Indian Institute of Technology Bombay

Popular Lecture Series in Mathematics

Mathematics Association of IIT Bombay and Bombay Mathematical Colloquium

Name of the Speaker : Prof.V.R. Sule

Dept of Electrical Engg., IIT Bombay

Title of the Talk:: Science of Automatic Control: Paradigms and

Puzzles

Day and date : 31st October 2003

Time : 5.05 p.m

Venue: Ramanujan Hall, Dept of Mathematics

Abstracts

The science of automatic control was formally inaugurated by the analysis of the Watt’s governor of steam engine by Maxwell. This is the stability paradigm of control systems which raises the question, how many complex roots (eigenvalues) does a polynomial (a matrix) have inside and on the boundary of a specified domain in the complex plane.  The associated problem of pole placement is to determine, given a pair of matrices (A,B) with same number of rows and A square, whenever it exists, a matrix K such that (A+BK) has specified eigenvalues. The more general problem of finding conditions under which there exists a matrix K such that (A+BKC) has a given set of eivenvalues (C not equal to identity) is open. Similarly  the problem of finding a common stabilizing controller, an element f/g of the field of fractions of an integral domain A such that fp1 +pgq0 is any  unit u1 in A for j=1,2,…,n for n > 2 is open. The linear quadratic problem is to determine the matrix K such that an integral of a given quadratic form in the state and input variables is minimized. The H-inifinty control problem is to determine a closed loop controller so as to minimize the maximum energy of the regulated variables, ||z||2  with respect to input energy ||w||2 £ 1. This theory led to the solution of the important roubust stabilization problem. In recent times, systems are being considered as kernels ker R(x) in C-infinityq f WHERE r(x) IS A POLYNOMIAL MATRIX IN THE DIFFERENTIAL OPERATOR x. This is the behavioral approach and leads to reformulation of many control problems in terms of natural models  of systems. This talk will take an overview of the central, mathematical problems of control theory and end with a brief description of the state space approach for behavioral systems.