Second variation method and optimal control in the presence of model-reality differences

This article concerns the finite horizon optimal control of continuous-time deterministic processes in the presence of model-reality differences. The latter occurs either due to the complexity of the real physical process and hence inability in accurate and exact modelling or the desire of the control engineers in employing simplified models, e.g. linear time varying models. A typical optimal control algorithm in the presence of model-reality differences may be as follows. A nominal control signal is applied to the process and all the states in the process are measured for the duration of time horizon (let us assume this is possible). Using the state measurements, one can construct a linear time-varying model for the small variations in state and control around the nominal trajectories, i.e. a small signal model. Now by solving a minimisation problem based on the linear time-varying model, one can calculate a new control signal which reduces the objective function compared to the previous step. The new control signal is applied to the process and all the calculations are repeated again. These calculations continue iteratively until the control signal converges to its optimal value. In this algorithm instead of using the exact physical model of the process, a linear time-varying model is used which is updated after each iteration. This makes the calculation of optimal control in each iteration much easier