EE 414/614 Control Systems I LabLab 5

Time-Domain Analysis for Control Systems Using MATLAB/Simulink

Department of Electrical EngineeringWright State University

1 Objective

The objective of this lab is to use MATLAB/Simulink for analysis of the velocity and position control systems modeled in previous labs.

2 Background

2.1 Time Domain Analysis

In control system is it desirable to measure the performance of a system with respect to time. The time response of the system is divided into two parts, the transient response and the steady state response. The transient response is the part of the time response that goes to zero as the time becomes large. The steady state response is what is left after the transient has died out. All real, stable control system exhibit a transient response before the steady state is reached. Mass, inertia, and inductance are present in real physical systems, therefore the system cannot respond to input changes instantaneously, and the transient should be observed. In addition, the dynamic behavior of systems is closely associated to the transient response so it must be carefully controlled. Some of the transient specifications as they relate to a unit-step input are listed below and shown in figure 1.

The steady state response of the system is also important, since it controls the overall accuracy of the system. Often it is desirable to have the output directly correspond to the input signal and any difference results in steady-state error.

The transient response specifications can be related to a 2nd order prototype system. While few systems are truly 2nd order systems, they can be approximated as 2nd order and/or can be related to the 2nd order prototype system to gain insight for analysis and design. The transfer function of the prototype 2nd order system can be written as follows:

G (s )=ωn

2

s2+2ξ ωn∗s+ωn2

where ξ is the damping ratio and ωn is the natural frequency.

2.1.1 Maximum Overshoot

Maximum overshoot is the difference between the maximum value of the signal and the steady state value of the signal.

Max Overshoot= ymax−1=e−πξ

√1−ξ2

2.1.2 Delay Time

The delay time, td, is the time required for the step response to reach 50 percent of the final value.

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t d ≌1+0.7 ξ

ωn

0<ξ<1.0

2.1.3 Rise Time

The rise time, tr, is the time required for the step response to rise from 10 percent to 90 percent of the final value.

t r ≌0.8+2.5 ξ

ωn

0<ξ<1.0

2.1.4 Settling Time

The settling time, ts, is the time required for the step response to reach and stay within a specified percentage of its final value. A commonly used percentage is 5 percent.

t s ≌3.2ξ ωn

0<ξ<0.69∨t s≌4.5 ξωn

ξ>0.69

2.1.5 Steady State ErrorThe steady state error, ess, of a response is the discrepancy between the output and the reference input when the steady state is reached.

1ess

−1=lims →0

G(s)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4Step Response

Time (sec)

Am

plitu

de

Figure 1 Time Response of a System with the Maximum Overshoot, Delay Time, Rise Time, and Settling Time Labeled.

2

ts

tr

td

Max OS

1S

VOUT(s)

K

Gear Ratio PotKp

Θ(s)

K

AmpK Servomotor System

Ω(s)

K

Va(s)

K

PreamplifierKPA+

K_

K

VIN(s)

K

Ve(s)

K

PotKp

2.2 Velocity Control System

A velocity control system is one method of controlling a dc motor. In this control system, a tachometer readout is compared to the reference voltage, which is the desired velocity to be obtained. The difference between these two voltage results in an error signal, which is amplified and drives the motor until the desired velocity is reached. The block diagram of a velocity control system can be seen in figure 2.

Figure 2 Block Diagram of Velocity Control System

2.3 Position Control System

A position control system is another method for controlling a dc motor. In this control system, a potentiometer is coupled to the output shaft of the motor; the displacement of the motor displaces the potentiometer and generates a voltage, which is compared to a reference voltage supplied another potentiometer. The differences in the two potentiometers generate an error signal, which is amplified and drives the motor until it reaches the desired position. The block diagram of a position control system can be seen in figure 3.

Figure 3 Block Diagram of Position Control System

3 Assignment

1) Construct the velocity control system in simulink, inserting the values obtained during the previous labs for the gains and transfer functions in the block diagram.

2) Set the value of K = 1, 1.5, 2, and 10 and capture the step response to a unit step input.3) Use MATLAB to calculate the open-loop transfer function of the velocity control system, using K = 1, 1.5,

2, and 10. 4) Take the step response of the closed-loop system for each value of K and measure the maximum overshoot,

rise time, settling time, and steady state error.5) Repeat steps 1-4 with the position control system and gains K = 0.1, 0.15, 0.2, and 1.6) For the velocity control system, calculate the value of K needed to meet the steady state error is equal to 5%

and record the step response using the calculated value of K. Show the steady state error requirement is met.

3

AmplifierK

TachometerKt

Servomotor System

+_

VIN(s) Ω(s) VOUT(s)Ve(s) Va(s)

7) For the position control system, calculate the value of K needed to meet the overshoot requirement equal to 10% and record the step response using the calculated value of K. Show the overshoot requirement is met.

4 Report

Include all the following in the report:Simulink models of the velocity control system and position control systemMATLAB codeAll plots from Section 3 with all the time domain specifications labeled, where applicable.Answer all postlab questions

5 Postlab Questions

1) For the both the velocity control system and position, explain how the following time response characteristics change as the value of K is increased:

Maximum overshoot Rise time Settling time Steady State error

2) Draw a block diagram of the position control system with velocity feedback.

References

1) Kou, Benjamin C. (2003). Automatic Control Systems. 8th ed. John Wiley & Sons, Inc.

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