dissipative dynamical systems - Semantic Scholar
DISSIPATIVE DYNAMICAL SYSTEMS Jan C. Willems ESAT-SCD (SISTA), University of Leuven, Belgium
SICE Conference on Control Systems, Kobe, Japan
May 28, 2003
THEME A dissipative system absorbs ‘supply’ (e.g., energy). How do we formalize this? Involves the storage function. How is it constructed? Is it unique? KYP, LMI’s, ARE’s. Where is this notion applied in systems and control?
OUTLINE 1. Lyapunov theory 2. !! Dissipative systems !! 3. Physical examples 4. Construction of the storage function 5. LQ theory
LMI’s, etc.
6. Applications in systems and control 7. Dissipativity for behavioral systems 8. Polynomial matrix factorization 9. Recapitulation
LYAPUNOV THEORY
LYAPUNOV FUNCTIONS
, the state space, . Denote the set of by , the ‘behavior’. The function
Equivalent to
if along
is said to be a Lyapunov function for
with solutions
Consider the classical ‘dynamical system’, the flow
V
Typical Lyapunov ‘theorem’:
for
&
%
%
there holds
$
"
#
for
!
and
X
‘global stability’
Refinements: LaSalle’s invariance principle. Converse: Kurzweil’s thm.
+,)
+*)
(
(Matrix) ‘Lyapunov equation’
,
(
-*
*)
%' (
for
'
LQ theory
! )
,
,
)
*
.
*
/
A linear system is (asymptotically) stable iff it has a quadratic positive definite Lyapunov function sol’n . Basis for most stability results in control, physics, adaptation, even numerical analysis, system identification.
Lyapunov functions play a remarkably central role in the field.
Aleksandr Mikhailovich Lyapunov (1857-1918) Studied mechanics, differential equations. Introduced Lyapunov’s ‘second method’ in his Ph.D. thesis (1899).
DISSIPATIVE SYSTEMS
26 7 6 8
4 1 5
0
321
A much more appropriate starting point for the study of dynamics are ‘open’ systems.
INPUT/STATE/OUTPUT SYSTEMS
; :
9 C
Let be a function, called the supply rate.
B@
:
9
: the input, output, state. B>
A
@
:
?
all sol’ns
Behavior
>
9
9
9
If equality holds: ‘conservative’ system.
9
C
9
.
9
for all
9
Equivalent to
This inequality is called the dissipation inequality.
Increase in storage
.
Dissipativity
SUPPLY
STORAGE
PSfrag replacements DISSIPATION
Supply.
D
dissipativity
C
Special case: ‘closed system’:
then
is a Lyapunov function.
Stability for closed systems
E
Dissipativity is a natural generalization of Lyapunov theory to open systems.
Dissipativity for open systems.
PHYSICAL EXAMPLES
Electrical circuit:
L G
G K
Dissipative w.r.t.
JIH G F
(potential, current)
(electrical power).
etc.
etc.
L
) L
Electrical circuit
Supply
System
voltage current
Storage energy in capacitors and inductors
etc.
Mechanical device: (position, force, angle, torque)
RQ)
G
G
G
G
M
-
P
ON)
K
JIH G F
Dissipative w.r.t. (mechanical power)
Supply
voltage current S
force, : angle,
etc.
Q
P
etc.
Q)
P
-
S ) T
L N N
Mechanical system
energy in capacitors and inductors
)
Electrical circuit
Storage
L
System
velocity torque
potential + kinetic energy
etc.
Thermodynamic system: (heatflow, temperature)
FV
W G
-
U
Y
SICE Conference on Control Systems, Kobe, Japan
May 28, 2003
THEME A dissipative system absorbs ‘supply’ (e.g., energy). How do we formalize this? Involves the storage function. How is it constructed? Is it unique? KYP, LMI’s, ARE’s. Where is this notion applied in systems and control?
OUTLINE 1. Lyapunov theory 2. !! Dissipative systems !! 3. Physical examples 4. Construction of the storage function 5. LQ theory
LMI’s, etc.
6. Applications in systems and control 7. Dissipativity for behavioral systems 8. Polynomial matrix factorization 9. Recapitulation
LYAPUNOV THEORY
LYAPUNOV FUNCTIONS
, the state space, . Denote the set of by , the ‘behavior’. The function
Equivalent to
if along
is said to be a Lyapunov function for
with solutions
Consider the classical ‘dynamical system’, the flow
V
Typical Lyapunov ‘theorem’:
for
&
%
%
there holds
$
"
#
for
!
and
X
‘global stability’
Refinements: LaSalle’s invariance principle. Converse: Kurzweil’s thm.
+,)
+*)
(
(Matrix) ‘Lyapunov equation’
,
(
-*
*)
%' (
for
'
LQ theory
! )
,
,
)
*
.
*
/
A linear system is (asymptotically) stable iff it has a quadratic positive definite Lyapunov function sol’n . Basis for most stability results in control, physics, adaptation, even numerical analysis, system identification.
Lyapunov functions play a remarkably central role in the field.
Aleksandr Mikhailovich Lyapunov (1857-1918) Studied mechanics, differential equations. Introduced Lyapunov’s ‘second method’ in his Ph.D. thesis (1899).
DISSIPATIVE SYSTEMS
26 7 6 8
4 1 5
0
321
A much more appropriate starting point for the study of dynamics are ‘open’ systems.
INPUT/STATE/OUTPUT SYSTEMS
; :
9 C
Let be a function, called the supply rate.
B@
:
9
: the input, output, state. B>
A
@
:
?
all sol’ns
Behavior
>
9
9
9
If equality holds: ‘conservative’ system.
9
C
9
.
9
for all
9
Equivalent to
This inequality is called the dissipation inequality.
Increase in storage
.
Dissipativity
SUPPLY
STORAGE
PSfrag replacements DISSIPATION
Supply.
D
dissipativity
C
Special case: ‘closed system’:
then
is a Lyapunov function.
Stability for closed systems
E
Dissipativity is a natural generalization of Lyapunov theory to open systems.
Dissipativity for open systems.
PHYSICAL EXAMPLES
Electrical circuit:
L G
G K
Dissipative w.r.t.
JIH G F
(potential, current)
(electrical power).
etc.
etc.
L
) L
Electrical circuit
Supply
System
voltage current
Storage energy in capacitors and inductors
etc.
Mechanical device: (position, force, angle, torque)
RQ)
G
G
G
G
M
-
P
ON)
K
JIH G F
Dissipative w.r.t. (mechanical power)
Supply
voltage current S
force, : angle,
etc.
Q
P
etc.
Q)
P
-
S ) T
L N N
Mechanical system
energy in capacitors and inductors
)
Electrical circuit
Storage
L
System
velocity torque
potential + kinetic energy
etc.
Thermodynamic system: (heatflow, temperature)
FV
W G
-
U
Y
