REDUCED ORDER MODELS FOR CONTROL STUDIES

Abstract

The aim of this thesis is two-fold; firstly-to present new methods for model reduction of single-input single-output and multivariable linear time-invariant dynamic systems, and secondly —to devise new schemes for designing controllers using model reduction techniques. The contributions in the area of reduced order modelling have been addressed to both continuous-time and discrete-time systems. The work included herein deals with frequency domain model reduction techniques and control system design based on a transfer function descrip tion of the system. The subject of model reduction of both continuous-time systems and discrete-time systems is first reviewed. Reduced order models obtained by applying the continued fraction expan sion or the Pade approximation technique suffer from three inherent deficiencies, viz. (1) the reduced-order model may be unstable (stable) although the parent system is stable (unstable). (2) the reduced model often shows poor matching in the transient zone; and (3) it may exhibit non-minimum phase characteristics. The methods presented in this thesis attempt to remove the above difficulties. Various mixed methods are given for obtaining stable reduced order models from high-order stable systems. The denominator poles are chosen from the Routh-Hurwitz stability array while the numerator dynamics are determined so that the initial few quotients of the continued fraction expansions of the original and reduced systems are the same or so that the -11- initial few time moments of the respective systems are identi cal in the Pade sense. As seen from the various examples, this guarantees the stability of the reduced model but may, however, lead to poor correspondence in the transient zone. This has been overcome by matching a few initial time moments and Markov parameters of the respective systems to obtain a good overall fit in the time-region of interest. A further restric tion of continued-fraction based reduction methods being only applicable to square transfer function matrices is also removed. It is shown that the method of Pade approximation about a gene ral point assures a good overall fit in the entire frequencyrange of interest and particularly removes the possibility of significant errors in the intermediate to high-frequency behaviour of Pade based reduced models. The difficulty of Pade approximants being occassionally of non-minimum phase is remo ved by incorporating an additional time-delay term in the reduced model. A comprehensive method of model reduction using a modified Routh Hurwitz criterion is given that allows one to retain the dominant as well as the far-off poles in the reduced model. Some of the above methods of model reduction are exten ded to the discrete-time case. These new methods remove certain inherent deficiencies of Pade-type model reduction techniques. A new and novel scheme for the design of singleinput single-output and multivarlable control systems using the Pade approximation technique for model order reduction is -ingiven. It is shown that simple controllers may be systematically designed to satisfy various industrial specifications. It is also shown that higher order plants can effectively follow a low order specification model. The above method is extended for the design of singleinput single-output systems with a transport lag. It is shown that, unlike the Smith predictor approach, this method leads to a simple controller configuration and does not require modelling of the plant with the time-delay term. The implementation of an optimal control law (minimising a quadratic performance index) for linear time-invariant plants usually require the complete measurement of the plant state vector. Hence the optimal control problem will most likely fail to have a closed loop solution if some of the state variables are not available for feedback. Statistical estimation of the missing state variables using Kalman filtering techniques or their reconstruction using Luenberger-type observers increase the order of the system and the resultant controllers become more complex. Using model reduction techniques two new methods are given for arriving at a suboptimal controller using the available states for partial state-feedback. It is shown that with the new methods, dynamic feedback as well as precompensators may be easily incorporated to have an acceptable though suboptimal response.