Limits of propriety for linear-quadratic regulator problems

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Abstract
The linear-quadratic-regulator (LQR) theory, as presented in existing textbooks, restricts the Q-matrix to be positive-definite (or positive-semidefinite). In the present paper it is shown that this restriction on Q is unwarranted because there exist practical applications of LQR theory involving sign-indefinite (and even negative-definite!) Q-matrices that lead to well-defined well-behaved optimal solutions. A general class of scalar-control LQR problems is examined in detail and the fundamental theorem governing ‘propriety’ of such problems is derived. A procedure is then described for finding the set Q of all Q-matrices (including indefinite and negative-definite Q) that satisfy the propriety condition, i.e. lead to well-behaved optimal solutions. The procedure is illustrated by several worked examples, including explicit general solutions for Q for the cases n = 1 and n = 2.