Robust Stabilization by Linear Output Delay Feedback

The main result establishes that if a controller C (comprising of a linear feedback of the output and its derivatives) globally stabilizes a (nonlinear) plant P , then global stabilization of P can also be achieved by an output feedback controller C[h] where the output derivatives in C are replaced by an Euler approximation with sufficiently small delay h > 0. This is proved within the conceptual framework of the nonlinear gap metric approach to robust stability. The main result is then applied to finite dimensional linear minimum phase systems with unknown coefficients but known relative degree and known sign of the high frequency gain. Results are also given for systems with non-zero initial conditions. Nomenclature C+, C− = {s ∈ C |Re s > 0}, {s ∈ C |Re s < 0}, respectively |x| = √ xTx, the Euclidean norm of x ∈ Rn |A| = max { |Ax| ∣∣ x ∈ Rm, |x| = 1 } , the induced matrix norm for A ∈ Rn×m [a1/a2/ . . . /am] = [ a1 , a T 2 , . . . , a T m ]T ∈ Rm×n for a1, a2, . . . , am ∈ R1×n e (n) k = [0, . . . , 0, 1, 0, . . . , 0] T , the k-th unit vector in Rn, for k, n ∈ N and k ≤ n spec(A) the spectrum of A ∈ Rn×n imA, kerA the image and kernel of A ∈ Rn×m ‖v‖V the norm of v ∈ V, for a normed vector space V map(E → F ) the set of all maps from the set E to the set F Cr(I → Rl) the set of r-times continuous differentiable functions y : I → Rl, where r ∈ N ∪ {∞} and I ⊂ R is an interval Cpw(I → Rl) the set of piecewise continuous functions y : I → Rl, I ⊂ R an interval Lp(I → Rl) the space of p-integrable functions y : I → Rl, 1 ≤ p < ∞, I ⊂ R an interval, with norm ‖y‖Lp(I→Rl) = (∫ I |y(t)|p dt ) 1 p L∞(I → Rl) the space of essentially bounded functions y : I → Rl, I ⊂ R an interval, with norm ‖y‖L∞(I→Rl) = ess sup t∈I |y(t)| CL(I → Rl) = C(I → Rl) ∩ Lp(I → Rl), 1 ≤ p ≤ ∞, I ⊂ R an interval, with norm ‖y‖CLp(I→Rl) = ‖y‖Lp(I→Rl) CW (I → Rl) = { y ∈ Cr(I → Rl) ∣∣ ∀ i ∈ {0, . . . , r} : y(i) ∈ Lp(I → Rl) } , r ∈ N, 1 ≤ p ≤ ∞, I ⊂ R an interval, with norm ‖y‖CW (I→Rl) = r ∑ i=0 ‖y‖Lp(I→Rl) School of Electronics and Computer Science, University of Southampton, SO17 1BJ, UK, [email protected] Institut für Mathematik, Technische Universität Ilmenau, Weimarer Straße 25, 98693 Ilmenau, DE, achim.ilchmann, [email protected]