Background

The system to be studied involves the positioning of a mechanical load characterised by

mass and frictional damping. The load is moved by a hydraulic actuator which responds to

a control voltage, and the coupling arrangement between the actuator and load has to be

able to cope with a certain amount of flexing $-$ so it is "springy".Hydraulic

Actuator

Coupling

and load

Control

voltage

u $($V$)$

Actuator

position

q $($mm$)$

Load

position

y $($mm$)$

Figure $1-$ The Open$-$Loop System

The Laplace transfer function $($LTF$)$ model of the hydraulic actuator is:$8$

$8$

$)($

$)($

$+$

$=$ ssU

sQ

The LTF model of the coupling and load is:PsssQ

sY

$+++$

$=204$

$5\mathrm{.}14$

$)($

$)(2,$ where P is $7.$

Your task is to investigate the open$-$loop behaviour of this system and design and test

closed$-$loop controller to change the behaviour, in an effort to meet the specification given

below.

Specification

The test input to the system $($both in open$-$ and closed$-$loop$)$ is to be a step of $3$ units.

The indicators to be used in evaluating the performance of the system are the usual time$-$

domain ones:

$$ The steady$-$state error between y and the applied step input must be zero. Thus, since e

$=$ r $-$ y$,$ a step input of h units at r must cause an ultimate output movement of h units at

y in the closed$-$loop system.

$$ The percentage overshoot in the step response must be less than $5\%($of the final

value$).$

$$ The rise time from $10\%$ to $90\%$ of the final value must be less than $400$ ms$.$

$$ The settling time to within $1\%$ of the final value must be less than $2$ s$.$

The assignment questions follow. Note that ques