Almost-inevitable exponential stability analysis method for nonlinear random system with random pulses and application of almost-inevitable exponential stability analysis method in chaotic synchronization

Abstract

The invention provides an almost inevitable exponential stability analysis method of a nonlinear random system with random pulses and application of the almost inevitable exponential stability analysis method in chaotic synchronization. The almost inevitable exponential stability analysis method comprises the following steps: establishing a mathematical model of a pulse random system influenced by random noise and random pulses; a Lyapunov method is used to provide a new almost inevitable index stability condition of a pulse random system driven by an updating process and a Markov chain under different conditions. The invention provides an analysis method capable of comprehensively considering random pulse intensity and density randomness, establishing a stability criterion suitable for random pulse control, and applying the stability criterion to a synchronization problem of a master-slave random chaotic system.

Description

Technical Field

[0001] The present invention relates to the technical field of pulse systems, and in particular to an almost exponential stability analysis method for nonlinear random systems and its application in chaos synchronization. Background Art

[0002] As a hybrid system of continuous dynamics coupled with discrete jumps, the theoretical basis of the pulse system can be traced back to the theory of pulse perturbations in differential equations. Under random perturbations, the existing technical system has established a random pulse modeling framework that includes a Poisson process and an update process. The Poisson process characterizes the randomness of the pulse interval through an exponential distribution, while the update process allows for a wider interval distribution but relies on a known probability density function. In the field of stability analysis, the academic community has developed tools such as the comparison principle, the average residence time method, and the Razumikhin technique. These methods have made significant progress in the moment stability analysis of deterministic pulse systems and some pulse random systems. It is particularly noteworthy that existing research has revealed the coupling relationship between the linear growth constraint of noise intensity and the independent and identically distributed condition of pulse intensity, providing a basic theoretical support for the stability analysis of pulse random systems.

[0003] In the field of stability analysis of pulsed stochastic systems, the prior art closest to the present invention is presented in Reference 1 [Tang Y, Wu X, Shi P, Qian F. Input-to-state stability for nonlinear systems with stochastic impulses. Automatica, 2020;113:108766]. This proposed method for joint analysis of stochastic pulse intensity and density models the randomness of pulse intervals through an update process, describes pulse intensity using independent and identically distributed random variables, and establishes a stability criterion based on average dwell time. The core approach consists of three key stages: first, building a pulse interval update process model; second, deriving moment stability conditions using Lyapunov functions; and finally, eliminating the influence of random pulses using probability density integrals. Structurally, this approach employs a discrete architecture, in which the pulse generator controls pulse density through an update process, and the intensity regulator modulates pulse amplitude using an independent random sequence. Existing methods for stability analysis of pulsed stochastic systems suffer from three technical bottlenecks: First, the discrete modeling architecture results in a lack of joint characterization of the randomness of pulse intensity and density. Although Reference 1 separately models pulse density and intensity through an update process and independent and identically distributed variables, it lacks a dynamic coupling mechanism, making it impossible to accurately characterize the combined effects of pulses in complex random environments. This deficiency directly results in a lack of dynamic adaptability in the pulse triggering mechanism used in chaotic synchronization control, leading to a slow convergence rate of synchronization errors. Second, the dependence on probability density limits its applicability in engineering. Existing methods require that the probability density function of the pulse interval be known and integrable. However, the timing characteristics of random pulses in actual engineering often exhibit non-exponential distributions. This limitation makes existing criteria unsuitable for direct application in most industrial scenarios. Third, noise processing constraints are too strong. Existing stability criteria require that the noise term satisfy a linear growth condition, but the noise term in actual nonlinear systems often exhibits polynomial growth characteristics. This contradiction leads to a high misjudgment rate in typical nonlinear systems.

[0004] Specifically, the present invention aims to solve the following technical problems:

[0005] (1) How to characterize the intensity and density of random pulses through the update process and Markov chain, respectively, so as to conduct a more extensive and accurate analysis of the stability of the pulse random system;

[0006] (2) How to propose an easily verifiable almost certain exponential stability condition under the combined effects of random noise and random pulses to reduce the restrictive assumptions of existing methods;

[0007] (3) How to solve the synchronization problem of master-slave random chaotic systems based on the random pulse control framework, and provide theoretical support and methodological guidance for the practical application of random pulse control. Summary of the Invention

[0008] In response to the technical problems in random pulse systems, such as discrete modeling of pulse intensity and density, excessive dependence on the probability density of pulse intervals, and strict noise constraints, which limit the engineering applicability of stability criteria, the present invention proposes an almost exponential stability analysis method for nonlinear random systems with random pulses and its application in chaotic synchronization. By constructing a dynamic coupling model of pulse intensity and density and designing a Lyapunov exponent analysis method that is independent of probability density, easy detection of almost exponential stability is achieved.

[0009] In order to achieve the above object, the technical solution of the present invention is implemented as follows: a method for analyzing the almost certain exponential stability of a nonlinear random system with random pulses, the steps of which are as follows:

[0010] Step 1: Establish a mathematical model of the pulse random system affected by random noise and random pulses;

[0011] Step 2: New almost certain exponential stability conditions for impulsive stochastic systems driven by renewal processes and Markov chains under different conditions are given by the Lyapunov method.

[0012] Preferably, the mathematical model of the pulse random system is:

[0013]

[0014] in, and are the initial value and the system state at time t, x(0) is the value of the system state at time 0, x(t - ) represents the left limit state value of the system at time t, dx(t) represents the differential increment of the state variable x(t), Represents a set of n-dimensional non-negative numbers; It is defined in q-dimensional Brownian motion on , Ω represents the sample space, Represents an event family, represents a filter, represents the probability measure, represents standard Brownian motion The differential increment of k is the kth one that satisfies 0=ζ0<ζ1<…<ζ k <… and lim k→∞ ζ k =∞; the pulse jump r(k) is taken from the set S = {1,2,…,r}, which determines which type of pulse jump will occur at the pulse time k, and r represents the total number of pulse jumps; the function and Continuous on t, locally Lipschitz continuous on x; and f(t,0)=0, g(t,0)=0; for each i∈S, the function is locally Lipschitz continuous on x, and h i (0) = 0, jump mapping h r(k) The type of is randomly selected by a Markov chain, represents a set of non-negative numbers, Represents the set of positive integers.

[0015] Preferably, the random effect of pulse jumping is that the pulse time and pulse jumping It is a random process, during the pulse is independent and identically distributed and expected θ represents the expected value of the pulse duration; the number of pulses that have occurred before time t It is an updating process, sup represents the supremum; pulse jump is a transition probability matrix Π=[π ij ] r×r The discrete irreducible Markov chain, π ij Represents the conditional probability of the pulse type jumping from state i to state j, standard Brownian motion and random noise σ(t) are both filters The adaptation process of the pulse is independent of each other; k , pulse jump r(k) and standard Brownian motion Are independent of each other.

[0016] Preferably, the new almost certain exponential stability condition in step 2 includes:

[0017] Case 1: Consider an impulsive stochastic system driven by an update process and a Markov chain, taking into account the drift term f9t,x(t))dt and the diffusion term The coupling effect and the quantitative relationship between continuous dynamics and random pulse intensity and density are shown in Figure 2. If there is a Lyapunov function That is, the Lyapunov function V(t,x) is piecewise continuous, first-order continuously differentiable for t and second-order for x, and the constants c>0, p>0, α i >0, Make

[0018] c||x(t)|| p ≤V(t,x),

[0019]

[0020] V(t,h i (x))≤αi V(t,x),

[0021]

[0022] For all i∈S, this impulsive random system is almost necessarily exponentially stable;

[0023] in, Indicates piecewise continuity, || || p represents the p-norm, represents the Ito generator of the Lyapunov function V(t,x), represents the jump operator of the Lyapunov function V(t,x) under the jump or pulse condition, h i (x) represents the state jump map under the action of the i-th type pulse, θ represents the average pulse interval, α i represents the intensity of the i-th pulse, π i represents the stationary distribution of a Markov chain.

[0024] Preferably, the new almost certain exponential stability condition in step 2 further includes:

[0025] Case 2: Consider an impulsive stochastic system driven by an update process and a Markov chain, without the drift term f(t,x(t))dt and the diffusion term Limit, if there exists a Lyapunov function and constants c>0,p>0, α i >0, Make

[0026] c||x(t)|| p ≤V(t,x),

[0027] V(t,h i (x))≤α i V(t,x),

[0028]

[0029] For all i∈S, this impulsive random system is almost necessarily exponentially stable; where, Represents a constant.

[0030] Preferably, the new almost certain exponential stability condition in step 2 further includes:

[0031] Case 3: If the pulse jumps is independent and identically distributed, that is p irepresents the probability that r(k) = i; considering an impulse random system driven by an update process and a Markov chain, if there exists a Lyapunov function and constants c>0,p>0,α i >0, Make

[0032] c||x(t)|| p ≤V(t,x),

[0033]

[0034] V(t,h i (x))≤α i V(t,x),

[0035]

[0036] For all i∈S, this impulsive random system is almost certainly exponentially stable.

[0037] Preferably, the new almost certain exponential stability condition in step 2 further includes:

[0038] Case 4: For the case of a single impulse jump, that is, the set S = {1}; consider an impulse random system driven by an update process and a Markov chain. If there exists a Lyapunov function and constants c>0,p>0,α i >0, Make

[0039] c||x(t)|| p ≤V(t,x),

[0040]

[0041] V(t,h i (x))≤α i V(t,x),

[0042]

[0043] If this holds, then this impulse random system is almost necessarily exponentially stable.

[0044] Preferably, the new almost certain exponential stability condition in step 2 further includes:

[0045] Case 5: If there is no pulse effect, that is, when a pulse jump occurs, the state x remains unchanged h r(k) (x(t - ))=x(t - ), the impulse stochastic system driven by the update process and the Markov chain will degenerate into a random system:

[0046]

[0047] For random systems, if there is a Lyapunov function and constants c>0,p>0, Make

[0048] c||x(t)|| p ≤V(t,x),

[0049]

[0050] λ+0.5γ 2 <0,

[0051] If holds, then the random system is almost certainly exponentially stable.

[0052] Preferably, consider an impulsive random system:

[0053]

[0054] Where, the system state x(t) = [x1(t), x2(t)] T , is a one-dimensional Brownian motion, the pulse jump r(k) is a Markov chain with a transition probability matrix π, d r(k) represents the transition increment generated by the kth pulse;

[0055] Calculate the invariant distribution π, random process ζ k It is driven by the update process of parameter θ; set the shock pulse jump parameters d1, d2, d3; select Lyapunov function V(t,x)=x T (t)x(t), estimate

[0056]

[0057] T represents the transpose of the vector, They represent the kth pulse moment and the state of the system before the pulse occurs respectively;

[0058] Case 1: Considering the coupling effect of drift and diffusion terms, constants γ and λ are selected so that the condition Established; select α1, α2, α3, calculate It can be concluded that the above-mentioned impulse random system is almost necessarily exponentially stable;

[0059] Case 2: Considering that there is no drift term and diffusion term restriction, there is no constant γ>0, so that Select Constant and Make the conditions and established; however, It cannot be used to verify the stability of pulse random systems.

[0060] Preferably, the application in chaos synchronization is:

[0061] Considering the main control system, the Chua circuit disturbed by random noise is:

[0062] Among them, the nonlinear voltage-current relationship function in the state space model of Cai's circuit is and the system state x(t)=[x1(t),x2(t),x3(t)] T , x1(t), x2(t), x3(t) represent the voltage across capacitor C1, the voltage across capacitor C2, and the current in inductor L in the main drive circuit respectively; then, select the Lipschitz parameter matrix as L=diag[27 / 7,0,0], select the parameter matrix

[0063]

[0064] Then the main control system can generate a double vortex attractor;

[0065] The corresponding slave systems are:

[0066] Among them, y(t)=[y1(t),y2(t),y3(t)] T , y1(t), y2(t), y3(t) represent the voltage across capacitor C1, the voltage across capacitor C2, and the current in inductor L in the response circuit, respectively. The pulse control input is: Where δ(·) is the Dirac function, e(t) = x(t) - y(t) is the synchronization error, and K represents the control gain matrix.

[0067] Then, the dynamic equation of synchronization error is:

[0068]

[0069] Among them, the error function F(e(t))=f(x(t))-f(y(t)), ζ k is the random pulse control moment, the pulse period are identically distributed and

[0070] The impulse control input u(t) is in the form of a Dirac function at a specific time t = ζ kApply "shock" to the slave system; the master system and the slave system are pulse random systems.

[0071] If there exists a number 0<α1<1,λ>0, Matrix P>0, and the diagonal matrix Q>0, so that

[0072]

[0073] If the master-slave random chaotic system is established, Pulse controller Index synchronization can be achieved under

[0074] Take constants γ and α1 to get the parameters λ and matrix P, and diagonal matrix Q; calculate the control gain Set the expected θ during the average pulse control period and calculate The designed pulses realize the synchronization of master and slave random chaotic systems.

[0075] Compared with the existing technology, the beneficial effects of the present invention are: for the problem of almost inevitable exponential stability of nonlinear random systems under the action of random noise and random pulses, an analysis method is proposed that can comprehensively consider the randomness of random pulse intensity and density, establish a stability criterion suitable for random pulse control, and apply it to the synchronization problem of master-slave random chaotic systems.

[0076] The present invention studies the problem of almost inevitable exponential stability of a class of pulse random systems, in which the update process and Markov chain are used to characterize the random pulse intensity and density respectively. The system studied in the present invention can be regarded as a generalization of the pulse system driven by the Poisson process. After being generalized to a more general update process and Markov jump pulses, it can simulate and analyze various types of pulse behaviors in reality more realistically and flexibly. For pulse random systems with random pulse jumps, easy-to-check conditions for almost inevitable exponential stability are given. The research results of the present invention not only conduct a comprehensive quantitative analysis of the impact of random noise and random pulse jumps on the stability of pulse random systems, but also the conditions are weaker than some previous research results, making it easier for the system to meet the stability criterion, significantly expanding the scope of application of theoretical results and the possibility of engineering practice.

[0077] The core advantage of the present invention is reflected in the dual breakthroughs of theoretical innovation and engineering applicability: First, by constructing a dual random drive architecture of the update process and the Markov chain, the timing characteristics of the pulse interval are dynamically associated with the random jumps of the pulse intensity, solving the problem of the lack of coupling effect characterization caused by discrete modeling in traditional methods. Based on the deep integration of Lyapunov function and stochastic analysis, the proposed stability criterion not only accurately quantifies the joint influence of random noise, pulse density and intensity on the stability of the system, but also reveals the dynamic modulation mechanism of the Markov modal jump of pulse intensity on the Lyapunov energy function, which is conducive to more targeted design and control of the system. Secondly, the new probability density-independent criterion construction method breaks through the traditional method's explicit dependence on the pulse timing distribution by converting the statistical characteristics of the pulse interval into a measurable Lyapunov exponent convergence condition, and significantly expands the application scope of stability theory in non-Poisson pulse scenarios. In addition, the dynamic compensation mechanism designed for nonlinear noise effectively suppresses the interference of noise polynomial growth on system stability, providing theoretical support for engineering problems such as chaotic synchronization control in high noise environments. BRIEF DESCRIPTION OF THE DRAWINGS

[0078] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0079] Figure 1 Flowchart of the present invention.

[0080] Figure 2 This is the state trajectory diagram of the pulse random system in Example 1 under one sample path.

[0081] Figure 3 This is the state trajectory diagram of the pulse random system in Example 1 under 40 sample paths.

[0082] Figure 4 The circuit diagram of the Chua circuit perturbed by random noise.

[0083] Figure 5 Schematic diagram of the double vortex attractor of random Chua circuit in two-dimensional and three-dimensional space;

[0084] Figure 6 State trajectory diagram of the master-slave random chaotic system. DETAILED DESCRIPTION

[0085] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without creative work are within the scope of protection of the present invention.

[0086] like Figure 1 As shown, a method for analyzing the almost inevitable exponential stability of a nonlinear random system with random pulses is proposed. The present invention aims to establish a triple coupling mechanism: 1) Through the update process-Markov chain joint modeling architecture, the dynamic correlation characterization of pulse intensity and density is realized to improve the random pulse control response speed; 2) By analyzing the coupling effect between random noise, random pulse intensity and random pulse density, a new easy-to-detect condition for almost inevitable exponential stability is obtained based on the Lyapunov method, improving the engineering application scenario; 3) A random pulse control framework is established to solve the synchronization problem of master-slave random chaotic systems and improve the stability of nonlinear random systems with random pulses. The present invention includes the following steps:

[0087] Step 1: Establish a mathematical model of the pulse random system affected by random noise and random pulses.

[0088] Furthermore, the steps for establishing the mathematical model of the pulse random system affected by random noise and random pulses in step 1 are as follows:

[0089] Consider the impulse random system as:

[0090]

[0091] in, and is the initial value and system state, x(0) is the value of the system state at time 0, x(t - ) represents the left limit state value of the system at time t, dx(t) represents the differential increment of the state variable x(t), Represents a set of n-dimensional non-negative numbers; It is defined in q-dimensional Brownian motion on , Ω represents the sample space, Represents an event family, represents a filter, represents the probability measure, represents standard Brownian motion The differential increment of k is the kth one that satisfies 0=ζ0<ζ1<…<ζ k <… and lim k→∞ ζ k=∞; the pulse jump r(k) is taken from the value in the set S = {1,2,…,r}, which determines what type of pulse jump will occur at the pulse time k, r is a positive integer; the function and Continuous on t, locally Lipschitz continuous on x; and f(t,0)=0, g(t,0)=0; for each i∈S, the function is locally Lipschitz continuous on x, and h i (0)=0. represents a set of non-negative numbers, represents a set of positive integers. It can be seen that when t≠ζ k When the system state x(t) changes, the differential increment The Brownian motion noise of the driver directly affects the k , the system state x(t) will undergo a "jump" or "mutation": x(t) = h r(k) (x(t - )), where the jump map h r(k) The type of is randomly selected by the Markov chain, representing the change in jump intensity or pattern, and the pulse size and occurrence time are random.

[0092] The random effect of pulse skipping, i.e. and is a random process. During the pulse is independent and identically distributed and expected θ represents the expected value of the pulse duration. In this case, the number of pulses that have occurred before time t It is an update process, sup represents the supremum. is a transition probability matrix Π=[π ij ] r×r The discrete irreducible Markov chain, π ij represents the conditional probability of the pulse type changing from state i to state j, and σ(t) are both filters The adaptation process of , and the two are independent of each other, σ(t) represents random noise. Assume ζ k , r(k) and Are independent of each other.

[0093] Step 2: New almost certain exponential stability conditions for impulsive stochastic systems driven by renewal processes and Markov chains under different conditions are given by the Lyapunov method.

[0094] Furthermore, in step 2, the new almost certain exponential stability condition for the impulsive stochastic system driven by the update process and the Markov chain under different conditions is given by the Lyapunov method:

[0095] Case 1: Consider the impulsive stochastic system driven by the update process and Markov chain in step 1, considering the drift term f(t,x(t))dt and the diffusion term The coupling effect and the quantitative relationship between continuous dynamics and random pulse intensity and density. If there is a Lyapunov function That is, the Lyapunov function V(t,x) is piecewise continuous, first-order continuously differentiable for t and second-order for x, and the constants c>0, p>0, α i >0, Make

[0096] c||x(t)|| p ≤V(t,x),

[0097]

[0098] V(t,h i (x))≤α i V(t,x),

[0099]

[0100] For all i∈S, this impulsive random system is almost certainly exponentially stable.

[0101] in, Indicates piecewise continuity, || || p represents the p-norm, represents the Ito generator of the Lyapunov function V(t,x), represents the jump operator of the Lyapunov function V(t,x) under the jump or pulse condition, h i (x) represents the state jump mapping under the action of the i-th type pulse, α i represents the intensity of the i-th pulse, π i represents the probability of the Markov chain staying in the i-th type of pulse; || || p The p in the equation is selected by the analysis objective and physical meaning, and h i (x) and α i Determined by the pulse physical mechanism, c||x(t)|| p ≤V(t,x) means that the Lyapunov function V(t,x) has a lower bound c||x(t)|| p ; Indicates that the rate of change under the influence of the continuous part and the pulse is controllable as a whole; V(t,hi V(t, x)) ≤ α i V(t, x) represents that for each pulse mapping, the jump increase of the Lyapunov function does not exceed α i times, represents the overall stability criterion.

[0102] Proof: Construct an auxiliary variable η(t) that satisfies:

[0103]

[0104] η(0) = η0,

[0105] where η0 > 0 and 0 < a ≤ |λ|. Then, for any t ≥ 0, the auxiliary variable can be obtained α r(k) is a jump factor corresponding to the k-th pulse moment ζ k and represents the number of pulses that have occurred before time t.

[0106] Select According to Itô's formula, for any t ∈ [ζ k , ζ k+1 ),

[0107]

[0108] where Then, we can obtain

[0109]

[0110] For any t ∈ [ζ k , ζ k+1 ) holds. At the pulse instant we have:

[0111]

[0112] From this, we can obtain:

[0113]

[0114] Using mathematical induction, we deduce that for any we have

[0115]

[0116] holds. Therefore, for any t ≥ 0,

[0117]

[0118] holds, where,

[0119] If is a discrete-time irreducible Markov chain, then it has a unique invariant distribution π = (π1, π2, ···, π r ) and It follows that:

[0120]

[0121] Furthermore, we have:

[0122]

[0123] Note that is a continuous local martingale with quadratic variation

[0124]

[0125] For any ε ∈ (0, 1), by the exponential martingale inequality, for any

[0126]

[0127] Since Applying the Borel-Cantelli lemma, for almost all ω, there exists an integer n0 = n0(ω) such that for any N > n0

[0128]

[0129] It follows that for any 0 ≤ t ≤ N,

[0130]

[0131] For 0 ≤ t ≤ N, using the condition we have

[0132]

[0133] Note that 0 < a ≤ |λ|, which implies

[0134] [[ID=6�]]

[0135] For N - 1 ≤ t ≤ N, we have

[0136]

[0137] Let N → ∞, we can obtain

[0138]

[0139] Based on conditions and ε>0 is arbitrary, we can get Using η(t)>0 and the condition c||x(t)|| p ≤V(t,x), derive Therefore, impulsive random systems are almost certainly exponentially stable.

[0140] Case 2: Consider the impulsive stochastic system driven by the update process and Markov chain in step 1. There is no drift term and diffusion term restriction. If there is a Lyapunov function and constants c>0,p>0, α i >0, Make

[0141] c||x(t)|| p ≤V(t,x),

[0142] V(t,h i (x))≤α i V(t,x),

[0143]

[0144] For all i∈S, this impulsive random system is almost certainly exponentially stable.

[0145] in, Represents a constant, which is obtained based on the above inequality combined with the actual system parameters.

[0146] Proof: According to and It can be obtained that for any t>0,

[0147]

[0148] Similar analysis as in case 1 is applied, with

[0149]

[0150] Combined with this It can be seen that the impulse random system is almost necessarily exponentially stable.

[0151] Case 3: If the pulse jumps is independent and identically distributed, that is p i Denotes the probability that r(k) = i. Consider the impulse random system driven by the update process and the Markov chain in step 1. If there exists a Lyapunov function and constants c>0,p>0,αi >0, Make

[0152] c||x(t)|| p ≤V(t,x),

[0153]

[0154] V(t,x i (x))≤α i V(t,x),

[0155]

[0156] For all i∈S, this impulsive random system is almost certainly exponentially stable.

[0157] The proof process is the same as that of Case 2.

[0158] Case 4: For a single impulse jump, that is, S = {1}. Consider the impulse random system driven by the update process and Markov chain in step 1. If there exists a Lyapunov function and constants c>0,p>0,α i >0, Make

[0159] c||x(t)|| p ≤V(t,x),

[0160]

[0161] V(t,h i (x))≤α i V(t,x),

[0162]

[0163] If this holds, then this impulse random system is almost necessarily exponentially stable.

[0164] The proof process is the same as that of Case 2.

[0165] Case 5: If there is no pulse effect, that is, when pulse jump occurs, h r(k) (x(t - ))≡x(t - ) The state x remains unchanged, and the impulse random system driven by the update process and Markov chain in step 1 will degenerate into a random system:

[0166]

[0167] For this random system, if there exists a Lyapunov function and constants c>0,p>0, Make

[0168] c||x(t)|| p ≤V(t,x),

[0169]

[0170] λ+0.5γ 2 <0,

[0171] If holds, then this random system is almost certainly exponentially stable.

[0172] The proof process is the same as that of Case 2.

[0173] Example 1:

[0174] The present invention provides a new almost certain exponential stability determination method for a nonlinear random system with random pulses, comprising:

[0175] Consider the impulsive random system:

[0176]

[0177] Where, the system state x(t) = [x1(t), x2(t)] T , is a one-dimensional Brownian motion, and the pulse jump r(k) is a transition probability matrix

[0178] Markov chain. r(k) Represents the transition increment generated by the kth pulse.

[0179] Its invariant distribution can be calculated Stochastic process ζ k It is driven by the update process of parameter θ = 0.35. The shock pulse jump parameters are artificially set to d1 = 0.25, d2 = 0.16, and d3 = 1.44.

[0180] Choose Lyapunov function V(t,x)=x T (t)x(t), we can estimate

[0181]

[0182] T represents the transpose of a vector. They respectively represent the kth pulse (or jump) moment and the state of the system before the pulse occurs (left limit value).

[0183] From case 1 Can get Because V(t,x)=x T(t)x(t)≤x 2 (t), so the formula Squaring both sides of the equal sign gives

[0184] Next, the following two cases will be discussed to compare in detail Case 1 (considering the coupling effect of drift and diffusion terms) and Case 2 (without drift and diffusion term restrictions).

[0185] Case 1: Considering the coupling effect of drift and diffusion terms, constants γ = 1 and λ = 2.25 are selected (randomly given, and then the optimal values ​​are obtained through continuous modification and debugging), so that the condition Valid. Note that α1=0.25, α2=0.16, α3=1.44, and the calculated Therefore, it can be concluded that the impulse random system in Example 1 is almost necessarily exponentially stable.

[0186] Case 2: Consider the absence of drift and diffusion restrictions. Note that there is no constant γ>0 such that Select Constant and Make the conditions and However, Therefore, this situation cannot be used to verify the stability of the pulse random system in Example 1. That is, there is no corresponding constant in Case 2 Meet the conditions.

[0187] Compared with Case 2, the above analysis shows that Case 1 provides a less conservative method for analyzing the almost certain exponential stability of impulsive random systems. Given the initial value x0 = [4, -2] T , Figure 2 and Figure 3 The state trajectory diagrams of the pulse random system of Example 1 under 1 sample path and 40 sample paths are respectively described. It can be seen that the final state value tends to 0, that is, the pulse random system is almost necessarily exponentially stable.

[0188] Example 2:

[0189] The present invention provides an application of an almost exponential stability analysis method for a nonlinear random system with random pulses in chaotic synchronization. The application of this determination method in the synchronization of a master-slave random chaotic system will be discussed, including:

[0190] Consider the master control system, described as Figure 4 Chua's circuit shown perturbed by random noise:

[0191]

[0192] Among them, the nonlinear voltage-current relationship function in the state space model of Cai's circuit is and the system state x(t)=[x1(t),x2(t),x3(t)] T , x1(t), x2(t), x3(t) represent the voltage across capacitor C1, the voltage across capacitor C2, and the current in inductor L in the main drive circuit, respectively. Then, the Lipschitz parameter matrix can be selected as L=diag[27 / 7,0,0]. Select the parameter matrix

[0193]

[0194] Then the main control system can generate a double vortex attractor, such as Figure 5 As shown, the distribution of the typical chaotic "double scroll attractor" of the noisy Chua circuit system in two-dimensional and three-dimensional space is demonstrated.

[0195] The corresponding slave system is given as:

[0196]

[0197] Among them, y(t)=[y1(t),y2(t),y3(t)] T , y1(t), y2(t), y3(t) represent the voltage across capacitor C1, the voltage across capacitor C2, and the current in inductor L in the response circuit, respectively. The pulse control input is designed as:

[0198]

[0199] Where δ(·) is the Dirac function, e(t) = x(t) - y(t) is the synchronization error, and K represents the control gain matrix. Then, the dynamic equation of the synchronization error can be written as:

[0200]

[0201] Among them, the error function F(e(t))=f(x(t))-f(y(t)), ζ k is the random pulse control moment, that is, the pulse interval are identically distributed and

[0202] From the system equation It can be seen that the control input u(t) is in the form of a Dirac function at a specific time t = ζ k Apply "shock" to the slave system; In addition, there are random disturbance terms in the system, such as or here It usually represents the increment of a random process such as Brownian motion or Wiener process, which introduces random fluctuations in the system state. Therefore, the master system and the slave system are impulsive random systems.

[0203] Then, we can get that if there exists a number 0<α1<1,λ>0, Matrix P>0, and the diagonal matrix Q>0, so that

[0204]

[0205] If the master-slave random chaotic system is established, Pulse controller Index synchronization can be achieved under

[0206] Taking constants γ = 0.26, α1 = 0.26, (randomly given, and then the optimal value is obtained through continuous modification and debugging) we can get λ = 4.4, the matrix

[0207]

[0208] Q=diag{0.0011,3.8559,2.0462},

[0209]

[0210] Then, the control gain can be calculated Therefore, the average pulse control interval can be designed to be θ = 0.3. It can be calculated that From this we can get, such as Figure 6 As shown, it can be seen that the designed pulse can achieve synchronization of the master-slave random chaotic system.

[0211] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for analyzing the almost exponential stability of a nonlinear random system with random pulses, characterized by: The steps are as follows: Step 1: Establish a mathematical model of the pulse random system affected by random noise and random pulses; Step 2: New almost certain exponential stability conditions for impulsive stochastic systems driven by renewal processes and Markov chains under different conditions are given by the Lyapunov method.

2. The almost-certain exponential stability analysis method for a nonlinear random system with random pulses according to claim 1, characterized in that: The mathematical model of the pulse random system is: in, and are the initial value and the system state at time t, x(0) is the value of the system state at time 0, x(t - ) represents the left limit state value of the system at time t, dx(t) represents the differential increment of the state variable x(t), Represents a set of n-dimensional non-negative numbers; It is defined in q-dimensional Brownian motion on , Ω represents the sample space, Represents an event family, represents a filter, represents the probability measure, represents standard Brownian motion The differential increment of k is the kth one that satisfies 0=ζ0<ζ1<…<ζ k <… and lim k→∞ ζ k =∞; the pulse jump r(k) is taken from the set S = {1,2,…,r}, which determines which type of pulse jump will occur at the pulse time k, and r represents the total number of pulse jumps; the function and Continuous on t, locally Lipschitz continuous on x; and f(t,0)=0, g(t,0)=0; for each i∈S, the function is locally Lipschitz continuous on x, and h i (0) = 0, jump mapping h r(k) The type of is randomly selected by a Markov chain, represents a set of non-negative numbers, Represents the set of positive integers.

3. The almost-certain exponential stability analysis method for a nonlinear random system with random pulses according to claim 2, characterized in that: The random effect of pulse jumping is, the pulse time and pulse jumping It is a random process, during the pulse is independent and identically distributed and expected θ represents the expected value of the pulse duration; the number of pulses that have occurred before time t It is an updating process, sup represents the supremum; pulse jump is a transition probability matrix Π=[π ij ] r×r The discrete irreducible Markov chain, π ij Represents the conditional probability of the pulse type jumping from state i to state j, standard Brownian motion and random noise σ(t) are both filters The adaptation process of the pulse is independent of each other; k , pulse jump r(k) and standard Brownian motion Are independent of each other.

4. The almost-certain exponential stability analysis method for a nonlinear random system with random pulses according to claim 2 or 3, characterized in that: The new almost certain exponential stability conditions in step 2 include: Case 1: Consider an impulsive stochastic system driven by an update process and a Markov chain, taking into account the drift term f(t,x(t))dt and the diffusion term The coupling effect and the quantitative relationship between continuous dynamics and random pulse intensity and density are shown in Figure 2. If there is a Lyapunov function That is, the Lyapunov function V(t,x) is piecewise continuous, first-order continuously differentiable for t and second-order for x, and the constants c>0, p>0, α i >0, Make c||x(t)|| p ≤V(t,x), V(t,h i (x))≤α i V(t,x), For all i∈S, this impulsive random system is almost necessarily exponentially stable; in, Indicates piecewise continuity, || || p represents the p-norm, represents the Ito generator of the Lyapunov function V(t,x), represents the jump operator of the Lyapunov function V(t,x) under the jump or pulse condition, h i (x) represents the state jump map under the action of the i-th type pulse, θ represents the average pulse interval, α i represents the intensity of the i-th pulse, π i represents the stationary distribution of a Markov chain.

5. The almost-certain exponential stability analysis method for a nonlinear random system with random pulses according to claim 4, characterized in that: The new almost certain exponential stability condition in step 2 also includes: Case 2: Consider an impulsive stochastic system driven by an update process and a Markov chain, without the drift term f(t,x(t))dt and the diffusion term Limit, if there exists a Lyapunov function and constants c>0,p>0, α i >0, Make c||x(t)|| p ≤V(t,x), V(t,h i (x))≤α i V(t,x), For all i∈S, this impulsive random system is almost necessarily exponentially stable; where, Represents a constant.

6. The almost-certain exponential stability analysis method for a nonlinear random system with random pulses according to claim 5, characterized in that: The new almost certain exponential stability condition in step 2 also includes: Case 3: If the pulse jumps is independent and identically distributed, that is p i represents the probability that r(k) = i; considering an impulse random system driven by an update process and a Markov chain, if there exists a Lyapunov function and constants c>0,p>0,α i >0, Make c||x(t)|| p ≤V(t,x), V(t,h i (x))≤α i V(t,x), For all i∈S, this impulsive random system is almost certainly exponentially stable.

7. The almost-certain exponential stability analysis method for a nonlinear random system with random pulses according to claim 6, characterized in that: The new almost certain exponential stability condition in step 2 also includes: Case 4: For the case of a single impulse jump, that is, the set S = {1}; consider an impulse random system driven by an update process and a Markov chain. If there exists a Lyapunov function and constants c>0,p>0,α i >0, Make c||x(t)|| p ≤V(t,x), V(t,h i (x))≤α i V(t,x), If this holds, then this impulse random system is almost necessarily exponentially stable.

8. The almost-certain exponential stability analysis method for a nonlinear random system with random pulses according to claim 7, characterized in that: The new almost certain exponential stability condition in step 2 also includes: Case 5: If there is no pulse effect, that is, when a pulse jump occurs, the state x remains unchanged h r(k) (x(t - ))=x(t - ), the impulse stochastic system driven by the update process and the Markov chain will degenerate into a random system: For random systems, if there is a Lyapunov function and constants c>0,p>0, Such that c||x(t)|| p ≤V(t,x), λ+0.5γ 2 <0, If holds, then the random system is almost certainly exponentially stable.

9. The method for analyzing the almost certain exponential stability of a nonlinear random system with random pulses according to any one of claims 5 to 8, characterized in that: Consider the impulsive random system: Where, the system state x(t) = [x1(t), x2(t)] T , is a one-dimensional Brownian motion, the pulse jump r(k) is a Markov chain with a transition probability matrix π, d r(k) represents the transition increment generated by the kth pulse; Calculate the invariant distribution π, random process ζ k It is driven by the update process of parameter θ; set the shock pulse jump parameters d1, d2, d3; select Lyapunov function V(t,x)=x T (t)x(t), estimate T represents the transpose of the vector, They represent the kth pulse moment and the state of the system before the pulse occurs respectively; Case 1: Considering the coupling effect of drift and diffusion terms, constants γ and λ are selected so that the condition Established; select α1, α2, α3, calculate It can be concluded that the above-mentioned impulse random system is almost necessarily exponentially stable; Case 2: Considering that there is no drift term and diffusion term restriction, there is no constant γ>0, so that Select Constant and Make the conditions and established; however, It cannot be used to verify the stability of pulse random systems.

10. The almost-certain exponential stability analysis method for a nonlinear random system with random pulses according to any one of claims 5 to 8, characterized in that: Applications in chaos synchronization are: Considering the main control system, the Chua circuit disturbed by random noise is: Among them, the nonlinear voltage-current relationship function in the state space model of Cai's circuit is and the system state x(t)=[x1(t),x2(t),x3(t)] T , x1(t), x2(t), x3(t) represent the voltage across capacitor C1, the voltage across capacitor C2, and the current in inductor L in the main drive circuit respectively; then, select the Lipschitz parameter matrix as L=diag[27 / 7,0,0], select the parameter matrix Then the main control system can generate a double vortex attractor; The corresponding slave systems are: Among them, y(t)=[y1(t),y2(t),y3(t)] T , y1(t), y2(t), y3(t) represent the voltage across capacitor C1, the voltage across capacitor C2, and the current in inductor L in the response circuit, respectively. The pulse control input is: Where δ(·) is the Dirac function, e(t) = x(t) - y(t) is the synchronization error, and K represents the control gain matrix. Then, the dynamic equation of synchronization error is: Among them, the error function F(e(t))=f(x(t))-f(y(t)), ζ k is the random pulse control moment, the pulse period are identically distributed and The impulse control input u(t) is in the form of a Dirac function at a specific time t = ζ k Apply "shock" to the slave system; the master system and the slave system are pulse random systems; If there exists a number 0<α1<1,λ>0, Matrix P>0, and the diagonal matrix Q>0, so that If the master-slave random chaotic system is established, Pulse controller Index synchronization can be achieved under Take constants γ and α1 to get the parameters λ and matrix P, and diagonal matrix Q; calculate the control gain Set the expected θ during the average pulse control period and calculate The designed pulses realize the synchronization of master and slave random chaotic systems.

Images