Geometric Results

A geometric study of parameter estimates’ limit sets of indistinguishable closed-loop dynamics is presented. It is shown that optimal LQ performance is conditional upon consistent parameter estimation. This is performed by showing that the parameterized, long-run quadratic cost has a unique minimum at the true parameter generating the observations. The projection is then defined of the control cost gradient onto the tangent space to the limit set of parameters associated with indistinguishable closed-loop dynamics. That projection is shown to have a unique stationary point at the true parameter. A scalar example is provided.