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Chapter 3hapter 3hapter 3hapter 3StateState--Space RealizationsSpace Realizations
Assoc. Prof. Dr. Mohamad Noh Ahmad.
Faculty of Electrical Engineering
Universiti Teknologi Malaysia
u a
[email protected] / [email protected] 012-7379299
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:
)()()( ttt BuAxx +=& )()()( tutt BxAx +=&
uy =)()( tt Pxx =
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&v - :
)()()(
ttt DuCxy += . .
xPxPxx-1
==&
PBuPAxxPx +== & PBuxPAP +=
DuxCPDuCxy +=+= 1
DDCPCPBBPAPA ==== ,,, --
(**))()()(;)()()( uDxCyuBxAx tttttt +=+=&
Let
(*) and (**) are said to be equivalent to each other
and the procedure from (*) to (**) is called an
equivalent transformation.
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The feedforward matrix D between the input and output has nothing
to do with the state space and is not affected by the equivalenttransformation.
The characteristic equation for (*) is:
For (**), we have
AI = )( = APPPP =
APPPIP11
)(
= PAIP )(1
=
AIPP = 1 AI =
and hence the same set of eigenvalues.
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Recall: othereachsimilar toareand- APAPA =
They have same eigenvalues, same stability perf.
Similar transfer functions!!
1 DBAICG += ss DBAICG += 1ssand
)()( ss GG =To verify,
DBAICG += 1)()( ss
DPBPAPPPCP += 111-1
)(s1111
)( = XYZXYZDPBPAIPCP += -1-1-1 ))-(( s
DPBPAIPCP += 111 )(s
)()1
s(s GDBAIC =+=
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.
x. . :
From the circuit (via observation):
11 xx =
=
2
1
2
1
11
01
x
x
x
x
1)( 212 = xxx)()( tt xPx =Or
=
1
1
1 01 xx = 101 x
22
11 xx
2
11 x
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Two state equations are said to be zero-state equivalent
if they have the same transfer matrix or
BAICDBAICD 11 )()( +=+ ss
Note that:
L+++= 23211
)( AAIAI ssss,
LL ++++=++++ 3221321 ssssss BACBACBCDBCACABCBD 2
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Theorem 3.1
are zero-state equivalent or have the same transfermatrix iff and
, ,, ,,,
DD =
,...2,1,0; == mmm BACBCA
- In order for two state equations to be equivalent, they
.- This is, however, not the case for zero-state equivalence.
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Consider:
)(5.0)( = tuty )()( = txtx&
0=== B
5.0;5.0;0;1
0.5u(t))(5.0)(
====
+=
DCBA
txty
5.0== DD
Note that:
0== BACBCA mm
The two s stems are zero-state e uivalent.~ Theorem 3.1
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0123
Consider: )()()(
1
0)(
134
012)( tuttutt bAxxx +=
+
=&
It can be shown that:
2
YES!,,
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en:
1700
Thus the representation of A w.r.t. the basis is:}{2bAAb,B,
=
510
1501A
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12)3(
1700 ny spec a
)1(34
0)1(2)(
23 ==
==
I
=
510
1501A
000 43214 Companion-form matrices:
(-1)
0100
0010;
100
010
1
2
3
(-1) -
Transpose
100
010;
1000
0100
3
2
1
All have similar
characteristic
00041234
43
2
2
3
1
4
)( ++++=
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Q. What is "realization"?
For a iven , find a corres ondin state-)()( ss NG =space equation
&
s
)()()( ttt DuCxy +=
Implicitly implies LTI systems
Shall start with multi-variable systems and willsometimes specialize to single-variable systems
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Q. If (s) is realizable, how many possible
realizations? Infinite ~ in view of equivalent transformations and the
possibility of adding un-controllable or un-observable
componentsQ. Which one is the "good" realization?
- A good realization is the one with the minimal order
~ Irreducible realization Will be discussed in Topic 6
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Q. Under what condition is (s) realizable by an
LTI system?-Recall that the transfer function of the dynamic equation is
DBAICDBAICG +=+= 11 sss
Theorem 3.2(s) is realizable by a dynamic equation iff it is a
proper rational function (order of numerator order ofdenominator)
In fact, the part contributed by C(sI - A)-1B iss r c y proper
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To determine the realization of , decompose it
as:
)( sG
)()()( ss spGGG +=
[ ]BAICAIBAICG ).()(:)(1
==
sAdjssssp
rr
rr ssssd ++++=
1
1
1)( LLet
ssp.e. eas common enom na or o a en r es o .
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Then the realization of is given by
IIIII L pprprpp 121
)(sG
u0
0
x00I0
000I
x
+
= L
L
& p
p
~Block companion form
00I00 L p
r r [ ] uGxNNNNy )(121 += rrL
(q rp)where Ip is thep p unit matrix and every 0 is ap p zero matrix.
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Proof:
Define, Z
Z
12
1
Z
r
M i .
Then,Simplifying, BZAI = )(s (*)BAZZ += s
0
I
Z
Z
000I
IIII
Z
Z
L
Lp
p
prprpp
s
s
2
1121
2
1
+
=
0I00
MM
L
MMMM
L
M
ps 33
rpr
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Using the shifting property of the companion form of A, we obtain
12312 ,,, === rrsss ZZZZZZ Lwhich implies
1,,
1,
1ZZZZZZ === L
Substituting these into the 1st block of equation (*) yields
s IZZZZ += L
sss
pr
r
ss
IZ +
+++=
112
1
L
or
r sds IZ ==
++++ 1
21
)( L
sss
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Thus we have,
rr
ss IZIZIZ 121
===
L
Then,
prppsdsdsd )()()(
)()(1
+
GBAIC s
[ ] )()(
2
2
1
1 ++++=
GNNN rrr
sssd L
)()( += GG ssp
sG=
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.3104 s
,
)2(
1
)2)(12(
1212
)(
2
+
+
++
++=
s
s
ss
sssGConsider a proper rational matrix
Determine a realization of this transfer matrix.Solution:
312
3104sDecompose into a strictly proper rational matrix)(sG
+
+
++
+
=
2)2(
1
)2)(12(
100
s
s
ss
ss)()( sspGG +=
+
+
++
=
2)2(
1
)2)(12(
1)(
s
s
ss
sssG
265.4)2)(5.0()(232 +++=++= ssssssd
The monic least common denominator of is)(sspG
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.
+++
+++
+++=
)5.0)(1()2(5.0
)5.0)(2(3)2(6
265.4
12
23
sss
sss
sss
spGThus
+++
++++=
)5.05.1()2(5.0
.
)(
12 sss
ssss
sd
22
+++=
5.05.115.0
.
)( 2 ssssd
3245.72436
+
+
=5.015.15.010)(
sssd
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Given,
,11
212)(
+
++
=s
ss
s
sG )(
sG)2()2)(12(
2 +++ sss
Realizationthe realization is,
10
01
20605.40
020605.4
+
=
2
1
00
00
000100
000010 uxx&
DC
=S
023245.72436 u
00001000 { }DC,B,A,=
+
=2005.015.15.010 u
xy
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Consider again the proper rational matrix in Ex. 3.3.
3104s
+
+
++
++=
2)2(
1
)2)(12(
1212
)(
s
s
ss
sssG
The 1st column is
+
=1
12)( 1
sscG
++
=1
)2)(12( ss
++
=
1
2522 ss
ss
++ )2)(12( ss ++ )2)(12( ss ++ 2522 ss
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. . .
yields the following realization for the 1st
column of :
s ng : n1= 4 2 20;0 0 1 ;d1= 2 5 2 ; a,b,c,d =tf2ss n1,d1
)(sG
11111110
1
01
15.2uu
+
=+= xbxAx&
111111105.00 uuc +
=+= xdxCy
Similarly, the function tf2ss can generate the following realization for the2nd column of :)( sG
2222222
063
001uu
+
=+= xbxAx&
2222222
011
uuc
+
=+= xdxCy
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. . .
ese two rea zat ons can e com ne as
0bx0Ax11111
+
= u
&
&
[ ] [ ]uddxCCyyy
xx
212121
22222
+=+= cc
u
or
00
01
0001
0015.2
10
01
20605.40
020605.4
uxx
+
=
00
10
0100
4400&
+
=2
1
00
00
000010
000001
u
uxx&
(4x4) (4x2) (6x6) (4x2)uxy
+
=
00
02
115.00
63126
00001000
1023245.72436 uC D 2005.015.15.010 ux x
(2x6) (2x2)
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Special Case: p = 1 (For simplicity, let r= 4 and q =2.)
Consider a 21 proper rational matrix:
+++
+++
+++++
=
2423
2
22
3
21
1413
2
12
3
11
43
2
2
3
1
4
2
1 1)(
sss
sss
ssssd
dsG
Then, the realization is 14321 Note: The
u
+
=
0
0
0010
0001xx&
controllable-canonical-form
ud
+
= 114131211 xy
rea za on can eread out from the
224232221
.