Synthesis of a robust controller with reduced dimension by the Loop Shaping Design Procedure and decomposition based on Laguerre functions

Abstract

International audienceIn this paper, we present the synthesis of a robust controller for uncertain discrete systems. The synthesis method of such a robust controller is the generalization of the Loop Shaping Design Procedure (LSDP) approach of McFarlane and Glover in the discrete case based on the work of Gu et al. We exploit the bilinear transform known as Tustin’s method in order to formulate the discrete loop shaping technique. A discrete weighting filter and a shaped discrete plant result from this technique. By taking into account the coprime factor uncertainty representation for the resulting shaped plant and by applying the small gain theorem, we define the concept of the robust stabilization of the discrete LSDP approach. This concept is based on the resolution of an optimization problem characterized by the maximum stability margin for the synthesis of the robust controller. To calculate the robust controller we transform this problem to a standard robust Formula controller design based on the resolution of the Riccati equations. Also, we present the gap metric theory to characterize the controller’s robustness. We note that the resulting final controller is the combination of the discrete weighting filter and the robust controller. We propose then to exploit the recent work of Bouzrara et al. in order to develop a reduced robust controller by expanding the final controller on two independent Laguerre orthonormal bases. The discrete LSDP and the reduced controller approaches were validated on a Continuous Stirred Tank Reactor chemical reactor for a set of different equilibrium points in order to take into account the nonlinearities

A multi-model control of nonlinear systems: A cascade decoupled design procedure based on stability and performance

“…where N ~ and M ~ are normalized left coprime factorization of P , that is, P ( s ) = M ~ − 1 N ~ and ‖ · ‖ H is Hankel norm. Inequalities in (6) and (8) have been used in Du and Johansen (2014b) and Salah et al (2016a, 2016b) to decompose the operation space of the nonlinear system.…”

“…The second stage, called combination, is to combine the designed local controllers as a global controller. There are two different approaches to build the global controller: hard switching (Salah et al, 2016a, 2016b) and soft switching (Du and Johansen, 2014a, 2014b; Du et al, 2012).…”

“…In the third category, both stability and performance are incorporated and considered concurrently to design the global multi-model controller (Ahmadi and Haeri, 2017, 2018; Du and Johansen, 2014b; Salah et al, 2016a, 2016b). In Salah et al (2016b), a new approach is presented to identify intermediate equilibrium points when the first and last ones are known.…”

“…In contrary to Ahmadi and Haeri (2017, 2018), Du and Johansen (2014b), Salah et al (2016a, 2016b), the integration of stability and performance was done in two independent steps which is a good deal with complexity of the decomposition step.…”

Robust feedback linearization of an isothermal continuous stirred tank reactor: H∞ mixed-sensitivity synthesis and DK-iteration approaches

“…In particular, robust techniques have been utilized to reduce the effect of parameter uncertainties (Palanki et al, 2003; Salehi and Shahrokhi, 2008). In Salah et al (2015a) the synthesis of a robust controller for uncertain discrete systems is proposed. The synthesis method of such a robust controller is the generalization of the loop shaping design procedure approach of McFarlane and Glover (1990) in the discrete case.…”