Stabilization of Continuous-Time Fractional Positive Systems by Using a Lyapunov Function

Benzaouia, Abdellah; Hmamed, Abdelaziz; Mesquine, Fouad; Benhayoun, Mohamed; Tadeo, Fernando

doi:10.1109/tac.2014.2303231

Cited by 59 publications

(27 citation statements)

References 25 publications

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“…Especially, in [26], a simple Lyapunov function V = λ T x, λ > 0 has been proposed to solve the stabilization problem for fractional-order linear positive systems. By constructing some suitable stochastic Lyapunov functions Agarwal et al [27] established some sufficient conditions for two types of stability of stochastic differential equations.…”

Section: Introductionmentioning

confidence: 99%

“…Hu et al [29] considered an integerorder derivative instead of the fractional-order derivative of a Lyapunov function to prove the revised Lyapunov stability theorems. Nevertheless, the proposed Lyapunov functions [21][22][23][24][26][27][28][29] are valid only for some fractional-order systems with special characteristics. In classic Lyapunov theory, the quadratic form is one of the most commonly used Lyapunov functions for general integer-order nonlinear systems.…”

Section: Introductionmentioning

confidence: 99%

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Stabilization of a class of fractional-order nonautonomous systems using quadratic Lyapunov functions

Zhuang

et al. 2018

Adv Differ Equ

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In this paper, we aim to solve the stabilization problem for a large class of fractional-order nonautonomous systems via linear state feedback control and adaptive control. By constructing quadratic Lyapunov functions and utilizing a new property for Caputo fractional derivative we derive some sufficient conditions for the global asymptotical stabilization of a class of fractional-order nonautonomous systems. We give two illustrative examples to validate the effectiveness of the theoretical results.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stabilization of a class of fractional-order nonautonomous systems using quadratic Lyapunov functions

Zhuang

et al. 2018

Adv Differ Equ

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“…The minimum energy control problem for positive fractional electrical circuits has been investigated in [24] and for positive fractional linear systems with two different fractional orders in [28]. Robust stability and stabilization of the continuous-time fractional positive systems has been considered in [29,30] and continuous-time fractional positive systems with bounded states in [31]. …”

Section: Introductionmentioning

confidence: 99%

Minimum energy control of fractional positive continuous-time linear systems using Caputo-Fabrizio definition

Kaczorek

2017

Bulletin of the Polish Academy of Sciences Technical Sciences

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In this paper, the Caputo-Fabrizio definition of the fractional derivative will be applied to the minimum energy control problem for fractional positive continuous-time linear systems with bounded inputs.The paper is organized as follows. In Section 2 the conditions for the reachability of the standard and positive fractional linear continuous-time systems will be given. The minimum energy control problem for the fractional positive continuous-time linear systems with bounded inputs is formulated and solved in Section 3. Procedure for computation of the optimal input that steers the state of the system from zero initial state to the desired final state is given and illustrated by example of positive fractional electrical circuit in Section 4. Concluding remarks are given in Section 5.The following notation will be used: ℜ -the set of real numbers, ℜ n×m -the set of n£m real matrices, ℜ + n£m -the set of n£m matrices with nonnegative entries and ℜ + n = ℜ + n£1 , M n -the set of n£n Metzler matrices, I n -the n£n identity matrix. Reachability of standard fractional systemsThe Caputo-Fabrizio definition of fractional derivative of order α of the function f(t) for 0 < α < 1 has the form [32,33] 1 BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. XX, No. Y, 2016 DOI: 10.1515/bpasts-2016 Minimum energy control of fractional positive continuous-time linear systems using Caputo-Fabrizio definition Abstract. The Caputo-Fabrizio definition of the fractional derivative is applied to minimum energy control of fractional positive continuous-time linear systems with bounded inputs. Conditions for the reachability of standard and positive fractional linear continuous-time systems are established. The minimum energy control problem for the fractional positive linear systems with bounded inputs is formulated and solved.

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“…In [38], the robust stabilization problem was explored for continuoustime FPS with bounded control. Stabilization problem was studied for continuous-time FPS by using a Lyapunov function in [39]. In [40], a H ∞ model reduction problem was studied for FPS.…”

Section: Introductionmentioning

confidence: 99%

Stabilization Control of Continuous-Time Fractional Positive Systems Based on Disturbance Observer

Shao

2016

IEEE Access

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This paper studies the problem of stabilization control for a class of continuous-time fractional systems subject to external constant disturbances based on a disturbance observer. To tackle unknown disturbances, a fractional disturbance observer (FDO) is introduced for the fractional system. A stabilization controller incorporating the disturbance observer is then designed for the fractional system. Under the developed control scheme and linear program (LP) method, the sufficient conditions are proposed to guarantee that the fractional systems with closed-loop positivity are asymptotically stable. Numerical simulation results further demonstrate the effectiveness of the results.

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Stabilization of Continuous-Time Fractional Positive Systems by Using a Lyapunov Function

Cited by 59 publications

References 25 publications

Stabilization of a class of fractional-order nonautonomous systems using quadratic Lyapunov functions

Stabilization of a class of fractional-order nonautonomous systems using quadratic Lyapunov functions

Minimum energy control of fractional positive continuous-time linear systems using Caputo-Fabrizio definition

Stabilization Control of Continuous-Time Fractional Positive Systems Based on Disturbance Observer

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