In the previous lectures an interesting new approach based on the special decomposition of the sensitivity function was introduced and discussed. This method optimizes the three major components (design loss, realizability loss and modeling loss) separately.
“Nothing is more practical than a good theory.” – Boltzmann
This summary lecture presents simple examples to show the efficiency of the method and ease the understanding.
“The examples are more useful than formulas.” – Newton
First the optimization of the design loss is shown by two examples, where an iterative redesign technique is used to find the best reachable reference model, i.e., the fastest control system can be ever reached under practical constraints. It is very rare then we have an analytic solution for this optimization, instead repeated simulation runs results in the best reference model.
Then the optimization of the realizability loss is presented, which is a pure mathematical problem as the examples will show. For the E2 (and H2) norms the solution is to solve a Diophantine equation which can be done by solving special linear system equations. For the E͚ (and H͚) norms the problem can be solved via the Navelinna-Pick approximation procedure, which results in a nonlinear system equation.
Finally the optimization of the modeling loss is treated. It will be shown in the examples that the best modeling loss can be obtained if the closed-loop control system is excited by a binary reference signal, where the switching condition provides the optimality. The example will show clearly that such optimal excitation concentrates the frequency content of the signal around the cutting frequency, this is the frequency domain where we must have the most accurate models.
