Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems

This paper presents a second order P-type iterative learning control (ILC) scheme with initial state learning for a class of fractional order linear distributed parameter systems. First, by analyzing the control and learning processes, a discrete system for P-type ILC is established, and the ILC design problem is then converted to a stability problem for such a discrete system. Next, a sufficient condition for the convergence of the control input and the tracking errors is obtained by introducing a new norm and using the generalized Gronwall inequality, which is less conservative than the existing one. Finally, the validity of the proposed method is verified by a numerical example.

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“…Furthermore, applying the (L 2 , λ, ξ)-norm to Inequality (35), it yields (17), which completes the proof.…”

Section: Lemmamentioning

confidence: 56%

ILC with Initial State Learning for Fractional Order Linear Distributed Parameter Systems

Lan

Cui

2018

Algorithms

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“…[2][3][4][5][6][7] Recently, fractional-order calculus has received a particular interest from physicists and engineers because it has some interesting and special properties compared with the integer-order one. [8][9][10][11][12][13][14][15][16][17][18][19][20] In reality, lots of actual systems can be described by fractional-order differential equations due to their unusual properties. On the other hand, fractional-order calculus has been employed to establish the system models in many domains, for example, biophysics, physics, engineering, aerodynamics, blood flow phenomena, biology, control theory, electron-analytical chemistry.…”

Section: Introductionmentioning

confidence: 99%

Prescribed performance synchronization controller design of fractional-order chaotic systems: An adaptive neural network control approach

Jiao

2017

AIP Advances

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In this study, an adaptive neural network synchronization (NNS) approach, capable of guaranteeing prescribed performance (PP), is designed for non-identical fractional-order chaotic systems (FOCSs). For PP synchronization, we mean that the synchronization error converges to an arbitrary small region of the origin with convergence rate greater than some function given in advance. Neural networks are utilized to estimate unknown nonlinear functions in the closed-loop system. Based on the integer-order Lyapunov stability theorem, a fractional-order adaptive NNS controller is designed, and the PP can be guaranteed. Finally, simulation results are presented to confirm our results.

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“…⊲ continuously differentiable Lyapunov functions (see, for example, the papers [1], [3], [6], [7], [12], [10]). Different types of stability are discussed using the Caputo derivative of Lyapunov functions.…”

Section: Introductionmentioning

confidence: 99%

“…Stability for the zero solution of fractional nonlinear equations is studied in [1], [6], which requires differentiability of the applied Lyapunov function. Also, the fractional derivative of the Lyapunov function depends significantly on any solution of the given fractional equation.…”

Section: Introductionmentioning

confidence: 99%

Stability of Caputo fractional differential equations by Lyapunov functions

2015

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Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems

Cited by 630 publications

References 20 publications

ILC with Initial State Learning for Fractional Order Linear Distributed Parameter Systems

ILC with Initial State Learning for Fractional Order Linear Distributed Parameter Systems

Prescribed performance synchronization controller design of fractional-order chaotic systems: An adaptive neural network control approach

Stability of Caputo fractional differential equations by Lyapunov functions

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