1. Håkon Dahl-Olsen, Sridharakumar Narasimhan and Sigurd Skogestad Optimal output selection for batch processes

2. Outline Batch optimization Implementation schemes Variable selection method Reactor case study Summary

3. Batch optimization Minimum time to given specification Maximum product in fixed time

4. Dynamic optimization

5. Implementation Online optimization: measurements used to update model Self-optimizing control: good outputs give near optimal performance

6. Variable selection Unconstrained degrees of freedom Based on Pontryagin’s minimum principle Look for variables which give small deviation from optimality when controlled at fixed reference, even under disturbances

7. Maximum gain rule Loss is defined in terms of value of the Hamiltonian The loss is time-varying

8. Maximum gain rule Need to relate variations in inputs to variations in outputs

9. Maximum gain rule for dynamic optimization Assume the model is scaled such that δy max ≈1 Select y’s to minimize the following expression along the nominal trajectory

10. How to obtain G How does variations in inputs map to the states? Neighboring optimal control gives δu(t) = K(t)δx(t) We estimate G by

11. Example Maximize production of product P in a fed-batch bioreactor with fixed final time of 150 hours Reaction is driven by the presence of a substrate S, which is consumed in the biomass generation The biomass concentration is constrained Bioreactor Srinivasan, B., D. Bonvin, et al. (2002). "Dynamic optimization of batch processes II. Role of measurements in handling uncertainty." Comp. chem. eng. 27: 27-44.

12. Example X < 3.7 Biomass concentration S Substrate concentration P Product concentration VVolume u < 1 Substrate feedrate Bioreactor u [L/h] S in =200 g/L Controller Measurements

13. Example X < 3.7 Biomass concentration S Substrate concentration P Product concentration VVolume u < 1 Substrate feedrate Bioreactor

14. Biomass growth rate Bioreactor

15. Consider gain magnitude Transformation of input: ξ=√u H ξξ is constant Minimizing of 1/K 2 corresponds to maximizing K 2 Bioreactor

16. S Comparison of gains X P V Bioreactor

17. Simulation results

18. Summary Maximum gain rule extended to dynamics Variational gain from neighboring optimal control theory Method works well for a small case study