1. Lectures 3&4: Non-Parametric System Identification: Impulse & Frequency Response^
y(t)
Model
u(t)
u(t)
y(t)
Controller
Plant
h(t), H(jw)
Dr Martin Brown
Room: E1k, Control Systems Centre
Telephone:
EE-M /7, EF L3&4

2. L3&4: Resources Main texts Chapter 2, Ljung Chapter 2, Norton
Chapters 1 & 2 in on-line notes.
Overview
In these two lectures, we’re looking at how LTI systems can be described in a non-parametric fashion
Look at system impulse response h(t)
Look at frequency response H(jw)
In both cases, we’ll investigate the representation, look at how you calculate the system response and consider how the system can be identified from data
This is sometimes known as classical system identification
We’re not doing an in-depth analysis for either case.
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3. Lecture 3: System Impulse Response
Non-parametric system impulse response
Impulse response d(t)→h(t)
Properties of impulse response (causal, stable)
Sifting property for signals
System linearity & superposition property
System response using convolution
Examples of calculating the output using convolution
Identifying the impulse response from data
NB, we’ll be doing the main part of the analysis for discrete time signals and LTI systems, but the work carries over to continuous time signals and systems as well.
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4. System Impulse Response
A very important way to analyse a system is to study the output signal when a unit impulse signal is used as an input
Loosely speaking, this corresponds to giving the system a kick at t=0, and then seeing what happens
This is so common, a specific notation, h(t), is used to denote the output signal, rather than the more general y(t)
This impulse response signal can be used to infer properties about the system’s structure (LHS of difference equation or unforced solution)
The system impulse response, h(t) completely characterises a linear, time invariant system
x(t)
d(t)
System
h(t)
y(t)
h(t)
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5. Properties of System Impulse Response
Stable
A system is stable if the impulse response is absolutely summable
Causal
A system is causal if
h(t)=0 when t<0
Finite/infinite impulse response
The system has a finite impulse response and hence no dynamics in y(t) if there exists T>0, such that:
h(t)=0 when t>T
Linear
ad(t)  ah(t)
Time invariant
d(t-T)  h(t-T)
h(t) t
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6. System Impulse Response Examples
Causal, stable, finite impulse response
y(t) = x(t) + 0.5x(t-1) x(t-2)
Looking at system impulse responses, allows you to determine certain system properties
h(t)
Causal, stable, infinite impulse response
y(t) = x(t) + 0.7y(t-1) t
h(t)
h(t) t t
Causal, unstable, infinite impulse response
y(t) = x(t) + 1.3y(t-1)
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7. Introduction to Convolution
Definition Convolution is an operator that takes an input signal and returns an output signal, based on knowledge about the system’s impulse response h(t).
The basic idea behind convolution is to use the system’s response to simple input signals to calculate the response to more complex input signals
This is possible for LTI systems because they possess the superposition property
x(t) = d(t)
LTI system
y(t) = h(t)
LTI system
h(t)
x(t)
y(t)
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8. Sifting Property
Basic idea: use a (infinite) set of of DT impulses to represent any DT signal.
Consider any discrete input signal x(t). This can be written as the linear sum of a set of unit impulse signals:
Therefore, the signal can be expressed as:
In general, any discrete signal can be represented as:
actual value
Impulse, time shifted signal
The sifting property
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9. Example: DT Signal Sifting
The discrete signal x(t)
Is decomposed into the following additive components
… + x(-4)d(t+4) +
x(-3)d(t+3) +
x(-2)d(t+2) +
x(-1)d(t+1) + …
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10. System Linearity
Let y1(t) and y2(t) be the responses to x1(t) and x2(t), respectively
A linear system must satisfy the two properties:
1 Additive: the response to x1(t)+x2(t) is y1(t) + y2(t)
2 Scaling: the response to ax1(t) is ay1(t) where aC
Combined: ax1(t)+bx2(t)  ay1(t) + by2(t)
Examples
y(t) = 3x(t). Consider x(t) = ax1(t)+bx2(t),
y(t) = 3(ax1(t)+bx2(t)) = a3x1(t) + b3x2(t) = ay1(t) + by2(t). The system is linear
2) y(t) = 3x(t)+2. Consider x(t) = 2x1(t) and use the scaling property
y(t) = 3*2*x1(t)+2 = 6x1(t)+2  2y1(t). The system is not linear
3) y(t) = 3*x2(t). Consider x(t) = 2x1(t) and use the scaling property
y(t) = 3*(2x1(t))2 = 12x1(t)2  2y1(t). The system is not linear
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11. Linear Systems and Superposition
Generalizing the scaling/additive properties, suppose an input signal x(t) is made of a linear sum of other (basis) signals xk(t):
then the response of a linear system is (where xk(t)yk(t)):
The basic idea is that if we understand how simple signals get affected by the system, we can work out how complex signals are processed, by expanding them as a linear sum
This is known as the superposition property which is true for linear systems in both CT & DT
Important for understanding convolution
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12. Convolution Sum for LTI Systems
As the system is time invariant, the response to the impulse signal d(t-k) is h(t-k).
Then from the superposition property of linear systems, the system’s response to a more general input signal x(t) can be derived as follows.
Input signal
System output signal is given by the convolution sum
i.e. it is the scaled sum of time shifted impulse responses.
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13. Convolution in a Nut-Shell
For any LTI discrete time system, the response to an input signal x(t) is given as follows.
Using the sifting property:
h(t) is the system impulse response to d(t).
Because the system is time invariant,
Using the superposition property (linear):
The system response of any LTI system can be calculated as a linear combination of impulse responses & written as
x(t) t
h(t) t
y(t) t
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14. Interpreting the Convolution Sum
Note that convolution can be interpreted in two ways:
As a sum of scaled, shifted impulse response signals
As an algebraic formula which is a function of t
Method 1) works well when h(t) or x(t) has finite duration, but method 2) is more flexible and more widely used. See following examples
x(t)
h(t)
y(t) t t t
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15. Example 1: LTI Convolution
Calculate the DT, LTI system response when:
h(t) = [ ],
x(t) = [ ]
Then:
h(t) = u(t)u(2-t)
h(t-k) = u(t-k)u(2-t+k)
and the convolution sum:
u(t)
u(2-t) 2 t
Very common trick!
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16. Example 2: LTI Convolution
Consider a DT LTI system that has a step response h(t) = u(t) to the unit impulse input signal (integrator)
What is the response when an input signal of the form
x(t) = atu(t)
where 0<a<1, is applied?
h(t) t
x(t) t
y(t) t
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17. Example 3: LTI Convolution
Find the system response when:
x(t) = (1/2)tu(t+1)
h(t) = (1/3)tu(t-1)
Using DT, LTI convolution
h(t) t
x(t) t
y(t) t
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18. System Identification and Prediction
Note that the system’s response to an arbitrary input signal is completely determined by its response to the unit impulse.
Therefore, if we need to identify a particular LTI system, we can apply a unit impulse signal, d(t), and measure the system’s response, h(t).
That data can then be used to predict the system’s response to any input signal
How to:
Design impulse signal
Collect the data (number of experiments and length)
Make sure assumptions are satisfied (linearity)
System: h(t)
y(t)
x(t)
n(t)~N(0,s2)
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19. How to Identify the Impulse Response
Impulse signal. Signal should be of large magnitude and small duration (unit area). How to reproduce with a physical device. NB, the system is generally continuous time
Data collection. The number of impulse tests should be large enough to reduce mean estimate, affected by measurement noise, to acceptable accuracy. The length of the signal should be large enough compared to the input signal length (FIR/IIR system). The sample period should be short enough to capture high-order dynamics (Nyquist rate)
Input amplitude. The input signal should be scaled to make the impulse response large compared to the magnitude of the noise signal. However, it should not be too large, otherwise the system may be non-linear (systems are generally only locally linear)
NB could consider step response data instead
u(t) t
h(t)
s(m)=s/n0.5 x x x x x x x x x x x x x x x x x x x x x x x x x x t
ah(t)
a<<1 x x x x x x x x x x t
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20. Lecture 4: System Frequency Response
Non-parametric system frequency response
System eigenfunctions
System transfer functions (stable systems)
Sinusoidal basis functions and Fourier transforms
Convolution in the frequency domain
Examples of calculating the output
Bode plots
Identifying the frequency response curve
NB, we’ll be doing the main part of the analysis for continuous time signals and LTI systems, but the work carries over to discrete time signals and systems as well.
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21. What is a System Eigenfunction?
Let’s imagine what (basis) signals fk(t) have the property that:
i.e. the output signal is the same as the input signal, multiplied by the constant “gain” lk (which may be complex)
For CT LTI systems, we also have that
Therefore, to make use of this theory we need: 1) system identification is determined by finding {fk,lk}.
2) prediction, we also have to decompose x(t) in terms of fk(t) by calculating the coefficients {ak}.
The {fk,lk} are analogous to eigenvectors/eigenvalues matrix decomposition
x(t) = fk(t)
y(t) = lkfk(t)
System
x(t) = Sk akfk(t)
y(t) = Sk aklkfk(t)
LTI
System
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22. Sinusoidal Signals & LTI Systems
Imaginary exponential (sinusoidal) signals and stable LTI systems are very closely related because:
The signal is unchanged by any LTI system apart from a magnitude gain and a phase shift (multiply by complex number l)
This means that by superposition, to calculate the system’s output, we can find the frequency content of each signal and sum the scaled responses, where fk(t)=ejkw0t
Sinusoidal signals are so-called eigenfunctions of LTI systems, i.e. they remain unchanged by the system apart from a gain and a phase shift
x(t) = ejwt
y(t) = l(w)ejwt
LTI system
x(t)
y(t)
x(t) = Sk akfk(t)
y(t) = Sk aklkfk(t)
LTI system t
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23. Proof of Eigenfunction Property
Consider a CT LTI system with impulse response h(t) and input signal x(t)=f(t) = est, for any value of sC:
Assuming that the integral on the right hand side converges to H(s) (transfer function), this becomes (for any sC):
Therefore f(t)=est is an eigenfunction, with eigenvalue l=H(s). When s=jw, this represents frequency response data, and the system must be stable (h(t) tends to zero)
Very important for Fourier/Laplace transforms
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24. Fourier Series
Fourier series theory says that nearly all continuous time, periodic signal, x(t), can be represented as complex sums of harmonic (sinusoidal) basis signals
Examples (w0=1, T=2p)
w0 is the
fundamental frequency
x(t) = sin(t)+0.2*sin(7*t)
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25. Fourier Transform
When x(t) is not periodic we can calculate the Fourier transform as
This is a continuous spectrum of frequencies
Example
NB the Fourier transform of periodic exists and it is a sum of delta signals, centred at the harmonic frequencies
x(t) t T1
-T1
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26. System Prediction using Frequency Response
Taking Fourier transforms of the input and impulse response signals allows us to derive the frequency domain convolution equation
Y(jw) = H(jw)X(jw)
This can then be inverted to obtain the time-domain system response, y(t) to a input signal x(t):
Calculate Fourier transform of input x(t)
Calculate Fourier transform of impulse response h(t)
Multiply together to obtain Y(jw)
Invert the Fourier transform (generally by expressing as partial fractions)
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27. Example 1: Solving a First Order ODE
Consider the response of an LTI system with impulse response:
to the input signal:
Transforming these signals into the frequency domain:
and the frequency response:
to convert this to the time domain, express as partial fractions:
Therefore, the time domain response is:
ba
Try calculating using convolution
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28. Example 2: Designing a Low Pass Filter
H(jw) wc
-wc
Lets design an ideal low pass filter in frequency domain:
The impulse response of this filter is the inverse Fourier transform
which is an ideal low pass filter in time domain
Non-causal, so this cannot be manufactured exactly
The time-domain oscillations may be undesirable
How to approximate the frequency selection characteristics?
Consider the 1st order LTI system impulse response:
Causal and non-oscillatory time domain response and performs a degree of low pass filtering. Higher order filters (ODEs) are usually used.
h(t) t
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29. Bode Plots
A Bode Plot for a system is simply plots of log magnitude and phase against log frequency
Both the log magnitude and phase effects are now additive
Widely used for analysis and design of filters and controllers
Example
Low pass, unity filter
Log mag v log freq
Phase v log freq
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30. Example 1: Bode Plot 1st Order System
Consider a LTI first order system described by:
Fourier transfer function is:
the impulse response is:
and the step response is:
Bode diagrams are shown as log/log plots on the x and y axis with t=2.
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31. Estimating Frequency Response
Use sin() wave testing to estimate H(jw) over the whole range of frequencies necessary for prediction
Y(jw) = H(jw)X(jw)
If x(t) only contains low frequencies, it is not necessary to estimate H(jw) for high frequency components as the product is zero whatever the value
With a bit of work, we can show that the system response to x(t) = sin(wt) where H(jw)=A+jB is
y(t)=(A2+B2)0.5sin(wt+tan-1(B/A))
where
gain=(A2+B2)0.5
phase advance=tan-1(B/A)
These can be estimated from steady state response
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32. L3&4 Summary
These lectures have concentrated on “conventional” system identification methods that can be applied to LTI systems. They are non-parametric and make few assumptions about the order of the underlying ODEs/difference equations.
Impulse response
A key concept for analyzing system properties and can be easily identified. System response is calculated using convolution
Frequency response
Derived as the Fourier transform of the impulse response signal. Provides information about the gain and phase of the system for different input frequencies.
The introduction provided on this course has been very basic and glosses over many practical details.
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33. L3&4 Laboratory
Theory
Using convolution, calculate the DT response for h(t)=u(t-2), x(t)=0.5tu(t). Sketch all signals.
Using convolution, calculate the DT response for h(t)=u(t), x(t)=2tu(-t). Sketch all signals.
Using Fourier transforms, calculate the CT response for h(t)=e-3(t-1)u(t-1) and x(t)=e-(t+1)u(t+1). Sketch all signals.
Matlab and Simulink
Change to your P: directory and turn your diary on!
Use the Matlab conv() function to verify the DT convolution in questions (1) and (2). Plot all the signals.
Use the fourier() and ifourier() functions in the symbolic toolbox to verify the answer to question (3).
By evaluating the equivalent transfer functions for each of the impulse responses in questions (1), (2) and (3), simulate the signals and systems in Simulink and verify that all the answers are the same.
Simulate the frequency response of a range of first and second order systems in Simulink for a range of different frequencies
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34. Appendix A: DT Eigenfunctions/values
In an analogous fashion, we can show the same for discrete-time systems, so consider a DT system with DT impulse response h(t) and input x(t).
Again, we have assumed that the (infinite) summation on the right hand side converges to H(z).
Therefore x(t)=f(t)=zt is a DT eigenfunction with eigenvalue l=H(z).
Very important for z-transforms
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35. Appendix B: FT of Exponential Signal
Consider the (non-periodic) signal (solution to first order ODE)
Then the Fourier transform is:
x(t)
a = 1
If x(t) is a real signal 1) X(0) is real, 2) Im(X(-jw)) = -Im(X(-jw))
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