Eigenvalue assignment

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The dynamics and stability of a LTI system are determined by the eigenvalues of the dynamics matrix (i.e. the poles of the transfer function).

Linear Control Systems Frequency domain: a reminder

Guillaume Drion Academic year 2018-2019

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Input-output representation of LTI systems

Can we mathematically describe a LTI system using the following relationship?

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We exploit the superposition principle (linear systems):

Response to a pulse: impulse response time-domain.

Response to an oscillatory signal at a specific frequency: frequency response frequency-domain.

The complex exponential

The frequency response of a LTI system determines how it transmits oscillatory signals?

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We use the complex exponential: .

The complex exponential has two components: an exponential growth/decay and oscillatory component:

Special case: , giving .

In discrete time: .

The complex exponential

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Outline

Transfert function

Frequency response - Bode plots

1st and 2nd order responses

Rational transfer functions

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Transmission of complex exponentials through LTI systems

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where is the transfer function of the LTI system.

LTI system

Continuous case:

Transmission of complex exponentials through LTI systems

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Transfer function Impulse response

The transfer function is a transformation of the impulse response.

This transformation is called the Laplace transform: the transfer function of a LTI system is the Laplace transform of its impulse response (continuous time).

The Laplace/Fourier transforms: from time domain to frequency domain

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Transmission of complex exponentials through LTI systems

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where is the transfer function of the LTI system.

Discrete case:

LTI system

Transmission of complex exponentials through LTI systems

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Transfer function Impulse response

The transfer function is a transformation of the impulse response.

This transformation is called the z-transform: the transfer function of a LTI system is the z- transform of its impulse response (discrete time).

Important properties of transforms: linearity

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If with ROC and with ROC then

with ROC

Important properties of transforms: convolution/multiplication

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Duality convolution/multiplication (continuous)

Duality convolution/multiplication (discrete)

Important properties of transforms: differentiation and integration

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Continuous:

Discrete:

Transform of decaying exponential/time shift

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Laplace transform of decaying exponential (continuous)

z-transform of time-shift (discrete)

The transfer function

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LTI system LTI system

Transfer function:

In practice, analysis and design of LTI systems is done using the transfer function.

Transfer function of LTI systems (continuous case)

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Let’s consider the discrete LTI system described by the ODE the Laplace transform gives

The transfer function is therefore given by

Transfer function of LTI systems (discrete case)

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Let’s consider the discrete LTI system described by the difference equation the Z-transform gives

The transfer function is therefore given by

Transfer function of LTI systems

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The transfer function of LTI systems has a specific form: it is rational.

The roots of are called the zeros of the transfer function.

The roots of are called the poles of the transfer function.

Transfer function of LTI systems: relationship with state-space representation

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Transfer function from state-space representation: which givesand therefore

(1)

(1) (2)

Block diagrams and transfer function

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The duality convolution/multiplication makes it easy to connect LTI systems using the transfer function.

H1

H2

H

U Y

H1 H2

H

U Y

H1

H2

H

U Y-

Parallel: H = H1 + H2

Series: H = H1H2

Feedback: H = H1/(1+ H1 H2)

Relationship between transfer function and systems response

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Outline

Transfert function

Frequency response - Bode plots

1st and 2nd order responses

Rational transfer functions

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How do we characterize the response of a LTI systems to an oscillatory signal at a specific frequency?

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Using the polar representation , we have

Change in amplitude Change in phase

When an oscillatory signal goes through a LTI systems, his amplitude (amplification/attenuation) and phase (advance, delay) are affected. Not his frequency!

Frequency response of LTI systems

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The frequency response of a LTI system can be fully characterize by , and in particular:

: GAIN (change in amplitude)

: PHASE (change in phase)

A change in phase in the frequency domain corresponds to a time delay in the time domain:

which gives

The slope of the phase curve corresponds to a delay in the time domain.

Frequency response of LTI systems

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The frequency response of a LTI system can be fully characterize by , and in particular:

: GAIN (change in amplitude)

: PHASE (change in phase)

A plot of and for all frequencies gives all the informations about the frequency response of a LTI system: the BODE plots.

In practice, we use a logarithmic scale for such that

becomes

The Bode plots

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The Bode plots graphically represent the frequency response of a LTI system.

They are composed of two plots:

The amplitude plot (in dB): .

The phase plot: .

For discrete time systems, we use a linear scale for the frequencies, ranging from to .

The Bode plots

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Examples of Bode plots of continuous (left) and discrete (right) LTI systems.

Outline

Transfert function

Frequency response - Bode plots

1st and 2nd order responses

Rational transfer functions

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How do we characterize the response of a LTI systems to an oscillatory signal at a specific frequency?

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Continuous case:

Using the polar representation , we have

Time and frequency responses of 1st order systems

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We consider the general 1st order system of the form

The transfer function of the system is given by

The frequency response ( ), impulse response ( ) and step response ( ) writes

H(j!) =1

j!⌧ + 1, h(t) =

1

⌧e�t/⌧ I(t), s(t) = (1� e�t/⌧ )I(t)

Time and frequency responses of 1st order systems

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Bode plots of 1st order systems: amplitude

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Amplitude plot:

If :

If :

Low frequencies: constant frequency response ( )

High frequencies: frequency response linear decays by -20dB/dec.

Cutoff frequency: .

Bode plots of 1st order systems: amplitude

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Amplitude plot: first order systems are low-pass filters! (but the slope at HF might be too low to achieve good filtering properties...).

Bode plots of 1st order systems: phase

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Phase plot:

Low frequencies: no phase shift.

Mid frequencies: phase response decays linearly (slope = time-delay = ).

High frequencies: phase-delay of .

Bode plots of 1st order systems: amplitude and phase

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Time and frequency responses of 2nd order systems

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We consider the general 2nd order system of the form

The transfer function of the system is given by

The frequency response writes

= natural frequency = damping factor

Time and frequency responses of 2nd order systems

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The transfer function of a 2nd order system is given by

The transfer function has two poles:

Case 1 ( ) : two real poles cascade of two first order systems.

Case 2 ( ) : two complex conjugates poles.

New behaviors (oscillations, overshoot, etc.)

Time and frequency responses of 2nd order systems

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Bode plots of 2nd order systems

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Amplitude plot:

Phase plot:

Bode plots of 2nd order systems

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Amplitude plot: second order systems are low-pass filters! (higher slope, possible resonant frequency with overshoot in the frequency response).

Bode plots of 2nd order systems: resonant frequency

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If , there is an overshoot in the frequency response at a resonant frequency .

The amplitude of the peak is given by

For , there is no peak in the frequency response.

Time and frequency responses of 2nd order systems

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Outline

Transfert function

Frequency response - Bode plots

1st and 2nd order responses

Rational transfer functions

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Frequency response of LTI systems

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Bode plots of first and second order systems are building blocks for the construction of Bode plots of any LTI systems.

Indeed, the transfer function of LTI systems is rational, and the denominator terms can all be expressed as

or

In other terms, the Bode plots of LTI systems can be sketched from the poles and zeros of the transfer function!

Frequency response of LTI systems: poles and zeros

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The Bode plots of LTI systems can be sketched from the poles and zeros of the transfer function!

Each real pole induce a first order system response where .

Each pair of complex conjugate poles induce a second order system response where

Zeros induce the opposite behavior.

Frequency response of LTI systems: poles and zeros

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Frequency response of LTI systems: Bode plots

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Amplitude:

any real pole induces a decrease in the slope of -20dB/dec.

any real zero induces an increase in the slope of 20dB/dec.

any pair of complex conjugate poles induces a decrease in the slope of -40dB/dec.

Phase:

any real pole induces a decrease in the phase of .

any real zero induces an increase in the phase of .

any pair of complex conjugate poles induces a decrease in the phase of .

Frequency response of LTI systems: Bode plots

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Example: DC gain of -20dB, zero in 10 K Hz and pole in 100 K Hz.

Frequency response of LTI systems: Bode plots

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Example: DC gain of -20dB, zero in 10 K Hz and pole in 100 K Hz.

Frequency response of LTI systems: poles and zeros

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