Linear Robust Control (Prentice Hall Information and System by Michael Green;David J. N. Limebeer
By Michael Green;David J. N. Limebeer
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To see this we observe that (I + ∆1 )G K = (I + ∆1 )GK and (I − ∆2 )−1 G K = (I − ∆2 )−1 GK. 8: ❄ ❢✛ Loop with multiplicative model error. 4 Let G and K be given rational transfer function matrices. 8 is internally stable if the following conditions are satisfied: 1. 4 ROBUST STABILITY ANALYSIS 2. The model error ∆1 is a rational transfer function matrix such that G and (I + ∆1 )G have the same number of poles in the closed-right-half plane; 3. The model error ∆1 satisfies σ ∆1 (s) < 1 σ GK(I − GK)−1 (s) for all s ∈ DR .
Consider the frequency weighted disturbance attenuation problem of finding a stabilizing controller that minimizes w(1 − gk)−1 ∞ . If g= s−α s+2 , w= s+4 2(s + 1) , in which α is real, show that when 0 ≤ α ≤ 2 there is no stabilizing controller such that |(1 − gk)−1 (jω)| < |w−1 (jω)|, for all ω. 3. Consider the command tracking problem in which g= (s − 1)2 (s + 2)(s + 3) , h= 1 . s+4 Show that the error e = h − gf must satisfy the interpolation constraints e(1) = 1 , 5 de −1 (1) = . ds 25 The construction of such an e requires the solution of an interpolation problem with derivative constraints.
2. Consider the unstable plant g= s−α s−1 , α = 1, which has a zero at α. As with the previous example, we require a= 1+s 1−s which gives s+1 s−α −ag −1 = . The only interpolation constraint is therefore ˜ (1) = −ag −1 q s=1 = 2 . 1−α ˜ = 2/(1 − α) as the optimal Invoking the maximum modulus principle yields q interpolating function. 3) gives k= 2 1+α as the optimal controller. The closed loop will therefore be stable for all stable δ such that δ ∞ < |(1 − α)/2|. From this we conclude that the stability margin measured by the maximum allowable δ ∞ vanishes as α → 1.