A ZERO COMPENSATION APPROACH TO SINGULAR H-2 AND N-INFINITY PROBLEMS

In this work we analyse singular H-2 and H-infinity problems, i.e. H-2 and H-infinity problems for which the usual Riccati equations become ill-posed owing to the existence of plant zeros at infinity. We adopt a two-step approach to the analysis. First we replace the usual Riccat i equations with two generalized eigenproblems; these problems are alw ays well-posed. Next we extract those structural elements which pertai n to the troublesome plant zeros. We do this by introducing pre-compen sators which cancel the offending zeros. In so doing, we temporarily r elax the controller properness constraint that is traditionally impose d in H-2 and H-infinity problems by allowing pole-zero cancellations b etween the plant and controller at infinity. Since no significant adde d complexity of analysis results, we also treat the case of singularit y due to finite j(omega)-axis plant zeros by relaxing the internal sta bility requirement and allowing finite j omega-axis pole-zero cancella tions. The resultant theory allows us to specify necessary and suffici ent conditions for the existence of solutions to singular H-2 and H-om ega problems. The existence conditions and the resultant control laws are expressed directly in terms of the eigenvalues and eigenvectors of two Hamiltonian matrices associated with the problem. The theory also gives some insight into the character of the subset of all proper, in ternally stabilizing solutions, including whether this set is nonempty . An example is included.