Multi-objective Synthesis Using LMI Formulations for Application of the Cutting Plane Algorithm

Abstract

The problem of designing a controller to meet different specifications or deciding that no such controller exists is addressed in this paper by linking three types of tools: the Youla parameterization allows searching for a controller in a convex set; formulations using Linear Matrix Inequalities (LMI) are proposed for different practical specifications; the corresponding convex problem is solved using a Cutting Plane Algorithm (CPA). Such an approach is developed by overcoming the problem of the huge number of additional variables which often occurs in the LMI framework, particulary when used in conjunction with the Youla parameterization. Its efficiency is discussed by considering two practical control problems.

Multi-objective controller design for a space launcher

“…Only the expression of the LMI for each constraint is given without specifying the procedure to get them, except for the delay margin: the derivation of the other constraints are presented in Ref. 13.…”

“…According to a slightly modified version of the Kalman-Yakubovich-Popov (KYP) lemma, 13 the closed-loop constraint σ(G zw ) < γ is satisfied for all s = jω in [jω 1 , jω 2 ] if and only if one of the two following equivalent propositions holds:…”

“…The manufacturer specifications contain three constraints on the stability margins: two of them concern the gain margins and one the delay margin. Using a suitable LFT form 13 enables to consider that the margin specification is satisfied if and only if the closed-loop plant remains stable for any scalar uncertainty δ ∈ [0, 1]. Thus the Nyquist criterion can be used on the closed-loop plant G zw looped again by −δ (figure 11).…”

“…12 However, if LMI formulations are well known for H 2 and H ∞ norm constraints or for pole placement for instance, 8 such constraints are only indirect translations of the specifications in most cases (for instance, pole placement handles imperfectly a requirement on the settling time). In this paper, recently proposed formulations 13 are used for time-domain templates, stability margins and constraints in a given frequency range, together with H 2norm constraints.…”

“…In this paper the problem of using the CPA is not detailed (more information can be found in Ref. 13), but a brief presentation of the algorithm is given, aiming to understand the different tools and results given in the paper. The paper is devoted to surround the control problem of a space launcher during the atmospheric stage, using new LMI formulations and a suitable analysis to decide the feasibility or unfeasibility of the specifications.…”