A non-orthogonal projection approach to characterization of almost positive real systems with an application to adaptive control

We develop a very general projection approach to obtain and unify necessary and sufficient conditions for making a linear time-invariant system positive real using static output feedback. Such systems are called "almost positive real". These conditions are that the open-loop system must be weakly minimum phase, have positive definite high frequency gain and the marginally stable zero system alone must be positive real. When dynamic output feedback is used the necessary and sufficient conditions reduce to weakly minimum phase with a positive definite high frequency gain. These results yield a stability proof for direct model reference adaptive control and disturbance cancellation using only almost positive real systems.