Second‐order optimal control algorithm for complex systems

The solution to large‐scale optimal control problems, characterized by complex dynamics and extended time periods, is often computationally demanding. We present a solution algorithm with favourable local convergence properties as a way to reduce simulation times. This method is based on using a trapezoidal direct collocation to convert the differential equations into algebraic constraints. The resulting constrained minimization problem is then solved with an augmented Lagrangian formulation to accommodate both equality and inequality constraints. In contrast to the prevalent optimal control software implementations, we calculate analytical first and second derivatives. We then apply a generalized Newton method to the augmented Lagrangian formulation, solving for all unknowns simultaneously. The computational costs of the Hessian formation and matrix solution remain manageable as the system size increases due to the sparsity of all tensor quantities. Likewise, the total iterations for convergence scale well due to the local quadratic convergence of the generalized Newton method. We demonstrate the method with an inverted pendulum problem and a neuromuscular control problem with complex dynamics and 18 forcing functions. The optimal control solutions are successfully found. In both examples, we obtain quadratic convergence rates in the neighbourhood of the solution. Copyright © 2002 John Wiley & Sons, Ltd.