1 數位控制（十）. 2 continuous time ss equations 3 discretization of continuous time ss...
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1
數位控制（十）
2
Continuous time SS equations
dBuextx
tBuetxedt
dtAxtxe
tButAxtx
AeAtAk
tAAtI
AetAk
tAAtIA
tAk
tAtAAedt
d
k
tAtA
ktAAtIe
t tA
AtAtAt
Atkk
Atkk
kkAt
k
kkkkAt
0)(At
22
22
1332
0
22
)()0(e)(
gives t,and 0between integrate
)()]([)]()([
)()()(
]!
1
!2
1[
]!
1
!2
1[
)!1(
1
!2
1
!!
1
!2
1
3
Discretization of continuous time SS equations
)()()(
)()()()(1)T)x((k
)(,eG(T) define weif
)( ,)()(e
)()(e1)T)x((k
)()(e1)T)x((k
)()0(ex(kT) and
)()0(e1)T)x((k
T period sampling theon depend H and G
H(T)u(kT)G(T)x(kT)1)T)x((k
tiondiscretiza do
0
AT
0
AT
0
AT
)1()1(AT
0
AkT
)1(
0
)1(1)TA(k
kTDukTCxkTy
kTuTHkTxTG
BdeTHand
tTdkTBuekTx
dtkTBueekTx
dBueekTx
dBueex
dBueex
DuCxy
BuAxx
T A
T A
T AtAT
Tk
kT
ATkA
kT AAkT
Tk ATkA
4
Liapunov Stability Analysis
eoo
eo
xtxt
xx
txfx
),,(
),( An equilibrium state x of the system is said to be stabl
e in the sense of Liapunov, if corresponding to each S(e), there is an S(d) such that trajectory starting in S(d) do not leave S(e) as t increase indefinite
5
Pole placement and observer design Controllability: to transfer the system from any arbitra
ry initial state to any desired state. Observability: every initial state x(0) can be determin
ed from the observation of y(kT). The controllability is the basis for the pole placement
problem. The concept of observability play an important role fo
r the design of state observer. Pole placement design technique feedback all state
variables so that all poles of the closed-loop system are placed at designed location.
6
Open-loop control system
Closed-loop control system with u(k)=-Kx(k)
7
Pole placement techniques Assuming all state variables are available for
feedback. To design the state observer that estimates all
state variables that are required for feedback.
8
Simple Pole Placement
0HKG-zI is equation sticcharacteri the
HK)x(k)-(G1)x(k becomes equation state the
system control loop-closeda becomes system thethen
matrix gainfeedbcak state theis K where
-Kx(k),u(k)
as u(k) signal control unbounded thechoose
)()()1(
kHukGxkx
9
Lower order system
234.0
34.0,2k
5.016.0,1k1
j0.5)0.5-j0.5)(z-0.5-(z16.0)1(
016.0)1(116.0
1
1
0
116.0
10
0
0HKG-zI
equation sticcharacteri the
let K
j0.50.5zat pole loop-closed thehave willsystem the
that suchmatrix K gainfeedcack statea determine
1
0 H,
116.0
10G
Hu(k)Gx(k)1)x(k system heconsider t
21
12
12
122
122
21
21
21
kkK
k
k
kzkz
kzkzkzk
z
kkz
z
kk
10
Pole placement basis Completely state controllable: the desired
closed-loop poles can be selected. All the state variables are feedback to place
the poles. In practical measurement of all state variables
may not be possible. Hence, not all state variables will be available for feedback.
Completely output observable: make all state variables be observable or feedback.
11
Controllability matrix
le.controllab state complete is system thespace, ldimensiona-n thespan can
HG ... GH, H, vectorsn then,rank if
)(
))2((
))1((
)0()(
))1(()1()0()0(
,3,2,1),()0()(
)()()(
)()()1(
1-n1
1
21
1
0
1
nHGGHH
ou
Tnu
Tnu
HGGHHxGnTx
TnHuHuGHuGxG
kjTHuGxGnTx
kDukCxky
kHukGxkx
n
nn
nnn
n
j
jnn
12
Rank of a matrix A matrix A is called of rank n if the maximum n
umber of linear independent rows (or columns) is n.
Important Properties The rank is invariant under the interchange of two
rows (or columns), or addition, or multiplication. For a n by n matrix A, for rank A=n imply det(A) is
not equal to zero. For a n by n matrix A, rank A*=rank A, or rank AT=
rank A.
13
Complete State Controllability
elements. zero all has Fofrow no ifonly and if
lecontrollab state completely is system the
define uslet
)()(ˆ))1((ˆ
)(ˆ)( define uslet
0
0
thatsuch matrix P mationa transfor find topossible isit then
distinct, are G of rseigenvecto theif
)()())1((
21
22221
11211
1
11
2
1
1
nrnn
r
r
n
-
fff
fff
fff
FHP
kTHuPkTxGPPTkx
kTxPkTx
GPP
kTHukTGxTkx
14
For multiple eigenvectors (Jordan form)
The system is completely state controllable if and only if No two Jordan blocks are associated with the sa
me eigenvalues, The element of any row of the transformed H that
corresponds to the last row of each Jordan block are not all zero,
The elements of each row of the transformed H that correspond to distinct eigenvalues are not all zero,
15
Example-completely state controllable
)(
)(
12
00
03
00
10
)(
)(
)(
)(
)(
50
15
2
12
012
)1(
)1(
)1(
)1(
)1(
)(3
2
)(
)(
20
01
)1(
)1(
2
1
5
4
3
2
1
5
4
3
2
1
2
1
2
1
ku
ku
kx
kx
kx
kx
kx
kx
kx
kx
kx
kx
kukx
kx
kx
kx
16
Example-not completely state controllable
)(
)(
00
12
00
03
10
)(
)(
)(
)(
)(
50
15
2
12
012
)1(
)1(
)1(
)1(
)1(
)(0
2
)(
)(
20
01
)1(
)1(
2
1
5
4
3
2
1
5
4
3
2
1
2
1
2
1
ku
ku
kx
kx
kx
kx
kx
kx
kx
kx
kx
kx
kukx
kx
kx
kx
17
Complete state controllability in the z plane No cancellation in the pulse transfer function. If cancellation occurs, the system cannot be controlled in the
direction of the canceled mode.
le.controllab state completelynot is 1, isrank the
,64.08.0
8.01since
)(
)(01)(
)(8.0
1
)(
)(
116.0
10
)1(
)1(
le.controllab state completelynot is ,)2.0)(8.0(
2.0
)(
)(
2
1
2
1
2
1
GHH
kx
kxky
kukx
kx
kx
kx
zz
z
zU
zY
18
Complete Output Controllability
t.independenlinear are C of rows m theifonly and if
ility controllaboutput complete impliesility controllab state Complete
le.controllaboutput complete is system thespace, ldimensiona-m thespan can
HCG ... CGH, CH, vectors the then,ctor)(output ve rank if
)(
))2((
))1((
)0()(
)()0(
)()(
,3,2,1),()0()(
)()(
)()()1(
1-n1
1
1
0
1
1
0
1
mHCGCGHCH
ou
Tnu
Tnu
HCGCGHHCxCGnTy
jTHuCGxCG
nTCxnTy
kjTHuGxGnTx
kCxky
kHukGxkx
n
nn
n
j
jnn
n
j
jnn
19
Complete Output Controllability (w/ D)
le.controllaboutput complete is system the
space, ldimensiona-m thespan canH CG ... CH, D, vectors thethen
,ctor)(output ve matrix 1)r (nm theofrank theif
)(
))1((
)(
)0()(
)()()0(
)()()(
,3,2,1),()0()(
)()()(
)()()1(
1-n
1
1
1
0
1
1
0
1
mHCGCHD
ou
Tnu
nTu
HCGCHDxCGnTy
nTDujTHuCGxCG
nTDunTCxnTy
kjTHuGxGnTx
kDukCxky
kHukGxkx
n
nn
n
j
jnn
n
j
jnn