Non-linear stabilisation and regulation via an optimal gain schedule

A nonlinear design method based on LQ optimal control theory is presented that applies to a wide class of nonlinear systems. The method gives an asymptotically optimal solution that has the potential to be implemented in real-time in the sense that the solution is causal. The key feature of the method is the introduction of state-dependence in the weight matrices of the usual LQ cost function, leading to a nonlinear control policy. We make use of a result that generalises the LQ theory to nonlinear systems to provide a nonlinear design method that overcomes some of the difficulties. This nonlinear quadratic method applies to systems having a broad class of nonlinear dynamics with state-dependent weighting matrices. It turns out that the infinite-time-horizon LQ regulator problem when solved afresh at every point on the state trajectory leads to an asymptotically optimal control policy. For admissible system dynamics, the weighting parameters can be made to be functions of the state variables. Thus, in addition to handling nonlinear dynamics, the design stage allows for the introduction of state-dependence in the weighting matrices, leading to a more flexible control strategy. Our method is causal but has considerable computational overhead. However, by using a solution to the Riccati equation based upon the matrix sign function, it is possible to derive a parallel algorithm that may be suitable for real-time implementation