LMI based time-invariant observers and controllers in H∞ problems with transients*

Attenuating both exogenous signals and initial disturbances caused by unknown initial conditions and estimating the state of a system using the past measurements are considered in the framework of a so called H∞ problem with transients. Applying an LMI approach, instead of the Riccati equations one, in characterizing the performance measure that is the worst-case norm of the regulated output or the output to be estimated over all exogenous signals and initial states allows one to synthesize time-invariant, instead of the time-varying, filters and output-feedback controllers for which the performance measure of the closed-loop system is less than a prescribed number. Necessary and sufficient condition in the form of a fundamental inequality for the weighting matrix is obtained under which the trade-off between filtering or control in the H∞ setting, being optimal under unknown exogenous disturbances and zero initial state, and linear-quadratic (LQ) filtering or control, being optimal under zero exogenous signal and unknown initial conditions, is possible. These results can be used as a methodology basis in studying H∞ control and filtering theory.