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Overapproximating the Reachable Sets of LTI Systems Through a
Similarity Transformation
Shahab Kaynama, Meeko Oishi
ACC, 2010-06-30 Hybrid Systems and Control Group, ECE, UBC
[email protected]/~kaynama
Introduction and Motivation
• Reachability analysis
• The “curse of dimensionality”
• Linear dynamics
• Efficient techniques
• Safety control, non-convexity
ACC, 2010-06-30 2
,x Ax Bu u= + ∈ U
Introduction and Motivation
• Reachability analysis
ACC, 2010-06-30 3
0X
(unsafe) (safe)
( , , ), ,x f x u d u d= ∈ ∈U D
Target set
Introduction and Motivation
• Reachability analysis
ACC, 2010-06-30 4
0X
(unsafe) (safe)
[ , 0], 0t τ τ∈ − >
(·)ξ
(·)ξ
( , , ), ,x f x u d u d= ∈ ∈U D
Introduction and Motivation
• Reachability analysis
ACC, 2010-06-30 5
{}, , (·), (·) 0
| (·) , [ , 0],
( ) ,
:
(·)
n
x u d
x d s
s uτ
τ
τ
ξ −
= ∈ ∃ ∈ ∃ ∈ −
∈ ∀ ∈
X D
X U
τX
(safe)
0X
,du
∃ ∈∀ ∈
DU
(unsafe)
,ud
∃ ∈∀ ∈
UD
Reach set
( , , ), ,x f x u d u d= ∈ ∈U D
Introduction and Motivation
• Reachability analysis
ACC, 2010-06-30 6
{}, , (·), (·) 0
| (·) ,
( ) , [ , 0], (
:
·)
n
x u d
x u
s s dτ τ
τξ τ−
= ∈ ∃ ∈
∈ ∀ ∈ − ∈
=
∀
C X U
X D
τX
(safe)
0X
,du
∃ ∈∀ ∈
DU
(unsafe)
,ud
∃ ∈∀ ∈
UD
τCControlled-invariant
set
( , , ), ,x f x u d u d= ∈ ∈U D
Introduction and Motivation
ACC, 2010-06-30 10
{}, , (·), (·) 0
| (·) ,
( ) , [ , 0] ·)
:
, (
n
x u d
x u
s s dτ
τξ τ−
= ∈ ∃ ∈
∈ ∀ ∈ − ∀ ∈
C U
X D
{}, , (·), (·) 0
| (·) , [ , 0],
( ) ,
:
(·)
n
x u d
x d s
s uτ
τ
τ
ξ −
= ∈ ∃ ∈ ∃ ∈ −
∈ ∀ ∈
X D
X U
Reachability:
Introduction and Motivation
ACC, 2010-06-30 11
{}, , (·), (·) 0
| (·) , [ , 0],
( ) ,
:
(·)
n
x u d
x d s
s uτ
τ
τ
ξ −
= ∈ ∃ ∈ ∃ ∈ −
∈ ∀ ∈
X D
X U
{}, , (·), (·) 0
| (·) , [ , 0],
( ) ,
:
(·)
n
x u d
x u s
s dτ
τ
τ
ξ −
= ∈ ∃ ∈ ∃ ∈ −
∈ ∀ ∈
A U
X D
Reachability:
Attainability:
{}, , (·), (·) 0
| (·) ,
( ) , [ , 0] ·)
:
, (
n
x u d
x u
s s dτ
τξ τ−
= ∈ ∃ ∈
∈ ∀ ∈ − ∀ ∈
C U
X D
Introduction and Motivation
• Reachability analysis
ACC, 2010-06-30 12
Girard, Le Guernic [2008, 2010]Kurzhanskiy, Varaiya [2007]Goncharova, Ovseevich [2007]Asarin, Dang, Frehse, Girard, Le Guernic, Maler [2006]Nasri, Lefebvre, Gueguen [2006]Habets, Collins, Van Schuppen [2006]Mitchell, Bayen, Tomlin [2005]Yazarel, Pappas [2004]Krogh, Stursberg [2003]Nenninger, Frehse, Krebs [2002]Botchkarev, Tripakis [2000]Vidal, Schaffert, Lygeros, Sastry [2000]Kurzhanski, Varaiya [2000]Greenstreet, Mitchell [1999]Chutinan, Krogh [1998]. . .
Reac
habi
lity
/ at
tain
abili
ty
Introduction and Motivation
• Reachability analysis
• The “curse of dimensionality”
• Linear dynamics
• Efficient techniques
• Safety control, non-convexity
ACC, 2010-06-30 13
,x Ax Bu u= + ∈ U
Introduction and Motivation
• Reachability analysis
• The “curse of dimensionality”• Set representation
– Kurzhanski, Varaiya [2006]; Girard, Guernic, Maler [2006]; Krogh, Stursberg [2003],…
• Model reduction, approximation, Hybridization, Projection, Structure decomposition
– Han, Krogh [2004, 05]; Girard, Pappas [2007]; Asarin, Dang [2004]; Mitchell, Tomlin [2003]; Stipanovic, Hwang, Tomlin [2003]; Kaynama, Oishi [2009]; …
• Combination– Han, Krogh [2006]
ACC, 2010-06-30 14
Introduction and Motivation
• Reachability analysis
• The “curse of dimensionality”
• Linear dynamics
• Efficient techniques
• Safety control, non-convexity
ACC, 2010-06-30 15
,x Ax Bu u= + ∈ U
Introduction and Motivation
• Reachability analysis
• The “curse of dimensionality”
• Linear dynamics
• Efficient techniques– Attainability: Ellipsoidal, zonotopes, support functions, …
Kurzhanskiy, Varaiya [2007]; Girard, Le Guernic, Maler [2006]; Girard, Le Guernic [2008, 2010]; …
• Safety control, non-convexity
ACC, 2010-06-30 16
,x Ax Bu u= + ∈ U
Introduction and Motivation
• Reachability analysis
• The “curse of dimensionality”
• Linear dynamics
• Efficient techniques (Attainability)
• Safety control, non-convexity– Level-set methods (Mitchell, Bayen, Tomlin [2005])
– Polytopic MPT (Kvasnica, Grieder, Baotic, Morari [2004])
– Viability algorithms (Cardaliaguet, Quincampoix, Saint-Pierre [2004]; Gao, Lygeros, Quincampoix [2006])
ACC, 2010-06-30 17
,x Ax Bu u= + ∈ U
Outline
• Introduction and motivation
• Structure decomposition
• Reachability in lower dimensions
• Example
ACC, 2010-06-30 18
Outline
• Introduction and motivation
• Structure decomposition
• Reachability in lower dimensions
• Example
ACC, 2010-06-30 19
Structure Decomposition
ACC, 2010-06-30 20
1 1
1
( , )kz
z g z u=
∈
2 2( )
2
( , )n k
z h u
z
z−
=
∈
( , )n
x f x u
x
=
∈
( , )n
z f z u
z
=
∈
Structure Decomposition
ACC, 2010-06-30 21
11 12 1
21 22 2
Bx
A Ax u
BA A
⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜= +⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠
( )11 1pB B…
( )21 2pB B…
Structure Decomposition
ACC, 2010-06-30 22
11 12 1
21 22 2
Bx
A Ax u
BA A
⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜= +⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠1
2
1
2 2
1 B
B
Az z u
A
⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜= +⎟ ⎟⎜ ⎜⎟ ⎟⎟⎟ ⎜⎜ ⎝ ⎠⎝ ⎠0
0T
ACC, 2010-06-30 23
Structure Decomposition(disjoint control input)
0
11
22 23
1
1 11 0 1 0
2 0 2 0
11
22
222
3
( ) (
( )
)
)
( )
( ) ( ( )
( )(
)()
( )
t
t
z t t t z t
u r
u r
z t t t z t
tdr
t
r
Br Bu r
B
⎛ ⎞ ⎛ ⎞⎛ ⎞−⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎜ ⎜=⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟−⎟ ⎟ ⎟⎜ ⎜ ⎜⎝ ⎠ ⎝ ⎠⎝ ⎠
⎛ ⎞− ⎟⎜ ⎟⎜ ⎟⎜ ⎟−
ΦΦ
Φ ⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠⎜ ⎠
+⎠⎝Φ ⎟⎝
∫
00
00
0 00
ACC, 2010-06-30 24
0
11 13
2
1 11 0 1 0
2 0 2 0
1
22
2
1
2
1
22
3
( ) (
( )
)
)
)
)
( )
(
(
) (
)
(
( )
)
(
(
t
t
z t t t z t
u r
u r
z t t
rr
B B
B
t z t
td
rr
ut
⎛ ⎞ ⎛ ⎞⎛ ⎞−⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎜ ⎜=⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟−⎟ ⎟ ⎟⎜ ⎜ ⎜⎝ ⎠ ⎝ ⎠⎝ ⎠
⎛ ⎞− ⎟⎜ ⎟⎜ ⎟⎜ ⎟−
ΦΦ
Φ ⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠⎜ ⎠
+⎠⎝Φ ⎟⎝
∫
00
00
00 0
Structure Decomposition (disjoint control input)
ACC, 2010-06-30 25
0
12 13
2
1 11 0 1 0
2 0 2 0
1
22
2
1
2
1
21
3
( ) (
( )
)
)
)
)
( )
(
(
) (
)(
( )
( )
(
t
t
z t t t z t
u r
u r
z t t
B B
B
t z t
tdr
u r
r
rt
⎛ ⎞ ⎛ ⎞⎛ ⎞−⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎜ ⎜=⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟−⎟ ⎟ ⎟⎜ ⎜ ⎜⎝ ⎠ ⎝ ⎠⎝ ⎠
⎛ ⎞− ⎟⎜ ⎟⎜ ⎟⎜ ⎟−
ΦΦ
Φ ⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠⎜ ⎠
+⎠⎝Φ ⎟⎝
∫
00
00
00 0
Structure Decomposition (disjoint control input)
ACC, 2010-06-30 26
Structure Decomposition(disjoint control input)
0 21 22 23
1
1 11 0 1 0
2 0 2 0
11
22
222
3
( ) (
( )
)
)
( )
(
( )
(
) ( ( )
(
()
)
)
t
t
z t t t z t
u r
u r
z t t
B
t z
B B
r
r
t
td
tu
r
r
⎛ ⎞ ⎛ ⎞⎛ ⎞−⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎜ ⎜=⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟−⎟ ⎟ ⎟⎜ ⎜ ⎜⎝ ⎠ ⎝ ⎠⎝ ⎠
⎛ ⎞− ⎟⎜ ⎟⎜ ⎟⎜ ⎟−
ΦΦ
Φ ⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠⎜ ⎠
+⎠⎝Φ ⎟⎝
∫
00
00
0 0 0
ACC, 2010-06-30 28
Structure Decomposition (non-disjoint control input)
0
1 11 0 1 0
2 0
122
22
113
21
2 22
23
0
11
2
( )
( )
)
)
)
( ) ( ( )
( ) ( ( )
(
)(
()
t
t
B
B
Bu r
u r
rr
ru
z t t t z t
z t t t z t
r dt B
t
ΦΦ
Φ+
Φ
⎛ ⎞⎟⎜⎛ ⎞ ⎟⎜ ⎟⎟⎜ ⎜ ⎟⎟⎜ ⎜ ⎟⎟⎜
⎛ ⎞ ⎛ ⎞⎛ ⎞−⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎜ ⎜=⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟−⎟ ⎟ ⎟⎜ ⎜ ⎜⎝ ⎠ ⎝
⎜ ⎟⎟⎟⎜ ⎜ ⎟⎝ ⎠⎜ ⎟⎟⎜⎝
⎠⎝ ⎠
⎛ ⎞− ⎟⎜ ⎟⎜ ⎟⎜ −⎜⎠
⎟⎟⎝ ⎠∫
00
00
00
ACC, 2010-06-30 29
0
111
22
22
1 11 0 1 0
2 0
12 132
21 22 23
2
3
0
11
( ) ( ( )
( ) ( ( )
(
(
)
)
)( )
( )
))
(
t
t
z t t t z t
z t
u rB B B
t t z t
td
t
rr
ru r
B B Bu r
⎛ ⎞ ⎛ ⎞⎛ ⎞−⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎜ ⎜=⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟−⎟ ⎟ ⎟⎜ ⎜ ⎜⎝ ⎠ ⎝
ΦΦ
Φ+
⎠⎝ ⎠
⎛ ⎞− ⎟⎜ ⎟⎜ ⎟⎜ ⎟
⎛ ⎞⎟⎜⎛ ⎞ ⎟⎜ ⎟⎟⎜ ⎜ ⎟⎟⎜ ⎜ ⎟⎟⎜ ⎜ ⎟⎟⎟⎜ ⎜ ⎟⎝ ⎠⎜ ⎟⎟⎜⎝ ⎠− ⎟⎜⎝ ⎠Φ∫
00
00
Structure Decomposition (non-disjoint control input)
ACC, 2010-06-30 30
Not enough for the dynamics to be decoupled; Inputs must be disjoint too!
Structure Decomposition (non-disjoint control input)
0
111
22
22
1 11 0 1 0
2 0
12 132
21 22 23
2
3
0
11
( ) ( ( )
( ) ( ( )
(
(
)
)
)( )
( )
))
(
t
t
z t t t z t
z t
u rB B B
t t z t
td
t
rr
ru r
B B Bu r
⎛ ⎞ ⎛ ⎞⎛ ⎞−⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎜ ⎜=⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟−⎟ ⎟ ⎟⎜ ⎜ ⎜⎝ ⎠ ⎝
ΦΦ
Φ+
⎠⎝ ⎠
⎛ ⎞− ⎟⎜ ⎟⎜ ⎟⎜ ⎟
⎛ ⎞⎟⎜⎛ ⎞ ⎟⎜ ⎟⎟⎜ ⎜ ⎟⎟⎜ ⎜ ⎟⎟⎜ ⎜ ⎟⎟⎟⎜ ⎜ ⎟⎝ ⎠⎜ ⎟⎟⎜⎝ ⎠− ⎟⎜⎝ ⎠Φ∫
00
00
Structure Decomposition
ACC, 2010-06-30 31
11 12 1
21 22 2
Bx
A Ax u
BA A
⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜= +⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠1
2
1
2 2
1 B
B
Az z u
A
⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜= +⎟ ⎟⎜ ⎜⎟ ⎟⎟⎟ ⎜⎜ ⎝ ⎠⎝ ⎠
00T
Structure Decomposition
ACC, 2010-06-30 32
11 12 1
21 22 2
A A B
A A B
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠x
11 12 1
222
A A B
BA
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠0y
(Schur)
1 111 12 1
222
A A B
BA
α α− −⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠0w
11
222
A
BA
⎛ ⎞Π ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠
00
z
20
: 0I
T Iα⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1 †1 2
3 :0I B B
TI
α−⎛ ⎞⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜⎝ ⎠1T
Structure Decomposition
ACC, 2010-06-30 33
11 12 1
21 22 2
A A B
A A B
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠x
11 12 1
222
A A B
BA
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠0y
(Schur)1T
1 111 12 1
222
A A B
BA
α α− −⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠0w
11
222
A
BA
⎛ ⎞Π ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠
00
z
20
: 0I
T Iα⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1 †1 2
3 :0I B B
TI
α−⎛ ⎞⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜⎝ ⎠
Structure Decomposition
ACC, 2010-06-30 34
11 12 1
21 22 2
A A B
A A B
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠x
11 12 1
222
A A B
BA
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠0y
(Schur)1T
1 111 12 1
222
A A B
BA
α α− −⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠0w
11
222
A
BA
⎛ ⎞Π ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠
00
z
20
: 0I
T Iα⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1 †1 2
3 :0I B B
TI
α−⎛ ⎞⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜⎝ ⎠
Structure Decomposition
ACC, 2010-06-30 35
11 12 1
21 22 2
A A B
A A B
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠x
11 12 1
222
A A B
BA
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠0y
(Schur)1T
1 111 12 1
222
A A B
BA
α α− −⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠0w
11
222
A
BA
⎛ ⎞Π ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠
00
z
20
: 0I
T Iα⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1 †1 2
3 :0I B B
TI
α−⎛ ⎞⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜⎝ ⎠
Structure Decomposition
ACC, 2010-06-30 36
11 12 1
21 22 2
A A B
A A B
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠x
11 12 1
222
A A B
BA
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠0y
(Schur)1T
1 111 12 1
222
A A B
BA
α α− −⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠0w
11
222
A
BA
⎛ ⎞Π ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠
00
z
20
: 0I
T Iα⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1 †1 2
3 :0I B B
TI
α−⎛ ⎞⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜⎝ ⎠
Structure Decomposition
ACC, 2010-06-30 37
11 12 1
21 22 2
A A B
A A B
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠x
11 12 1
222
A A B
BA
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠0y
(Schur)
1 112 111
222
A
B
B
A
Aα α− −⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠0w
20
: 0I
T Iα⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1T
11
222
A
BA
⎛ ⎞Π ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠
00
z
1 †1 2
3 :0I B B
TI
α−⎛ ⎞⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜⎝ ⎠
Structure Decomposition
ACC, 2010-06-30 42
{ }12| ,w w T y y−= = ∈W Y
20
,0 1 1000Tα
α⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ =⎟⎝ ⎠
WY
Structure Decomposition
ACC, 2010-06-30 43
11 12 1
21 22 2
A A B
A A B
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠x
11 12 1
222
A A B
BA
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠0y
(Schur)
1 112 111
222
A
B
B
A
Aα α− −⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠0w
20
: 0I
T Iα⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1T
11
222
A
BA
⎛ ⎞Π ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠
00
z
1 †1 2
3 :0I B B
TI
α−⎛ ⎞⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜⎝ ⎠
Structure Decomposition
ACC, 2010-06-30 44
11 12 1
21 22 2
A A B
A A B
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠x
11 12 1
222
A A B
BA
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠0y
(Schur)
1 112 111
222
A
B
B
A
Aα α− −⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠0w
11
222
A
BA
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝
Π
⎠
00
z
20
: 0I
T Iα⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1 †1 2
3 :0I
TIB Bα−⎛ ⎞⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜⎝ ⎠
1T
Structure Decomposition
ACC, 2010-06-30 45
11 12 1
21 22 2
A A B
A A B
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠x
11 12 1
222
A A B
BA
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠0y
(Schur)
1 112 111
222
A
B
B
A
Aα α− −⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠0w
11
222
A
BA
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝
Π
⎠
00
z
20
: 0I
T Iα⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1 †1 2
3 :0I
TIB Bα−⎛ ⎞⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜⎝ ⎠
1T
† 12 2 2 2: ( )T TBBBB −=
Structure Decomposition
ACC, 2010-06-30 46
11 12 1
21 22 2
A A B
A A B
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠x
11 12 1
222
A A B
BA
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠0y
(Schur)
1 112 111
222
A
B
B
A
Aα α− −⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠0w
11
222
A
BA
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝
Π
⎠
00
z
20
: 0I
T Iα⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1 †1 2
3 :0I
TIB Bα−⎛ ⎞⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜⎝ ⎠
1T
1 † †11 1 2 1 2 22 12): (A B B AB B Aα− − +Π =
Structure Decomposition
ACC, 2010-06-30 47
11 12 1
21 22 2
A A B
A A B
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠x
11 12 1
222
A A B
BA
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠0y
(Schur)
1 112 111
222
A
B
B
A
Aα α− −⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠0w
11
222
A
BA
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝
Π
⎠
00
z
20
: 0I
T Iα⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1 †1 2
3 :0I
TIB Bα−⎛ ⎞⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜⎝ ⎠
1T
{ }† †
11 1 2 1 2 122
12
2max 1,
A B B B B A A
Aα = +
− +ε
Structure Decomposition
ACC, 2010-06-30 48
11 12 1
21 22 2
A A B
A A B
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠x
11 12 1
222
A A B
BA
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠0y
(Schur)
1 112 111
222
A
B
B
A
Aα α− −⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠0w
11
222
A
BA
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝
Π
⎠
00
z
20
: 0I
T Iα⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1 †1 2
3 :0I
TIB Bα−⎛ ⎞⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜⎝ ⎠
1T
{ }† †
11 1 2 1 2 122
12
2max 1,
A B B B B A A
Aα = +
− +ε
Structure Decomposition
ACC, 2010-06-30 49
{ }† †
11 1 2 1 2 122
12
2max 1,
A B B B B A A
Aα = +
− +εBut how large is ?
Structure Decomposition
ACC, 2010-06-30 50
{ }† †
11 1 2 1 2 22 12
12
max 1,A B B B B A A
Aα = +
− +εBut how large is ?
Structure Decomposition
ACC, 2010-06-30 51
† †11 1 2 1 2 22 12
12
11 22
12
1A B B B B A A A
A
A
A
⎛ ⎞+ ⎟⎜ ⎟⎜≤ Δ +⎟⎜ ⎟⎟⎜⎝ ⎠
−⋅
+
{ }† †
11 1 2 1 2 22 12
12
max 1,A B B B B A A
Aα = +
− +ε
( )
†1 2 11 22 12: argmin
k n kQB B A Q QA A
× −∈−Δ = − +
R
But how large is ?
Structure Decomposition
ACC, 2010-06-30 52
† †11 1 2 1 2 22 12
12
11 22
12
1A B B B B A A A
A
A
A
⎛ ⎞+ ⎟⎜ ⎟⎜≤ Δ +⎟⎜ ⎟⎟⎜⎝ ⎠
−⋅
+
{ }† †
11 1 2 1 2 22 12
12
max 1,A B B B B A A
Aα = +
− +ε
( )
†1 2 11 22 12: argmin
k n kQB B A Q QA A
× −∈−Δ = − +
R
0lim 1αΔ →
∴ = + ε
But how large is ?
Structure Decomposition
ACC, 2010-06-30 53
11 12 1
21 22 2
A A B
A A B
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠x
11
222
A
BA
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝
Π
⎠
00
z
1 2 3T TTT=
ACC, 2010-06-30 54
1 11
2 22 2 2 2
1 2z A z
z A z
z
B u
+ Π=
= +
Structure Decomposition
unidirectionally weakly-coupled
subsystems
ACC, 2010-06-30 55
1 11
2 22 2 2 2
1 2z A z
z A z
z
B u
+ Π=
= +
Structure Decomposition
trivially-uncontrollable
unidirectionally weakly-coupled
subsystems
ACC, 2010-06-30 56
disturbance
Structure Decomposition
1 11
2 22 2 2 2
1 2z A z
z A z
z
B u
+ Π=
= +
Outline
• Introduction and motivation
• Structure decomposition
• Reachability in lower dimensions
• Example
ACC, 2010-06-30 57
Reachability in Lower Dimensions
1: transform target set
2: project onto each subspace
3: for lower subsystem: i) [isolated]
4: for upper subsystem:i) compute upper-bound
ii) [perturbed]
5: back-project, intersect, reverse-transform
ACC, 2010-06-30 58
1 22 1( ) ( ): ( )Tτ τ τ τ= × ∩ × ⊇X Z R Z R X
10 0T−←Z X
0 0,Proj( ) 1,2,i i i =←Z Z
2 20Reach( )τ ←Z Z
22
2 2, : supz
z zτ
ζ ζ ∈Π ≤ Π = Z1 1
0conserv.
Reach( )τ ←⎯⎯Z Z
ACC, 2010-06-30 59
1 11 2
2 22 2 2 2
1z A z
z A z B
z
u
= + Π
= +
Reachability in Lower Dimensions
ACC, 2010-06-30 60
2 22 2 2
1 11 1 2
2z
z A
A z
z
B
z
u=
= + Π
+
Reachability in Lower Dimensions
Reachability in Lower Dimensions
ACC, 2010-06-30 61
1111( )
1 1,0 0
r AA drz e z eτ ττ ζ−− Π− ≤ ∫
1 11,0 0 1Let and .z z τ∈ ∈Z Z
111
1
)( ( )lim :
!
N i i
N i
A k
i
τ σζ μ
−
→∞ =≤ Π =∑
Formulating a bound on conservatism of : 1τZ
effect ofinput
Reachability in Lower Dimensions
ACC, 2010-06-30 62
11
[ ,0]
1 10 (0, )( )
s
A seτ
τ μ∈ −
∴ ⊆ ⊕∪Z Z B
1111( )
1 1,0 0
r AA drz e z eτ ττ ζ−− Π− ≤ ∫
1 11,0 0 1Let and .z z τ∈ ∈Z Z
111
1
)( ( )lim :
!
N i i
N i
A k
i
τ σζ μ
−
→∞ =≤ Π =∑
Formulating a bound on conservatism of : 1τZ
Reachability in Lower Dimensions
ACC, 2010-06-30 63
11
[ ,0]
1 10 (0, )( )
s
sAeτ
τ μ∈ −
∴ ⊆ ⊕∪Z Z B
1111( )
1 1,0 0
r AA drz e z eτ ττ ζ−− Π− ≤ ∫
1 11,0 0 1Let and .z z τ∈ ∈Z Z
111
1
( )( )lim :
!
N i i
N i i
A kμ
τ σζ
−
→∞ =≤ Π =∑
Formulating a bound on conservatism of : 1τZ
Outline
• Introduction and motivation
• Structure decomposition
• Reachability in lower dimensions
• Example
ACC, 2010-06-30 64
ACC, 2010-06-30 65
ExampleLongitudinal aircraft dynamics (4D)
0.0030 0.0390 0 0.3220 0.0100
0.0650 0.3190 7.7400 0 0.1800,
0.0200 0.1010 0.4290 0 1.1600
0 0 1 0 0
+ +
A B
⎡ ⎤ ⎡ ⎤− −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− − + −⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥+ − − −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥+⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
Source: A. Bryson, Control of Spacecraft and Aircraft. Princeton Univ. Press, 1994.
Full-order: 17352.0 sTransformation-based: 33.5 s
Produced using theLevel-Set Toolbox
{ }4 10 00.15,| ,| | || zz T xz x−= ∈ => ∈Z X
{ }[ 13.3 ,13.| 3 ]u u= ∈ ∈ − ° °U
Implications
• Formal verification:
• Safety-preserving control synthesis:
ACC, 2010-06-30 66
1 0 0 0T ττ τ− / / /∩ = ⇔ ∩ = ⇒ ∩ =Z I X I X I
{ } 21
1Proj( , ) 0 0
ii
iT iτ τ
=−=
/ /∩ = ⇒ ∩ =Z I X I
{ *
( , ), ( ) int
( , ), ( )( )
pu x t x t
u x t x tu t
τ
τ
∈
∈ ∂=
X
X
Summary
• Computations scale poorly with dimension
• Safety control, severe non-convexity
• Appropriate coordinate transformation
• Significant computational advantage
• Easily extendible to hybrid systems
• Motivated by safety-based control of anesthesia
ACC, 2010-06-30 67
Overapproximating the Reachable Sets of LTI Systems Through a
Similarity Transformation
Shahab Kaynama, Meeko Oishi
ACC, 2010-06-30 Hybrid Systems and Control Group, ECE, UBC
[email protected]/~kaynama
ACC, 2010-06-30 69
ACC, 2010-06-30 70
{ }
11 12
21 22
111 12
2
0
( , ) : diag( , ), ,
{ | ( ) 0}
( , )min 0, ( , ( , )) 0,
[ , 0], ( , 0) ( )
( , ) sup ( )
B B
B B
x
T Tu
A B A A A B u
x g x
x tH x x t
tt x g x
H x
u
u
p p Ax p Bu
φφ
τ φ
∈
⎛ ⎞⎛ ⎞ ⎟⎜⎟⎜ ⎟⎜= = = ∈⎟⎜ ⎟⎜⎟⎜ ⎟⎟⎝ ⎠ ⎜⎝ ⎠= ≤
−−−−−−−−−−−−−−−−−−−−−−−−−−−∂
+ ∇ =∂
∈ − =
= +U
S U
X
HJI PDE(non-disjoint control input)
ACC, 2010-06-30 71
11 11 1
2 2
11
2 22 2 1 12 2
1
1
2 2
1
2 2( , ) sup ()
TT
uT T
T
T
p B u
p
H x p p
B up B u p B u
A x p A x∈
= + +
+ + +
SU
HJI PDE(non-disjoint control input)
ACC, 2010-06-30 72
HJI PDE(non-disjoint control input)
11 11 1
2 2
11
2 22 2 1 12 2
1
1
2 2
1
2 2( , ) sup ()
TT
uT T
T
T
p B u
p
H x p p
B up B u p B u
A x p A x∈
= + +
+ + +
SU
ACC, 2010-06-30 73
2 22
1 11 1 2 22 2
2 1 12
1 11 1
222
1
1
2 1
( , ) sup
( , ) ( , ).
()
k
T
T T
u
k
T
kk
T
T
p B u
H x p p A x p A x
H x p
p B
p B
H x p
B
u
u up∈
=
=/
= + +
+ + +
∑
SU
S S
HJI PDE(non-disjoint control input)
ACC, 2010-06-30 74
1 11 1 1 12 12 12 2( , ) sup ( T T T
uH x pp p A x p BA x u
∈= + +S
U
1 12 22 22 2 2 21 1T T Tp B u p B p B uu+ + +
2
1
( , ) ( , ).
)
k k kk
H x p H x p=
= ∑S S
HJI PDE(disjoint control input)
ACC, 2010-06-30 75
{ }
0 { | ( ) 0}, [ , 0]
( , ) min 0, ( , ( , )) 0, ( , 0) ( ),xt
x g x t
x t H x x t x g x
τ
φ φ φ
= ≤ ∈ −
∇ + ∇ = =
X
Attainability vs. Reachability (HJI PDE)
( , ) inf sup ( , , )T
u dH x p p f x u d
∈ ∈=
U D( , ) sup inf ( , , )T
duH x p p f x u d
∈∈=
DU
Reachability Attainability
