Quantitative design of robust multivariable control systems

By a systematic use of the theory of non-negative matrices, and the associated theory of M-matrices, it is possible to derive measures of robustness which overcome the undue conservatism inherent in the use of singular values as measures of robustness for particular types of structured perturbations. Using these ideas, it is shown that for nominal diagonal closed loop transfer matrices, the controller which maximises robustness is the one that minimises the Perron root (maximum eigenvalue) of a certain non-negative matrix. From this, a simple criterion for robustness based on the maximum magnification (M_p) of the closed loop transmission functions, and the Perron root of the uncertainty matrix is derived. This is then used to give a design scheme for simultaneous stability and performance robustness. That is to say, the final control scheme guarantees satisfaction of given performance bounds in the time domain in the face of any given plant stable, but bounded uncertainty. This is in contrast to robust multivariable control where emphasis is on the satisfaction of frequency domain upper bounds on the error and disturbance response, respectively.