H∞ preview control and fixed-lag smoothing I: matrix ARE solutions in continuous-time systems

Preview control and fixed-lag smoothing allow an anti-causal component in the controller/estimator. Time domain variational analysis is used in a reduction to an open loop differential game, leading to a complete, necessary and sufficient characterization of suboptimal values and an explicit state space design, in terms of a parameterized (non-standard) algebraic matrix Riccati equation in a general continuous time linear system setting. The solution offers insight into the appropriate structure of the associated Hamiltonian, where the state and co-state are not the usual state of the original dynamic system and that of its adjoint. Rather, the state and co-state are selected to capture the respective lumped effects of initial data and future input selection in the allied game.