A lower bound for limiting time delay for closed-loop stability of an arbitrary SISO plant

This paper concerns the stabilization of linear timeinvariant (LTI) systems subject to uncertain, possibly timevarying delays. The fundamental issue under investigation, referred to as the delay margin problem, addresses the question: What is the largest range of delay such that there exists a single LTI feedback controller capable of stabilizing all the plants for delays within that range? Drawing upon analytic interpolation and rational approximation techniques, we derive fundamental bounds on the delay margin, within which the delay plant is guaranteed to be stabilizable by a certain LTI output feedback controller. Our contribution is threefold. First, for single-input single-output (SISO) systems with an arbitrary number of plant unstable poles and nonminimum phase zeros, we provide an explicit, computationally efficient bound on the delay margin, which requires computing only the largest real eigenvalue of a constant matrix. Second, for multi-input multi-output (MIMO) systems, we show that estimates on the variation ranges of multiple delays can be obtained by solving LMI problems, and further, by finding bounds on the radius of delay variations. Third, we show that these bounds and estimates can be extended to systems subject to time-varying delays. When specialized to more specific cases, e.g., to plants with one unstable pole but possibly multiple nonminimum phase zeros, our results give rise to analytical expressions exhibiting explicit dependence of the bounds and estimates on the pole and zeros, thus demonstrating how fundamentally unstable poles and nonminimum phase zeros may limit the range of delays over which a plant can be stabilized by a LTI controller.

Section: Introductionmentioning

confidence: 99%

Fundamental Limits on Uncertain Delays: When Is a Delay System Stabilizable by LTI Controllers?

Zhu

Chen

2017

“…Here, we wish to obtain comparable results on the use of LTI controllers for the delay margin problem. Indeed, for the LTI control case, it has been conjectured that there is an upper bound on the achievable delay margin for unstable plants (see, for example, [6], in which a lower bound on the upper bound is given). Until now, to the best of our knowledge, there have been no firm results to this effect.…”

Section: Introductionmentioning

confidence: 99%

On the Achievable Delay Margin Using LTI Control for Unstable Plants

Middleton

Miller

2007

109

Abstract-Handling delays in control systems is difficult and is of long-standing interest. It is well known that, given a finite-dimensional linear time-invariant (FDLTI) plant and controller forming a strictly proper stable feedback connection, closed-loop stability will be maintained under a small delay in the feedback loop, although most closed loop systems become unstable for large delays. One previously unsolved fundamental problem in this context is whether, for a given FDLTI plant, an arbitrarily large delay margin can be achieved using LTI control. Here, we adopt a frequency domain approach and demonstrate that, for a strictly proper real rational plant, there is a uniform upper bound on the delay that can be tolerated when using an LTI controller, if and only if the plant has at least one closed right half plane pole not at the origin. We also give several explicit upper bounds on the achievable delay margin, and, in some special cases, demonstrate that these bounds are tight.

“…We suspect, as does the author of [4], that for many LTI models the bound is finite. Even if it is infinite, to the best of our knowledge there are no design techniques available to make the delay margin arbitrarily large.…”

Section: Problem Formulationmentioning

confidence: 89%

“…3 Following standard practice, we can obtain an input-output description by considering the system behavior when t = k = 0;v = 0, and z = 0. 4 We do not requite that p be the fundamental period of (H; J; L; M ), so T may not be the fundamental period of (3).…”

Section: Problem Formulationmentioning

confidence: 99%

“…Moreover, the fundamental question of whether or not there exists an upper bound on the delay 0018-9286/$20.00 © 2005 IEEE margin that is achievable by a LTI controller has not even been addressed in the literature. However, because there is an upper bound on the gain margin which is achievable by an LTI controller if the plant has poles and zeros in the open right-half plane [10], it is reasonable to expect, as does the author of [4], that the delay margin which is achievable by an LTI controller is also fundamentally bounded in certain cases.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Scanning the issue

2005

Handling delays in control systems is difficult and is of long-standing interest. It is well known that, given a finite-dimensional linear time-invariant (LTI) plant and a finite-dimensional LTI stabilizing controller, closed-loop stability will be maintained under a small delay in the feedback loop. However, in some situations there is a large delay, perhaps arising from a slow communications link or a large but variable computational delay (e.g., in a system which uses image processing). While there are techniques available to design a controller to handle a known delay, there is no general theory for designing a controller to handle an arbitrarily large uncertain delay. Here we prove constructively that, given a finite-dimensional LTI plant and an upper bound on the admissible time delay, there exists a linear periodic controller which robustly stabilizes the plant. Index Terms-Delaymargin, linear periodic control, robust control, uncertain time delay. Daniel E. Miller (S'82-M'90-SM'03) received the B.Sc. degree in electrical engineering from the University of New Brunswick, Fredericton, NB, Canada, in 1984, and the M.A.Sc. and Ph.D. degrees in electrical engineering from the University of