A Smooth Asynchronous Switching Control Method and System for Amplitude-Bounded Interference

Abstract

The present invention discloses a smooth asynchronous switching control method and system for amplitude-bounded disturbances, which relates to the technical field of switching system control and aims to solve problems such as damage to the actuators of the controlled object caused by jitter of the system state and control input in the existing switching system. The technical key points of the present invention include: establishing a discrete-time asynchronous switched linear system model subject to amplitude-bounded disturbances; designing a smooth transition control; solving the smooth transition controller to obtain a control input that is stable in the sense of invariant set; substituting the solved control input into the smooth transition controller, and using the smooth transition controller to control the smooth asynchronous switching of the discrete-time asynchronous switched linear system containing amplitude-bounded disturbances. The present invention can suppress the jitter of the switching system state and control input in the presence of continuous amplitude-bounded disturbances from the outside, ensuring the smooth transition control performance of the system; restricting the state trajectory of the error system, enabling the disturbed system to achieve stability in the sense of invariant set.

Description

Technical Field

[0001] The present invention relates to the technical field of switching system control, and particularly relates to a smooth asynchronous switching control method and system for amplitude-bounded disturbances. Background Art

[0002] As an effective model for describing multi-modal systems, switching systems have been widely studied and applied in recent decades. A major feature of switching systems is the switching behavior, which can occur spontaneously due to changes in the system's dynamic model or be artificially introduced to optimize control performance. However, this switching behavior often leads to drastic changes in the system state and control input, namely, the chattering phenomenon. Chattering deteriorates the expected performance of the system, reduces the service life of the equipment, and may even lead to system instability. To solve this problem, scholars at home and abroad have proposed several smooth transition control methods.

[0003] Early attempts at smooth transition control tried to overcome the chattering problem by introducing a low-pass filter at the controller output, but this increased the complexity of system design. In recent years, some smooth transition control methods have achieved chattering suppression by directly designing the controller gain, resulting in methods such as co-designing the switching law and a controller with rate saturation, normalizing the controller gain, and introducing a transition controller.

[0004] However, these smooth transition control methods mainly target switching systems without external disturbances or with bounded external disturbance energy. In practice, external disturbances usually persist and do not decay to zero over time. Such disturbances are not energy-bounded but amplitude-bounded. When the system is affected by such amplitude-bounded disturbances, the design of smooth transition control becomes more complex. In addition, switching systems usually encounter asynchronous phenomena in practice, that is, due to the delay in mode identification, there is a mismatch between the controller and the mode, which exacerbates chattering and reduces the robustness of the system. However, the smooth transition control method for asynchronous switching systems affected by amplitude-bounded disturbances is difficult and rarely studied. Summary of the Invention

[0005] In view of the above problems, the present invention proposes a smooth asynchronous switching control method and system for amplitude-bounded disturbances, aiming to solve the problems in the existing smooth transition control technology of switching systems that lack consideration of amplitude-bounded disturbances and asynchronous switching, and the problem that the existing methods cannot simultaneously achieve smooth transition control of the system state and control input.

[0006] According to one aspect of the present invention, a smooth asynchronous switching control method for amplitude-bounded disturbances is proposed, and the method includes:

[0007] Establish a discrete-time asynchronous switching linear system model including amplitude-bounded disturbances;

[0008] Design a smooth transition controller based on the discrete-time asynchronous switched linear system model; the smooth transition controller includes a stabilizing controller and a smooth controller;

[0009] Solve the smooth transition controller to obtain a control input that is stable in the sense of invariant set;

[0010] Substitute the obtained control input into the smooth transition controller, and use the smooth transition controller to control the smooth asynchronous switching of the discrete-time asynchronous switched linear system with amplitude-bounded disturbances.

[0011] Furthermore, the discrete-time asynchronous switched linear system model with amplitude-bounded disturbances is established as follows:

[0012]

[0013] where, x k represents the system state vector at time k; u k represents the control input at time k; w k represents the amplitude-bounded disturbance input; σ k represents the asynchronous switching signal at time k corresponding to N modes; respectively represent the system matrix, input matrix, and disturbance matrix.

[0014] Furthermore, the smooth transition controller is designed as follows:

[0015]

[0016] where, k s represents the switching time, κ ij represents the asynchronous duration from mode i to mode j; K i represents the stabilizing controller corresponding to mode i; represents the set of the number of smooth controllers corresponding to the transition from mode i to mode j, and l is the label of the smooth controller; K j represents the stabilizing controller corresponding to mode j; represents the l-th smooth controller from mode i to mode j.

[0017] Furthermore, the solving of the smooth transition controller to obtain a control input that is stable in the sense of invariant set includes:

[0018] Establish a mathematical description of the smooth transition control problem for the asynchronous switching system with amplitude-bounded disturbances;

[0019] Establish the globally uniformly asymptotically stable conditions for a closed-loop nominal system with a smooth transition control structure; where the closed-loop nominal system is a discrete-time asynchronous switched linear system without external disturbances;

[0020] Solve the robust invariant set for a discrete-time asynchronous switched linear system with a smooth transition control structure;

[0021] Give sufficient conditions to ensure the smooth transition control performance.

[0022] Furthermore, the mathematical description of the smooth transition control problem for the asynchronous switched system subject to amplitude-bounded disturbances is established as follows:

[0023] For a discrete-time asynchronous switched linear system subject to amplitude-bounded disturbances, find a set of smooth controllers and stabilizing controllers such that the jitters of the system state and control input of the system can be suppressed by the smooth transition controller structure;

[0024] Define the nominal system corresponding to the discrete-time asynchronous switched linear system subject to amplitude-bounded disturbances as:

[0025]

[0026] where, and represent the state vector and control input of the nominal system, respectively;

[0027] Then the error system is:

[0028]

[0029] where, represents the closed-loop system matrix;

[0030] Assume that there is no error in the system at the initial time k0, there is:

[0031]

[0032] Then

[0033] Ensure the stability of the nominal system and find the error set E k , which can ensure the stability of the discrete-time asynchronous switched linear system subject to amplitude-bounded disturbances in the sense of the invariant set;

[0034] Define the smooth transition control performance δ r of the system state to satisfy: ||x k+1 -x k || ≤ δ r ||x k ||, and the smooth transition control performance ε of the system control inputr Satisfied: ||u k+1 - u k || ≤ ε r ||x k ||.

[0035] Furthermore, the globally uniformly asymptotically stable condition of the closed - loop nominal system with a smooth - transition control structure is established as follows:

[0036] Theorem 1: For the nominal system corresponding to a discrete - time asynchronous switched linear system with amplitude - bounded disturbances: α > 1, β l > 1, 0 < μ < 1 are given constants, and satisfy Suppose there exists a family of matrices R i and as well as a family of positive - definite matrices S i and such that for there is:

[0037]

[0038] where, denotes the matrix [A, C; C T , B], denotes the set denotes the set {X, Y, Z, θ} take values from {S i , S i , R i , α}, {S j , S j , R j , μ} respectively, and denote Then the nominal system corresponding to the discrete - time asynchronous switched linear system with amplitude - bounded disturbances is globally uniformly asymptotically stable under the smooth - transition controller structure, and

[0039] Furthermore, the solution of the robust invariant set for the discrete - time asynchronous switched linear system with a smooth - transition control structure includes:

[0040] Represent a switching sequence of length k and meeting the dwell - time τ as where and satisfy

[0041] The corresponding sequence of closed - loop system matrices is:

[0042]

[0043] Among them:

[0044]

[0045] For the said error system, under zero initial conditions, a given set Let be the set of all possible system states evolved one step backward from E. Then the set of system states evolved q steps backward is:

[0046]

[0047] Wherein, That is Therefore, for a discrete-time asynchronous switched linear system subject to amplitude-bounded disturbances, there is

[0048] Theorem 2: For a discrete-time asynchronous switched linear system subject to amplitude-bounded disturbances, given a non-empty set Then the necessary and sufficient condition for E to be a robust invariant set of the corresponding error system is

[0049] Suppose E ∞ = lim k→∞ E k exists. Replacing the mode-dependent subscript k with the stage-dependent subscript p, we get And That is Performing the convex operation on the above process, denoted as Then there is And Therefore is a robust invariant set of the error system; According to Theorem 2, use the following Algorithm 1 to calculate and

[0050] 1) Initialize p = 0, 2) Loop τ ≤ q ≤ 2τ - 1, 1 ≤ j ≤ N, 1 ≤ i ≤ N, and calculate 3) If Then we get Stop the calculation; otherwise let p = p + 1 and repeat step 2).

[0051] Furthermore, use the following Theorem 3 to pre-determine whether there exists a robust invariant set of the error system:

[0052] Theorem 3: For a discrete-time asynchronous switched linear system subject to amplitude-bounded disturbances, α > 1, β l > 1, 0 < μ < 1 are given constants, and satisfy Suppose there exists a family of matrices R i and and a family of positive definite matrices S i and such that for the following equations all hold, then there must exist:

[0053]

[0054] Furthermore, the sufficient condition to ensure the smooth transition control performance is:

[0055] Theorem 4: For a discrete-time asynchronous switched linear system subject to amplitude-bounded disturbances, given ν1 > 0, ν2 > 0, ν3 > 0, if there exist scalars ε r > 0, δ r > 0, matrices R i , positive definite matrices S i , such that for all have

[0056]

[0057] where {X, Y, Z, W} are taken from {S i , S i , R i , R i}, {S j , S j , R j , R j} respectively, and denote then the discrete-time asynchronous switched linear system subject to amplitude-bounded disturbances has the smooth transition control performance of the system state and the control input and

[0058] According to another aspect of the present invention, a smooth asynchronous switching control system for amplitude-bounded disturbances is proposed, and the system includes:

[0059] A system model establishment module configured to establish a discrete-time asynchronous switched linear system model including amplitude-bounded disturbances;

[0060] A controller design module configured to design a smooth transition controller based on the discrete-time asynchronous switched linear system model; the smooth transition controller includes a stabilizing controller and a smoothing controller;

[0061] A control quantity solving module, configured to solve the smooth transition controller to obtain a control input that is stable in the sense of invariant set;

[0062] A control module, configured to substitute the obtained control input into the smooth transition controller, and use the smooth transition controller to control the smooth asynchronous switching of a discrete-time asynchronous switched linear system with amplitude-bounded disturbances.

[0063] The beneficial technical effects of the present invention are:

[0064] The present invention takes an asynchronous switching system with amplitude-bounded disturbances as the research object, and proposes a smooth asynchronous switching control method that can achieve anti-shake of system states and control inputs, which is called a tube-based bumpless transfer control (TBBTC) method. The present invention can simultaneously solve the following two problems existing in some existing smooth transition control strategies: 1) When the system is affected by external disturbances, it is assumed that the external disturbances are energy-bounded and will decay to zero over time, which cannot be applied to some actual systems; 2) When designing a smooth transition controller, it is assumed that there is no asynchronous phenomenon in the switched system. This assumption is too ideal, and when there is asynchrony in the actual system due to mode identification, the stability and smoothness of the system cannot be achieved.

[0065] The present invention proposes a new smooth transition control method for an asynchronous switching system with amplitude-bounded disturbances, and gives a suppression strategy for system state and control input jitter. This method can suppress the jitter of states and control inputs in the switched system in the presence of continuous amplitude-bounded external disturbances, and ensure the smooth transition control performance of the system. By introducing a robust invariant set, the state trajectory of the error system is restricted, so that the perturbed system achieves stability in the sense of invariant set.

[0066] In addition, in an actual switched system, the identification of system modes often takes a certain amount of time, resulting in a phenomenon of mismatch between system modes and controllers, that is, an asynchronous phenomenon. The present invention studies the smooth transition control problem under asynchronous switching conditions to ensure the stability and robustness of the system under asynchronous conditions. Description of the Drawings

[0067] The present invention can be better understood by referring to the descriptions given in conjunction with the accompanying drawings in the following text. The accompanying drawings, together with the following detailed description, are included in this specification and form a part of this specification, and are used to further illustrate the preferred embodiments of the present invention and explain the principles and advantages of the present invention.

[0068] Figure 1 It is the smooth transition control structure diagram proposed in the embodiment of the present invention; the horizontal axis in the figure represents the sampling time.

[0069] Figure 2 This is the program flowchart for the smooth transition control in the embodiments of the present invention.

[0070] Figure 3 This is the effect diagram of the smooth transition control in the embodiments of the present invention, including the system state and control input; from top to bottom in the legend are the nominal system signal without applying the method proposed in the present invention, the disturbed system signal without applying the method proposed in the present invention, the nominal system signal applying the method proposed in the present invention, and the disturbed system signal applying the method proposed in the present invention.

[0071] Figure 4 This is the system state and control input in the embodiments of the present invention under 100 groups of bounded-amplitude random disturbances; from top to bottom in the legend are the nominal system signal, the disturbed system signal, and the threshold boundary for signal limitation.

[0072] Figure 5 This is the jitter situation of the system state and control input when the system with different numbers of smooth controllers in the embodiments of the present invention is applied with 100 groups of bounded-amplitude random disturbances. Detailed implementation manners

[0073] In order to enable those skilled in the art of this technology to better understand the solution of the present invention, the exemplary embodiments or examples of the present invention will be described below in conjunction with the accompanying drawings. Obviously, the described embodiments or examples are only part of the embodiments or examples of the present invention, rather than all of them. Based on the embodiments or examples in the present invention, all other embodiments or examples obtained by those of ordinary skill in the art without creative efforts shall fall within the scope of protection of the present invention.

[0074] The embodiments of the present invention propose a smooth asynchronous switching control method for bounded-amplitude disturbances, and the method includes:

[0075] Step 1: Establish a discrete-time asynchronous switching linear system model including bounded-amplitude disturbances;

[0076] Step 2: Design a smooth transition controller based on the discrete-time asynchronous switching linear system model; the smooth transition controller includes a stabilizing controller and a smooth controller;

[0077] Step 3: Solve the smooth transition controller to obtain a control input that is stable in the sense of invariant set;

[0078] Step 4: Substitute the obtained control input into the smooth transition controller, and use the smooth transition controller to control the smooth asynchronous switching of the discrete-time asynchronous switching linear system including bounded-amplitude disturbances.

[0079] The method starts from Step 1. In Step 1, a discrete-time asynchronous switched linear system model containing amplitude-bounded disturbances is established.

[0080] According to an embodiment of the present invention, consider the following discrete-time asynchronous switched linear system subject to amplitude-bounded disturbances:

[0081]

[0082] where x k is the system state vector, u k is the control input, is the amplitude-bounded disturbance input. σ k is the asynchronous switching signal corresponding to N subsystems (modes), which takes values in The switching instants of the system modes are k1, k2, …. Due to the asynchronous phenomenon, the controller switching instant is

[0083] The system (1) is called the perturbed system, the system corresponding to the system (1) without external disturbances is called the nominal system, and the difference between the perturbed system and the nominal system is called the error system.

[0084] Then Step 2 is executed. In Step 2, a smooth transition controller is designed based on the discrete-time asynchronous switched linear system model.

[0085] According to an embodiment of the present invention, according to Step 1, assume that the perturbed system (1) switches from mode i to mode j at the switching instant k s Then the following smooth transition control structure can be established:

[0086]

[0087] where K and G are the stabilizing controller and the smooth controller respectively, κ is the asynchronous duration, M is the number of smooth controllers, τ is the dwell time of the switched system. The variable subscripts i, j, and ij in the variables respectively mean that the variable is related to mode i, related to mode j, and related to the process of the mode changing from i to j. After the system mode changes, due to the existence of the asynchronous phenomenon, the stabilizing controller K i of the previous mode i will continue to operate for κ ij , and then it will turn to the smooth control stage, adopt M ij smooth controllers G, and then adopt the stabilizing controller K j of mode j. The controller structure is as Figure 1 shown.

[0088] Then, perform Step 3. In Step 3, solve the smooth transition controller to obtain a control input that is stable in the sense of invariant set. Here, stability means convergence to 0, and stability in the sense of invariant set means convergence to a set, which is called an invariant set. The solution process includes:

[0089] Step 3-1: Establish a mathematical description of the smooth transition control problem for an asynchronous switching system subject to amplitude-bounded disturbances.

[0090] Step 3-2: Establish the global uniform asymptotic stability conditions for a closed-loop nominal system with a smooth transition control structure. Among them, the closed-loop nominal system is a discrete-time asynchronous switching linear system without external disturbances.

[0091] Step 3-3: Solve the robust invariant set for a discrete-time asynchronous switching linear system with a smooth transition control structure.

[0092] Step 3-4: Give a sufficient condition to ensure the smooth transition control performance.

[0093] According to the embodiments of the present invention, first, in Step 3-1, establish a mathematical description of the smooth transition control problem for an asynchronous switching system subject to amplitude-bounded disturbances.

[0094] According to Steps 1 and 2, the smooth transition control can be described as the following problem: for the perturbed system (1), find a set of smooth controllers and stabilizing controllers such that the jitter of the system state and control input of the perturbed system (1) can be suppressed by the control structure (2).

[0095] For the subsequent steps, the following explanations are made on the mathematical relationships among the perturbed system, the nominal system, and the error system:

[0096] Define the nominal system corresponding to the perturbed system (1) as

[0097]

[0098] where and are the state and control input of the nominal system, respectively. Then the error system is

[0099]

[0100] where Assume that there is no error in the system at the initial time k0, then there is

[0101]

[0102] Then Therefore, it is only necessary to ensure the stability of the nominal system (3) and find the error set Ek , it can ensure the stability of the perturbed system in the sense of invariant set. Therefore, the problem of smooth transition control for asynchronous switched systems subject to amplitude-bounded disturbances can be transformed into: deriving the stability conditions of the nominal system, the stability conditions of the perturbed system in the sense of invariant set, and the performance constraints of smooth transition control.

[0103] Define the smooth transition control performance of the system state as ||x k+1 - x k || ≤ δ r ||x k ||, and the smooth transition control performance of the system control input as ||u k+1 - u k || ≤ ε r ||x k ||.

[0104] Then, in Step 3.2, establish the globally uniformly asymptotically stable conditions of the closed-loop nominal system with a smooth transition control structure.

[0105] Use the Lyapunov method to describe the stability of the system. The energy function of the system can be expressed as:

[0106]

[0107] Theorem 1: For the asynchronous switched discrete-time nominal system (3), α > 1, β l > 1, 0 < μ < 1 are given constants, and satisfy Suppose there exists a family of matrices R i and as well as a family of positive definite matrices S i and such that for there is

[0108]

[0109] where denotes the matrix [A, C; C T , B], denotes the set denotes the set {X, Y, Z, θ} take values from {S i , S i , R i , α}, {S j , S j , R j , μ}, and denote Then the nominal system (3) is globally uniformly asymptotically stable under the control structure (2), and

[0110] Proof: Take the case where k s ≤ k < k s + κ ij - 1 as an example. At this time, {X, Y, Z, θ} takes {S i , S i , R i , α}. Then, from (6) and by using the Schur complement theorem, we can obtain

[0111] Y((A i S i + B i R i ), S i ) - αS i ≤ 0 (7) where Y(A, B):=A T B -1 A. At the same time, multiply (7) on the left and right by According to the controller structure (2), we get

[0112] Y((A i + B i K i ), P i -1 ) - αP i ≤ 0

[0113] According to the energy function (5) of the system, we can obtain

[0114]

[0115] Similarly, let We can obtain

[0116]

[0117] For There are

[0118]

[0119] There must exist two types of functions and such that

[0120]

[0121] Then we can get

[0122]

[0123] That is Let s:=s(k). Then s(k) is an increasing function of k. Then is Class function, we can get

[0124]

[0125] where is class function. Therefore, the nominal system (3) is globally uniformly asymptotically stable. Q.E.D.

[0126] Then, in Step 3, for the discrete-time asynchronous switched linear system with a smooth transition control structure, solve for the robust invariant set.

[0127] Represent the switching sequence of length k that satisfies the dwell time τ as where (there exists r j = r j+1 (it is possible)), and satisfies The above switching sequence is divided into stages, which facilitates subsequent analysis and can represent all switching sequences with a length not exceeding (p + 1)(2τ - 1). At this time, the corresponding sequence of closed-loop system matrices is as follows:

[0128]

[0129] where

[0130]

[0131] For the error system (4), under zero initial conditions, given the set Let be the set of all possible system states evolved one step backward from E. Then the set of system states evolved q steps backward (i.e., one stage) is

[0132]

[0133] where

[0134]

[0135] That is Therefore, for the perturbed system (1), we have

[0136] Theorem 2: For the perturbed switched system (1), given a non-empty set Then E is the robust invariant set of the corresponding error system (4) if and only if

[0137] Proof: (i) Sufficiency: If Then let It can be obtained that

[0138]

[0139] That is, for there is always e k ∈E. When k < τ, the dwell-time switching condition is not satisfied. Therefore, E is a robust invariant set of the error system (4).

[0140] (ii) Necessity: If E is a robust invariant set of the corresponding error system (4), then for there is always e k ∈E. Here, we use proof by contradiction. Suppose there exist and such that holds, which means that there is at least one element in that does not belong to the set E. However, {j′} q′ (with i′ as the previous mode) itself is indeed a switching sequence that satisfies the conditions. According to the concept of the robust invariant set, any element in the set should belong to the set E, which leads to a contradiction. Therefore, the assumption is not valid, that is, for there is always Q.E.D.

[0141] Theorem 2 provides a way to find the robust invariant set. Assume that E ∞ = lim k→∞ E k exists. By replacing the mode-dependent subscript k with the stage-dependent subscript p, it can be obtained that and That is Performing the convex operation on the above process, denoted as then there is and Therefore is a robust invariant set of the error system (4). Algorithm 1 given below can calculate and

[0142] Algorithm 1:

[0143] 1) Initialize p = 0,

[0144] 2) Loop for τ ≤ q ≤ 2τ - 1, 1 ≤ j ≤ N, 1 ≤ i ≤ N, and calculate

[0145] 3) If then obtain Stop the calculation. Otherwise, let p = p + 1 and repeat step 2).

[0146] Algorithm 1 attempts to calculate through an iterative method but cannot guarantee its existence. To make Algorithm 1 meaningful, the following theorem is also needed to ensure its existence.

[0147] Theorem 3: For the perturbed system (1), α > 1, β l > 1, 0 < μ < 1 are given constants, and satisfy Suppose there exists a family of matrices R i and as well as a family of positive definite matrices S i and such that for Equation (6) holds for all, then must exist.

[0148] Proof: Let For the nominal system (3), we have

[0149]

[0150] From Theorem 1, we can obtain is a compact set,[[]] is a compact set and a convex set. Therefore, there exist constants γ, a spherical set such that Then we have

[0151]

[0152] So exists. Q.E.D.

[0153] Therefore, for the perturbed system (1), we have (p - 1)(2τ - 1)+1 ≤ k ≤ p(2τ - 1), p ≥ 1.

[0154] Then, in steps three and four, sufficient conditions for ensuring the smooth transition control performance are given.

[0155] Theorem 4: For the perturbed system (1), given ν1 > 0, ν2 > 0, ν3 > 0, if there exist scalars ε r > 0, δ r > 0, a matrix R i , a positive definite matrix S i , such that for all have

[0156]

[0157] where {X, Y, Z, W} are respectively selected from {S i , S i , R i , R i}, {S j , S j , R j , R j}, and let Then the perturbed system (1) has the smooth transition control performance of the system state and the control input and

[0158] Proof: Take the case where k s ≤ k < k s + κ ij - 1 as an example. At this time, {X, Y, Z, W} takes {S i , S i , R i , R i}. Then, applying Lemma 1, let X0 = S i , Y0 = ε, Z0 = ρ2I. From (10), we can obtain

[0159]

[0160] According to the Schur complement theorem, we can get

[0161]

[0162] Multiply the left and right sides of equation (11) by Applying the Schur complement theorem again, we can get

[0163]

[0164] That is

[0165]

[0166] Similarly, from Lemma 1 and equation (9), we can obtain

[0167]

[0168] Multiply the left and right sides of equation (13) by Applying the Schur complement theorem, we can get

[0169]

[0170] Multiply the left sides of equations (12) and (14) by and right multiply by where k s ≤k<k s +T ij -1, we can obtain

[0171]

[0172] From (1) and (2), we know that

[0173]

[0174] Similarly, equations (15) and (16) hold for all. According to the definition of the smooth transition control performance, the perturbed system (1) has the smooth performance of the system state and control input and Proof completed.

[0175] Combining Theorems 1 - 4 above, we can solve the smooth transition control problem of the asynchronous switching system subject to amplitude - bounded disturbances, obtain the corresponding smooth controller and stabilizing controller, and ensure the stability and smooth transition control performance of the system in the sense of invariant set.

[0176] Finally, execute Step 4. In Step 4, substitute the obtained control input into the smooth transition controller, and use this smooth transition controller to control the smooth asynchronous switching of the discrete - time asynchronous switching linear system with amplitude - bounded disturbances.

[0177] Further, verify the technical effects of the present invention through experiments.

[0178] According to Figure 2 the shown process to conduct experiments. Taking the switching system with two modes as an example, Figure 3 the response diagrams of the system state and control input of the switching system (1) before and after using the TBBTC method proposed in the present invention are given. It can be seen that the jitters of both the system state and control input are significantly reduced. Figure 3 In the above figure, the abscissa is the sampling time, the ordinate x1 and the vertical coordinate x2 are the state components of the system respectively; in the following figure, the abscissa is the sampling time, and the ordinate u is the control input of the system. From Figure 3 it can be seen that for the given discrete - time asynchronous switching linear system with amplitude - bounded disturbances, after giving appropriate parameters, the corresponding smooth controller and stabilizing controller can be generated to control the system. Compared with not adopting the proposed smooth transition control measures, both the system state and control input of the present invention have achieved significant smoothing.

[0179] Figure 4The system state trajectories and control inputs under 100 sets of randomly bounded disturbances are given. In the upper figure, the abscissa is the sampling time, the ordinate x1 and the vertical coordinate x2 are the state components of the system respectively; in the lower figure, the abscissa is the sampling time and the ordinate u is the control input of the system. From Figure 4 It can be seen that the system state trajectories and control inputs are always restricted within a certain range, achieving stability in the sense of invariant set.

[0180] Figure 5 The jitter situations of the system state and control input when 100 sets of randomly bounded disturbances are applied to the system with different numbers of smoothing controllers. From left to right are the cases where the number M of smoothing controllers is 2, 3, 4, and 5 respectively. The abscissa is the sampling time for each figure. From top to bottom in each figure are the jitter situations of the system control input and the system state respectively. From Figure 5 it can be seen that the jitter situations of the system state and control input are both restricted within a certain threshold, and the corresponding upper bound of jitter decreases with the increase in the number of smoothing controllers, achieving smooth transition control of the system.

[0181] Another embodiment of the present invention proposes a smooth asynchronous switching control system for bounded-amplitude disturbances, and the system includes:

[0182] A system model establishment module configured to establish a discrete-time asynchronous switching linear system model including bounded-amplitude disturbances;

[0183] A controller design module configured to design a smooth transition controller based on the discrete-time asynchronous switching linear system model; the smooth transition controller includes a stabilizing controller and a smoothing controller;

[0184] A control quantity solving module configured to solve the smooth transition controller to obtain a control input that satisfies stability in the sense of invariant set;

[0185] A control module configured to substitute the solved control input into the smooth transition controller and use the smooth transition controller to control the smooth asynchronous switching of the discrete-time asynchronous switching linear system including bounded-amplitude disturbances.

[0186] The functions of the smooth asynchronous switching control system for bounded-amplitude disturbances described in this embodiment can be illustrated by the aforementioned smooth asynchronous switching control method for bounded-amplitude disturbances. Therefore, for the parts not detailed in this embodiment, reference can be made to the above method embodiments and will not be elaborated here.

[0187] Although the present invention has been described in terms of a limited number of embodiments, those skilled in the art, having the benefit of the foregoing description, will appreciate that other embodiments can be contemplated within the scope of the invention as thus described. The disclosure of the present invention is illustrative, not restrictive, of the scope of the invention, which is defined by the appended claims.

Claims

1. A smooth asynchronous switching control method for amplitude bounded interference, characterized in that: include: A discrete-time asynchronously switched linear system model with bounded amplitude disturbances is established; Designing a smooth transition controller based on the discrete-time asynchronous switching linear system model; The smooth transition controller includes a stabilization controller and a smoothing controller; The smooth transition controller is solved to obtain a stable control input that satisfies the invariant set; including: establishing a mathematical description of the smooth transition control problem of an asynchronous switching system with bounded amplitude interference; establishing a globally consistent asymptotic stability condition for a closed-loop nominal system with a smooth transition control structure; wherein the closed-loop nominal system is a discrete-time asynchronous switching linear system without external interference; solving a robust invariant set for the discrete-time asynchronous switching linear system with a smooth transition control structure; and providing sufficient conditions to ensure smooth transition control performance as follows: Theorem 4: For a discrete-time asynchronously switched linear system subject to bounded-amplitude disturbance, given If there is a scalar ε r >0,δ r >0, matrix R i , Positive definite matrix S i , Make for Both Among them, {X,Y,Z,W} are respectively derived from {S i ,S i ,R i ,R i }, {S j ,S j ,R j ,R j }, and remember Then the discrete-time asynchronously switched linear system with bounded amplitude disturbance has the smooth transition control performance of system state and control input and The obtained control input is substituted into a smooth transition controller, which is used to control the smooth asynchronous switching of a discrete-time asynchronously switched linear system with bounded amplitude disturbance.

2. A method for smooth asynchronous switching control for amplitude-bounded interference according to claim 1, characterized in that: The discrete-time asynchronous switching linear system model containing amplitude-bounded interference is established as follows: Among them, x k represents the system state vector at time k; u k represents the control input at time k; w k represents the disturbance input with bounded amplitude; σ k represents the asynchronous switching signal at time k corresponding to N modes; They represent the system matrix, input matrix and interference matrix respectively.

3. A method for smooth asynchronous switching control for amplitude-bounded interference according to claim 2, characterized in that: The smooth transition controller is designed as follows: Among them, k s represents the switching time, κ ij K represents the asynchronous duration of switching from mode i to mode j; i represents the stabilizing controller corresponding to mode i; represents the set of the number of smoothing controllers corresponding to the switch from mode i to mode j, l is the label of the smoothing controller; K j represents the stabilizing controller corresponding to mode j; Indicates the switch from mode i to mode j of the lth smoothing controller.

4. A method for smooth asynchronous switching control for amplitude-bounded interference according to claim 3, characterized in that: The mathematical description of the smooth transition control problem of the asynchronous switching system subject to bounded amplitude disturbance is established as follows: For a discrete-time asynchronously switched linear system subject to bounded amplitude disturbances, a set of smoothing controllers and stabilizing controllers are found such that jitters of the system state and control inputs of the system can be suppressed by the smooth transition controller structure; The nominal system corresponding to the discrete-time asynchronous switching linear system with bounded amplitude disturbance is defined as: in, and denote the state vector and control input of the nominal system respectively; The error system is: in, represents the closed-loop system matrix; Assuming that the system has no error at the initial time k0, we have: but Ensure that the nominal system is stable and find the error set ε k , which can ensure the stability of discrete-time asynchronously switched linear systems subject to bounded amplitude disturbances in the sense of invariant sets; Define the smooth transition control performance of the system state δ r Satisfies: ||x k+1 -x k ||≤δ r ||x k ||, smooth transition control performance of system control input ε r Satisfy: ||u k+1 -u k ||≤ε r ||x k ||.

5. A method for smooth asynchronous switching control for amplitude-bounded interference according to claim 4, characterized in that: The globally consistent asymptotic stability condition of the closed-loop nominal system with smooth transition control structure is established as follows: Theorem 1: For the discrete-time asynchronously switched linear system subject to bounded amplitude disturbances, the corresponding nominal system is: is a given constant and satisfies Assume that there is a family of matrices R i and And a family of positive definite matrices S i and Make for have: in, represents the matrix [A, C; C T ,B], Representing a collection Representing a collection {X,Y,Z,θ} are respectively derived from {S i ,S i ,R i ,α}, {S j ,S j ,R j ,μ}, and record Then the nominal system corresponding to the discrete-time asynchronously switched linear system with bounded amplitude disturbance is globally uniformly asymptotically stable under the smooth transition controller structure, and 6. A method for smooth asynchronous switching control for amplitude-bounded interference according to claim 5, characterized in that: The method for solving a robust invariant set for a discrete-time asynchronous switching linear system with a smooth transition control structure comprises: The switching sequence of length k and the dwell time τ is expressed as in And meet The corresponding closed-loop system matrix sequence is: in: For the error system, under zero initial conditions, given the set make is the set of all possible system states that evolve one step backward from ε, then the set of system states that evolve q steps backward is: in, Right now Therefore, for a discrete-time asynchronously switched linear system subject to a disturbance of bounded amplitude, we have Theorem 2: For a discrete-time asynchronously switched linear system subject to a bounded-amplitude disturbance, given a nonempty set Then the necessary and sufficient condition for ε to be the robust invariant set of the corresponding error system is Assume ε ∞ =lim k→∞ ε k Exists, replace the modal dependency subscript k with the phase dependency subscript p, and we get and Right now The above process is subjected to convex operation, which is recorded as Then there is and therefore is a robust invariant set of the error system; according to Theorem 2, the following Algorithm 1 is used to calculate and 1) Initialization 2) Loop τ≤q≤2τ-1,1≤j≤N,1≤i≤N, calculate 3) If Then Stop calculation; otherwise set p=p+1 and repeat step 2).

7. A method for smooth asynchronous switching control for amplitude-bounded interference according to claim 6, characterized in that: The robust invariant set of the error system is determined in advance using the following Theorem 3 Does it exist: Theorem 3: For a discrete-time asynchronously switched linear system subject to bounded amplitude disturbance, α>1, β l >1,0<μ<1 is a given constant and satisfies Assume that there is a family of matrices R i and And a family of positive definite matrices S i and Make for The following equations are all true, then Must exist:

8. A smooth asynchronous switching control system for amplitude bounded disturbances, characterized in that: include: a system model building module configured to build a discrete-time asynchronously switched linear system model including a disturbance with bounded amplitude; A controller design module, configured to design a smooth transition controller based on the discrete-time asynchronous switching linear system model; the smooth transition controller includes a stabilization controller and a smoothing controller; A control quantity solving module is configured to solve the smooth transition controller to obtain a stable control input that satisfies the invariant set; including: establishing a mathematical description of the smooth transition control problem of an asynchronous switching system with bounded amplitude interference; establishing a globally consistent asymptotic stability condition for a closed-loop nominal system with a smooth transition control structure; wherein the closed-loop nominal system is a discrete-time asynchronous switching linear system without external interference; solving a robust invariant set for the discrete-time asynchronous switching linear system with a smooth transition control structure; and providing sufficient conditions to ensure smooth transition control performance as follows: Theorem 4: For a discrete-time asynchronously switched linear system subject to bounded-amplitude disturbance, given If there is a scalar ε r >0,δ r >0, matrix R i , Positive definite matrix S i , Make for Both Among them, {X,Y,Z,W} are respectively derived from {S i ,S i ,R i ,R i }, {S j ,S j ,R j ,R j }, and remember Then the discrete-time asynchronously switched linear system with bounded amplitude disturbance has the smooth transition control performance of system state and control input and The control module is configured to substitute the control input obtained by solving the problem into a smooth transition controller, and utilize the smooth transition controller to control the smooth asynchronous switching of a discrete-time asynchronous switching linear system containing amplitude-bounded disturbances.

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