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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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SKETCH OF PROOF

piecewise constant

’s and ’s don’t accumulate

Can prove:

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EXAMPLES

1)

For ,

For ,

2)

and

So can take

if we choose as in Example 1

For ,

Can take

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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

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