A stability analysis method and system for an exponentially uncertain switched time-delay system

Abstract

This invention discloses a stability analysis method and system for exponentially uncertain switching time-delay systems. A novel state-dependent switching strategy based on the energy decay principle is designed, and the maximum allowable time delay of the system is obtained by solving its stability conditions. First, a state-dependent switching strategy is designed based on the energy decay principle. Second, a functional is constructed, and sufficient conditions for the asymptotic stability of the exponentially uncertain switching time-delay system are given and proved based on the generalized free matrix integral inequality method, Jensen's inequality method, and the negative definite lemma of quadratic functions. Finally, simulation examples verify the effectiveness of the designed switching strategy and the correctness of the obtained stability conditions. The switching strategy and stability conditions designed in this invention are applicable to control systems with complex switching dynamics, state-space delay dynamics, and uncertain disturbances, and have significant theoretical and practical application value.

Description

Technical Field

[0001] This invention belongs to the field of complex system modeling and control, and specifically relates to a stability analysis method and system for exponentially uncertain switching time delay systems. Background Technology

[0002] Switched systems, as a special type of hybrid system, consist of a series of continuous or discrete subsystems and switching strategies that govern the operation of these subsystems. With the rapid development of information technology and complexity science, time delays and uncertainties have made switched dynamic control systems increasingly complex.

[0003] In recent years, switched-time-delay systems have attracted considerable attention from scholars due to their wide range of applications, such as network control systems and power electronic systems. A switched-time-delay system refers to a system in which a time delay exists within a subsystem of a switched system. Switched-time-delay systems differ from general switched dynamic systems because they simultaneously exhibit switching dynamics and state-space delay dynamics. Therefore, designing appropriate switching strategies and reducing the impact of time delays on the system to maintain good stability are key research areas for switched-time-delay systems. On the one hand, most current research on switched-time-delay systems assumes sufficient conditions for system stability given the known magnitude of the time delay, while few studies address the maximum allowable time delay, i.e., the upper bound of the time delay. Only a few studies have found the upper bound, but their stability or stabilization conditions are highly conservative, meaning the maximum allowable time delay is relatively small. On the other hand, existing research on switched dynamic systems also lacks consideration of both uncertainty and time delay, which are prevalent in real-world systems. Therefore, research on switched-time-delay systems with uncertainty is of great significance.

[0004] Compared to time-delay-free and uncertainty-free switching dynamic systems, uncertain switching time-delay systems not only have complex switching dynamics and state-space delay dynamics, but are also subject to interference from uncertain factors. Therefore, constructing a suitable functional and switching strategy is crucial for the analysis and control of uncertain switching time-delay systems. Currently, there are many well-developed methods for constructing functionals for time-delay systems, such as time-delay partitioned functionals and augmented functionals. Applying these methods to switching time-delay systems can greatly enrich the theory of complex systems. In addition, there are many classic methods for designing switching strategies, such as the residence time method, the average residence time method, and the continuous residence time method. However, these methods have certain limitations. For example, the residence time method and the average residence time method are no longer applicable to some systems whose internal structure or properties may undergo random mutations. In recent years, the continuous residence time method proposed by relevant scholars has incorporated both the residence time method and the average residence time method. However, under this method, the number of switching operations is limited within a finite time interval, and this method is no longer applicable to some complex systems that require frequent switching. Therefore, a novel switching method is needed to obtain sufficient conditions for system stability under the influence of time delay and uncertainty, and to derive the maximum allowable time delay of the system by solving these sufficient conditions. This invention proposes a novel state-dependent switching strategy based on energy decay. Under this switching strategy, by constructing a functional, sufficient conditions for asymptotic stability of the system are given and proved using the generalized free matrix integral inequality method and Jensen's inequality method. The switching strategy and system stability conditions designed in this invention are applicable to control systems that simultaneously possess complex switching dynamics, state-space delay dynamics, and uncertain disturbances, and have significant practical implications for the modeling and control of complex systems. Summary of the Invention

[0005] In view of this, in order to expand the research on switching dynamic systems and address the shortcomings of existing switching methods, this invention proposes a stability analysis method and system for exponentially uncertain switching time-delay systems, and designs a novel state-dependent switching strategy. Under this switching strategy, by constructing a functional, sufficient conditions for the asymptotic stability of the exponentially uncertain switching time-delay system are obtained using the generalized free matrix integral inequality method, Jensen's inequality method, and the negative definite lemma of quadratic functions. By solving these sufficient conditions, the maximum allowable time delay of the system is obtained.

[0006] The present invention solves the above problems through the following technical means:

[0007] In a first aspect, the present invention provides a stability analysis method for an exponentially uncertain switching time-delay system, comprising the following steps:

[0008] Step 1: Give the state-space expression of the exponentially uncertain switching time-delay system;

[0009] Step 2: Based on the state-space expression of the exponentially uncertain switching time-delay system, and combined with the energy decay principle, design a state-dependent switching strategy;

[0010] Step 3: Prove that the exponentially uncertain switching delay system is asymptotically stable under the designed switching strategy;

[0011] Step 4: Verify the effectiveness of the designed switching strategy and the correctness of the system stability conditions through simulation examples.

[0012] Preferably, in step one, the state-space expression of the exponentially uncertain switching time-delay system is given as follows:

[0013]

[0014]

[0015] In the above formula, t is a continuous time variable. Let be the system state vector. Let it be an n-dimensional Euclidean space. Let θ be the initial state vector, and θ be a constant. The switching signal is ω(t); ω(t) is an exponentially uncertain parameter and satisfies... in ω and A is a constant; i (ω(t))=exp{E i ·ω(t)} and Let E be the exponential uncertainty matrix, where E i B i Let s be a constant coefficient matrix of appropriate dimension, where s is a constant. It is a positive integer; It is a continuous time-varying time delay and satisfies Here And d > 0 is a constant;

[0016] Based on the convex combination lemma and the property of exponential matrices Where D is a constant matrix, and λ,j,s are constants; therefore, the above exponentially uncertain switching time delay system can be rewritten in the following form;

[0017]

[0018]

[0019] Here,

[0020]

[0021]

[0022]

[0023] Here, h is a constant, I is the identity matrix, and B... σ =B i E σ =E i Let ω(t) be a constant matrix and... The following relationship exists:

[0024]

[0025] ΔA σ (h,ω(t)) and ΔA dσ (h,ω(t)) is A σ (h,ω(t)) and A dσ The higher-order terms of the Taylor series expansion of (h,ω(t)) satisfy:

[0026]

[0027]

[0028] Preferably, in step two, a state-dependent switching strategy based on energy decay is designed. Under this switching strategy, the system selects the operating mode at the next moment according to the energy value of the subsystem, specifically including the following sub-steps:

[0029] Step 2-1: Consider the following i-th exponentially uncertain switching delay system:

[0030]

[0031] Suppose there exists a The Herwitz convex combination H satisfies the following condition:

[0032]

[0033] Here, 0 < α i <1, Since H is a Herwitz matrix, for a given positive definite matrix M and P, the matrix equation H... T P + PH = -M holds true;

[0034] Step 2-2: When the system exhibits exponential uncertainty, the switching region becomes:

[0035]

[0036] When the state trajectory of an exponentially uncertain switching time-delay system moves to the boundary of the switching region, the system is prone to fast switching.

[0037] Steps 2-3: To avoid fast switching, construct the following switching region:

[0038]

[0039] In the aforementioned switching domain, ξ > 1 is a constant; therefore, the right-hand side of the inequality will increase. This means that more system states will satisfy the above condition, i.e., the switching region will become larger, resulting in overlapping regions. Therefore... In this switching domain, the subsystems of the exponentially uncertain switching delay system are determined by the system's state trajectory. When the state trajectory enters region i, subsystem i is activated and runs.

[0040] Steps 2-4: For systems with uncertain exponential switching time lags, the following switching strategy is proposed:

[0041]

[0042] In the above formula, let This represents the system energy of each exponentially uncertain switching delay system; under this switching strategy, the system switches according to its energy value without needing to construct a switching region for the system state; therefore, the system can avoid the occurrence of fast switching phenomena under this switching strategy.

[0043] Preferably, in step three, a functional containing a triple integral term is constructed, and the time-delay part of the system is handled using the generalized free matrix integral inequality method, Jensen's inequality method, and the negative definite lemma of quadratic functions to prove the asymptotic stability of the exponentially uncertain switching time-delay system; the specific steps are as follows:

[0044] Step 3-1: First, define the following Lyapunov-Krasovsky functional:

[0045] V i (x t ) = V 1i (x t )+V 2i (x t )+V 3i (x t )+V 4i (x t )

[0046] in

[0047] V 1i (x t )=x T (t)Px(t)

[0048]

[0049]

[0050]

[0051] In the above formula, P, Q 1i Q 2i ,S 1i ,S 2i Given a symmetric positive definite real matrix, the i-th subsystem of the exponentially uncertain switching time-delay system is obtained using the Newton-Leibniz formula:

[0052]

[0053] Step 3-2: Differentiating the above functional yields:

[0054]

[0055]

[0056] Using the generalized free matrix integral inequality method and Jensen's inequality method, we obtain and as follows:

[0057]

[0058]

[0059] Step 3-3, obtained from the above:

[0060]

[0061] By applying the negative definite lemma of quadratic functions, we get therefore The exponentially uncertain switching delay system asymptotically stabilizes under the designed switching strategy.

[0062] Preferably, after step three, the following steps are also included:

[0063] The maximum allowable time delay of an exponentially uncertain switching time delay system can be solved using the following linear matrix inequality:

[0064] H T P+PH<0

[0065]

[0066]

[0067]

[0068] in

[0069]

[0070]

[0071]

[0072]

[0073]

[0074]

[0075]

[0076]

[0077]

[0078]

[0079]

[0080] e i =[0 m×(i-1)m I m 0 m×(5-i)m ], E1 = e2 - e5

[0081] E2 = e1 - e4, Ξ i =Ξ 1i +Ξ 2i +Ξ 3i ,

[0082]

[0083] Λ1(0)=[I -I 0], Λ1(1)=[II -2I]

[0084] In the above formula, P, Q 1i Q 2i ,S 1i ,S 2i N is a symmetric positive definite real matrix; 1i N 2i Let be any matrix; the maximum allowable time delay of the exponentially uncertain switching time delay system can be obtained by solving the above linear matrix inequalities.

[0085] In a second aspect, the present invention provides a stability analysis system for an exponentially uncertain switching time-delay system, comprising:

[0086] The expression-giving module is used to give the state-space expression of an exponentially uncertain switching time-delay system;

[0087] The switching strategy design module is used to design state-dependent switching strategies based on the state-space expression of an exponentially uncertain switching time-delay system and the principle of energy decay.

[0088] The asymptotic stability proof module is used to prove the asymptotic stability of the exponentially uncertain switching delay system under the designed switching strategy;

[0089] The correctness verification module is used to verify the effectiveness of the designed switching strategy and the correctness of the system stability conditions through simulation examples.

[0090] Preferably, the state-space expression for the exponentially uncertain switching time-delay system is given as follows:

[0091]

[0092]

[0093] In the above formula, t is a continuous time variable. Let be the system state vector. Let it be an n-dimensional Euclidean space. Let θ be the initial state vector, and θ be a constant. The switching signal is ω(t); ω(t) is an exponentially uncertain parameter and satisfies... in ω and A is a constant; i (ω(t))=exp{E i ·ω(t)} and Let E be the exponential uncertainty matrix, where E i B i Let s be a constant coefficient matrix of appropriate dimension, where s is a constant. It is a positive integer; It is a continuous time-varying time delay and satisfies Here And d > 0 is a constant;

[0094] Based on the convex combination lemma and the property of exponential matrices Where D is a constant matrix, and λ,j,s are constants; therefore, the above exponentially uncertain switching time delay system can be rewritten in the following form;

[0095]

[0096]

[0097] Here,

[0098]

[0099]

[0100]

[0101]

[0102] Here, h is a constant, I is the identity matrix, and B... σ =B i E σ =E i Let ω(t) be a constant matrix and... The following relationship exists:

[0103]

[0104] ΔA σ (h,ω(t)) and ΔA dσ (h,ω(t)) is A σ (h,ω(t)) and A dσ The higher-order terms of the Taylor series expansion of (h,ω(t)) satisfy:

[0105]

[0106]

[0107] Preferably, in designing a state-dependent switching strategy based on the principle of energy decay, a state-dependent switching strategy based on energy decay is designed. Under this switching strategy, the system selects the operating mode at the next moment according to the energy value of the subsystem, specifically including the following sub-steps:

[0108] Step 2-1: Consider the following i-th exponentially uncertain switching delay system:

[0109]

[0110] Suppose there exists a The Herwitz convex combination H satisfies the following condition:

[0111]

[0112] Here 0 < α i <1, Since H is a Herwitz matrix, for a given positive definite matrix M and P, the matrix equation H... T P + PH = -M holds true;

[0113] Step 2-2: When the system exhibits exponential uncertainty, the switching region becomes:

[0114]

[0115] When the state trajectory of an exponentially uncertain switching time-delay system moves to the boundary of the switching region, the system is prone to fast switching.

[0116] Steps 2-3: To avoid fast switching, construct the following switching region:

[0117]

[0118] In the aforementioned switching domain, ξ > 1 is a constant. Therefore, the right side of the inequality will increase, which means that more system states will satisfy the above condition, i.e., the switching region will become larger, resulting in overlapping regions. In this switching domain, the subsystems of the exponentially uncertain switching delay system are determined by the system's state trajectory. When the state trajectory enters region i, subsystem i is activated and runs.

[0119] Steps 2-4: For systems with uncertain exponential switching time lags, the following switching strategy is proposed:

[0120]

[0121] In the above formula, let This represents the system energy of each exponentially uncertain switching delay system; under this switching strategy, the system switches according to its energy value without needing to construct a switching region for the system state; therefore, the system can avoid the occurrence of fast switching phenomena under this switching strategy.

[0122] Thirdly, the present invention provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the steps of the stability analysis method for an exponentially uncertain switching time delay system as described in the first aspect of the present invention.

[0123] Fourthly, the present invention provides a non-transitory computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the stability analysis method for an exponentially uncertain switching time-delay system as described in the first aspect of the present invention.

[0124] Compared with the prior art, the beneficial effects of the present invention include at least the following:

[0125] This invention focuses on exponentially uncertain switching time-delay systems. First, a state-dependent switching strategy based on energy decay is designed. This strategy eliminates the need to construct switching regions for the system state; the system switches to the next mode based on the energy value of the subsystem. This strategy effectively avoids fast switching phenomena, and there is no limit to the number of switching operations. Second, sufficient conditions for the asymptotic stability of exponentially uncertain switching time-delay systems are obtained by constructing a functional and utilizing the generalized free matrix integral inequality method, Jensen's inequality method, and the negative definite lemma of quadratic functions. Solving for these sufficient conditions yields the maximum allowable time delay of the system. The switching strategy and stability conditions designed in this invention are applicable to systems with complex switching dynamics, state-space delay dynamics, and uncertain disturbances, possessing significant theoretical and practical application value. Attached Figure Description

[0126] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0127] Figure 1 This is a flowchart of the stability analysis method for an exponentially uncertain switching time-delay system according to the present invention;

[0128] Figure 2 This is a schematic diagram of the switching domain constructed in Reference 1;

[0129] Figure 3 This is a schematic diagram of the switching domain constructed in Reference 2;

[0130] Figure 4 This is a schematic diagram of the switching design improved by the present invention;

[0131] Figure 5 For switching signals when the system has a constant time delay;

[0132] Figure 6 The state response of the system when it has a constant time delay;

[0133] Figure 7 For systems with time-varying and time-delay switching signals;

[0134] Figure 8 The system has a time-varying and time-delay state response;

[0135] Figure 9 This is a structural diagram of the stability analysis system for the exponentially uncertain switching time delay system of the present invention;

[0136] Figure 10This is a block diagram of the electronic device structure of the present invention. Detailed Implementation

[0137] To make the above-mentioned objects, features, and advantages of the present invention more apparent and understandable, the technical solutions of the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that the described embodiments are merely some embodiments of the present invention, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0138] Example 1

[0139] This invention provides a stability analysis method for exponentially uncertain switching time-delay systems, addressing the switching strategies designed for such systems and the sufficient conditions for their asymptotic stability. Figure 1 As shown, it includes the following steps:

[0140] Step 1: Give the state-space expression of the exponentially uncertain switching time-delay system.

[0141]

[0142]

[0143] In the above formula, t is a continuous time variable. Let be the system state vector. Let it be an n-dimensional Euclidean space. Let θ be the initial state vector, and θ be a constant. The switching signal is ω(t); ω(t) is an exponentially uncertain parameter and satisfies... in ω and A is a constant; i (ω(t))=exp{E i ·ω(t)} and Let E be the exponential uncertainty matrix, where E i B i Let s be a constant coefficient matrix of appropriate dimension, where s is a constant. It is a positive integer; It is a continuous time-varying time delay and satisfies Here And d > 0 is a constant;

[0144] Based on the convex combination lemma and the property of exponential matrices Where D is a constant matrix, and λ,j,s are constants; therefore, the above exponentially uncertain switching time delay system can be rewritten in the following form;

[0145]

[0146]

[0147] Here,

[0148]

[0149]

[0150]

[0151]

[0152] Here, h is a constant, I is the identity matrix, and B... σ =B i E σ =E i Let ω(t) be a constant matrix and... The following relationship exists:

[0153]

[0154] ΔA σ (h,ω(t)) and ΔA dσ (h,ω(t)) is A σ (h,ω(t)) and A dσ The higher-order terms of the Taylor series expansion of (h,ω(t)) satisfy:

[0155]

[0156]

[0157] Step 2: Based on the state-space expression of the exponentially uncertain switching time-delay system, and combined with the energy decay principle, design a state-dependent switching strategy.

[0158] Consider the following i-th uncertain switching delay system:

[0159]

[0160] Suppose there exists a (A) i +A di The Herwitz convex combination H of ) satisfies the following condition:

[0161]

[0162] Here 0 < α i <1. Since H is a Herwitz matrix, for a given positive definite matrix M and P, the matrix equation H... TP+PH=-M holds true. Therefore, reference 1 ([1] Kim S, Campbell SA, Liu X. Stability of a class of linear switching systems with time delay[J]. IEEE Transactions on Circuits and Systems I: Regular Papers, 2006, 53(2): 384-393.) constructs the following switching region based on the Herwitz convex combination H and the system matrix of the switching time delay system:

[0163]

[0164] Combining M > 0 and the Herwitz convex combination H, it can be concluded that at least one subsystem of the switching time-delay system satisfies α. i x T [(A i +A di ) T P+P(A i +A di x < 0, therefore the entire vector space Divided by the aforementioned switching domain, i.e. Under this switching domain, as long as the system's state trajectory enters the constructed switching domain, the subsystem corresponding to that switching domain will be activated and run. A schematic diagram of this switching method is shown below. Figure 2 As shown.

[0165] However, it is worth noting that when the state trajectory of the switching time-delay system moves to the boundary of the switching region, that is, when the subsystem switches out and in, the system is prone to fast switching. Fast switching refers to the phenomenon that the number of switching from one subsystem to another within a specific time period is large, which causes the state trajectory of the system to exhibit slippage and chattering phenomena at the boundary of the switching region, thereby causing the system performance to deteriorate or even become unstable. In order to better avoid fast switching, reference 2 ([2] Zonouz AZ, Badamchizadeh MA, Ghiasi A RA new control approach for a class of linear switched systems with time-varying delay[J]. Transactions of the Institute of Measurement and Control, 2021(10):43.) constructs the following switching region:

[0166]

[0167] In the switching domain constructed above, ξ > 1 is a constant, therefore the right side of the inequality will increase. This means that more system states will satisfy the above condition, resulting in a larger switching region and overlapping regions. Therefore... In this switching domain, the operation of the subsystems in the switching time-delay system is determined by the system's state trajectory. When the state trajectory enters region i, subsystem i will be activated. A schematic diagram of this switching method is shown below. Figure 3 As shown.

[0168] The aforementioned switching method is an improvement upon the switching method in Reference 1. It increases the switching region to create overlapping areas between the switching regions of the subsystems, thereby reducing the likelihood of fast switching to some extent. However, under this switching method, when the state trajectory of the switching time-delay system enters the overlapping area of ​​the switching regions of multiple subsystems, it becomes difficult to determine which subsystem is currently running or which subsystem should be activated next.

[0169] Based on the above discussion, this invention improves the switching method for exponentially uncertain switching time-delay systems. A state-dependent switching strategy based on energy decay is designed. Under this strategy, the system selects the next operating mode based on the energy value of the subsystem.

[0170] According to Lyapunov's stability theory, when a control system tends to stabilize, its internal "energy" will decrease over time, meaning the system's state will gradually approach its equilibrium point. Therefore, consider the following i-th exponentially uncertain switching time-delay system:

[0171]

[0172] Similarly, suppose there exists a The Herwitz convex combination H satisfies the following condition:

[0173]

[0174] Here 0 < α i <1, Since H is a Herwitz matrix, for a given positive definite matrix M and P, the matrix equation H... T The statement P + PH = -M holds true.

[0175] As can be seen from the switching region constructed in Reference 1, when the system has exponential uncertainty, the switching region becomes:

[0176]

[0177] However, it is worth noting that when the state trajectory of an exponentially uncertain switching delay system moves to the boundary of the switching region, the system is prone to fast switching. That is, the system switches from one subsystem to another a large number of times within a finite time period, causing the system's state trajectory to exhibit slippage and chattering phenomena at the boundary of the switching domain, which in turn leads to deterioration or even instability of system performance.

[0178] To avoid fast switching, as shown in the switching region constructed in Reference 2, when the system has exponential uncertainty, the switching region becomes:

[0179]

[0180] In the aforementioned switching domain, ξ > 1 is a constant; therefore, the right-hand side of the inequality will increase. This means that more system states will satisfy the above condition, i.e., the switching region will become larger, resulting in overlapping regions. Therefore... In this switching domain, the subsystems of the exponentially uncertain switching delay system are determined by the system's state trajectory. When the state trajectory enters region i, subsystem i is activated and runs.

[0181] To better avoid rapid switching phenomena in the system, according to Lyapunov stability theory, the internal energy of the system decreases over time; that is, as the system's state trajectory approaches the equilibrium point, the system's "energy" gradually decreases. Therefore, for exponentially uncertain switching delay systems, this invention proposes the following switching strategy:

[0182]

[0183] The switching strategy switching diagram is shown below. Figure 4 As shown, where Q i Let represent the system energy of each exponentially uncertain switching time-delay subsystem. Under this switching strategy, there is no need to construct a switching region for the system state. Regardless of the system's state trajectory, the system always activates the subsystem with lower energy. Therefore, exponentially uncertain switching time-delay systems can effectively avoid the occurrence of fast switching phenomena.

[0184] Step 3: Prove the asymptotic stability of the exponentially uncertain switching time-delay system under the designed switching strategy. Step 3 constructs a functional containing a triple integral term. The asymptotic stability of the exponentially uncertain switching time-delay system is proven by using the generalized free matrix integral inequality method, Jensen's inequality method, and the negative definite lemma of quadratic functions to handle the time-delay part of the system.

[0185] First, the Lyapunov-Krasovsky functional is defined as follows:

[0186] V i (x t ) = V 1i (xt )+V 2i (x t )+V 3i (x t )+V 4i (x t )

[0187] in

[0188] V 1i (x t )=x T (t)Px(t)

[0189]

[0190]

[0191]

[0192] In the above formula, P, Q 1i Q 2i ,S 1i ,S 2i Given a symmetric positive definite real matrix, the i-th subsystem of the exponentially uncertain switching time-delay system is obtained using the Newton-Leibniz formula:

[0193]

[0194] Differentiating the above functional yields:

[0195]

[0196]

[0197] Using the generalized free matrix integral inequality method and Jensen's inequality method, we obtain and as follows:

[0198]

[0199]

[0200] From the above, we can conclude that:

[0201]

[0202] By using the negative definite lemma of quadratic functions, we can obtain... therefore The system asymptotically stabilizes under uncertain exponential switching time lags.

[0203] Step 4: Solve for the maximum allowable time delay of the exponentially uncertain switching time delay system using the following linear matrix inequality:

[0204] H T P + PH < 0

[0205]

[0206]

[0207]

[0208] where

[0209]

[0210]

[0211]

[0212]

[0213]

[0214]

[0215]

[0216]

[0217]

[0218]

[0219]

[0220] e i = [0 m×(i-1)m I m 0 m×(5-i)m , E1 = e2 - e5

[0221] E2 = e1 - e4, Ξ i = Ξ 1i + Ξ 2i + Ξ 3i ,

[0222]

[0223] Λ1(0) = [I -I 0], Λ1(1) = [I I -2I]

[0224] In the above formula, P, Q 1i , Q 2i,S 1i ,S 2i N is a symmetric positive definite real matrix; 1i N 2i Let be any matrix. The maximum allowable time delay of an exponentially uncertain switching time delay system can be solved using the above linear matrix inequalities.

[0225] This embodiment presents a stability analysis method for an exponentially uncertain switching time-delay system. Using Matlab 2020b software, the method verifies the invented state-dependent switching strategy and the obtained sufficient conditions for system stability. Let... The system matrices from References 1 and 2 are selected as the exponential uncertainty coefficient matrix, i.e., B is selected. i With E i as follows:

[0226]

[0227] A σ(t) (ω(t))=exp{E σ(t) (ω(t))}, Let ω = 0.1, The system matrix of the exponentially uncertain switching time-delay system can be obtained as follows:

[0228]

[0229] make The Herwitz matrix H can be obtained as follows:

[0230]

[0231] Next, consider the following two cases:

[0232] Case 1: Exponentially uncertain switching delay system with constant time delay.

[0233] Solving the linear matrix inequality yields the upper bound of the maximum time delay. To verify the effectiveness of the designed switching strategy, we can obtain:

[0234]

[0235] Therefore, the switching strategy can be obtained as follows: Figure 5 As shown. Figure 6 constant time delay The state response of an exponentially uncertain switching time-delay system with a constant time delay is given. Therefore, an exponentially uncertain switching time-delay system with a constant time delay is asymptotically stable.

[0236] Case 2: Exponentially uncertain switching delay system with time-varying delay.

[0237] Selecting the time-varying delay from reference 2 Solving the linear matrix inequality yields the upper bound of the maximum time delay. Similarly, to verify the effectiveness of the designed switching strategy, we can obtain:

[0238]

[0239] Therefore, the switching strategy can be obtained as follows: Figure 7 As shown. Figure 8 Time-varying and time-delayed The state response of an exponentially uncertain switching delay system. Therefore, an exponentially uncertain switching delay system with time-varying delays is asymptotically stable.

[0240] Table 1 shows the maximum time delay that the system can tolerate.

[0241] Methods used constant time delay Time-varying and time-delay Reference 1 0.001573 — Reference 2 0.849 0.849 This invention 2.310 2.021

[0242] Table 1 lists the maximum time delay that the systems in Reference 1, Reference 2 and the present invention can tolerate. As can be seen from Table 1, the maximum time delay that the system considered in the present invention can tolerate is larger, that is, the switching strategy and the resulting stability conditions designed in the present invention have less conservatism.

[0243] This invention focuses on exponentially uncertain switching time-delay systems. First, a state-dependent switching strategy based on energy decay is designed. This strategy eliminates the need to construct switching regions for the system state; the system switches to the next mode based on the energy value of the subsystem. This strategy effectively avoids fast switching phenomena, and there is no limit to the number of switching operations. Second, sufficient conditions for the asymptotic stability of exponentially uncertain switching time-delay systems are obtained by constructing a functional and utilizing the generalized free matrix integral inequality method, Jensen's inequality method, and the negative definite lemma of quadratic functions. Solving these sufficient conditions yields the maximum allowable time delay of the system. The switching strategy and stability conditions designed in this invention are applicable to systems with complex switching dynamics, state-space time-delay dynamics, and uncertain disturbances, possessing significant theoretical and practical application value.

[0244] Example 2

[0245] like Figure 9 As shown, the present invention provides a stability analysis system for an exponentially uncertain switching time delay system, including an expression giving module, a switching strategy design module, an asymptotic stability proof module, and a correctness verification module;

[0246] The expression-giving module is used to give the state-space expression of the exponentially uncertain switching time-delay system;

[0247] The switching strategy design module is used to design state-dependent switching strategies based on the state-space expression of an exponentially uncertain switching time-delay system and the principle of energy decay.

[0248] The asymptotic stability proof module is used to prove the asymptotic stability of the exponentially uncertain switching delay system under the designed switching strategy;

[0249] The correctness verification module is used to verify the effectiveness of the designed switching strategy and the correctness of the system stability conditions through simulation examples.

[0250] Other features in this embodiment are the same as in Embodiment 1, so they will not be repeated here.

[0251] Example 3

[0252] Based on the same concept, the present invention also provides a schematic diagram of a physical structure, such as... Figure 10 As shown, the server may include a processor 810, a communications interface 820, a memory 830, and a communication bus 840. The processor 810, communications interface 820, and memory 830 communicate with each other via the communication bus 840. The processor 810 can call logical instructions in the memory 830 to execute the steps of the stability analysis method for the exponentially uncertain switching time-delay system.

[0253] Furthermore, the logical instructions in the aforementioned memory 830 can be implemented as software functional units and, when sold or used as independent products, can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, essentially, or the part that contributes to the prior art, or a part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0254] Example 4

[0255] Based on the same concept, the present invention also provides a non-transitory computer-readable storage medium storing a computer program containing at least one piece of code that can be executed by a master control device to control the master control device to implement the steps of the stability analysis method for the exponentially uncertain switching time delay system.

[0256] In the above embodiments, implementation can be achieved, in whole or in part, through software, hardware, firmware, or any combination thereof. When implemented in software, it can be implemented, in whole or in part, as a computer program product. The computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, all or part of the processes or functions described in this application are generated. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions can be stored in a computer-readable storage medium or transmitted from one computer-readable storage medium to another. For example, the computer instructions can be transmitted from one website, computer, server, or data center to another website, computer, server, or data center via wired (e.g., coaxial cable, fiber optic, digital subscriber line) or wireless (e.g., infrared, wireless, microwave, etc.) means. The computer-readable storage medium can be any available medium accessible to a computer or a data storage device such as a server or data center that integrates one or more available media. The available medium can be a magnetic medium (e.g., floppy disk, hard disk, magnetic tape), an optical medium (e.g., DVD), or a semiconductor medium (e.g., solid-state drive).

[0257] Those skilled in the art will understand that all or part of the processes in the methods of the above embodiments can be implemented by a computer program instructing related hardware. This program can be stored in a computer-readable storage medium, and when executed, it can include the processes described in the above method embodiments. The aforementioned storage medium includes various media capable of storing program code, such as ROM or random access memory (RAM), magnetic disks, or optical disks.

[0258] The embodiments described above are merely illustrative of several implementations of the present invention, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention. Therefore, the scope of protection of this patent should be determined by the appended claims.

Claims

1. A stability analysis method for an exponentially uncertain switching time-delay system, characterized in that, Includes the following steps: Step 1: Give the state-space expression of the exponentially uncertain switching time-delay system; Step 2: Based on the state-space expression of the exponentially uncertain switching time-delay system, and combined with the energy decay principle, design a state-dependent switching strategy; Step 3: Prove that the exponentially uncertain switching delay system is asymptotically stable under the designed switching strategy; Step 4: Verify the effectiveness of the designed switching strategy and the correctness of the system stability conditions through simulation examples; In step two, a state-dependent switching strategy based on energy decay is designed. Under this switching strategy, the system selects the operating mode at the next moment according to the energy value of the subsystem, specifically including the following sub-steps: Step 2-1: Consider the following i-th exponentially uncertain switching delay system: Suppose there exists a The Herwitz convex combination H satisfies the following condition: Here, Since H is a Herwitz matrix, for a given positive definite matrix M and P, the matrix equation H... T P + PH = -M holds true; Step 2-2: When the system exhibits exponential uncertainty, the switching region becomes: When the state trajectory of an exponentially uncertain switching time-delay system moves to the boundary of the switching region, the system is prone to fast switching. Steps 2-3: To avoid fast switching, construct the following switching region: In the aforementioned switching region, ξ > 1 is a constant; therefore, the right-hand side of the inequality will increase. This means that more system states will satisfy the above condition, i.e., the switching region will become larger, resulting in overlapping regions. Therefore... In this switching region, the subsystems of the exponentially uncertain switching time delay system are determined by the system's state trajectory. When the state trajectory enters region i, subsystem i is activated and runs. Steps 2-4: For systems with uncertain exponential switching time lags, the following switching strategy is proposed: In the above formula, let This represents the system energy of each exponentially uncertain switching delay system; under this switching strategy, the system switches according to its energy value without needing to construct a switching region for the system state; therefore, the system can avoid the occurrence of fast switching phenomena under this switching strategy.

2. The stability analysis method for an exponentially uncertain switching time-delay system according to claim 1, characterized in that, In step one, the state-space expression of the exponentially uncertain switching time-delay system is given as follows: In the above formula, t is a continuous time variable. Let be the system state vector. For an n-dimensional Euclidean space, Let θ be the initial state vector, and θ be a constant. The switching signal is ω(t); ω(t) is an exponentially uncertain parameter and satisfies... in ω and A is a constant; i (ω(t))=exp{E i ·ω(t)} and Let E be the exponential uncertainty matrix, where E i B i Let s be a constant coefficient matrix of appropriate dimension, where s is a constant. It is a positive integer; It is a continuous time-varying time delay and satisfies Here And d > 0 is a constant; Based on the convex combination lemma and the property of exponential matrices Where D is a constant matrix, and λ,j,s are constants; therefore, the above exponentially uncertain switching time delay system can be rewritten in the following form; Here, Here, h is a constant, I is the identity matrix, and B... σ =B i E σ =E i Let ω(t) be a constant matrix and... The following relationship exists: ΔA σ (h,ω(t)) and ΔA dσ (h,ω(t)) is A σ (h,ω(t)) and A dσ The higher-order terms of the Taylor series expansion of (h,ω(t)) satisfy:

3. The stability analysis method for an exponentially uncertain switching time-delay system according to claim 1, characterized in that, In step three, a functional containing a triple integral term is constructed. The time-delay component of the system is handled using the generalized free matrix integral inequality method, Jensen's inequality method, and the negative definite lemma of quadratic functions to prove the asymptotic stability of the exponentially uncertain switching time-delay system. The specific steps are as follows: Step 3-1: First, define the following Lyapunov-Krasovsky functional: V i (x t )=V 1i (x t )+V 2i (x t )+V 3i (x t )+V 4i (x t ) in V 1i (x t )=x T (t)Px(t) In the above formula, P, Q 1i Q 2i ,S 1i ,S 2i Given a symmetric positive definite real matrix, the i-th subsystem of the exponentially uncertain switching time-delay system is obtained using the Newton-Leibniz formula: Step 3-2: Differentiating the above functional yields: Using the generalized free matrix integral inequality method and Jensen's inequality method, we obtain and as follows: Step 3-3, obtained from the above: By applying the negative definite lemma of quadratic functions, we get therefore The exponentially uncertain switching delay system asymptotically stabilizes under the designed switching strategy.

4. The stability analysis method for an exponentially uncertain switching time-delay system according to claim 1, characterized in that, Following step three, the following steps are also included: The maximum allowable time delay of an exponentially uncertain switching time delay system can be solved using the following linear matrix inequality: H T P+PH<0 in have been i =[0 m×(i-1)m I m 0 m×(5-i)m ], E1=e2-e5 E2=e1-e4,X i =Ξ 1i +Ξ 2i +Ξ 3i , Λ1(0)=[I-I 0],Λ1(1)=[I I-2I] In the above formula, P, Q 1i Q 2i ,S 1i ,S 2i N is a symmetric positive definite real matrix; 1i N 2i Let be any matrix; the maximum allowable time delay of the exponentially uncertain switching time delay system can be obtained by solving the above linear matrix inequalities.

5. A stability analysis system for an exponentially uncertain switching time-delay system, characterized in that, include: The expression-giving module is used to give the state-space expression of an exponentially uncertain switching time-delay system; The switching strategy design module is used to design state-dependent switching strategies based on the state-space expression of an exponentially uncertain switching time-delay system and the principle of energy decay. The asymptotic stability proof module is used to prove the asymptotic stability of the exponentially uncertain switching delay system under the designed switching strategy; The correctness verification module is used to verify the effectiveness of the designed switching strategy and the correctness of the system stability conditions through simulation examples. The state-dependent switching strategy based on the energy decay principle is designed as follows: Under this switching strategy, the system selects the next operating mode based on the energy value of the subsystem; specifically, it includes the following sub-steps: Step 2-1: Consider the following i-th exponentially uncertain switching delay system: Suppose there exists a The Herwitz convex combination H satisfies the following condition: Here 0 < α i <1, Since H is a Herwitz matrix, for a given positive definite matrix M and P, the matrix equation H... T P + PH = -M holds true; Step 2-2: When the system exhibits exponential uncertainty, the switching region becomes: When the state trajectory of an exponentially uncertain switching time-delay system moves to the boundary of the switching region, the system is prone to fast switching. Steps 2-3: To avoid fast switching, construct the following switching region: In the aforementioned switching region, ξ > 1 is a constant; therefore, the right side of the inequality will increase. This means that more system states will satisfy the above condition, i.e., the switching region will become larger, resulting in overlapping regions. In this switching region, the subsystems of the exponentially uncertain switching time delay system are determined by the system's state trajectory. When the state trajectory enters region i, subsystem i is activated and runs. Steps 2-4: For systems with uncertain exponential switching time lags, the following switching strategy is proposed: In the above formula, let This represents the system energy of each exponentially uncertain switching delay system; under this switching strategy, the system switches according to its energy value without needing to construct a switching region for the system state; therefore, the system can avoid the occurrence of fast switching phenomena under this switching strategy.

6. The stability analysis system for an exponentially uncertain switching time-delay system according to claim 5, characterized in that, The state-space expression for an exponentially uncertain switching time-delay system is given as follows: In the above formula, t is a continuous time variable. Let be the system state vector. For an n-dimensional Euclidean space, Let θ be the initial state vector, and θ be a constant. The switching signal is ω(t); ω(t) is an exponentially uncertain parameter and satisfies... in ω and A is a constant; i (ω(t))=exp{E i ·ω(t)} and Let E be the exponential uncertainty matrix, where E i B i Let s be a constant coefficient matrix of appropriate dimension, where s is a constant. It is a positive integer; It is a continuous time-varying time delay and satisfies Here And d > 0 is a constant; Based on the convex combination lemma and the property of exponential matrices Where D is a constant matrix, and λ,j,s are constants; therefore, the above exponentially uncertain switching time delay system can be rewritten in the following form; Here, Here, h is a constant, I is the identity matrix, and B... σ =B i E σ =E i Let ω(t) be a constant matrix and... The following relationship exists: ΔA σ (h,ω(t)) and ΔA dσ (h,ω(t)) is A σ (h,ω(t)) and A dσ The higher-order terms of the Taylor series expansion of (h,ω(t)) satisfy:

7. An electronic device, characterized in that, It includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the steps of the stability analysis method for an exponentially uncertain switching time-delay system as described in any one of claims 1-4.

8. A non-transitory computer-readable storage medium, characterized in that, It stores a computer program that, when executed by a processor, implements the steps of the stability analysis method for an exponentially uncertain switching time delay system as described in any one of claims 1-4.

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