Data-Based Modelling for Control

1. Data-Based Modelling for Control Paul M.J. Van den Hof www.dcsc.tudelft.nl/~pvandenhof/publications 2006 IEEE Workshop Advanced Process Control Applications for Industry (APC2006), Vancouver, Canada, May 8-10 2006.

2. Contents • Introduction • Basic facts on system identification • Example from a MSW incineration plant • Models for control • Model uncertainty and model validation • Basis functions model structures • Cheapest experiments • Discussion and prospects

3. “obtaining process models is the single most time- consuming task in the application of model-based controllers” (Ogunnaike, An Rev Control, 1996; Hjalmarsson, Automatica, 2005) Introduction Costs distribution in an advanced process control project: • Feasibility study 10% • Pre-tests 10% • Model identification 40% • Controller tuning 15% • Commissioning and training 25% (Zhu, IFAC SYSID, 2006) Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

4. Which kind of models to consider? First principles / rigorous models Process design; planning and scheduling; off-line • large number of equations (PDE,ODE,DAE) • high computational complexity • question of validation • nonlinear Data-based models • compact model structures • computational feasible • validated by data • often linearized Advanced control; on-line operations; on-line For advanced process control data-based models seem dominant Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

5. On-line use of first principle models • State dimension >> • f and h nonlinear • For monitoring/diagnosis problems, state variables have clear • physical interpretation, which has to be retained • Full models in general too complex for on-line evaluations • Input-output model reduction destroys the state structure • State-based model reduction techniques (POD,…) only help computationally in the case of linearf and h • The (nonlinear) mappings have to be approximated/simplified Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

6. Model structures Black box Emphasis, for the moment Well sorted out in linear case Not mature in nonlinear case Physics-based Problem of accurate parametrization (where to put the unknowns?) Identifiability Data-based models (identification) • Relatively easily obtained • Model costs are related to experiments on the plant Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

7. “Here is a dynamical process with which you are allowed to experiment (preferably cheap). Design and implement a high-performance control system”. • Issues involved: • Experiment design • Modelling / identification • Characterization of disturbances and uncertainties • Choice of performance measure • Control design and implementation • Performance monitoring Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

8. t 1.4 p 1.2 M p ± 1% 1 90% 0.8 0.6 y(t) 0.4 0.2 10% 0 t r t s time Classical experiments for finding control-relevant dynamics • Ziegler/Nichols tuning rules • for PID-controllers • Relay feedback: amplitude and frequency at -180° phase Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

9. Ad-hoc simple cases to be extended to • general methodology for model-based control, • including issues of robustness induced by • model uncertainties Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

10. Identification for Control (1990-…) • Basic principles for identifying models, well sorted out • Relation with control through • Certainty equivalence principle: • “Controller based on exact model is suited for • implementation on the plant” However: • Identification had been extended to identify • approximatemodels • Control design had been evolved to robust control • taking account of model uncertainties Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

11. Data Model Feedback control system Feedback control system disturbance reference input + output Model Controller controller process controller process - Experiments: Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

12. r + C u G0 y - When is a model suitable for control? For a given controller C: r Ĝ + C u y - Designed loop Achieved loop • Both loops should be “close” (r  y): should be small • Disturbance effects on y should be similar Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

13. wb When is a model suitable for control? plant model1: accurate for w<wb model2: accurate for wwb Model quality becomes dependent on control bandwidth (to be designed) Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

14. Control bandwidth is based on model + .. exp. design Experiment Experiment Experiment data If models are uncertain/approximate due to limited experiment, achievable performance needs to be discovered Identificatie Identification model Control design Regelaarontwerp controller ! modelling for control is learning (Schrama, 1992; Gevers, 1993) Implementatie Implementatie Implementation evaluation Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

15. high-order model high-order controller experiment data low-order model low-order controller From experiment to control: validation and uncertainty • Current opinion: • Extract all information from data, but • Keep experiments simple Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

16. Identification Control 1990 - : Schrama, Gevers, Bitmead, Anderson, Åström, Rivera, …. • control-relevant • nominal model • nominal control 1994 - : Hakvoort, de Vries, Ninness, Bitmead, Gevers, Bombois, … • nominal model + • uncertainty bound • nominal control + • stab/perf robustness • control-relevant • model uncertainty set • robust control; worst-case • performance optimiz. 1997 - : de Callafon & vdHof,Douma • design of “cheap” • experiments for id of • uncertainty sets 2002 - : Bombois, Gevers,Hjalmarsson, vdHof, • control under performance • guarantees Development trend: Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

17. Contents • Introduction • Basic facts on system identification • Example from a MSW incineration plant • Models for control • Model uncertainty and model validation • Basis functions model structures • Cheapest experiments • Discussion and prospects

18. Basic facts on system identification Identification of parametric models through prediction error identification (open-loop) Data generating system: Predictor model: e is realization of stochastic white noise process From measured data {u(t),y(t) }, t=1,..,N to estimated model Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

19. {u(t),y(t) }, t=1,..,N fractions of polynomials Convex or non-convex optimization Prediction error framework: (Ljung, 1987) Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

20. 2. If and u is sufficiently exciting then provided that G and H are parametrized independently. Asymptotic variance typically dependent on (frequency-dependent noise to signal ratio) Classical consistency results 1. If and u is sufficiently exciting then Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

21. Since parameter estimates are asymptotically normally distributed (cental limit theorem), the variance expression can be converted to parameter confidence regions, e.g. 3s-bounds (99.7%). Using the mappings • the uncertainty bounds • can be converted to • frequency response • step response • etc. Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

22. Computational issues 1. In general situation: non-convex optimization(with risk of local minima) 2. Convex optimization if prediction error is affine in the parameters:property of model structure: FIR: ARX: ORTFIR: with A,B,Fpolynomials in q-1 : Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

23. 1. If then 2. If and (fixed) then Design variables in general case: model structure Characterization of asymptotic estimate Limiting parameter estimate: i.e. minimizing the power in the weighted residual signal Substituting the expressions from the signal block diagram delivers Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

24. Same approach can be followed (direct method), on the basis of measurements u(t),y(t) Consistency result: provided that and either: • r is sufficiently exciting, or • C is sufficiently complex (high order / time-varying) Accurate noise modelling is necessary for identifying G Closed-loop situation Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

25. In direct closed-loop identification, possibilities for separately identifying G0 and H0 are lost. In a MIMO situation this happens already when a single loop is closed: Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

26. Asymptotic variance in closed-loop identification where now because of the closed-loop. Writing a simple analysis leads to reference part noise part (only the reference part of the input signal contributes to variance reduction) Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

27. 2. Two-stage method • Identify transfer r  u • Simulate • Identify G0 as transfer Input signal u is denoised Alternative indirect methods When focussing on plant model only Several options, among which: 1. Indirect Method • Identify transfer r  y • Retrieve plant model, with knowledge of C Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

28. Properties indirect methods 1. If then 2. If then provided that r is sufficiently exciting and C is linear General expression for the asymptotic estimate (with slight variations) Closed-loop properties of the plant are approximated. Note that: separate identification of G0 and H0 is possible. Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

29. |.| T 0 10 S -1 10 -2 10 -1 0 ω 10 10 red blue Low frequencies are hidden; frequencies around bandwidth are emphasized Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

30. Alternative closed-loop ID methods • IV methods, using r as instrumental variable • Coprime factor identification (related to gap, nu-gap metric) • Dual-Youla identification Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

31. Wrap-up PE identification • Mature framework for system ID • Open-loop and closed-loop data can be handled • Stochastic noise framework • Extensions to multivariable situation available • Analysis is available but mainly for infinite data • Analysis much more explicit than e.g. for subspace ID /state-space models approximate models – design variables Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

32. Contents • Introduction • Basic facts on system identification • Example from a MSW incineration plant • Models for control • Model uncertainty and model validation • Basis functions model structures • Cheapest experiments • Discussion and prospects

33. Municipal Solid Waste Combustion (Martijn Leskens) IntroSystem ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

34. (Nolinear) Model Predictive Control of MSWC Plants • Aim: NMPC of furnace and boiler part of MSWC plant: NMPC requires good dynamic model of MSWC plant  MODEL VALIDATION Simulation results IntroSystem ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

35. Closed-loop identification of MSWC plants • Closed-loop experimental configuration typically encountered in MSWC plants: IntroSystem ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

36. Experimental model is fit in the same i/o structure as the first principles model “PARTIAL” closed-loop identification: u1 = “open-loop” inputs y1 = “open-loop” outputs Etc. IntroSystem ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

37. Goal • Identification of linear models in 2 working points:Tprim = 70, 120 °C • Use these models to validate/calibrate a simple first-principles model Identification setup • RBS excitation of all controlled inputs • Closed-loop identification with (indirect) two-stage method • Use of high-order ARX models and model-reduction • Enforcements of static gains to improve low-frequent behaviour • Sample time of 1 minute • Identified model validated through correlation tests • 8 scalar transfers identified with order between 2 – 5. Simplified physical model (5th order NL) tuned to identified models. IntroSystem ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

38. Considerable disturbances on output data: dashed is measured data, solid is simulated data IntroSystem ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

39. Estimated model (dashed) and NL-physical model (solid) upon excitation of the waste inlet IntroSystem ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

40. Estimated model (dashed) and NL-physical model (solid) upon excitation of the primary air flow IntroSystem ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

41. Results • Identification and validation results for Tprim = 70 (I): good to very good agreement: Responses on step from waste inlet flow IntroSystem ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

42. Results • Identification and validation results for Tprim = 120 (I): moderate to reasonable agreement: Responses on step from waste inlet flow IntroSystem ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

43. Results • Closed-loop identification strategy is ‘easily’ applicable in an industrial setting and works well • Fitting of first-principles model is still rather ad-hoc • Models are accurate enough for model-based MPC IntroSystem ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

44. Contents • Introduction • Basic facts on system identification • Example from a MSW incineration plant • Models for control • Model uncertainty and model validation • Basis functions model structures • Cheapest experiments • Discussion and prospects

45. wb How poor can models be? plant model1: accurate for w<wb model2: accurate for wwb IntroSystem ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

46. Controlled with 5th order controller, with I-action, bandwidth 0.5 rad/s Model quality becomes dependent on control bandwidth (to be designed) IntroSystem ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

47. In general terms: • Need for a structured way to measure control relevance of models, and • methods to identify them from data What looks like a good model in open-loop may be poor in closed-loop and vice versa • Rule of thumb: models need to be accurate around control bandwidth IntroSystem ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

48. r + C u G0 y - When is a model suitable for control? For a given controller C: r Ĝ + C u y - Designed loop Achieved loop Performance measure for model quality could be: The power of the difference signal: In frequency domain: IntroSystem ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

49. Indirect closed-loop ID delivers: Can this performance measure be minized through identification? Requested: Conclusion: A C-relevant model is identified by indirect closed-loop ID, by choosing IntroSystem ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

50. Can this be achieved by open-loop identification? OL-expression (OE-case): Required integrand: This requires: which is unfeasible because of lack of knowledge of IntroSystem ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion