MEETING THE NEED FOR ROBUSTIFIED NONLINEAR SYSTEM THEORY CONCEPTS Daniel Liberzon Coordinated...
-
Upload
russell-west -
Category
Documents
-
view
214 -
download
0
Transcript of MEETING THE NEED FOR ROBUSTIFIED NONLINEAR SYSTEM THEORY CONCEPTS Daniel Liberzon Coordinated...
MEETING THE NEED FOR ROBUSTIFIED
NONLINEAR SYSTEM THEORY CONCEPTS
Daniel Liberzon
Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign
Plenary lecture, ACC, Seattle, 6/13/08 1 of 36
TALK OVERVIEW
ContextConcept
Control withlimited information
Input-to-statestability
Adaptive controlMinimum-phaseproperty
Stability of switched systems
Observability
2 of 36
TALK OVERVIEW
ContextConcept
Control withlimited information
Input-to-statestability
Adaptive controlMinimum-phaseproperty
Stability of switched systems
Observability
2 of 36
Plant
Controller
INFORMATION FLOW in CONTROL SYSTEMS
3 of 36
INFORMATION FLOW in CONTROL SYSTEMS
• Limited communication capacity
• Need to minimize information transmission
• Event-driven actuators
• Coarse sensing
3 of 36• Theoretical interest
[Brockett, Delchamps, Elia, Mitter, Nair, Savkin, Tatikonda, Wong,…]
• Deterministic & stochastic models
• Tools from information theory
• Mostly for linear plant dynamics
BACKGROUND
Previous work:
• Unified framework for
• quantization
• time delays
• disturbances
Our goals:
• Handle nonlinear dynamics
4 of 36
Caveat:
This doesn’t work in general, need robustness from controller
OUR APPROACH
(Goal: treat nonlinear systems; handle quantization, delays, etc.)
• Model these effects as deterministic additive error signals,
• Design a control law ignoring these errors,
• “Certainty equivalence”: apply control
combined with estimation to reduce to zero
Technical tools:
• Input-to-state stability (ISS)
• Lyapunov functions
• Small-gain theorems
• Hybrid systems
5 of 36
QUANTIZATION
Encoder Decoder
QUANTIZER
finite subset
of
is partitioned into quantization regions
6 of 36
QUANTIZATION and INPUT-to-STATE STABILITY
7 of 36
– assume glob. asymp. stable (GAS)
QUANTIZATION and INPUT-to-STATE STABILITY
7 of 36
QUANTIZATION and INPUT-to-STATE STABILITY
7 of 36
no longer GAS
quantization error
Assume
class
QUANTIZATION and INPUT-to-STATE STABILITY
7 of 36
Solutions that start in
enter and remain there
This is input-to-state stability (ISS) w.r.t. measurement errors
In time domain: [Sontag ’89]
quantization error
Assume
class
QUANTIZATION and INPUT-to-STATE STABILITY
7 of 36class , e.g.
Solutions that start in
enter and remain there
This is input-to-state stability (ISS) w.r.t. measurement errors
In time domain: [Sontag ’89]
quantization error
Assume
class
QUANTIZATION and INPUT-to-STATE STABILITY
7 of 36class , e.g.
LINEAR SYSTEMS
Quantized control law:
9 feedback gain & Lyapunov function
Closed-loop:
(automatically ISS w.r.t. )
8 of 36
DYNAMIC QUANTIZATION
9 of 36
DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
9 of 36
DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
9 of 36
Zoom out to overcome saturation
DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
9 of 36
After the ultimate bound is achieved,recompute partition for smaller region
DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
Can recover global asymptotic stability
9 of 36
QUANTIZATION and DELAY
Architecture-independent approach
Based on the work of Teel
Delays possibly large
QUANTIZER DELAY
10 of 36
QUANTIZATION and DELAY
where
hence
Can write
11 of 36
SMALL – GAIN ARGUMENT
if
then we recover ISS w.r.t. [Teel ’98]
Small gain:
Assuming ISS w.r.t. actuator errors:
In time domain:
12 of 36
FINAL RESULT
Need:
small gain true
13 of 36
FINAL RESULT
Need:
small gain true
13 of 36
FINAL RESULT
solutions starting in
enter and remain there
Can use “zooming” to improve convergence
Need:
small gain true
13 of 36
EXTERNAL DISTURBANCES [Nešić–L]
State quantization and completely unknown disturbance
14 of 36
EXTERNAL DISTURBANCES [Nešić–L]
State quantization and completely unknown disturbance
14 of 36
Issue: disturbance forces the state outside quantizer range
Must switch repeatedly between zooming-in and zooming-out
Result: for linear plant, can achieve ISS w.r.t. disturbance
(ISS gains are nonlinear although plant is linear; cf. [Martins])
EXTERNAL DISTURBANCES [Nešić–L]
State quantization and completely unknown disturbance
After zoom-in:
14 of 36
TALK OVERVIEW
ContextConcept
Control withlimited information
Input-to-state stability
Adaptive controlMinimum-phase property
Stability of switched systems
Observability
15 of 36
OBSERVABILITY and ASYMPTOTIC STABILITY
Barbashin-Krasovskii-LaSalle theorem:
(observability with respect to )
observable=> GAS
Example:
is glob. asymp. stable (GAS) if s.t.
• is not identically zero along any nonzero solution
• (weak Lyapunov function)
16 of 36
SWITCHED SYSTEMS
Want: stability of for classes of
Many results based on common and multiple
Lyapunov functions [Branicky, DeCarlo, …]
In this talk: weak Lyapunov functions
• is a collection of systems
• is a switching signal
• is a finite index set
17 of 36
SWITCHED LINEAR SYSTEMS [Hespanha ’04]
Want to handle nonlinear switched systems
and nonquadratic weak Lyapunov functions
Theorem (common weak Lyapunov function):
• observable for each
Need a suitable nonlinear observability notion
Switched linear system is GAS if
• infinitely many switching intervals of length
• s.t. .
18 of 36
OBSERVABILITY: MOTIVATING REMARKS
Several ways to define observability
(equivalent for linear systems)
Benchmarks:
• observer design or state norm estimation
• detectability vs. observability
• LaSalle’s stability theorem for switched systems
No inputs here, but can extend to systems with inputs
Joint work with Hespanha, Sontag, and Angeli
19 of 36
STATE NORM ESTIMATION
This is a robustified version of 0-distinguishability
where
(observability Gramian)
Observability definition #1:
where
20 of 36
OBSERVABILITY DEFINITION #1: A CLOSER LOOK
Initial-state observability:
Large-time observability:
Small-time observability:
Final-state observability:
21 of 36
DETECTABILITY vs. OBSERVABILITY
Observability def’n #2: OSS, and can decay arbitrarily fast
Detectability is Hurwitz
Observability can have arbitrary eigenvalues
A natural detectability notion is output-to-state stability (OSS):
[Sontag-Wang]where
22 of 36
and
Theorem: This is equivalent to definition #1 (small-time obs.)
Definition:
For observability, should have arbitrarily rapid growth
OSS admits equivalent Lyapunov characterization:
OBSERVABILITY DEFINITION #2: A CLOSER LOOK
23 of 36
STABILITY of SWITCHED SYSTEMS
Theorem (common weak Lyapunov function):
• s.t.
Switched system is GAS if
• infinitely many switching intervals of length
• Each system
is observable:
24 of 36
MULTIPLE WEAK LYAPUNOV FUNCTIONS
Example: Popov’s criterion for feedback systems
Instead of a single , can use a
family with additional
hypothesis on their joint evolution:
linear system observable
positive real
See also invariance principles for switched systems in: [Lygeros et al., Bacciotti–Mazzi, Mancilla-Aguilar, Goebel–Sanfelice–Teel]
Weak Lyapunov functions:
25 of 36
TALK OVERVIEW
ContextConcept
Control withlimited information
Input-to-state stability
Adaptive controlMinimum-phase property
Stability of switched systems
Observability
26 of 36
MINIMUM-PHASE SYSTEMS
[Byrnes–Isidori]
Nonlinear (affine in controls): zero dynamics are (G)AS
Robustified version [L–Sontag–Morse]: output-input stability
Linear (SISO): stable zeros, stable inverse
• implies minimum-phase for nonlinear systems (when applicable)
• reduces to minimum-phase for linear systems (SISO and MIMO)27 of 36
where for some ;
UNDERSTANDING OUTPUT-INPUT STABILITY
uniform over :OSS (detectability) w.r.t. extended output,
28 of 36
Output-input stability detectability + input-bounding property:
Sufficient Lyapunov condition for this detectability property:
CHECKING OUTPUT-INPUT STABILITY
For systems affine in controls,
can use structure algorithm for left-inversion
to check the input-bounding property
equation for is ISS w.r.t.
Example 1:
can solve for and get a bound on
can get a bound onOU
TPU
T-IN
PU
T S
TAB
LE
29 of 36
Detectability:
Input bounding:
CHECKING OUTPUT-INPUT STABILITY
Example 2:
30 of 36
For systems affine in controls,
can use structure algorithm for left-inversion
to check the input-bounding property
Does not have input-bounding property:
can be large near
while stays bounded
zero dynamics , minimum-phase
Output-input stability allows to distinguish
between the two examples
FEEDBACK DESIGN
Output stabilization state stabilization
( r – relative degree)
...
Output-input stability guarantees closed-loop GAS
In general, global normal form is not needed
Example: global normal form
GAS zero dynamics not enough for stabilization
Apply to have with Hurwitz
It is stronger than minimum-phase: ISS internal dynamics
31 of 36
CASCADE SYSTEMS
For linear systems, recover usual detectability
If: • is detectable (OSS)
• is output-input stable
then the cascade system is detectable (OSS)
through extended output
32 of 36
ADAPTIVE CONTROL
Linear systems
Controller
Plant
Designmodel
If: • plant is minimum-phase• system inside the box is output-stabilized• controller and design model are detectable
then the closed-loop system is detectablethrough (“tunable” [Morse ’92])
33 of 36
ADAPTIVE CONTROL
If: • plant is minimum-phase• system inside the box is output-stabilized• controller and design model are detectable
then the closed-loop system is detectablethrough e (“tunable” [Morse ’92])
Nonlinear systems
Controller
Plant
Designmodel
34 of 36
ADAPTIVE CONTROL
If: • plant is output-input stable• system inside the box is output-stabilized• controller and design model are detectable
then the closed-loop system is detectablethrough e (“tunable” [Morse ’92])
Nonlinear systems
Controller
Plant
Designmodel
34 of 36
ADAPTIVE CONTROL
Nonlinear systems
If: • plant is output-input stable• system in the box is output-stabilized• controller and design model are detectable
then the closed-loop system is detectablethrough e (“tunable” [Morse ’92])
Controller
Plant
Designmodel
34 of 36
ADAPTIVE CONTROL
Nonlinear systems
If: • plant is output-input stable• system in the box is output-stabilized• controller and design model are detectable
then the closed-loop system is detectablethrough and its derivatives (“weakly tunable”)
Controller
Plant
Designmodel
34 of 36
TALK SUMMARY
ContextConcept
Control withlimited information
Input-to-statestability
Adaptive controlMinimum-phaseproperty
Stability of switched systems
Observability
35 of 36
ACKNOWLEDGMENTS
• Roger Brockett
• Steve Morse
• Eduardo Sontag
• Financial support from NSF and AFOSR
• Colleagues, students and staff at UIUC
http://decision.csl.uiuc.edu/~liberzon
• Everyone who listened to this talk
Special thanks go to:
36 of 36