Stabilization of fractional order systems using a finite number of state feedback laws

Abstract

In this paper, the stabilization of linear timeinvariant systems with fractional derivatives using a limited number of available state feedback gains, none of which is individually capable of system stabilization, is studied. In order to solve this problem in fractional order systems, the linear matrix inequality (LMI) approach has been used for fractional order systems. A shadow integer order system for each fractional order system is defined, which has a behavior similar to the fractional order system only from the stabilization point of view. This facilitates the use of Lyapunov function and convex analysis in systems with fractional order 1 < q < 2. To this end, an extremumseeking method is used for obtaining Lyapunov function and defining a suitable sliding sector in order to enable switching between available control gains for system stabilization. Consequently, using the LMI approach in fractional order systems, necessary and sufficient conditions are provided for stabilization of systems with fractional order 1 < q < 2 using a limited number of available state feedback gains which lead to variable structure control.

Observer‐based stabilisation of a class of fractional order non‐linear systems for 0 < α <2 case

“…For control systems, the stability problem is essential. The stability and stabilisation problem of fractional order control systems have been investigated in [10][11][12][13][14][15][16][17]. The stabilisation problem of fractional order linear time invariant switched systems and the stabiliser switching law were proposed in [10,11].…”

“…The stability and stabilisation problem of fractional order control systems have been investigated in [10][11][12][13][14][15][16][17]. The stabilisation problem of fractional order linear time invariant switched systems and the stabiliser switching law were proposed in [10,11]. The stability problems of fractional order interval systems were investigated in [12][13][14][15][16], and the sufficient and necessary stability conditions obtained in [14,15] were proposed in terms of linear matrix inequality, which was very convenient to be utilised in practice.…”

Stabilization of autonomous linear time invariant fractional order derivative switched systems with different derivative in subsystems

“…In [32], some stabilization issue for fractional order switching systems of fractional order linear systems has been addressed. In [17] the stabilization of linear time invariant systems with fractional derivatives using a limited number of available state feedback gains, using switching method is studied. Author in [5] has established the sufficient conditions for the asymptotic stability of positive fractional switched continuous-time linear systems for any switchings.…”

Fractional order modelling of fractional-order holds

“…A huge range of potential applications emerges by embedding the FC concepts with physics and engineering, and presently active research is being pursued [7][8][9][10][11][12][13][14][15][16][17][18]. In spite of this state of affairs, the formalism of FC is not yet widely known, there remain many phenomena whose study can be more deeply accomplished in the light of FC.…”