Dynamics and Control of Nonlinear Variable Order Oscillators

Abstract

The denomination Fractional Order Calculus has been widely used to describe the mathematical analysis of differentiation and integration to an arbitrary non-integer order, including irrational and complex orders. First proposed around three hundred years ago, it has attracted much interest during the past three decades (Oldham & Spanier (1974), Miller & Ross (1993), Podlubni (1999)). The increased interest in fractional systems in the past few decades is due mainly to a large body of physical evidence describing fractional order behavior in diverse areas such as fluid mechanics, mechanical systems, rheology, electromagnetism, quantitative finances, electrochemistry, and biology. Fractional order modeling provides exceptional capabilities for analysing memory-intense and delay systems and it has been associated with the exact description of complex transport phenomena such as fractional history effects in the unsteady viscous motion of small particles in suspension (Coimbra et al. 2004, L’Esperance et al. 2005). Although fractional order dynamical and control systems were studied only marginally until a few decades ago, the recent development of effective mathematical methods of integration of non-integer order differential equations (Charef et al. (1992); Coimbra & Kobayashi (2002), Diethelm et al. (2002); Momany (2006), Diethelm et al. (2005)) has resulted in a number of control schemes and algorithms, many of which have shown better performance and disturbance rejection compared to other traditional integer-order controllers (Podlubni (1999); Hartly & Lorenzo (2002), Ladaci & Charef (2006), among others). Variable order (VO) systems constitute a generalization of fractional order representations to functional order. In VO systems the order of the derivative changes with respect to either the dependent or the independent variables (or both), or parametrically with respect to an external functional behavior (Samko & Ross, 1993). Compared to fractional order applications, VO systems have not received much attention, although the potential to characterize complex behavior by the functional order of differentiation or integration is clear. Variable order formulations have been utilized, among other applications, to describe the mechanics of an oscillating mass subjected to a variable viscoelasticity damper and a linear spring (Coimbra, 2003), to analyze elastoplastic indentation problems (Ingman & Suzdalnitsky (2004)), to interpolate the behavior of systems with multiple fractional terms (Soon et al., 2005), and to develop a statistical mechanics model that yields a macroscopic constitutive relation for a viscoelastic composite material undergoing compression at varying strain rates (Ramirez & Coimbra, 2007). Concerning the dynamics and control of VO