willems - dissipative dynamical structures
TRANSCRIPT
-
7/26/2019 Willems - Dissipative Dynamical Structures
1/57
DISSIPATIVE DYNAMICAL SYSTEMS
Jan C. Willems
ESAT-SCD (SISTA)
SISTA seminar, January 9, 2002
1
-
7/26/2019 Willems - Dissipative Dynamical Structures
2/57
THEME
Adissipative systemabsorbs supply (e.g., energy).
How do we formalize this?
Involves thestorage function.
How is it constructed? Is it unique?
KYP,LMIs,AREs.
Where is this notion applied in systems and control?
2
-
7/26/2019 Willems - Dissipative Dynamical Structures
3/57
OUTLINE
Lyapunov theory
Dissipative systems
Physical examples
Construction of the storage function
LQ theory LMIs, etc.
Applications in systems and control
Extensions
Polynomial spectral factorization
Recapitulation
3
-
7/26/2019 Willems - Dissipative Dynamical Structures
4/57
LYAPUNOV THEORY
4
-
7/26/2019 Willems - Dissipative Dynamical Structures
5/57
LYAPUNOV FUNCTIONS
Consider the classical dynamical system, the flow
with , thestate space, . Denote the set of
solutions by , thebehavior.The function
is said to be a Lyapunov function for if along
Equivalent to
5
-
7/26/2019 Willems - Dissipative Dynamical Structures
6/57
Typical Lyapunov theorem:
and for
there holds for global stability
Refinements:LaSalles invariance principle.
Converse: Kurzweils thm.
LQ theory Lyapunov (matrix)
equation. A linear system is stable iff it has a quadratic positive
definite Lyapunov function.
Basis for most stability results in control, physics, adaptation,
even numerical analysis, system identification.
6
-
7/26/2019 Willems - Dissipative Dynamical Structures
7/57
V
7
-
7/26/2019 Willems - Dissipative Dynamical Structures
8/57
V
X
Aleksandr Mikhailovich Lyapunov (1857-1918)
Studied mechanics, differential equations.
Introduced Lyapunovssecond methodin his Ph.D. thesis (1899).
8
-
7/26/2019 Willems - Dissipative Dynamical Structures
9/57
DISSIPATIVE SYSTEMS
9
-
7/26/2019 Willems - Dissipative Dynamical Structures
10/57
A much more appropriate starting point for the study of dynamics
areopen systems.
SYSTEM outputinput
10
-
7/26/2019 Willems - Dissipative Dynamical Structures
11/57
INPUT/STATE/OUTPUT SYSTEMS
Consider the dynamical system
: the input, output, state.
Behavior all solns
Let
be a function, called thesupply rate.
11
-
7/26/2019 Willems - Dissipative Dynamical Structures
12/57
models something like thepowerdelivered to the system
when the input value is and output value is .
supply
input
output
SYSTEM
12
-
7/26/2019 Willems - Dissipative Dynamical Structures
13/57
DISSIPATIVITY
is said to be dissipative w.r.t. the supply rate if
called the storage function, such that
along input/output/state trajectories ( .
This inequality is called the dissipation inequality.
Equivalent to
for all .
If equality holds:conservative system.
13
-
7/26/2019 Willems - Dissipative Dynamical Structures
14/57
Dissipativeness Increase in storage Supply.
SUPPLY
DISSIPATION
STORAGE
14
-
7/26/2019 Willems - Dissipative Dynamical Structures
15/57
Special case: closed system:
then dissipativeness is a Lyapunov function.
Dissipativity is a natural generalization of LF to open systems.
Stabilityfor closed systems Dissipativityfor open systems.
15
-
7/26/2019 Willems - Dissipative Dynamical Structures
16/57
PHYSICAL EXAMPLES
16
-
7/26/2019 Willems - Dissipative Dynamical Structures
17/57
System Supply Storage
Electrical
circuit voltage
current
energy in
capacitors and
inductors
Mechanical
system force, velocity: angle, torque
potential +
kinetic energy
Thermodynamic
system heat, work
internal
energy
Thermodynamic
system heat, temp.
entropy
etc. etc. etc.
17
-
7/26/2019 Willems - Dissipative Dynamical Structures
18/57
Electrical circuit:
(potential, current)
Dissipative w.r.t. (electrical power).
18
-
7/26/2019 Willems - Dissipative Dynamical Structures
19/57
Mechanical device:
(position, force, angle, torque)
Dissipative w.r.t. (mech. power).
19
-
7/26/2019 Willems - Dissipative Dynamical Structures
20/57
Thermodynamic system:
(work)
(heatflow, temperature)
Conservative w.r.t. ,
Dissipative w.r.t.
20
-
7/26/2019 Willems - Dissipative Dynamical Structures
21/57
THE CONSTRUCTION OF STORAGE FUNCTIONS
21
-
7/26/2019 Willems - Dissipative Dynamical Structures
22/57
Central question:
Given (a representation of ) ,the dynamics,and
given ,the supply rate,
is the system dissipative w.r.t. , i.e.,
does there exista storage function such that the
dissipation inequality holds?
Assume henceforth that a number of (reasonable) conditions hold:
;
Maps and functions (including ) smooth;State space of connected:
every state reachable from every other state;
Observability.
22
-
7/26/2019 Willems - Dissipative Dynamical Structures
23/57
Thm:Let and be given.
Then is dissipative w.r.t. iff
for allperiodic .
23
-
7/26/2019 Willems - Dissipative Dynamical Structures
24/57
Two universal storage functions:
1. The available storage
2. The required supply
Storage fns form convex set, every storage function satisfies
24
-
7/26/2019 Willems - Dissipative Dynamical Structures
25/57
LINEAR SYSTEMS with QUADRATIC SUPPLY RATES
25
-
7/26/2019 Willems - Dissipative Dynamical Structures
26/57
Assume linear, time-invariant,finite-dimensional:
and quadratic: e.g.,
E.g., for circuits , etc.
Assume controllable, observable., the transfer function of .
26
-
7/26/2019 Willems - Dissipative Dynamical Structures
27/57
Theorem:The following are equivalent:
1. is dissipative w.r.t. (i.e., there exists a storage function ),
2. ,
,
3. for all ,
4. a quadratic storage fn, ,
27
-
7/26/2019 Willems - Dissipative Dynamical Structures
28/57
5. there exists a solution to the
Linear Matrix Inequality (LMI)
6. there exists a solution to the
Algebraic Riccati Inequality (ARIneq)
7. there exists a solution to the
Algebraic Riccati Equation (ARE)
28
-
7/26/2019 Willems - Dissipative Dynamical Structures
29/57
Solution set (of LMI, ARineq) is convex, compact, and attains
itsinfimumand itssupremum:
These extreme solns and themselves satisfy the ARE.
Extensive theory, relation with other system representations,
many applications, well-understood (also algorithmically).
Connection with optimal LQ control, semi-definite programming.
29
-
7/26/2019 Willems - Dissipative Dynamical Structures
30/57
Important refinement:
Existence of a (i.e., bounded from below) (energy?)
In LQ case
,
Note def. of -norm !
soln to LMI, ARineq, ARE.
KYP-lemma.
30
-
7/26/2019 Willems - Dissipative Dynamical Structures
31/57
APPLICATIONS
31
-
7/26/2019 Willems - Dissipative Dynamical Structures
32/57
Synthesis of RLC-circuits
Robust stability and stabilization
Norm estimation
32
-
7/26/2019 Willems - Dissipative Dynamical Structures
33/57
Interconnection laws:
Interconnected system:
33
-
7/26/2019 Willems - Dissipative Dynamical Structures
34/57
Assume
dissipative, storage fn ,
dissipative, storage fn
.
For example,
,
;
or , .
Then is a Lyapunov functionfor the interconnected system .
34
-
7/26/2019 Willems - Dissipative Dynamical Structures
35/57
Proof:
Small gain theorem, Positive operator theorem,Robust stability.
35
-
7/26/2019 Willems - Dissipative Dynamical Structures
36/57
ROBUST STABILITY
: linear, time-invariant, transfer fn ,
:uncertain system,
e.g. memoryless: with
Then
stable
Quadratic LF: , from LMI, ARIneq, or ARE.
36
-
7/26/2019 Willems - Dissipative Dynamical Structures
37/57
Leads to:
SYSTEM
sensor output
control input
UNCERTAINPLANT
!! Stabilize robustly
Find a controller that stabilizes for a whole class of systems at once.
37
-
7/26/2019 Willems - Dissipative Dynamical Structures
38/57
sensor output
control input
output
CONTROLLER
input
PLANT
CLOSED LOOP SYSTEM
!! Givenplant , findcontroller such that
-control theory, synthesis of dissipative systems.
38
-
7/26/2019 Willems - Dissipative Dynamical Structures
39/57
Application:
SYSTEM NORM ESTIMATION
Model reduction:
Given a (linear, time-invariant) system , find a system ,
with a low dimensional state space, that approximates well.
Well small -norm of tf fn :
39
-
7/26/2019 Willems - Dissipative Dynamical Structures
40/57
+
_
40
-
7/26/2019 Willems - Dissipative Dynamical Structures
41/57
Some beautiful results have been obtained, in particular:
Balanced reductionof linear systems:
Let be a strictly proper, transfer function.
Then admits a representation
with controllable, observable, Hurwitz.
Moreover, can be made to bebalanced(controllability grammian = observability grammian):
41
-
7/26/2019 Willems - Dissipative Dynamical Structures
42/57
. . . . . .
. . . . . .
with theHankel SVsof the system.
Assume large, , small.
42
-
7/26/2019 Willems - Dissipative Dynamical Structures
43/57
Neglect (heuristic: these are the state components that
are bothmost difficult to reach and most difficult to observe).
With the obvious partitioning of as
we obtain the -orderreduced system
Call its transfer fn .
Question:
How close is to ?
!! Estimate = .
43
-
7/26/2019 Willems - Dissipative Dynamical Structures
44/57
Theorem (Glover):
with such that , where are
thedistinct Hankel SVsof the system.
bound sum of the neglected SVswithout repetition.
44
-
7/26/2019 Willems - Dissipative Dynamical Structures
45/57
Proof using dissipative systems:
Step 1: Neglect ONE (possibly repeated) SV:
. . .
45
-
7/26/2019 Willems - Dissipative Dynamical Structures
46/57
Consider the error system:
+
_
46
-
7/26/2019 Willems - Dissipative Dynamical Structures
47/57
Its dynamics:
47
-
7/26/2019 Willems - Dissipative Dynamical Structures
48/57
Relations between system parameters due to balancing:
48
-
7/26/2019 Willems - Dissipative Dynamical Structures
49/57
Now verify (straightforward, tedious):
Whence
Conclude,using LMI-theory,
49
-
7/26/2019 Willems - Dissipative Dynamical Structures
50/57
Step 2: Triangle inequality.In the obvious notation
where = the balanced representation truncated at :
neglect ; .
Whence
Combine step 1 and step 2:
Open problems: improve bound, find storage fn for .
50
-
7/26/2019 Willems - Dissipative Dynamical Structures
51/57
Generalizations
51
-
7/26/2019 Willems - Dissipative Dynamical Structures
52/57
Drawback 1: requires separation of interaction variables in
inputs and outputs
Behavioral systems.
Drawback 2: imposes storage function = state function.This is something one would like to prove!
Drawback 3: limited to dynamical (as opposed to distributed,
PDE) systems.
52
-
7/26/2019 Willems - Dissipative Dynamical Structures
53/57
Recap
53
-
7/26/2019 Willems - Dissipative Dynamical Structures
54/57
The notion of adissipative system:
Generalization of Lyapunov function toopensystems
Central concept in control theory: many applications to
feedback stability, robust ( -) control, adaptive control,
system identification, passivation control
Other applications: system norm estimates
passive electrical circuit synthesis procedures
Natural systems concept for the analysis of physical systems
Notable special case:second law of thermodynamics
Forms a tread through modern system theory
54
-
7/26/2019 Willems - Dissipative Dynamical Structures
55/57
55
-
7/26/2019 Willems - Dissipative Dynamical Structures
56/57
More info, ms, copy sheets? Surf to
http://www.math.rug.nl/ willems
56
-
7/26/2019 Willems - Dissipative Dynamical Structures
57/57
THANK YOU !