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Title: Dynamic polynomial combinants and linear systems
Author: Galanis, Georgios
ISNI:       0000 0004 2694 9118
Awarding Body: City University
Current Institution: City, University of London
Date of Award: 2010
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The theory of polynomial combinants is intimately linked to problems of frequency assignment, stabilization and presence of almost fixed modes phenomena in linear systems and control. The thesis develops the fundamentals of the theory of Dynamic Polynomial Combinants (DPC) which underpins the development of the Determinantal Assignment Problem (DAP) approach to dynamic pole-zero assignment and sta› bilization problems by dynamic compensators. The results in the thesis extend the theory of Polynomial Combinants from the constant case, which has been the main developments so far to the dynamic case. The thesis reviews first the fundamentals of the DAP approach in Control Theory that motivate the developments in the the› sis and then identifies the open issues for the required theory of Dynamic Polynomial Combinants. The theory of Constant Polynomial Combinants is also reviewed and this indicates the range of topics which require development, when it comes to the dynamic case. The fundamental problem driving this research is the spectrum assignability of Dynamic Polynomial Combinants and the basic theory is developed around this. We consider first the problem of representation of DPCs and two alternative forms are given, the Toeplitz and the Generalised Resultant representations. The problem of parameterisation is considered according to the order and degree and the results are interpreted as matrix operations generating the entire family of the corresponding generalised resultants. The arbitrary assignment of spectrum of DPCs is then considered and its solvability to the coprimeness of the set generating the combinant is established. This result is then further developed by determining the family of assignable combinants. The open problem of "minimal design" I that is defining the least degree and order combinant that is spectrum assignable is considered and an algorithmic solution terminating in a small number of steps is given using the rank properties of Generalised Resultants. The least order and degree guaranteeing spectrum assignability also defines the boundaries of the family of DPCs for which the spectrum cannot be assigned. The notion of Strongly Non-Assignability DPCs is introduced as those where not all roots can be assigned at s = 00 and the entire family is parameterised. This family has its spectrum restricted to a finite region of the complex plain and the spectrum properties are linked to the characterisation of almost zeros in the DPC case. For the family of Strongly Non-Assignability DPCs we consider the problem of dynamic stabilisation and new criteria for existence of solutions are given. The link of the spectral properties to the Almost GCD of the polynomial set generating the combinant is discussed and a new procedure for the characterisation of Almost GCD is given using as a tool the Structured Singular Value. A research agenda for the further development of the DPC theory is finally given.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available