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A zero compensation approach to singular H 2 and H ∞ problems
Author(s) -
Copeland B. R.,
Safonov M. G.
Publication year - 1995
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.4590050202
Subject(s) - eigenvalues and eigenvectors , mathematics , zero (linguistics) , singularity , hamiltonian (control theory) , finite set , controller (irrigation) , constraint (computer aided design) , control theory (sociology) , mathematical analysis , control (management) , mathematical optimization , computer science , physics , geometry , quantum mechanics , artificial intelligence , agronomy , biology , philosophy , linguistics
In this work we analyse singular H 2 and H ∞ 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 Riccati equations with two generalized eigenproblems; these problems are always well‐posed. Next we extract those structural elements which pertain to the troublesome plant zeros. We do this by introducing pre‐compensators which cancel the offending zeros. In so doing, we temporarily relax the controller properness constraint that is traditionally imposed in H 2 and H ∞ problems by allowing pole‐zero cancellations between the plant and controller at infinity. Since no significant added complexity of analysis results, we also treat the case of singularity due to finite jω‐axis plant zeros by relaxing the internal stability requirement and allowing finite jω‐axis pole‐zero cancellations. The resultant theory allows us to specify necessary and sufficient conditions for the existence of solutions to singular H 2 and H ∞ 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, internally stabilizing solutions, including whether this set is nonempty. An example is included.