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NORM - CONTROLLABILITY, or
How a Nonlinear System Responds to Large Inputs
Daniel Liberzon
Univ. of Illinois at Urbana-Champaign, U.S.A.
Workshop in honor of Andrei Agrachev’s 60th birthday, Cortona, Italy
Joint work with Matthias Müller and
Frank Allgöwer (Univ. of Stuttgart, Germany)
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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:
• can check controllability via matrix rank conditions
• Kalman contr. decomp. controllable modes
Nonlinear control-affine systems:
similar results exist but more difficult to apply
General nonlinear systems:
controllability is not well understood
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NORM - CONTROLLABILITY: BASIC IDEA
Possible application contexts:
• process control: does more reagent yield more product ?
• economics: does increasing advertising lead to higher sales ?
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 will define directions)
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NORM - CONTROLLABILITY vs. NORM - OBSERVABILITY
where
For dual notion of observability, [Hespanha–L–Angeli–Sontag ’05]
followed a conceptually similar path
Instead of reconstructing precisely from measurements of ,
norm-observability asks to obtain an upper bound on
from knowledge of norm of :
Precise duality relation – if any – between norm-controllability
and norm-observability remains to be understood
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NORM - CONTROLLABILITY vs. ISS
Lyapunov characterization[Sontag–Wang ’95]:
Norm-controllability (NC):
large large
Lyapunov sufficient condition:
(to be made precise later)
Input-to-state stability (ISS) [Sontag ’89]:
small / bounded small / bounded
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FORMAL DEFINITION
(e.g., )
Definition: System is norm-controllable (NC) if
where is nondecreasing in and class in
fixed scaling function id
reachable set at
from using
radius of smallest ball
around containing
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LYAPUNOV - LIKE SUFFICIENT CONDITION
Here we use
•
where continuous,
where
and
• set s.t.
is norm-controllable if s.t. :
Theorem:
[CDC’11, extensions in CDC’12]
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SKETCH OF PROOF
For simplicity take and
•
•
•
1) Pick
For time s.t.
, where is arb. small
Repeat for
time s.t.
until
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SKETCH OF PROOF
2) Pick
For simplicity take and
•
•
•
For time s.t.
Repeat for
until we reach
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EXAMPLES
3)
Since we can’t have
with as for fixed
(although we do have as if ) x
Not norm-controllable with
but norm-controllable with
4)
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EXAMPLES
5)
For ,
For ,
Can take
Can take
s.t.
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2
2.5
3
3.5
4
t
x1
0 2 4 6 8 10 12 14 16 18 20-4
-3
-2
-1
0
1
2
3
4
t
u
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EXAMPLES6) Isothermal continuous stirred tank reactor with
irreversible 2nd-order reaction from reagent A to product B
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
This system is NC from initial conditions satisfying
but need to extend the theory to prove this:
• Several regions in state space need to be analyzed separately
• Relative degree = 2 need to work with
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LINEAR SYSTEMS
More generally, consider Kalman controllability decomposition
If is a real left eigenvector of , then system is NC
w.r.t. from initial conditions
( )
Corollary: If is controllable and has real
eigenvalues, then is NC
Let be real. Then is controllable if and only if
system is NC w.r.t. left eigenvectors of
If is a real left eigenvector of not orthogonal to
then system is NC w.r.t.
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CONCLUSIONS
• 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
Contributions:
Future work:
• Identify classes of systems that are norm-controllable
(in particular, with identity scaling function)
• Study alternative (weaker) norm-controllability notions
• Develop necessary conditions for norm-controllability
• Treat other application-related examples