Decoupling in the design and synthesis of multivariable control systems

Necessary and sufficient conditions for the "decoupling" of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> -input, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> -output time-invariant linear system using state variable feedback are determined. Given a system which satisfies these conditions, i.e., which can be decoupled by state variable feedback, the class φ of all feedback matrices which decouple the system is characterized. The characterization of φ is used to determine the number of closed-loop poles which can be specified for the decoupled system and to develop a synthesis technique for the realization of desired closed-loop pole configurations. Transfer matrix consequences of decoupling are examined and practice implications discussed through numerical examples.