טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentMalka Omer Jacob
SubjectRational Approximation of Distributed-Delay Control
Laws via Moment-Matching
DepartmentDepartment of Mechanical Engineering
Supervisor Professor Emeritus Zalman Palmor
Full Thesis textFull thesis text - English Version


Abstract

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 H2, L1 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.