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Unformatted text preview: Problem Set 11 Solutions April 5, 2010 Problem 1: Robust Tracking Problem (a) 1. Feasibility . The problem is feasible because no zero of the plant is a pole of the exosystem. 2. Preparation . We must find the state equations of the plant ˙ x = Ax + Bu, x ∈ R n y = Cx. and of the exosystem ˙ w = Sw , w ∈ R q y d = C d w . In this case we have ˙ x = x + u y = x. ˙ w = bracketleftbigg 1 bracketrightbigg w y d = [ 1 0 ] w . 3. Pole placement for K . We want to find K such that σ ( A + BK ) = { 1 } . We obtain K = 2. 4. Regulator equations . We must solve Π S = A Π + B Γ C Π = C d for the unknowns Π ∈ R n × q and Γ ∈ R 1 × q . This yields Π = bracketleftbig 1 bracketrightbig , Γ = bracketleftbig 1 1 bracketrightbig . Therefore the (nonrobust) asymptotic tracking controller is: u = Γ w + K ( x Π w ) = 1 + t 2 x. Note that this controller contains a feedforward and feedback piece. 5. Pole placement for G . We must find G such that σ parenleftbiggbracketleftbigg A S bracketrightbigg bracketleftbigg G 1 G 2 bracketrightbigg bracketleftbig C C d bracketrightbig parenrightbigg = { 1 , 2 , 3 } , where G 1 ∈ R n and G 2 ∈ R q . Using the method of matching coefficients this yields G =  24 17 6 . 1 6. Robust regulator . The robust regulator is ˙ ξ = bracketleftbigg A + BK + G 1 C B (Γ K Π) G 1 C d G 2 C S G 2 C d bracketrightbigg ξ + Ge u = Γ ξ 2 + K ( ξ 1 Π ξ 2 ) . This yields ˙ ξ =  25 25 1 17 17 1 6 6 ξ +  24 17 6 e u = bracketleftbig 2 1 1 bracketrightbig ξ . In transfer function form this is C ( s ) = U ( s ) E ( s ) = 25 s 2 + 17 s + 6 s 2 ( s + 8) . Notice that the internal model principle is satisfied because C ( s ) has a copy of the poles of the exosystem. (b) 1. Feasibility . The problem is feasible because no zero of the plant is a pole of the exosystem. 2. Preparation . We must find the state equations of the plant ˙ x = Ax + Bu, x ∈ R n y = Cx. and of the exosystem ˙ w = Sw , w ∈ R q y d = C d w . In this case we have ˙ x = x + u y = x. ˙ w = bracketleftbigg 2 2 bracketrightbigg w y d = [ 1 0 ] w ....
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 Spring '11
 Francis
 Pole Placement, robust regulator

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