, and θ i (t) ∈ Ω i0 , where Ω i0 is an unknown compact set, and the "radius" of Ω i0 , denoted by β θi is assumed to be bounded, but not necessarily known; Hypothesis 3: Function and ψ i (x i ), i = 1, …, N satisfy the global Lipschitz continuity condition, such that wherein and are unknown bounded constants; S2: Coordinate transformation: z i,1 = x i,1 (4) z i,k = x i,k - α i,k-1 , k = 2, …, n i (5) where z i,1 represents the coordinate transformation of the system state x i,1 in the continuous state; z i,k represents the coordinate transformation of the state signal x i,k in the continuous state; α i,k-1 represents the virtual controller in the continuous state; S3: The decentralized adaptive backstepping controller under continuous state feedback is designed as: where α i,1 , α i,k and α i,ni are virtual controllers in the continuous state; c i,1 , c i,k , and are positive design parameters; is the partial derivative of α i,k-1 with respect to x i,l , which is a constant depending on c i,k , and ; x i,l+1 represents the continuous state information of the i-th subsystem at the (l + 1)-th moment; z i,k-1 represents the coordinate transformation of the state information x i,k-1 in the continuous state; The update law is designed as follows: where σ i is a positive design parameter, is an estimate of θ i , and Γ i is a positive definite design matrix, represents the state information after coordinate transformation at the end time (n i ) and in the continuous state; S4: Based on S3, a decentralized adaptive backstepping controller under event-triggered conditions is constructed: S41: Event-triggering mechanism Let and u i , i, j = 1, …, N (j ≠ i), k = 1, …, n i be represented as local state information, other subsystem state information, and drive signal information respectively, and transmit their information according to the designed event-triggering mechanism. and represent the l-th event time for system i, other subsystem j, and drive signal to transmit their information respectively. Then we can obtain and u i remain unchanged as and where l = 0, 1, 2, …; The following triggering conditions that only depend on locally available information are proposed: where Δx i,k , Δx j,k and Δu i are positive triggering thresholds, and respectively represent the first instants when (21)-(23) are satisfied; S42: Controller design Since in the case of state triggering, only local intermittent state signals are available in the control system the coordinate transformation defined by Eqs. (4)-(5) is modified to: where and represent the coordinate transformation of the system state in the intermittent state; represents the system state information in the intermittent state; represents the virtual controller in the intermittent state; Based on intermittent state feedback, a decentralized event-triggered adaptive backstepping controller is constructed as: where c i,1 , c i,k , and are positive design parameters, and represent the virtual controller in the intermittent state; v i represents the designed controller in the intermittent state; The update law is designed as: where σ i is a positive design parameter, Γ i is a positive definite design matrix, represents the system state after coordinate transformation at the end time (n i ) and in the intermittent state.

2. The decentralized event-triggered adaptive backstepping control method according to claim 1, characterized in that The state vector x i and its transformation vector z i obey the following relationship where A i and B i are constant matrices.

3. The decentralized event-triggered adaptive backstepping control method according to claim 1, characterized in that The influence of the event triggering is shown as follows: where Δz i,k and Δα i,k are positive constants that depend on the trigger threshold Δx i,k , Δx j,k and Δu i , as well as the design parameter c i,k , and k = 1, …, n i .

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