Problem 1. Consider the unstable plant . (a) Find a characterization of the set of all controllers stabilizing the feedback system (C,P):
where N,D,X,Y,∈ H∞ are such that N(s)X(s) + D(s)Y (s) = 1 and P(s) = N(s)/D(s). (b) Find a controller C(s) stabilizing (C,P), and satisfying the following steady state performance conditions: • steady state error for a unit step reference input is zero • steady state error for a sinusoidal input of the form r(t) = sin(3t), t ≥ 0, is zero. Implement this feedback system in Simulink and illustrate that performance conditions are satisfied. Problem 2. For the nominal plant given in Problem 1 consider the following set of uncertain plants: P = {P∆ = P(1 + ∆m) : P∆ has 2 poles in C+ , |∆m(jω)| < |Wm(jω)| , ∀ ω } where Wm(s) = δ (s + 1). (a) Find the largest δ 0 for which there exists a controller C stabilizing (C,P∆) for all P∆ ∈ P; and determine the corresponding optimal controller, Copt. (b) With the largest δ computed above, pick an arbitrary element P∆ 6= P in the set P, and prove that (Copt,P∆) is indeed stable (find the location of the closed loop system poles and determine the stability margins of this system i.e. gain, phase, delay and vector margins).
