Projection Scheme and Adaptive Control for Symmetric Matrices With Eigenvalue Bounds

A projection scheme to handle eigenvalue bounds for adaptive control with uncertain symmetric matrix parameters is introduced. Conventional parameter projection techniques are generally unable to handle explicit eigenvalue bounds. The continuous projection scheme presented here maintains the closed-loop stability properties for adaptive controllers while simultaneously satisfying
a priori
available eigenvalue bounds of the uncertain symmetric matrix valued parameters. The projection scheme uses the eigen decomposition of the symmetric matrix parameter to project its eigenvalues to lie within the prescribed bounds. The eigenvalues of the symmetric matrix may be lower bounded or upper bounded or both. A direct adaptation over the eigenvalues and the eigen projections of the symmetric matrix parameter is also derived to help circumvent expensive eigen decomposition calculations. The new projection here shows improved performance in numerical simulations of rigid body attitude tracking control and trajectory tracking of robotic manipulators with unknown inertia parameters.