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1© 2012 The MathWorks, Inc.
Introduction to System Identification
Lennart Ljung
April 4, 2012
43
22
ss
st
y(t
)
Data to Model
2
Modeling Dynamic Systems
Data-Driven ModelingFirst-Principles Modeling
SimscapeSimMechanicsSimHydraulics
SimPowerSystemsSimDriveline
SimElectronicsAerospace BlocksetSimulink
Tools for Modeling Dynamic Systems
Modeling Approaches
Neural NetworkToolbox
SimulinkDesign
Optimization
SystemIdentification
Toolbox
3
The System
ruddersaileronthrust
velocitypitch angle
Input Output
4
The Model
ruddersaileronthrust
velocitypitch angle
Input Outputu 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 dryerInput signal: heating powerOutput 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.
Err
or
Number of parameters
Estimation data
Validation dataDEMO: “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
Input data Output dataReal System
Gather Sets of Experimental Data
Input data
Outut data
Estimated SystemSystem Identification Toolbox
Estimate Mathematical Model of System
Input data
Validate Mathematical Model of System
Estimated System
Real System
Compare Real and Estimated
Outputs
13
The Identification Process
Collect the input-output data
Select a model structure
Find best model in a structure
Evaluate the resulting model
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
𝑥ሺ𝑡+ 1ሻ= 𝐴𝑥ሺ𝑡ሻ+ 𝐵𝑢(𝑡) 𝑦ሺ𝑡ሻ= 𝐶𝑥ሺ𝑡ሻ
Both are just ways of writing a linear differential equation for relationship between input (u) and output (y) (take n=1)𝑦ሺ𝑡+ 1ሻ= 𝐶𝑥ሺ𝑡+ 1ሻ= 𝐶𝐴𝑥ሺ𝑡ሻ+ 𝐶𝐵𝑢ሺ𝑡ሻ= 𝐴𝑦ሺ𝑡ሻ+ 𝐶𝐵 𝑢(𝑡)
𝑌ሺ𝑧ሻ= 𝐶𝐵ሺ𝑧− 1ሻ𝐴𝑈(𝑧)
Transfer function
B-order: nb (zeros)F-order: nf (poles)
𝑌ሺ𝑧ሻ= 𝐵(𝑧)𝐹(𝑧) 𝑈(𝑧)
17
Delays in TF and SS models
There could also be a delay (dead time) in the system: It takes nk samples before a change in u is visible in y.
y
u
nk
time
𝑦ሺ𝑡ሻ= 𝐶𝑥ሺ𝑡ሻ 𝑥ሺ𝑡+ 1ሻ= 𝐴𝑥ሺ𝑡ሻ+ 𝐵𝑢ሺ𝑡− 𝑛𝑘ሻ
𝑌ሺ𝑧ሻ= 𝐵(𝑧)𝐹(𝑧) 𝑧−𝑛𝑘𝑈(𝑧)
state space
transfer function
Linear parametric model structures are characterized by a few integers: n, or (np nz) and nk.
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
systemu y
input output
e: disturbance source
𝑌ሺ𝑧ሻ= 𝐵ሺ𝑧ሻ𝐹ሺ𝑧ሻ𝑈ሺ𝑧ሻ+ 𝐶ሺ𝑧ሻ𝐷ሺ𝑧ሻ𝐸(𝑧)
𝑥ሺ𝑡+ 1ሻ= 𝐴𝑥ሺ𝑡ሻ+ 𝐵𝑢ሺ𝑡ሻ+ 𝐾𝑒(𝑡)
𝑦ሺ𝑡ሻ= 𝐶𝑥ሺ𝑡ሻ+ 𝑒(𝑡)
state spacetransfer 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.
systemu y
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(jωTs) or s = jω.
Estimation: From experimental input-output data, it may be estimated directly by using various Fourier-transform inspired techniques. SPA, SPAFDR, ETFE
systemu y
𝐻ሺ𝜔ሻ= 𝐵(𝑒𝑖𝜔𝑇𝑠)𝐹(𝑒𝑖𝜔𝑇𝑠)
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.
𝑌ሺ𝑠ሻ= 𝐾𝑒−𝑠𝑇𝑑1+ 𝑠𝑇𝑈(𝑠)
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
+
-
errorinput e(t)
t
acceptable
Whiteness TestIndependence Test
Residuals = ”Leftovers” = 1-step-ahead prediction errors
Check correlation functions!
Should be uncorr-elated with known things
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
Data Model
uy
• 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 Model
NPosition Error Control
Estimate plant with parameter uncertainties
Estimate noise model
ControllerControl System ToolboxSimulink Control DesignRobust Control ToolboxModel Predictive Control Toolbox
System Identification Toolbox
29
More Information
Product page on mathworks.com:
http://www.mathworks.com/products/sysid/– Reach demos, webinars and documentation from here
Tech Support: http://www.mathworks.com/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(t1) and u(t2) and also between ε(t1) and ε(t2), t1≠ t2
𝜖ሺ𝑡ሻ= 𝑦ሺ𝑡ሻ− 𝑦ො��(𝑡|𝑡− 1)