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ECE4520/5520: Multivariable Control Systems I. 5–1
OBSERVABILITY AND CONTROLLABILITY
5.1: Continuous-time observability: Where am I?
■ We describe dual ideas called observability and controllability.
■ Both have precise (binary) mathematical descriptions, but we need to
be careful in interpreting the result.
■ We develop some other techniques to help quantify the concepts.
Continuous-time observability
■ If a system is observable, we can determine the initial condition of the
state vector x.0/ via processing the input to the system u.t/ and the
output of the system y.t/.
■ Since we can simulate the system if we know x.0/ and u.t/ this also
implies that we can determine x.t/ for t ! 0. e.g., for LTI,
x.t/ D eAtx.0/C
Z t
0
eA.t"!/Bu.!/ d!:
■ Consider the LTI SISO system LCCODE
«y.t/C a1 Ry.t/C a2 Py.t/C a3y.t/ D b0«u.t/C b1 Ru.t/C b2 Pu.t/C b3u.t/:
■ If we have a realization of this system in state-space form
Px.t/ D Ax.t/C Bu.t/
y.t/ D Cx.t/CDu.t/;
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–2
and we have initial conditions y.0/, Py.0/, Ry.0/, how do we find x.0/?
y.0/ D Cx.0/CDu.0/
Py.0/ D C.Ax.0/C Bu.0/„ ƒ‚ …
Px.0/
/CD Pu.0/
D CAx.0/C CBu.0/CD Pu.0/
Ry.0/ D CA2x.0/C CABu.0/C CB Pu.0/CD Ru.0/:
■ In general,
y.k/.0/ D CAkx.0/C CAk"1Bu.0/C $ $ $C CBu.k"1/.0/CDu.k/.0/;
or,2
64
y.0/
Py.0/
Ry.0/
3
75 D
2
64
C
CA
CA2
3
75
„ ƒ‚ …
O.C;A/
x.0/C
2
64
D 0 0
CB D 0
CAB CB D
3
75
„ ƒ‚ …
T
2
64
u.0/
Pu.0/
Ru.0/
3
75 ;
where T is a (block) “Toeplitz matrix”.
■ Thus, if O.C; A/ is invertible, then
x.0/ D O"1
8
ˆ<
ˆ:
2
64
y.0/
Py.0/
Ry.0/
3
75 " T
2
64
u.0/
Pu.0/
Ru.0/
3
75
9
>=
>;
:
■ We say that fC; Ag is an observable pair if O is nonsingular.
CONCLUSION: For a SISO system, if O is nonsingular, then we can
determine/estimate the initial state of the system x.0/ using only u.t/
and y.t/ (and therefore, we can estimate x.t/ for all t ! 0).
EXTENSION: For a MIMO system, if O is full rank, then we can
determine/estimate the initial state of the system x.0/ using only u.t/
and y.t/ (and therefore, we can estimate x.t/ for all t ! 0).
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–3
EXAMPLE: Observability canonical form:
Px.t/ D
2
64
0 1 0
0 0 1
"a3 "a2 "a1
3
75 x.t/C
2
64
ˇ1
ˇ2
ˇ3
3
75u.t/
y.t/ Dh
1 0 0i
x.t/:
■ Then
O D
2
64
C
CA
CA2
3
75 D
2
64
1 0 0
0 : : : 0
0 0 1
3
75 D In:
■ This is why it is called observability form!
EXAMPLE: Two unobservable networks
(Redrawn)
1"
1"1"1"
1"
1"1F1F
1H1H
2"2"
x
x1x1
x2x2y yyuu
■ In the first, if u.t/ D 0 then y.t/ D 0 8 t . Cannot determine x.0/.
■ In the second, if u.t/ D 0, x1.0/ ¤ 0 and x2.0/ D 0, then y.t/ D 0 and
we cannot determine x1.0/. [circuit redrawn for u.t/ D 0].
Observers
■ An observer is a device that has as inputs u.t/ and y.t/—the input
and output of a linear system. The output of the observer is the
(estimated) state of the linear system.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–4
■ The observer “observes”
the internal state x
(estimated as Ox) from
external signals u and y.
u.t/ y.t/A; B; C; D
Observer
x
Ox
■ Note that our equations yield an observer:
u.t/ y.t/A; B; C; D
u
Pu
Ru
y
Py
Ry
11
ss
s2s2
"T
O"1
x
Ox
■ Later, we’ll design more practical observers that don’t use
differentiators.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–5
5.2: Continuous-time controllability: Can I get there from here?
■ Can we generate an input u.t/ to set an initial condition quickly?
Px.t/ D Ax.t/C Bu.t/
y.t/ D Cx.t/CDu.t/:
■ If u.t/ D ı.t/ and x.0"/ D 0, then
X.s/ D .sI " A/"1BU.s/ D .sI " A/"1B:
■ So, via the Laplace initial-value theorem,
x.0C/ D lims!1
sX.s/
D lims!1
s.sI " A/"1B
D lims!1
!
I "A
s
""1
B
D B:
■ Thus, an impulse input brings the state to B from 0.
■ What if u.t/ D ı.k/.t/?
■ Then
X.s/ D .sI " A/"1Bsk D1
s
!
I "A
s
""1
Bsk
D1
s
!
I CA
sC
A2
s2C : : :
"
„ ƒ‚ …
holds for large s
Bsk
D Bsk"1 C ABsk"2 C A2Bsk"3 C $ $ $CAkB
sC
AkC1B
s2C $ $ $
■ The first terms are impulsive: they have zero value for t > 0.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–6
■ Thus,
x.0C/ D lims!1
s
!
AkB
sC
AkC1B
s2C $ $ $
"
D AkB:
So, if u.t/ D ı.k/.t/ then x.0C/ D AkB.
■ Now, consider the input
u.t/ D g1ı.t/C g2Pı.t/C $ $ $gnı.n/.t/:
Since x.0"/ D 0, x.0C/ D g1B C g2AB C $ $ $C gnAn"1B, or
x.0C/ D#
B AB $ $ $ An"1B$
„ ƒ‚ …
C
2
64
g1
g2
g3
3
75
where C is called the “controllability matrix.”
CONCLUSION: For a SISO system, if C is nonsingular, then there is an
impulsive input u such that x.0C/ is any desired vector if x.0"/ D 0.
EXTENSION: For a MIMO system, if C is full rank, then there is an
impulsive input u such that x.0C/ is any desired vector if x.0"/ D 0.
■ In fact, we may use
u.t/ Dn"1X
iD0
giı.i/.t/
where 2
64
g1
g2
g3
3
75 D
#
B AB $ $ $ An"1B$"1
xd
where xd is the desired x.0C/ vector.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–7
■ If C is nonsingular, we say fA; Bg is a controllable pair and the system
is controllable.
EXAMPLE: Controllability canonical form:
Px.t/ D
2
64
0 0 "a3
1 0 "a2
0 1 "a1
3
75x.t/C
2
64
1
0
0
3
75u.t/
y.t/ Dh
ˇ1 ˇ2 ˇ3
i
x.t/:
■ Then
C D ŒB AB $ $ $ An"1B#
D
2
64
1 0 0
0 : : : 0
0 0 1
3
75 D In:
■ This is why it is called controllability form!
■ If a system is controllable, we can instantaneously move the state
from any known state to any other state, using impulse-like inputs.
■ Later, we’ll see that smooth inputs can effect the state transfer (not
instantaneously, though!).
DUALITY: fA; B; C; Dg controllable”fAT ; C T ; BT ; DT g observable.
EXAMPLE: Two uncontrollable networks.
1" 1"
1"
1"
1"
1"
1F 1Fx
x1 x2y
u u
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–8
■ In the first one, if x.0/ D 0 then x.t/ D 0 8 t . Cannot influence state!
■ In the second one, if x1.0/ D x2.0/ then x1.t/ D x2.t/ 8 t . Cannot
independently alter state.
Diagonal systems, controllability and observability
■ Recall the diagonal form
Px.t/ D
2
66664
$1 0
$2
: : :
0 $n
3
77775
x.t/C
2
66664
%1
%2:::
%n
3
77775
u.t/
y.t/ Dh
ı1 ı2 $ $ $ ın
i
x.t/Ch
0i
u.t/:
u.t/ y.t/
x1.t/
xn.t/
%1
%n
ı1
ın
$1
$n
R
R
:::
■ When controllable? When observable?
O D
2
66664
C
CA:::
CAn"1
3
77775
D
2
66664
ı1 ı2 $ $ $ ın
$1ı1 $2ı2 $ $ $ $nın
: : :
$n"11 ı1 $n"1
2 ı2 $ $ $ $n"1n ın
3
77775
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–9
D
2
66664
1 1 $ $ $ 1
$1 $2 $ $ $ $n
: : :
$n"11 $n"1
2 $ $ $ $n"1n
3
77775
„ ƒ‚ …
Vandermonde matrixV
2
66664
ı1 0
ı2
: : :
0 ın
3
77775
:
■ Singular?
detfOg D .ı1 $ $ $ ın/ detfVg D .ı1 $ $ $ ın/Y
i<j
.$j " $i /:
CONCLUSION: Observable” $i ¤ $j , i ¤ j and ıi ¤ 0 i D 1; $ $ $ ; n.
u.t/ u.t/ y.t/y.t/
x1.t/x1.t/
x2.t/x2.t/
1
s C 1
1
s C 1
1
s C 1
1
s C 2
■ If $1 D $2 then not observable. Can only “observe” the sum x1 C x2.
■ If ık D 0 then cannot observe mode k.
■ What about controllability? Use duality and switch ıs and %s.
CONCLUSION: Controllable” $i ¤ $j , i ¤ j and %i ¤ 0 i D 1; $ $ $ ; n.
u.t/u.t/ y.t/ y.t/
x1.t/x1.t/
x2.t/ x2.t/
1
s C 1
1
s C 1
1
s C 1
1
s C 2
■ If $1 D $2 then not controllable. Can only “control” the sum x1 C x2.
■ If %k D 0 then cannot control mode k.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–10
5.3: Discrete-time controllability and observability
Discrete-time controllability
■ Similar concept for discrete-time.
■ Consider the problem of driving a system to some arbitrary state xŒn#
xŒk C 1# D AxŒk#C BuŒk#
xŒ1# D AxŒ0#C BuŒ0#
xŒ2# D A ŒAxŒ0#C BuŒ0##C BuŒ1#
xŒ3# D A#
A2xŒ0#C ABuŒ0#C BuŒ1#$
C BuŒ2#
:::
xŒn# D AnxŒ0#Ch
B AB A2B $ $ $ An"1Bi
„ ƒ‚ …
C
2
64
uŒn " 1#:::
uŒ0#
3
75 :
■ Which leads to2
64
uŒn " 1#:::
uŒ0#
3
75 D C"1 ŒxŒn# " AnxŒ0## :
If C has no inverse (det.C/ D 0; C is not full-rank) then these control
signals don’t exist. In that case, the input is only partially effective in
influencing the state.
■ If C is full-rank, then the input can move the system to any arbitrary
state for any xŒ0#.
NOTE I: State transition is not instantaneous. Takes n time steps.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–11
NOTE II: In continuous-time, we used input u.t/ D g0ı.t/C g1Pı.t/C $ $ $ , a
signal we could only approximate in practice. Here, the input is a
perfectly good input signal.
Discrete-time reachability
■ In the literature, there are three different controllability definitions:
1. Transfer any state to any other state.
2. Transfer any state to zero, called controllability to the origin.
3. Transfer the zero state to any state, called controllability from the
origin, or reachability.
■ In continuous time, because eAt is nonsingular, the three definitions
are equivalent.
■ In discrete time, if A is nonsingular, the three definitions are also
equivalent.
■ However, if A is singular, (1) and (3) are equivalent but not (2) and (3).
EXAMPLE:
xŒk C 1# D
2
64
0 1 0
0 0 1
0 0 0
3
75xŒk#C
2
64
0
0
0
3
75uŒk#:
■ Its controllability matrix has rank 0 and the equation is not controllable
in (1) or (3).
■ However, Ak D 0 for k ! 3 so xŒ3# D A3xŒ0# D 0 for any initial state
xŒ0# and any input uŒk#.
■ Thus, the system is controllable to the origin but not controllable from
the origin or reachable.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–12
■ Definition (1) encompasses the other two definitions, so is used as
our definition of controllable.
Discrete-time observability
■ Can we reconstruct the state xŒ0# from the output yŒk# and input uŒk#?
yŒk# D CxŒk#CDuŒk#
yŒ0# D CxŒ0#CDuŒ0#
yŒ1# D C ŒAxŒ0#C BuŒ0##CDuŒ1#
yŒ2# D C#
A2xŒ0#CABuŒ0#C BuŒ1#$
CDuŒ2#
:::
yŒn" 1# D C#
An"1xŒ0#CAn"2BuŒ0#C $ $ $C BuŒn " 1#$
CDuŒn" 1#:
■ In vector form, we can write2
66664
yŒ0#
yŒ1#:::
yŒn" 1#
3
77775
D
2
66664
C
CA:::
CAn"1
3
77775
„ ƒ‚ …
O
xŒ0#C
2
66664
D 0 $ $ $ 0
CB D $ $ $ 0
CAB CB $ $ $ 0::: ::: : : : D
3
77775
„ ƒ‚ …
T
2
66664
uŒ0#
uŒ1#:::
uŒn " 1#
3
77775
:
■ So,
xŒ0# D O"1
2
64
2
64
yŒ0#:::
yŒn " 1#
3
75 " T
2
64
uŒ0#:::
uŒn " 1#
3
75
3
75 :
■ If O is full-rank or nonsingular, xŒ0# may be reconstructed with any
yŒk#; uŒk#. We say that fC; Ag form an “observable pair.”
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–13
■ Do more measurements of yŒn#; yŒnC 1#; : : : help in reconstructing
xŒ0#? No! (Caley–Hamilton theorem: next section).
■ So, if the original state is not “observable” with n measurements, then
it will not be observable with more than n measurements either.
■ Since we know uŒk# and the dynamics of the system, if the system is
observable we can determine the entire state sequence xŒk#, k ! 0
once we determine xŒ0#
xŒn# D AnxŒ0#Cn"1X
iD0
An"1"iBuŒk#
D AnO"1
2
64
2
64
yŒ0#:::
yŒn" 1#
3
75 " T
2
64
uŒ0#:::
uŒn" 1#
3
75
3
75C C
2
64
uŒn" 1#:::
uŒ0#
3
75 :
■ A perfectly good observer (no differentiators...)
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–14
5.4: Cayley–Hamilton Theorem
■ A square matrix A satisfies its own characteristic equation. That is, if
&.$/ D det.$I " A/ D 0
then
&.A/ D 0:
■ We can easily show this if A is diagonalizable. Let
A D V "1ƒV:
■ Then
A2 D V "1ƒV V "1ƒV
D V "1ƒ2V
Ak D V "1ƒkV:
■ The characteristic polynomial is:
&.$/ D $n C an"1$n"1 C $ $ $C a1
so if we replace $ with A we get
&.A/ D An C an"1An"1 C $ $ $C a1I
D V "1#
ƒn C an"1ƒn"1C $ $ $C a1I
$
V:
■ To “prove” the Cayley–Hamilton theorem, we just need to show that
the quantity inside the brackets is zero.
■ It is a diagonal matrix, and each diagonal element has the form
$ni C an"1$
n"1i C $ $ $C a1 D 0
because $i is an eigenvalue of A.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–15
■ So, each diagonal element is zero, and we have shown the proof.
■ If A is not diagonalizable, the same proof may be repeated using the
Jordan form and Jordan blocks:
A D T "1JT:
■ Consider a sketch of the proof for a Jordan block of size 2 and
&.$i / D $3i C a2$
2i C a1$i C a0 D 0:
■ Then
Ji D
"
$i 1
0 $i
#
&.Ji/ D
"
$3i 3$2
i
0 $3i
#
C a2
"
$2i 2$i
0 $2i
#
C a1
"
$i 1
0 $i
#
C a0
"
1 0
0 1
#
:
■ We can easily see that the diagonal and lower-diagonal components
are zero so
&.Ji/ D
"
0 ˛
0 0
#
where
˛ D 3$2i C 2a2$i C a1
but ˛ Dd
d$&.$/ D 0 which completes the sketch.
SIGNIFICANCE: The Cayley–Hamilton theorem shows us that An is a
function of matrix powers An"1 down to A0. Therefore, to compute any
polynomial of A it suffices to compute only powers of A up to An"1
and appropriately weight their sum. A lot of proofs use the
Cayley–Hamilton theorem.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–16
■ As we just saw with the section on discrete-time observability, the
Cayley–Hamilton theorem implies that if we cannot observe the state
with n measurements, we cannot observe it with more measurements
either.
EXAMPLE: With A D
"
1 2
3 4
#
we have &.$/ D det.$I " A/, so
&.$/ D det
"
$ 0
0 $
#
" A
!
&.$/ D $2 " 5$ " 2
&.A/ D A2 " 5A " 2I
D
"
7 10
15 22
#
" 5
"
1 2
3 4
#
" 2
"
1 0
0 1
#
D
"
0 0
0 0
#
:
Disclaimer: Does observability/controllability matter, practically?
■ The singularity of C has only one “bit” of information: Is the realization
mathematically controllable or not? This may not tell the whole story.
A D
"
1 0
0 1C '
#
; B D
"
1
1
#
; C D
"
1 1
1 1C '
#
■ fA; Bg are a controllable pair, but barely.
EXAMPLE: Controlling an airplane. (Ideas only, no details). System state
x4Dh
( P( ) P)iT
; ( D Pitch; ) D Roll:
■ Control with elevator?
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–17
Px D
"
F( '
0 F)
#
x C
2
66664
0
1
0
0
3
77775
ıe
where ıe is the elevator angle. C is singular ➠ can’t influence roll with
elevators.
■ Control with ailerons?
Px D
"
F( '
0 F)
#
x C
2
66664
0
0
0
1
3
77775
ıa
where ıa is the aileron angle. C is nonsingular! So, we can control
both pitch AND roll with ailerons.
■ THIS IS NONSENSE ! Physically think of the system! Do you want to
roll plane over every time you need to pitch down?
■ Physical intuition can be better than finding C. Other tools can help. . .
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–18
5.5: Continuous-time Gramians
Continuous-time controllability Gramian
■ If a continuous-time system is controllable, then
Wc.t/ D
Z t
0
eA!BBT eAT ! d!
is nonsingular for t > 0.
SIGNIFICANCE: Consider
x.t1/ D eAt1x.0/C
Z t1
0
eA.t1"!/Bu.!/ d!:
■ We claim that for any x.0/ D x0 and any x.t1/ D x1 the input
u.t/ D "BT eAT .t1"t /W "1c .t1/
#
eAt1x0 " x1
$
will transfer x0 to x1 at time t1.
PROOF: Substitute the expression for u.t/ into the convolution
expression:
x.t1/ D eAt1x.0/ "
Z t1
0
eA.t1"!/BBT eAT .t1"!/ d! W "1c .t1/
#
eAt1x0 " x1
$
D eAt1x.0/ "
Z t1
0
eAˇBBT eAT ˇ dˇ W "1c .t1/
#
eAt1x0 " x1
$
D eAt1x.0/ "Wc.t1/W"1
c .t1/#
eAt1x0 " x1
$
D eAt1x.0/ " eAt1x0 C x1 D x1:
■ Therefore, we can compute the input u.t/ required to transfer the
state of the system from one state to another over an arbitrary interval
of time. The solution is also the minimum-energy solution.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–19
EXAMPLE: Consider the system in diagonal form
Px.t/ D
"
"0:5 0
0 "1
#
x.t/C
"
0:5
1
#
u.t/:
■ The controllability matrix is:
C D
"
0:5 "0:25
1 "1
#
which has rank 2, so the system is controllable.
■ Consider the input required to move the system state from
x.0/ D Œ10 " 1#T to zero in two seconds.
Wc.2/ D
Z 2
0
"
e"0:5! 0
0 e"!
#"
0:5
1
#h
0:5 1i"
e"0:5! 0
0 e"!
#!
d!
D
"
0:2162 0:3167
0:3167 0:4908
#
;
and
u.t/ D "h
0:5 1i"
e"0:5.2"t / 0
0 e".2"t /
#
Wc.2/"1
"
e"1 0
0 e"2
#"
10
"1
#
D "58:82e0:5t C 27:96et :
■ If a continuous-time system is controllable, and if it is also stable, then
Wc D
Z 1
0
eA!BBT eAT ! d!
can be found by solving for the unique (positive-definite) solution to
the (Lyapunov) equation
AWc CWcAT D "BBT :
Wc is called the controllability Gramian.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–20
PROOF: We proved this identity when considering Lyapunov stability.
■ Wc measures the minimum energy required to reach a desired point
x1 starting at x.0/ D 0 (with no limit on t)
min
(Z t
0
ku.!/k2 d!
ˇˇˇˇˇ
x.0/ D 0; x.t/ D x1
)
D xT1 W "1
c x1:
■ In fact, for any specific “t”, the minimum energy is xT1 W "1
c .t/x1.
■ If A is stable, W "1c > 0 which implies “we can’t get anywhere for free”.
■ If A is unstable, then W "1c can have a nonzero nullspace W "1
c ´ D 0 for
some ´ ¤ 0 which means that we can get to ´ using u’s with energy
as small as you like! (u just gives a little kick to the state; the
instability carries it out to ´ efficiently).
■ Wc may be a better indicator of controllability than C.
Continuous-time observability Gramian
■ If a system is observable, Wo.t/ is nonsingular for t > 0 where
Wo.t/ D
Z t
0
eAT !C T CeA! d!:
SIGNIFICANCE: We can prove that
x.0/ D W "1o .t1/
Z t1
0
eAT tC T Ny.t/ dt
where
Ny.t/ D y.t/" C
Z t
0
eA.t"!/Bu.!/ d! "Du.t/ D CeAtx.0/:
■ Therefore, we can determine the initial state x.0/ given a finite
observation period (and not use differentiators!).
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–21
PROOF: We prove that the above equations are correct by substitution:
W "1o .t1/
Z t1
0
eAT tC T Ny.t/ dt D W "1o .t1/
Z t1
0
eAT tC T CeAt dt x.0/
D W "1o .t1/Wo.t1/x.0/
D x.0/:
■ If a continuous-time system is observable, and if it is also stable, then
Wo D
Z 1
0
eAT !C T CeA! d!
is the unique (positive-definite) solution to the (Lyapunov) equation
AT Wo CWoA D "C T C:
Wo is called the observability Gramian.
■ This relationship can be proven in the same way we proved the
similar relationship for the controllability gramian.
■ If measurement (sensor) noise is IID N .0; *2I / then Wo is a measure
of error covariance in measuring x.0/ from u and y over longer and
longer periods
limt!1
E k Ox.0/ " x.0/k2 D *x.0/T W "1o x.0/:
■ If A is stable, then W "1o > 0 and we can’t estimate the initial state
perfectly even with an infinite number of measurements u.t/ and y.t/
for t ! 0 (since memory of x.0/ fades).
■ If A is not stable then W "1o can have a nonzero nullspace W "1
o x.0/ D 0
which means that the covariance goes to zero as t !1.
■ Wo may be a better indicator of observability than O.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–22
5.6: Discrete-time Gramians
Discrete-time controllability Gramian
■ In discrete-time, if a system is controllable, then
WdcŒn " 1# Dn"1X
mD0
AmBBT .AT /m
is nonsingular. In particular,
Wdc D1X
mD0
AmBBT .AT /m
is called the discrete-time controllability Gramian and is the unique
positive-definite solution to the Lyapunov equation
Wdc " AWdcAT D BBT :
■ As with continuous-time, Wdc measures the minimum energy required
to reach a desired point x1 starting at xŒ0# D 0 (with no limit on m)
min
(mX
kD0
kuŒk#k2
ˇˇˇˇˇ
xŒ0# D 0; xŒm# D x1
)
D xT1 W "1
dc x1:
ASIDE: When considering discrete-time stability, we showed that this
form of equation is indeed a Lyapunov equation.
Discrete-time observability Gramian
■ In discrete-time, if a system is observable, then
WdoŒn " 1# Dn"1X
mD0
.AT /mCC T Am
is nonsingular. In particular,
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–23
Wdo D1X
mD0
.AT /mCC T Am
is called the discrete-time observability Gramian and is the unique
positive-definite solution to the Lyapunov equation
Wdo " AT WdoA D C T C:
■ As with continuous-time, if measurement (sensor) noise is IID
N .0; *I / then Wdo is a measure of error covariance in measuring xŒ0#
from u and y over longer and longer periods
limt!1
E k OxŒ0# " xŒ0#k2 D *xŒ0#T W "1do xŒ0#:
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–24
5.7: Computing transformation matrices
Transformation to controllability form
■ Given a system
Px.t/ D Ax.t/C Bu.t/
y.t/ D Cx.t/CDu.t/;
can we find a transformation (similarity) matrix T to transform this
system into controllability form? Recall, this looks like:
Pxco.t/ D
2
66664
0 $ $ $ 0 "an
1 ::: "an"1::: : : : :::
0 $ $ $ 1 "a1
3
77775
xco.t/C
2
66664
1
0:::
0
3
77775
u.t/
y.t/ Dh
ˇ1 ˇ2 $ $ $ ˇn
i
xco.t/CDu.t/:
■ Note that Cco D I so it is controllable. Thus, our original system must
be controllable.
■ The transformation is accomplished via
x D T xco:
■ Thus
T "1AT D Aco; T "1B D Bco
C T D Cco; D D Dco:
■ Let’s find T explicitly; let
T D Œt1; : : : ; tn#:
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–25
■ Note that B D TBco or
B D T
2
66664
1
0:::
0
3
77775
D t1:
■ So, t1 D B. From AT D TAco we have
AŒt1; : : : ; tn# D Œt1; : : : ; tn#
2
66664
0 $ $ $ 0 "an
1 ::: "an"1::: : : : :::
0 $ $ $ 1 "a1
3
77775
:
■ So
ŒAt1; : : : Atn# D
"
t2; : : : ; tn;"nX
kD1
aktn"kC1
#
:
■ By induction,
At1 D t2 D AB
At2 D t3 D A2B; and so forth : : :
so
T D ŒB AB $ $ $ An"1B# D C:
CONCLUSION: A system fA; B; C; Dg can be transformed to controllability
canonical form if and only if it is controllable, in which case the
change of coordinates is
x D Cxco:
EXTENSION I: If xold D T xnew then T D ColdC"1new. That is, to convert
between any two realizations, T is a combination of the controllability
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–26
matrices of the two different realizations.
Cnew D#
Bnew AnewBnew $ $ $ An"1new Bnew
$
D#
T "1Bold T "1AoldT T "1Bold $ $ $ T"1An"1
old T T "1Bold
$
D T "1Cold;
or
T D ColdC"1new:
EXTENSION II: If xold D T xnew then T D O"1oldOnew. This can be shown in a
similar way.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–27
5.8: Canonical (Kalman) decompositions
■ What happens if fA; Bg not controllable or if fA; C g not observable?
% Is “part” of the system controllable?
% Is “part” of the system observable?
■ Given a system with fA; Bg not controllable, Px.t/ D Ax.t/C Bu.t/,
y.t/ D Cx.t/, let’s try to transform to controllability canonical form.
■ Let t1 D B; t2 D AB; : : : tr D Ar"1B, and suppose t1; t2; : : : ; tr are
independent but
trC1 D ArB D "˛rt1 " $ $ $ " ˛1tr
for some constants ˛1; : : : ; ˛r .
■ Then rank.C/ D r since the vectors ArB; ArC1B; : : : ; An"1B can all be
expressed as a linear combination of t1; t2; : : : ; tr .
■ Let srC1; : : : ; sn be your favorite vectors for which
NCold Dh
t1 $ $ $ tr srC1 $ $ $ sn
i
is invertible.
% If we were able to change coordinates to controllability form via
xold D T xnew, we would use T D ColdC"1new.
% But, the system is not controllable, so we use T D NColdC"1new D NCold.
■ We get (in the new coordinate system)"
Pxc.t/
Px Nc.t/
#
D
"
Ac A12
0 A Nc
#"
xc.t/
x Nc.t/
#
C
"
Bc
0
#
u.t/
y.t/ Dh
Cc C Nc
i"
xc.t/
x Nc.t/
#
CDu.t/:
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–28
■ Ac is a right-companion matrix and Bc is of the
controllability-canonical form.
■ We see that the uncontrollable modes x Nc are completely decoupled
from u.t/.
EXAMPLE: Consider1
s C 1D
s " 1
.s C 1/.s " 1/D
s " 1
s2 " 1:
■ In observer-canonical form,
Px.t/ D
"
0 1
1 0
#
x.t/C
"
1
"1
#
u.t/
y.t/ Dh
1 0i
x.t/:
■ So,
t1 D
"
1
"1
#
and t2 D At1 D
"
"1
1
#
D "t1:
■ So, rank.C/ D 1. Let s1 D Œ1 0#T . The converted state-space form is
PNx.t/ D
"
"1 "1
0 1
#
Nx.t/C
"
1
0
#
u.t/
y.t/ Dh
1 1i
Nx.t/:
■ Now suppose that we desire to transform a system that is not
observable. The dual form separates out the unobservable states.
■ Let t1 D C; t2 D CA; : : : tr D CAr"1, and suppose t1; t2; : : : ; tr are
independent but
trC1 D CAr D "˛rt1 " $ $ $ " ˛1tr
for some constants ˛1; : : : ; ˛r .
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–29
■ Then, let srC1; : : : ; sn be your favorite row vectors for which
NOold D
2
6666666664
t1:::tr
srC1
:::sn
3
7777777775
is invertible.
% If we were able to change coordinates to observability form via
xold D T xnew, we would use T D O"1oldOnew.
% But, the system is not observable, so we use T D NO"1oldOnew D NO
"1old.
■ Then, we get the Kalman decomposition"
Pxo.t/
Px No.t/
#
D
"
Ao 0
A21 A No
#"
xo.t/
x No.t/
#
C
"
Bo
B No
#
u.t/
y.t/ Dh
Co 0i"
xo.t/
x No.t/
#
CDu.t/:
■ Note: No path from x No to y!
Full Kalman decomposition
■ We can do both transformations in sequence to get full Kalman
decomposition. See text chapter 16.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–30
5.9: Popov–Belevitch–Hautus controllability/observability tests
PBH EIGENVECTOR TEST: fC; Ag is an unobservable pair iff a non-zero
eigenvector v of A satisfies C v D 0. (i.e., C and v are perpendicular).
PROOF:) Suppose Av D $v and C v D 0 for v ¤ 0. Then
CAv D $C v D 0 and so forth up to CAn"1v D $n"1C v D 0. So,
Ov D 0 and since v ¤ 0 this means that O is not full rank and fC; Ag
is not observable.
■( Now suppose that O is not full rank (fC; Ag unobservable). Let’s
extract the unobservable part. That is, find T such that
T "1AT D
"
Ao 0
A21 A No
#
C T Dh
Co 0i
;
where Ao is size r where r D rank.O/ and therefore A No is size n " r .
% Let v2 ¤ 0 be an eigenvector of A No. Then
T "1AT
"
0
v2
#
D
"
Ao 0
A21 A No
#"
0
v2
#
D $
"
0
v2
#
so we have found an eigenvector ´ of A
A´ D $´ where ´ D T
"
0
v2
#
:
% Now, we just need to show that C ´ D 0 (note: ´ ¤ 0).
C ´ D C T„ƒ‚…
NC
"
0
v2
#
Dh
Co 0i"
0
v2
#
D 0
and we are done.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–31
DUAL: fA; Bg is uncontrollable iff there is a left eigenvector wT of A such
that wT B D 0.
INTERPRETATION: In modal coordinates, homogeneous response is
x.t/ D
nX
iD1
e$i tvi.wTi x.0// and y.t/ D Cx.t/:
Or,
y.t/ D
nX
iD1
e$i tC vi.wTi x.0//:
If fC; Ag is unobservable, then it has an unobservable mode, where
Avi D $ivi and C vi D 0.
■ If fA; Bg is uncontrollable, then it has an uncontrollable mode, namely
the coefficients of the state along that mode is independent of the
input u.t/.
% The coefficients of x in the mode associated with $ are wT x.
d
dt.wT x/ D wT .Ax C Bu/ D $.wT x/
or
wT x.t/ D e$t .wT x.0//
regardless of the input u.t/!
PBH RANK TESTS: The following two tests are often easier to perform
1. fA; Bg controllable iff rankh
sI " A Bi
D n for all s 2 C.
) If rankh
sI " A Bi
D n for all s 2 C then there can be no nonzero
vector v such that vTh
sI " A Bi
Dh
vT .sI " A/ vT Bi
D 0:
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–32
■ Consequently, there is no nonzero vector v such that vT s D vT A
and vT B D 0. By the PBH eigenvector test, the system will
therefore be controllable
( If the system is controllable, then there is no nonzero vector v
such that vT s D vT A and vT B D 0 by the PBH eigenvector test.
■ Therefore, rankh
sI " A Bi
D n for all s 2 C.
2. fC; Ag observable iff rank
"
C
sI " A
#
D n for all s 2 C. Proof similar.
COMMENTS: rankh
sI " A Bi
D n for all s not eigenvalues of A, so the
test is really fA; Bg controllable iff rankh
$iI " A Bi
D n for $i ,
i D 1; : : : ; n the eigenvalues of A. (Dual argument for observability).
■ Ifh
sI " A Bi
drops rank at s D $ then there is an uncontrollable
mode with exponent (frequency) $.
■ If
"
C
sI " A
#
drops rank at s D $ then there is an unobservable
mode with exponent (frequency) $.
Summary
■ Therefore, we can label individual modes of a system as either
controllable or not, or observable or not.
■ The overall picture is:
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–33
replacements Controllableand
observable
Controllablebut not
observable
Observablebut not
controllable
Neitherobservable nor
controllable
u.t/ y.t/
■ Some other definitions:
STABILIZABLE: A system whose unstable modes are controllable.
DETECTABLE: A system whose unstable modes are observable.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–34
5.10: Minimal realizations—Why a system isn’t control/observ-able
■ To realize H.s/ D1
s C 1we could use
1. Px.t/ D "x.t/C u.t/, y.t/ D x.t/. This gives
A D Œ"1#, B D Œ1#, C D Œ1#, D D Œ0#.
C adj.sI " A/B
det.sI " A/D
1
s C 1:
This realization is both controllable and observable.
2. Observer realization of1
s C 1
s " 1
s " 1D
s " 1
s2 " 1. This gives
A D
"
0 1
1 0
#
, B D
"
1
"1
#
, C Dh
1 0i
, D D Œ0#.
C adj.sI " A/B
det.sI " A/D
s " 1
s2 " 1D
1
s C 1:
O D
"
C
CA
#
D
"
1 0
0 1
#
. Observable.
C Dh
B ABi
D
"
1 "1
"1 1
#
. Not controllable.
3. Controller realization of1
s C 1
s C 10
s C 10D
s C 10
s2C 11s C 10.
A D
"
"11 "10
1 0
#
, B D
"
1
0
#
, C Dh
1 10i
, D D Œ0#.
C adj.sI " A/B
det.sI " A/D
s C 10
s2 C 11s C 10D
1
s C 1:
C Dh
B ABi
D
"
1 "11
0 1
#
. Controllable.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–35
O D
"
C
CA
#
D
"
1 10
"1 "10
#
. Not observable.
TREND: Non-minimal realizations of transfer functions will either be
uncontrollable, unobservable, or both.
■ Four equivalent statements:
I: There exist common roots of C adj.sI " A/B and det.sI " A/.
II: There exist eigenvalues of A which are not poles of G.s/, counting
multiplicities.
III: The system is either unobservable or uncontrollable.
IV: There exist extra (unnecessary) states—non minimal.
DEFINITION: We say a system is minimal if no system with the same
transfer function has fewer states.
PROOF: I”II
IH)II: The transfer function
G.s/ D C.sI " A/"1B CD
DC adj.sI " A/B CD det.sI " A/
det.sI " A/:
If there are common roots in C adj.sI " A/B and det.sI " A/ they will
cancel out of the transfer function. But, eigenvalues of A are
det.sI " A/ D 0, so poles of G.s/ will not contain all eigenvalues of A.
IIH)I: Eigenvalues of A are det.sI " A/ D 0. Poles of G.s/, from above,
are det.sI " A/ D 0 unless canceled. The only way to cancel a pole is
to have common root in C adj.sI " A/B.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–36
PROOF: I”IV fA; B; C; Dg is minimal iff C adj.sI " A/B and det.sI " A/
are coprime (have no common roots).
IH)IV: Suppose C adj.sI " A/B and det.sI " A/ have common roots.
■ Cancel to get
G.s/ DC adj.sI " A/B
det.sI " A/CD D
br.s/
ar.s/
where br.s/ and ar.s/ are coprime (“r” means “reduced”).
■ Because of cancellation, k D deg.ar/ < deg.det.sI " A// D n.
■ Consider controller canonical form realization of br.s/=ar.s/, for
example.
■ It has k states, but same transfer function as fA; B; C; Dg,
contradicting that fA; B; C; Dg minimal.
IVH)I: If there are extra states (non-minimal) then n > k (n D# states,
k D# poles in G.s/).
G.s/ D C.sI " A/"1B CD
DC adj.sI " A/B CD det.sI " A/
det.sI " A/
G.s/ has n poles unless some cancel with C adj.sI " A/B. Therefore,
if n > k, C adj.sI " A/B and det.sI " A/ are not coprime.
PROOF: III”IV Controllable and observable iff minimal.
IIIH)IV: Uncontrollable or unobservable H) not minimal.
Perform Kalman decomposition to split system into co, c No, Nco and Nc No
parts.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–37
u.t/
y.t/
xc.t/
xc.t/
Ac
A Nc
A12
Bc Cc
C Nc
R
RUncontrollablepart ofrealization
NA D
"
Ac A12
0 A Nc
#
, NB D
"
Bc
0
#
, NC Dh
Cc C Nc
i
, ND D Œ0#.
Same transfer function using Ac, Bc, Cc as NA, NB, NC . Therefore
uncontrollable and/or unobservable means not minimal.
IVH)III: Non-minimal means uncontrollable or unobservable.
■ Suppose fA; B; C; Dg is non-minimal.
C.sI " A/"1B CD„ ƒ‚ …
n states
D NC .sI " NA/"1 NB C ND„ ƒ‚ …
r<n states
■
CB
sC
CAB
s2C
CA2B
s3C $ $ $ D
NC NB
sCNC NA NB
s2CNC NA2 NB
s3C $ $ $
■ Consider
OC D
2
6666664
C
CA
CA2
:::
CAn"1
3
7777775
h
B AB A2B $ $ $ An"1Bi
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett
ECE4520/5520, OBSERVABILITY AND CONTROLLABILITY 5–38
D
2
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CB $ $ $ CAn"1B::: :::
CAn"1B $ $ $ CA2n"2B
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NC NB $ $ $ NC NAn"1 NB::: :::
NC NAn"1 NB $ $ $ NC NA2n"2 NB
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NC NA::: 0
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■ Therefore det.O/ det.C/ D 0, so the system is either unobservable,
or uncontrollable, or both.
■ The four equivalences have now been proven.
FACT: All minimal realization of G.s/ are related by a unique change of
coordinates T . Can you prove this?
Where to from here?
■ Have seen important topics of observability and controllability.
■ Now understand how a system could be unobservable or
uncontrollable.
% If physically true, may need to add sensors or actuators.
■ Important to work with minimal realizations, when possible.
■ Now, time to consider: How do I build a controller?
% Very powerful tools in next chapter.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright c# 2015, 2011, 2009, 2007, 2005, 2003, 2001, 2000, Gregory L. Plett