An Lmi–Based Heuristic Algorithm for Vertex Reduction in LPV Systems

Abstract

The linear parameter varying (LPV) approach has proved to be suitable for controlling many non-linear systems. However, for those which are highly non-linear and complex, the number of scheduling variables increases rapidly. This fact makes the LPV controller implementation not feasible for many real systems due to memory constraints and computational burden. This paper considers the problem of reducing the total number of LPV controller gains by determining a heuristic methodology that combines two vertices of a polytopic LPV model such that the same gain can be used in both vertices. The proposed algorithm, based on the use of the Gershgorin circles, provides a combinability ranking for the different vertex pairs, which helps in solving the reduction problem in fewer attempts. Simulation examples are provided in order to illustrate the main characteristics of the proposed approach.

A recursive LMI-based algorithm for efficient vertex reduction in LPV systems

“…(iv) Polytope size reduction, by discarding infeasible combinations of time-varying parameters based on their interdependency (Robert, Sename, & Simon, 2009). (v) LMI-based reduction, by designing a common controller gain for a set of vertices based on analysing how much the LMIs, which represent a set of specifications, are modified by this common controller (Sanjuan et al, 2019).…”

“…This last category of approaches is still premature, since the recent (and sole available) work (Sanjuan et al, 2019) only set the basis for its development towards a more mature stage. More specifically, (Sanjuan et al, 2019) provided a heuristic methodology to combine two vertices of an LPV polytopic model.…”

“…This last category of approaches is still premature, since the recent (and sole available) work (Sanjuan et al, 2019) only set the basis for its development towards a more mature stage. More specifically, (Sanjuan et al, 2019) provided a heuristic methodology to combine two vertices of an LPV polytopic model. Hence, the number of vertex controllers could be reduced from n v to n v −1, which was deemed to be a first necessary step to develop an iterative algorithm for achieving a bigger reduction.…”

“…Hence, the number of vertex controllers could be reduced from n v to n v −1, which was deemed to be a first necessary step to develop an iterative algorithm for achieving a bigger reduction. The heuristic algorithm proposed in (Sanjuan et al, 2019) was based on the use of Gershgorin circles to analyse the perturbation of LMIs when two vertices are combined together, in the sense of a common controller gain being used for both.…”

“…The main goal of the present work is to develop an algorithm that allows considering generic tuples of vertices, for which a common controller gain is used, in order to achieve an overall reduction from n v vertices to a much smaller number n c . This is attained by starting with the algorithm in (Sanjuan et al, 2019), but with two important modifications, that are the main contributions of this paper:…”