טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentAlterman Igal
SubjectRobustness to Delay Uncertainties in Sampled-data Systems
DepartmentDepartment of Mechanical Engineering
Supervisor Professor Leonid Mirkin
Full Thesis textFull thesis text - English Version


Abstract

Time delays are one of the real world phenomena that might have a destabilizing effect on control systems. They are met as transport delays in chemical and mechanical systems, raised due to digital processing in control applications, etc. The analysis of the effect of time delays on the stability of continuous-time systems is knotty because the delay element has infinite-dimensional dynamics. In many applications loop delays are partly unknown or uncertain, which renders their effect even more harmful and analysis more intricate.


This work studies the robustness to uncertain analog delays in the context of

sampled-data (SD) systems. These are control systems in which a continuous-time plant is controlled by a discrete-time controller. The analysis of SD systems is complicated by their hybrid, analog/digital, nature and periodically time-varying dynamics in continuous time. From the stability analysis point of view, these complications can be circumvented by discretizing the system. This converts a SD system to an equivalent pure discrete-time (DT) shift-invariant system, which is easier to analyze.


Applying to the robustness analysis of SD systems with respect to uncertainties in analog delays, the discretization-based approach might lead to a hidden problem

though. Namely, the value of the analog loop delay affects not only the discretized delay, but also parameters of the delay-free part of the DT system. Consequently, delay uncertainty can no longer be treated as independent of uncertainties in plant parameters in the DT analysis. This fact, however, is rarely appreciated in the literature. Conventionally, uncertain delays in DT models are not linked with plant parameters, which can only be justified if analog delays are assumed to be a multiple of the sampling period. This work demonstrates that this assumption might be misleading. It is shown that there are systems, which are destabilized by arbitrarily small continuous-time delays despite being delay-independent stable with respect to such ``integer'' delays.


We then propose a robustness analysis procedure, which is based on embedding continuous-time delays into unstructured analog uncertainty with the subsequent

reduction of the problem to a standard sampled-data H problem. Toward this end, a novel nominal model for the uncertain delay is put forward. It yields a tighter

unstructured uncertainty covering than in the existing approaches, thus having a potential to reduce the conservatism of the method. We demonstrate, by numerical

examples, that this technique indeed normally reduces the conservatism and might be advantageous in the pure continuous-time case as well.