Mitigation of symmetry condition in positive realness for adaptive control

Feasibility of nonlinear and adaptive control methodologies in multivariable linear time-invariant systems with state-space realization {A,B,C}{A,B,C} is apparently limited by the standard strictly positive realness conditions that imply that the product CB must be positive definite symmetric. This paper expands the applicability of the strictly positive realness conditions used for the proofs of stability of adaptive control or control with uncertainty by showing that the not necessarily symmetric CB is only required to have a diagonal Jordan form and positive eigenvalues. The paper also shows that under the new condition any minimum-phase systems can be made strictly positive real via constant output feedback. The paper illustrates the usefulness of these extended properties with an adaptive control example.