Uncertainty model based on partial linearization and robust control

A an uncertainty model is presented that is useful to the robust control design. Under-actuated state affine systems are considered. A partially linear model is gotten taking into account the second and third order partial derivatives of the Taylor series expansion around a constant equilibrium point. This model includes the linear model and constant matrices multiplied by the quadratic and cubic of the state. These constant matrices are known and are included in the stability and performance criterions and the quadratic and cubic terms of the state are proposed as an uncertainty model. The finite gain of the uncertainty model is obtained on line and the Small Gain Theorem is used to guarantee stability. A stabilizing controller is designed for the nominal plant and its free parameter is actualized on line solving a mixed sensitivity problem. A robust controller is designed for this type of uncertainty and is applied to the under-actuated non-linear model of a Pendubot. The considered equilibrium point is at top position and the uncertainty model has only a cubic state term. A non-linear control is used to go from low position to around the top position when this control commutes to the robust controller getting the desired performance. The results are shown in simulations.