Scilab/C4/Discrete-systems/English

Title of script: Discrete Time System

Author: Manas, Aditya, Shamika

Keywords: State space, Discrete time

Visual Cue	Narration
Slide 1	Dear Friends, Welcome to the Spoken Tutorial on “Discrete Time System”
Slide 2-Learning 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 state-space 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. Close the graphic window
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 Watch the video available at http://spoken-tutorial.org/What_is_a_Spoken_Tutorial 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
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On previous slide	This is Anuradha Amrutkar from IITB signing off. Thanks for joining. Thank you.

Contributors and Content Editors

Lavitha Pereira