Dynamic and Time Series Modeling for Process Control

Transcription

1 Dynamic and ime Series Modeling for Process Control Sachin C. Patwardhan Dept. of Chemical Engineering Why Mathematical Modeling? Key Component of All Advanced Monitoring, Control and Optimization Schemes Process Synthesis and Design offline Operation scheduling and planning Process Control - Soft sensing / Inferential measurement - Optimal control batch operation - On-line optimization continuous operation - On-line control Single loop / multivariable Online performance monitoring Fault diagnosis / fault prognosis /8/6 System Identification

2 Plant Wide Control Framewor Layer 4 Long erm Scheduling and Planning Layer 3 On-line Optimizing Control Setpoints PV Layer Model Predictive Control Setpoints PV Layer PID & Operating constraint Control MV Plant PV Maret Demands / Raw material availability Plant-Model Mismatch Load Disturbances /8/6 System Identification 3 Models for Plant-wide Control Layer 4 Layer 3 Aggregate Production Rate Models Steady State / Dynamic First Principles Models Layer Layer Dynamic Multivariable ime Series Models SISO ime Series Models, ANN/PLS/Kalman Filters Soft Sensing /8/6 System Identification 4

3 Mathematical Models Qualitative Qualitative Differential Equation Qualitative signed and directed graphs Expert Systems Quantitative Differential Algebraic systems Mixed Logical and Dynamical Systems Linear and Nonlinear time series models Statistical correlation based PCA/PLS Mixed Fuzzy Logic based models /8/6 System Identification 5 White Box Models First Principles / Phenomenological / Mechanistic Based on energy and material balances physical laws, constitutive relationships Kinetic and thermodynamic models heat and mass transfer models Valid over wide operating range Provide insight in the internal woring of systems Development and validation process: difficult and time consuming /8/6 System Identification 6 3

4 Example: Quadruple an System Pump V an3 an an 4 an Pump V dh a dt A dh a dt A dh3 a3 dt A 3 dh4 a4 dt A 4 a3 gh + A Manipulated Inputs : v Measured Outputs : h a4 gh + A γ gh3 + v A γ gh4 + v A γ gh3 + v A γ gh4 + v A 3 4 and v and h /8/6 System Identification 7 Example: Non-isothermal CSR dca V F C dt Reaction : A B Material Balance C V exp E / R C Enery Balance d Vρc p ρc pf Q dt + H V exp E / R C Heat ranfer to Cooling Jacet b + afc Q b F + af / ρ C c A A rxn c /8/6 System Identification 8 pc cin A A 4

5 Example: Fed-Batch Fermenter d XV µ S, S XV X : Biomass Conc. dt d SV FS F σ S, S XV S : Substrate - Conc. dt d PV π S, S XV PV P : Product Conc. dt dv F V : Reactor Volume dt.86s S µ S, S. + S +.33S σ S, S µ S, S /.5 ; π S, S 7.7e.3S µ S, S /8/6 System Identification 9 Fixed Bed Reactor Material Balances Distributed Parameter System C t C t B A C v e C z Energy Balances A E/R r l A C v + e C e C z H B E / Rr E / Rr l A B H r v + e C t z ρ C r r E/R r l A m pm U + e C + ρ r E /Rr w B j r mcpm ρmcpmvr..reactant A..Product B..Reactor emp. j j Uwj u + t z ρ C V mj pmj j r j..jacet emp. /8/6 System Identification 5

6 Grey Box Models Semi-Phenomenological Part of model developed from the first principles and part developed from data Example: dynamic model for reactor model using energy and material balance and Reaction inetics modeled using neural networ Better choice than complete blac box models /8/6 System Identification Example: Stirred an Heater-Mixer Cold Water Flow CV- Cold Water Flow 4- ma Input CV- - Heater Coil L 4- ma Input Signal hyrister Control Unit - -3 Experimental Setup: Schematic Diagram /8/6 System Identification 6

7 Example: Stirred an Heater-Mixer d F Q I i + dt V V ρc dh dt A U 39.5J / m [ F + F I F ] d F + F i dt ha Q I 7.979I +.989I F I I.7I I I Ks p ; UA ρc.73i +.93I F h :% current input to control valve /8/6 System Identification h h p atm :% current input to thyrister power controller Example: Stirred an Heater-Mixer 5 Heat Input to an 5 Charecteraisation of Control Valve CV- Heat supplied to water in tan one process predicted flow rate in ml/min data cubic % Current input to the power module % valve opening hyrister Power Controller Characterization Control Valve Characterization /8/6 System Identification 4 7

8 Model Validation: Input Excitations Heater inputma Valve two inputma Input Perturbations Sampling instant Sampling instant /8/6 System Identification 5 Model Validation: Level Variations Heightcm Model Simulation Predictions Measured Validation Output data Sampling instant /8/6 System Identification 6 8

9 Model Validation: emperature Profiles 345 an empk S im M odel u la tiopredictions n Validation M easured data O utput Sam pling Instant M odel Predictions M easured O utput an tempk Sam pling instat /8/6 System Identification 7 Dynamic Models for Control Linear perturbation models: Regulatory operation around fixed operating point of mildly nonlinear processes operated continuously. Developed using Local linearization of white/gray box models Identification from input output data Why use approximate Linear Models? Linear control theory for controller synthesis and closed loop analysis is very well Developed For small perturbations near operating point, processes exhibit linear dynamics Nonlinear dynamic models: strongly nonlinear systems, operation over wide operating range, batch / semi-batch processes /8/6 System Identification 8 9

10 Local Linearization Given a lumped parameter model dx/dt F X, U, D ; Y G X and steady state operating point X, U, D, we apply aylor series expansion around X, U, D to develop linear perturbation model dx/dt Ax + Bu + Hd ; y Cx Perturbation variables xt Xt - X ;yt Yt - Y ; ut Ut -U ; dt Dt -D ; /8/6 System Identification 9 Local Linearization A H where [ F / X ] ; B [ F / U ] ; [ F / D] ; C [ G / X ] computed at X, U, D ransfer Function Matrix: Can be obtained by taing Laplace transform together with assumption x i.e. initial state of the process corresponds to operating steady state y s Gp s u s + Gd s d s G s C [ si A] B ; G s C [ si A] H u d /8/6 System Identification

11 Perturbation Model for CSR Consider non-isothermal CSR dynamics States X dca f CA,, F, Fc, C dt d f CA,, F, Fc, C dt Manipulated Inputs U A A,, [ C ] MeasuredOutput Y [ ] A [ F UnmeasuredDisturbances D MeasuredDisturbances D m cin cin F ] c [ CA ] [ ] u cin feed flow rate coolant flow rate Feed conc. Cooling water emp. /8/6 System Identification CSR: Model Parameters and Steady state Operating Point 3 V Reactor volume m 3 CA Inlet concentration of A. mol/m ; 3 Inlet temperature 5 C ; F Coolant flow 5 m /min ; Cp Specific heat of reacting mixture cal/g K ; cin Coolent Inlet emperature 9 Cpc Inlet flow C ; spacific heat of coolent cal /g K ; ; F 3 m /min ; ρ Reacting liquid density g/m ; ρc Coolent density g/m ; 6 - Hrnx Heat of reaction 3 x cal/mol ; 6 a.678 x cal / min ; b. 5 ; E/ R 833. K 3 CAConcentration of A.65mol/m Reactor emperature C Operating Steady State /8/6 System Identification

12 Discrete Dynamic Models Computer control relevant discrete models x + Φx + Γu Φ exp A y Cx ; Γ Definition Φ exp A I + Φ + Φ! exp Aτ B dτ +... Note: Assumption of piece-wise constant inputs holds only for manipulated inputs and NO for the disturbances or any other input /8/6 System Identification 3 ransfer Function Matrix q-ransfer Function Matrix: Can be obtained by taing q-transform together with assumption x q{f} f + Alternatively, taing z - transform on bothe sides of difference equation G z p y G q u G q C p [ qi Φ] q : Shift Operator ; p q - {f} f - zx z x Φx z + Γu z When x xz [ zi - Φ] Γu z y z Cx z [ zi - Φ] Γu z C [ zi Φ] Γ : Pulse ransfer Function Γ /8/6 System Identification 4

13 Computation of System Matrices Compouation Method : - Let A ΨΛΨ where Λ is diagonal matrix with eigenvalues appearing on maion diagonal Ψ : matrix with eigenvectors of A as columns - Φ Ψexp Λ Ψ - Γ Ψ exp Λτ dτ Ψ B Compouation Method : Φ t exp At is solution of ODE - IVP dφ AΦ t ; Φ I dt aing Laplace ransform [ ] sφs - Φ AΦs Φs [ si Φ] Φ L si Φ Γ exp Aτ dτ B When Α is invertible matrix Γ [ exp A I ] A B [ Φ I ] A B /8/6 System Identification 5 t CSR: Continuous Perturbation Model Continuous linear state space model C A t CA F t F x t ; u t ; t F t F c c dx / dt x t + u t y t [ ] x t Laplace ransfer Function s s s s +.79 s s +.79 s G p /8/6 System Identification 6 3

14 Models for Computer Control Computer controlled system / Distributed Digital Control system Digital o Analog Converter Process Analog o Digital Converter Manipulated Inputs From Control Computer Control Computer /DCS Measured Outputs o Control Computer /8/6 System Identification 7 Digital Control: Measured Outputs Output measurements are available only at discrete sampling instant t :,,,... Where represents sampling interval { } Measured Output ADC Measured Output Sampling Instant Sampling Instant Continuous Measurement from process Sampled measurement sequence to computer /8/6 System Identification 8 4

15 Digital Control: Manipulated Inputs In computer controlled digital systems Manipulated inputs implemented through DAC are piecewise constant u t u t u for t t t Manipulated Input Sequence DAC Manipulated Input Sampling Instant Sampling Instant Input Sequence Generated by computer Continuous input profile generated by DAC /8/6 System Identification 9 CSR: Discrete Perturbation Model Discrete linear state space model Sampling ime. min.85 x x y [ ] x Discrete q-transfer function model q Gp q q +.79 q q Gp q +.79 q q q q q +.79 q q +.79 q u q q - - /8/6 System Identification 3 5

16 Blac Box Models Data Driven / Blac Box Models Static maps correlations/ dynamic models difference equations developed directly from historical input-output data Valid over limited operating range Provide no insight into internal woring of systems Development process: much less time consuming and comparatively easy /8/6 System Identification 3 Blac Box Models Dynamic Models: Given observed data Set of past Inputs : U Measured Outputs : Y [ u u... u ] [ y y... y ] we are looing for relationship y Ω U, Y, θ + e such that noise residuals e are as small as possible d θ R represents parameter vector /8/6 System Identification 3 6

17 ools for Blac Box Modeling Linear Difference equation time series models Principle component analysis PCA / Projection Of latent structures PLS / Statistical models based on linear correlation analysis of historical data Artificial Neural Networs/Wavelet Networs Excellent for capturing arbitrary nonlinear maps Fuzzy Rule Based Models Quantification of qualitative process nowledge /8/6 System Identification 33 Steps in Model Development Selection of model structure Planning of experiments for estimation of unnown model parameters Design of input perturbation sequences Open loop / closed loop experimentation Estimation of model parameters from experimental data using optimization techniques Model validation Prediction capabilities Steady state behavior /8/6 System Identification 34 7

18 Model Structure Selection Issues in Model Selection Process application batch / continuous ime scale of operation ype of application scheduling/optimization/mpc/fault Diagnosis Availability of physical nowledge / historical data Development time and efforts Model granularity decides how well we can mae control / planning moves or diagnose / analyze process behavior /8/6 System Identification 35 Data Driven Models Development of linear state space/transfer models starting from first principles/gray box models is impractical proposition. Practical Approach Conduct experiments by perturbing process around operating point Collect input-output data Fit a differential equation or difference equation model Difficulties Measurements are inaccurate Process is influenced by unnown disturbances Models are approximate /8/6 System Identification 36 8

19 Discrete Model Development Excite plant around the desired operating point by injecting input perturbations.9.8 Measurement Noise 3. 3 Manipulated Input Process Measured Output Sampling Instant Input excitation for model identification Unmeasured Disturbances Sampling Instant Measured output response /8/6 System Identification 37 CSR: Input Excitation PRBS: Pseudo Random Binary Signal 7 Manipulated Input Excitation Coolent Flow Sampling Instant Reactant Inflow Sampling Instant /8/6 System Identification 38 9

20 CSR: Identification Experiments Effect of Inlet Concentration Fluctuations On Reactor emperature Inlet Conc.mod/m3 Measured emperature K Sampling Instant Sampling Instant Dotted Line: Data without noise Continuous Line: Data with noise /8/6 System Identification 39 CSR: Noise Component 3 Unmeasured Disturbance Component in Measured Outputs v y - Gq u, Deg K Sampling Instant /8/6 System Identification 4

21 wo Non-Interacting ans Setup an CV an L CV SISO System Output: Level in tan Manipulated Input : Valve Position CV- Disturbance: Valve Position CV- Pump Sump Pump Non Interacting an Level Control setup /8/6 System Identification 4 Input Output Data 6.5 Raw Input and output signals Output ma Input ma ime /8/6 System Identification 4

22 Perturbation Data for Identification Input and output signals Output Input ime Mean values removed from Input and Output data /8/6 System Identification 43 Impulse Response Model y s Consider.F. g s with impulse input u s y impulse y g τ dτ u t gt L [ g s ] Convolution Integral : y t g τ u u τ dτ For piece - wise constant inputs [ ] + g τ dτ u [ ] y g τ dτ u j g j u j j j Impulse Response Coefficients : g j g τ dτ /8/6 System Identification 44

23 Impulse Response Model y Impulse Response Model j g u j j j Defining transfer operator Gq y G q u g q j u j g q j Current output y is viewed as weighted sum of all past inputs moves. Impulse response coefficients determine weighting of each past move Gq is open loop BIBO stable if g j < j j j /8/6 System Identification 45 Discrete Model Forms Finite Impulse Response FIR Model For open loop stable systems g as N y g u j j Discrete ransfer Function Model j - - y q b q + bq +...bn q - n u q + a q +..a q Example - - y q b q + bq - u q + a q + a q which is equivalent to y -a y - a y - + b u - + b u - /8/6 System Identification 46 n - 3

24 Output Error OE Model Data collected through experiments Set of N output Measurements Y U N N { y : y, y, y,..., y N } Set of Input Sequence { u : u, u, u,..., u N } Output / Measurement Error Model y G q u + v Measured Value of Output Deterministic component Residue: unmeasured disturbances + measurement noise /8/6 System Identification 47 Estimation of FIR Model Consider FIR model with n coeffieints yn g un g un -n + vn yn u n yn + u n..... yn u N y g u g u -n + v Using experimental data we can write yn g un g u + vn yn + g un g u + vn +... Arranging in matrix form u n u n n n n n u g vn u g + vn u N n gn vn /8/6 System Identification 48 4

25 Least square estimation Let the noise sequence Resulting model is linear in parameters min min θˆ V V θ θ θˆ E θˆ { v } Y Aθ + V Least square parameter estimation [ A A] Y Aθ [ Y Aθ] [ Y Aθ] AY have zero mean and let θ + V [ A A] AY [ A A] A [ Aθ + V] θ + [ A A] A V [ θˆ ] θ + [ A A] A E [ V] θ true value of the parameter vector, i.e. /8/6 System Identification 49 represent Estimated FIR Model Impulse Response Coefficients Comparison of Impulse Responses FIR 69 coeff ARMAX,,, ARX6,6, OE,, /8/6 System Identification 5 5

26 FIR Model Fit y Plant Model. 5 5 v Sampling ime /8/6 System Identification 5 Estimated step Response Step response can be estimated from impulse response coefficients Unit Step Response Coefficient : a i g j i j.6 Unit Step Response Unit Step Response y y ime ime Unit Delay /8/6 System Identification 5 6

27 Features of estimation unbiased estimate of model parameters when If hus, the least square estimation generates { v } Cov Cov [ V] E [ VV ] {} θˆ E θˆ θ θˆ θ [ A A] A E [ VV ] A[ A A] σ [ A A] [ V] is white noise sequence with variance σ, then Ĉov σ I σ can be estimated as ˆ ˆ ˆ σ V V Y Aθˆ Y Aθˆ N N hus, estimated parameter covariance matrix is /8/6 System Identification 53 { ˆ } ˆ ˆ θˆ, θ V V[ A A] N n E an Difficulties with FIR Model Advantages: Method can be easily extended to multiple input case Difficulty: Variance Errors in FIR Model Parameters var g i α [ /N -n] Variances of parameter estimates can be reduced by increasing data length N Disadvantages: Large number of parameters for MIMO case Large data set required to get good parameter estimates, which implies long time for experimentation. Alternate Model Form - - b q b q n d y q u + v - + a q +... a q + n Output Error /8/6 System Identification 54 7

28 Parameterized OE model wo tan system under consideration is expected to have second order dynamics p x s u s τ s + τ s + which is equivalent to 'nd order discrete time model Since time delay dead time was found to be d - - b q + bq x - + a q + a q which is equivalent to x -a x - a x - + b u - + b u - 3 bq + bq x + a q + a q y x + v u q u /8/6 System Identification 55 Parameter Estimation x : true value Y : measured value v : measurement noise / disturbance Difficulty: Only {y} sequence is nown. Sequence {x} is unnown Consequence: Linear least square method can t be used for parameter estimation Given a, a, b, b, x, x, x and d we can recursively estimate x as x 3 ax ax + bu + bu x 4 ax 3 ax + bu + bu... x N ax N ax N + bu N + bu N 3 v y x for 3,4,... N /8/6 System Identification 56 8

29 Parameter Estimation Nonlinear Optimization Problem Estimate a, a, b, b, x, x such that N [ e,... e ] [ v ] is minimized with respect to a, a, b, b, x, x v y x x a x a x + b u + b u 3 Ψ Simplification : Choose x x Identified Model Parameters y [Bq/Aq]u + v Bq 4.567e-6 q^ q^-3 Aq q^ q^- /8/6 System Identification 57 OE Model OE,,: Measured and Simulated Outputs.5 y v /8/6 System Identification 58 9

30 OE Model : Autocorrelation Auto Cotrelation for OE,,.8 Sample Autocorrelation Lag 95% Confidence Interval /8/6 System Identification 59 Unmeasured Disturbance Modeling he measured output y contains contributions due to Measurement errors noise Unmeasured disturbances In additions modeling equation errors arise while developing approximate linear perturbation models hus, in order to extract true model parameters from the data, we need to carry out modeling of unmeasured disturbances or noise Noise is modeled as a stochastic process sequence of random variables, which are correlated in time /8/6 System Identification 6 3

31 Noise Modeling y G q u + v Deterministic component Residue: unmeasured disturbances + measurement noise Note: Information about unmeasured disturbances in the past is contained in the output measurement record. hus, an obvious choice of model structure is [ u -,...,u - m, y -,..., y - p ] e y f + /8/6 System Identification 6 Equation Error Model A discrete linear model, which captures the effect of past unmeasured disturbances, can be proposed as y b u d b u d m a y... a y n + e d : ime delay / dead time How many past outputs do we include in the model? We can choose n such that error e becomes uncorrelated with y and contains no information about past disturbances Error e is lie a random variable uncorrelated with e-, e-, How do we mathematically state above requirement? n m /8/6 System Identification 6 3

32 White Noise Let us define auto-correlation in a random process {e:,, } as Rv τ cov[ e, e τ ] lim N e e τ N N τ τ Equation error sequence e in ARX model should be independent and equally distributed random variable sequence, i.e. σ for τ r ee τ for τ ±, ±,... Such sequence is called discrete time white noise /8/6 System Identification 63 Example: White Noise Mean Variance e Sampling Instant /8/6 System Identification 64 3

33 White Noise: Autocorrelation Sample Autocorrelation Function ACF of White Noise.8 Sample Autocorrelation Lag /8/6 System Identification 65 ARX Model Development Consider 'nd order ARX model with d y ay ay + bu + bu 3 + e Advantages: Sequences {y} and u} are nown Linear in parameter model optimum can be computed analytically We can recursively estimate ŷ as ˆ3 y ay ay + bu + bu ˆ4 y ay 3 ay + bu + bu... ˆ y N ay N ay N + bu N + bu N 3 e y ˆ y for 3,4,... N /8/6 System Identification 66 33

34 ARX : Parameter Identification Arranging in matrix form yn y y u u a e yn + y y u u a e b... yn b y N y N u N u N en Resulting model is linear in parameters Y Aθ + e Least square parameter estimation min min θˆ e e θ θ θˆ [ Y Aθ] [ Y Aθ] [ A A] AY Choose model order n such that sequence {e} becomes white noise /8/6 System Identification 67 ARX: Order Selection 5.5 x -4 ARX Order Selection Objective Function Value ime Delay d Model Order /8/6 System Identification 68 34

35 ARX: Order Selection Sample Autocorrelation Sample Autocorrelation.5 Auto Cotrelations for ARX,, Auto Cotrelations for ARX 4,4, Lag Model Residuals are not white /8/6 System Identification 69 ARX: Order Selection Auto Cotrelation for ARX6,6, Sample Autocorrelation Model Residuals white Lag /8/6 System Identification 7 35

36 ARX: Identification Results ARX6,6,: Measured and Simulated Output y e ime /8/6 System Identification 7 6 th Order ARX Model Identified ARX Model Parameters Aqy Bqu + et Aq q q q q q q -6 Bq.4 q q q q q q -7 Error statistics Estimated Mean : E{e} Estimated Variance : ˆ.5496 λ {e} is practically a zero mean white noise sequence -4-3 /8/6 System Identification 7 36

37 ARX: Estimated Parameter Variances a Value σˆ Value b σˆ a a 3 a 4 b b b a 5 a 6 b b 6 /8/6 System Identification 73 ARX Model Auto Regressive with Exogenous input ARX y b u b u m a y... a y n + e m n Using shift operator q, ARX model can be expressed as B q d y q u + e A q A q A q + a q B q b q a q b n n m mq where e is white noise sequence Disadvantage Large model order required to get white residuals Noise Model /8/6 System Identification 74 37

38 Noise Models e : Zero mean white noise process with variance v a v... a v n + e Alternatively, if poles of Aq are inside unit circle, A q v A q Auto Regressive AR Model then, by long division + hq v H q e e +... H q h e Moving Average MA Process v e + h e +... h e n + h q i i /8/6 System Identification 75 n n λ ARMA Model v a v... a v m + e + c e c e m e AR and MA models can be combined to formulate a more general ARMA model n or C q v A q e : Zero mean white noise process with variance If poles of Aq are inside unit circle, then, by long division C q + h q + hq A q +... H q Advantage: Parsimonious in parameters significantly less number of model parameters required than AR or MA models for capturing noise characteristics m λ /8/6 System Identification 76 38

39 Example: Colored Noise 4. 8q. 4q e. 5q +. 8q v White Noise Passing hrough Filter Filter Output 3 - Properties Of e Mean Variance Sampling Instant /8/6 System Identification 77 Colored Noise: Autocorrelation Sample Autocorrelation Function ACF.8 Sample Autocorrelation Lag /8/6 System Identification 78 39

40 Parameterized Models ARMAX: Auto Regressive Moving Average with exogenous input ARMAX y b u d b u d m Or a y... a y n + e + c e c e r B q y A q q d n m r C q u + A q e Box-Jenins BJ model: most general representation of time series models B q y A q q C q u + D q d e {e} is white noise sequence in both the cases /8/6 System Identification 79 Parameter Identification Problem Given input output data collected from plant Y U N N { y : y, y, y,..., y N } { u : u, u, u,..., u N } Choose a suitable model structure for the time series model and estimate the parameters of the model coefficients of Aq, Bq, Cq polynomials such that some objective function of the residual sequence e [, e,... e N ] Ψ e is minimized. he resulting residual sequence {e} should be a white noise sequence /8/6 System Identification 8 4

41 4 /8/6 System Identification 8 ARMAX: One Step Prediction 3 with d Consider 'nd order ARMAX model 3 e q a q a q c c q u q a q a b q b q y e c c e e b u b u y a y a y Difficulties: Sequences {y} and {u} are nown but {e} is unnown Non-Linear in parameter model optimum can t be computed analytically Solution Strategy Problem solved numerically using nonlinear optimization procedures /8/6 System Identification 8 Inevitability of Noise Model < < - ~ q is stable i.e. H ~ Hq is stable i.e. i i i i i i i i h i h e v q H e h i h e e q H v Crucial Property of Noise Model : Noise model and its inverse are stable and all its poles and zeros are inside unit circle Key problem in identification is to find such Hq and a white noise sequence {e} h i.e. always 'monic' polynomial :Hq is Note

42 Example: A Moving Average Process i.e. Hq Consider a first order MA process + cq - v e + ce - where {e} is a white noise sequence q + c has a pole at q and zero at q - c q - hen, H q + cq e i i i c q i c v i i if c < and e can be recovered from measurements of v Inversion of Noise Model plays a crucial role in the procedure for model identification /8/6 System Identification 83 One Step Prediction Suppose we have observed vt upto t v i i h e ˆ v ˆ v i e + h e i ˆ v v e h e [ H q ] [ H q ] v H q v H q i - and we want to predict v based on measurements upto time - i i i i i e as e has zero mean ˆ v : Conditional expectation of v based on information upto - e ~ hv i i /8/6 System Identification 84 4

43 One Step Output Prediction Suppose we have observed yt and ut upto t - and We have y Gqu + v and we want to predict y based on information upto time - ˆ y G q u + ˆ v G q u + [ H q ] v However, v y - Gqu ˆ y G q u + [ H q ][ y - Gqu ] Rearranging we have ˆ y H q G q u + [ H q ] y or H q ˆ y G q u + [ H q ]y /8/6 System Identification 85 ARX: One Step Predictor Consider 'nd order ARX model with d 3 bq + bq y u + e + a q + a q + aq + aq One step ahead predictor for this model is 3 bq + bq aq a q ˆ y + u y which is equivalent to difference equation ˆ y a y a y + b u + b u 3 Advantage : All terms in RHS are nown Residual at 'th instant can be estimated as e y ˆ y /8/6 System Identification 86 43

44 ARMAX: One Step Predictor Consider 'nd order ARMAX model with d 3 bq + bq + cq + cq y + u e + a q + a q + aq + aq One step ahead predictor for this model is 3 bq + bq c a q + c a q ˆ y + u + cq + cq + cq + cq which is equivalent to difference equation ˆ y c ˆ y c ˆ y 3 + b u + b u 3 + c a y + c a Residual at 'th instant can be estimated as ε y ˆ y y y /8/6 System Identification 87 ARMAX: One Step Predictor Alternatively, using residuals ε ε [ y ˆ y ] [ y ˆ y 3 ] we can rearrange one step predictor as ˆ y a y a y + b u + b u 3 + c ε + c ε ε y ˆ y We can start prediction with initial guesses ε ε at previous instances and, given model parameters a, a, b, b, c, c, we can generate sequence { ε} using sequences {y} and {u}. /8/6 System Identification 88 44

45 nd Order ARMAX Model Optimization formulation Estimate a, a, b, b,, c, c such that objective function N Ψ 3 Identified Model Parameters Aq -.65 q^ q^- Bq.748 q^ q^-3 Cq q^- +.5 q^- Residual {e} Statistics Estimated Mean : E{e} 4.36e - 3 Estimated Variance : ˆ λ N [ e ] [ y ˆ y ] 3 is minimized with respect to a, a, b, b, c, c.683e - 4 /8/6 System Identification 89 nd Order ARMAX Model ARMAX,,,: Measured and Simulated Ouputs y e ime /8/6 System Identification 9 45

46 ARMAX: Autocorrelation Auto Cotrelation for ARMAX,,,.8 Sample Autocorrelation Lag /8/6 System Identification 9 Comparison of Model Predictions.6 Measured and simulated model output y Best Fit % ARMAX : ARX : OE : ARMAX,,, ARX Plant OE,, ime /8/6 System Identification 9 46

47 Comparison of Models Step Response y.4. y 3 Sampling Instant 4 5 Impulse Response.3.. OE,, ARX6,6, ARMAX,, Sampling Instant /8/6 System Identification 93 Comparison of Models: Nyquist Plots OE,, ARX6,6, ARMAX,, /8/6 System Identification 94 47

48 Prediction Error Method Z N ˆ y H { yk,u :,... N }, Model y G q, θ u + H q, θ e Optimal - step predictor Given data set q, θ G q, θ u + V θ, Z N N ε N, θ [ H q, θ ] y One step prediction error is defined as ε, θ y ˆ y, θ Parameter Estmimation by Prediction Error Method Find θ that minimizes objective function /8/6 System Identification 95 PEM: Parameter Estimation Min ˆ θ N V θ, Z θ N ypically, the resulting parameter estimation problem is solved numerically using a Nonlinear optimization b Gauss Newton Method If it is desired to emphasize certain frequency of interest, then, we can minimize N V θ, Z N εf, θ N where ε F q F ε F q represents a filter Alternate Choice of Objective Function N V θ, Z N ε, θ N /8/6 System Identification 96 48

49 Model Order Selection Model order determined by minimizing Aaie Information Criterion AIC N ˆ ˆ AIC θn N ln, ε θn + n N n : Number of model parameters AIC {Prediction erm} + { Model Order term} Prediction erm: estimate of how ell the model fits data Model Order erm: measure of model complexity required to obtain the fit AIC stries a balance between low residual variance and excessive number of model parameters, with smaller values indicating more desirable models Basic Idea: Penalize model complexity measured by n and obtain a model, which is reasonable w.r.t. variance errors and model complexity /8/6 System Identification 97 ARMAX: State Realization a a Φ... a a n n B q G q C A q x + Φx + Γu + L e y Cx + e... b c a... b c a ; Γ... ; L bn cn an C [... ] [ qi Φ] Γ ; H q C [ qi Φ] Interpretation as a State Observer x + Φx + Γu + L C q A q L + I [ y Cx ] /8/6 System Identification 98 49

50 Connection with Steady State Kalman Estimator Steady state form of Kalman prediction estimator for large time is given as ˆ x Φ ˆ x + Γu + L e + L ΦP C R + CP C P ΦP Φ e y Cxˆ + R L CP Φ hus, development of time series model can be viewed as identification of steady state Kalman estimator without requiring explicit nowledge of noise covariance matrices R,R Steady state Kalman gain L is parameterized through Hq and estimated directly from data. /8/6 System Identification 99 MIMO System Identification ARX Model: Method for ARX parameter identification can be extended to deal directly with multivariate data OE / ARMAX / BJ Models: ypically, an n x m MIMO system is modeled as n MISO Multi Input Single Output systems yi Gi q u Gim q um + Hi q ei i,,... n MISO models are combined to form a one MIMO model Input excitation: Inputs can be perturbed sequentially or simultaneously /8/6 System Identification 5

51 Identification Experiments on 4 an Setup Input Input Output Output /8/6 System Identification 4 an Setup: Input Excitations Manipulated Input Sequence Input ma Input ma ime sec /8/6 System Identification 5

52 4 an Setup: Measured Output Level ma Level ma 5 Meaured Output ime sec /8/6 System Identification 3 Splitting Data for Identification and Validation 5 Input and output signals y -5 5 u Identification Data 5 Samples Validation data /8/6 System Identification 4 5

53 MISO OE Model MISO nd Order Model y t [B q/a q]u t + [B q/a q]u t + e t Bq.393 q q - Bq.375 q q - Aq q q - Aq q q - Estimated using Prediction Error Method Loss function.479 Sampling interval: 3 /8/6 System Identification 5 OE Model: Validation Measured and simulated model output 3 y oe Fits 87.7% Validation data ime /8/6 System Identification 6 53

54 Residual Autocorrelation: ARX Sample Autocorrelation Function ACF of arx for y Sample Autocorrelation Lag /8/6 System Identification 7 Residuals Autocorrelations ARMAX 4 th Order B-J 3 rd Order Sample Autocorrelation Function ACF of armx44 for y Sample Autocorrelation Function ACF of bj3333 for y Sample Autocorrelation Lag 5 Sample Autocorrelation Lag /8/6 System Identification 8 54

55 Model Validation 4 Measured and simulated model output y - bj % armax % -4 Validation data arx 86.83% Samples /8/6 System Identification 9 ARMAX Model ARMAX 4 th Order Aqy t B qu t + B qu t + Cqe t Aq q q q q -4 B q.834 q q q q -4 B q.445 q q q q -4 Cq q q q q -4 Loss function.4377 /8/6 System Identification 55

56 Box-Jenins Model yt [Bq/Fq]ut + [Cq/Dq]et Bq.896 q q q -3 Bq.97 q q q -3 Cq q q q -3 Dq -.96 q q q -3 Fq q q q -3 Fq q q q -3 Loss function.3939 /8/6 System Identification ARMAX:State Realization x+ x + x + L e Φ Γ Y C x + e Φ [ ] Γ [.83.4 L [ ] -.86 ] ; C [ ] /8/6 System Identification 56

57 Recursive Parameter Estimation Consider 'nd order ARX model with d y a y a y + b u + b u 3 + e ϕ [ y y u u 3 ] Arranging in matrix form y ϕ a e yn + 3 a e3 ϕ b... yn N b ϕ en Y N A N θ + e N Least square parameter estimation θˆ N y ϕ θ + e [ A N A N ] A N Y N introduced as formal parameter to indicate that data up to instant N has been used /8/6 System Identification 3 RLS: Problem Formulation Nhen an additional measurementis obtained on - line A N Y N A N +, + Y N ϕ N + y N + New estimate ˆ θ N + can be written as ˆ θ N +, matrix A becomes [ A N + A N + ] A N + Y N + hus, AN eeps growing in size as new data arrives. Instead of inverting [ A N + A N + ] at every instat, Can we rearrange calculations at N+ so that solution at Instant N can be used to compute solution at instant N+? ˆ θ N + [ A N A N + ϕ N + ϕ N + ] [ A N Y N + ϕ N + y N + ] /8/6 System Identification 4 57

58 RLS: Solution Using matrix inversion lemma [ A BCD] A A B[ C + DA B ] + DA and some rearrangement Solution to above problem is given by following recursive set of equatiosns ˆ θ N + ˆ θ N + L N L N P N ϕ N [ y N + ˆ y N + N ] where ˆ y N + N ϕ N + ˆ θ N Estimator gain matrix LN is computed by solving following Riccati equations P N + + [ + ϕ N + P N ϕ N + ] [ I L N ϕ N + ] P N /8/6 System Identification 5 RLS: Initialization In order to obtain an initial conditions it is necessary to choose N N such that A N A N is nonsingular. Alternatively, recursive equations P αi where α is chosen large we have not trust in the initial parameter estimate i.e. ˆ θ his choice ensures PN P N [ A N A N ] Recursive canculations can be used for N N the initial covariance matrix and ˆ θ are begun with indicating that [ A N A N ] as N increases /8/6 System Identification 6 4 ˆ θ N P N A N Y N to start RLS, 58

59 ime Varying Systems For time varying systems, it is necessary to eliminate the influence of old data. his can be achieved using an exponential weighting in the loss function. J θ ˆ θ + ˆ θ + K K P ϕ N λ N [ y ϕ θ ] λ is called as forgetting factor RLS for this system is given by, [ y + ϕ + ˆ θ ] + [ λ + ϕ + P ϕ + ] [ I K ϕ + ] P λ P + Asymptotic Data Length ASL / - λ λ.999 : ASL ; λ.95: ASL /8/6 System Identification 7 RLS Application to 4 an Data y a y a y + b u + b u + d u + d u + e /8/6 System Identification 8 59

60 Model Predictions /8/6 System Identification 9 Parameter Variations Model Parameters a a b b d d Sampling Instant /8/6 System Identification 6

61 Extended Recursive Formulations ELS ϕ x a x + a x + b u + b u 3 Recursive OE Formulation uses regressor vector ϕ OE Model y x + v [ ˆ x ˆ x u u ] x a x + a x + b u + b u 3 + e + c e + c e Extended Recursive Formulation uses regressor vector [ y y u u ε ε ] where ε - andε - represent model residuals ARMAX Model /8/6 System Identification at previous instants Frequency Domain Analysis ime domain formulations of parameter estimation problem Useful for carrying out parameter estimation Does not provide any insight into internal woring of optimization problem Frequency domain power spectrum analysis Based on Fourier transform of auto-correlation and cross correlation function of signals Powerful tool for analysis analogous to use of Laplace transforms in linear control theory Provides insight into various aspects of optimization formulation Can be used for perturbation signal design and estimation error analysis /8/6 System Identification 6

62 Stationary Process where { e } R τ E v Consider noise modelv is a zero meanwhite noise process with variance λ E { v } h E { e i } i i and auto - covariance is { v,v τ } h t h j E { e t,e j τ } t j t j t Note : h r if r <. Covariance R τ is independent of and is uniquely defined by { h } h e i and λ. Such stochastic process is called'stationary' since it has zero mean and auto - covariance is independent of time. /8/6 System Identification 3 v i i λ h t h j δ j τ λ h t h t τ Quasi-Stationary Process E y G q u + H q e deterministic + stochastic Since E { e }, we have { y } G q u and { y } is not a stationary process Quasi-stationary process A signal { s } i E { s } ms ii E { s s t } R τ s is said to be quasi stationary if it is subject to R, t s m C R, t C Lim N Rs, τ Rs τ N N : auto - correlation function of signal { s } s s /8/6 System Identification 4 6

63 Signal Spectrum and Cross Spectrum Φ Defining E We define power spectrum of signal ω Rs τ e τ and cross sprctrum between sw s Φ { f } R τ E s R τ E sw Φ ω sw s Lim N f we have N N { s s τ } { s w τ } Rsw τ { s } { s } and { w } τ e ω is, in general, complex valued function of ω /8/6 System Identification 5 jωτ jωτ provided the infinite sum exists. Φ ω is always a real function of ω as Power Spectrum Contd. Note: Spectrum of signal st represents Fourier transform of auto-covariance function Inverse ransform: By definition of inverse Fourier transform E [ s ] Parseval's heorem π Rs Φ π -π s ω dω Fundamental Modeling Problem Given a disturbance with spectrum Φ ω Can we find a transfer function Hq such that the random process v Hqe has same spectrum with {e} being a white noise? v /8/6 System Identification 6 63

64 Spectral Factorization heorem : Suppose that Φ or e iω.hen there exists z, Rz, with no poles Main Result v ω > is a rational a monic rational transfer function of and zeros outside unit circle such that v iω Φ ω λ R e function of cos ω If the residue signal v is wealy stationary, i.e. cov[v,vs] is function of only -s for any pair,s, then spectral factorization theorem states that such a random sequence can be thought of being generated by a stable linear transfer function driven by white noise /8/6 System Identification 7 Spectrum of White Noise r ee τ λ for τ for τ ±, ±,... Case : λ Pxx - X Power Spectral Density heoretical Estim ated White Noise: Uniform power spectrum density at all frequencies Frequency /8/6 System Identification 8 64

65 Example: Autocorrelation for AR Numerical Estimate N ˆ R v τ y t y t τ N τ t τ + heoretical with nowledge of λ E Example : y t.5y t + e t [ y t y t τ ].5E [ y t y t τ ] + E [ e t y t τ ] R τ.5r τ + R τ But, R τ E ye y [ e t y t τ ] For τ : R.5R + λ y For τ : R.5R y y 4 τ Ry τ.5 λ 3 ifτ > λ ifτ /8/6 System Identification 9 y ye y Example: Autocorrelation Function y t.5y t + e t Case : λ /8/6 System Identification 3 65

66 Spectrum of a Stochastic Process { e } Consider a stationary stochastic process v H q : zero mean white noise process with variance λ v Example : y.5q e iω Φ ω λ H e e Φ y ω λ.5 exp jω λ [ +.5 cos ω] +.5 sin ω ω [, π / ] ; π / :Nyquist Frequency /8/6 System Identification 3 Spectrum: Colored Noise 5 4 Pxx - X Power Spectral Density heoretical Estimated from data 3 y t.5y t + e t Case : λ Frequency /8/6 System Identification 3 66

67 67 /8/6 System Identification 33 Spectra for Linear Systems { } { } Zero mean white noise process with variance : and deterministic signal Quasi - stationary : ω ω λ ω ω λ ω ω ω u i yu i u i y e G e H e G e u e q H u q G y Φ Φ + Φ Φ + { } { } [ ] [ ] [ ] [ ] : Stationary stochastic process with spectrum with spectrum and deterministic signal Quasi - stationary : v u s Spectrum of mixed deterministic and stochastic signal v u + + Φ Φ + τ τ τ τ τ τ ω ω v u E as R R v v E u u E s s E v u v u /8/6 System Identification 34 Errors Analysis [ ] [ ] { } { } Variance Error Bias Error of Estimation Error otal,, Parameters estimated by minimizing ˆ ˆ ˆ ˆ Prediction error ˆ ˆ Proposed / Identified Model rue Behavior N N N Z V e q H u q G q G q H u q G y q H e q H u q G y e q H u q G y θ ε θ ε

68 Variance Errors Asymptotic variance of estimates using PEM are iω [ ] iω [ ] Var Ĝe Var Ĥe n : Model Order n Φ N Φ i e i e n iω He N N :Data Length Implications: Variance errors can be reduced by increasing the data length N choosing high signal to noise ratio Signal to Noise Ratio SNR Models with large number of parameters require relatively larger data set for better parameter estimation. /8/6 System Identification 35 v u ω ω v i Φu e i Φ e ω ω Noise to Signal Ratio Example: Input Selection Random Binary Signal in frequency range [,.] u Random Binary Signal in frequency range [.3,.5] Generated By MALAB Command idinput u Sampling Instant /8/6 System Identification 36 68

69 Input Spectrum Pxx - X Power Spectral Density RBS in [.] RBS in [.3.5] Frequency /8/6 System Identification 37 Bias Error: Concept Real systems are of very high order and model is always chosen of lower order hus, bias errors are always present in any identification exercise Classic Example in Process Control Process Dynamics : Identified FOPD model : G s 8 s + Ĝs e 5s + 36s /8/6 System Identification 38 69

70 Bias Errors: Concepts Step Response.8 Amplitude.6.4. FOPD Model Step Response PLant Step Response ime sec /8/6 System Identification 39 Bias Error: Concept Nyquist Diagram.5 Imaginary Axis Model Real Axis Plant /8/6 System Identification 4 7

71 Bias Errors R E ε ε Hˆ [ G q Gˆ q u + H q e ] Parameters estimated by minimizing N V θ, ZN ε, θ N When data length N is large, we can write lim N N N [ ε ] ε, θ [ V θ, Z ] lim q N Prediction error By Parseval's heorem [ V θ, Z ] iω Φ ε ω G e Gˆ e N iω u π π ω Hˆ e iω /8/6 System Identification 4 ε Φ ω + Φ v lim N Φ ω dω N Bias Error: Interpretations lim N π iω iω Φ + Φ N G e Gˆ e u ω v ω π iω [ θ ] iω iω Bias distribution of G e Gˆ e in frequency domain is weighted by Signal o Noise Ratio Input spectrum can be chosen intelligently to reduce variance errors in certain frequency regions of interest For Output Error model i.e. Hq, i hus, Gˆ e lim π iω iω [ θ ] ˆ N G e G e Φ N ω i G e ω π Hˆ e ω dω dω if model is not under - parameterized u /8/6 System Identification 4 7

72 Summary Grey box models Better choice for representing system dynamics. Provide insight into internal woring of the system Development process time consuming and difficult Blac Box Models Relatively easy to develop Provide no insight into internal woring of systems Limited extrapolation abilities. Blac Box Model Development Noise modeling is necessary to be able to extract the deterministic component of the model properly Prediction error method used for parameter estimation /8/6 System Identification 43 Summary Blac Box Model Development FIR and ARX models are relatively easy to develop but require large data set for reducing variance errors OE, ARMAX or BJ models provide parsimonious description of model dynamics but require application nonlinear optimization for parameter estimation Variance errors are directly proportional to number of model parameters and inversely proportional to data length Frequency domain analysis provides insight into woring of PEM. Variance errors can be reduced by appropriately selecting Signal to Noise Ratio Bias errors in certain frequency region of interest can be reduced by appropriately choosing the spectrum of perturbation input sequences /8/6 System Identification 44 7

73 References Ljung, L., System Identification: heory for Users, Prentice Hall, 987. Sodderstorm and Stoica, System Identification, 989. Astrom, K. J., and B. Wittenmar, Computer Controlled Systems, Prentice Hall India 994. /8/6 System Identification 45 73