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Stabilisation using a stable controller and robust stability and performance

A class of strictly proper multi-input multi-output (MIMO) causal linear time invariant (LTI) systems with a detectable and stabilisable realisation in a proposed observer-controller scheme, is considered. This class is characterised by the rank of the observability matrix in closed loop. The considered scheme is based on an ℋ∞ pre-compensator and on an ℋ∞ dual post-compensator stabilising a full actuation full information plant and on pseudo inverses of the input and of the output matrices. The pre-compensator and post-compensator are of reduced complexity and belong to the family of all stabilising controllers. The separation principle is satisfied and necessary and sufficient stability conditions are presented for the overall system getting a stable ℋ∞ controller. An algebraic approach and a state space approach are used to find this stability condition. The estimated error is zero in stationary state. Also, the control parameters allow the stationary state error and the complementary sensitivity function in high frequencies to approach zero asymptotically. So, robust stability and performance are achieved. A compromise exists between the stability of the compensators and the robust performance objective. The results are illustrated on a mechanical system.