Multimodel Discrete Second Order Sliding Mode Control: Stability Analysis and Real Time Application on a Chemical Reactor

The variable structure control is principally characterized by its robustness with respect to the system’s modeling uncertainties and external disturbances (Decarlo et al. (1988); Filippov (1960); Lopez & Nouri (2006); Utkin (1992)). Sliding Mode Systems are a particular case of the Variable Structure Systems (VSS). They are feedback systems with discontinuous gains switching the system’ s structure according to the state evolution, in order to maintain the trajectory within some specified subspace called the sliding surface (Utkin (1992)). However, the application of this control law is confronted to a serious problem. In fact, sliding mode necessitates an infinite switching frequency which is impossible to realize in numerical applications because of the calculation time and of the sensors dynamics that can not be neglected. The discontinuous control generates in that case oscillations on the state and on the switching function (Utkin (1992)). Owing to the many advantages of the digital control strategy (Ben Abdennour et al. (2001)), the discretization of the sliding mode control (SMC) has become an interesting research field. Unfortunately, the chattering phenomenon is more obvious in this case, because the sampling rate is more reduced. Many approaches have been suggested in order to resolve this last problem. Most of them propose a reduction in the oscillation amplitude at cost of the robustness of the control law (Utkin et al. (1999)). In the eighties, a new control technique, called high order sliding mode control, have been investigated. Its main idea is to reduce to zero, not only the sliding function, but also its high order derivatives. In the case of the r-order slidingmode control, the discontinuity is applied on the (r-1) derivative of the control. The effective control is obtained by (r-1) integrations and can, then, be considered as a continuous signal. In other words, the oscillations generated by the discontinuous control are transferred to the higher derivatives of the sliding function. This approach permits to reduce the oscillations amplitude, the notorious sliding mode systems robustness remaining intact (Levant (1993)). Another problem of the SMC is its vulnerability to external disturbances, parametric variations and non linearity, essentially, during the reaching phase. A solution to this problem, based on the multimodel approach, was proposed by the authors in (Mihoub et al. (2009a)). The combination of the multimodel approach and the second order discrete sliding mode control (2-DSMC) allows resolving both the chattering problem and the vulnerability during Multimodel Discrete Second Order Sliding Mode Control: Stability Analysis and Real Time Application on a Chemical Reactor 25