signals & system using matlab - rit pampady

“FEAR OF THE LORD IS THE BEGINNING OF WISDOM”

SIGNALS & SYSTEM ANALYSIS USING MATLAB

By

Mr. Anish BennyAsst. Professor

SIGNALS & SYSTEM

• Signals and systems deal with various types of signals and the analysis of the system.• Signals can be elementary type or derived type. • Elementary signals are impulse signal, step signal

and ramp signal.• Derived signals are sinusoidal signals,

exponential signals, pulse etc.• Matlab can be used for generating all types of

signals.• Signals and systems toolbox is employed for

theses purpose

KITES - 2014, RIT PAMPADY

SYSTEM

• A is an interconnection of components forming a particular configuration to provide a desired system response

Mobile phone

Computer

Robotic arm

Automobile cruise control system etc.


Basic Signals & System Components

System

- The device which will generate response according to the given input

ProcessSystem Input

signal

Output

CONCEPT OF A SYSTEM


SIGNALS & SYSTEM MODELLING

• Time domain modeling is done on both the signals and system to get the mathematical equation.

• Time domain model (both continuous and discrete)is needed for further analysis.

• By applying a frequency domain transformation on the above time domain model will result in Transfer Function (TF) of the system and frequency spectrum of the signal.

• By using the model of a system we can analyze the system for various properties like linearity, time invariance, stability etc.


STEP SIGNAL


VARIOUS FORMS OF STEP SIGNAL


EXAMPLE


RAMP FUNCTION


IMPULSE SIGNAL


EXAMPLE – SYSTEM MODELING


• Find the model of the system? Take v1(t) as input and v2(t) as the output.

• Solution

EXAMPLE – CIRCUIT MODELING

• Find the model of the system? Take v1(t) as input and v2(t) as the output.

• Solution


TRANSFER FUNCTION OF THE MODEL

• Case 1• By applying Laplace transform we get,

• RCSV2(s)+V2 (s) = V1 (s)

• Output/Input = V2 (s)/ V1 (s) = 1/(SRC + 1)

• Case 2• By applying Laplace transform we get,

• LCS2V2 (s)+RCSV2(s)+V2 (s) = V1 (s)

• Output/Input = V2 (s)/ V1 (s) = 1/(S2LC + SRC + 1)


MATLAB COMMANDS

• ss - Specify state-space models or convert LTI model to state space

• tf - Create or convert to transfer function model• zpk - Create or convert to zero-pole-gain model• bodeplot - Plot Bode frequency response and return plot

handle• impulseplot - Plot impulse response and return plot handle• pzplot - Plot pole-zero map of LTI model and return plot

handle• stepplot - Plot step response of LTI systems and return

plot handle


• conv - Convolution and polynomial multiplication• deconv - Deconvolution and polynomial division• fourier – Finds the Fourier transform• ifourier - Inverse Fourier transform• laplace – Finds the Laplace transform• ilaplace - Inverse Laplace transform• ztrans – Finds the Z transform• iztrans - Inverse Z transform• sawtooth - Sawtooth or triangle wave• square - Square wave• gensig - Generate test input signals for lsim


MATLAB COMMANDS

• allmargin - All crossover frequencies and corresponding stability margins

• bode - Bode diagram of frequency response• bodemag - Bode magnitude response of LTI models• evalfr - Evaluate frequency response at given frequency• freqresp - Evaluate frequency response over frequency

grid• frd - Create or convert to frequency-response data models• margin - Gain and phase margins and associated crossover

frequencies• impulse - Impulse response of LTI model• step - Step response of LTI systems


MATLAB COMMANDS

MATLAB COMMANDS

• bandwidth - Frequency response bandwidth• lti/order - LTI model order• pole - Compute poles of LTI system• zero - Transmission zeros of LTI model• pzmap - Compute pole-zero map of LTI models• ss2tf - Convert state-space filter parameters to transfer

function form• tf2ss - Convert transfer function filter parameters to state-

space form• feedback - Feedback connection of two LTI models


CONTINUOUS TIME SYSTEM ANALYSIS

• Transfer Function Representation• Time Simulations• Frequency Response Plots• Control Design • State Space Representation

TRANSFER FUNCTION REPRESENTATION

• Tf2zp• Zp2tf• Feedback• Parallel• series

CONT…

• Transfer functions are defined in MATLAB by storing the coefficients of the numerator and the denominator in vectors. Given a continuous-time transfer function

B(s) H(s) = --------- A(s)

CONT…

• Where B(s) = bMsM+bM-1sM-1+…+b0

and A(s) = aNsN+aN-1sN-1+…+a0

Store the coefficients of B(s) and A(s) in the vectors

num = [bM bM-1 … b0]

den = [aN aN-1 … a0]

EXAMPLE

5s+6 H(s) = ---------------- s3+10s2+5• num = [5 6]; • den = [1 10 0 5];

• all coefficients must be included in the vector, even zero coefficients

CONT…

• To find the zeros, poles and gain of a transfer function from the vectors num and den which contain the coefficients of the numerator and denominator polynomials:

[z,p,k] = tf2zp(num,den)

EXAMPLE

• num = [5 6]; • den = [1 10 0 5];

• [z,p,k] = tf2zp(num,den)

• z = -1.2000• p = -10.0495 0.0248+0.7049i 0.0248-

0.7049i• k = 5

EXAMPLE CONT…

(s-z1)(s-z2)...(s-zn)

H(s) = K -------------------------- (s-p1)(s-p2)...(s-pn)

5*(s+1.2) ---------------------------------------------------------- (s+10.0495)(s-{0.0248+0.7049i})(s-{0.0248-

0.7049i})

CONT…

• To find the numerator and denominator polynomials from z, p, and k:

[num,den] = zp2tf(z,p,k)

• To reduce the general feedback system to a single transfer function: T(s) = G(s)/(1+G(s)H(s)) [numT,denT]=feedback(numG,denG,numH,denH);

CONT…

• To reduce the series system to a single transfer function, T(s) = G(s)H(s)

[numT,denT] = series(numG,denG,numH,denH);• To reduce the parallel system to a single transfer

function, T(s) = G(s) + H(s)[numT,denT] = parallel(numG,denG,numH,denH);

TIME SIMULATIONS

• residue • Step• Impulse• lsim

CONT…

• [R,P,K] = residue (B,A) finds the residues, poles and direct term of a partial fraction expansion of the ratio of two polynomials B(s)/A(s)• The residues are stored in r, the corresponding

poles are stored in p, and the gain is stored in k.

EXAMPLE

• If the ratio of two polynomials is expressed as• b(s) 5s3+3s2-2s+7 ------- = ------------------- a(s) -4s3+8s+3

• b = [ 5 3 -2 7]• a = [-4 0 8 3]

EXAMPLE CONT…

• r = -1.4167 -0.6653 1.3320• p = 1.5737 -1.1644 -0.4093• k = -1.2500

• B(s) R(1) R(2) R(n)• ------ = -------- + --------- + ... + --------- + K(s)• A(s) s - P(1) s - P(2) s - P(n)

FIND THE RESPONSE OF A SYSTEM TO A PARTICULAR INPUT

• First store the numerator and denominator of the transfer function in num and den, respectively.• To plot the step response: step(num,den)• To plot the impulse response: impulse(num,den)

CONT…

• For the response to an arbitrary input, use the command lsim (linear simulation)• Create a vector t which contains the time values

in seconds t = a:b:c;• Define the input x as a function of time, for

example, a ramp is defined as x = t lsim(num,den,x,t);

FREQUENCY RESPONSE PLOTS

• Freqs• Bode• Logspace• Log10• Semilogx

CONT…

• To compute the frequency response H of a transfer function, store the numerator and denominator of the transfer function in the vectors num and den.

• Define a vector w that contains the frequencies for which H) is to be computed, for example w = a:b:c where a is the lowest frequency, c is the highest frequency and b is the increment in frequency.

H = freqs(num,den,w)

CONT…

• To draw a Bode plot of a transfer function which has been stored in the vectors num and den:

bode(num,den)

CONT…

• To customize the plot, first define the vector w which contains the frequencies at which the Bode plot will be calculated.

• Since w should be defined on a log scale, the command logspace is used.

• For example, to make a Bode plot ranging in frequencies from 0.1 to 100, define w by

w = logspace(-1,2);• The magnitude and phase information for the Bode

plot can then be found by: [mag,phase] = bode(num,den,w);

CONT…

• To plot the magnitude in decibels, convert mag using the following command:

magdb = 20*log10(mag);• To plot the results on a semilog scale where the

y-axis is linear and the x-axis is logarithmic: semilogx(w,magdb)• For the log-magnitude plot : semilogx(w,phase)

VERY SHORT SIMULINK TUTORIAL

• In the Matlab command window write simulink.• The window that has opened is the Simulink Library

Browser. • It is used to choose various Simulink modules to use in your

simulation.

• From this window, choose the File menu, and then New (Model). • Now we have a blank window, in which we will build our model. • This blank window and the library browser window, will be the

windows we’ll work with.

• We choose components from the library browser, and then drag them to our work window. • We’ll use only the Simulink library (also called toolbox) for now.


SIMULINK TUTORIAL

As we can see, the Simulink library is divided into several categories:

1. Continuous – Provides functions for continuous time, such as integration, derivative, etc.

2. Discrete – Provides functions for discrete time.

3. Functions & Tables – Just what the name says.

4. Math – Simple math functions.

5. Nonlinear – Several non-linear functions, such as switches, limiters, etc.

6. Signals & Systems – Components that work with signals.

7. Sinks – Components that handle the outputs of the system (e.g. display it on the screen).

8. Sources – Components that generate source signals for the system.


EXAMPLE OF SIMULINK


Two main libraries for manipulating signals in Simulink:

• Sources: generate a signal• Sink: display, read or store a signal

EXAMPLE OF SIMULINK


signals & system using matlab - rit pampady

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