[Solved] EE444/544 Homework #3

Consider the unstable plant

• 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).

• 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(j)| < |Wm(j)| , }

where

Wm(s) = (s + 1).

• Find the largest > 0 for which there exists a controller C stabilizing (C,P) for all P P; and determine the corresponding optimal controller, C_opt.
• With the largest computed above, pick an arbitrary element P 6= P in the set P, and prove that (C_opt,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).