On Asymptotic Stabilizability of Discrete-time Linear Systems with Delayed Input

Abstract

Abstract. This paper examines the asymptotic stabilizability of discrete-time linear systems with delayed input. By explicit construction of stabilizing feedback laws, it is shown that a stabilizable and detectable linear system with an arbitrarily large delay in the input can be asymptotically stabilized by either linear state or output feedback as long as the open loop system is not exponentially unstable (i.e., all the open loop poles are on or inside the unit circle.) It is further shown that such a system, when subject to actuator saturation, is semi-globally asymptotically stabilizable by linear state or output feedback.

Stabilization of polytopic delay difference inclusions via the Razumikhin approach

“…Remark 6. SDP based synthesis solutions for quadratic cLRFs were also proposed in, e.g., Lin (2007) (discrete-time) and Cao et al (2002) (continuous-time). The results in Lin (2007) apply to systems where A 0 is a singleton, A 0 is strictly stable and…”

“…A translation of the Razumikhin approach to discretetime systems was obtained in Liu and Marquez (2007) by requiring the candidate function to be less than the maximum over its past values within a time window. So far, the application of cLRFs has been limited to particular sub-classes of polytopic DDIs, see, e.g., Lin (2007), where a synthesis method is developed for linear timeinvariant systems subject to input delays with a strictly stable matrix multiplying the current state.…”

Stabilisation of linear delay difference inclusions via time-varying control Lyapunov functions

“…In this case, the number of generators of the set AB increases exponentially with h, that is L = 0 i=−h L i L i , and hence existing control schemes, such as [1,[5][6][7][8][9][10][11][12][13][14] are not computationally tractable for large delays. In [15], to make the on-line computation of the control update independent of h, the set-valued map (2) was replaced bŷ…”

“…Interestingly, the control scheme proposed in [15] consists of an online optimisation-based component that is tractable for large delays because of an on-line Minkowski addition of sets. Unfortunately, the method also requires the off-line computation of a local static state-feedback controller and thus, as it is also the case for the control schemes proposed in [1,14], remains computationally demanding. Furthermore, as the Razumikhin approach provides [1] sufficient but not necessary conditions for stability, any method based on this approach is inherently conservative.…”

“…Based on the Razumikhin approach a state feedback controller was obtained for systems with delay in [14] and for uncertain systems with delay in [1]. Furthermore, a control strategy that takes constraints into account was developed for uncertain systems with delay in [15].…”