Limitations of robust stability of a linear delayed feedback control

In this paper we consider the issue of robust stability of a linear delayed feedback control (DFC) mechanism. In particular we consider a DFC for stabilizing fixed points of a smooth function f : R^m → R^m of the form x(k+1) = f(x(k))+u(k), where u(k) is given by the formula u(k) = − ^N_j=1⁻¹_j(x(k−j)−x(k−j+1)). We associate with each fixed point of f an explicit polynomial whose Schur stability corresponds to the local asymptotic stability of the DFC at that fixed point. This polynomial is the characteristic polynomial of the Jacobian matrix of an auxiliary map from R^mN to R^mN and May be given in terms of the eigenvalues of the Jacobian of f at the fixed point. This enables us to evaluate the robustness of the control by considering over what possible sets of eigenvalues of the Jacobian of f the associated characteristic polynomials are Schur stable. We will show that, for a given control of the above form, stability is guaranteed only if the set of eigenvalues lies in a set in C of diameter less than or equal to 16 and whose connected components all have diameters less than or equal to 4. An explicit example indicating the sharpness of the diameter of connected components is provided.