A globally fixed-time lyapunov stability analysis method

Abstract

The application discloses a global fixed-time Lyapunov stability analysis method, comprising the following steps: S1, a mathematical model of a general nonlinear system suitable for the method is given; S2, a Lyapunov function meeting specific conditions and used for judging the convergence performance of the system is constructed; S3, convergence time estimation is carried out for three different cases of a parameter Δ meeting the conditions of the Lyapunov function; different from convergence time estimation in the traditional fixed-time Lyapunov stability theory, the global fixed-time Lyapunov stability analysis method has less conservativeness in convergence time estimation.

Description

Technical Field

[0001] This invention relates to the technical field of nonlinear systems and fixed-time control, specifically to a global fixed-time Lyapunov stability analysis method. Background Technology

[0002] In the performance metrics of control systems, convergence performance is a crucial indicator. From the perspective of optimizing the convergence time of a control system, the control method that enables the closed-loop system to converge within a finite time is the time-optimal control method. Besides its optimal convergence performance, finite-time control, due to the presence of fractional power terms in the controller, gives finite-time closed-loop control systems better robustness and disturbance rejection compared to non-finite-time closed-loop control systems. Although finite-time control can enable the system to converge within a finite time, the convergence time is highly dependent on the initial state and increases with the increase of the initial state, which greatly limits its applicability.

[0003] Compared to finite-time control, fixed-time control retains the advantages of fast convergence speed and high convergence accuracy of finite-time control, while also possessing the characteristic that the convergence time does not depend on the initial state of the system. Therefore, it has attracted widespread attention from scholars. Due to these numerous advantages, the fixed-time control problem for nonholonomic systems has become a hot topic in recent years. Consequently, the fixed-time stability analysis of systems has also received widespread attention. The traditional Lyapunov fixed-time stability theorem only applies to the case where Δ = 0 for V(x) in S2; that is, when Δ = 0, for... This invention guarantees that V(x) = 0. Unlike the traditional Lyapunov fixed-time stability theorem, the global fixed-time Lyapunov stability analysis method proposed in this invention considers the Δ value under different conditions. Furthermore, it is shown later that the estimate of the system convergence time when Δ = 0 is smaller than that estimated in the traditional Lyapunov fixed-time stability theorem. This means that the time estimate of the practical fixed-time stability Lyapunov criterion proposed in this invention is less conservative than the time estimate in the traditional Lyapunov fixed-time stability theory. This invention proposes a global fixed-time Lyapunov stability analysis method with clear theoretical significance and high practical value. Simulation verification of this method is then presented. Summary of the Invention

[0004] To address the limitations of the traditional Lyapunov fixed-time stability theory, this invention proposes a global fixed-time Lyapunov stability analysis method. This invention considers the Δ value under different conditions in S2 and estimates the convergence time for each case. Through rigorous mathematical derivation, the optimal estimate of the system's convergence time is achieved when Δ = 0. This invention uses a general nonlinear system as an example for analysis. The technical solution of this invention is a global fixed-time Lyapunov stability analysis method, with the following specific steps:

[0005] A global fixed-time Lyapunov stability analysis method, the method comprising the following steps:

[0006] S1. For general nonlinear systems

[0007]

[0008] Where x∈R n It represents the system state, and f(t, x) is a smooth nonlinear function;

[0009] S2. Construct a positive definite, continuous, and radially unbounded Lyapunov function V(x) such that it satisfies:

[0010]

[0011] Where α, β, p, q, and Δ satisfy α > 0, β > 0, p < 1, q > 1, and Δ ≥ 0, respectively;

[0012] S3, targeting There are three scenarios, each allowing for estimation of the system's convergence time.

[0013] Furthermore, step S3 estimates the system convergence time for different Δ cases;

[0014] When Δ = 0, then the system's solution x(t) satisfies the condition for... The following holds true: V(x) = 0, where It is independent of the initial value and satisfies:

[0015]

[0016] When Δ > 0, then for any constant θ ∈ [0, 1) and The solution x(t) of the system satisfies the following condition for The following holds true: V(x)≤ε It is independent of the initial value and satisfies:

[0017]

[0018] when When the system's solution x(t) satisfies the condition for The following holds true: V(x)≤ε It is independent of the initial value and satisfies:

[0019]

[0020] Compared with the traditional Lyapunov fixed-time stability theorem, this invention has the following significant advantages:

[0021] The global fixed-time Lyapunov stability analysis method proposed in this invention considers the Δ value under different conditions compared with the traditional Lyapunov fixed-time stability theory, and therefore has a wider range of applications.

[0022] The global fixed-time Lyapunov stability analysis method proposed in this invention demonstrates that the estimated convergence time of the system when Δ=0 is smaller than that estimated in the traditional Lyapunov fixed-time stability theory. This is proven through rigorous mathematical derivation. This means that the time estimate of the actual fixed-time stability Lyapunov criterion proposed in this invention is less conservative than the time estimate in the traditional Lyapunov fixed-time stability theory, and has clear theoretical significance and high practical value. Attached Figure Description

[0023] Figure 1 This is a flowchart of a global fixed-time Lyapunov stability analysis method according to the present invention.

[0024] Figure 2 The simulation results show the system convergence time estimation under three initial states when Δ=0 according to the present invention.

[0025] Figure 3 The system convergence time of this invention under different initial states Detailed Implementation

[0026] To make the design concept and proof process of this invention clearer, the following discussion covers Lyapunov function construction and convergence time estimation, along with related appendices. Figure 1 The design process will be described in detail.

[0027] According to the design concept of this invention, Lyapunov fixed-time stability theory is used to determine system stability and estimate convergence time. Taking a general nonlinear system as an example, the specific technical solution is a global fixed-time Lyapunov stability analysis method, the specific steps of which are: 1) For a general nonlinear system, construct a positive definite, continuous, and radially unbounded Lyapunov function V(x), and make it satisfy specific conditions; 2) For the Lyapunov function constructed above, respectively in… The system convergence time is estimated in three cases; 3) Specific simulation examples verify that when Δ = 0, this invention achieves the optimal estimation of the system convergence time. This invention is mainly used for fixed-time analysis and convergence time estimation of nonlinear systems. To achieve the above objectives, the specific implementation steps of this invention are as follows:

[0028] S1. For general nonlinear systems

[0029]

[0030] Where x∈R n It represents the system state, and f(t, x) is a smooth nonlinear function;

[0031] S2. Construct a positive definite, continuous, and radially unbounded Lyapunov function V(x) such that it satisfies:

[0032]

[0033] Where α, β, p, q, and Δ satisfy α > 0, β > 0, p < 1, q > 1, and Δ ≥ 0, respectively;

[0034] S3, targeting There are three scenarios, each allowing for estimation of the system's convergence time.

[0035] Furthermore, step S3 estimates the system convergence time for different Δ cases;

[0036] When Δ = 0, then the system's solution x(t) satisfies the condition for... The following holds true: V(x) = 0, where It is independent of the initial value and satisfies:

[0037]

[0038] When Δ > 0, then for any constant θ ∈ [0, 1) and The solution x(t) of the system satisfies the following condition for The following holds true: V(x)≤ε It is independent of the initial value and satisfies:

[0039]

[0040] when When the system's solution x(t) satisfies the condition for The following holds true: V(x)≤ε It is independent of the initial value and satisfies:

[0041]

[0042] In step S3, the system convergence time is estimated through mathematical derivation and simulation verification.

[0043] First, we estimate the system convergence time when Δ = 0, assuming θ > 0 is any positive constant. It is important to note that:

[0044]

[0045] Integrating both sides of the above inequality, we can obtain:

[0046] β(q-1)t≤V 1-q (x)-V 1-q (x0)≤V 1-q (x)

[0047] Since 1-q<0, we can further obtain:

[0048]

[0049] According to the above inequality, when Then V(x)≤θ holds true.

[0050] Once V(x)≤θ holds, according to the finite-time stability theorem:

[0051]

[0052] We can obtain that V(x) = 0 holds true within a finite time t2, where t2 satisfies:

[0053]

[0054] Therefore, the system convergence time can be estimated as follows:

[0055]

[0056] It is important to note that when we take θ = 1, we can obtain:

[0057]

[0058] The above equation is consistent with the system convergence time estimate in the traditional Lyapunov fixed-time stability theory.

[0059] It is also important to note that we obtain... You can get S3

[0060] Next, we will demonstrate the findings of this invention. It should be smaller than the system convergence time T in the traditional Lyapunov fixed-time stability theory. max (θ=1), that is

[0061]

[0062] First, regarding T max Differentiating (θ) yields:

[0063]

[0064] It is important to note that when hour, Similarly, it is also necessary to pay attention to when Sometimes, when Sometimes, Therefore, it is easy to conclude:

[0065]

[0066] The above steps complete the estimation of the system convergence time and the proof of optimal convergence time when Δ = 0.

[0067] Next, the system convergence time for the case Δ>0 is estimated, and here is defined as follows: Then we can get:

[0068]

[0069] From the above equation, we can see that when V(x)≥ε, we have This holds true. This means that if the initial condition satisfies V(x0)≤ε, then V(x) will always remain in the set V(x)≤ε thereafter. Next, we only need to prove that when V(x0)≥ε, the set V(x)≤ε can be reached in a fixed amount of time.

[0070] Integrating both sides of the inequality in 66, we get:

[0071] β(q-1)t≤V 1-q (x)-V 1-q (x0)≤V 1-q (x), where 1-q<0, therefore we can obtain:

[0072]

[0073] From the above inequality, we can obtain:

[0074] for The following holds true: V(x)≤ε.

[0075] The above steps complete the estimation of the system convergence time for the case where Δ > 0.

[0076] Finally, Estimate the system convergence time under certain conditions, and define a constant. Easy to find Dangdang Sometimes, This means that if the initial conditions are satisfied... Established and If this holds true, then V(x) will always remain in the set. Next, we only need to prove that when At that time, the set can be reached within a fixed period of time.

[0077] Define a function as well as Then we can get The derivative:

[0078]

[0079] in Therefore, we can further conclude that:

[0080]

[0081] The above equation is similar to the case when Δ = 0. Using the same method, we can deduce that there exists a time... Make Established. Note: express Indicates any Satisfying set Established.

[0082] Note that We can obtain:

[0083]

[0084] This means that for any The statement V(x) ≤ ε holds true. Also note... It also satisfies the condition Δ>0, which means that for any The following holds true: V(x)≤ε.

[0085] Therefore, we can obtain that for any The following holds true: V(x)≤ε.

[0086] The above steps have completed the process of... Estimation of system convergence time under the given conditions.

[0087] Based on the analysis of the above stages, we will give the estimate of the system convergence time in the form of the following theorem:

[0088] Theorem 1: For general nonlinear systems

[0089]

[0090] Where x∈R n It represents the system state, and f(t, x) is a smooth nonlinear function;

[0091] Construct a positive definite, continuous, and radially unbounded Lyapunov function V(x) such that:

[0092]

[0093] Where α, β, p, q, and Δ satisfy α > 0, β > 0, p < 1, q > 1, and Δ ≥ 0, respectively;

[0094] against There are three scenarios, each allowing for estimation of the system's convergence time.

[0095] When Δ = 0, then the system's solution x(t) satisfies the condition for... The following holds true: V(x) = 0, where It is independent of the initial value and satisfies:

[0096]

[0097] When Δ > 0, then for any constant θ ∈ [0, 1) and The solution x(t) of the system satisfies the following condition for The following holds true: V(x)≤ε It is independent of the initial value and satisfies:

[0098]

[0099] when When the system's solution x(t) satisfies the condition for The following holds true: V(x)≤ε It is independent of the initial value and satisfies:

[0100]

[0101] Example

[0102] To verify the effectiveness of this invention, this section performs simulation verification on the following system with Δ = 0:

[0103]

[0104] Where a1, a2, a3, and a4 are the selected system parameters, respectively selected as a1 = 1, a2 = 1, a3 = 0.5, and a4 = 1.5, and d = 0 is the system disturbance.

[0105] The Lyapunov function V(x) of the system is selected as

[0106] Taking the derivative of V(x) and combining it with

[0102] , we can obtain

[0107]

[0108] Therefore, when Δ = 0, the parameters in

[0032] are respectively p = 3 / 4, q = 5 / 4, system convergence time estimation Convergence time estimates for three different initial states are as follows: Figure 2 .from Figure 2 It can be seen that the design method proposed in this invention can achieve a convergence time of no more than a fixed time under different initial states when Δ=0. from Figure 3 It can also be easily seen that as the initial state increases, the system convergence time gradually stabilizes, thus achieving an accurate estimate of the system convergence time.

[0109] This invention proposes a global fixed-time Lyapunov stability analysis method. It demonstrates that the estimated convergence time of the system when Δ = 0 is smaller than that estimated in traditional Lyapunov fixed-time stability theory, and this is proven through rigorous mathematical derivation. This means that the time estimate of the practical fixed-time stability Lyapunov criterion proposed in this invention is less conservative than the time estimate in traditional Lyapunov fixed-time stability theory, possessing clear theoretical significance and high practical value.

[0110] It should be understood that the above description is only a general procedure 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 shall be protected within the scope of protection of the present invention.

Claims

1. A global fixed-time Lyapunov stability analysis method, characterized in that: The method includes the following steps: S1. For general nonlinear systems Where x∈R n It represents the system state, and f(t, x) is a smooth nonlinear function; S2. Construct a positive definite, continuous, and radially unbounded Lyapunov function V(x) such that it satisfies: Where α, β, p, q, and Δ satisfy α>0, β>0, p<1, q>1, and Δ≥0, respectively; S3. Estimate the system convergence time for different Δ cases; 1) When Δ = 0, then the system's solution x(t) satisfies the following condition: The following holds true: V(x) = 0, where It is independent of the initial value and satisfies: 2) When Δ>0, then for any constant θ∈[0,1) and The solution x(t) of the system satisfies the following condition for The following holds true: V(x)≤ε It is independent of the initial value and satisfies: 3) When When the system's solution x(t) satisfies the condition for The following holds true: V(x)≤ε It is independent of the initial value and satisfies:

2. A global fixed-time Lyapunov stability analysis method according to any one of claims 1: characterized in that... The proposed global fixed-time Lyapunov stability analysis method proposes a more general Lyapunov theorem regarding practical fixed-time stability.

Images