Path-wise control of stochastic systems: overcoming the curse of causality

In this thesis we address the topic of path-wise control of stochastic systems defined by stochastic differential equations. By path-wise control we mean that the controller's decisions are not intended to regulate the moments of the state or the output (or a function of them), as customary in stochastic control. Instead, we aim at designing a controller that achieves a desired, specific, trajectory of the state (or the output) itself, for all possible realisations of the noise affecting the system. We show that path-wise control is cursed by insuperable causality issues, because in order to perfectly attain a predefined trajectory for each realisation of the noise, the controller needs to access measurements of the noise itself, which is not possible in practice. Therefore, we approach path-wise control in two steps. Firstly, we design idealistic controllers, which achieve exact regulation by employing a feedback of the noise. Although unrealistic, these designs are preliminary to the second step, i.e. the construction of practical controllers, which estimate the noise from measurements of available quantities (state or output) and use such estimates to perform approximate path-wise control in a hybrid way. We show that the performance of the practical controllers can retrieve the idealistic ones in a limit behaviour. In this framework we address two classical control problems. Firstly, we consider output regulation of linear stochastic systems. We show that the idealistic controllers achieve a zero steady-state tracking error, while the practical controllers allow for a nonzero steady-state error, which, however, can be made arbitrarily small by tuning a design parameter. Secondly, we consider the control of stochastic systems defined by nonlinear, control-affine, stochastic differential equations. In this case, we show that the idealistic controllers achieve exact feedback linearisation and output tracking, while the practical controllers achieve state (and output) trajectories which can be made close to the idealistic ones by tuning a design parameter, thus obtaining approximate feedback linearisation and tracking.Open Acces