Impulse Elimination And Observer Design For Descriptor Linear Systems

Descriptor-system models are more convenient and natural than normal systemsin the description of interconnected large-scale systems, mechanical multi-body mo-tion systems, or electrical networks. In addition, as an approach the model of descrip-tor linear systems has been utilized to develop robust stability condition and robustperformance index for some complex systems, such as time delay systems and neu-tral systems. Therefore, study on descriptor linear systems is both theoretically andpractically important.A big difference between a descriptor linear system and a conventional linearsystem is that the response of a descriptor linear system may contain impulsive terms.Such infinite jumps are obviously destructive as they could completely destroy the sys-tem instantaneously. In addition, in practice the state of a system is usually not directlyavailable, so the state feedback control usually can not be realized directly. A feasiblemethod is to firstly asymptotically estimate the state vector by using an observer, andthen use the estimation of the state to construct the state feedback control. In orderto guarantee the accuracy of estimation, it is required that the designed observer sys-tem should be impulse free. Due to the above reasons, this dissertation systematicallyand deeply investigates the problem of impulse elimination and observer design. Thedetailed results are as follows.1 Regularization via dynamic output feedback is investigated for square descriptorlinear systems. Necessary and sufficient conditions of regularizability via dy-namic output feedback are established. In addition, it is shown that the set of allregularizing gain matrix groups is a Zariski open set for regularizable systems.2 The problems of impulse elimination via state feedback and proportional-derivative (PD) state feedback are dealt with. Based on a canonical equivalentform for descriptor linear systems, a general parametrization of all state feedbackgains which make the resulted closed-loop system impulse-free is presented. Inorder to deal with impulse elimination via PD state feedback, the concepts ofI-controllablizability and impulsive-mode controllablizability are introduced forsquare and nonsquare systems, respectively. Two class of criteria are given for the I-controllablizability and impulsive-mode controllablizability in terms of initialsystem matrices. General parametrization of all I-controllablizing and impulsive-mode controllablizing controllers are presented. With the above preparation, thesolution to impulse elimination via PD state feedback can be completed by atwo-step method.3 Several types of observers are proposed for regular descriptor linear systems, andparametric design approach for these observers are established. The proposedobservers include generalized proportional-integral (PI) observers, proportional-multiple-integral (PMI) observers, generalized proportional-integral-derivative(PID) observers, proportional-multiple-integral-derivative (PMID)observers, andso on. The proposed approaches give parameterizations of all observer gain ma-trices in terms of some free parameters which represent the degrees of designfreedom, and can be further utilized to achieve additional specifications and per-formances. The proposed approach, which guarantees the regularity of the ob-server system, realizes the elimination of impulsive behaviors of the observersystem.4 The problem of observer based robust control is investigated. The robustnessis characterized by the property of loop transfer recovery (LTR). The involvedobservers include Luenberger observers, full-order state observers, PI observersand PMI observers. Using the concept of recovery error, simple necessary andsufficient conditions for the controller are derived for exact LTR in terms of theso-called recovery matrices.5 Generalized PI observer based state feedback controller is designed with thefunction of LTR for ?exible joint robots. By combining the proposed parametricexpression of observer gains and the recovery condition, an index is proposed tocharacterize the function of LTR. By adopting genetic optimization algorithm,controllers with LTR are provided. Compared with the results in the case ofPI observers, it is also shown that the extra freedom degrees of generalized PIobservers can provide better LTR performance.