Isolation of small parameters in controlled systems by the criterion of controllability measures (Q1311504)

In a single-input, linear lumped parameter system with state \(x\in \mathbb{R}^ n\) \[ \dot x= Ax+bu, \] certain controllability measures are introduced -- via a nonsingular transformation relative to the controllability matrix \([b Ab\dots A^{n-1} b]\). Small values of the measures correspond to high energy expenditure for reducing the state to zero, i.e., stabilization in finite time. A procedure for their determination is developed. In the multivariate case, the above procedure is applied to isolating a poorly controlled subspace. A two-dimensional example containing a small parameter illustrates the results.