Description

15MH302J – LINEAR AND DIGITAL CONTROL SYSTEMS
EXERCISE 3
TIME DOMAIN SPECIFICATIONS OF 1ST AND 2ND ORDER SYSTEMS
LAB PREREQUISITES:
Exercise 1 to 2
PREREQUISITE KNOWLEDGE:
Fundamentals of MATLAB programming OBJECTIVES:
The objective of this exercise will be to study the performance characteristics of first and second order systems using MATLAB and SIMLINK PRELAB:
Automatic cruise control is an excellent example of a feedback control system found in many modern vehicles. The purpose of the cruise control system is to maintain a constant vehicle speed despite external disturbances, such as changes in wind or road grade. This is accomplished by measuring the vehicle speed, comparing it to the desired or reference speed, and automatically adjusting the throttle according to a control law.

Use the following values for the parameters:

• (m) vehicle mass 1000 kg
• (b) damping coefficient 50 N.s/m • (u) nominal control force 500 N
Derive the transfer function of the system where the velocity of the vehicle is the controlled (output) variable.
On a graph sheet, obtain the time domain response of the system for a step input force of 500N. Take sufficient number of data points.
Obtain the following parameters. Show all necessary calculations.
• steady state value of the system
• rise time
• settling time
• time constant
• DC gain READINGS:
Overview of First Order Systems:
An electrical RC-circuit is the simplest example of a first order system. It comprises of a resistor and capacitor connected in series to a voltage supply as shown below on the figure. Assume the values a. R=2KΩ and C=0.01F b. R=2.5KΩ and C=0.003F

For the RC-circuit as shown in Fig. 1, the equation governing its behavior is given by

where vc (t)is the voltage across the capacitor, R is the resistance and C is the capacitance. The constant τ=RC is the time constant of the system and is defined as the time required by the system output i.e. vc (t)to rise to 63% of its final value (which is E). Hence the above equation can be expressed in terms of the time constant as:

Obtaining the transfer function of the above differential equation, we get

where τ is time constant of the system and the system is known as the first order system. The performance measures of a first order system are its time constant and its steady state.
Overview of Second Order Systems:
force:
Then the transfer function representation of the system is given by

The performance measures could be described as follows:
Rise Time: The time for a system to respond to a step input and attains a response equal to a percentage of the magnitude of the input. The 0-100% rise time, Tr, measures the time to 100% of the magnitude of the input. Alternatively, Tr1, measures the time from 10% to 90% of the response to the step input.
Peak Time: The time for a system to respond to a step input and rise to peak response.
Overshoot: The amount by which the system output response proceeds beyond the desired response. It is calculated as
where MPt is the peak value of the time response, and fv is the final value of the response.
Settling Time: The time required for the system’s output to settle within a certain percentage of the input amplitude (which is usually taken as 2%). Then, settling time, Ts, is calculated as

LTI Viewer
The LTI ( Linear Time Invariant) Viewer is an interactive graphical user interface (GUI) for analyzing the time and frequency responses of linear systems and comparing such systems. In particular, some of the graphs the LTI Viewer can create are step and impulse responses, Bode, Nyquist, Nichols, pole-zero plots, and responses with arbitrary inputs. In addition, the values of critical measurements on these plots can be displayed with a click of the mouse. The following table shows the critical measurements that are available for each plot.

• Create LTI transfer functions
• Access the LTI Viewer
• Select LTI transfer functions for the LTI Viewer from Import menu.
• Select the LTI objects for the next response plot
• Select the plot type by right clicking anywhere in the plot and selecting Plot Types
• Select the characteristics by right clicking anywhere in the plot and selecting Characteristics
PROGRAMS, OBSERVATIONS AND INFERENCES FIRST ORDER SYSTEMS
1. Find the step response for the above mentioned cruise and electrical first order systems with given set of values by using the transfer function.
2. Find the time constant value for two systems at which the system reaches the 63 % of its final output.
3. Obtain the steady state value of the system, rise time and settling time.
4. Perform first three tasks using LTI Viewer.

DERIVATION OF TRANSFER FUNCTION FOR FIRST ORDER SYSTEMS

PROGRAMS FOR FIRST ORDER SYSTEMS MECHANICAL SYSTEM
clc; clear all; close all;
F=input(‘Enter the value for input force F= ‘);
M=input(‘Enter the value for Mass M=’);
B=input(‘Enter the value for Damping coefficient B=’); num=[F]; den=[M B]; sys=tf(num,den)
ltiview(sys);

ELECTRICAL SYSTEM
clc; clear all; close all;
V=input(‘Enter the value for input voltage V=’);
R=input(‘Enter the value for resistance R= ‘); C=input(‘Enter the value for capacitance C=’); s=tf(‘s’); sys=V/((s*R*C)+1)
ltiview(sys);

SECOND ORDER SYSTEMS
1. Find all time domain specifications of the second order systems which are derived in the Exercise number 2 using Matlab codes.

TRANSFER FUNCTION OF SECOND ORDER SYSTEMS (from experiment-2) MECHANICAL

ELECTRICAL

DERIVATION OF TRANSFER FUNCTION FOR RLC CIRCUIT

PROGRAMS FOR SECOND ORDER SYSTEM MECHANICAL
clc; clear all; close all;
F=input(‘Enter the value for input force F= ‘);
M=input(‘Enter the value for Mass M=’);
B=input(‘Enter the value for Damper B=’); K=input(‘Enter the value for Spring constant K=’); num=[F]; den=[M B K]; sys=tf(num,den) ltiview(sys);
ELECTRICAL RLC CIRCUIT clc; clear all; close all;
V=input(‘Enter the value for input voltage V=’);
R=input(‘Enter the value for resistance R= ‘);
C=input(‘Enter the value for capacitance C=’); L=input(‘Enter the value for inductance L=’); s=tf(‘s’); sys=V/(((s^2)*L*C)+(s*R*C)+1) subplot(2,1,1); impulse(sys);
title(‘Impulse response for RLC circuit’); ylabel(‘output voltage’); subplot(2,1,2); step(sys);
title(‘Step response for RLC circuit’); ylabel(‘output voltage’); ltiview(sys);

2. Effect of damping ratio  on performance measures: For a single-loop second order feedback system given below

Using MATLAB find the step response of the system for values of n = 1 and  = 0.1, 0.4, 0.7, 1.0 and 2.0.Plot all the results in the same figure window and fill the following table.
Damping
Ratio Rise time Peak Time % Overshoot Settling time Steady state value
0.1
0.4
0.7
1.0
2.0

PROGRAM
clc; clear all; close all;
z=input(‘Enter the value for zeta z=’); w=input(‘Enter the value for omega n w=’); num=[w*w]; den=[1 2*w*z w*w]; sys=tf(num,den)
ltiview(sys);

Evaluation
Component Maximum Marks Marks Obtained
Pre-lab Tasks 10
In-Lab Tasks 20
Post-lab Tasks 10
Bonus Tasks 10
15MH302J LINEAR AND DIGITAL CONTROL SYSTEMS
RESULTS & INFERENCES:
(This page must be the last page of the exercise)
PRE-LAB TASKS:
1. Derive the transfer function of first order systems
a. Mechanical system
b. Electrical system
2. Derive the transfer function of second order systems
a. Mechanical system
b. Electrical system

IN-LAB TASKS
Task 1: Create m file for first order system to obtain the time domain specifications for both mechanical and electrical system.
Task 2: Create m file for the second order systems and obtain the time domain specifications.
Task 3: Tabulate the Effect of damping ratio ζ on performance measures for the second order system.

ADDITIONAL TASKS/LEARNING FOR BONUS MARKS: