Lyapunov method for nonlinear fractional differential systems with delay

Wen, Yanhua; Zhou, Xian‐Feng; Zhang, Zhixin; Liu, Song

doi:10.1007/s11071-015-2214-y

Cited by 61 publications

(28 citation statements)

References 29 publications

(42 reference statements)

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“…Lemma (Razumikhin‐type stability) Assume that

u, v, w : R^{+} \to R^{+}

are continuous nondecreasing functions, u ( s ) and v ( s ) are positive for s >0, and u (0)= v (0)=0, and q >1. If there exists a continuous function

V false(t, x false(t false) false) : R^{+} \times R^{n} \to double-struckR

such that (i)

u false(false‖ x false‖ false) \leq V false(t, x false) \leq v false(false‖ x false‖ false), t \in R^{+}, x \in R^{n}

and (ii) D α V ( t , x ( t ))≤− w (‖ x ( t )‖) if V ( t + s , x ( t + s ))< q V ( t , x ( t )),∀ s ∈[−τ,0], t ≥0, then the zero solution of system is asymptotically stable. The following theorem provides a sufficient condition guaranteeing that system is asymptotically stable.…”

Section: Resultsmentioning

confidence: 99%

“…Various results on fractional‐order systems have been reported. () It was found that a variety of physical and biological systems can be well characterized by fractional‐order differential equations, such as the fractional‐order Schrodinger equation in quantum mechanics and fractional‐order oscillator in damping vibration. () Fractional calculus offers more advantages than integer‐order calculus.…”

Section: Introductionmentioning

confidence: 99%

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Design of unknown‐input reduced‐order observers for a class of nonlinear fractional‐order time‐delay systems

Huong

Thuận

2017

Adaptive Control & Signal

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This paper considers the design of reduced-order state observers for fractional-order time-delay systems with Lipschitz nonlinearities and unknown inputs. By using the Razumikhin stability theorem and a recent result on the Caputo fractional derivative of a quadratic function, a sufficient condition for the asymptotic stability of the observer error dynamic system is presented. The stability condition is obtained in terms of linear matrix inequalities, which can be effectively solved by using existing convex algorithms. Numerical examples and simulation results are given to illustrate the effectiveness of the proposed design approach. KEYWORDS fractional-order systems, Lipschitz nonlinearities, reduced-order observers, time-delay systems INTRODUCTIONDuring the past decades, fractional-order systems have attracted considerable attention. Various results on fractional-order systems have been reported. 1-6 It was found that a variety of physical and biological systems can be well characterized by fractional-order differential equations, such as the fractional-order Schrodinger equation in quantum mechanics and fractional-order oscillator in damping vibration. 7-12 Fractional calculus offers more advantages than integer-order calculus. For example, there are many dynamical systems described by high integer-order systems that can be expressed more compactly into fractional operators. 13 This fact can be useful not only in the field of control systems but also in areas involving digital signal processing. Moreover, fractional-order systems have been found by directly generalizing integer-order derivatives into the corresponding fractional-order ones, in which way chaotic dynamics has been explored, such as fractional-order Chen's system and fractional-order cellular neural networks. 14 Furthermore, fractional-order controllers have so far been implemented to enhance the robustness and the performance of the closed-loop control systems.As it is well known, in many practical applications, the states of the considered systems are not easily obtained due to technical or economic reasons. In this case, the estimation of actual states and output feedback control law are very necessary. Therefore, the problem of designing state observers for dynamical systems with or without time delays has attracted considerable attention in the literature. [15][16][17][18][19][20][21][22][23][24][25] Observers theory for systems with integer order has been extensively investigated (see, for example, other works 17,18,19,25,26 ). For fractional-order systems, some interesting results on the observer design for fractional-order systems were also reported in other works. 16,20,21,23,24 For instance, a nonfragile fractional-order nonlinear observer design for a class of fractional-order nonlinear systems was derived in the work of Boroujeni and Momeni 16 ; reduced-order fractional descriptor observers for fractional-order descriptor continuous-time linear systems were reported in the work of Kaczorek 20 ; reduced-order perfect nonlinear observe...

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“…Lemma (Razumikhin‐type stability) Assume that

u, v, w : R^{+} \to R^{+}

are continuous nondecreasing functions, u ( s ) and v ( s ) are positive for s >0, and u (0)= v (0)=0, and q >1. If there exists a continuous function

V false(t, x false(t false) false) : R^{+} \times R^{n} \to double-struckR

such that (i)

u false(false‖ x false‖ false) \leq V false(t, x false) \leq v false(false‖ x false‖ false), t \in R^{+}, x \in R^{n}

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Design of unknown‐input reduced‐order observers for a class of nonlinear fractional‐order time‐delay systems

Huong

Thuận

2017

Adaptive Control & Signal

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“…□ Remark 5 For the case

A = 0

, similar stability conditions were given in [12–14] by using singular value decomposition of the characteristic equations. Moreover, it is worth noting that the asymptotic stability of the LFDDEs (1) can be verified by the Lyapunov stability theorem (Theorem 4.1 in [19] or Theorem 4 in [22]) using Lyapunov function method. However, the solution leads to solving LMIs depending either on the trace of the system matrices or on the positive definite matrix solution, which is not easy to solve and to verify the Lyapunov stability conditions (see, e.g.…”

Section: Resultsmentioning

confidence: 99%

“…In fact, if any Lyapunov functional is applied to the system (e.g. by Theorem 4.1 in [19] or by Theorem 4 in [22]), then the stability condition leads to an LMI of the form

(][, \begin{matrix} A P + P A^{normalT} + q P & B P \\ P B^{normalT} & - I \end{matrix}) < 0

where

q > 1, P

is a symmetric positive definite matrix. By the Schur complement lemma, this LMI implies

A^{normalT} P + P A < 0

, and hence by the Lyapunov equation stability theorem, matrix A is Hurwitz.…”

Section: Resultsmentioning

confidence: 99%

Stability analysis of fractional differential time‐delay equations

Thành

Trinh

Phát

2017

IET Control Theory & Applications

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“…In the following, for simplicity, we use a notation D α instead of C 0 D α t . Lemma 2 (Razumikhin-type stability Wen et al 2015) Assume that u, v, w : R + −→ R + are continuous nondecreasing functions, u(s), v(s) are positive for s > 0, and u(0) = v(0) = 0, and q > 1. If there exists a continuous function V (t, x(t)…”

Section: Problem Statement and Preliminariesmentioning

confidence: 99%

Distributed functional observers for fractional-order time-varying interconnected time-delay systems

Huong

2020

Comp. Appl. Math.

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Our main purpose in this paper is to design distributed functional observers for a class of fractional-order time-varying interconnected time-delay systems. The contributions of this paper conclude three aspects. First, we propose novel functional observers for subsystems of the fractional-order time-varying interconnected time-delay systems. The designed distributed functional observers in this paper work for a wider class of interconnected time-delay systems in the sense that when some existing design methods cannot be applied to estimate linear functions of the state vector, the distributed functional observers in this paper can still estimate linear functions of the original state vector. Then, we establish existence conditions for the proposed functional observers. Finally, we provide effective algorithms for computing unknown observer matrices.

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Lyapunov method for nonlinear fractional differential systems with delay

Cited by 61 publications

References 29 publications

Design of unknown‐input reduced‐order observers for a class of nonlinear fractional‐order time‐delay systems

Design of unknown‐input reduced‐order observers for a class of nonlinear fractional‐order time‐delay systems

Stability analysis of fractional differential time‐delay equations

Distributed functional observers for fractional-order time-varying interconnected time-delay systems

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