Input-restricted robust feedback control of nonlinear dynamical systems

This proposal investigates fundamental robust feedback control theory of nonlinear dynamical systems via input-output methods. The input-output approach in control theory describes dynamical systems as operators mapping an input to an output. Feedback stability in this input-output framework means that endogenous signals around the feedback loop are well-behaved for all exogenous inputs – in an input space – injected into the feedback loop. Influential input-output feedback stability results which can be applied to nonlinear systems to guarantee robust feedback stability include the classical small-gain, input-output passivity, input-output dissipativity, and absolute stability criteria, as well as more modern input-output stability approaches such as the Integral Quadratic Constraint framework and dynamic dissipativity. However, all these important methods guarantee feedback stability for all exogenous inputs within a specified input space. If a closed-loop system is well behaved for a subset of inputs within an input space, but not well-behaved for every input in that space, these methods cannot guarantee closed-loop stability in this scenario. However, no feedback system will ever encounter every possible exogenous input. In fact, in many applications, some basic prior information is known about the exogenous inputs, such as the maximum energy or the maximum magnitude of the possible input excitations, that the feedback loop will encounter. If a controller can be designed to guarantee feedback stability only for the exogenous inputs a feedback loop is expected to experience, instead of guaranteeing feedback stability for all possible inputs, this controller would be less conservative and therefore can provide high levels of closed-loop performance as a trade-off for this reduced conservatism in stability. It is therefore the aim of the proposed research to develop theory which can guarantee stability of a closed-loop nonlinear dynamical system for a restricted subset of exogenous inputs, even if such a system would be destabilised when subjected to inputs outside of this restricted subset. We call this type of stability ‘input-restricted stability’. Using graph-separation techniques, we intend to show how restricting the exogenous inputs of a feedback system restricts the regions of the system graphs which we are interested in, such that only separation of the restricted graphs is required to prove input-restricted stability. Once such theory has been established, we aim to develop input-restricted feedback stability synthesis theory to enable the design of controllers which achieve a certain specified level of input-restricted stability. Such controllers would enable an engineer to guarantee input-restricted stability for a given norm restriction on the exogenous inputs, such as the maximum energy or maximum magnitude of external disturbance/noise excitations, thereby allowing greater freedom in controller design compared to standard controller synthesis methods. This then leads to higher levels of closed-loop performance in exchange for reduced conservatism in stability. After developing the input-restricted stability theory and input-restricted controller synthesis methods, we shall develop both simulation and hardware experiments to demonstrate the power of these new methods. These examples will be physically motivated and designed to showcase the applicability of input-restricted feedback stability control theory.