Continuous finite-time stabilization for some classes of fractional order dynamics

This paper analyzes some basic issues involving the application of discontinuous control techniques for controlling fractional order systems (FOS). With reference to a simple class of SISO perturbed processes, we compare the performance of two distinct approaches, namely the “fractional sliding manifold” (FSM) approach, and the “fractional dynamic input extension” (FDIE) approach. The latter, which is the main new contribution of this paper, is shown to provide for the finite time convergence property of the systems' state. Remarkably, this approach also attenuates the chattering phenomenon by leading to a continuous, although obviously non-smooth, control action. The performance of the two methodologies under investigation are determined by means of Lyapunov approach. Additionally, we derive two new results involving the application of second-order sliding mode control techniques in the context of SISO FOS, and the application of the unit-vector methodology for controlling uncertain multivariable FOS. The presented analysis is supported by simulation results.

[1]  Alessandro Pisano,et al.  Second-order sliding mode approaches to disturbance estimation and fault detection in fractional-order systems , 2011 .

[2]  Y. Q. Chen,et al.  Using Fractional Order Adjustment Rules and Fractional Order Reference Models in Model-Reference Adaptive Control , 2002 .

[3]  Alexander I. Saichev,et al.  Distributions in the physical and engineering sciences , 1997 .

[4]  Samir Ladaci,et al.  On Fractional Adaptive Control , 2006 .

[5]  Duarte Valério,et al.  Fractional sliding mode control , 2012 .

[6]  Maamar Bettayeb,et al.  A sliding mode control for linear fractional systems with input and state delays , 2009 .

[7]  A. Levant.  Sliding order and sliding accuracy in sliding mode control , 1993 .

[8]  Cosku Kasnakoglu,et al.  A fractional adaptation law for sliding mode control , 2008 .

[9]  I. Podlubny.  Fractional differential equations , 1998 .

[10]  Mehmet Önder Efe,et al.  Fractional Order Sliding Mode Controller Design for Fractional Order Dynamic Systems , 2010 .

[11]  O. Agrawal,et al.  Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering , 2007 .

[12]  S. V. Baida.  Unit sliding mode control in continuous- and discrete-time systems , 1993 .

[13]  Jaime A. Moreno,et al.  A Lyapunov approach to second-order sliding mode controllers and observers , 2008, 2008 47th IEEE Conference on Decision and Control.

[14]  Alessandro Pisano,et al.  Sliding mode control approaches to the robust regulation of linear multivariable fractional‐order dynamics , 2010 .

[15]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[16]  Christopher Edwards,et al.  Sliding mode control : theory and applications , 1998 .