A new multirate LQ optimal regulator for linear time-invariant systems and its stability robustness properties
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In the present paper, the LQ optimal regulation problem for continuous-time systems is solved by using a new class of multirate controllers, called Two-Point-Multirate Controllers (TPMRCs). In such a type of controllers, the control is constrained to a certain piecewise constant signal, while each of the controlled plant outputs is detected many times over a fundamental sampling period To. The proposed control strategy is readily applicable in the cases where the state variables of the controlled plant are not available for feedback, since TPMRCs provide the ability to reconstruct exactly the action of static state feedback controllers from input-output data, without resorting to state estimators, and without introducing high-order exogenous dynamics in the control loop. On the basis on this strategy, the original problem is reduced to a discrete-time LQ regulation problem for the performance index with cross-product terms (LQRCPT), for which a fictitious static state-feedback controller is needed to be computed. Moreover, the stability robustness properties of the TPMRC-based LQ regulator are analysed. In particular, guaranteed stability margins for TPMRC-based LQ optimal regulators are derived on the basis of a fundamental spectral factorization equality called the Modified Return Difference Equality. The suggested guaranteed stability margins are expressed directly in terms of the singular values of the elementary cost and system matrices associated with the equivalent discrete-time LQRCPT optimal design. Sufficient conditions to guarantee the suggested stability margins are established. Finally, the connection between the suggested stability margins and the selection of cost weighting matrices is presented.
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