Transcript NORM-CONTROLLABILITY, or How a Nonlinear System Responds to Large Inputs Daniel Liberzon Univ.

NORM-CONTROLLABILITY, or
How a Nonlinear System Responds to Large Inputs
Daniel Liberzon
Univ. of Illinois at Urbana-Champaign, U.S.A.
Joint work with Matthias Müller
and
Frank Allgöwer, Univ. of Stuttgart, Germany
NOLCOS 2013, Toulouse, France
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TALK OUTLINE
• Motivating remarks
• Definitions
• Main technical result
• Examples
• Conclusions
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CONTROLLABILITY: LINEAR vs. NONLINEAR
Point-to-point controllability
Linear systems:
x_ = A x + B u
• can check controllability via matrix rank conditions
• Kalman contr. decomp.
controllable modes
Nonlinear control-affine systems:
x_ = f ( x) + g( x) u
similar results exist but more difficult to apply
General nonlinear systems:
x_ = f ( x; u)
controllability is not well understood
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NORM-CONTROLLABILITY: BASIC IDEA
x_ = f ( x; u)
Instead of point-to-point controllability:
• look at the norm
• ask if can get large
by applying large
for long time
This means that can be steered far away at least in some
directions (output map y = h( x) will define directions)
Possible application contexts:
• process control: does more
reagent yield more product ?
• economics: does increasing
advertising lead to higher sales ?
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NORM-CONTROLLABILITY vs. ISS
x_ = f ( x; u)
Input-to-state stability (ISS)
[Sontag ’89]:
small / bounded
small / bounded
ISS gain:
upper bound on for all
Lyapunov characterization
[Sontag–Wang ’95]:
jxj ¸ ½( juj) ) V_ < 0
Norm-controllability (NC):
large
large
NC gain: lower bound on
for worst-case
Lyapunov sufficient
condition:
jxj · ½( juj) ) V_ > 0
(to be made precise later)
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NORM-CONTROLLABILITY vs. NORM-OBSERVABILITY
For dual notion of observability, [Hespanha–L–Angeli–Sontag ’05]
followed a conceptually similar path
x_ = f ( x) ;
y = h( x)
Instead of reconstructing precisely from measurements of ,
norm-observability asks to obtain an upper bound on
from knowledge of norm of :
³
´
where ° 2 K 1
jx( 0) j · ° kyk[0;¿]
Precise duality relation – if any – between norm-controllability
and norm-observability remains to be understood
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FORMAL DEFINITION
x_ = f ( x; u) ; x( 0) = x 0
y = h( x) (e.g., h( x) = x )
Ua;b := f u( ¢) : ju( t) j · b 8t 2 [0; a]g
R af x 0 ; Ua;bg := reachable set at
from x( 0) = x 0 using u( ¢) 2 Ua;b
R ha ( x 0 ; Ua;b) := radius of smallest ball
around y = 0 containing h( R a)
Definition: System is norm-controllable (NC) if
R ha ( x 0 ; Ua;b) ¸ ° ( a; b)
where
is nondecreasing in
8a; b > 0
and class
in
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LYAPUNOV-LIKE SUFFICIENT CONDITION
Theorem:
x_ = f ( x; u) ; y = h( x)
is NC if 9 V : Rn !
R¸ 0 satisfying
• ®1 ( j! ( x) j) · V ( x) · ®2 ( j! ( x) j)
º ( j! ( x) j) · jh( x) j
where ! : Rn !
•
(e.g., ! ( x) = h( x) )
R` continuous, ®1 ; ®2 ; º 2 K 1
9 ½; Â 2 K 1 : 8 b> 0; 8 x s.t. j! ( x) j · ½( b)
9 u , juj · b that gives V 0( x; f ( x; u) ) ¸ Â( b)
where V 0( x; h) :=
lim inf
t& 0; ¹h ! h
V ( x+ t ¹h ) ¡ V ( x)
t
See paper for extension using higher-order derivatives
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IDEA of CONTROL CONSTRUCTION
For simplicity take h( x) = x and ! ( x) = x . Fix a; b > 0 .
• ®1 ( jxj) · V ( x) · ®2 ( jxj)
9u
juj · b
( u is “good” for x )
s.t. V 0( x; f ( x; u) ) ¸ Â( b)
• when jxj · ½( b) ,
with
1) Given x 0 , pick a “good” value u 0
For u ´ u 0 9 time
s.t.
V ( x( t) ) ¸ V ( x 0 ) + ( 1 ¡ " ) tÂ( b)
8t 2 [0; t 1 ] , where
is arb. small
Repeat for x( t 1 ) ; x( t 2 ) ; : : : until
time
s.t. V ( x( ·t 1 ) ) = ®1 ( ½( b))
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IDEA of CONTROL CONSTRUCTION
For simplicity take h( x) = x and ! ( x) = x . Fix a; b > 0 .
• ®1 ( jxj) · V ( x) · ®2 ( jxj)
9u
juj · b
( u is “good” for x )
s.t. V 0( x; f ( x; u) ) ¸ Â( b)
• when jxj · ½( b) ,
with
· 1 for x( ·t 1 )
2) Pick a “good” u
· 1 9 time ·t 2 > ·t 1 s.t.
For u ´ u
V ( x( t) ) ¸ ®1 ( ½( b)) 8t 2 [·t 1 ; ·t 2 ]
Repeat for x( ·t 2 ) ; x( ·t 3 ) ; : : :
until we reach
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IDEA of CONTROL CONSTRUCTION
®1 ( jxj) · V ( x) · ®2 ( jxj)
u( ¢) 2 Ua;b piecewise constant
Can prove:
’s and
’s don’t accumulate
n
o
V ( x( a) ) ¸ m in ( 1 ¡ " ) aÂ( b) + V ( x 0 ) ; ®1 ( ½( b) )
³
n
R a( x 0 ; Ua;b) ¸ ®¡2 1 m in aÂ( b) + V ( x 0 ) ; ®1 ( ½( b) )
o´

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EXAMPLES
1) x_ = ¡ x 3 + u; y = x
Take V ( x) = jxj
For
,
V_ = ¡ jxj 3 + sgn( x) u
= ¡ jxj 3 + µsgn( x) u + ( 1 ¡ µ) sgn( x) u
where µ 2 ( 0; 1) is arbitrary
For each b > 0 , “good” u are
juj = b and xu ¸ 0 :
V_ ¸ ( 1 ¡ µ) b = : Â( b) when jxj · ( µb) 1=3 = : ½( b)
For
so any
, V 0( 0; u) = juj
u with juj = b is “good”: V 0( 0; u) = b ¸ Â( b)
V non-smooth at 0 can be increased from 0 at desired speed
n
p3 o
System is NC with ° ( a; b) = m in ( 1 ¡ µ) ab + jx 0 j; µb 
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EXAMPLES
4
2)
x_1 = ( 1 + sin( x 2 u)) juj ¡ x 1
x_2 = x 1 ¡ 0:2x 2
x1
3.5

3
2.5
2
y = x1
Assume x 1 ( 0) ¸ 0 ) x 1 ( t) ¸ 0 8 t
Take V ( x) = jx 1 j
1.5
1
0.5
0
0
2
4
6
8
10
12
14
16
18
20
12
14
16
18
20
t
4
6 0 , V_ = x_1
For x 1 =
3
2
“Good” inputs: juj = b; sin( x 2 u) ¸ 0
V_ ¸ b ¡ x 1 ¸ ( 1 ¡ µ) b = : Â( b)
when x 1 · µb = : ½( b) ,
µ 2 ( 0; 1)
u
1
0
-1
-2
-3
-4
0
2
4
6
8
10
t
For x 1 = 0, V 0( 0; u) = x_1 – same as above
System is NC with ° ( a; b) = m inf ( 1 ¡ µ) ab + jx 1 ( 0) j; µbg 
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EXAMPLES
3) Isothermal continuous stirred tank reactor with
irreversible 2nd-order reaction from reagent A to product B
kx 21 +
x_1 = ¡ cx 1 ¡
x_2 = kx 21 ¡ cx 2
cu
y = qx 2
x 1 ; x 2 = concentrations of species A and B
concentration of reagent A in inlet stream
amount of product B per time unit
flow rate,
reaction rate
reactor volume
c = q=V ,
This system is (globally) NC, but showing this is not straightforward:
• Lyapunov sufficient condition only applies when initial conditions
satisfy x 2 ( 0) · ( k=c) x 2
1 ( 0) – meaning that enough of
reagent A is present to increase the amount of product B
• Relative degree = 2
need to work with
• Several regions in state space need to be analyzed separately
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LINEAR SYSTEMS
x_ = A x + B u
S := span( B ; A B ; : : : ; A n¡ 1 B )
If
is a real left eigenvector of
not orthogonal to
then system is NC w.r.t. h( x) = ` > x
(V ( x) = j` > xj ) V_ = sgn( ` > x) ` > x_ = ¸ j` > xj + sgn( ` > x) ` > B u )
Corollary: If
is controllable and
has
eigenvalues, then x_ = A x + B u; y = x is NC
real
Let
be real. Then
is controllable if and only if
system is NC w.r.t. h( x) = ` > x 8 left eigenvectors
of
More generally, consider Kalman controllability decomposition
x_1 = A 11 x 1 + A 12 x 2 + B~ u;
x_2 = A 22 x 2
If
is a real left eigenvector of
, then system is NC
w.r.t. h( x) = ( `~> 0) x from initial conditions
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CONCLUSIONS
Contributions:
• Introduced a new notion of norm-controllability for general
nonlinear systems
• Developed a Lyapunov-like sufficient condition for it
(“anti-ISS Lyapunov function”)
• Established relations with usual controllability for linear systems
Future work:
• Identify classes of systems that are norm-controllable
• Study alternative (weaker) norm-controllability notions
• Develop necessary conditions for norm-controllability
• Treat other application-related examples
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