Polynomial Chaos reformulation in Nonlinear Stochastic Optimal Control with application on a drivetrain subject to bifurcation phenomena

This paper discusses a method enabling optimal control of nonlinear systems that are subject to parametric uncertainty. A stochastic optimal tracking problem is formulated that can be expressed in function of the first two stochastic moments of the state. The proposed formulation allows to penalize system performance and system robustness independently. The use of polynomial chaos expansions is investigated to arrive at a computationally tractable formulation expressing the stochastic moments in function of the polynomial expansion coefficients rigorously. It is then demonstrated how the stochastic optimal control problem can be reformulated as a deterministic optimal control problem in function of these coefficients. The proposed method is applied to find a robust control input for the start-up of an eccentrically loaded drive train that is inherently prone to bifurcation behaviour. A reference trajectory is chosen to deliberately provoke a bifurcation. The proposed framework is able to avoid the bifurcation behaviour regardlessly.