# Computational optimal control : tools and practice by S Subchan; R Żbikowski; Wiley InterScience (Online service)

By S Subchan; R Żbikowski; Wiley InterScience (Online service)

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Xn , . ), and the operations of vector addition and scalar multiplication generalise immediately. The question of extending the definition of vector length x from ( ni=1 xi2 )1/2 to ∞ 2 1/2 is less straightforward. This is because the infinite series 2 ( ∞ i=1 xi ) i=1 xi may not n 2 converge, while the finite sum i=1 xi is always a number. A useful infinite-dimensional vector space only allows those infinite sequences which are quadratically convergent. It then 26 COMPUTATIONAL OPTIMAL CONTROL: TOOLS AND PRACTICE holds that x = ∞ xi2 1/2 <∞ y = and i=1 ⇒ x+y = ∞ yi2 1/2 <∞ i=1 ∞ (xi + yi )2 1/2 < ∞, i=1 so that the finite length property is preserved under vector addition, despite the infinite dimension.

38 COMPUTATIONAL OPTIMAL CONTROL: TOOLS AND PRACTICE Direct Collocation Approach The basic approach for solving optimal control problems by direct collocation is to transform the optimal control problem into a sequence of nonlinear constrained optimisation problems by discretising the state and/or control variables. Two approaches will be considered. The first approach is based on the discretisation of both the state and control variables. The following derivation is based mainly on that of von Stryk and Bulirsch (1992).

G. Teo et al. (1999) have studied control parameterisation by introducing variable switching time into equivalent standard optimal control problems involving piecewise constant or piecewise linear control functions with pre-fixed switching times. Control parameterisation with the direct shooting method has been studied and applied for mechanical multi-body systems by Gerdts (2003) where the control is parameterised by the B-spline function. Further references include Goh and Teo (1988), Teo et al.