Robust periodic reference tracking for uncertain linear systems subject to control saturations

Abstract

This paper addresses the problem of tracking periodic references for uncertain linear systems subject to control saturation. Accordingly to the internal model principle, a control loop containing the modes of both the references and additive disturbances is considered. From this structure, conditions in a "quasi" LMI form are proposed to simultaneously compute a stabilizing state feedback gain and an anti-windup gain. Provided that the references and disturbances belong to a certain admissible set, these gains guarantee that the trajectories of the closed-loop system starting in a certain invariant ellipsoidal set contract to the linearity region of the closed-loop system, ensuring therefore the perfect reference tracking. Based on these conditions, an optimization problem aiming at the maximization of the invariant set of admissible states and/or the maximization of the set of admissible references/disturbances is proposed. Numerical examples to illustrate the method are provided.

Gain Scheduling Resonant Control with Anti-Windup Compensation for LPV Systems

“…In addition, the maximum values of reference/disturbances must be admissible, i.e., the saturation limits should not be violated in steady-state. To address these issues, it is proposed an ultimate boundedness framework based on the following sets (Flores et al, 2009): a set of initial conditions Z 0 ; a set of admissible references/disturbances signals Q 0 and a positively invariant terminal set J 0 ⊂ Z 0 included in the closed-loop linearity region S(F, u 0 ) {z ∈ R n+nc ; |F z| ≤ u 0 }. Hence, since inside J 0 the closed-loop system is linear, then the tracking/rejection is ultimately guaranteed by the IMP.…”

“…In this context, restrictions on the allowable control signal can significantly deteriorate transient performance, in the phenomenon of windup (Tarbouriech and Turner, 2009), or even lead the closed-loop system to instability. In terms of steady-state tracking performance, saturation imposes constraints on the amplitude of references/disturbances signals and initial conditions for which the tracking/rejection is guaranteed (Flores et al, 2009). Regarding IMP based controllers able to deal with tracking/rejection of periodic signals taking into account small frequency variation and input saturation we can cite (Ramos and Costa-Castelló, 2013) and (Flores and Flores, 2018).…”

Repetitive Control Design for MIMO Systems With Saturating Actuators

“…In this case, the designer must be aware that some reference signals can lead to trajectories that do not ensure the perfect tracking/rejection or can even lead to divergent trajectories. These particular issues are in part addressed in [18]- [20] for constant references, and [21], [22] for time-varying references. Regarding the repetitive control in the presence of constraints and anti-windup design we can cite [23], where the periodic nature of the steady state in the presence of constraints is ensured by an iterative Picard process and then an anti-windup controller is synthesized in order to recover the closed-loop performance under saturation.…”

Robust repetitive control with saturating actuators: a LMI approach

“…In the presence of constraints, the designer must be aware that some reference signals can lead to trajectories that does not ensure the perfect tracking/rejection or can even lead to divergent trajectories. These particular issues are in part addressed in [11], [12] and [13] for constant references, and [14] and [5] for time-varying references. Regarding the repetitive control in the presence of constraints and antiwindup design we can cite [15], where the periodic nature of the steady state in the presence of constraints is ensured by an iterative Picard process and then an anti-windup controller is synthetised in order to recover the closed-loop performance under saturation.…”

“…Regarding the problem of periodic reference tracking, the sinusoidal internal model is commonly used in the so called resonant approach, to ensure the sinusoidal tracking in power converters [3], [4]. Also, this sinusoidal model can be used to describe any periodic reference as a sum of sinusoids, at the expense of high order controllers [5].…”