Nonlinear Systems and Control
Lecture # 38
Observers
High-Gain Observers
Motivating Example
– p. 1/1
ẋ1 = x2 ,
ẋ2 = φ(x, u),
y = x1
Let u = γ(x) stabilize the origin of
ẋ1 = x2 ,
ẋ2 = φ(x, γ(x))
Observer:
x̂˙ 1 = x̂2 + h1 (y − x̂1 ),
x̂˙ 2 = φ0 (x̂, u) + h2 (y − x̂1 )
φ0 (x, u) is a nominal model φ(x, u)
x̃1 = x1 − x̂1 ,
x̃˙ 1 = −h1 x̃1 + x̃2 ,
x̃2 = x2 − x̂2
x̃˙ 2 = −h2 x̃1 + δ(x, x̃)
δ(x, x̃) = φ(x, γ(x̂)) − φ0 (x̂, γ(x̂))
– p. 2/1
Design H =
"
h1
h2
#
such that Ao =
"
−h1 1
−h2 0
#
is Hurwitz
Transfer function from δ to x̃:
Go (s) =
1
s2 + h1 s + h2
"
1
s + h1
#
Design H to make supω∈R kGo (jω)k as small as possible
h1 =
Go (s) =
α1
ε
,
h2 =
α2
ε2
,
ε
(εs)2 + α1 εs + α2
ε>0
"
ε
εs + α1
#
– p. 3/1
Go (s) =
ε
(εs)2 + α1 εs + α2
"
ε
εs + α1
#
Observer eigenvalues are (λ1 /ε) and (λ2 /ε) where λ1 and
λ2 are the roots of
λ2 + α1 λ + α2 = 0
sup kGo (jω)k = O(ε)
ω∈R
– p. 4/1
η1 =
x̃1
εη̇1 = −α1 η1 + η2 ,
ε
,
η2 = x̃2
εη̇2 = −α2 η1 + εδ(x, x̃)
Ultimate bound of η is O(ε)
η decays faster than an exponential mode e−at/ε ,
a>0
Peaking Phenomenon:
x1 (0) 6= x̂1 (0) ⇒ η1 (0) = O(1/ε)
The solution contains a term of the form
1
ε
1
ε
e−at/ε
e−at/ε approaches an impulse function as ε → 0
– p. 5/1
Example
ẋ1 = x2 ,
ẋ2 = x32 + u,
y = x1
State feedback control:
u = −x32 − x1 − x2
Output feedback control:
u = −x̂32 − x̂1 − x̂2
x̂˙ 1 = x̂2 + (2/ε)(y − x̂1 )
x̂˙ 2 = (1/ε2 )(y − x̂1 )
– p. 6/1
0.5
0
SFB
OFB ε = 0.1
OFB ε = 0.01
OFB ε = 0.005
x
1
−0.5
−1
−1.5
−2
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
0.01
0.02
0.03
0.04
0.05
t
0.06
0.07
0.08
0.09
0.1
1
0
x
2
−1
−2
−3
0
u
−100
−200
−300
−400
– p. 7/1
ε = 0.004
0.2
1
0
x
−0.2
−0.4
−0.6
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0
0.01
0.02
0.03
0.04
t
0.05
0.06
0.07
0.08
0
x
2
−200
−400
−600
2000
u
1000
0
−1000
– p. 8/1
u = sat(−x̂32 − x̂1 − x̂2 )
SFB
OFB ε = 0.1
OFB ε = 0.01
OFB ε = 0.001
0.15
x1
0.1
0.05
0
−0.05
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
0.01
0.02
0.03
0.04
0.05
t
0.06
0.07
0.08
0.09
0.1
0.05
x2
0
−0.05
−0.1
u
0
−0.5
−1
– p. 9/1
Region of attraction under state feedback:
2
x2
1
0
−1
−2
−3
−2
−1
0
x1
1
2
3
– p. 10/1
Region of attraction under outputfeedback:
1
x2
0.5
0
−0.5
−1
−1.5
−1
−0.5
0
x1
0.5
1
1.5
ε = 0.1 (dashed) and ε = 0.05 (dash-dot)
– p. 11/1
Analysis of the closed-loop system:
ẋ1 = x2
εη̇1 = −α1 η1 + η2
ẋ2 = φ(x, γ(x − x̃))
εη̇2 = −α2 η1 + εδ(x, x̃)
η6
q
D
D
D
D
D
O(1/ε)
q
D
D
D
D
D
D
D
O(ε)
D
D
W W
Ωb Ωc
-
x
– p. 12/1
What is the effect of measurement noise?
The high-gain observer is an approximate differentiator
Transfer function from y to x̂ (with φ0 = 0):
"
#
" #
α2
1 + (εα1 /α2 )s
1
→
as ε → 0
2
s
s
(εs) + α1 εs + α2
Differentiation amplifies the effect of measurement noise
y = x1 + v,
kn = sup |v(t)| < ∞
t≥0
– p. 13/1
1.1
1
0.9
the error bound
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
εopt = O
s
0
kn
kd
0.02
!
,
0.04
0.06
0.08
0.1
0.12
the high−gain parameter epsilon
kd = sup |ẍ1 (t)|,
t≥0
0.14
0.16
kn = sup |v(t)|
t≥0
– p. 14/1