A method for solving maximum allowed sampling interval based on integral combination closed loop function

Abstract

The application discloses a method for solving the maximum allowed sampling interval of a sampling control system based on integral combination closed loop function, and first establishes a sampling control system model. Secondly, the integral combination closed loop function is designed by using the integral inequality of the sampling interval. The function relaxes the restrictions of continuity, positive definiteness and monotone non-decreasing of the sampling point. Finally, according to the closed loop function theorem, the system stability criterion and the method for calculating the maximum allowed sampling interval are given. Compared with other existing methods, the analysis method of the application can obtain a lower conservative stability condition and a larger allowed sampling interval, which is beneficial to improve the anti-interference ability of the system and the utilization rate of the controller and reduce the design cost of the sampling control system.

Description

Technical Field

[0001] This invention relates to a method for solving the maximum permissible sampling interval based on an integral combination closed-loop function, and particularly to a method for solving the maximum permissible sampling interval of a sampled control system based on an integral combination closed-loop function, belonging to the technical field of sampled control systems. Background Technology

[0002] In recent years, with the rapid development of communication and digital technologies, the research on sampled-sample control systems has received widespread attention. Compared with continuous-time control systems, sampled-sample control relies solely on system information at discrete sampling times, thus offering advantages such as high efficiency and high reliability. A key issue in the research of sampled-sample control systems is estimating the maximum permissible sampling interval while ensuring the desired system performance, which is crucial for reducing the communication burden in networks. To address this problem, several analytical methods based on Lyapunov stability theory have been proposed, such as the input-output method, the discrete-time system method, and the input time-delay method.

[0003] To obtain a larger feasible sampling interval, numerous studies have focused on constructing different Lyapunov-like functions to reduce the conservatism of stability conditions. For example, technical literature 1 (A. Seuret and F. Gouaisbaut. Wirtinger-based integral inequality: Application to time-delay systems, Automatica, 2013, 49(9):2860–2866.) proposes a closed-loop function stability analysis method for linear sampling systems. Considering the boundary information of the sampling interval, technical literature 2 (H. Zeng, K. Teo, and Y. He. A new looped-functional for stability analysis of sampled-data systems, Automatica, 2017, 82:325–331.) further presents a method for constructing a two-sided closed-loop function. Compared to traditional Lyapunov functions, the closed-loop function does not need to consider positive definite constraints, which is the main reason for reducing conservatism. In addition, technical document 3 (T. Lee and J. Park. Stability analysis of sampled-data systems via free-matrix-based time-dependent discontinuous Lyapunov approach, IEEE Transactions on Automatic Control, 2017, 62(7):3653–3657.) constructs a Lyapunov function that is discontinuous at discrete time points. This function retains the positive definite constraint of the traditional Lyapunov function, but relaxes the requirement for function continuity. Technical document 4 (Z. Sheng, C. Lin, B. Chen, and Q. Wang. Stability analysis of sampled-data systems via novel Lyapunov functional method, Information Science, 2022, 585:559–570.) introduces multiple integral terms to construct an augmented discontinuous Lyapunov function.Recently, technical document 5 (J. Park and P. Park. An extended looped-functional forstability analysis of sampled-data systems, International Journal of Robustand Nonlinear Control, 2020, 30(18): 7962–7969.) presents an extended closed-loop function that relaxes the requirement for the closed-loop function to be continuous at the sampling points, as well as the positive definite constraint on the discontinuous Lyapunov function.

[0004] While the aforementioned technical document 5 has achieved significant results in reducing the conservatism of the stability criterion, the limitation of the single extended closed-loop function being monotonically non-decreasing at the sampling points, and the loss of integral information during the construction of the function, still introduce a certain degree of conservatism, thereby increasing the cost of control system design. Therefore, constructing a more relaxed Lyapunov-like function, obtaining a lower conservatism stability condition, and solving for the maximum permissible sampling interval have become a pressing technical challenge in this field. Summary of the Invention

[0005] To address the aforementioned problems, this invention proposes a method for solving the maximum permissible sampling interval of a sampled control system based on an integral combination closed-loop function. This method can reduce the conservatism of stability analysis of the sampled system and estimate the maximum permissible sampling interval more accurately.

[0006] The computational solution of this invention is:

[0007] A method for solving the maximum permissible sampling interval of a sampled control system based on an integral combination closed-loop function, the method comprising the following steps:

[0008] S100, Establish a sampling control system model;

[0009] S200, establish the integral combination closed-loop function;

[0010] S300 uses the established integral combination closed-loop function to obtain the stability conditions of the sampled control system;

[0011] S400, calculates the maximum permissible sampling interval of the sampling control system under stable conditions.

[0012] The sampling control system model established in step S100 is as follows:

[0013]

[0014] Where, t∈[t k ,t k+1(k = 0, 1, ...), Indicates the status of the sampling control system, A c A s For a fixed system matrix, t k+1 -t k =h k It is the sampling interval, and satisfies t k Let t represent the k-th sampling time. k+1 This represents the (k+1)th sampling time. h This represents the minimum sampling interval. Indicates the maximum value of the sampling interval;

[0015] In step S200, the established integral combination closed-loop function is:

[0016]

[0017] in,

[0018] d1(t)=tt k

[0019] d2(t)=t k+1 -t

[0020]

[0021]

[0022]

[0023]

[0024] matrix G ij H ij (i = 1, 2, j = 1, 2, 3) satisfies in

[0025]

[0026] 's' represents time;

[0027]

[0028]

[0029]

[0030]

[0031] θ1=[I n -I n 0n 0 n ]

[0032] θ2=[I n n 2I n 0 n ]

[0033] θ3=[I n -I n -6I n 12I n ]

[0034] In step S300, when the sampling control system is stable:

[0035] Integral combination closed-loop function The derivative satisfies:

[0036]

[0037] in,

[0038] p≥1, c2>c1>0

[0039]

[0040] The stability condition of the sampling control system is:

[0041] Existence matrix G ij H ij M j N j (i = 1, 2, j = 1, 2, 3) and all satisfy The following linear matrix inequalities hold:

[0042]

[0043] in,

[0044]

[0045]

[0046]

[0047]

[0048]

[0049]

[0050] Γ=A c L1+A s L2

[0051]

[0052]

[0053]

[0054]

[0055]

[0056]

[0057]

[0058]

[0059]

[0060]

[0061]

[0062] L0=0 n×7n

[0063] L1=[I n 0 n×6n ]

[0064] L2=[0 n×n n 0 n×5n ]

[0065] L3=[0 n×2n I n 0 n×4n ]

[0066] L4=[0 n×3n I n 0 n×3n ]

[0067] L5=[0 n×4n I n 0 n×2n ]

[0068] L6=[0 n×5n I n 0 n×n ]

[0069] L7=[0n×6n I n ]

[0070] In step S400, the maximum allowable sampling interval of the sampling control system is the upper limit of the sampling interval of the sampling control system obtained under stable conditions.

[0071] Beneficial effects

[0072] This invention proposes a method for solving the maximum permissible sampling interval of a sampled-sampled control system based on an integral combination closed-loop function. The combined closed-loop function relaxes the restrictions of continuity, positive definiteness, and monotonically non-decreasing sampling points, and introduces more sampling-related information in the stability analysis by constructing a combined integral term. Based on this closed-loop function method, a stability condition with low conservatism is derived. Furthermore, the maximum permissible sampling interval of the system is calculated, reducing the cost of control system design.

[0073] This invention discloses a method for solving the maximum permissible sampling interval of a sampled-sampling control system based on an integral combination closed-loop function. This method specifically includes: first, establishing a model of the sampled-sampling control system; second, designing an integral combination closed-loop function using the sampling interval integral inequality. This function relaxes the restrictions of continuity, positive definiteness, and monotonically non-decreasing sampling points; finally, based on the closed-loop function theorem, providing a system stability criterion and a method for calculating the maximum permissible sampling interval. Compared to other existing methods, the analytical method proposed in this patent can obtain a lower conservative stability condition and a larger permissible sampling interval, which is beneficial for improving the system's anti-interference capability and controller utilization, and reducing the design cost of the sampled-sampling control system. Attached Figure Description

[0074] Figure 1 This invention provides a method for solving the maximum allowable sampling interval of a sampled control system based on an integral combination closed-loop function;

[0075] Figure 2 This is the state response of the sampling control system in one embodiment of the present invention. Detailed Implementation

[0076] The embodiments of the present invention will be described in detail below with reference to the accompanying drawings and examples.

[0077] In the design of sampled-sample control systems, obtaining a larger allowable sampling interval is beneficial to improving controller utilization, thereby reducing system design costs. To derive a low-conservatism stability criterion and more accurately estimate the upper bound of the system's sampling interval, this paper presents an integral combination closed-loop function method. This method relaxes the restrictions of continuity, positive definiteness, and monotonically non-decreasing sampling points, and introduces integral information over the sampling interval, thus deriving a low-conservatism stability condition.

[0078] The following symbols: Represents n-dimensional Euclidean space; This represents the transpose of matrix P; The expression indicates that matrix p is positive definite (semi-positive definite) and symmetric; diag{…} denotes a block diagonal matrix; Sym{P} denotes… ||·|| denotes the Euclidean vector norm; Indicates t k+1 The left limit.

[0079] To simplify the description, this example defines the following vectors and matrices:

[0080] d1(t) := tt k d2(t) := t k+1 -t

[0081]

[0082]

[0083]

[0084]

[0085]

[0086]

[0087] like Figure 1 The diagram shown illustrates the process of this invention, a method for solving the maximum allowable sampling interval of a sampled control system based on an integral combination closed-loop function, comprising the following steps:

[0088] S100, Establish a sampling control system model;

[0089] In one embodiment, the sampling control system model described in step S100 is:

[0090]

[0091] Where, t∈[t k ,t k+1 (k = 0, 1, ...), Indicates the status of the sampling control system, A c A s For a fixed system matrix, t k+1 -t k =h k It is the sampling interval, and satisfies t k Let t represent the k-th sampling time. k+1 This represents the (k+1)th sampling time. hThis represents the minimum sampling interval. This indicates the maximum sampling interval.

[0092] S200, Derive the combinatorial closed-loop function theorem;

[0093] In one embodiment, the derivation process of the combined closed-loop function theorem described in step S200 is as follows:

[0094] S211, the stability theorem for closed-loop functions states: Assume there exist scalars p ≥ 1, c2 > c1 > 0, such that a given function satisfies Then, given a closed-loop function satisfy Simultaneously satisfy

[0095]

[0096] The system state described in sampling step S100 is asymptotically stable.

[0097] S212, Design the combined closed-loop function as follows:

[0098] Given two differentiable functions that satisfy the following conditions:

[0099]

[0100]

[0101] Therefore, the combination function It is a closed-loop function.

[0102] S213. Based on the closed-loop function theorem in S211 and the combined closed-loop function in S212, the combined closed-loop function theorem is derived as follows:

[0103] Consider scalars p ≥ 1, c2 > c1 > 0, and functions It satisfies:

[0104]

[0105] If a differentiable function exists The following conditions must be met:

[0106]

[0107]

[0108] At the same time, the combination function The derivative satisfies:

[0109]

[0110] Therefore, the system state described in sampling step S100 is asymptotically stable.

[0111] S300, design an integral combination closed-loop function;

[0112] In one embodiment, the design steps for the integral combination closed-loop function described in step S300 are as follows:

[0113] S311, define functions respectively and for:

[0114]

[0115]

[0116] in,

[0117]

[0118]

[0119] f1(s,t): = 2s - tt k g1(s,t) := 2s-t k+1 -t

[0120] f2(s,t): = 6(st k (st)+(tt) k ) 2

[0121] g2(s,t):=6(st)(st k+1 )+(t k+1 -t) 2

[0122]

[0123] S312, when When, the estimated value of the function is:

[0124]

[0125] in,

[0126] S313, Based on S311 and S312, design the integral combination closed-loop function as follows:

[0127] Given If a matrix exists G ij H ij (i = 1, 2, j = 1, 2, 3) such that Therefore, the following integral combination closed-loop function is constructed:

[0128]

[0129] S400, obtain the stable conditions of the sampling system;

[0130] In one embodiment, the derivation steps for the sampling system stability condition in step S400 are as follows:

[0131] S411, the constructor V(x,t) is:

[0132]

[0133] in,

[0134]

[0135]

[0136]

[0137] S412, the derivative of the function V(x,t) is:

[0138]

[0139]

[0140]

[0141] in,

[0142]

[0143]

[0144]

[0145]

[0146] L i :=[0 n×(i-1)n I n 0 n×(7-i)n ], i = 1, ..., 7, L0: = 0 n×7n

[0147] S413, by defining the terms of the integral inequality, we obtain:

[0148]

[0149] Among them, M j N j(j=1,2,3) is an arbitrary matrix of suitable dimensions, and

[0150]

[0151] θ2:=[I n I n 2I n 0 n ],θ3:=[I n -I n -6I n 12I n ]

[0152]

[0153] S414, construct the zero equation as follows:

[0154]

[0155]

[0156]

[0157]

[0158] S415, Based on S411, S412, S413, and S414, the estimated value of the derivative of the function V(x,t) is:

[0159]

[0160] in,

[0161]

[0162]

[0163]

[0164]

[0165]

[0166]

[0167]

[0168]

[0169]

[0170] S416, based on S415 and Schur's complement lemma, the stability conditions for the sampled-sample control system are obtained as follows:

[0171] For a given If a matrix exists G ij H ij M j N j (i = 1, 2, j = 1, 2, 3) and all satisfy and the following linear matrix inequalities

[0172]

[0173] Therefore, the system state described in sampling step S100 is asymptotically stable.

[0174] S500, calculates the maximum allowable sampling interval;

[0175] In one embodiment, the maximum allowable sampling interval in step S500 is calculated from the stability condition in S400.

[0176] In one embodiment, the effectiveness and superiority of the criterion obtained in this invention are verified through simulation experiments.

[0177] Consider the system parameter matrix as follows:

[0178]

[0179] Set the lower bound of the sampling interval to h =10 -5 Based on the linear matrix inequality in step S416, the upper bound of the maximum sampling interval that guarantees system stability is calculated as follows: h =9.85.

[0180] Set the system initial state and the sampling interval h∈[10 -5 [9.85], obtain the state trajectory diagram of the system in time t∈[0,150], x(t)=[x1(t),x2(t)] T ,like Figure 2 As shown, the system state gradually approaches zero over time, demonstrating the effectiveness of the method of this invention.

[0181] Table 1 lists the lower bound of the sampling interval. h =10 -5 At that time, the upper bound of the sampling interval obtained by this example method and other methods h In this embodiment h=9.85;

[0182] Table 1 shows the lower bound of the sampling interval. h =10 -5 Maximum allowable sampling upper bound h

[0183]

[0184] The comparative examples shown in Table 1 are from technical documents 1, 2, 3, 4, and 5 in the background section. As can be seen from the table, the results obtained in this embodiment are significantly greater than those of other methods; that is, a larger sampling interval can be obtained under stable system conditions, demonstrating the superiority of the method of this invention.

[0185] The above description is merely a preferred embodiment of the present invention, and the present invention includes, but is not limited to, the content disclosed in this embodiment and the accompanying drawings. Any equivalent or modified versions made without departing from the spirit of the present invention fall within the scope of protection of the present invention.

Claims

1. A method for solving the maximum permissible sampling interval of a sampled control system based on an integral combination closed-loop function, characterized in that... The steps of this method include: S100, Establish a sampling control system model; S200, establish the integral combination closed-loop function; S300 uses the established integral combination closed-loop function to obtain the stability conditions of the sampled control system; S400, calculates the maximum permissible sampling interval of the sampling control system under steady-state conditions; The sampling control system model established in S100 is as follows: in, , Indicates the status of the sampling control system. For a fixed system matrix, It is the sampling interval, and satisfies ; This indicates the k-th sampling time. This represents the (k+1)th sampling time. This represents the minimum sampling interval. Indicates the maximum value of the sampling interval; The integral combination closed-loop function established in S200 is as follows: in, symbol This operation represents the addition of a square matrix and its transpose. satisfy , ,in Indicates time; 。 2. The method for solving the maximum permissible sampling interval of a sampled control system based on an integral combination closed-loop function according to claim 1, characterized in that: In S300, when the sampling control system is stable, the integral combination closed-loop function The derivative satisfies: in, ， ， ， matrix , ， Indicates the sampling time The left limit.

3. The method for solving the maximum permissible sampling interval of a sampled control system based on an integral combination closed-loop function according to claim 2, characterized in that: The stability condition of the sampling control system is: Existence matrix , , And all ,satisfy The following linear matrix inequalities hold: in, ; 。 4. The method for solving the maximum permissible sampling interval of a sampled control system based on an integral combination closed-loop function according to claim 3, characterized in that: In S400, the maximum allowable sampling interval of the sampling control system is the upper limit of the sampling interval of the sampling control system obtained under stable conditions.

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