A Contribution to the Design of Linear Multivariable Systems

Abstract

This paper studies some basic problems in the linear time-invariant multivariable systems by using the polynomial fraction description Ĝ(s)=A <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> (s)B(s)=N(s)D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> (s). The following results are presented: (1) In forming the generalized resultant, it is shown that if the coefficient matrices of N(s) and D(s) are arranged in the order of ascending power, then all rows formed from D(s) are linearly independent, and the degree of Ĝ(s) is equal to the total number of linearly independent rows formed from N(s). (2) An algorithm is introduced so that A(s) and B(s) computed from the resultant of N(s) and D(s) are left coprime and A(s) is in the polynomial echelon form. It is also shown that the row degrees of A(s) are invariant of N(s) and D(s) used in the resultant. (3) The result in the state feedback and state estimator is extended and a simple design procedure is presented. The combination of the results in (1) and (3) can be viewed as parameterizations of compensators. Hence the results are useful in the design of control systems to achieve, in addition to arbitrary pole-placement, other design objectives such as sensitivity and steady-state errors. The design in this paper requires only solving linear algebraic equations, hence it is hoped that the method can be readily adopted by practicing engineers.