1. Structured H ∞ control of a continuous crystallizer L. Ravanbod, D. Noll, P. Apkarian Institut de Mathématiques de Toulouse IFAC Workshop on Control Distributed Parameter Systems Toulouse, France July 20-24, 2009

2. 2 Outline Industrial Crystallizer : presentation, physical model. Why H ∞ control? H ∞ control : structured controller, structured + time constraints. Simulation results : application to continuous crystallizer.

3. 3 Continuous Industrial Crystallizer used for mass production of high-purity solids from liquids. This crystallizer produces hundreds of tons of amonium sulfate per day. (a) (b) (c) (d) (e) a: body b: settling region of fine crystals c: slurry is withdrawn d: slurry heated and combined with product feed e: solvent evaporates

4. 4 q, c f q, h f,n, c q, h p, n, c Continuous Crystallizer : Our hypotheses ► q: feed rate ► C f : solute concentration in feed ► C: solute concentration ► h f : classification function of fines dissolution ► h p : classification function of product removal ► n: number density function  ideal mixing,  isothermal operation,  constant overall volume,  nucleation at negligible size,  size-independence growth rate,  no breakage, no agglomeration

5. 5 Continuous Crystallizer: equations Population balance : nL,t t Gc nL,t L q v h f Lh p LnL,t nL,0n 0 L,n0,t Bc Gc GcK g ctc s g,BcK b ctc s b with initial and boundary conditions: and the classification functions: nucleation (birth) rategrowth rate =(number of crystals/crystal length L)/volume at time t

6. 6 Continuous Crystallizer : equations mole balance : with initial condition: M dcdc dt qMcMc v McMcd q Mc f v q v 1K v 0 h p L1nL,tL 3 dL c0c 0 Crystal size distribution is represented by mass density function: and by overall crystal mass and where: t1K v 0 nL,tL 3 dL mL,tK v nL,tL 3 mt 0 K v nL,tL 3

7. 7 Continuous Crystallizer: Why feedback control? Nucleation, crystal growth, fines dissolution, classified product removal … Undesirable oscillatory behaviour As in solute concentration:

8. 8 Continuous Crystallizer: Why feedback control? Or as in mass density function:

9. 9 Control strategies P: plant (crystallizer),K: controller P K y w u z w z guarantying internal stability, minimizing impact of on Find structured controller i.e. K(s) ĸ (s) : Two families of linear regulators: if w white noise: if w of finite energy: w(t) : u(t) : z(t) : y(t) : ( Sup ω )

10. 10 H ∞ control of crystallizer Population balance finite or infinite dim.model finite dim. H ∞ controller (SISO) (Chiu et al 1999, Bosgra et al 1995) (Vollmer and Raisch 2001) Population balance Large state linear model small dim. H ∞ controller We propose: Advantage:  selection of controller structure,  easily extendable to MIMO,  time constraints conveniently added. Previous works:

11. 11 min K T w z K Constrained structured H ∞ control Minimize: Subject to z l tzK,tz u t,forallttt zK,sT w z K,s w 0 s KK Θ Time domain constraints, ( w 0 (t) step, ramps, sinusoid) Θ decision variable

12. 12 Multistage H ∞ synthesis Smooth optimization (SQP) Non smooth optimization stabilizing 2 nd order H ∞ controller Closed-loop interconnection Non smooth optimization stabilizing 2 nd order H ∞ controller: time constraints are approximately satisfied stabilizing 2 nd order H ∞ controller: time constraints satisfied

13. 13 H ∞ control Numerical method  H ∞ synthesis is minmax  nonsmooth and nonconvex techniques is proposed: minimized by Cutting-Plane Algorithm.  Smooth optimization (SQP) accelerates creation of good starting points.  Closed-loop stability is guaranteed by constraint: Efficient for large systems due to possibility of structure selection ( Apkarian, Noll, Bompart, Rondepierre,…2006, 2007, 2008) is handled through a progress function:

14. 14 Continuous Crystallizer: modelling 1 Choosing the parameter values : KCl laboratory crystallizer used by : U. Vollmer, J. Raisch, Control Engineering Practice 2001 solute concentration in the feed C f (t) solute concentration in the liquid C(t) C f (t) and disolution rate R 1 (t) C(t) and overall crystal mass M(t) Model input, output choice SISO MIMO discretization of n(L,t) w.r.t L 0 mm 2 mm 2 3

15. 15 Continuous Crystallizer: modelling 4 Linearization at an equilibrium point Linear model (for synthesis) 5 Nonlinear model (for validation) with state space representation Equidistant discretization with N=250, et mollifying the classification functions Equidistant discretization with N=1000

16. 16 Simulation results (Input-Output Model precision) SISO MIMO

17. 17 Simulation results SISO : solute concentrations Second order H ∞ controller+ time domain constraints

18. 18 Simulation results MIMO: solute concentration, overall mass Second order H ∞ controller+ constraints

19. 19 Simulation results Mass density evolution in open-loop and in closed-loop evolution from one equlibrium point to another one

20. 20  New control methodology allows great flexibility of controller structure: small controllers for large systems.  Time constraints can be added and allow to include features of nonlinear systems.  Problem is genuinely nonsmooth and specific algorithm has to be developed. Conclusion

21. 21 Simulation results MIMO: solute concentration, overall mass PID H ∞ controller+ time constraints

22. 22 Control strategies P: plant (crystallizer),K: controller P K y w u z w z internal stability, minimizing impact of on Objectif find K(s) ĸ (s) : Two families of linear regulators can be found: if w white noise: if w of finite energy: w(t) : u(t) : z(t) : y(t) :