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Transcript of Multivariable Control Systems
Multivariable Control Multivariable Control SystemsSystems
Ali Karimpour
Assistant Professor
Ferdowsi University of Mashhad
2
Ali Karimpour Jan 2010
Chapter8Chapter 8
Multivariable Control System Design: LQG Method
Topics to be covered include:
• LQG Control
• Robustness Properties
• Loop transfer recovery (LTR) procedures
- Recovering robustness at the plant output
- Recovering robustness at the plant input
• Some practical consideration
- Shaping the principal gains (singular values)
3
Ali Karimpour Jan 2010
Chapter8
LQG Control
• LQG Control
• Robustness Properties
• Loop transfer recovery (LTR) procedures
- Recovering robustness at the plant output
- Recovering robustness at the plant input
• Some practical consideration
- Shaping the principal gains (singular values)
4
Ali Karimpour Jan 2010
Chapter8
LQG Control
In traditional LQG Control, it is assumed that the plant dynamics are linear and known
and that the measurement noise and disturbance signals (process noise) are stochastic
with known statistical properties.
vCxy
wBuAxx
That is, wd and wn are white noise processes with covariances
0
0
VvvE
WwwET
T
00 and TT vwEwvE
The problem is then to devise a feedback-control law which minimizes the ‘cost’
T TT
TdtRuuQzz
TEJ
0
1lim 0 and 0 , where TT RRQQMxz
5
Ali Karimpour Jan 2010
Chapter8
LQG Control
The solution to the LQG problem is prescribed by the separation theorem, which
states that the optimal result is achieved by adopting the following procedure.
• First, obtain an optimal estimate of the state x
Optimal in the sense that xxxxE T ˆˆ is minimized
• Then use this estimate as if it were an exact measurement of the state to solve
the deterministic linear quadratic control problem.
6
Ali Karimpour Jan 2010
Chapter8
LQG Control: Optimal state feedback
0
dtRuuQzzJ TTr
The optimal solution for any initial state is
)()( txKtu r
where
XBRK Tr
1
Where X=XT ≥ 0 is the unique positive-semidefinite solution of the
algebraic Riccati equation
01 QMMXBXBRXAXA TTT
0 and 0 , where TT RRQQMxz
7
Ali Karimpour Jan 2010
Chapter8
LQG Control: Kalman filter
The Kalman filter has the structure of an ordinary
state-estimator or observer, as
)ˆ(ˆˆ xcyKBuxAx f
:is ˆˆ minimizes which of choice optimal The xxxxEK Tf
1 VYCK Tf
Where Y=YT ≥ 0 is the unique positive-semidefinite solution of the
algebraic Riccati equation
01 TTT WCYVYCAYYA
8
Ali Karimpour Jan 2010
Chapter8
LQG Control: Combined optimal state estimation and optimal state feedback
KLQG(s)
00)(
1
111
XBR
VYCCVYCXBBRA
K
KCKBKAsK
T
TTT
r
ffrLQG
Exercise 1: Proof the relation of KLQG(s) according to above figure.
9
Ali Karimpour Jan 2010
Chapter8
Robustness Properties
• LQG Control
• Robustness Properties
• Loop transfer recovery (LTR) procedures
- Recovering robustness at the plant output
- Recovering robustness at the plant input
• Some practical consideration
- Shaping the principal gains (singular values)
10
Ali Karimpour Jan 2010
Chapter8
Robustness Properties
For an LQR-controlled system (i.e. assuming all the states are available and no
stochastic inputs) it is well known (Kalman, 1994; Safonov and Athans, 1997) that,
if the weight R is chosen to be diagonal, the sensitivity function
11 BAsIKIS rsatisfies ,1)( jS
-1
From this it can be shown that the system will
have a gain margin equal to infinity, a gain
reduction margin (lower gain margin) equal
to 0.5 and a (minimum) phase margin of 60˚
in each plant input control channel.
Nyquist plot in MIMO case
1-at center circleunit
theoutside lie will So 1 BAsIKr
-1
11
Ali Karimpour Jan 2010
Chapter8
Robustness Properties
mikand ii ,...,2,1,5.00
miandk ii ,...,2,1,601
12
Ali Karimpour Jan 2010
Chapter8
Robustness Properties
Example 8-1: LQR design of a first order process.
assG
1)(
Consider a first order process
xy
uaxx
For a non-zero initial state the cost function to be minimized is
dtRuxJ r
0
22
The algebraic Riccati equation becomes
0201 21 RaRXXXXRXaaX
RaRaRX 2RaaRXK r /1/ 2
xRauaxx /12
13
Ali Karimpour Jan 2010
Chapter8
Robustness Properties
Example 8-1: LQR design of a first order process.
assG
1)(
Consider a first order process
xy
uaxx
xRaaxKu r /12 xRauaxx /12
RLet ap :is pole loop Closed
0a if 0
0a if 2axaaxKu r
0Let R - ....... :is pole loop Closed ap
14
Ali Karimpour Jan 2010
Chapter8
Robustness Properties
So, an LQR-controlled system has good stability margins at the plant inputs,
1-at center circleunit
theoutside lie will So 1 BAsIKr
Arguments dual to those employed for the LQR-controlled system can then be used to
show that, if the power spectral density matrix V is chosen to be diagonal, then at the
input to the Kalman gain matrix Kf there will be an infinite gain margin, a gain
reduction margin of 0.5 and a minimum-phase margin of 60˚.
1-at center circleunit
theoutside lie will So 1fKAsIc
15
Ali Karimpour Jan 2010
Chapter8
Robustness Properties
So, an LQR-controlled system has good stability margins at the plant inputs,
And Kalman filter has good stability margins at the inputs to Kf
This was brought starkly to the attention of the control community by Doyle (1978 )
(in a paper entitled “Guaranteed Margins for LQR Regulators” with a very compact
abstract which simply states “There are none”).
For an LQG-controlled system with a combined Kalman filter and LQR control law are there any guaranteed stability margins?
Unfortunately there are no guaranteed stability margins.
Doyle showed, by an example, that there exist LQG combinations with
arbitrarily small gain margins.
16
Ali Karimpour Jan 2010
Chapter8
Robustness Properties
Why there are no guaranteed stability margins in LQG controller.
BsCKCKBKsKsGsKsL ffrrLQG )()()()()(11
1
)()()(2 sKsGsL LQG
BsKsL r )()(3
(Regulator transfer function)
fKsCsL )()(4
(Kalman Filter transfer function)
guaranteed stability margins
guaranteed stability margins
The most important loop but no guaranteed stability margins
L3
L4
L2
17
Ali Karimpour Jan 2010
Chapter8
Loop Transfer Recovery
• LQG Control
• Robustness Properties
• Loop transfer recovery (LTR) procedures
- Recovering robustness at the plant output
- Recovering robustness at the plant input
• Some practical consideration
- Shaping the principal gains (singular values)
18
Ali Karimpour Jan 2010
Chapter8
Loop transfer recovery (LTR) procedures
Assume that the plant model G(s) is minimum-phase and that it has at least as many inputs as outputs.
The LQG loop transfer function
fKsCsL )()( 4 Guaranteed stability margins
If Kr in the LQR problem is designed to be large using the sensitivity recovery
procedure of Kwakernaak (1969).
The LQG loop transfer function
BsKsL r )()( 3 Guaranteed stability margins
If Kf in the Kalman filter to be large using the robustness recovery procedure of
Doyle and Stein (1979).
)()()(1 sGsKsL LQG
)()()(2 sKsGsL LQG
19
Ali Karimpour Jan 2010
Chapter8
Loop transfer recovery (LTR) procedures
Assume that the plant model G(s) is minimum-phase and that it has at least as many inputs as outputs.
The LQG loop transfer function
BsKsLsGsKsL rLQG )()()()()( 31
The LQG loop transfer function
fLQG KsCsLsKsGsL )()()()()( 42
L2
Recovering robustness at the plant output
Recovering robustness at the plant input
L1
20
Ali Karimpour Jan 2010
Chapter8
Loop transfer recovery (LTR) procedures
Assume that the plant model G(s) is minimum-phase and that it has at least as many inputs as outputs.
BsKsLsGsKsL rLQG )()()()()( 31
L1
Recovering robustness at the plant input
Step I: First, design the linear quadratic problem whose transfer function KrΦ(s)Bis desirable.
This is done, in an iterative fashion, by manipulate the matrices Q and R, emphasis of The design is on aspects such as gains, possibly ‘balancing’ the principal gains, and
adjusting the low frequency behavior.
Step II: When the singular values of KrΦ(s)B are thought to be satisfactory, LTR isachieved by designing Kf in the Kalman filter by setting Г=B, W=I and V= ρI ,where
ρ is a scalar. As ρ tends to zero
BsKsLsGsKsL rLQG )()()()()( 31
21
Ali Karimpour Jan 2010
Chapter8
Loop transfer recovery (LTR) procedures
Assume that the plant model G(s) is minimum-phase and that it has at least as many inputs as outputs.
fLQG KsCsLsKsGsL )()()()()( 42
L2
Recovering robustness at the plant output
Step I: First, we design a Kalman filter whose transfer function CΦ(s)Kf is desirable.
By choosing the power spectral density matrices W and V so that the minimum singular
value of CΦ(s)Kf is large enough at low frequencies for good performance and its
maximum singular value is small enough at high frequencies for robust stability.
Step II: When the singular values of CΦ(s)Kf are thought to be satisfactory, loop
transfer recovery is achieved by designing Kr in an LQR problem with M=C, Q=I
and R= ρI, where ρ is a scalar. As ρ tends to zero
fLQG KsCsLsKsGsL )()()()()( 42
22
Ali Karimpour Jan 2010
Chapter8
Loop transfer recovery (LTR) procedures
• If RHP zeros exist in the plant the procedure may still work, particularly if these zeros lie beyond the operation bandwidth of the system as finally designed.
• Since it relies on the ‘cancellation’ of some of the plant dynamics by the filter Dynamics)
LTR procedure guaranteed to work only with minimum-phase plants.
L1L2
23
Ali Karimpour Jan 2010
Chapter8
Loop transfer recovery (LTR) procedures
1
1
)()(
)(Let
rBKAsIs
AsIs
The LQG loop transfer function at the plant input is:
Proof: Recovering robustness at the plant input
BCKCKKsGsKsL ffrLQG 111 )()()(
By matrix-inversion lemma we have
)()()()( 1111 IBCKCIKKBCKCKKsGsKsL ffrffrLQG
Exercise: Derive equation I .
Г=B, W=I and V= ρI ,As ρ tends to zero
Now the algebraic Riccati equation is:
01 TTT WCYVYCAYYA qqWW 0Let ?
?L1
BAsICKCKBKAsIKsGsKsL ffrrLQG11
1 )()()(
24
Ali Karimpour Jan 2010
Chapter8
L1
Loop transfer recovery (LTR) proceduresProof: Recovering robustness at the plant input
)()()()( 1111 IBCKCIKKBCKCKKsGsKsL ffrffrLQG
Г=B, W=I and V= ρI ,As ρ tends to zero
Now the algebraic Riccati equation is:
01 TTT WCYVYCAYYA qqWW 0Let
001
TTTT
q
W
q
CYVYC
q
AY
q
YA
0lim q
Yq
It can be shown (Kwakernaak and Sivan, 1973) that, if• C(sI-A)-1ГW1/2 has no RHP zero• and if it has at least as many outputs as rank(Σ), then
TT qCYVYC 1
2/12/1lim
VYCq TT
q
1 VYCK Tf
qasVqK f2/12/12/1
2/12/1 VV
25
Ali Karimpour Jan 2010
Chapter8
Loop transfer recovery (LTR) proceduresProof: Recovering robustness at the plant input
Г=B, W=I and V= ρI ,As ρ tends to zero
001
TTTT
q
W
q
CYVYC
q
AY
q
YA
It can be shown (Kwakernaak and Sivan, 1973) that, if• C(sI-A)-1ГW1/2 has no RHP zero• and if it has at least as many outputs as rank(Σ), then
qasVqK f2/12/12/1
In particular if we choose IB ,
and provided C(sI-A)-1B has no zeros in LHP, then qasBVqK f2/12/1
Substituting this in the LQG loop transfer function at the plant input leads to:
qasBCBVCqIBVKq r 12/12/12/12/1
)()()()( 1111 IBCKCIKKBCKCKKsGsKsL ffrffrLQG
L1
26
Ali Karimpour Jan 2010
Chapter8
Loop transfer recovery (LTR) proceduresProof: Recovering robustness at the plant input
Г=B, W=I and V= ρI ,As ρ tends to zero
qasBCBVCqIBVKqsGsKsL rLQG 12/12/12/12/1
1 )()()(
qasBCBVCBVKsGsKsL rLQG 12/12/1
1 )()()(
qasBCBCBK r 1 1 rBKI
qasBCBBKICBBKIKsGsKsL rrrLQG
1111 )()()(
By push-through rule:
qasBCBKIBCBKIBKsGsKsL rrrLQG
1111 )()()(
Finally we have:
qassLBKsGsKsL rLQG )()()()( 31 qassLsL )()( :So 31
L1
27
Ali Karimpour Jan 2010
Chapter8
Shaping the Principal Gains (Singular Values)
• LQG Control
• Robustness Properties
• Loop transfer recovery (LTR) procedures
- Recovering robustness at the plant output
- Recovering robustness at the plant input
• Some practical consideration
- Shaping the principal gains (singular values)
28
Ali Karimpour Jan 2010
Chapter8
Shaping the principal gains (singular values)
In order to exploit LTR technique, we must to know:
• How to modify W and V in order to bring about desirable changes in C(sI-A)-1Kf
• How to modify Q and R in order to bring about desirable changes in Kr(sI-A)-1B
In order to obtain an intuitive grasp of this, consider the Kalman filter. (let u=0)
vCxy
wBuAxx
)ˆ(ˆˆ xcyKxAx f
This is now looks like a feedback system which is
to:track (in a sense) the
‘reference input’ z, while rejecting the measurement
errors v.
29
Ali Karimpour Jan 2010
Chapter8
Shaping the principal gains (singular values)
To shape the principal-gain plots we can do one of two things:
• Modify the matrices ГWГT and V in a more sophisticated way,
• Modify the plant model by augmenting it with additional dynamics
For example adding integrator in each loop.
We can use ГWГT to increase the smallest principal gain of the sensitivity matrix, or decrease the largest one near some particular frequency.
30
Ali Karimpour Jan 2010
Chapter8
Shaping the principal gains (singular values)
Let Ff as the return difference of Kalman filter,
ff KAsICIsF 1)( and define 1)( AsICsG f
Then we can show that
(I) )()()()( sWGsGVsVFsF Tff
Tff
Exercise I : Derive equation I . (Hint Maciejowski 1989 pp. 227-231)
Modify the matrices W and V
Suppose we choose V=I. Then
)()()()( jWGjGIjFjF Hff
Hff
from which it follows that
)()(1)(2/12/12 IIWjGjF fifi
Exercise II: Derive equation II .
31
Ali Karimpour Jan 2010
Chapter8
2/12/121 )(1)( WjGjF ff 2/12/121 )(1)(
WjGjF ff
So we can reduce by increasing , etc. )(1 jFf 2/1)( WjG f
But the point is not merely to reduce all the singular values of , but to reduce the largest one, relative to smallest.
)(1 jFf
One way is: Suppose we need adjustment at ω1
m
i
Hiii
Hf uyUYWjG
1
2/11)(
Now let
)(2/12/1 Hjj uuIWW
Hjj
m
jij
Hiii
HHjjf uyuyUYuuIWjG
)1()()( 2/11
So
so the jth singular value has been changed by a factor (1+α), while all the other singular values have been left unchanged.
Shaping the principal gains (singular values)Modify the matrices W and V
32
Ali Karimpour Jan 2010
Chapter8
)(2/12/1 Hjj uuIWW H
jj
m
jij
Hiii
HHjjf uyuyUYuuIWjG
)1()()( 2/11
Example: Let
Shaping the principal gains (singular values)Modify the matrices W and V
1600
090
004
100
016
031
WG f
Singular value of GfW1/2 is:
We want to change 7.6 to 3*7.6 so we change W1/2 by
100
086.051.0
051.086.0
400
060.70
0043.13
100
047.088.0
088.047.02/1WG f
)2( 222/12/1 HuuIWW
100
051.086.0
086.051.0
400
043.130
0080.22
100
088.047.0
047.088.02/1WG f
33
Ali Karimpour Jan 2010
Chapter8
Now let)(2/12/1 Hjj uuIWW
Hjj
m
jij
Hiii
HHjjf uyuyUYuuIWjG
)1()()( 2/11
So
so the jth singular value has been changed by a factor (1+α), while all the other singular values have been left unchanged.
The problem with this approach is that uj is usually a complex vector, whereas we wish to keep W1/2 real.
Once again we are faced with the problem of approximating a complex matrix by a real matrix, and as before we can employ the align algorithm.
In this case other algorithms may be more appropriate, however, since we really want to approximate uj rather than align it.
In particular, Re{uj} is sometimes an adequate approximation.
A further possibility is to approximate uj by the output direction of the matrix [Re{uj} Im{uj}] which corresponds to its largest singular value.
Shaping the principal gains (singular values)Modify the matrices W and V
34
Ali Karimpour Jan 2010
Chapter8
Some Practical Consideration
• LQG Control
• Robustness Properties
• Loop transfer recovery (LTR) procedures
- Recovering robustness at the plant output
- Recovering robustness at the plant input
• Some practical consideration
- Shaping the principal gains (singular values)
35
Ali Karimpour Jan 2010
Chapter8Some practical consideration
LTR procedures are limited in their applicability.
Their main limitation is to minimum phase plants.
This is because the recovery procedures work by canceling the plant
zeros, and a cancelled non-minimum phase zero would lead to instability.
The cancellation of lightly damped zeros is also of concern because of
undesirable oscillations at these modes during transients.
A further disadvantage is that the limiting process
)0( For full recovery
Introduces high gains which may cause problems with unmodelled dynamics.
The recovery procedures are not usually taken to their limits.
The result is a somewhat ad-hoc design procedure.
36
Ali Karimpour Jan 2010
Chapter8
Design example
0732.005750.1
6650.104190.4
000
00000.11200.0
000
,
6859.00532.102909.00
0130.18556.000485.00
00000.1000
0705.001712.00538.00
0000.101320.100
BA
000
000
000
,
00100
00010
00001
DC
the model has three inputs, three outputs and five states.
Consider the aircraft model AIRC described in the following state-space model.
DuCxy
BuAxx
Loop transfer recovery (LTR) procedures
37
Ali Karimpour Jan 2010
Chapter8
Design example
• We shall attempt to achieve a bandwidth of about l0 rad/sec for each loop.
THE SPECIFICATION
• Integral action in each loop, little interaction between outputs.
• Good damping of step responses and zero steady-state error in the face of step
demands or disturbances.
Loop transfer recovery (LTR) procedures
PROPERTIES OF THE PLANT
• The time responses of the plant to unit step signals on inputs 1 and 2 exhibit very
severe interaction between outputs.
• The poles of the plant (eigenvalues of A) are jj 1826.00176.0,03.178.0,0
so the system is stable (but not asymptotically stable).
• Thus this plant has no finite zeros, and we do not expect any limitations on
performance to be imposed by zeros.
38
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
The LQG loop transfer function
fLQG KsCsLsKsGsL )()()()()( 42
Recovering robustness at the plant output
The LQG loop transfer function
BsKsLsGsKsL rLQG )()()()()( 31
Recovering robustness at the plant input
Here we use Recovering robustness at the plant output
39
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Kalman filter design
1 VYCK Tf
01 TTT WCYVYCAYYA
),,,,( VWCALQEK f We shall write
and obtain the Kalman-filter gain Kf from
It is generally advisable to start with simple choices of Г, W, V, inspect L4
We need to choose the matrices Г, W, V, which appear in
Then adjust Г, W, V accordingly, and so gradually improve L4
One of the simplest possible choices is Г=B, W=I3 and V=I3.
40
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Kalman filter design
2215.22483.08076.0
6190.10485.04934.0
7807.10642.02507.0
0642.09436.00732.0
2507.00732.09897.0
),,,,(1 VWCBAlqeK f
So try with
The loop transfer function is: 11
4 )( fKAsICsL
41
Ali Karimpour Jan 2010
Chapter8
10-3
10-2
10-1
100
101
102
-60
-40
-20
0
20
40
60
80
100Singular Values
Frequency (rad/sec)
Sin
gula
r V
alue
s (d
B)
Design example
Loop transfer recovery (LTR) procedures
Kalman filter design 1
14 )( fKAsICsL
BW around 1 rad/sec
Constant gain at low frequencies
Decreasing with 20 db/dec at low frequencies
42
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Kalman filter design
The first thing to do is to insert integral action, by augmenting the plant model.
Placing poles of the augmented model at the origin leads to problems in the recovery step later, so in this case we place them at -0.001, which is virtually at the origin, when compared to the required bandwidth 10 rad/sec
3,3333 0001.0 wwww DICIBIA
We could also have chosen Cw more carefully, with the aim of adjusting the low frequency gains. The augmented model is
w
waaaa
w
wa B
DDCC
BB
A
CAA 3,300
00
43
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Kalman filter design
Now we have
w
waaaa
w
wa B
DDCC
BB
A
CAA 3,300
00
6463.00576.07573.0
0624.09964.00225.0
7587.00329.06484.0
4314.22823.02024.1
1018.20062.04556.0
0429.20539.01648.0
0539.03501.10909.0
1648.00909.04129.1
),,,,(2 VWCAlqeK aaaf
44
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Kalman filter design
The loop transfer function is: 24 )( faa KAsICsL
10-3
10-2
10-1
100
101
102
-60
-40
-20
0
20
40
60
80
100
120
140Singular Values
Frequency (rad/sec)
Sin
gula
r V
alue
s (d
B)
Kf2 Kf1
The gain was added around 60 db at low frequency according to integrator
60 db
We want to increase the gain at low frequency.
By tuning W one can manipulate gains.
45
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Kalman filter design
By tuning W one can manipulate gains. But how?
01 TTT WCYVYCAYYA
In a case of diagonal system every diagonal element of W corresponds to a singular value.But in non-diagonal system we must use singular value decomposition.
Haaa UYWAIjC 2/11001.0
Which gives
j
ju
0005.0846.0
0005.0158.0
509.0
3
651.211900
046370
00107942.4 6
46
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Kalman filter design
By tuning W one can manipulate gains. But how?
846.0
158.0
509.0
3u
TuuIW 3332/1
3 Re9
Now let α=9 (α+1=10) so we have (better value for α+1 is 4637/651=7.12)
TWWW 2/13
2/133
so
05.316.324.7
71.048.188.0
89.264.194.3
42.421.161.4
27.354.011.1
51.215.050.0
15.054.137.0
50.037.077.2
),,,,( 33 VWCAlqeK aaaf
47
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Kalman filter design
Singular Values
Frequency (rad/sec)
Sin
gula
r V
alue
s (d
B)
10-3
10-2
10-1
100
101
102
-60
-40
-20
0
20
40
60
80
100
120
140
The loop transfer function is: 34 )( faa KAsICsL
Kf2
Kf1
Kf3 Band width problem?
We need at least 7 rad/sec.
W3 must increase.
48
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Kalman filter design
10-3
10-2
10-1
100
101
102
-40
-20
0
20
40
60
80
100
120
140
160Singular Values
Frequency (rad/sec)
Sin
gula
r V
alue
s (d
B)
Kfx
Kf3
Kf4
W3 must increase.
W3 and 10W3 and 100W3 are considered
),,,,( 33 VWCAlqeK aaaf
),10,,,( 3 VWCAlqeK aaafx
),100,,,( 34 VWCAlqeK aaaf
Maximum singular value of 4,,34 )( xfaa KAsICsL
Which one is ok?
49
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Kalman filter design
),100,,,( 34 VWCAlqeK aaaf
The loop transfer function is: 44 )( faa KAsICsL
10-3
10-2
10-1
100
101
102
-40
-20
0
20
40
60
80
100
120
140
160Singular Values
Frequency (rad/sec)
Sin
gula
r V
alue
s (d
B)
We need to find closed loop transfer functions
1
411
4 )()( faaf KAsICILIsS
)()( sSIsT ff
50
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Kalman filter design
We need to find closed loop transfer functions
1
411
4 )()( faaf KAsICILIsS )()( sSIsT ff
100
101
102
-20
-15
-10
-5
0
5Singular Values
Frequency (rad/sec)
Sin
gula
r V
alue
s (d
B)
-3
rad/sec 12 till6.5between is BT
rad/sec 5.5 till2.5between is B
)dB4(6.1fT
We could therefore terminate the Kalman filter design and move on to the recovery step.
However we shall suppose that we wish to improve the sensitivity further.
51
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Kalman filter design
However we shall suppose that we wish to improve the sensitivity further.
Haaa UYWAIjC 2/11 )100(5.5
W51/2 can be shape as follows:
HHH uuIuuIuuIWW 3322112/1
32/1
5 Re8039.8Re9567.1Re2382.010
TWWW 2/15
2/155
jjj
jjjU
05.077.0003.012.004.062.0
06.039.010.085.003.032.0
4979.049.071.0
1020.000
03382.00
008076.0
18076.0
1 1
3382.0
1 1
1020.0
1
52
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Kalman filter design
HHH uuIuuIuuIWW 3322112/1
32/1
5 Re409.7Re639.1Re069.010
TWWW 2/15
2/155
37.13369.1355.312
50.102.3172.12
75.1228.465.119
31.627.164.52
00.5100.179.6
08.1012.057.0
12.077.719.0
57.019.031.10
),,,,( 55 VWCAlqeK aaaf
So we have
53
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Kalman filter design
The loop transfer function is: 54 )( faa KAsICsL
The principle gains have clearly been “squeezed together” near 10 rad/sec, and at higher frequency.
10-3
10-2
10-1
100
101
102
-50
0
50
100
150
200Singular Values
Frequency (rad/sec)
Sin
gula
r V
alue
s (d
B)
54
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Kalman filter design
We need to find closed loop transfer functions
1
511
4 )()( faaf KAsICILIsS )()( sSIsT ff
100
101
102
-20
-15
-10
-5
0
5Singular Values
Frequency (rad/sec)
Sin
gula
r V
alue
s (d
B)
55
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Kalman filter design
The characteristic loci of is: 54 )( faa KAsICsL
All the loci remain outside or on the boundary of the unit circle as predicted by Kalman filter theory, from which we know that:
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0-5
-4
-3
-2
-1
0
1
2
3
4
5
Re
Im
1)( fi S
56
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Recovery at the plant output
Since the system has no transmission zero in the RHP so arbitrary good recovery should be possible.
To obtain LTR, we solve the following Riccati equation
01 QMMXBXBRXAXA TTT
With M=Ca, Q=I and R=ρI
),,,( 3ICCBAlqrK aT
aaar We shall write
When Kr has been find the controller realization is:
3,355 0 LTRrLTRfLTRafraaLTR DKCKBCKKBAA
Now we need
fLQG KsCsLsKsGsL )()()()()( 42
57
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Recovery at the plant output
Now we need 542 )()()()()( fLQG KsCsLsKsGsL ρ=10-2
10-3
10-2
10-1
100
101
102
-100
-50
0
50
100
150
200=1e-2
Frequency (rad/sec)
Sin
gula
r V
alue
s (d
B)
L2
L4
ρ=10-4
10-3
10-2
10-1
100
101
102
-100
-50
0
50
100
150
200=1e-4
Frequency (rad/sec)
Sin
gula
r V
alue
s (d
B)
L2
L4
58
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Recovery at the plant output
Now we need 542 )()()()()( fLQG KsCsLsKsGsL ρ=10-6
10-3
10-2
10-1
100
101
102
-50
0
50
100
150
200=1e-6
Frequency (rad/sec)
Sin
gula
r V
alue
s (d
B)
L2
L4
ρ=10-8
10-3
10-2
10-1
100
101
102
-50
0
50
100
150
200=1e-8
Frequency (rad/sec)
Sin
gula
r V
alue
s (d
B)
L2
L4
59
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Recovery at the plant output
1001804.7962058.78000
0108.93.49109933700
001823.36785311585960
),,,( 3ICCBAlqrK aT
aaar
We have for ρ=10-8
60
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Recovery at the plant output
-7 -6 -5 -4 -3 -2 -1 0-4
-3
-2
-1
0
1
2
3
Re
Im
ρ=10-6
ρ=10-8
Characteristic loci at the output of compensated plant, for ρ=10-6 and ρ=10-8
61
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Recovery at the plant output
ρ=10-8
100
101
102
-20
-15
-10
-5
0
5
10=1e-8
Frequency (rad/sec)
Sin
gula
r V
alue
s (d
B)
ST
ρ=10-6
100
101
102
-20
-15
-10
-5
0
5
10=1e-6
Frequency (rad/sec)
Sin
gula
r V
alue
s (d
B)
ST
Principle gains of S and T, for ρ=10-6 and ρ=10-8
62
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Recovery at the plant output
0
0.5
1
1.5From: In(1)
To:
Out
(1)
0
0.5
1
1.5
To:
Out
(2)
0 1 2
0
0.5
1
1.5
To:
Out
(3)
From: In(2)
0 1 2
From: In(3)
0 1 2
Step Response
Time (sec)
Am
plitu
deClosed-loop step responses to step responses to different inputs.
63
Ali Karimpour Jan 2010
Chapter8
Design example
Loop transfer recovery (LTR) procedures
Recovery at the plant output
The closed-loop poles are located at
10072
90.390.362.474.2
92.474.216.048.5
14.4814.4893.15693.156
jj
jj
jj