By Michael Green;David J. N. Limebeer

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**Example text**

128. 3 This lack of stability robustness is not evident from the eigenvalue loci of G. We conclude that the eigenvalue loci of a multivariable plant are not good robust stability indicators. 9363) minimizes the distance from the line 1 2 3 − 50k2 + 47k1 = 0 to the point (−1, −1). 0874. 26 MULTIVARIABLE FREQUENCY RESPONSE DESIGN since they cannot always account for loop interactions. For example, suppose a transfer function has the form 1 0 G= φ(s) 1 . The eigenvalues of G are independent of the off-diagonal term φ(s), so they provide no indication of the fact that φ(s) may cause significant inter-loop coupling between the second input and the first output.

5: The equivalent open loop. yc yo = Gt K(I − Gt K)−1 r = Gt K(I − GK)−1 r, it follows that yc = yo for all r if the system Gt and the model G are identical. Suppose the plant Gt depends on a parameter δ, so that Gt becomes Gt (δ). The effect of changes in δ on yc and yo can now be evaluated: ∂Gt ∂yc = (I − Gt K)−1 K(I − Gt K)−1 , ∂δ ∂δ while ∂Gt ∂yo = K(I − GK)−1 . ∂δ ∂δ Assume now that the model G is obtained by using a nominal value for δ, so that G = Gt (δnom ). Then ∂yc ∂δ δ=δnom = (I − GK)−1 ∂yo ∂δ .

Consider the case of 1 . 1) degenerates to 1. Suppose we ignore this difficulty for the moment and restrict our attention to constant controllers given by k ≤ 0. This gives g= q = (1 − kg)−1 k = with (1 − kg)−1 k ∞ ks s−k sk s−k = |k|. = s=∞ To solve the problem we observe that if we want to stabilize the closed loop for any stable δ such that δ ∞ < 1/ , we simply set k = − ; may be arbitrarily small! In problems such as this one, which has an interpolation constraint on the imaginary axis, it is not possible to achieve the infimal value of the norm.