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Lyapunov functions for nabla discrete fractional order systems
Yiheng Wei, Yuquan Chen, Tianyu Liu, Yong Wang
TL;DR
Available Lyapunov-function fractional-difference inequalities leave gaps across definitions, while fractional derivatives do not satisfy the well-known Leibniz rule used in stability analysis. The paper proposes a new way to examine the asymptotic stability of nabla discrete fractional systems. The inequalities apply across the three definitions and support Lyapunov stability analysis of discrete fractional nonlinear systems, with three examples reported as demonstrating the theory’s effectiveness.
Problem
Available Lyapunov-function fractional-difference inequalities leave gaps across definitions, while fractional derivatives do not satisfy the well-known Leibniz rule used in stability analysis.
Source: §1 · pp. 1–2Method
The paper proposes a new way to examine the asymptotic stability of nabla discrete fractional systems.
Source: §1 · p. 2Results
The inequalities apply across the three definitions and support Lyapunov stability analysis of discrete fractional nonlinear systems, with three examples reported as demonstrating the theory’s effectiveness.
Source: §5 · pp. 17–18Takeaways & Limitations
The results provide a tool for analyzing discrete fractional nonlinear systems with or without delays.
Source: §5 · p. 18For newly defined time-varying-order differences, similar results are given for Caputo and Grunwald–Letnikov definitions, but the corresponding Riemann–Liouville result is hard to develop.
Source: §3.2 · p. 14
Abstract
from arXiv · showhide
This paper focuses on the fractional difference of Lyapunov functions related to Riemann-Liouville, Caputo and Grunwald-Letnikov definitions. A new way of building Lyapunov functions is introduced and then five inequalities are derived for each definition. With the help of the developed inequalities, the sufficient conditions can be obtained to guarantee the asymptotic stability of the nabla discrete fractional order nonlinear systems. Finally, three illustrative examples are presented to demonstrate the validity and feasibility of the proposed theoretical results.
1. Introduction
The introduction frames Lyapunov stability analysis as difficult for fractional-order systems and identifies gaps in discrete fractional-difference inequalities. It proposes a new stability-analysis approach and five inequalities applicable across three fractional-difference definitions.
1. Introduction
- Fractional calculus is studied in science and engineering for properties including nonlocality, long memory, and self-similarity.Prior work spans system modeling, automatic control, and adaptive filtering.Source: §1 · p. 1
- Fractional stability analysis is difficult because the Leibniz rule does not hold for fractional derivatives, and a prior two-step approach relies on an unproven convergence expression.That approach constructs a Lyapunov function using the exact state, then takes an integer-order derivative.Source: §1 · pp. 1–2
- Prior inequalities largely address Caputo derivatives, Riemann–Liouville results remain incomplete, and Grunwald–Letnikov Lyapunov-difference inequalities are scarce.The introduction also identifies a need for more approaches to designing the required Lyapunov functions.Source: §1 · p. 2
- The paper proposes a new way to examine asymptotic stability and develops five Lyapunov-function fractional-difference inequalities for three definitions.The definitions are Riemann–Liouville, Caputo, and Grunwald–Letnikov.Source: §1 · p. 2
2. Preliminaries
The preliminaries introduce discrete fractional-calculus definitions and foundational facts used in the paper. They include backward-difference identities, fractional difference and sum definitions, and Young’s inequality.
2. Preliminaries
- The preliminaries introduce the integer-order backward difference and state its first-order Leibniz rule in equivalent forms.The rule is used to derive summation-by-parts formulas.Source: §2 · p. 3
- The section defines the Grunwald–Letnikov fractional difference and sum, and introduces the nabla Riemann–Liouville and Caputo fractional differences.It also gives initial-value information and cites a related result for these definitions.Source: §2 · pp. 3–4
- Young’s inequality is included as a preliminary because it plays an important role in the work.The stated lemma gives a condition for equality.Source: §2 · p. 4
3. Main Results
The section establishes Lyapunov inequalities for nabla discrete fractional differences and extends related results to modified definitions with shifted initial instants, fixed memory, and time-varying order. The three main theorems share the same form, and the authors connect the inequalities to stability analysis.
3.1. Lyapunov inequalities
- The three theorems establish Lyapunov inequalities for the considered fractional-difference definitions, with related results also established for Riemann–Liouville differences.Theorem 1’s inequalities include scalar and multivariable cases under the stated conditions.Source: §3.1 · pp. 4, 10, 11
- The authors present alternatives to address drawbacks they identify in prior formulas and relate their results to discrete extensions and generalizations of earlier work.They describe some conditions in as harsh and impractical, and results in as questionable.Source: §3.1 · p. 9
3.2. Natural extension
- Modified fractional differences with a shifted initial instant retain the corresponding Theorems 1–3, and similar results hold for definitions with fixed memory.For α < 0, the modified Grünwald–Letnikov formula equals the Riemann–Liouville sum.Source: §3.2 · p. 14
- For time-varying order, similar results hold for the Grünwald–Letnikov and Caputo definitions, while the corresponding Riemann–Liouville result is difficult to develop and omitted.The authors do not provide the related context for that Riemann–Liouville result.Source: §3.2 · p. 14
3.1. Lyapunov inequalities
- Theorems 1–3 share the same form and build a bridge from fractional differences of Lyapunov functions to system equations for stability analysis.The authors describe this uniformity across the considered definitions as useful for analyzing the related systems.Source: §3.1 · pp. 4, 10, 11, §3.2 · p. 15
4. Numerical Simulation
Three examples apply the proposed Lyapunov method to nabla discrete fractional systems and an optimization algorithm. The reported outcomes are asymptotic stability, Lyapunov stability, and convergence of the algorithm to (1, 1).
4. Numerical Simulation
- The examples use the proposed method, which the authors say can check a system that methods in and cannot.The section presents three examples to demonstrate the method's convenience and efficiency for validating stability.Source: §4 · p. 15
- The system in (86) is asymptotically stable, with the origin as its equilibrium point.Its fractional difference of the Lyapunov function is reported as negative definite; Fig. 1 shows the state response.Source: §4 · p. 15, Figure 1 · p. 16
- The analysis concludes that the system in (89) is stable in the Lyapunov sense, and Fig. 2 illustrates this result with the system’s state response.Source: §4 · p. 16, Figure 2 · p. 17
- The algorithm in (93) is reported to find the desired point (1, 1) asymptotically.The corresponding curves in Fig. 3 illustrate the result.Source: §4 · p. 17
5. Conclusions
The paper investigates fractional differences of Lyapunov functions across three fractional-difference definitions and applies the resulting inequalities to stability analysis. Examples illustrate the approach, while several system classes remain directions for future work.
5. Conclusions
- The inequalities apply to Riemann–Liouville, Caputo, and Grünwald–Letnikov definitions and support classical Lyapunov analysis of discrete fractional nonlinear systems with or without delays.The paper reports three examples as evidence that the proposed theory is effective for checking stability.Source: §5 · pp. 17–18
- Figure 3 shows the search process of the algorithm in (93); three examples show the proposed theory is effective for checking stability of discrete fractional nonlinear systems, without and with delays.Future research directions include inequalities on distributed-order, incommensurate-order, singular, and time-delay systems.Source: Figure 3 · p. 18, §5 · p. 18