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![Page 1: 1 of 16 NORM - CONTROLLABILITY, or How a Nonlinear System Responds to Large Inputs Daniel Liberzon Univ. of Illinois at Urbana-Champaign, U.S.A. NOLCOS.](https://reader038.fdocuments.in/reader038/viewer/2022110116/551c569d550346a66a8b4eb6/html5/thumbnails/1.jpg)
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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