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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.

NOLCOS 2013, Toulouse, France

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

_x = Ax + Bu

Nonlinear control-affine systems:

similar results exist but more difficult to apply

_x = f (x) + g(x)u

General nonlinear systems:

controllability is not well understood

_x = f (x;u)

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NORM - CONTROLLABILITY: BASIC IDEA_x = f (x;u)

• 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)y = h(x)

Possible application contexts:

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NORM - CONTROLLABILITY vs. ISS

_x = f (x;u)

Lyapunov characterization[Sontag–Wang ’95]:

jxj ¸ ½(juj) ) _V < 0

Lyapunov sufficient condition:

(to be made precise later)

jxj · ½(juj) ) _V > 0

Input-to-state stability (ISS) [Sontag ’89]:

small / bounded small / bounded ISS gain: upper bound on for all

Norm-controllability (NC):

large large

NC gain: lower bound on for worst-case

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NORM - CONTROLLABILITY vs. NORM - OBSERVABILITY

_x = f (x); y = h(x)

jx(0)j · °³kyk[0;¿]

´° 2 K1where

For dual notion of observability, [Hespanha–L–Angeli–Sontag ’05]

followed a conceptually similar path

Precise duality relation – if any – between norm-controllability

and norm-observability remains to be understood

Instead of reconstructing precisely from measurements of ,

norm-observability asks to obtain an upper bound on

from knowledge of norm of :

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FORMAL DEFINITION

_x = f (x;u); x(0) = x0y = h(x) (e.g., )h(x) = x

Definition: System is norm-controllable (NC) if

Rah(x0;Ua;b) ¸ °(a;b) 8a;b> 0

where is nondecreasing in and class in

reachable set at

from usingx(0) = x0 u(¢) 2 Ua;b

R af x0;Ua;bg :=

radius of smallest

ball around containingy = 0Rah(x0;Ua;b) :=

h(R a)

Ua;b := f u(¢) : ju(t)j · b 8t 2 [0;a]g

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LYAPUNOV - LIKE SUFFICIENT CONDITION

where V 0(x; h) := lim inft& 0;¹h! h

V (x+t¹h)¡ V (x)t

• ®1(j! (x)j) · V (x) · ®2(j! (x)j)

º(j! (x)j) · jh(x)j

where continuous, ! : Rn ! R` ®1;®2; º 2 K1

_x = f (x;u); y = h(x)

is NC if satisfying9V : Rn ! R¸ 0

Theorem:

9½;Â 2 K1V 0(x; f (x;u)) ¸ Â(b)

• : s.t.8b>0; 8x j! (x)j · ½(b)

9u , that givesjuj · b

See paper for extension using higher-order derivatives

(e.g., )! (x) = h(x)

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IDEA of CONTROL CONSTRUCTION

For time s.t.u ´ u0 9

, where is arb. small8t 2 [0; t1]

V (x(t)) ¸ V (x0) + (1 ¡ ")tÂ(b)

Repeat for x(t1);x(t2); : : :

time s.t. V(x(·t1)) = ®1(½(b))

until

•

• ®1(jxj) · V (x) · ®2(jxj)

when , withjxj · ½(b)

V 0(x; f (x;u)) ¸ Â(b)

9us.t.

juj · b

For simplicity take and . Fix .h(x) = x ! (x) = x a;b> 0

1) Given , pick a “good” valuex0 u0

( is “good” for )u x

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2) Pick a “good” for·u1

until we reach

For time s.t.u ´ ·u1 9 ·t2 > ·t1

V(x(t)) ¸ ®1(½(b)) 8t 2 [·t1; ·t2]

Repeat for x(·t2);x(·t3); : : :

IDEA of CONTROL CONSTRUCTION

x(·t1)

For simplicity take and . Fix .h(x) = x ! (x) = x a;b> 0

•

• ®1(jxj) · V (x) · ®2(jxj)

when , withjxj · ½(b)

V 0(x; f (x;u)) ¸ Â(b)

9us.t.

juj · b

( is “good” for )u x

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V (x(a)) ¸ minn(1 ¡ ")aÂ(b) + V (x0);®1(½(b))

o

piecewise constantu(¢) 2 Ua;b

®1(jxj) · V (x) · ®2(jxj)

’s and ’s don’t accumulate

Can prove:

Ra(x0;Ua;b) ¸ ®¡ 12³min

naÂ(b)+V (x0);®1(½(b))

o´

IDEA of CONTROL CONSTRUCTION

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EXAMPLES

1) _x = ¡ x3 + u; y = x

For , _V = ¡ jxj3 + sgn(x)u

V (x) = jxjTake

For , V 0(0; u) = juj

System is NC with °(a;b) = minn(1 ¡ µ)ab+ jx0j;

3p µbo

= ¡ jxj3 + µsgn(x)u + (1 ¡ µ)sgn(x)u

where is arbitraryµ2 (0;1)

_V ¸ (1 ¡ µ)b=: Â(b) jxj · (µb)1=3 =: ½(b)when

For each , “good” are and : u juj= b xu ¸ 0b> 0

non-smooth at 0 can be increased from 0 at desired speedV

so any with is “good”: u juj= b V 0(0; u) = b¸ Â(b)

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EXAMPLES

2)

For ,x1 6= 0 _V = _x1

For , – same as abovex1 = 0 V 0(0; u) = _x1

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

_x1 = (1 + sin(x2u))juj ¡ x1_x2 = x1 ¡ 0:2x2y = x1

Assume x1(0) ¸ 0 ) x1(t) ¸ 0 8t

V(x) = jx1jTake

juj = b; sin(x2u) ¸ 0“Good” inputs:

_V ¸ b¡ x1x1 · µb=: ½(b)when ,

¸ (1 ¡ µ)b=: Â(b)

°(a;b) = minf (1¡ µ)ab+ jx1(0)j;µbg System is NC with

µ2 (0;1)

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EXAMPLES3) Isothermal continuous stirred tank reactor with

irreversible 2nd-order reaction from reagent A to product B

_x1 = ¡ cx1¡ kx21+cu

_x2 = kx21¡ cx2y = qx2

, reactor volumec= q=V

x1;x2 = concentrations of species A and Bconcentration of reagent A in inlet streamamount of product B per time unitflow rate, reaction rate

This system is (globally) NC, but showing this is not straightforward:

• Several regions in state space need to be analyzed separately

• Relative degree = 2 need to work with

x2(0) · (k=c)x21(0)

• Lyapunov sufficient condition only applies when initial conditions

satisfy – meaning that enough of

reagent A is present to increase the amount of product B

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LINEAR SYSTEMS

_x = Ax + Bu S := span(B ;AB ; : : : ;An¡ 1B)

More generally, consider Kalman controllability decomposition

_x1 = A11x1 + A12x2 + ~Bu; _x2 = A22x2If is a real left eigenvector of , then system is NC

w.r.t. from initial conditions h(x) = (~̀> 0)x

( )V(x) = j`>xj ) _V = sgn(`>x)`> _x = ¸j`>xj + sgn(`>x)`>Bu

Corollary: If is controllable and has real

eigenvalues, then is NC_x = Ax + Bu; y = x

Let be real. Then is controllable if and only if

system is NC w.r.t. left eigenvectors of

h(x) = `>x 8

If is a real left eigenvector of not orthogonal to

then system is NC w.r.t. h(x) = `>x

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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

• Study alternative (weaker) norm-controllability notions

• Develop necessary conditions for norm-controllability

• Treat other application-related examples