M.Sc Student | Malka Omer Jacob |
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Subject | Rational Approximation of Distributed-Delay Control Laws via Moment-Matching |

Department | Department of Mechanical Engineering |

Supervisor | Professor Emeritus Zalman Palmor |

Full Thesis text |

Time delay systems (TDS) are feedback control systems
containing loop delays. Delays appear commonly in almost any engineering
application. They usually appear in one of two forms, as a part of the physical
process (actuation delays, measurement delays, communication delays, delays in
data processing etc.) or as a part of a simplified model of a complicated
physical phenomena. Since the late 1950’s and up to nowadays the most widely
used control scheme that deals with TDS is the so-called Dead-Time-Compensator
(DTC). When designed for unstable plants, the DTC often contains an infinite
dimensional element called distributed-delay (DD). DD element is
infinite-dimensional, it is an entire function (contains no singularities) and
it has a finite impulse response thus it is a stable element. Control laws
involving DD elements are used extensively in control of TDS. In addition to
their existence in various configurations of the DTC, they arise also in the
solutions to the *H*^{2}, *L*^{1} and *H*^{∞}
optimization problems, they enable finite spectrum assignment, preview
tracking, etc. The problem of digital implementation of DD control laws is not
trivial. At the beginning of this century it was considered to be one of the
open problems in the area of TDS. The literature offered some solutions since
then. One of the approaches to deal with this problem is to use rational
approximations of the infinite-dimensional DD element. In this work a novel
method for rational approximation of DD control laws is offered. This research
was inspired by the work of Prof. L.Mirkin in his technical note [33].
This method is based on the idea of model reduction by moment matching that
enables perfect approximation of a system and its derivatives (up to the
defined order) at desired points of interest in the complex domain.
Traditionally, model reduction by moment matching is done using projections or
by solving Sylvester equations thus it is applicable only to rational systems.
In this work a different approach was considered, namely matching the steady
state response of the system and its approximation to some pre-defined input
(as explained in detail in this work). This approach enabled to use model
reduction by moment matching to irrational systems such as the DD element.
Besides demonstrating superior accuracy (comparing to other known methods from
the literature), the novelty of this method lies in its degrees of freedom that
can be exploited to guarantee stability of the approximation regardless of its
order and even more important is the fact that it enables the designer to
preserve some pre-designed closed-loop (CL) properties. In other words, it
allows the designer to approximate the DD element whilst taking into account CL
considerations. These advantages help in achieving an approximation that meets
the CL performance specifications using a lower order of approximation
(comparing to other rational methods). This work presents a formula for the
approximation of the modified Smith predictor and suggests guidelines for using
the method. The advantages of the proposed method over existing rational
approximations are demonstrated through examples and simulations.