Visual Cue

Narration

Slide 1

Dear Friends,
Welcome to the Spoken Tutorial on “Discrete Time System”

Slide 2Learning Objectives

At the end of this tutorial, we will learn how to:
 Convert between state space and transfer function descriptions
 Define a discrete time system and plot its step response
 Discretize a continuous time system

Slide 3 System Requirement slide

* I am using Ubuntu 12.04 operating system with Scilab 5.3.3 for demonstation

Slide 4 Prerequisite slide

* To practise this tutorial, you should have basic knowledge of Scilab.
 If not, please refer to the Scilab tutorials available on the Spoken Tutorial website.

Slide 5 State Space Model

* The state space model
 x dot is equal to A x plus B u
 y is equal to c x plus D u
 is specified by sys three is equal to syslin into bracket into quotes c comma A comma B comma C comma D close bracket
 for prespecified matrices A, B, C and D of suitable sizes.

Switch to Scilab and type this on your Scilab Console
sys3=syslin(’c’,4,3,6,9) Press Enter
Press Enter again
clc

* Start Scilab on your computer
 Type
 sys three is equal to syslin into bracket into quotes c comma four comma three comma six comma nine close bracket and press Enter. Press enter to continue the display.
 This is an example for single state, Single Input Single Output
 The output will have matrices A, B, C and D and initial state x zero
 Type clc to clear the console

Slide 6 State Space Model

* Define for example matrices A, B, C, D on Scilab console as you see

Switch to Scilab and type this on your Scilab Console
A = [2 3;4 5]
Press enter
B = [1;2]
Press enter
C = [3 6]
Press enter
D = 2
Press Enter
sys4=syslin('c',A,B,C,D)
Press enter
Press enter again

* Type this on your Scilab Console
 A is equal to open square bracket two space three semicolon four space five close square bracket
 Press enter
 B is equal to open square bracket one semicolon two close square bracket
 Press enter
 C is equal to open square bracket minus three space minus six close square bracket
 Press enter
 D is equal to two
 Press enter
 Let us substitute these matrices in the previous command
 sys four is equal to sys lin into brackets into quotes c comma A comma B comma C comma D close the bracket and press enter
 You will get the following output.
 Press enter to continue display.
 The output will have matrices A B C D and initial state x zero

Slide 7 State Space Model

* Check whether poles of sys4 are same as eigenvalues of A .
 For this you can use the function p l z r function and the spec function

Slide 8 State Space Model

* The s s two t f command can be used to obtain a transfer function of a statespace system sys S S.

Switch to Scilab and type on your Scilab Console
clc
sysTF = ss2tf(sys4)
Press Enter

* Type on your Scilab Console
 clc to clear it
 Type sys capital T capital F is equal to s s two t f open bracket sys four close bracket
 Press enter
 You see this output

Slide 8 State Space Model

* It is in the form sys TF equal to ss two tf into bracket sys of SS

Slide 9 State Space Model

* Use ss two tf function for sys three defined earlier
 sys T F is a new variable for which 'denom' command is applicable and not applicable to sys four as it is in state space form

Slide 10, 11, 12 Exercise

* Solve the following exercise
 Find a state space realization of the second order transfer function defined below
 Use t f two s s command
 For the new system in state space form, say sys S S, check if the eigenvalues of the matrix A and the poles of the transfer function G of s are the same.
 Use the A, B, C, D matrices of the system sys S S to obtain the transfer function and check if the answer is the original one.

Slide 13, 14 Discrete Time System

* We now define a discrete time system.
 It is customary to use ’z’ for the variable in the numerator and denominator polynomials.
 Recall that the variable ’z’ has a shortcut
 Instead of z is equal to poly into bracket zero comma inside quotes z : use z is equal to percentage z

Type on your Scilab Console
clc
z=%z
Press Enter

* Go to Scilab console.
 Type clc to clear
 Type z is equal to percentage z.
 Press enter

Slide 15 Discrete Time System

* We now define a first order discrete time system

Type on your Scilab Console
DTSystem = syslin(’d’, z/(z – 0.5))
Press Enter

* Type on your Scilab Console
 D T System is equal to syslin into bracket into quotes small d comma z divided by inside bracket z minus zero point five close the bracket close outer bracket .
 Press enter

Slide 15 Discrete Time System

* We use the same ’syslin’ function as before.
 This time, we specify the domain to be discrete time, instead of continuous time.

Slide 16 Discrete Time System

* For checking the step response, we have to define the input explicitly as ones, for example, for 50 points.

Type this on your Scilab Console
u = ones(1, 50);
Press Enter

* Type on your Scilab Console
 u is equal to ones open bracket one comma fifty close bracket semicolon
 Press enter

Slide 16 Discrete Time System

* Instead of csim, the function we have to use the ’flts’ function to simulate this system.

Type this on your Scilab Console
clc
y = flts(u, DTSystem);
Press Enter

* Type this on your Scilab Console
 clc to clear console
 y is equal to f l t s open bracket u comma D T System close bracket semi colon
 Press enter

Type on your Scilab Console
plot(y)
Press Enter

* Type on your Scilab Console
 plot of y and then press Enter

Display output

* The output will be plotted.

Slide 17 Discrete Time System

* It is helpful to discretize a given continuous time system.
 This is done using the dscr function.

Type on Scilab Console
s=%s
sysG=syslin('c', 2/(s^2+2*s+9))
Press enter

* Let us define a continuous system s is equal to percent s
 sys G is equal to syslin into bracket into quotes c comma two divided by into bracket s square plus two multiplied by s plus nine close bracket close outer bracket and press enter

Slide 18 Discrete Time System

* Let us discretize the system sys G with a sampling period of zero point one.

Type on your scilab console
clc
sys5=dscr(sysG, 0.1)
Press Enter

* Type on your Scilab Console
 clc to clear
 sys five is equal to d s c r into bracket capital sys G comma zero point one close bracket and then press Enter
 Press enter to continue display
 As you see system is discretized as A B C D matrices and inital state x zero

Slide 19 Discrete Time System

* Notice that we obtain the discretized system in state space representation.
 We can convert this to a transfer function representation in discrete time using the s s to t f function

Type on your Scilab Console
clc
sys6 = ss2tf(sys5, 0.1)
Press Enter

* Type on your Scilab Console
 clc and clear it
 sys six is equal to s s two t f into bracket sys five comma zero point one close the brackets and press enter

Display output

* The output gives the transfer function.

Slide 20 Summary

In this tutorial we have learnt to:
 Convert between state space and transfer function descriptions
 Define a discrete time system and plot its step response
 Discretize a continuous time system.

Show Slide 21
Title: About the Spoken Tutorial Project
 It summarises the Spoken Tutorial project
 If you do not have good bandwidth, you can download and watch it

* Watch the video available at the following link
 It summarises the Spoken Tutorial project
 If you do not have good bandwidth, you can download and watch it

Show Slide 22
Title: Spoken Tutorial Workshops
The Spoken Tutorial Project Team
 Conducts workshops using spoken tutorials
 Gives certificates for those who pass an online test
 For more details, please write to contact@spokentutorial.org

The Spoken Tutorial Project Team
 Conducts workshops using spoken tutorials
 Gives certificates for those who pass an online test
 For more details, please write to contact at spoken hyphen tutorial dot org

Show Slide 23
Title: Acknowledgement
 Spoken Tutorial Project is a part of the Talk to a Teacher project
 It is supported by the National Mission on Education through ICT, MHRD, Government of India
 More information on this Mission is available at

* Spoken Tutorial Project is a part of the Talk to a Teacher project
 It is supported by the National Mission on Education through ICT, MHRD, Government of India
 More information on this Mission is available at
 spoken hyphen tutorial dot org slash NMEICT hyphen Intro

On previous slide

This is Anuradha Amrutkar from IITB signing off. Thanks for joining. Thank you.
