Date of final exam: 29/01/1999

E-mail: franze@si.deis.unical.it

Tutor: Prof.  L. Carotenuto and A. Eisinberg, Università della Calabria

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Pole placement by output gain feedback   ___________________________________________________________________________________________________________

Advisor:

Prof.  L. Carotenuto, Università della Calabria

Summary of the thesis

The pole placement problem for a linear time invariant multi-input-multi-output system using output feedback has been widely investigated in the past 25 years. However, after the first results due to Davison and Wang and Kimura, published in 1975, where a simple sufficient assignability condition, which involves the dimensions of the system (m+p >n), is proved, the successive analysis focused either on "technical" aspects, as the genericity of the assumptions on the system, exact or approximate assignability, or particular cases, characterized by the condition mp>= n >= m+p. Also the computational aspect has received minor attention: in particular, the algorithms known in the literature seem to be "conceptuald", i.e. they aim at explaining and/or validating the theoretical results, rather than at providing an efficient tool to solve the problem.
In this context, the thesis proposes a novel formulation of the problem, which allows to obtain a self-evident relationship between the open-loop and closed-loop eigenvalues and eigenvectors, pointing out the entire information which characterizes the problem. The more evident advantage consists in a non-iterative procedure for the computation of the gain matrix: indeed, the steps of the proposed algorithm, in the case of problems of practical dimensions, can be performed by the tools of symbolic algebra and a search procedure, namely the Newton-Raphson algorithm, is used only to solve a set of n-max(m,p) non-linear equations. Moreover the algorithm can be used without restrictions on location and multiplicity of open and closed-loop eigenvalues.
The solution of the assignment problem is constrained by the condition mp>= n, which is less restrictive with respect to m+p>n; it doesn't seem to involve particular problems from a computational point of view, and the proposed applications show the effectiveness of the method. Moreover, the comparison with the other algorithms are comforting, taking into account that they are based on the solution of Sylvester equations with non linear constraints or involve iterative search procedures.
It is worth to notice that the explicit parameterization of the whole class of gain matrices, which assign a fixed spectrum, enables to satisfy also other design requirements (for example, minimizing the control effort).
Finally, the algebraic approach has allowed to obtain interesting results in line with the ones by Brockett e Byrnes, (IEEE Trans. Aut. Contr., 1981.) Indeed, in the case min(m,p)=2, the algebraic re-formulation of the pole placement problem involves a simpler representation of the assignment equations, from which interesting properties of such equations are shown. The numerical experiments seem to put in evidence that, in the case n=4 and m=p=2, the assertion of Brockett and Byrnes on the "probability" to obtain a real gain matrix is an under estimate.
 

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