António Pascoal 2011 Instituto Superior Tecnico [email protected] Loop Shaping (SISO case) 0db ) ( 1 rads ) ( ) ( j K j G n 10 20 log 2 n 1 n d r r 10 20 log d 10 20 log
António Pascoal 2011 Instituto Superior Tecnico [email protected] Loop Shaping (SISO case) 0db.
Apr 01, 2015
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António Pascoal
2011
Instituto Superior Tecnico
Loop Shaping (SISO case)
0db)( 1rads
)()( jKjG
n1020 log
2n1
n
dr
r1020 log
d1020 log
G(s)_
K(s)
dy
n
r e u
Controller Plant
r – reference signal ( to be tracked by the output y)
d – external perturbation (referred to the output)
n – sensor noise
e – error
y – output signal
u – actuation signal
Feedback Control structure
Key objectives
i) K(s) stabilizes G(s)
ii) The output y follows the reference signals r.
iii) The system reduces the effect of external disturbance d and noise n on the output y.
v) The system meets stability and performance requirements in the face of plant parameter uncertainty and unmodeled dynamics (robust stability and robust performance).
Design the controller K(s) such that
iv) The actuation signal u is not driven beyond limits imposed by saturation values and bandwith of the plant´s actuator.
Control objectives
External disturbance attenuation (reducing the impact of d on y)
G(s)_
K(s)
dy
Linear system superposition principle
e
)()()()(
)(sS
sKsGsD
sY
1
1
)()()()(
)()()()()(
sYsKsGsD
sEsKsGsDsY
yeGKedy ;
)())()()(( sDsKsGsY 1
)(sSD(s) Y(s)
S(s) – Sensitivity Function
Disturbance attenuation
)()()()(
)(sS
sKsGsD
sY
1
1
S(s) – possible Bode diagram
0db
)( jS
)( 1radsd
-x db
xdbjS )(
below the ‘barrier’ of –x db for
d ,0
Attenuation of at least
–x db
Attenuation of sinusoidal disturbances
)( 1radsd
)(sS
0
d y
d – sinusoidal signals
Performancespecs on disturbance attenuation djS ,;)( 0
Upper limit on
Performance bandwith
Upper limit –x db and performance bandwith
d,0 are problem dependent
What happens when d is not a sinusoid?
d- modeled as a stationary stochastic process with spectral density
)(d
djSd
0
2)()(Energy y2
y - stationary stochastic process with spectral density
2)()()( jSdy
Disturbance attenuation
d
)(d
)( 1rads
If dd ,)( 0
spectral contents of d concentrated in the
frequency band d,0
Basic technique to reduce the energy of y:
ddjS ,,)( 0
reduce djS ,,)( 0
Its is up to the system designer to selectthe level of attenuation d
Disturbance attenuation
Disturbance attenuation: constraints on the Loop Gain GK
ddjS ,,)( 01
djKjG
)()(1
1
11
1 d
jKjG
)()(
11
d
If
)()()()( jKjGjKjG 1
1d
Disturbace attenuation
11
)()( dd
jKjG
d ,0
11
)()( dd
jKjG
d ,0
0db
)( 1radsd
djKjG )()(d1020 log
Lower bound (“barrier”) on
)()( jKjG
shaped by proper choice of controller K(s)
Disturbance attenuation: constraints on the Loop Gain GK
Reference following
G(s)_
K(s)r ye
)()()()()( sEsKsGsRsE
)()()()(
)(sS
sKsGsR
sE
1
1
GKeyyre ;
GKere
)()())()(( sRsEsKsG 1
djSd
2
0
)()(Energia e2
e - stationary stochastic process with spectral density
)()()( jSre
)(r
r- modeled as a stationary stochastic process with spectral density
Reference following
r
)(r
)( 1rads
If rr ;)( 0
spectral contents of d concentrated in the
frequency band r,0
Reference following
Technique to reduce the energy of the tracking error e
rrjS ,,)( 0
Reduce
rjS ,,)( 0
rIts is up to the system designer to selectthe level of error reduction
rrjS ,,)( 01
0db
)( jS
)( 1radsr
rjS 1020 log)(
below the “barrier” of db for
r ,0
Geometric constraint
r1020 log
r1020 log
db
Reference following
rrjS ,,)( 01
rjKjG
)()(1
1
11
1 r
jKjG
)()(
11
r
If
)()()()( jKjGjKjG 1
1r
reference following:
11
)()( rr
jKjG
r ,0
Reference following: constraints on the Loop Gain GK
11
rr
jKjG
)()(
r ,0
0db
)( 1radsr
rjKjG )()(
r1020 log
Reference following: constraints on the Loop Gain GK
Lower bound (“barrier”) on
)()( jKjG
shaped by proper choice of controller K(s)
Noise reduction
G(s)_
K(s)y
n
e u
)()()(
)()(
)(
)(sT
sKsG
sKsG
sN
sY
1
)(, yneGKey
GKyGKny
)()()(
)())()((
sNsKsG
sYsKsG
1
y - stationary stochastic process with spectral density
2)()()( jTny
)(n
n- modeled as a stationary stochastic process with spectral density
Energy y2 djTn
0
2)()(
Noise reduction
Noise reduction (high frequency noise)
1n
)(r
)( 1rads
If 21,;0)( nnr
spectral contents of n concentrated in the frequency band 21, nn
2n
Technique to reduce the energy of y caused by the noise n:
21,,1)( nnnjT
Reduce
21,,)( nnjT
nIts is up to the system designer to selectthe level of error reduction
0db
)( 1radsnjT )(
n1020 log
upper bound (“barrier”) on
)( jT
shaped by proper choice of controller K(s)
21,,1)( nnnjT
2n1
n
Noise reduction (high frequency noise)
21,,1)( nnnjT
rjKjG
jKjG
)()(
)()(
1
If
)()()()(
)()(
jKjGjKjG
jKjG
1
1r
noise reduction
1)()( njKjG
21, nn
Noise reduction: constraints on the Loop Gain GK
1)()( njKjG
21, nn
0db
)( 1radsnjKjG )()(
n1020 log
2n1
n
Upper bound (“barrier”) on loop gain
shaped by proper choice of K(s)
)()( jKjG
Noise reduction: constraints on the Loop Gain GK
Actuator limits
G(s)_
K(s)r ye u
)()(1
1
)(
)(
sKsGsR
sE
)()(
)(
)(
)()(
)(
)(
sKsG
sK
sR
sEsK
sR
sU
1
)(
1
)()(1
)()(
)(
)(
sGsKsG
sKsG
sR
sU
Suppose1pjG )(
p (plant gain rolls off at high frequencies)
Actuator limits
Suppose
1)()( jKjG
qp ,
111
1
1
pjG
jGjKjG
jKjG
jR
jU
)(
)()()(
)()(
)(
)(
Actuation signals too high unless the loop gain starts rolling off at frequencies below
p
Golden rule: never try to make the closed loop bandwidth extend well above the region where there the plant gain starts to roll off below 0db.
1ljKjG )()(
1 kk p ;
0db
)( 1radsljKjG )()(
l1020 log
pk
Upper bound (“barrier”) on loop gain
shaped by proper choice of K(s)
)()( jKjG
Actuator limits
Technique for limiting actuation signals
kl ,Its is up to the system designer to select the parameters
Putting it all together
Loops Gain restrictions
0db
)( 1rads
)()( jKjG
n1020 log
2n1
n
dr
r1020 log
d1020 log
Low frequency barriersr, d
High frequency barriersn, u
Goal: Shape (by appropriate choice of K(s) the LOOP GAIN G(s)K(s)so that it will meet the barrier constraints while preserving closed loop stability.
pk
Loop Shaping – Design examples
Exemple 1
2
1
s
G(s)
. Plant (system to) be controlled
. Control specifications
G(s)_
K(s)
dy
n
r e u
Controller Plant
Design K(s) so as to stabilize G(s) and meet the following performance specifications:
Specifications
i) Reduce by at least –80db the influence of d on y in the frequency band
11000 radsd .,,
ii) Follow with error less than or equal to -40db the reference signals r in the frequency band
1100 radsr ,,
iii) Attenuate by at least –20db the noise n in the frequency band
13221 10,10, radsnn
iv) Static error in response to a unit parabola reference
020.)( pare
v) Phase Margin 045 M
vi) Gain Margin dbGM 20
Loop Shaping – Design examples
Geometrical constraints; conditions i), ii), iii)
i) 1110080 radsdbjS ,,)(
1110080 radsdbjKjG ,,)()(
ii) 11040 radsdbjS ,,)(
11040 radsdbjKjG ,,)()(
iii 132 101020 radsdbjT ,,)(
132 101020 radsdbjKjG ,,)()(
Loop Shaping – Design examples
)()( jKjG
0db
)( 1rads
db20
310210
10. 1
db40
db80
Loop Gain Constraints
Low frequency barriersr, d
High frequency barriern
Loop Shaping – Design examples
Condition iv)
)()(
)()(
sKsG
sRsE
1
)()()(
sKsGssE
1
123
Let
10 )(~
);(~
)( KsKksK
0202
1
12
2
30
0
.lim
)(lim)(
k
s
ks
s
ssEe
s
spar
100k
Loop Shaping – Design examples
020.)( pare
(possible to achieve, because G(s) has two poles at the origin)
Static error in response to a unit parabola reference
A simple controller candidate:
1001 ksKsKksK ;)(~
);(~
)(
Checking the constraints on Loop Gain
0db
)( 1rads
db20
310210
10. 1
db40
db80
2
100
)()()(
jjKjG
)( 1rads
)()( jKjG0180
Phase of
The constraints are met but …..
00 M
00 M !
10
Loop Shaping – Design examples
It is necessary to introduce some phase lead
045 desMMinimum phase margin (specs):
realM
desM
Additional phase required :
security factorreal phase margin =0 graus
(start by trying security factor = 0).
Additional phase required: 450
Pure “PHASE LEAD” network
110;)(
radszz
zsksK
z z
1,)( kjK
odb
)( 1rads )( 1rads
090045
Phase of )( jK
Loop Shaping – Design examples
Phase lead
110;100;)(
)(~
);(~
)(
radszkz
zssKsKksK
0db
)( 1rads
db20
310210
10. 1
db40
db80 )()( jKjG
)( 1rads
)()( jKjG
0180
Phase of
Loop Gain constraints are met and ….. 045M
New
)(~ jK
0900135
Loop Shaping – Design examples
Checking the constraints on Loop Gain
Phase lead
NOTICE: phase lead “opens-up” the loop gain! The newloop gain barely avoids violating the noise-barrier!
Final check on stability and Gain Margin
10
)10(100)()(
~)()(
2
s
ssGsKksKsG
Use Nyquist’s Theorem
Nyquist contour
xx
Number of open loop polesinside the Nyquist contour
P=0
x-1
Number of encirclementsaround –1
N=0
Stable!Gain Margin equals infinity!
Loop Shaping – Design examples
Phase lead
Example 2
1
1
s
G(s)
. Plant (simple torpedo model)
. Control objectives
G(s)_
K(s)
dy
n
r e u
Controller Plant
Design K(s) so as to stabilize G(s) and meet the following performance specifications:
Loop Shaping – Design examples
Specifications
ii) Attenuate by at least –40db the signals d in the frequency band
1210,0,0 radsd
iii) Follow with error smaller than or equal to -100db the signals r in the frequency band
1310,0,0 radsr
iv) Attenuate by at least –40db the noise n in the frequency band
13221 10,10, radsnn
v) Phase Margin 045 M
vi) Gain Margin dbGM 20
i) Static position error = 0.
Loop Shaping – Design examples
Geometrical constraints; conditions i), ii), iii)
ii) 1210,0,40)( radsdbjS
1210,0,40)()( radsdbjKjG
iii) 1310,0,100)( radsdbjS
1310,0,100)()( radsdbjKjG
iv) 132 10,10,40)( radsdbjT
132 10,10,40)()( radsdbjKjG
Loop Shaping – Design examples
Condition i)
Static position error
0)( escalãoe
1)0(~
);(~
)( KsKs
ksK
(1 pure integrator in the direct path)
A simple controller candidate:
0;)( ks
ksK
Loop Gain
0;1
1)()( k
ss
ksKsG
Loop Shaping – Design examples
Checking the constraints on Loop Gain
0db
db40
310
100;)()( kjKjG
)()( jKjG
0180
Phase of
The constraints on the loop gain are met, but … 00 M !
)( 1rads210310
1 10210 110
+40db
+80db
)( 1rads210310
1 10210 110
0180
090
Loop Shaping – Design examples
Notice! Now it is not possible to use a phase-lead networkbecause the open-loop plot would “open-up” and violate the noise barrier!
0db
db40
310
)( 1rads210310
1 10210 110
+40db
+80db
)( 1rads210310
1 10210 110
0180
090
)(~ jK
)()( jKjGNew
)()( jKjGPhase of
use 1113 10;10 radszradsp
045M
Loop Shaping – Design examples
The high frequency barrier does not allow for the use of a lead network – use a lag network (“gain-loss” network)!
Force a new 0dB crossing point such that if the phase were not changed, the gain margin would meet the specifications (must loose -40dB at 1.0 rads-1)!
0db
db40
310
)( 1rads210310
1 10210 110
+40db
+80db
)( 1rads210310
1 10210 110
0180
090
)(~ jK
)()( jKjGPhase of
045M
Loop Shaping – Design examples
NOTICE: the LAG network must introduce a loss of -40dB at 1 rads-1. But .. the zero is introduced at -10-1rads-1, not -1rads-1!WHY?So that the extra phase introduced by the lag network will not “interfere too much” around 1 rads-1.
Final check on stability and Gain Margin
1
3
3
1
10
10
10
10
)1(
100)()(
~)()(
s
s
sssGsK
s
ksKsG
Nyquist Theorem
Nyquist Contour
x
Number of open loop polesinside the Nyquist contour
P=0
x-1
Number of encirclementsaround -1
N=0
Stable!Gain Margin equals infinity!
xx
-p-z-1
Loop Shaping – Design examples
Example 3 (Lunar Excursion Module – LEM)
Loop Shaping – Design examples
Example 3 (Lunar Excursion Module – LEM)
G(s)
. Plant (vehicle controlled in attitude by gas jets and actuator; J=100 Nm/(rads-2))
. Control objectives (attitude control)
G(s)_
K(s)y
n
r e u
Controller Plant
Design K(s) so as to stabilize G(s) and meet the following performance specifications:
Loop Shaping – Design examples
TorqueInput Voltage
Attitude
Specifications
ii) Follow with error smaller than or equal to -40db the signals r in the frequency band
iii) Attenuate by at least –40db the noise n in the frequency band
13221 10,10, radsnn
iv) Gain Margin dbGM 20
i) Static position error = 0.
Loop Shaping – Design examples
v) Phase Margin 045 M
v) Robustness of stability with respect to a total delayin the control channel of up to 0.5 sec
Geometrical constraints; conditions ii), iii)
ii)
iii) 132 10,10,40)( radsdbjT
132 10,10,40)()( radsdbjKjG
Loop Shaping – Design examples
Condition i)
Static position error
0)( escalãoe
(there are already two integrators in the direct path)
A simple controller candidate:
Loop Gain
Loop Shaping – Design examples
Candidate Loop Gain
Checking the constraints on Loop Gain (with )
0db
)( 1rads
310210
10. 1
db40
0180)()( jKjGFase de
10
Loop Shaping – Design examples
-40db
Checking the stability of the closed-loop system
Use Nyquist’s Theorem
Nyquist contour
xx
Number of open loop polesinside the Nyquist contour
P=0
x-1
Number of encirclementsaround –1
N=+2
Unstable!
Loop Shaping – Design examples
x
Possible strategy: introduce some phase lead
Pure Phase Lead network
z z
0 dB
)( 1rads )( 1rads
090045
Phase of
Loop Shaping – Design examples
Phase lead
What value of z should be adopted?
Try z =1 rads-1; that is, frequency at which
New candidate Loop Gain
Checking the constraints on the Loop Gain
0db
)( 1rads
310210
10. 1
db40
0180
10
Loop Shaping – Design examples
-40db
“old” loop gain
“old” loop gain
“new” loop gain
“new” loop gain
Final check on stability and Gain Margin
Use Nyquist’s Theorem
Nyquist contour
xx
Number of open loop polesinside the Nyquist contour
P=0
x-1
Loop Shaping – Design examples
Phase lead
Number of encirclementsaround –1
N=0
Stable!Gain Margin equals infinity!
Phase Margin
New candidate Loop Gain
Checking the constraints on the Loop Gain
0db
)( 1rads
310210
10. 1
db40
0180
10
Loop Shaping – Design examples
-40db
“old” loop gain
“old” loop gain
“new” loop gain
“new” loop gain
Robustness of stability with respect to a delayin the control channel
Loop Shaping – Design examples
Transfer function of a pure delay exp (-s)
Only change in the Bode diagram!
0180
Danger: if the gain margin of 45º is completely lost!
Maximum allowed is app. 0.75s >0.5 sec!
G(s)_
K(s)
dy
n
r e u
Controller Plant
Intrinsic Limitations on Achievable Performance
Simple algebraic limitation
Find (if at all possible) a controller K(s) that willstabilize G(s) and such that
010.)(
)(
)(
jSjR
jE
050.)()(
)(
jTjN
jY
(reference following spec)
(noise attenuation spec)
Notice:
111
1
)()(
)()(
)()()()(
sKsG
sKsG
sKsGsTsS
G(s)_
K(s)
dy
n
r e u
Controller Plant
Intrinsic Limitations on Achievable Performance
1 )()( sTsS
)()( sTsS 1 )()( jTjS 1
If 050.)( jT
then!..
.)(
010950
0501
jS
There is no controller that will meet the specs!
(cannot expect good performance over a frequency bandwhere there is significant sensor noise: buy a better sensor, or relax the specs)
G(s)_
K(s)yr e u
Controller Plant
Intrinsic Limitations on Achievable Performance
Analytic Limitation
Find (if at all possible) a controller K(s) that willstabilize G(s) and such that the sensitivity function S(s) will “ acquire a desired target shape”.
)( jS
0db
)( 1rads
High performance
-xdb
+ydb
Intrinsic Limitations on Achievable Performance
Analytic Limitation
)( jS
0db
)( 1rads
High performance
-40db
+20db“Barrier” approximation )( j
3
1010
zs
zss)(
)()( jjS
Objective: design a stabilizing controller K(s) such that
z 10z
)(s stable with a stable inverse
db400100 .)(
db2010 )(
)()( sWs 1
)(sW is analytic in the right half complex plane (RHP)
Intrinsic Limitations on Achievable Performance
11 )()( jjS
If K(s) stabilizes G(s), then S(s) is analytic in the RHP
)()( jjS
CssWzS ;)()( 1
(maximum modulus principle)
Suppose the plant G(s) has an “unstable” zero1
0 1 radsz
1)()(1
1)(
000
zKzGzS
(no unstable pole-zero cancellations)
100 )()( zWzS
10 )(zW
condition to be satisfied!
Intrinsic Limitations on Achievable Performance
Case 1. Z=2rad/s (plant zero “inside” the high performance bandwidth region)
3
1010
zs
zss)(
!)( 12
20
10
11
3
s
sW
Impossible to meet the specifications!
Case 1. Z=0.05rad/s (plant zero “outside” the high performance bandwidth region)
3
1010
zs
zss)(
1050
50
10
11
3
.
.)(
s
sW
The specs are met.
Intrinsic Limitations on Achievable Performance
Analytic Limitations (extension)
Case 1. Z=2rad/s (plant zero “inside” the high performance bandwidth region)
Impossible to meet the specifications!
Possible strategies:
i) Reduce the performance bandwith and / or relax the level of performance
0db 0z
plant zero
original spec
Intrinsic Limitations on Achievable Performance
ii) Allow for increased gain over the complementary range of frequencies
0db 0z
plant zero
original spec
Waterbed effect
0db 0z
plant zero
original spec
Intrinsic Limitations on Achievable Performance
Open loop (unstable) zeros and poles placefundamental restrictions on what can be donewith feedback! (not “textbook” examples)
Freudenberg and Looze, “Right half plane polesand zeros and design tradeoffs in feedback systems,”IEEE Trans. Automatic Control, Vol. 39(6), pp. 55-565, 1985.
Before designing a controller, take a step back .. examine the system physics.
Open loop unstable system
Must maintain a given closed loopbandwith (dangerous!)
António Pascoal
2011
Loop Shaping (SISO case)
0db)( 1rads
)()( jKjG
n1020 log
2n1
n
dr
r1020 log
d1020 log
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