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TRANSCRIPT
Dr. Ali Karimpour May 2017
Lecture 8
Multivariable Control
Systems
Ali Karimpour
Associate Professor
Ferdowsi University of Mashhad
Lecture 8
References are appeared in the last slide.
Dr. Ali Karimpour May 2017
Lecture 8
2
Multivariable Control System Design(Sequential loop closing, Characteristic-locus method and PI controller)
Topics to be covered include:
• Sequential loop closing
• The characteristic-locus method
• PI control for MIMO systems
• Sequential loop closing
Remark: Evaluation of this lecture can be done before end of class under special circumstances.
Dr. Ali Karimpour May 2017
Lecture 8
3
Sequential loop closing
The simplest approach to multivariable design is to ignore its multivariable nature.
• A SISO controller is designed for one pair of input and output variables.
• When this design has been successfully completed another SISO controller is
designed for a second pair of variables and so on.
How input and output variables should be paired?
loop-assignment problem or input-output pairing
Dr. Ali Karimpour May 2017
Lecture 8
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• The transfer function matrix of
such a controller is diagonal.
• It also has the advantage that it
can be implemented by closing
one loop at a time.
Benefits:
Drawbacks
• It allows only a very limited class of controllers to be designed, and the design must
proceed in a very ad hoc manner.
• Interactions are very important.
• The only means available for the reduction of interaction (if this is a requirement) is
to use high loop gains
• Control difficulty in the case of element zeros.
Sequential loop closing
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Lecture 8
5
A more sophisticated version of sequential loop closing is called
sequential return-difference method
A cross-coupling stage of compensations should be introduced
This stage should consist of either a constant-gain matrix
or a sequence of elementary operations. )()(
1)( sU
sdsCp
Sequential loop closing
Dr. Ali Karimpour May 2017
Lecture 8
6
Example 8-1:
ss
ss
ssG
123
11
)1(
1)(
2
If we try to design a SISO controller for either the first or second loop here, we have
difficulties, if the required bandwidth is close to unity, or greater, because the transfer
function ‘seen’ for the design, namely the (1,1) or (2,2) element of G(s), has a zero at 1.
However, G(s) itself has a transmission zero at - 1 only, so there should be no inherent
difficulty of this kind.
If we choose
12
01pC
ss
ss
sCsGsQ p
13
1
3
1
)1(
1)()(
2
We see that no right half-plane zero ‘appears’ when a SISO compensator is being
designed for the first loop.
Sequential loop closing
Dr. Ali Karimpour May 2017
Lecture 8
7
ss
ss
sKsGsQ a
13
1
3
1
)1(
1)()(
2
Once the first loop has been
closed by
The transfer function ‘seen’ in
the second loop is
)(1 sk
222
1
22)1(
3
1
)()1()1(
1)(
s
s
shs
s
s
ssq )(1/
3
11
)()( where
1
2
1
skss
sksh
Example 8-1 Continue:
Sequential loop closing
Dr. Ali Karimpour May 2017
Lecture 8
8
222
1
22)1(
3
1
)()1()1(
1)(
s
s
shs
s
s
ssq )(1/
3
11
)()( where
1
2
1
skss
sksh
Now, if we assume high gain in the first loop
s
ssk
3
1
)1()(
2
1s
ssh
3
1
)1()(
2
and hence
)13)(1(
1)(1
22
ss
sq
so that no right half-plane zero is seen when the compensator for the second loop
is designed.
Example 8-1 Continue:
Sequential loop closing
Dr. Ali Karimpour May 2017
Lecture 8
9
• The main idea that of using a first stage of compensation to make subsequent
loop compensation easier
• The main weakness of the method is that little help is available for choosing that
first stage of compensation.
The available analysis relies on the assumption that there are high gains in the loops
which have already been closed, and such an assumption can rarely be justified,
except at low frequencies.
One rather special case, in which the assumption of high gains is justified, arises when
different bandwidth is required for each loop, and all the bandwidths are well separated
from each other.
Sequential loop closing
Dr. Ali Karimpour May 2017
Lecture 8
10
Mayne (1979) suggests that, if the plant has a state-space realization (A, B, C), the
product CB being non-singular (and the matrix D being zero), then the first stage of
compensation can be chosen to be
1 CBCp
sass
CBsG )( Since sas
s
ICsG p)( so
so that each loop looks like a first-order SISO system at high frequencies.
However, an alternative choice, such as
)(1
bp jGC )(C or 1
bbp jGj
Sequential loop closing
Dr. Ali Karimpour May 2017
Lecture 8
11
• Sequential loop closing
• The characteristic-locus method
• PI control for MIMO systems
• The characteristic-locus method
Multivariable Control System Design(Sequential loop closing, Characteristic-locus method and PI controller)
Dr. Ali Karimpour May 2017
Lecture 8
12
The characteristic-locus method
Approximate commutative compensators
• How this can be done. • Why it should be useful
Suppose we have a square transfer-function matrix G(s), with m input and outputs
iondecomposit spectral is )()()()( 1 sWssWsG
W(s) is a matrix whose columns are the eigenvectors, or characteristic directions, of G(s)
)(,.....,)(,)()( 21 sssdiags m
where the λi(s) are the eigenvalues, or characteristic functions, of G(s).
Commutative compensators
)()()()( sGsKsKsG
)()()()( sGsKsKsG
Dr. Ali Karimpour May 2017
Lecture 8
13
iondecomposit spectral is )()()()( 1 sWssWsG
Let following structure as controller
)()()()( 1 sWsMsWsK where
)(,...,)(,)()( 21 sssdiagsM m
then the return ration (loop transfer function) is
)()()()()()()()()()()( 11 sGsKsWsNsWsWsMssWsKsG
)(,.....,)(,)()( 21 svsvsvdiagsN mwhere
)()()( sssv iii
• How this can be done. • Why it should be useful
A compensator having this structure is called a commutative compensator.
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
14
The strategy which suggests itself is to obtain graphical displays of the characteristic
loci of the plant,
miji ,.....,2,1:)(
One would then obtain the compensator as the series connection of three systems
corresponding toW(s)sMsW and )(,)(1
Then design a compensator
using the well-established single-loop techniques.
)( ji for each )( ji
There are some main drawback in designing commutative controller
1- Robust stability may not be derived.
2- It is not realizable in general case.
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
15
The characteristic-locus method
Example 8-2: The transfer-function matrix
25042
56472
21
1)(
ss
ss
sssG
1- Robust stability may not be derived.
has the following spectral decomposition
76
87
2
20
01
1
76
87)()()()( 1
s
ssWssWsG
and so W(s) and Λ(s) is
2
20
01
1
)(76
87)(
s
sssW
Dr. Ali Karimpour May 2017
Lecture 8
16
The characteristic-locus method
Choose controller as
1- Robust stability may not be derived.
)k(-1 constant aby stabilized is 1
1 clear that isIt 11
ks
)k(-1 constant aby stabilized is 2
2 clear that also isIt 22
ks
so we have good stability margin in each channel but what about the stability margin
in multivariable system?
76
87
0
0
76
87)()()()(
2
11
k
ksWsMsWsK
2
20
01
1
)(
s
ss
Now we can choose k 1 =k 2 =1 we have
10
01
76
87
10
01
76
87)(sK
Dr. Ali Karimpour May 2017
Lecture 8
17
The characteristic-locus method
1- Robust stability may not be derived.
1)( GKIKTu
Suppose that we use above controller in our system but there is some
uncertainty in inputs so let the value of controller as:
10
01)(sK
so we have good stability margin in each channel but what about the stability margin
in multivariable system?
25042
56472
21
1)(
ss
ss
sssG
b
asK
0
0)(
To check the stability we need to check the stability of
Dr. Ali Karimpour May 2017
Lecture 8
18
The characteristic-locus method
1- Robust stability may not be derived.
1)( GKIKTu
so we have good stability margin in each channel but what about the stability margin
in multivariable system?
We need to check the stability of
Since K is a constant matrix we need to check the stability of1)( GKI
1
1
250)2)(1(42
56472)2)(1(
)2)(1(
1)(
bsssas
bsasss
ssGKI
Closed loop characteristic equation is:
0)2222()47503()2)(1( 2 abbaabssss
For stability we need:
02222&047503 abbaab
Dr. Ali Karimpour May 2017
Lecture 8
19
1- Robust stability may not be derived.
so we have good stability margin in each channel but what about the stability margin
in multivariable system?
For stability we need:
02222&047503 abbaab
For a=1 we need b>44/50=0.88
For b=1 we need a<53/47=1.128
For a=50.5/47=1.07 and b=0.95 we have lost stability.
So robust stability may not be derived.
Is it possible to know this weak stability margin by some measure?
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
Perturbed PlantUncertainty M in MΔ-structure
20
structureM
Suitable for robust stability analysis
Additive uncertainty 2
1
1 )( wGKIKwM 12 wwGGp
Multiplicative input uncertainty2
1
1 )( GwGKIKwM )( 12 wwIGGp
Multiplicative output uncertainty2
1
1 )( wGKIGKwM GwwIGp )( 12
Inverse additive uncertainty 2
1
1 )( wKGIGwM 1
12
GwwIGGp
Inverse multiplicative input uncertainty2
1
1 )( wKGIwM 1
12
wwIGGp
Inverse multiplicative output uncertainty2
1
1 )( wGKIwM GwwIGp
1
12
1- Robust stability may not be derived.
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
21
Example 8-3: The transfer-function matrix
2
1
2
21
2
1
1
)(
ss
sssG
33481632)2)(1(2
1)(,)( 2
21
sssss
ss
has characteristic functions
3348161
84
)(
)(
22
1
ss
s
sw
sw
i
i
The characteristic directions are given by
and so W(s) is
11
3348161
84
3348161
84
)(
22 ss
s
ss
s
sW
2- It is not realizable in general case.
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
22
A practical alternative is to give the compensator the structure
)()()()( sBsMsAsK )()()()( 1 sWsMsWsK
)()( sWsA )()( 1 sWsB where
are chosen to be realizable, and such that
Whichever approximation technique is chosen, we obtain an
approximate commutative compensator
2- It is not realizable in general case.
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
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Design procedure
• Manipulating the characteristic loci
• Stable design by satisfying the generalized Nyquist stability theorem
• Large magnitudes at low frequencies
• Stay outside the 2M
All the loci look satisfactory-when judged against classical SISO criteria.
Do one ensure satisfactory performance of the multivariable feedback system? NO.
m
i
T
iiii ssvswsWsvdiagsWsKsG1
1 )()()()()()()()(Let
)()(1
1)()(
)(1
1)()(
1
1 ssv
swsWsv
diagsWsS T
i
i
m
i
i
i
)()(1
)()()(
)(1
)()()(
1
1 ssv
svswsW
sv
svdiagsWsT T
i
i
im
i
i
i
i
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
24
Design procedure
)()(1
)()()(
)(1
)()()(
1
1 ssv
svswsW
sv
svdiagsWsT T
i
i
im
i
i
i
i
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
25
Design procedure
)()()()( 11 swsrssrT
)()()()( 22 swsrssrT
.......
)()()()( swsrssr m
T
m
)(sy
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
26
Design procedure
The performance of a feedback system depends essentially on the singular values
(principal gains) of functions such as S(s) and T(s).
)(1
1
svi )(1
)(
sv
sv
i
i
These transfer functions have eigenvalue functions
We know that, for any matrix X,
)()()( XXX
If the characteristic loci of GK
have low magnitudes
The sensitivity may be large,
in some signal directions
If some characteristic locus
penetrates the circle2M
At least one principal gain of T(s)
will exhibit a resonance peak, of
magnitude greater than 2
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
27
Design procedure
If the eigenvectors are nearly
orthogonal to each other
Magnitudes of the smallest and largest
eigenvalues are close to the smallest
and largest singular values
Good system performance
This situation happens if the return ratio has low skewness
)( sviShaping
Note that the discrepancies between characteristic loci and singular values (principal
gains) do not make the shaping of characteristic loci a fruitless activity.
On the contrary, manipulating the characteristic loci is often the most straightforward
and productive way of designing a multivariable feedback system, or at least of
initiating the design. Of course, the properties of the resulting design must be checked
by methods more revealing than examination of the characteristic loci.
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
28
Design procedure
The only thing that can be done is to insert a series compensator whose main purpose is
to obtain a ‘decoupled’ (that is, diagonal) return ratio of -G(s)K(s) at (some of) these
frequencies.
One way of attempting to decouple the return ratio at one or several frequencies is to
attempt to invert G(s) at these frequencies by ALIGN algorithm. .
The characteristic-locus method
)( Let 1
bh jGK
1)( seigenvalue hb KjG
It may be advantageous to adjust the signs of the columns of Kh at this stage.
In such a case changing the sign of the corresponding column of Kh may bring this
characteristic locus into a region in which subsequent compensation becomes easier.
In which case one of the characteristic loci will be very far from -1
Dr. Ali Karimpour May 2017
Lecture 8
29
Design procedure of characteristic-locus method
1- Compute a real )(1
bh jGK
2- Design an approximate commutative controller Km(s) at some frequency bm
for the compensated plant G(s)Kh, such that asIjKm )(
3- If the low-frequency behavior is unsatisfactory (typically because there are excessive
steady-state errors), design an approximate commutative controller Kl(s) at low
frequency, for the compensated plant G(s)KhKm(s), such that asIjK l )(
(the red blocks are some attempt to ensure that the decoupling effected by Kh is not
disturbed too much).
4- Realize the complete compensator as )()()( sKsKKsK lmh
high, medium and low frequency
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
30
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.
Example 8-4: Consider the aircraft model AIRC described in the following
state-space model.
DuCxy
BuAxx
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
31
Design example
• We shall attempt to achieve a bandwidth of about l0 rad/sec for each loop.
THE SPECIFICATION
• Little interaction between outputs.
• Good damping of step responses and zero steady-state error in the face of step
demands or disturbances.
• We assume a one-degree-of-freedom control structure.
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
32
Design example
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. since
the number of finite zeros of the plant can be at most
)(2 CBrankmn 013.25
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
33
Negative feedback were
applied around the plant
111 NPZ
111 Z
21 Z
Closed loop would be unstable.
(Two RHP poles)
2M
Characteristic loci of G , with logarithmic calibration
02.173.069.159.090.0:arevaluesEigen 53,41,2 jj
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
34
Design example
ALIGNMEIVT AT 10 rad/sec
378.690065.044.189
5376.09984.95375.8
669.30036.0535.71
))10(( jGalignK h
jjj
jjj
jjj
KjG h
086.09962.00005.00000.0100.0009.0
0003.00008.0010.1005.00004.0063.0
009.00005.0003.00001.0068.0983.0
)10(
From input 1 to output 2 and 3, and that these interactions have been reduced to 10% or
less (at that frequency).
378.690065.044.189
5376.09984.95375.8
669.30036.0535.71
hK
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
35
Characteristic loci of GKh , with logarithmic calibration
)10,2
3
2
1:( jes
111 NPZ
111 Z
01 Z
Negative feedback were
applied around the plantClosed loop would be stable.
(No RHP poles)
05.1099.953.096.924.0:arevaluesEigen 53,41,2 jj
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
36
Design example
APPROXIMATE COMMUTATIVE COMPENSATOR
-180 -160 -140 -120 -100 -80 -60 -40 -20 0-40
-20
0
20
40
60
80
100
120
Phase (deg)
Gain
(dB
)
Characteristic loci of GKh , on Nichols chart
As might be expected from
hKjG )10(
however, two of the loci pass
extremely close to -1
1)10( WWKjG h
0009.09526.08829.0
9999.00076.00124.0
0015.02049.03483.0
)( 1WalignA
0024.00001.10391.0
6633.00026.07310.1
3728.00031.08165.1
)(WalignB
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
37
APPROXIMATE COMMUTATIVE COMPENSATOR
In this case it is the (1,1) and (2,2) elements of Λ that require compensation.
100
010933.0
2175.00933.00
0010933.0
2175.00933.0
)(s
ss
s
sM BsAMsKm )()(
To evaluate the interactions at this frequency it is appropriate to examine
6731.00012.00763.0
0025.09999.00265.0
0071.00023.06811.0
)10()10( jKKjGabs mh
This shows that decoupling has not been destroyed at l0 rad/sec.
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
38
APPROXIMATE COMMUTATIVE COMPENSATOR
Characteristic loci of GKhKm , on Nichols chart
LOW FREQUENCY PROBLEM
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
39
LOW FREQUENCY COMPENSATION
)()()( XXX
10))0()0((
mh KKG
Which will certainly not achieve the objective of zero steady-state error in the face of
step disturbances.
IsT
sTsK l
1)(
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
40
APPROXIMATE COMMUTATIVE COMPENSATOR
Characteristic loci of GKhKm
on Nichols chart
-240 -220 -200 -180 -160 -140 -120 -100 -80 -60-50
0
50
100
150
Phase (deg)
Gain
(dB
)
Characteristic loci of GKhKmKl
on Nichols chart
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
41
Design example
10-2
10-1
100
101
102
-50
0
50
100
150
Frequency ( rad/sec )
Gain
(dB
)
Largest and smallest open-loop singular values (principal gains).
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
42
Design example
10-1
100
101
102
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
5
Frequency (rad/sec)
Gain
(dB
)
Largest and smallest closed-loop singular values (principal gains).
Bandwidth is ok
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
43
Design example
0 0.5 1 1.5 2 2.5 3-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Step Response
Time (sec)
Am
plit
ude
Closed-loop step responses to step demand on output 1 (solid curves),
output 2 (dashed curves) and output 3 (dotted curves).
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
44
Design example
REALIZATION OF THE COMPENSATOR
The controller can be realized as a five-state linear system (two states for the
phase-lead transfer functions in Km, and three for the integrators in Kl).
5.000
05.00
005.0
653.20106.0924.6
491.10126.0266.7
,
00000
00000
00000
653.20106.0924.672.100
491.10126.0266.7072.10
KK BA
52.651204.09.183
4372.0996.9369.8
091.30129.045.70
,
52.651204.09.1832.578.266
4372.0996.9369.8752.2489.7
091.30129.045.704.3303.59
KK DC
The characteristic-locus method
Dr. Ali Karimpour May 2017
Lecture 8
45
• Sequential loop closing
• The characteristic-locus method
• PI control for MIMO systems
Multivariable Control System Design(Sequential loop closing, Characteristic-locus method and PI controller)
Dr. Ali Karimpour May 2017
Lecture 8
46
P: Proportional, Output of controller is proportional to error.
I: Integral action, Remove steady state error.
D: Derivative action, Anticipate the performance.
Aim of PID in MIMO systems: Derive following specification with minimum
information of system.
* Stability of closed loop system. * Minimum value of interaction.
* Reference tracking. * Disturbance rejection.
PI control for MIMO systems
* PID Controller :
r(t) u(t)
Dr. Ali Karimpour May 2017
Lecture 8
47
Design procedure
a) Design based on step response.
* System is stable.
* System model is unknown.
* System is regular1 or irregular.
-----------------------------------------------
1 A system is regular if CB has full rank otherwise its irregular.
b) Design based on model.
* No transmission zero on origin.
* Suitable for industrial plants.
* System model is known.
* System is regular or irregular.
* No transmission zero on origin.
PI control for MIMO systems
Dr. Ali Karimpour May 2017
Lecture 8
48
v P-controller : The idea of the proportional feedback controller is to speed
up the transient. In the multivariable case, this is carried out by choosing
the m × l matrix K so that the effect of the i th component of the reference
signal is strong on the i th component of the output y but weak on the j thcomponent of y when i ≠ j.
v Robust I-controller : The robust multivariable I-controller construction
follows Davison (1976) is a robust controller which eliminates the steady-
state error in the output.
v Robust PI-controller : The robust multivariable PI-controller can now be
constructed (if it exists) based on P-controller and I-controller in the case
that inverse of two important matrices CB and G(0) are exist.
PI control for MIMO systems
Multivariable Tuning Regulators for Unknown Systems
(Penttinen and Koivo, 1980)
Design based on step response (Regular System)
Dr. Ali Karimpour May 2017
Lecture 8
49
PI control for MIMO systems
----------------------------------------------------------------------------------------Multivariable Tuning Regulators for Unknown Systems (Penttinen And Koivo, 1980)
)(
;;;;)()()()(
tCxy
mlRdRyRuRxtDdtButAxtx plmn
Consider the following system.
32211)()( BsCACABsCBsBAsICsG
Suppose:
* A, B, C and D are unknown. * System is stable.
* It is possible to apply individual input and derive output.
* The order n is unknown. * The disturbance d(t) might not be measurable.
Design based on step response (Regular System)
Dr. Ali Karimpour May 2017
Lecture 8
50
r(t)BKC) x(t)(A-BK(t)x
Cx(t)r(t)-BKBKAx(t))(r(t)-y(t)BKAx(t)Bu(t)Ax(t)(t)x
e(t)Ku(t)
pp
ppp
p
0if d(t)
PI control for MIMO systems
----------------------------------------------------------------------------------------Multivariable Tuning Regulators for Unknown Systems (Penttinen And Koivo, 1980)
K P
G(s)r(s) e(s) u(s)
P-controller Design
K I /S
Suppose : 0IK
-
y(s)
Dr. Ali Karimpour May 2017
Lecture 8
51
PI control for MIMO systems
----------------------------------------------------------------------------------------Multivariable Tuning Regulators for Unknown Systems (Penttinen And Koivo, 1980)
K P
G(s)r(s) e(s) u(s)
P-controller Design
r(t)dτBKe e : x(t))if x( p
tC)τ-(A-BKC)t(A-BK pp
0
00
1
0
I]M[edτe Mt
t
Mτ
0
1
0
1 rBKC)(A-BKerBKC) (A-BKx(t) pp
C)t(A-BK
ppp
r(t)BKC) x(t)(A-BK(t)x pp
-
y(s)
0
0
0 r)BKde (e x(t) :r r(t) p
tC)τ-(A-BKC)t(A-BK pp
Dr. Ali Karimpour May 2017
Lecture 8
52
trCBK : y(t)t
r]BKk!
tC)(A-BKC[trCBKy(t)
p
p
k
kk
pp
0
0
2
1
0
0
0
1
0
1 rBKC)(A-BKerBKC) (A-BKx(t) pp
C)t(A-BK
ppp
0
1
0
1 rBKC)(A-BKCerBKC) C(A-BKy(t) pp
C)t(A-BK
ppp
...An!
t...A
!
ttAIe n
nAt 2
2
2
PI control for MIMO systems
----------------------------------------------------------------------------------------Multivariable Tuning Regulators for Unknown Systems (Penttinen And Koivo, 1980)
P-controller Design
Dr. Ali Karimpour May 2017
Lecture 8
53
trCBK : y(t) t p 00
12
1
00
0
0
00
)C(CBBCB;(CB)
k
k
k
(CB)K TTTT
m
p
C)]P(BKPC)PA-[(BKPAC)P (A-BKPC)(A-BK p
T
p
T
p
T
p
By following Kp , y(t) is diagonal.
For checking stability:
PI control for MIMO systems
----------------------------------------------------------------------------------------Multivariable Tuning Regulators for Unknown Systems (Penttinen And Koivo, 1980)
P-controller Design
r(t)BKC) x(t)(A-BK(t)x pp
Dr. Ali Karimpour May 2017
Lecture 8
54
1
2121 ]...][...[ mm uuuyyyCB
))=CBu( )+CBu()=CAx((y=> and d(t)=)=x(Suppose:
d(t)+CBu(t)+CD(t)=CAx(t)x(t)=Cy
0000000
… u uuy … y y
=CBuy
=CBuy
=CBuy
mm
mm
]CB[=] [ 2121
22
11
The following algorithm gives an experimental way for determining the matrix
CB for Unknown Systems :
)(
)()()()(
tCxy
tDdtButAxtx
Step 2. Repeat step 1 and let :
u1, u2 , … um are independent then:
PI control for MIMO systems
----------------------------------------------------------------------------------------Multivariable Tuning Regulators for Unknown Systems (Penttinen And Koivo, 1980)
P-controller Design
Step 1. Let . When the steady-state is achieved, choose a constant input
and apply it to the plant. Then ( this value can
be computed e.g. graphically.)
0u(t)01 uu(t) 110 =CBuy)(y
Dr. Ali Karimpour May 2017
Lecture 8
55
;
00
0
0
00
2
1
m
p
k
k
k
(CB)K
PI control for MIMO systems
----------------------------------------------------------------------------------------Multivariable Tuning Regulators for Unknown Systems (Penttinen And Koivo, 1980)
P-controller Design
Remark. The existence of the P-controller follows if rank (CB)= k.
If the P-controller exists, the tuning parameters can be found
experimentally.
Dr. Ali Karimpour May 2017
Lecture 8
56
PI control for MIMO systemsI-controller Design
The robust multivariable I-controller construction follows:
G(s)
K P
r(s) e(s) u(s)
K I /S-
Suppose : 0pK
d(s)
0
I )(K=u(t) dtte
Multivariable tuning regulators: The feedforward and robust control of a general
servomechanism problem (Davison ,1976).
BCA)G(
);(εG); K(Gε K; KK II
10
00
Let :
----------------------------------------------------------------------------------------Multivariable Tuning Regulators for Unknown Systems (Penttinen And Koivo, 1980)
y(s)
Dr. Ali Karimpour May 2017
Lecture 8
57
PI control for MIMO systemsI-controller Design
G(s)r(s) e(s) u(s)
K I /S-
d(s)
Davison shows that the controller where is the
scalar tuning parameter, is a robust controller which eliminates the steady-state
error in the output. He also shows that there exists so that the closed-
loop system is stable if satisfies .
][ 1BCAKKI
0*
*0
)(
)()()()(
tCxy
tDdtButAxtx
0
e(t)dtKu(t)=
0
e(t)dtz(t)=
d(t)D
r(t)z(t)
x(t)
C
εBKA
(t)z
(t)x
01
0
0
0limlim e(t)(t)z tt
y(s)
----------------------------------------------------------------------------------------Multivariable Tuning Regulators for Unknown Systems (Penttinen And Koivo, 1980)
Dr. Ali Karimpour May 2017
Lecture 8
58
steady state gain of system
1
2121 ]...][...[)0( mmssssss uuuyyyG
B;CA r
yG -ss 1
0
(0)
PI control for MIMO systems
----------------------------------------------------------------------------------------Multivariable Tuning Regulators for Unknown Systems (Penttinen And Koivo, 1980)
The next algorithm of Davison (1976) provides a way to experimentally
determine the parameter for Unknown Systems. Again assume the
disturbance d(t)=0.
)0(G
Step 2. Repeat step 1 and let :
u1, u2 , … um are independent then:
I-controller Design
Step 1. Let . When the steady-state is achieved, choose a constant
input and apply it to the open loop plant. Then
(Measure e.g. graphically.)
0u(t)
01 uu(t) 1ss1 (0)= uGy
ss1y
]...)[0(]...[
0)((0)
(0)
(0)
2121
22
11
mssmssss
mmssm
ss
ss
u uuG y yy
ut uu=Gy
u=Gy
u=Gy
Dr. Ali Karimpour May 2017
Lecture 8
59
PI control for MIMO systemsPI-controller Design
G(s)r(s) e(s) u(s)
K I /S-
d(s)
;εGKI (0)
----------------------------------------------------------------------------------------Multivariable Tuning Regulators for Unknown Systems (Penttinen And Koivo, 1980)
Remark. Davison's necessary and sufficient condition for the existence
of the robust I-controller which eliminates the steady-state error in the
output for all constant disturbances and constant reference signals is
rank (G(0))= k. The tuning parameter can be found experimentally.
y(s)
Dr. Ali Karimpour May 2017
Lecture 8
60
PI control for MIMO systemsPI-controller Design
G(s)
K P
r(s) e(s) u(s)
K I /S-
d(s)
;εGKI (0)
----------------------------------------------------------------------------------------Multivariable Tuning Regulators for Unknown Systems (Penttinen And Koivo, 1980)
;
k
k
k
(CB)K
m
p
00
0
0
00
2
1
y(s)
Dr. Ali Karimpour May 2017
Lecture 8
61
v So we can determine the gain matrices and (if they exist) .
v The step-inputs needed can be the same for both algorithms.
v Tune the P-controller for the plant by adjusting the tuning parameters,
[starting with small positive values so that the closed-loop response for a
step-input reference signal is as good as possible.]
v Tune the PI-controller by adjusting the tuning parameter of the I-
controller [starting with small positive values so that the closed-loop (Pl-
controlled) response for a step-input reference signal has the maximum
speed of response.]
IKPK
PI control for MIMO systems
----------------------------------------------------------------------------------------Multivariable Tuning Regulators for Unknown Systems (Penttinen And Koivo, 1980)
PI-controller Design
Dr. Ali Karimpour May 2017
Lecture 8
62
This proposed method is quite simple, no model is constructed, and one
only needs steady-state information and derivatives of the step-responses
which are easily evaluated graphically.
Example 8-5:
)(010
001)(
)(
1
5
1
1
1
3
)(
0
1
1
1
1
1
)(
300
020
001
)(
txty
tdtrtxtx
PI control for MIMO systemsDesign example
Dr. Ali Karimpour May 2017
Lecture 8
63
11
11
10
01
11
111
CB
5.05.0
11
10
01
5.05.0
11)0(
1
G
Let u=[ 1 0]
1
1CB
5.0
1)0(G
PI control for MIMO systemsDesign example
Dr. Ali Karimpour May 2017
Lecture 8
64
11
11
10
01
11
111
CB
5.05.0
11
10
01
5.05.0
11)0(
1
G
Let u=[ 1 0]
11
11CB
5.05.0
11)0(G
Let u=[ 0 1]
PI control for MIMO systemsDesign example
Dr. Ali Karimpour May 2017
Lecture 8
65
m
p
k
k
k
(CB)K
00
0
0
00
2
1
B]CAε[)(εGKI
10
2
1
1
0
0
11
11
k
kK p
1
5.05.0
11
KK I
121 kk 5.0
PI control for MIMO systemsDesign example
Dr. Ali Karimpour May 2017
Lecture 8
66
121 kk 5.0
5.1121 kk
PI control for MIMO systemsDesign example
Dr. Ali Karimpour May 2017
Lecture 8
67
PI control for MIMO systems
The second control design method is based on :
Self-tuning and adaptive control : theory and applications (Harris and Billings, 1981)This method can be applied on regular or irregular systems.
G(s)
K P
r(s) e(s) u(s)
K I /S-
d(s)
)(
)()()()(
tCxy
tDdtButAxtx
Consider the following system and assuming the same conditions exist :
R )0(KK I KK p
)()()( tKztKetu
0
e(t)dtz(t)=
By following Kp and KI
Design based on step response (Irregular System)
Dr. Ali Karimpour May 2017
Lecture 8
Harris and Billings show that if
68
)()()( 1 sNsDsG
Let R )0(KK I KK p
PI control for MIMO systems
----------------------------------------------------------------------------------------Self-tuning and adaptive control : theory and applications (Harris and Billings, 1981)
KsNsDsIsDs lc )()()1()()( 1
Remark. The existence of the PI-controller follows if rank (G(0))= k. If the
PI-controller exists, the tuning parameters can be found experimentally.
PI-controller Design
and for small value of the closed loop system will be stable.
the characteristic equation of the closed loop system is
1)( TT GGGK )0(;
000
0
00
00
2
1
i
m
)0()0()0( 1 NDGG
by following K
Dr. Ali Karimpour May 2017
Lecture 8
69
33
33
5454
5454
11
11
35110
02
1
K,..
..αεK
Σ)(GGGK
,,ε.,αΣ
TT
Example 8-5:
)(010
001)(
)(
1
5
1
1
1
3
)(
0
1
1
1
1
1
)(
300
020
001
)(
txty
tdtrtxtx
PI control for MIMO systems
----------------------------------------------------------------------------------------Self-tuning and adaptive control : theory and applications (Harris and Billings, 1981)
Design example
Dr. Ali Karimpour May 2017
Lecture 8
70
Penttinen and Koivo’s Method
Harris and Billings’s Method
PI control for MIMO systemsDesign example
Dr. Ali Karimpour May 2017
Lecture 8
71
Design based on step response(regular systems). rank full has CB
1
2
2
1
1 0
00
0
0
00
)(XXXX),(GεK,
k
k
k
(CB)K TT
m
Design based on step response(irregular systems). rank fullnot has CB
mm k
k
k
)(GεK,
k
k
k
)(GεαK
00
0
0
00
0
00
0
0
00
02
1
2
2
1
1
Design based on step response.
PI control for MIMO systems
Dr. Ali Karimpour May 2017
Lecture 8
72
Design based on model.
Design based on the model (regular systems). rank full has CB
Exercise 8-1 : Design a PI controller for following system.
a) Derive the eigenvalues of the open loop system.
b) Derive the eigenvalues of the closed loop system.
c) Draw the step response of the closed loop system.
d) If we have input multiplicative uncertainty check the robust margin.
xy
ux
0010
1101
0136.1
146.3136.1
0679.5
00
104.2343.1273.4048.0
893.5654.6273.4067.1
675.0029.45814.0
676.5715.62077.038.1
x
PI control for MIMO systems
Dr. Ali Karimpour May 2017
Lecture 8
73
Design based on model.
Design based on the model (irregular systems). rank fullnot has CB
Exercise 8-2: Design a PI controller for following system.
a) Derive the eigenvalues of the open loop system.
b) Derive the eigenvalues of the closed loop system.
c) Draw the step response of the closed loop system.
d) If we have input multiplicative uncertainty check the robust margin.
xy
ux
011
001
284.742.11
143.0717.0
00
964.039.90
121.10
100
x
PI control for MIMO systems
Dr. Ali Karimpour May 2017
Lecture 8
74
Exercise 8-3: Design a PI controller for following system.
a) Derive a PI controller for system for a=0.
c) Compare the result of part a and part b and find the transmission zeros and
analysis the results.
xa
y
ux
11
111
10
11
00
300
016
113
x
b) Derive a PI controller for system for a=1.
PI control for MIMO systems
Dr. Ali Karimpour May 2017
Lecture 8
75
References
• Control Configuration Selection in Multivariable Plants, A. Khaki-Sedigh, B. Moaveni, Springer Verlag, 2009.
References
• Multivariable Feedback Control, S.Skogestad, I. Postlethwaite, Wiley,2005.
• Multivariable Feedback Design, J M Maciejowski, Wesley,1989.
• http://saba.kntu.ac.ir/eecd/khakisedigh/Courses/mv/
Web References
• http://www.um.ac.ir/~karimpor
• تحليل و طراحی سيستم های چند متغيره، دکتر علی خاکی صديق
Dr. Ali Karimpour May 2017
Lecture 8
76
References
• Control Configuration Selection in Multivariable Plants, A. Khaki-Sedigh, B. Moaveni, Springer Verlag, 2009.
References
• Multivariable Feedback Control, S.Skogestad, I. Postlethwaite, Wiley,2005.
• Multivariable Feedback Design, J M Maciejowski, Wesley,1989.
• http://saba.kntu.ac.ir/eecd/khakisedigh/Courses/mv/
Web References
• http://www.um.ac.ir/~karimpor
• تحليل و طراحی سيستم های چند متغيره، دکتر علی خاکی صديق