12-23 high gain observer introduction where (a,c) is observable. consider the following system this...
TRANSCRIPT
12-1
High Gain Observer
Introduction
Cxy
uygAxx
),(
where (A,C) is observable.
Consider the following system
This form is special because g depends only on y and u.
Taking the observer as
)ˆ(),(ˆˆ xCyHuygxAx
we obtain that satisfiesxxx ˆ~
xHCAx ~)(~
Therefore, designing C such that AHC is Hurwitz guarantees asymptoticerror convergence.
12-2
High Gain Observer (Continued)
)ˆ(),(ˆˆ 0 xCyHuygxAx
However, any error in modeling g will be reflected in the estimation errorequation. Thus,
where is a nominal model of g. Hence0g
We give a special design of the observer gain that makes the observer robust
to uncertainties in modeling the nonlinear functions. The technique, called as
high-gain observers, works for a wide class of nonlinear systems and guarantees
that the output feedback controller recovers the performance of the state feedback
controller when the observer gain is sufficiently high.
),(),(~)(~0 uyguygxHCAx
12-3
High Gain Observer (Continued)
The main result is a separation principle that allows us to separate the design into
two tasks. First, design a state feedback controller that stabilizes the system and
meets other design specifications. Then, obtain an output feedback controller by
replacing x by provided by the high-gain observer. A key property that makes
this separation possible is the design of the state feedback controller to be globally
bounded in x.
x̂
12-4
Example
Ex:
1
2
21
),(
xy
uxx
xx
Assume that u = (x) is a local state feedback control law that stabilizes the origin.
To implement this control law using only y, we use the observer
)ˆ(),ˆ(ˆ
)ˆ(ˆˆ
1202
1121
xyhuxx
xyhxx
where is a nominal model of the nonlinear function),(0 ux ).,( uxThen
)~,(~~
~~~
122
2111
xxxhx
xxhx
where
))ˆ(,ˆ())ˆ(,()~,(
ˆ~ ,ˆ~222111
xxxxxx
xxxxxx
12-5
Example (Continued)
We want to design such that ThhH 21 .0)(~lim
txt
In the absence of , asymptotic error convergence is achieved by designingH such that
0
1
2
10 h
hA
is Hurwitz. In the presence of , we need to design H with the goal of rejectingthe effect of on .~x
This is ideally achieved, for any , if the transfer function
121
20
11)(
hshshssG
from to is ideally zero. x~
12-6
Example (Continued)
)(sup 0 jwGRw
While this is impossible, we can make arbitrarily small by choosing .112 hh
Taking ,10 and 0,for , 2122
21
1
hh it can be shown that
121
20 )()(
ssssG
Hence .0)(lim 00
sG
Define the scaled estimation errors
221
1~ ,
~x
x
Then the newly defined variables satisfy the singularly perturbed equation.
)~,(122
2111
xx
This equation shows clearly that reducing diminishes the effect of .
12-7
Example (Continued)
Notice, however, that will be whenever )0(1 )/1( O ).0(ˆ)0( 11 xx
Consequently, the solution contains a term of .0 with )/1(
aeat
In fact, .0 asfunction impulsean approaches )/(
at
ea
This behavior is known as the peaking phenomenon.
12-8
Globally Stabilized by State Feedback Controller
1
322
21
xy
uxx
xx
Let’s consider the system
which can be globally stabilized by the state feedback controller
2132 xxxu
The output controller is taken as
)ˆ)(1(ˆ
)ˆ)(2(ˆˆ
ˆˆˆ
122
121
2132
xyx
xyxx
xxxu
where the observer gain assigns the eigen values of .1 and 1at 0 A
12-9
State Feedback Controller
The above figure shows a counter intuitive behavior as 0. Since decreasing
causes the estimation error to decay faster toward zero, one would expect the
response under output feedback to approach the response under state feedback as
decreases. This is the impact of peaking phenomenon. Fortunately, we can
overcome the peaking phenomenon by saturating the control outside a compact
region of interest in order to create a buffer that protects the plant from peaking.
12-10
State Feedback Controller
Suppose the control is saturated as)ˆˆˆ(sat 21
32 xxxu
The above figure shows the performance of the system under saturated state
and output feedback. The control u is shown on a shorter time interval that
exhibits control saturation during peaking. The peaking period decreases with .
12-11
Stabilization
),(
),,(
),,(
zxq
Cxy
uzxz
uzxBAxx
Consider the MIMO nonlinear system
(1)
bygiven are ,, The s. variablestate the
esconsititut and and outputs, measured are
and input, control theis where
CBA
RzRx
RRyRul
SmP
12-12
Stabilization (Continued)
,diagblock 1 mAAA
ii
iA
00
10
0100
0010
,diagblock 1 mm BBB
11
0
0
i
iB
,diagblock 1 mm CCC iiC 1001
. and ,,1 where 21 mmi
12-13
Output Feedback Controller
The functions , and q are locally Lipschitz and (0,0,0)=0, (0,0,0)=0,
q(0,0)=0. Our goal is to find an output feedback controller to stabilize the
origin.
We use a two-step approach to design the output feedback controller.
(i) A partial state feedback controller using x and is designed
to asymptotically stabilize the origin.
(ii) A high-gain observer is used to estimate x from y.
The state feedback controller can be shown as
),,(
),,(
xru
x
where r, are locally Lipschitz in their arguments over the domain of interestand globally bounded functions of x. Moreover, r(0,0,0)=0 and (0,0,0)=0.
12-14
Output Feedback Controller (Continued)
For convenience, we write the closed-loop system under state feedback as
)(XfX (2)
where X = (x, z, ).
The output controller is taken as
),ˆ,(
),ˆ,(
xru
x
(3)
observergain high by the generated is ˆ where x
)ˆ(),,ˆ(ˆˆ 0 xCyHuxBxAx
12-15
Output Feedback Controller (Continued)
The observer gain H is chosen as
,diagblock 1 mHHH
i
ii
i
i
iH
2
2
1
.,,for OLHP in the are
0
of roots thesuch thatchosen
are and specified be toconstant positive a is where
11
1
mii
sss iii
ij
ii
ii
0. )0,0,0( and in boundedglobally andinterest of
domain over the arguments itsin Lipschitzlocally iswhich
),,( of model nominal a is ),,(function The
0
0
x
uxux
12-16
Theorem
Theorem: Consider the closed-loop system of the plant (1) and the output
feedback controller (3). Suppose the origin of (2) is asymptotically
stable and RR is its region of attraction. Let S be any compact set in the
interior of RR and Q be any compact subset of Then,.R
.0 allfor bounded are ,in starting system, loop-closed theof
))(ˆ),(( solutions the,0every for ,such that 0 exists there *1
*1
tQS
txtX
2
*2
2*2
,)(ˆ and )(
satisfy ,in
starting system, loop-closed theof solutions the,0every for that,
such ,on dependent both ,0 and 0exist there,0any given
TttxtX
QS
T
12-17
Theorem (Continued)
.attraction ofregion its ofsubset a is and stablelly exponentia
is system loop-closed theoforigin the,0every for ,such that
0 exists re then thestable,lly exponentia is oforigin theif*4
*4
QS
).0(at starting ofsolution theis )( where
0 ,)()(
satisfy ,in
starting system, loop-closed theof solutions the,0every for
,such that ,on dependent ,0exist there,0any given *3
*3
XtX
ttXtX
QS
r
r
(2)
Proof: See Ch 14.5
(2),
12-18
Results
The theorem shows that the output feedback controller recovers the
performance of the state feedback controller for sufficiently small .Note that
(i) recovery of exponential stability
(ii) recovery of the region of attraction in the sense
that we can recover any compact set in its interior
(iii) the solution under output feedback approaches the
solution under the state feedback as 0.
12-19
Example
Ex: Consider the following plant:
1
213
232
121
sin
sin
xy
uxx
uxxx
txxx
Note : the given system is in triangular form. Thus, it is stabilizableby
33221121 xkxkxkxu
The output feedback controller is
)ˆˆˆˆ(sat 332211213 xkxkxkxu
12-20
Example (Continued)
with
)ˆ(ˆˆ
)ˆ(ˆˆ
)ˆ(ˆˆ
1332
13
122
32
11
21
xyh
uxx
xyh
xx
xyh
xx
For simulation:
All system eigenvalues at 5.
All observer eigenvalues at 5/ with = 0.05.
)15 ,75 ,125( 321 kkk
)125 ,75 ,15( 321 hhh
12-21
Simulation Results
ix
ix̂