When implementing a nonlinear model predictive controller, the control engineer is faced with a number of different, but essentially mathematically equivalent, algorithms. However, computing power is limited and it is well-known that a particular choice of algorithm for solving an optimal control problem has a significant impact on the accuracy and speed of the implementation. This presentation will compare the numerical conditioning of two popular methods for solving an optimal control problem via the use of nonlinear programming solvers, namely "Direct Single Shooting" and "Direct Multiple Shooting". We will show via some simple examples that, whereas the former method results in an optimization problem with a smaller number of decision variables, the latter method is much better conditioned. The lessons learnt from these examples suggest that Direct Multiple Shooting should probably be the preferred method of choice for systems that are "difficult" to control, such as unstable systems with input and state constraints.
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