1 © 2012 The MathWorks, Inc. Introduction to System Identification Lennart Ljung April 4, 2012 t y(t) Data to Model

2 Modeling Dynamic Systems Data-Driven ModelingFirst-Principles Modeling Simscape SimMechanics SimHydraulics SimPowerSystems SimDriveline SimElectronics Aerospace Blockset Simulink Tools for Modeling Dynamic Systems Modeling Approaches Neural Network Toolbox Simulink Design Optimization System Identification Toolbox

3 The System rudders aileron thrust velocity pitch angle Input Output

4 The Model rudders aileron thrust velocity pitch angle Input Output u y u, y: measured time or frequency domain signals

5 The System and the Model System Model + - Minimize errorMeasured input

6 Fitting and Comparing Models  Core feature: Estimating models by tuning its parameters such that model outputs is close to measured output COMPARE function: Plots measured and model output curves together Shows a numerical measure of fit: the percent of the output variation reproduced by the model DEMO: Model for data collected from a hair dryer Input signal: heating power Output signal: air temperature

7 Estimation and Validation Go Together  A large enough model can reproduce a measured output arbitrarily well. We must verify that model is relevant for other data – data that was not used for estimation, but was collected for the same system. Error Number of parameters Estimation data Validation data DEMO: “Validate” the hair-dryer models on new data set

8 Process of Building Models from Data  Gather experimental data  Estimate model from data –Select a structure –Find a model in it  Validate model with independent data

9  Various forms of data –Sampled inputs and outputs in discrete time IDDATA –Frequency Domain data –Frequency function measurements IDFRD  Requires thought about how to “excite” the process –Sampling interval should fit the process dynamics –The input should have a suitable spectrum –The toolbox may generate suitable inputs IDINPUT  Necessary to look at and “polish” data –Handle trends and offsets DETREND –Look for bad measurements (“outliers”) MISDATA –Is filtering required to reduce disturbances? IDFILT Collect the input-output data Select a model structure Find best model in a structure Evaluate the resulting model

10  The most difficult problem –There are a confusing multitude of possibilities. –The quality of the resulting model depends crucially on the chosen model structure  Many structures to choose from –Transfer functions, state-space, ARX, nonlinear networks,… IDTF, IDSS, IDROC, IDNLARX, IDNLHW, IDPOLY, IDGREY,…  How to know what to test? –May have prior insights. –May have preferences for the intended application. –Test many! Select a model structure Collect the input- output data Select a model structure Find best model in a structure Evaluate the resulting model

11 Collect the input- output data Select a model structure Find best model in a structure Evaluate the resulting model  Find the model parameter values in the chosen model structure –Main feature of the toolbox! TFEST, SSEST, ARX, ARMAX, OE, PEM, NLARX, … –Conceptually easy but can be computationally intensive Find best model in a structure

12 Collect the input- output data Select a model structure Find best model in a structure Evaluate the resulting model  Make a final choice of model structure –Includes choice model size (model orders and delays)  Twist and turn it to see if it is good enough for the intended application –Simulate and look at model output for validation data SIM, COMPARE –Residual analysis, Uncertainty evaluation, … RESID, SIMSD, BODE, STEP, PZMAP, … Evaluate the resulting model

13 The Identification Process Collect the input- output data Select a model structure Find best model in a structure Evaluate the resulting model

14 Model Structures  Do we know anything about the system apriori? –Black-Box models: Flexible structures with considerable approximation power –Grey-Box models: Taylor-made structures, made to incorporate prior knowledge: Structured differential equations with (some) parameters unknown.  Is the output a linear or nonlinear function of the input?  Do we want to describe also how disturbances affect the output? system uy inputoutput e: disturbance source

15 Model Structures in System Identification Tool (GUI) IDENT Linear Parametric Models  Input-Output models (transfer functions)  State-space models Linear Nonparametric models Impulse response model IMPULSE, STEP Frequency Response SPA, SPAFDR, ETFE Process Models PEM Nonlinear Models NLARX, NLHW

16 Transfer Functions and State Space Models  Linear models are typically described by state-space models IDSS (SSST) or transfer functions IDTF (TFEST) State-space Number of states: n  Both are just ways of writing a linear differential equation for relationship between input (u) and output (y) (take n=1) Transfer function B-order: n b (zeros) F-order: n f (poles)

17 Delays in TF and SS models  There could also be a delay (dead time) in the system: It takes n k samples before a change in u is visible in y. y u nknk time state space transfer function Linear parametric model structures are characterized by a few integers: n, or (n p n z ) and n k. Commands such as n4sid, ssest, tfest use these integers to “create” models from data.

18 Handling Disturbance  Knowledge of nature of disturbance is useful: –Essential for predicting future system outputs, by understanding correlations between disturbances  Handling technique for identification: treat a disturbance source e as an extra unmeasured input.  e is not measured; its properties (white noise) are characterized statistically – mean, covariance system uy inputoutput e: disturbance source state space transfer function DEMO: Models with disturbance component for hair-dryer data.

19 Non-Parametric Methods  Linear systems can also be described by non- parametric models that is curves that capture the system properties: –Transient responses –Frequency response  They can be estimated directly from data  Often useful to take a first look at these before parametric model estimation to get a feel for system’s basic properties.

20 Transient Response  A system’s transient response is its output to a transient like an impulse or step in its input.  Can be found by special experimentation with such inputs  For an existing model, its transient response is obtained by simulation with such inputs. IMPULSE, SIM, STEP  Estimation: From experimental input/output data, transient response is typically estimated via a flexible (high order) linear model. system uy

21 Frequency Response  A system’s frequency response H(ω) is its response to a sinusoidal input sin(ωt). The output has same frequency ω but a different amplitude and a phase shift y(t) = A(ω) sin(ωt+ ϕ (ω))  Plotting A(ω) and ϕ (ω) vs. ω gives Bode plot.  Can be found experimentally by subjecting the system to sinusoidal inputs of various frequencies.  For an existing model, it is obtained from the z- or Laplace transform of the transfer function using z = exp(jωT s ) or s = jω.  Estimation: From experimental input-output data, it may be estimated directly by using various Fourier-transform inspired techniques. SPA, SPAFDR, ETFE system uy DEMO: Direct frequency and transient response estimation for dryer data

22 Process Models  Use a combination of gain (K), delay (Td) and one or more time constants (T) to describe the model.  Such forms are popular in process industry, hence called “process models”. IDPROC  Can be expanded to contain more poles, zeros and integrators.  Structure choices: –number/nature of poles –whether a delay element, a zero and/or an integrator should be included.

23 Use of Disturbance Model for Simulation and Prediction  Measured outputs contain disturbances. If there is a correlation between disturbances it is possible to better predict the future outputs from observing the past ones. DEMO: Models with disturbance component for hair-dryer data.

24 Residual Analysis DEMO: Residual analysis on hair-dryer models System Model + - error input e(t) t acceptable Whiteness Test Independence Test Residuals = ”Leftovers” = 1-step-ahead prediction errors Check correlation functions! Should be uncorr- elated with known things

25 Model Uncertainty  Estimated model is influenced by the disturbances –If we repeat identification on another data set with the same input u, we will get a different model since e is not the same. This is true even if no disturbance model is built.  The model is an uncertain quantity. The amount of disturbance induced variations in model can be estimated. –We can create models with its uncertainty region; we can determine the uncertainty information on a model’s parameters. –We can visualize the uncertainty on model’s response in time and frequency domains (“confidence bounds”) DEMO: Studying model uncertainty

26 Putting the Model to Work  Use them for understanding a system’s behavior, and predicting future response of a system  Import estimated models into Simulink using dedicated blocks for simulation and code generation  Controller design: Import into SISOTOOL and MPC design task

27 Simplifying Complex Systems DataModel u y Simplify complex Simulink model using simulation data. Use Identified model to describe a component of a larger system

28 Using Models for Control System Design Dynamic Model P + ∆P Current Position Noise ModelN Position Error Control  Estimate plant with parameter uncertainties  Estimate noise model Controller  Control System Toolbox  Simulink Control Design  Robust Control Toolbox  Model Predictive Control Toolbox System Identification Toolbox

29 More Information  Product page on mathworks.com: –Reach demos, webinars and documentation from here  Tech Support:  Textbooks –System Identification – Theory for the User, Lennart Ljung –System Identification – A frequency domain approach, Rik Pintelon, Johan Schoukens –Others…

30 Residual Analysis  Technique for validating a model’s quality.  Works by analyzing the residues which are differences between model’s response and measured values: where is the model’s best prediction of y(t) made at time t-1.  ε(t) should ideally be uncorrelated with information known at time t-1. Hence we test correlation between ε(t 1 ) and u(t 2 ) and also between ε(t 1 ) and ε(t 2 ), t 1 ≠ t 2