By F. J. W. Hahne

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7 Periodic Control of Time-invariant Systems δ 2J = ∞ ∑ k=−∞ 43 Uk∗ Π (kΩ )Uk , where Uk∗ is the conjugate transpose of Uk . This expression is at the basis of the design. Suppose indeed that for some value Ωˆ of ω the matrix Π (ω ) be positive definite, in other words X ∗ Π (Ωˆ )X > 0 , for some vector X = 0. Then take ˆ ˆ δ u(t) = ε (Xe jΩ t + X ∗ e− jΩ t ) . For ε small enough, the first variation of J around the optimal steady-state can be neglected whereas the second variation is δ 2 J = ε 2 X ∗ Π (Ωˆ )X .

We are now in a position to introduce the notion of relative degree, [126]. 20 1 Introduction Suppose that at time t there exists an integer k such that the following two conditions hold: 1. ci (τ )B(τ ) = (LA ci (τ ))B(τ ) = . . = (LAk−2 ci (τ ))B(τ ) = 0 for all τ in a neighborhood of t. 2. in every neighborhood of t there exists τ such that (LAk−1 ci (τ ))B(τ ) = 0. Then, ri (t) = k is named relative degree of the i-th output signal at time t. If such a ri (t) exists for each i = 1, 2, · · · , p, the vector r(t) = [r1 (t) r2 (t) .

12sin2 (6t) + 9sin(6t)cos(6t) −1 − 9sin2 (6t) − 12sin(6t)cos(6t) 4 For simplicity, in the above statement we used the word stable in place of the word asymptotically stable, adopted in various textbooks. The same terminology will be used in the rest of the book. 16 1 Introduction This matrix, periodic with period T = π /3, has constant eigenvalues equal to −1 and −10. 2 e2T + 4e−13T 2e2T − 2e−13T 2e2T − 2e−13T 4e2T + e−13T whose eigenvalues are e2T and e−13T . Since e2T > 1, the system is unstable.