**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 : **31 ^{st} 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 fp_{1 }+pgq_{0
}is any unit u_{1 }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-infinity^{q} 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._{}