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

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Delft Center for Systems and Control 2 Contents 1.Introduction 2.Basic facts on system identification 4.Models for control 5.Model uncertainty and model validation 6.Basis functions model structures 7.Cheapest experiments 8.Discussion and prospects 3.Example from a MSW incineration plant

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Delft Center for Systems and Control 3 Introduction Feasibility study10% Pre-tests10% Model identification40% Controller tuning15% Commissioning and training25% Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Costs distribution in an advanced process control project: (Zhu, IFAC SYSID, 2006) 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)

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

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Delft Center for Systems and Control 5 Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion 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 State-based model reduction techniques (POD,…) only help computationally in the case of linear f and h The (nonlinear) mappings have to be approximated/simplified Input-output model reduction destroys the state structure

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

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Delft Center for Systems and Control 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

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

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Delft Center for Systems and Control 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

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Delft Center for Systems and Control 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 approximate models 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

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

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Delft Center for Systems and Control 12 When is a model suitable for control? For a given controller C: CG0G0 + - r u y Achieved loop C + - r u y Designed loop Both loops should be close (r y): should be small Ĝ Disturbance effects on y should be similar Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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

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

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Delft Center for Systems and Control 15 data high-order controller high-order model low-order model low-order controller experiment 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

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

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Delft Center for Systems and Control 17 Contents 1.Introduction 2.Basic facts on system identification 4.Models for control 5.Model uncertainty and model validation 6.Basis functions model structures 7.Cheapest experiments 8.Discussion and prospects 3.Example from a MSW incineration plant

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Delft Center for Systems and Control 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

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

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

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Delft Center for Systems and Control 21 Since parameter estimates are asymptotically normally distributed (cental limit theorem), the variance expression can be converted to parameter confidence regions, e.g. 3 -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

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Delft Center for Systems and Control 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,F polynomials in q -1 : Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 23 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 1. If then 2. If and (fixed) then Design variables in general case: model structure Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 24 Closed-loop situation 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 Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 25 Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion In direct closed-loop identification, possibilities for separately identifying G 0 and H 0 are lost. In a MIMO situation this happens already when a single loop is closed:

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Delft Center for Systems and Control 26 Asymptotic variance in closed-loop identification where now because of the closed-loop. Writing a simple analysis leads to (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 reference partnoise part

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Delft Center for Systems and Control 27 Alternative indirect methods When focussing on plant model only 1. Indirect Method Identify transfer r y Retrieve plant model, with knowledge of C Several options, among which: 2. Two-stage method Identify transfer r u Simulate Identify G 0 as transfer Input signal u is denoised Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 28 Properties indirect methods General expression for the asymptotic estimate (with slight variations) 1. If then 2. If then provided that r is sufficiently exciting and C is linear Closed-loop properties of the plant are approximated. Note that: separate identification of G 0 and H 0 is possible. Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 29 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

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

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Delft Center for Systems and Control 31 Wrap-up PE identification Mature framework for system ID Extensions to multivariable situation available Stochastic noise framework 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 Open-loop and closed-loop data can be handled

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Delft Center for Systems and Control 32 Contents 1.Introduction 2.Basic facts on system identification 4.Models for control 5.Model uncertainty and model validation 6.Basis functions model structures 7.Cheapest experiments 8.Discussion and prospects 3.Example from a MSW incineration plant

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Delft Center for Systems and Control 33 Municipal Solid Waste Combustion Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion (Martijn Leskens)

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Delft Center for Systems and Control 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 Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 35 Closed-loop experimental configuration typically encountered in MSWC plants: Closed-loop identification of MSWC plants Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 36 PARTIAL closed-loop identification: u 1 = open-loop inputs y 1 = open-loop outputs Etc. Experimental model is fit in the same i/o structure as the first principles model Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 37 Goal Identification of linear models in 2 working points: T prim = 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 (5 th order NL) tuned to identified models. Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 38 Considerable disturbances on output data: dashed is measured data, solid is simulated data Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 39 Estimated model (dashed) and NL-physical model (solid) upon excitation of the waste inlet Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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

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Delft Center for Systems and Control 41 Results Identification and validation results for T prim = 70 (I): good to very good agreement: Responses on step from waste inlet flow Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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

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Delft Center for Systems and Control 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 Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 44 Contents 1.Introduction 2.Basic facts on system identification 4.Models for control 5.Model uncertainty and model validation 6.Basis functions model structures 7.Cheapest experiments 8.Discussion and prospects 3.Example from a MSW incineration plant

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Delft Center for Systems and Control 45 How poor can models be? plant model1: accurate for < b model2: accurate for b b Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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

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Delft Center for Systems and Control 47 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 In general terms: Need for a structured way to measure control relevance of models, and methods to identify them from data Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 48 When is a model suitable for control? For a given controller C: CG0G0 + - r u y Achieved loop C + - r u y Designed loop Ĝ Performance measure for model quality could be: The power of the difference signal: In frequency domain: Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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

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Delft Center for Systems and Control 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 Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 51 For a known controller C, the best control relevant model is naturaly identified on the basis of a closed-loop experiment when C is implemented on the plant However: the actual aim is to build models for a to-be-designed controller…. Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 52 Optimal models for control design Consider control performance cost function: J is a general function, possibly including signal spectra, weightings etc. Optimal controller: E.g. tracking weighted sensitivity Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 53 Optimal models for control design Bounding the achieved control performance cost Triangle inequality: achieved performance designed performanceperformance degradation nominal control design (when fixed) Identification of nominal model (when fixed) Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 54 Implementatie Implementation Regelaarontwerp Control design controller IdentificatieIdentification model Experiment data Experiment evaluation exp. design Successive iteration of nominal (closed-loop) identification nominal control design Appropriate convergence requires cautious/robust tuning, (taking account of model uncertainty) Experiment design is not critically incorporated yet (later) Problem evolves to data/model- based controller tuning Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 55 Including model uncertainty for controller robustness 10 -2 10 10 0 -4 10 -2 10 0 2 -2 10 10 0 -600 -400 -200 0 amplitude phase frequency 1)In situation convert probabilistic parameter uncertainty bound to the frequency domain (Bode, Nyquist) Determine and use this as f-dependent (hard) upper bound on additive error to be used in classical robust stability and performance checks Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 56 Including model uncertainty for controller robustness 1) Some work has been done to extend this to the situation by including a deterministic bounding term on the bias (Goodwin, Gevers & Ninness, 1992; de Vries & Van den Hof, 1995; Hakvoort & Van den Hof, 1997) relying on FIR/ORTFIR model structures Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 57 Including model uncertainty for controller robustness 2)Retain the parametric structure in the model uncertainty by considering U is parameter ellipsoid determined by cov.matrix A particular robust stability test is available for this structure Worst-case frequency response of any closed-loop transfer of (C,G) can be exactly calculated by solving an LMI problem. (Bombois, Gevers, Scorletti, Anderson, 2001) Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 58 Including model uncertainty for controller robustness 3)In stead of looking at explicit bounds for the model error there are options for formulating the bounds directly on the level of the performance function: e.g. by considering bounds on the closed-loop system rather than on the plant. Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion For virtually all closed-loop measures J, performance degradation is affine in dual-Youla parameter R

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Delft Center for Systems and Control 59 Controller tuning (Double Youla parametrization) Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion A plant uncertainty structure based on the double Youla parametrization delivers a larger set of controllers guaranteed to be robustly stabilizing than an uncertainty based on gap metrics. Stability guaranteed if Control-dependent plant uncertainty and plant-dependent controller deviation

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Delft Center for Systems and Control Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 61 Contents 1.Introduction 2.Basic facts on system identification 4.Models for control 5.Model uncertainty and model validation 6.Basis functions model structures 7.Cheapest experiments 8.Discussion and prospects 3.Example from a MSW incineration plant

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Delft Center for Systems and Control 62 Uncertainty regions with probability 10 -2 10 10 0 -4 10 -2 10 0 2 -2 10 10 0 -600 -400 -200 0 amplitude phase frequency Frequeny response parameter region Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 63 Classical reasoning for quantifying uncertainty Let identified models pass a validation test Assume that the real system belongs to the model set Use analytical (variance) expressions for quantifying parameter variance and resulting model variance Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 64 Residual tests When model passes the test, there is no evidence in the data that the model is wrong Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 65 Question: Classical reasoning for quantifying uncertainty Let identified models pass a validation test Assume that the real system belongs to the model set Use analytical (variance) expressions for quantifying parameter variance and resulting model variance When a model passes the validation test is it justified to assume that ? If YES: validation test is crucial If NO: how to justify the assumption? Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 66 Intriguing example – 4 th order process; 2 nd order model; white input log Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 67 Intriguing example – 4 th order process; 2 nd order model Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 68 Intriguing example – 4 th order process; 2 nd order model No undermodelling is detected Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 69 Problems: Bounds for testing are dependent on is estimated by Due to unmodelled dynamics the noise variance is over-estimated Bounds in the correlation test are too large In Output Error case: Additionally: Pointwise test on does not detect any structure in the signal Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 70 Remedies 1.Improve noise model for validating 2.Replace pointwise test on by vector valued test on Accurate noise model, e.g. through high-order auxiliary plant model and time-series modeling on the basis of the output residual (suggested before by Söderström, 1989; Hjalmarsson, 1993) Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 71 Example (continued) Same 4 th order system as before; white output noise N = 256 Input signal is white noise Auxiliary high-order OE plant model of order N/5 Time-series model on output residual for noise modelling Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 72 Example (continued) effect of model error effect of noise classical bound improved bound Improved noise model leads to falsification of 2 nd order model Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 73 Vector-valued test over 128 lags Exact noise model Improved estimate From residual All models are validated Example (continued) Noise cov estim from: Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 74 However Even if we can improve the validation test, it remains dependent on input-data (no conclusions possible about frequency areas that are not excited) Best case scenario (validation implies no-bias) is questionable Argument for incorporating bias-term in quantifying model ucertainty There are limitations in validating plant models without having accurate noise models Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 75 Contents 1.Introduction 2.Basic facts on system identification 4.Models for control 5.Model uncertainty and model validation 6.Basis functions model structures 7.Cheapest experiments 8.Discussion and prospects 3.Example from a MSW incineration plant

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Delft Center for Systems and Control 76 Basis functions model structures In identification the use of FIR models is very attractive: FIR: 2. Output Error structure: plant/noise model parametrized independently Simple computations and analysis of model uncertainty 1. Linear regression LS criterion is convex 3. Covariance matrix P is explicitly available and not dependent on G 0 holding true even for finite N 4. Parameter maps linearly to f-response Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 77 Disadvantage: High number of parameters is required to cover fast and slow dynamics Alternative: Use pre-chosen generalized functions, tailored to the dynamics Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 78 Laguerre-case: Expansionbecomes particularly fast for systems with dominating pole around z=a Generalized situation through Gramm-Schmidt orthogonalization on (Takenaka-Malmquist) Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 79 (Takenaka-Malmquist) Practical use: choose any finite sequence of stable pole locations i for i=1,..,n b (preferably in neighborhood of dominating system poles) use model expansion: where usually the pole locations are repeated to extend the expansion for n 1 all stable LTI systems can be represented basis functions are orthogonal in l 2 sense Central role is played by the stable all-pass function: Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 80 Generalization of tapped-delay line Generation of the OBFs By a set of stable poles By a stable all-pass (inner) function: Choice of poles determines rate of convergence of the series expansion Heuberger, Van den Hof & Wahlberg (2005) Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 81 For a given system G 0 with poles in p 1, p n the convergence rate of the series expansion is determined by its slowest eigenvalue: If basis poles and system poles approach each other, the convergence rate becomes very fast Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 82 Example basis pole selection Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 83 ORTFIR: ORTFIR Model Structure Least squares solution: Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 84 ORTFIR extension to multivariable case 1. Scalar functions, matrix coefficients (compare MIMO FIR) 2. Matrix functions, e.g. basis poles are reflected in input balanced pair A,B Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 85 Basis functions wrap-up Attractive model structure for computations and analysis Prior knowledge of system poles can be incorporated The more accurate the prior info, the more efficient is the structure (small number of coefficients bias/variance trade-off) No loss of generality (all systems can be representeed) Full analysis equipment of PE identificaiton is available Simple extension to MIMO case Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Looking for solutions in the neighborhood where you expect them!

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Delft Center for Systems and Control 86 Contents 1.Introduction 2.Basic facts on system identification 4.Models for control 5.Model uncertainty and model validation 6.Basis functions model structures 7.Cheapest experiments 8.Discussion and prospects 3.Example from a MSW incineration plant

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Delft Center for Systems and Control 87 Cheapest Identification Experiment for Control Excitation of r is desirable to improve accuracy of identified model Excitation of r disturbs process operation and efficiency Find the smallest r such that an identified model is sufficiently accurate to robustly enhance the controller C Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 88 Excitation experiment is determined by N and For fixed N, the costs of identification are determined by Problem Set-up with the power of signal and the power of signal and are scalar design variables Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 89 Accuracy of identified model is characterized by its covariance matrix It appears that is affine in N and in : Note: is dependent on plant, and on asymptotic theory Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 90 Experiment design problem: Under the constraint For fixed N, solve Formulated as a result of control performance specs Problem can be solved by an LMI, provided that excitation spectrum is parametrized linearly system knowledge in is replaced by initial model estimate Signal power and data length are related to each other Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 91 Reduction of required data length in process application Minimum identification cost as a function of N For N = 4901, minimum cost is 0: no external excitation is necessary! rbs FIR filtered rbs (Bombois et al. 2004) Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 92 designed bandwidth Reduction of required data length in process application Compared with white noise PRBS signal of same power Gain in experiment time reduction from PRBS to dedicated periodic signal (Jansson, 2005) Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion

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Delft Center for Systems and Control 93 Plant Designed closed-loop Optimal input Example results Inclusing constraints

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Delft Center for Systems and Control 94 Input constraint

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Delft Center for Systems and Control 95 Experiment design New framework for control-relevant experiment design Economic objective is attractive for industrial processes Results –on simulations based- look promising Finite-time perspectives to be taken into account Excite the important dynamics and not more Towards dedicated experiments: Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Generalization of the idea: If you need the static gain, apply a step signal

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Delft Center for Systems and Control 96 Contents 1.Introduction 2.Basic facts on system identification 4.Models for control 5.Model uncertainty and model validation 6.Basis functions model structures 7.Cheapest experiments 8.Discussion and prospects 3.Example from a MSW incineration plant

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Delft Center for Systems and Control 97 Discussion and Prospects Wandering through the field Data-based modelling in open-loop and closed-loop Tools become mature, but Paradigm dominantly linear! Have we solved the problem? Picture of high-level automated plants controlled by `autonomous controllers

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Delft Center for Systems and Control 98 Towards fully automated plant control Continuous performance monitoring Performance benchmarking for detecting changes Event-driven model update through dedicated experiments (economic criterion: will a new model pay…) Including model uncertainty bounding and disturbance analysis Controller update for improved performance Monitoring Control design Identification Experiment Continuously active loop

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Delft Center for Systems and Control 99 A challenging problem field Reservoir Engineering with Smart Wells / Fields

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Delft Center for Systems and Control 100 E&P activity domains production operations days years decades time space OU asset field well reservoir management field dev. planning portfolio management business planning historic data & forecasts objectives & constraints historic data & forecasts

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Delft Center for Systems and Control 103 Thanks to: Xavier Bombois Sippe Douma Roland Tóth Martijn LeskensGijs van Essen Robert Bos Peter Heuberger, Maarten Zandvliet,

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Delft Center for Systems and Control 104 Selection of References P. Albertos and A. Sala (Eds.). Iterative Identification and Control. Springer Verlag, London, UK, 2002. X. Bombois, M. Gevers, G. Scorletti and B.D.O. Anderson (2001). Robustness analysis tools for an uncertainty set obtained by prediction error identification. Automatica, 37, pp. 1629-1636. X. Bombois, G. Scorletti and P. Van den Hof (2005). Least disturbing closed-loop identification experiment for control. Proc. 16 th IFAC World Congress, Prague, 2005, paper Tu-E02-TO/6. R.A. de Callafon and P.M.J. Van den Hof (1997). Suboptimal feedback control by a scheme of iterative identification and control design. Mathem. Modelling of Systems, 3, pp. 77-101. S. Douma, X. Bombois and P.M.J. Van den Hof (2005). Validity of the standard cross-correlation test for model structure validation. Proc. 16 th IFAC World Congress, Prague, 2005, paper We-M13-TO/6. M. Gevers (1993). Towards a joint design of identification and control. In: Essays on Control: Perspectives in the Theory and its Applications, pp. 111--115. Birkhäuser, Boston, 1993. M. Gevers (2005). Identification for control: from the early achievements to the revival of experiment design. European J. Control, 11, 335-352. P.S.C. Heuberger, P.M.J. Van den Hof and B. Wahlberg (Eds.). Modelling and Identification with Rational Orthogonal Basis Functions. Springer Verlag, 2005. H. Hjalmarsson, M. Gevers and F. De Bruyne (1996). For model-based control design, closed-loop identification gives better performance. Automatica, 32, 1659-1673. H. Hjalmarsson (2005). From experiment design to closed-loop control. Automatica, 41, 393-438. M. Leskens, L.B.M. van Kessel and P.M.J. Van den Hof (2002). MIMO closed-loop identification of an MSW incinerator. Control Engineering Practice, 10, 315-326. P.M.J. Van den Hof and R.J.P. Schrama (1995). Identification and control -- closed-loop issues, Automatica, 31, 1751-1770.

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