On adaptive optimal input design

Abstract

The problem of optimal input design for a specific fedbatch bioreactor case study is solved recursively. Hereto an adaptive receding horizon optimal control problem, involving the so-called E-criterion, is solved ‘on-line’, using the current estimate of the parameter vector θ at each sample instant {t <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> , k = 0,…, N − h}, where N marks the end of the experiment and h is the control horizon for which the input design problem in solved. The optimal feed rate F∗ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf> (t <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> ) thus obtained is applied and the observation y(t <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k+1</inf> ) that becomes available is subsequently used in a recursive prediction error algorithm in order to find an improved estimate of the parameter estimate θ(t <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> ). The case study involves an identification experiment with a Rapid Oxygen Demand TOXicity device for estimation of the biokinetic parameters μ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</inf> and K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</inf> in a Monod type of growth model. It is assumed that the dissolved oxygen probe is the only instrument available which is an important limitation. Satisfactory results are presented and compared to a ‘naive’ input design in which the system is driven by an independent binary random sequence with switching probability p = 0.5. This comparison shows that, indeed, the optimal input design approach yields improved uncertainty bounds on the parameter estimates.