Controller and observer design for first order LTI systems with unknown dynamics

Abstract

The design of controllers and observers often relies on first order models of the system in question. These models are often obtained either through step-response tests, through on-line or off-line identification or through development of a mathematical model. When the system in question has unknown or uncertain parameters, the developed model also contains uncertainties and the controller/observer design may result in bad performance or even instability. In this paper we present a combined design of a controller and an observer for scalar linear time-invariant systems with unknown parameters. We combine a model reference adaptive controller, which does not require a model of the system, with a Luenberger observer which uses the desired closed-loop dynamics as its model. We show through Lyapunov theory and by application of Barbǎlat's lemma that all error states in the closed-loop system converge to zero and that all signals are bounded. CCS CONCEPTS • Computing methodologies → Control methods; Modeling and simulation; • Applied computing → Engineering;

Adaptive Controller and Observer Design Using Open and Closed-Loop Reference Models for Linear Time-Invariant Systems With Unknown Dynamics

“…This ensures a more predictable behaviour (in the sense of closer to that of the desired reference model) of the closedloop system. This work is a continuation of the work found in [15], where a solution for first-order systems is presented. Our method is referred to as a model reference adaptive controller and observer (MRACO) and the controller structure is depicted in Fig.…”

“…We now state the main result of this paper: Theorem 1 (Main result): For the systems (A, B, C) and (Am, B, C) satisfying Assumptions 1-3, also assume that ρ, P and Q 1 are chosen such that (13)- (15) hold.…”

“…Adjusting the eigenvalues of P , Q 1 , and M appropriately will reduce all bounds on the norms discussed in Corollary 1. However, doing so is difficult in light of also needing to satisfy the matching criterion (13)- (15). Even so, optimizing the choice of P , Q 1 , and M w.r.t.…”

“…It is now possible to find all P, ρ that satisfy (13)- (15) for any ΛK * T in some set numerically, as the solutions to the LMI problem is a convex set (Remark 2).…”

“…To verify if (15) has a solution, we now do a two-step procedure. The first step is to find candidate ρ and p that satisfies (13)- (15), for values of ΛK * T on some domainD. The second step is to verify that the candidates are suitable for the desired range of values of ΛK * T (D of (48)).…”