Extensions in adaptive model tracking with mitigated passivity conditions

Feasibility of nonlinear and adaptive control methodologies in multivariable linear time-invariant systems with state space realization {A, B, C} has apparently been limited by the standard strict passivity (or positive realness) conditions that imply that the product CB must be positive definite symmetric. More recently the symmetry condition has been mitigated, requiring instead that the not necessarily symmetric matrix CB be diagonalizable and with positive real eigenvalues. However, although the mitigated conditions are useful in proving pure stabilizability with Adaptive Controllers, the Model Tracking question has remained open and counterexamples seem to demonstrate total divergence of standard model reference adaptive controllers when the regular passivity conditions are not fully satisfied. Therefore, this paper further extends the previous results, showing that the new passivity conditions do guarantee stability with adaptive model tracking. Examples show how the new conditions solve the case of flexible structures with unknown parameters when perfect collocation is not possible. Also, the so-called counterexamples become simple, well-behaved, examples.