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 (APC 2006), Vancouver, Canada, May 8 -10 2006. 1 Delft Center for Systems and Control
Contents 1. Introduction 2. Basic facts on system identification 3. Example from a MSW incineration plant 4. Models for control 5. Model uncertainty and model validation 6. Basis functions model structures 7. Cheapest experiments 8. Discussion and prospects 2 Delft Center for Systems and Control
Introduction Costs distribution in an advanced process control project: • • • Feasibility study Pre-tests Model identification Controller tuning Commissioning and training 10% 40% 15% 25% (Zhu, IFAC SYSID, 2006) “obtaining process models is the single most timeconsuming task in the application of model-based controllers” (Ogunnaike, An Rev Control, 1996; Hjalmarsson, Automatica, 2005) Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 3
Which kind of models to consider? First principles / rigorous models • • large number of equations (PDE, ODE, DAE) high computational complexity question of validation nonlinear Process design; planning and scheduling; off-line 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 Delft Center for Systems and Control 4
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 linear f 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 Delft Center for Systems and Control 5
Data-based models (identification) • Relatively easily obtained • Model costs are related to experiments on the plant Model structures Black box Well sorted out in linear case Not mature in nonlinear case Emphasis, for the moment Physics-based Problem of accurate parametrization (where to put the unknowns? ) Identifiability Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 6
“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 Delft Center for Systems and Control 7
Classical experiments for finding control-relevant dynamics tp 1. 4 • Ziegler/Nichols tuning rules for PID-controllers 1. 2 Mp ± 1% 1 90% 0. 8 y(t) 0. 6 0. 4 0. 2 10% 0 tr ts time • Relay feedback: amplitude and frequency at -180° phase Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 8
• 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 Delft Center for Systems and Control 9
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 Delft Center for Systems and Control 10
Experiments: Data Model Feedback control system disturbance Model Controller reference input + controller process output - Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 11
When is a model suitable for control? For a given controller C: r+ - C u Ĝ y r+ C - Designed loop u G 0 y 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 Delft Center for Systems and Control 12
When is a model suitable for control? wb plant model 1: accurate for w<wb model 2: 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 Delft Center for Systems and Control 13
Control bandwidth is based on model +. . If models are uncertain/approximate due to limited experiment, achievable performance needs to be discovered exp. design Experiment data Identification Identificatie model Control design Regelaarontwerp controller ! modelling for control is learning (Schrama, 1992; Gevers, 1993) Implementation Implementatie evaluation Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 14
From experiment to control: validation and uncertainty high-order model low-order model experiment high-order controller low-order controller data 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 Delft Center for Systems and Control 15
Development trend: Identification Control • control-relevant nominal model • nominal control 1990 - : Schrama, Gevers, Bitmead, Anderson, Åström, Rivera, …. • nominal model + uncertainty bound • nominal control + stab/perf robustness 1994 - : Hakvoort, de Vries, Ninness, Bitmead, Gevers, Bombois, … • control-relevant • robust control; worst-case model uncertainty set performance optimiz. 1997 - : de Callafon & vd. Hof, Douma • design of “cheap” experiments for id of uncertainty sets 2002 - : Bombois, Gevers, Hjalmarsson, vd. Hof, • control under performance guarantees Intro Sysid ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 16
Contents 1. Introduction 2. Basic facts on system identification 3. Example from a MSW incineration plant 4. Models for control 5. Model uncertainty and model validation 6. Basis functions model structures 7. Cheapest experiments 8. Discussion and prospects 17 Delft Center for Systems and Control
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 Delft Center for Systems and Control 18
Prediction error framework: (Ljung, 1987) {u(t), y(t) }, t=1, . . , N fractions of polynomials Convex or non-convex optimization Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 19
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. Asymptotic variance typically dependent on (frequency-dependent noise to signal ratio) Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 20
Since parameter estimates are asymptotically normally distributed (cental limit theorem), the variance expression can be converted to parameter confidence regions, e. g. 3 s-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 Delft Center for Systems and Control 21
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 Delft Center for Systems and Control 22
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 Delft Center for Systems and Control 23
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 Delft Center for Systems and Control 24
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: Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 25
Asymptotic variance in closed-loop identification where now Writing because of the closed-loop. 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 Delft Center for Systems and Control 26
Alternative indirect methods When focussing on plant model only Several options, among which: 1. Indirect Method • Identify transfer r y 2. Two-stage method • Identify transfer r u • Simulate • Retrieve plant model, with knowledge of C • 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 Delft Center for Systems and Control 27
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 G 0 and H 0 is possible. Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 28
red blue |. | 10 T 0 S 10 -1 10 -2 10 -1 10 0 ω Low frequencies are hidden; frequencies around bandwidth are emphasized Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 29
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 Delft Center for Systems and Control 30
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 Delft Center for Systems and Control 31
Contents 1. Introduction 2. Basic facts on system identification 3. Example from a MSW incineration plant 4. Models for control 5. Model uncertainty and model validation 6. Basis functions model structures 7. Cheapest experiments 8. Discussion and prospects 32 Delft Center for Systems and Control
Municipal Solid Waste Combustion (Martijn Leskens) Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 33
(Nolinear) Model Predictive Control of MSWC Plants • Aim: NMPC of furnace and boiler part of MSWC plant: Simulation results NMPC requires good dynamic model of MSWC plant MODEL VALIDATION Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 34
Closed-loop identification of MSWC plants • Closed-loop experimental configuration typically encountered in MSWC plants: Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 35
• Experimental model is fit in the same i/o structure as the first principles model “PARTIAL” closed-loop identification: u 1 = “open-loop” inputs y 1 = “open-loop” outputs Etc. Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 36
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 (5 th order NL) tuned to identified models. Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 37
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 Delft Center for Systems and Control 38
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 Delft Center for Systems and Control 39
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 Delft Center for Systems and Control 40
Results • Identification and validation results for Tprim = 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 Delft Center for Systems and Control 41
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 Delft Center for Systems and Control 42
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 Delft Center for Systems and Control 43
Contents 1. Introduction 2. Basic facts on system identification 3. Example from a MSW incineration plant 4. Models for control 5. Model uncertainty and model validation 6. Basis functions model structures 7. Cheapest experiments 8. Discussion and prospects 44 Delft Center for Systems and Control
How poor can models be? wb plant model 1: accurate for w<wb model 2: accurate for w wb Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 45
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 Delft Center for Systems and Control 46
What looks like a good model in open-loop may be poor in closedloop 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 Delft Center for Systems and Control 47
When is a model suitable for control? For a given controller C: r+ - C u Ĝ y Designed loop r+ C - u G 0 y Achieved loop 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 Delft Center for Systems and Control 48
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 Delft Center for Systems and Control 49
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 Delft Center for Systems and Control 50
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 Delft Center for Systems and Control 51
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 Delft Center for Systems and Control 52
Optimal models for control design Bounding the achieved control performance cost Triangle inequality: achieved performance designed performance degradation nominal control design Identification of nominal model (when fixed) Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 53
Successive iteration of • nominal (closed-loop) identification • nominal control design Appropriate convergence requires cautious/robust tuning, exp. design Experiment data Identification Identificatie (taking account of model uncertainty) Experiment design is not critically incorporated yet (later) Problem evolves to data/modelbased controller tuning model Regelaarontwerp Control design controller Implementation Implementatie evaluation Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 54
Including model uncertainty for controller robustness 1) In situation convert probabilistic parameter uncertainty bound to the frequency domain (Bode, Nyquist) 2 10 amplitude 0 10 -2 10 -4 10 -2 10 phase -1 10 0 -200 Determine -400 -600 -2 10 -1 10 0 10 frequency 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 Delft Center for Systems and Control 55
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 relying on FIR/ORTFIR model structures (Goodwin, Gevers & Ninness, 1992; de Vries & Van den Hof, 1995; Hakvoort & Van den Hof, 1997) Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 56
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 Delft Center for Systems and Control 57
Including model uncertainty for controller robustness 3) In stead of looking at explicit bounds for the model error there are options 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. For virtually all closed-loop measures J, performance degradation is affine in dual-Youla parameter R Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 58
Controller tuning (Double Youla parametrization) Stability guaranteed if Control-dependent plant uncertainty and plant-dependent controller deviation 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. Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 59
Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control
Contents 1. Introduction 2. Basic facts on system identification 3. Example from a MSW incineration plant 4. Models for control 5. Model uncertainty and model validation 6. Basis functions model structures 7. Cheapest experiments 8. Discussion and prospects 61 Delft Center for Systems and Control
Uncertainty regions with probability Frequeny response 2 10 parameter region amplitude 0 10 -2 10 -4 10 -2 10 phase -1 0 10 10 0 -200 -400 -600 -2 10 -1 10 0 10 frequency Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 62
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 Delft Center for Systems and Control 63
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 Delft Center for Systems and Control 64
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 Question: 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 Delft Center for Systems and Control 65
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 Delft Center for Systems and Control 66
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 Delft Center for Systems and Control 67
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 Delft Center for Systems and Control 68
Problems: • Bounds for testing • are dependent on is estimated by In Output Error case: • Due to unmodelled dynamics the noise variance is over-estimated • Bounds in the correlation test are too large 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 Delft Center for Systems and Control 69
Remedies 1. Improve noise model for validating 2. Replace pointwise test on by vector valued test on (suggested before by Söderström, 1989; Hjalmarsson, 1993) Accurate noise model, e. g. through high-order auxiliary plant model and time-series modeling on the basis of the output residual Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 70
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 Delft Center for Systems and Control 71
Example (continued) classical bound improved bound effect of noise effect of model error 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 Delft Center for Systems and Control 72
Example (continued) Vector-valued test over 128 lags Noise cov estim from: Exact noise model Improved estimate From residual All models are validated Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 73
There are limitations in validating plant models without having accurate noise models 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 Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 74
Contents 1. Introduction 2. Basic facts on system identification 3. Example from a MSW incineration plant 4. Models for control 5. Model uncertainty and model validation 6. Basis functions model structures 7. Cheapest experiments 8. Discussion and prospects 75 Delft Center for Systems and Control
Basis functions model structures In identification the use of FIR models is very attractive: FIR: LS criterion is convex 1. Linear regression 2. Output Error structure: plant/noise model parametrized independently 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 Simple computations and analysis of model uncertainty Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 76
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 Delft Center for Systems and Control 77
Laguerre-case: Expansion becomes 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 Delft Center for Systems and Control 78
(Takenaka-Malmquist) Practical use: • choose any finite sequence of stable pole locations i for i=1, . . , nb (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 Delft Center for Systems and Control 79
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 Delft Center for Systems and Control 80
For a given system G 0 with poles in p 1, L pn 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 Delft Center for Systems and Control 81
Example basis pole selection Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 82
ORTFIR Model Structure ORTFIR: Least squares solution: Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 83
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 Delft Center for Systems and Control 84
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 Looking for solutions in the neighborhood where you expect them! Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 85
Contents 1. Introduction 2. Basic facts on system identification 3. Example from a MSW incineration plant 4. Models for control 5. Model uncertainty and model validation 6. Basis functions model structures 7. Cheapest experiments 8. Discussion and prospects 86 Delft Center for Systems and Control
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 Delft Center for Systems and Control 87
Problem Set-up • Excitation experiment is determined by N and • For fixed N, the costs of identification are determined by with 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 Delft Center for Systems and Control 88
Accuracy of identified model is characterized by its covariance matrix It appears that Note: is affine in N and in : is dependent on plant, and on asymptotic theory Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 89
Experiment design problem: For fixed N , solve Under the constraint 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 Delft Center for Systems and Control 90
Minimum identification cost as function of N (Bombois et al. Reduction of required dataalength in process application 2004) rbs FIR filtered rbs For N = 4901, minimum cost is 0: no external excitation is necessary! Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 91
Gain. Reduction oftime reduction from PRBS in process application in experiment required data length to dedicated periodic signal (Jansson, 2005) Compared with white noise PRBS signal of same power designed bandwidth Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 92
Example results Inclusing constraints Plant Designed closed-loop Optimal input 93 Delft Center for Systems and Control
Input constraint 94 Delft Center for Systems and Control
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: Generalization of the idea: ‘’If you need the static gain, apply a step signal’’ Intro System ID MSW-example Models for control Uncertainty&validation Basis functions Experiment design Discussion Delft Center for Systems and Control 95
Contents 1. Introduction 2. Basic facts on system identification 3. Example from a MSW incineration plant 4. Models for control 5. Model uncertainty and model validation 6. Basis functions model structures 7. Cheapest experiments 8. Discussion and prospects 96 Delft Center for Systems and Control
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 97 Delft Center for Systems and Control
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 Experiment • Controller update for improved performance Identification Continuously active loop Control design Monitoring 98 Delft Center for Systems and Control
A challenging problem field Reservoir Engineering with Smart Wells / Fields 99 Delft Center for Systems and Control
E&P activity domains space OU historic data & forecasts asset historic data & forecasts field well portfolio management business planning reservoir management objectives & constraints field dev. planning production operations days objectives & constraints years decades time 100 Delft Center for Systems and Control
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102 Delft Center for Systems and Control
Thanks to: Xavier Bombois Martijn Leskens Sippe Douma Roland Tóth Robert Bos Gijs van Essen Peter Heuberger, Maarten Zandvliet, 103 Delft Center for Systems and Control
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-E 02 -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-M 13 -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. 104 Delft Center for Systems and Control
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