Lecture 4 Supplemental Material

2. Direct Adaptive Control of Scalar Plant. Let us begin with a simple scalar adaptive system, ẋp(t) = apxp(t) + kpu(t) (1) where xp(t) ∈ R is the plant state, u(t) ∈ R is the control input, ap ∈ R is an unknown scalar and only the sign of kp ∈ R is known. For the direct adaptive controller we choose a reference model as ẋm(t) = amxm(t) + kmr(t)− l(x(t)− xm(t)). (2) All of the parameter above are known and scalar, xm(t) is the reference model state, r(t) is a bounded reference input. We choose am, l < 0 so that the reference model and the subsequent error dynamics will be stable. Compared to the direct adaptive controller we presented in Lecture 3, we now need to distinguish between Open-loop Reference Models (ORM) and Closed-loop Reference Models (CRM): ORM : ẋm(t) = amxm(t) + kmr(t) (3) CRM : ẋm(t) = amxm(t) + kmr(t)− l(x(t)− xm(t)). (4) The control law is chosen as u(t) =θ(t)xp(t) + k(t)r(t) =θ̄ (t)φ(t) (5)