Control design for nonlinear systems: trading robustness and performance with the model predictive control approach

control laws with robustness properties for nonlinear systems without the need to solve a Hamilton-Jacobi-Isaacs (HJI) equation, which is a difficult computational task. In particular, in an H1 setting, RH schemes have been introduced in [11, 12], to derive a state-feedback control law with the same robustness properties of an auxiliary control law kð�Þ previously defined while enlarging the invariant set where kð�Þ solves the H1 problem. The results reported in [13] show that a proper selection of the design parameters guarantees that the region of attraction of the RH law can be made larger than the largest one associated with kð�Þ without a significant computational burden due to the optimisation procedure implied by the MPC approach. The technique proposed here shares many ideas both with the previously developed MPC algorithms for robust control and with the method proposed in [9]. Specifically, an auxiliary control law kð�Þ solving the H1 problem in a given invariant set is assumed to be known, and then an H2 optimisation problem is formulated and solved under the additional constraint that its solution provides at least the same level of attenuation guaranteed by kð�Þ in the same invariant set or in a larger one. This result is achieved by extending the algorithm described in [14] and by properly selecting the free MPC design parameters, such as the control and prediction horizons. The proposed algorithm calls for the solution of a min=max optimisation problem, where minimisation must be performed with respect to control law strategies, while maximisation is over disturbance sequences. As such, the proposed solution is analytically and=or computationally almost intractable. However to this regard two levers can be used to reduce the size of the optimisation space. Firstly, the control law strategies can be suitably parametrised with respect to a fixed number of parameters. Secondly, the control horizon, adopted in the MPC formulation, can be definitively smaller than the prediction horizon.