Diskretne PID regulátory
45
Discrete-time controllers structures and tuning Š. Kozák 2000, Department of Automatic Control Systems Vienna TUniversity
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Transcript of Diskretne PID regulátory
Discrete-time controllersstructures and tuning
Š. Kozák 2000, Department of Automatic Control Systems
Vienna TUniversity
PID regulátory – spätnoväzbové štruktúry
• Riadenie (CO) u(t) :Proportional gain
Integral gain Derivative gain
dt
deKdeKteKtu d
t
Ip )()()(0
Proportional feedback gain KP
• Proportional control :
• Feedback control c(t) is linearly proportional to the error :
• Steady state error will decrease
• Faster response
• Too much gain will make the system unstable
)()( teKtu pp
Integral feedback gain KI
• Integral control:
• Penalty on the past error
• Zero steady state error
• Destabilizing influence – It gets oscillatory as KI increases
deKtut
Ii 0
)()(
Derivative feedback gain KD
• Derivative control:
• Stabilize the system: – reduce oscillatory behavior
• Create a damping effect in the system dynamics
• It makes system slow down
dt
deKtu DD )(
Continuous regulator: principle of PID
set-point plant
commandvariable
Ki
Kp
Kd
d
ddt
PID
The proportional factor Kp generates an output proportional to the error, it requires a non-zero error to produce the command variable.
Increasing the amplification Kp decreases the error, but may lead to instability
The integral factor Ki produces a non-zero control variable even when the error is zero, but makes response slower.
The derivative factor Kd speeds up response by reacting to an error step with a control variable change proportional to the step.
error
process value
integral factor
derivative factor
proportional factor
measurement
PID response
asymptotic error: proportional only
too much proportional factor: unstable
no remaining error, but sluggish response:
integral only
differential factorincreases responsiveness
Performance specifications of the closed loop
system (step response)
• Steady state error:
• Maximum overshoot:
• Delay time:
• Rise time:
• Settling time:
Digital control systemsDigital control systems
Digital realisation of an “analogue type” controller
ADC- Digital controller - DAC should behave the same as an analogue controller (e.g. PID type), which implies the use of a high sampling frequency (the algorithm implemented is very simple)
Bad use of the potentialities of the digital controller
SystemDigitalController
y(t)
Perturbations
et)yref(t) +-
DACADC
ek) u(k) u(t)
Ts
“Il ne suffit pas de mettre un TIGRE (microprocesseur ou DSP) dans son régulateur, il faut rajouter de l’intelligence”
SystemDigitalController
y(t)
Perturbations
Yref(k)+-
DACe(k) u(k) u(t)
Ts
The sampling frequency is chosen in accordance with the bandwidth desired for the closed-loop systemIntelligent use of the “computer” : high sampling period and then implementation of complex algorithms requiring greater computation time.
Not only a copy of analogue control : BRAINWARE
The sampling frequency is chosen in accordance with the bandwidth desired for the closed-loop systemIntelligent use of the “computer” : high sampling period and then implementation of complex algorithms requiring greater computation time.
Not only a copy of analogue control : BRAINWARE
ADCy(k)
Discretized System
Digital control systemsDigital control systems
Discrete-time system models and digital control algorithms
y(k) = f[y(k-i), u(k-j)]or
G(z-1) = z -d B(z -1)/A(z -1)
Choice of sampling frequencyChoice of sampling frequency
fs = 1/Ts = (6 to 25) * fCLB
fs : sampling period
fCLB : bandwidth of the closed-loop
system
fs = 1/Ts = (6 to 25) * fCLB
fs : sampling period
fCLB : bandwidth of the closed-loop
system
If fs is fixed => limit for fCLB ( fs /15 )
No more
Diskrétne algoritmy riadenia
• Ideálne „textbook“ PID regulátory• Neideálne formy a opisy PID
regulátorov• Podmienky ekvivalentnosti spojitých a
diskrétnych PID regulátorov vzhľadom na periódu vzorkovania
• Rekurentné formy – diferenčné rovnice diskrétnych PID regulátorov
• PID regulátory s ohraničením riadiaceho zásahu
dt
tdeTde
TteKtu d
t
i
)()(
1)()(
0
srs
rrsT
sTKsG d
iR 1
10)(
11)(
d
i
KTr
T
Kr
Kr
1
1
0
Neidealizovaný (reálny) PID regulátor obsahuje v derivačnej zložke oneskorovací člen (zabezpečujúci realizovateľnosť derivačnej zložky).
sT
sT
sTKsG
f
d
iR 1
11)(
Základné spojité formy PID regulátorov
Neideálna forma opisu PID (realizovateľná)
))(()( / fTt
f
d
i
edt
de
T
Tdte
TteKtu
1
ff =Td*(1/Tf*Dirac(t)-1/Tf2*exp(-t/Tf))
Základné diskrétne formy opisu PID regulátora Prenosová funkcia diskrétneho regulátora v s a z-oblasti :
)()(
)()()(
)()(zE
zU
z
zqzqqzG
sE
sUsT
sTKsG Rd
iR
1
22
110
11
1
)()(
)()()(
)()(zE
zU
zpzp
zqzqqzG
sE
sU
sT
sT
sTKsG R
f
d
iR
22
11
22
110
111
1
s
sG
z
zzGGzG R
RTCR
)()()( 1
if
d
f
d
T
T
T
TDKq
T
TKq
21
1
11
0
fT
T
i
d
f
d
eD
DT
T
T
TKq
1
12 1)(
12
11 1
Dp
Dp
)2()1()()2()1()( 21021 keqkeqkeqkupkupku
)()()()()( 211 210 keqkeqkeqkuku
2.
1.
)(
)()(
)()(
f
f
f
d
f
ff
f
d
f
f
d
f
d
f
d
iR
Ts
T
T
TL
Ts
TTs
T
TL
Ts
s
T
TL
sT
sTĹ
sE
sU
sT
sT
sTKsG
1
1
11
11
11
11
1
1111
ff =Td*(1/Tf*Dirac(t)-1/Tf2*exp(-t/Tf))
Doplnok :
Ako určiť originál k derivačnej zložke
1. Ak nahradíme integrál v spojitej verzii sumou (obdlžníková náhrada) deriváciu diferenciou prvého rádu, potom v k-tom diskrétnom kroku riadiaci zásah je vyjadrený
kde P - je koeficient zosilnenia odpovedajúci proporcionálnemu zosilneniu spojitého PID regulátora, Ti – resp. Td sú koeficienty odpovedajúce integračnej resp. derivačnej časovej konštante spojitého regulátora
k
i
d
i
kekeT
Tie
T
TkeKku
1
11 )()()()()(
Rekurentný vzťah pre riadiaci zásah sa určí rozdielom u(k)-u(k-1)
1
1
21111k
i
d
i
kekeT
Tie
T
TkeKku )()()()()(
k
i
d
i
kekeT
Tie
T
TkeKku
1
11 )()()()()( k
k-1
)()()()()()()()()( 212111 kekeke
T
Tke
T
TkekeKkukuku d
i
- odčítaním
Základné diskrétne formy PID regulátorov (DPID)
)()()()()(
)()()()()()()()()(
21211
212111
keT
TKke
T
T
T
TKke
T
TK
kekekeT
Tke
T
TkekeKkukuku
d
i
dd
d
i
q0 q1 q2
)()()()()( 211 210 keqkeqkeqkuku
q0 > 0
q1>- q0
-(q0+q1) < q2< q0
Podmienky ekvivalentnosti :
T
TKq d10
T
T
T
TKq d
i
211
T
TKq d2
q0
u(k)
q0-q22q0+q1
t=kT
q0+q1 + q2
PCH - PID REGULÁTORA-PODM.EKVIVALENTNOSTI
q0 > 0
q1>-q0
-(q0+q1) < q2< q0
K(1+cd)
u(k)
KK(1+ci)
t=kT
Kci
PCH - PID REGULÁTORA-PODM.EKVIVALENTNOSTI
cd > 0
ci > 0
ci < cd
u(k)
q0
t=kT
PCH - PI REGULÁTORA-PODM.EKVIVALENTNOSTI
q0+q1
)]()()([)()]()([)(
()()()()()(
)(
)()()()(
)()(
212111
221212
1
11
1
kykykyT
TKke
T
TKkykyKku
kykwkykwkykwT
T
keT
T
kykwkykw
Kkuku
d
i
d
i
)()()()()(
)()()()()()()()()(
21211
212111
keT
TKke
T
T
T
TKke
T
TK
kekekeT
Tke
T
TkekeKkukuku
d
i
dd
d
i
w(k)=w(k-1)=w(k-2)
)]()()([)()]()([)()( 212111 kykykyT
TKke
T
TKkykyKkuku d
i
Iné formy a vyjadrenia diskrétneho PID regulátora
Takahashiho vzťah (feedforward forma diskrétneho PID-u):
)()()]())()(
[()()(
)()(])())()(
[()()(
212
1011
12
0
2
1
1
1
kekeT
Tie
kee
T
TkeKku
kekeT
Tie
kee
T
TkeKku
dk
ii
dk
ii
T
Tke
T
T
T
Tke
T
T
T
TkeKkuku d
i
dd
i
)())(())(()()( 22
2112
11
q0 q1 q2
)()()()()( 211 210 keqkeqkeqkuku
-
2. Ak nahradíme integrál v spojitej verzii sumou (lichobežníková náhrada) deriváciu diferenciou prvého rádu, potom v k-tom a k-1 diskrétnom kroku riadiaci zásah je vyjadrený
T
T
T
TKq d
i210
i
d
T
T
T
TKq
2211
T
TKq d2
)2()1()()( 210 keqkeqkeqku
)}2({)}1({)}({)}1({)}({ 210 keqZkeqZkeqZkuZkuZ
)()()()()( 22
110
1 zEzqzEzqzEqzUzzU
)()()1)(( 22
110
1 zEzqzqqzzU
)(
)(
)(
)(
1)(
1
22
110
zE
zU
zP
zQ
z
zqzqqzGR
Prenosová funkcia diskrétneho PID regulátora
)()()()()( 211 210 keqkeqkeqkuku
0k1k
210210
210210
1010
0
111
23122
2011
00
qkkqqk qqqk u ku
qqq qqq u uk
qq qq u uk
quk
)(
............................................................................................
)()(
)()(
)(
21 k pre k-uku )(
00 q0110010 0201 q qqqqqq uu )()(
102210 021 qq q alebo qq qk pre kuku )(
0210010 0 qqqqq qq
Podmienky ekvivalentnosti PID a PSD regulátora
Podmienky „ekvivalentnosti“:
00
011
kpre
kprekke )()(
q0
u(k)
q0-q22q0+q1
t=kT
q0+q1 + q2
q0 > 0
q1>-q0
-(q0+q1) < q2< q0
K
qqqK
qc
q
d
)(c
qK
210i
2
20
1
22
1101
1
1
1
21
1)(
)()1(
11
1
)12()1()(
z
zqzqq
ZE
zUzc
z
zcK
z
zczcccKzG
di
ddidR
T
Tc d
d dd
d KcT
TKK
ii T
Tc
ii
i KcT
TKK
Podmienky ekvivalentnosti PSD regulátora s PID regulátorom
diid cc 0c 0c
Iná ekvivalentná forma vyjadrenia diskrétneho PID regulátora
1
22
1101
1
1
1
21
1)(
)()1(
11
1
)12()1()(
z
zqzqq
ZE
zUzc
z
zcK
z
zczcccKzG
di
ddidR
(rkoz0)
1
22
1101
1
1
21
11
11
1
1
121
z
zqzqq
ZE
zUzc
zcK
z
zczcccKzG
di
ddidR
)()(
)(
)()()(
Veľmi často sa odchýlka nahrádza e(k) e(k-1), čím sa dosahuje okamžité pôsobenie riadiaceho zásahu na proces. Táto zmena sa prejaví aj v prenosovej funkcii regulátora na integračnej zložke, ktorá neobsahuje v čitateli člen z-1.
)()()( 20 keqqkeKku p Integračná
zložka: 1111)( 210 k eqqqkukeKckuku iiii
)1()(1)( 2 keke q kecKkeKcku ddd
Riadiaci zásah podľa (rkoz0) je potom tvorený súčtom jednotlivých zložiek
)()(u(k)u)( ip kukku d
Derivačná zložka:
Proporcionálna zložka:
K
qqqK
qc
q
d
)(c
qK
210i
2
20
Paralelná forma diskrétneho PID regulátora:
Upravený tvar DPID
Vynechávaním jednotlivých koeficientov qi, pre i=0,1,2 dostaneme rôzne štruktúry diskrétnych regulátorov.
Ak vo vzťahu (rkoz0) položíme q2=0, dostaneme prenosovú funkciu diskrétneho regulátora v tvare:
)(
)(
1)(
1
110
zE
zU
z
zqqzGR
Diskrétny regulátor opísaný vzťahom (rkoz1) voláme diskrétny regulátor prvého rádu (PS-regulátor).
Riadiaci zásah diskrétneho PI regulátora je vyjadrený diferenčnou rovnicou
)1()()1()( 10 keqkeqk u ku
)()())(( zUzzqqzE 1110 1
(rkoz1)
Podmienky ekvivalentnosti sú odvodené podobne ako u PID regulátora
0(0)(1) 0 q a uu0110 0 q q aleboqq
u(k)
q0
t=kT
PCH - PI REGULÁTORA-PODM.EKVIVALENTNOSTI
q0+q1
Iné vyjadrenie PS regulátora je možné pomocou koeficientov K, ci a cd. Pre q2=0 je zosilnenie K a koeficienty cd a ci vyjadrené
)(
0
10
0
qqKc
c
qK
i
d
0)(q 0K 0
0c i
Prenosová funkcia diskrétneho PI regulátora (DPI) použitím koeficientov K, ci :
)(
)(
1
)1(1)(
1
1
zE
ZU
z
zcKzG i
R
Riadiaci zásah
)1()1()()1()( kecKkKekuku i
Diskrétny I regulátor získame ak položíme q0=0, q2=0. Prenosová funkcia diskrétneho I regulátora je v tvare:
Riadiaci zásah určíme z prenosovej funkcie
Ak položíme ci=0, dostaneme diskrétny PD regulátor s prenosovou funkciou:
)(
)(
1)(
1
11
zE
zU
z
zqzGR
)1()1()( 1 keqkuku
)1(1)(
)()( 11
20 zcK
zE
zUzqqzG dR
K
qc
qqK
d2
20 )(
0)( qzGR Prenosová funkcia diskrétneho P regulátora
Riadiaci zásah P regulátora: u(k) = q0e(k) ?
)()()()()()()()()( 2121111 kekeke
T
Tke
T
TkekePkukuku d
i
)]()([)()( 11 kekeKkuku
Modifikácia PID regulátorov úpravou derivačného člena
• jednoduchá náhrada derivácie diferenciou prvého rádu vnáša nepresnosti do rekurzívnzych a nerekurzívnzych foriem PSD regulátorov a môže spôsobiť, že riadiaci zásah nadobúda veľké a prudké zmeny.
• Aby sa tomu predišlo, využíva sa náhrada derivácie priemernou hodnotou napr. zo štyroch hodnôt odchýlky:
• Ak použijeme nerekurzívnu formu PID regulátora, potom deriváciu nahradíme vzťahom:
4
)()1()2()3()(
4
1)(
3
kekekekeieke
k
kis
)( keT
T
dt
deT s
dd
T
keke
T
keke
T
keke
T
kekeT ssssd
5.1
)3()(
5.0
)3()(
5.0
)()1(
5.1
)()(
4
)]3()2(3)1(3)([6
kekekekeT
Td
pre rekurentnú formu: )4()3(2)2(6)1(2)([6
kekekekekeT
T
dt
deT d
d
PID formy „neidealizovaného“ diskrétneho regulátora • Ak spojitý PID regulátor obsahuje v derivačnej
zložke oneskorovací člen, môžeme jeho diskrétny opis určiť niekoľkými spôsobmi. Prakticky sa
využívajú dva spôsoby prepočtu: • Prvý spôsob prepočtu je realizovaný na základe
určenia z-obrazu zo spojitého opisu, t.j.
s
sG
z
zzGGzG R
RTCR
)()()( 1
sT
sT
sTKsG
f
d
iR 1
11)(
sT
T
sTsLK
z
zKzG
fi
d
iR 1
1112
1)(
1
11
11
Dz
z
T
T
zT
TK
f
d
i
11
1
21
11
0
11
1
1
1
11
1211
11
11
zDz
zDT
T
T
Tz
T
T
T
TDz
T
TK
zD
z
T
T
z
z
T
TK
if
d
if
d
f
d
f
d
i
)(
)(
1 22
11
22
110
zE
zU
zpzp
zqzqq
11
1
21
11
0
11
1
1
1
11
1211
11
11
zDz
zDT
T
T
Tz
T
T
T
TDz
T
TK
zD
z
T
T
z
z
T
TK
if
d
if
d
f
d
f
d
i
)(
)(
1 22
11
22
110
zE
zU
zpzp
zqzqq
11
1
21
11
0
11
1
1
1
11
1211
11
11
zDz
zDT
T
T
Tz
T
T
T
TDz
T
TK
zD
z
T
T
z
z
T
TK
if
d
if
d
f
d
f
d
i
11
1
21
11
0
11
1211
zDz
zDT
T
T
Tz
T
T
T
TDz
T
TK
zGif
d
if
d
f
d
R )()(
)(
1 22
11
22
110
zE
zU
zpzp
zqzqq
if
d
f
d
T
T
T
TDKq
T
TKq
21
1
11
0
fT
T
i
d
f
d
eD
DT
T
T
TKq
1
12 1)(
12
11 1
Dp
Dp
)2()1()()2()1()( 21021 keqkeqkeqkupkupku
Druhý spôsob výpočtu parametrov PID neideálneho diskrétneho PID regulátora môžeme určiť aproximatívnym spôsobom podľa Tustinového vzťahu.
)()(
112
zT
zs
sT
sT
sTKsG
f
d
iR 1
11)(
Dosadením za
)(
)(
1)(
22
11
22
11
zE
zU
zpzp
zqzqqzG o
R
1
11110 21
)]21(5.0(21[
p
ppdp
c
ccccKq
1
1111 21
)](4[
p
dpp
c
cccKq
1
11112 21
15022
p
idip
c
ccccKq
].)([
T
Tc f
p 1
ii T
Tc 1
T
Tc d
d 1)21(4 111 pp ccp
1
12 21
12
p
p
c
cp
)(
)(
1)(
22
11
22
11
zE
zU
zpzp
zqzqqzG o
R
)2()1()()2()1()( 21021 keqkeqkeqkupkupku
Riadiaci zásah PID(NI) regulátora
Typ Koeficienty PSD regulátora 2
21
10
22
110)(
zpzpp
zqzqqzGR
0q 1q 2q
0p 1p 2p
T
TK d1
i
d
T
T
T
TK 21
T
TK d PSD
(1) 1 1 1
T
T
T
TK d
i21
i
d
T
T
T
TK
2
21
T
TK d PSD
(2) 1 1 0
01
T
TK d
)/exp(
21
1
01
i
i
d
TTD
T
T
T
TDK
10
1 DT
T
T
TK
i
d
PSD
(3)
1 11 D 1D
1
1111
21
)21(2/)(21
p
pidp
c
ccccK
1
111
21
)(4
p
dpi
c
cccK
1
1111
21
12/2)2(
p
idip
c
ccccK
PSD (4)
TTcp /
1
01
ii
pp
TTc
cc
/
)21/(4
1
11
TTc
cc
dd
pp
/
)21/()12(
1
11
Doplnok – kvalita regulácie frekvenčná oblasť
Closed LoopClosed Loop
Open loop : system H(s) ; system with controller F(s).H(s)
Closed loop : HCL(s) = F(s).H(s) / [ 1 + F(s). H(s) G(s) ]
G(s)
SystemReg.
Transducer
F(s) H(s)
Yref Y
DisturbancesPerturbations
+-
Feed P
Steady state error : F(s) must contains the internal model of the reference (the transfer function that generates Yref(t) from the Dirac impulse ; e.g. step = (1/s) * Dirac ; ramp = (1/s2) * Dirac,...
Closed Loop : Perturbation rejection
Closed Loop : Perturbation rejection
Perturbation-output sensitivity function :
Syp(s) = Y(s) / P(s) = 1 / [1 + F(s).H(s).G(s)]
Perturbation-output sensitivity function :
Syp(s) = Y(s) / P(s) = 1 / [1 + F(s).H(s).G(s)]
Perturbation rejection :
Syp(0) = 0 to get a perfect rejection of the perturbation insteady state (controller must contain the classes ofperturbation)
and|Syp() | < G ; [ Example : |Syp() | < 2 (6dB) ;
If the energy of the perturbation is concentrated in a given frequency band, the |Syp() | should be limited in this band.
Perturbation rejection :
Syp(0) = 0 to get a perfect rejection of the perturbation insteady state (controller must contain the classes ofperturbation)
and|Syp() | < G ; [ Example : |Syp() | < 2 (6dB) ;
If the energy of the perturbation is concentrated in a given frequency band, the |Syp() | should be limited in this band.
Controller DesignController Design
In order to design and tune a controller :1) To specify the desired control loop performances
Regulation and tracking : rise time and max overshoot or bandwidth and resonance
2) To choose a suitable controller design method
3) To know the dynamic model of the plant to be controlled => control model
In order to design and tune a controller :1) To specify the desired control loop performances
Regulation and tracking : rise time and max overshoot or bandwidth and resonance
2) To choose a suitable controller design method
3) To know the dynamic model of the plant to be controlled => control model
Control model :- Non parametric models : e.g. frequency response, step response,…- Parametric models : e.g. transfer function, differential eq., state eq.
To get the model :- knowledge type model (based on the physic laws) ; used for plant simulation and design- identification models (from experimental data)
Continuous - time Models : Frequency Domain
Continuous - time Models : Frequency Domain
Linear system
Systemu(t) = ejt
u(t) = est
y(t) = G(j) . ejt
y(t) = G(s) . est
or f
20.log(|G|)Gain
Phase
or fdeg
Bode Diagram
x
xx o
Root locus : poles and zeroes
Nyquist, Nichols,...
Note:State EquationDifferential Eq.Transfer function Observability, Controlablity
Continuous - time Models : Time responsesContinuous - time Models : Time responses
Response of a dynamic system for a step input
t
Final Value(Steady state)
Maximum overshoot (M)
tR
ts
0.9 FV
tR : Rise Time ; define as the time needed to attain 90% of the final value ; or as the time needed for the output to pass from 10 to 90% of the final valuetS : Settling Time ; define as the time needed for the output to reach and remain within a tolerance zone around the final value (±10%, ± 5%, ±1%,…)FV : Final Value ; a fixed output value obtained for t M : Maximum Overshoot ; expressed as a percentage of the final value
Example : 1st OrderH(s) = G/(1+sT)FV = GtR = 2.2 TtS = 2.2 T (for 10% FV)tS = 3 T ( for 5% FV)M = 0
Continuous - time Models : Frequency responses
Continuous - time Models : Frequency responses
fB : Bandwidth ; the frequency from which the zero-frequency (steady state) gain G(0) is attenuated by more than 3 dB ; G(wB) = G(0) - 3dB or G(wB) = 0.707 . G(0)
fC : Cut-off frequency ; the frequency from which the attenuation is more than N dB ; G(wC) = G(0) - NdB
Q : Resonance factor ; the ratio between the gain corresponding to the maximum of the frequency response curve and the value G(0)
fB : Bandwidth ; the frequency from which the zero-frequency (steady state) gain G(0) is attenuated by more than 3 dB ; G(wB) = G(0) - 3dB or G(wB) = 0.707 . G(0)
fC : Cut-off frequency ; the frequency from which the attenuation is more than N dB ; G(wC) = G(0) - NdB
Q : Resonance factor ; the ratio between the gain corresponding to the maximum of the frequency response curve and the value G(0)
20 log[ H(jw) ]
w= 2 p f
Resonance
- 3 dB
wB
(fB)wC
(fC)
(p - m) x 20 dB/dec
Nb of polesNb of zeroes
N dB
Reciprocity : Time / FrequencyReciprocity : Time / Frequency
- 3 dB
fB = 1/(2 T)
fB 0.35 / tR
0.1
0.9
tR = 2.2 T
time
frequency
Closed Loop : MarginsClosed Loop : Margins
Im H(j)
-1
Re H(j)
1/G
Gain Margin : DG = 1 / |H(jw180)| for F(w180) = -180Typical : G 2 (6dB) [min: 1.6 (4dB)]
Gain Margin : DG = 1 / |H(jw180)| for F(w180) = -180Typical : G 2 (6dB) [min: 1.6 (4dB)]
Phase Margin : = 180 - (cr) for |(jcr) = 1cr : crossing pulsation
Typical : 30 60
Phase Margin : = 180 - (cr) for |(jcr) = 1cr : crossing pulsation
Typical : 30 60
cr
Delay Margin : = 0.cr
additional delay that could be tolerate by the open loop system without instability for the closed loop system
Delay Margin : = 0.cr
additional delay that could be tolerate by the open loop system without instability for the closed loop system
Module Margin : M = |1 + H(j)|min = |S-1
yp(j)|min
Measure of perturbation rejection and robustness of non linearity and time variable parametersTypical : M 0.5 (-6dB) [min: 0.4 (-8dB)]
Module Margin : M = |1 + H(j)|min = |S-1
yp(j)|min
Measure of perturbation rejection and robustness of non linearity and time variable parametersTypical : M 0.5 (-6dB) [min: 0.4 (-8dB)]
