Cheap control tracking performance for non-right-invertible systems

Abstract

SUMMARYThere exists a well-known fundamental limitation upon the achievable setpoint tracking performance of a non-right-invertible plant. This limitation manifests itself, for example, in the cost associated with the cheap control tracking problem. In this paper, we provide a new interpretation of this limitation. We show that the cheap control cost may be decomposed into the sum of two terms. The first of these depends upon certain non-minimum phase zeroes that include the non-minimum phase plant zeroes. The second term depends upon the extent to which the plant direction varies with frequency. To state these results, we first develop a co-ordinate transformation that may be used to define the notion of frequency dependent plant direction and to display the relevant non-minimum phase zeroes. We also show that the cheap control cost is connected to an integral relation that constrains the performance of any stable closed-loop system (not necessarily under cheap control) for which the plant has a single control input and two performance outputs.

Limits of Performance in Reference-Tracking and Path-Following for Nonlinear Systems

“…It is its role as a stabilizing control input that prevents the output y(t) from perfect tracking. Extensions to non-right-invertible systems are given in [Woodyatt et al, 2002;Braslavsky et al, 2002]. In this paper we show that these results hold for the problem of tracking any reference signal generated by a known exosystem.…”

“…Considering, as in [Qiu and Davison, 1993;Woodyatt et al, 2002;Braslavsky et al, 2002;Su et al, 2003], that x(0) = 0, prove:…”

Optimal Preview Control of Markovian Jump Linear Systems

“…In particular, more general performance metrics, such as H ∞ , have been considered [11], [24]. There is also a substantial collection of results on fundamental limits of optimal reference tracking [25] for a variety of metrics [30], constraints [9], [8], [7] and plant classes [29], [6], [20], [34]. All of these results, in one way or another, conclude that reference look-ahead (preview) may lead to a substantial increase in the tracking performance.…”

Path-following for nonminimum phase systems removes performance limitations

“…As in [4], [6], [19], [20], we assume that initially the system is at rest, that is, we let x(t0) = 0. (A, B) is stabilizable, then for the geometric pathfollowing problem and any given positive constant δ there exist constant matrices K and L, and a timing law θ(t) such that the feedback law…”

“…It is its role as a stabilizing control input that prevents the output y(t) from perfect tracking. Extensions to non-right-invertible systems are given in [19], [20]. Control energy constraints as another source of fundamental performance limitations have been investigated in [21].…”