matlab step

7/30/2019 MATLAB STEP

1/12

Time Response Analysis

Nader Sadegh

George .W. Woodruff School of Mechanical Engineering

Georgia Institute of technology

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Under Damped 2nd System

10),(222


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Step Response Characteristics

Mp=Maximum Overshoot=ts= Settling Time= 4/

tp= Peak Time=/dTd= Period of oscillations= 2/d

21

e

Mp

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Relationship between Mp and

22

1

)ln(

|)ln(|

2

p

p

p

M

M

eM

+=

=

c

Maximum overshoot is directly related to the

damping ratio:


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Design Example

Find kand b of the vibration isolatorshown such that its step response meets

the following specifications:

Maximum overshoot=5%

Settling time = 0.5 sec

Solution: ts=0.5 and Mp=0.05. Thus

69.03

3

)ln(

|)ln(|

2222=

+=

+=

p

p

M

M

= = = =

48

81159

tsn .

=== 222 )59.11)(10(nn mkm

kN/m1344=k

= = = b

Mb M

n22 2 10 8( )( ) N.sec/m160=b

k

m=10

yb

f

fkyybym =++ &&&

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Resulting Step Response

Time (sec.)

Amplitude

Step Response

0 0.14 0.28 0.42 0.56 0.70

0.2

0.4

0.6

0.8

1

1.2

1.4

From: U(1)

To:Y(1)

*f/k


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Higher order systems

Higher order systems can be expressedassuperposition of 1st and 2nd order systems

Example:

( )( ) 5215211

22 +++

++

=+++ ss

cbs

s

a

sss

3

rd

tf=1

st

order+2

nd

order transfer functions

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Matlab Tutorial and Time

Response Analysis using

Matlab


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What is Matlab?

The name MATLAB is an abbreviation for MATrixLABoratory . MATLAB is an interactive, matrix-based package for scientific and engineeringcalculations.

It can be used for performing all sorts of numericalcomputations

And for dynamic system simulation and controldesign

In this course we mainly use the control systems

toolbox and basic programming commands youlearned in 2016.

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Matlab Basics: Vectors and Matrices

Vectors and arrays are defined and

manipulated with ease in Matlab

You can perform most matrix and vector

computations in Matlab

Linear dynamic systems may be definedusing vector and matrices too!

Lets review basic array and matrix

assignments and operations


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Examples of Array

Assignments

r=[1 2 3] sets r to a row vector

v=[1;2;3] sets v to a column vector

r=1:10 is equivalent to r=[1 2 10]

r=[1:0.2:2] is equivalent to r=[1 1.2 2]

A=[1 2 3;4 5 6] sets A to the 2x3 matrix

=

654

321A

Note: semicolon (;) separate rows and colon (:) is

used create and incrementally increasing array.

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Extraction Commands

Vectors:

v(i) is the i-th element of v, e.g., if v=[3;4;5]

then v(2)=4

Matrices:

A(i,j) is the ij-th element of A, e.g., if A=[1 2 3;

4 5 6] then A(2,3)=6

A(i,:) is the i-th row of A, e.g., A(2,:)=[4 5 6]

A(:,j) is the j-th column of A, e.g., A(:,3)=[3;6]


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Some Array Commands

length(v) returns the length of vector v

[m,n]=size(A) returns the number of rows (m)

and columns (n) of matrix A

zeros(m,n) returns mxn matrix of zeros

A=eye(n) returns an nxn identity matrix

A is the transpose of matrix A, e.g.,

=

=63

52

41

654

321AA

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Matrix Algebra Examples

Addition

A + B A + B =0 5

8 11

Subtraction

A B A B =2 1

2 3

Multiplication

A * B A * B = 9 1717 37

Forward Division

A / B = A * B1

A / B =0 .1364 0. 2273

0 . 0455 0 . 5909

Backward Division

A \ B = A1

* B A \ B =7 1

4 1

Transpose

A A =1 3

2 4

A =1 2

3 4

, B =

1 3

5 7

Operation Example


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Plotting Example

%Script file PlotSin

t=[0:0.1:2*pi]'; %x=[0;0.12*pi]

N=length(x); %length of vector x

y=zeros(N,1); %initialize vector y to zero

for i=1:N

y(i)=sin(t(i));

end

plot(t,y);

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Linear Dynamic System Definition

A dynamic system can either be defined

based on a

single transfer function or differential

equation, or

A system of 1st order differential

equations (state-model)


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Transfer Function System Creation

Command: TF Creation of transfer functions or

conversion to transfer function.

Creation:

SYS = TF(NUM,DEN) creates a continuous-time

transfer function SYS with numerator(s) NUM and

denominator(s) DEN. The output SYS is a TF object.

Example: Y(s)/U(s)=1344/(s2+160s+1344)

num=[0 0 1344]; den=[1 160 1344];

msd=tf(num,den);

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Some Forced Response Commands

Plots the unit impulseresponse of sys

impulse(sys) orimpulse(num,den)

Plots the unit step response

of sys

step(sys) or

step(num,den)

Plots the time response of

sys to input U specified at

time instants T.

lsim(sys,U,T) or

lsim(num,den,U,T)

DescriptionCommand


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State-Space System Creation

Any linear dynamic system can be expressed as a

system of first order differential equations called the

state-space model dx/dt=Ax+Bu, y=Cx+Du

Where

u and y are the input and output vectors

X is the state vector

A,B,C,D are matrices of appropriate dimensions

SYS = SS(A,B,C,D) creates a continuous-time state-

space (SS) model SYS with matrices A,B,C,D. The

output SYS is a SS object. You can set D=0 to mean

the zero matrix of appropriate dimensions.

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Why State-Space Model

Many dynamic systems (e.g., multidegree of

freedom system) are more conveniently

modeled using a system of differential

equations rather than a single equation

State models may be used to simulate the

natural as well as the forced or complete timeresponse

Matlab allows for easy conversion from state

to transfer function models


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More Time Response Commands

Plots the initial condition

response specified by

initial state x0

initial(sys,x0) or

initial(A,B,C,D,x0)

Plots the time response of

sys to input U specified at

time instants T and initial

state x0.

lsim(sys,U,T,x0) or

lsim(A,B,C,D,U,T,x0)

DescriptionCommand

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Mass Spring-Damper Example

Solve d2y/dt2+2dy/dt+10=u with initial conditions

y(0)=1 and dy(0)/dt=2, and zero input.

Defining x1=y and x2=dy/dt the system can be

expressed as

dx1/dt=x2, dx2/dt=-10x1-2x2+u or

[ ] ux

xy

ux

x

x

x

001

1

0

210

10

2

1

2

1

2

1

+

=

+

=

&

&

Matlab commands: msd=ss(A,B,C,D); initial(msd,[1;2])


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Other Utility Functions

)3()(

)2()2(

)1()1(

DENNUM

psnr

psr

psr

-++

-+

-=

MATLAB Functions Description

p=roots(DEN) roots of the polynomial equation DEN=0

[z,p]=tf2zp(NUM,DEN) z=roots of NUM=zeros of (NUM/DEN)

p=roots of DEN=ploes of (NUM/DEN)

[r,p]=residue(NUM,DEN) Calculates the partial fraction expansion of

(NUM/DEN):

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Inverse Laplace Transform

syms s

g=(s^2+4*s+1)/(s*(s^2+4*s+5))

g =

(s^2+4*s+1)/s/(s^2+4*s+5)

ilaplace(g)

ans =

1/5+4/5*exp(-2*t)*cos(t)+8/5*exp(-2*t)*sin(t)

matlab step

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