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

In this paper, the stabilization of linear time-invariant 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 extremum-seeking 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.