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Modern Control SystemEKT 308
Root Locus and PID controllers
PID Controllers
.controller Derivative iii)
controller Integral ii)
controller alProportion i)
:components threehasit implies, name theAs
.controller derivative plus integra plus alproportionfor stands PID
control process industrialin used Widely *
Proportional Controller
Pp
P
KsE
sUsT
teKtu
teu(t)
)(
)()(
)( )(
)( input , output Assume
PID Controllers (contd…)
s
K
sE
sUsT
s
sEKsU
dtteKtu
teu(t)
II
I
I
)(
)()(
)conditions initial (zero )(
)(
)( )(
)( input , output Again,
Integral controller
)(
)()(
)conditions initial (zero )( s)(
)( )(
sKsE
sUsT
sEKsUdt
tdeKtu
DD
D
D
Derivative controller
PID controller
sKsKKsEsUsG
ssEKs
sEKsEKsU
dt
tdeKdtteKteKtu
DIPc
DIP
DIP
/)(/)()(
)()(
.)()(
)()()()(
PID Controllers (contd…)
/ and / ,
)(
/)(
analyzing,Further
2
2
DIDP
D
IPD
DIPc
KKbKKaWheres
bassK
s
KsKsK
sKsKKsG
plane.-s in the anywhere located zeros twoandorigin at the pole one
ith function w transfer a introduces controller PID a Therefore,
))((
)( 21
s
zszsKsG D
c
PID Controllers (contd…)Effect of a PID controller on the system
1. Figin as controller PID aby controlled is system The
)3)(2(
1)(
function, transfer following thehas system a Suppose,
sssG
)3)(2()ˆ)((
1
)3)(2()ˆ)((
)()(1
)()()(
function, transfer Overall
conjugate)(complex ˆ ,13-
zeros,complex has controller PID theAssume,
11
11
121
ssszszsK
ssszszsK
sGsG
sGsGsT
zzjz
D
D
c
c
)ˆ)()((
)ˆ)((
)ˆ)(()3)(2(
)ˆ)((
112
11
11
11
rsrsrs
zszsK
zszsKsss
zszsK
D
D
D
PID Controllers (contd…)
Whereas the original system had two poles and no zeros, now the controller + system has three poles and two zeros. The root locus of the system is shown in figure 1 below.
Fig 1: root locus of system + controller
point,Breakaway e)
axis.imaginary of crossing no So
.z- and z- zerosat endmust 2- and 0 poles from Locus d)
infinity. to3- from locus the toscorrespond This
0.kfor 18018023
)12( angle Asymptote c)
infinity. to3- from axis real on the Locus b)
plane-s in the zeros and 3)- 2,- (0, poles Place a)
1. figfor hints drawing locusRoot
11
0
ok
PID Controllers (contd…)
)ˆ)((
)3)(2( ,0
)3)(2(
)ˆ)((1
11
11
zszs
sssK
sss
zszsKD
D
f) Angle of arrival at –z1 and –z2
The resulting PID+System has i) Percentage overshoot to step input , less than 2% ii) Steady state error for step input will be zero. iii) Settling time will be approax 1 sec. Lower settling time can be obtained by further adjusting Kd, Kp and KI
Frequency Response
By the term frequency response, we mean the steady-state response of a system to sinusoidal input. In frequency response methods, we vary the frequency of the input signal over a certain range and study the resulting response.
Outline for Bode plot
