Stable controllers for robust stabilization of systems with infinitely many unstable poles

Abstract

a b s t r a c tThis paper studies the problem of robust stabilization by a stable controller for a linear time-invariant single-input single-output infinite dimensional system. We consider a class of plants having finitely many simple unstable zeros but possibly infinitely many unstable poles. First we show that the problem can be reduced to an interpolation-minimization by a unit element. Next, by the modified Nevanlinna-Pick interpolation, we obtain both lower and upper bounds on the multiplicative perturbation under which the plant can be stabilized by a stable controller. In addition, we find stable controllers to provide robust stability. We also present a numerical example to illustrate the results and apply the proposed method to a repetitive control system.

Robust control of infinite dimensional systems

“…Other robust control problems that are not discussed here include simultaneous stabilization (control of finitely many plant models by a single robust controller) and strong stabilization (robust control with the added restriction that the controller must be stable) of infinite-dimensional systems. Stable robust controller design techniques for different types of systems with time delays are illustrated in Özbay (2010) and Wakaiki et al (2013); see also their references.…”

On the real, rational, bounded, unit interpolation problem in ℋ∞ and its applications to strong stabilization

“…This paper studies the Real, Rational, Bounded, Unit Interpolation Problem (RRBUIP), which is closely related to strong stabilization and simultaneous stabilization problems in feedback control theory (Abdallah et al, 1995; Bredemann, 1995; Wakaiki et al, 2013; Xin and Liu, 2013). In Hara and Vidyasagar (1990), the bounded unit interpolation problem is defined and its connections with sensitivity shaping and robust stabilization by a stable controller are discussed.…”

“…To the best of our knowledge, robust stabilization by a stable controller for infinite dimensional systems is an open research problem and one of the most recent contributions has been made in Wakaiki et al (2013), where it is shown that robust stabilization of an infinite dimensional system by an infinite dimensional stable controller can be reduced to a bounded unit interpolation problem. In general, it is easy to show that for the finite dimensional case, robustly stabilizing stable controller design can be reduced to bounded unit interpolation.…”

“…is required, where W is the multiplicative plant uncertainty (Doyle et al, 1992). Following the ideas of Wakaiki et al (2013) it is easy to see that for ∥ WT ∥ ∞ < 1 it is sufficient to have…”

Sensitivity Reduction by Stable Controllers for MIMO Infinite Dimensional Systems via the Tangential Nevanlinna-Pick Interpolation

“…For SISO infinite dimensional systems, the Nevanlinna-Pick interpolation [9], [10] enables us to design stable controllers providing the minimum sensitivity [11] or robust stability [12]. The point of this approach is that a stable controller stabilizes the plant if and only if a unit element in satisfies certain interpolation conditions at the unstable zeros of the plant.…”