when is a time -delay system stable and...
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When is a Time-Delay System Stable and Stabilizable?
--A Third-Eye View
36th Chinese Control Conference
Dalian, July 2017
陈 杰 (Jie Chen)
City University of Hong Kong
Hong Kong, China
Those’re such happy times not so long ago …
(CCC’03, Yichang)
Research Areas2003 CCC, Yichang
Jie Chen
Electrical Engineering
University of California
Riverside, CA 92506
To be, or not to be?
鱼，我所欲也，熊掌，亦我所欲也，二者不可得兼 。。。
Fundamental Limitation and
Tradeoff of Feedback
When is a Time-Delay System Stable and Stabilizable?
--A Third-Eye View by an Amateur
36th Chinese Control Conference
Dalian, July 2017
陈 杰 (Jie Chen)
City University of Hong Kong
Hong Kong, China
Why Delay? Classical Examples
Power blackout, Ukraine, 2015
Turkish oil pipeline, 2008
US water distribution system
UK power plant, etc
Classical Gyroscopic Systems
Cutting/Milling Process
Classical Economic Growth Model
Why Delay? Contemporary Examples
Power blackout, Ukraine, 2015
US water
distribution
system
TCP/AQM Network
Donor Blood Cell Dynamics
Platoon
The Dujiangyan (都江堰) Irrigation System
US water distribution system
UK power plant, etc
• Built by Li Bing (李冰) in 256 BC,
for flood control and water
distribution.
• An isle constructed in the river
center to divide the flow into two
parts.
• Mountainside cut by fire and
water to yield a water channel.
Dujiangnyan as a Delay System
US water distribution system
UK power plant, etc
• Fish Mouth (鱼嘴）divides the water. Deeper inner river and curvature leads
to slower flow. Outer river flows much faster. This creates a de facto delay at
Mouth of Treasure Bottle (宝瓶口）.
• Delay leads to accumulation of silt and sediment.
• Flying Sand Levee (飞沙堰) acts as an actuator, reducing the accumulation.
Fish Mouth
Mouth of Treasure Bottle
Flying Sand Levee
Where are we? A Long Journey
1800 1950 1960 1970 1980 1990
Euler/Volterra/Picard
Pontryagin/Myshkis/Minorsky
Differential equations
Quasipolynomials
Functional differential equations Robust control
LMI
Bellman/Krasovskii/Razumikhim/Hale
and Rise of LMI
Classical Frequency Domain Methods
Two-Variable Criterion and many other variations: Elimination of one variable from two polynomial equations
Advantage: Exact, necessary and sufficient stability conditions.
Disadvantage: Require symbolic computation and hence are computationally inefficient; largely applicable to low-order systems with a small number (a couple) of delays.
Contemporary Time Domain Method
LMI (Linear Matrix Inequality)
conditions: Construct Lyapunov
functionals
Advantage: Numerically efficient via
convex optimization; can handle
nonlinear time-varying systems, though
essentially solve an augmented linear
problem.
Disadvantage: Inherently conservative,
incremental improvements.
01
,
,0) ,(
zsa
ezzsa s
Classical Methods
• A mathematical problem: Eigenvalue series of matrix-
valued functions.
• A control problem: Stability of delay systems.
• Stabilization and robust stabilization of delay systems.
A Third Eye
An Operator-Theoretic, Non-LMI Perspective
Research AreasThe Myth of the Third Eye
Hindu God: Shiva Dao Deity: Erlang
GOT: Three-Eye Raven
Stability: Linear Delay Systems
• Characteristic Function
• Linear Delay Systems with Commensurate Delays
• The system is stable if and only if
q
k
sk
k
s eAsIesa0
det) ,(
CRHP ,0) ,( sesa s
• Delay Stability Intervals: For what ranges of delay, is the
system stable?
Fundamental Issues-Stability
• Delay Stabilization Margin: For a delay system
or in state space form
Fundamental Issues-Feedback Stabilization
)()(
)()()(
tCxty
tButAxtx
yu
Rethink the Delay Interval Problem
• Idea—Continuity and Analyticity:
1. To determine the critical parameter values, and the imaginary zeros, where the zeros may cross the imaginary axis.
2. To determine the asymptotic analytical behavior of critical characteristic zeros on the imaginary axis, and hence how the imaginary zeros may vary with the parameter.
• Question—Stability Switch Problem: When will an imaginary zero/eigenvalue cross from one half plane into another as the system parameter varies?
• Quasipolynomial
in
i
kik
in
i
i
n
q
k
sk
k
s
sasasassa
esaesa
1
0
1
0
00
0
)( ,)(
)() ,(
Stability Switch: An Illustration
The Art of Operator Perturbation
E. Schrödinger
F. Rellich
T. Kato …
• Matrix Operator of a real variable .
• is analytic in the neighborhood of , namely,
• Eigenvalue of : Can be expanded in a power series or
a Puiseux series.
Eigenvalue Perturbation
...!2
)0(")0(')0()( 2
TxxTTxT
Semisimple Eigenvalues
Kato’s Theory
Kato’s Reduction Procedure-Semisimple
Define the reduced resolvent
)](~
[)( )0( xTxx ii
An iterative procedure
Non-Semisimple Eigenvalues
2/1)]([
01
00
00
10
0
10)(
xxT
xx
xT
• Singularity occurs. The eigenvalue is multi-valued, and hence not analytic.
• No longer expandable in a power series, but instead in a fractional Puiseux series.
Reduction Procedure: Non-Semisimple Eigenvalues
)]([)]([
)]([)(
1
11
)1(
1
1
11
)0(
xTxxT
xTxx
kmi
mkki
mi
mi
A doubly iterative procedure
• A non-semisimple eigenvalue can be expanded via an
iteration of continuous m-th roots of unity, in a
fractional Puiseux series.
• The result provides a complete answer!
First-Order Analysis
Semisimple
Non-Semisimple
• Let be an eigenvalue of with multiplicity m.
• Let be ordered as the first eigenvalue.
Eigen-Decomposition
if semisimple if is non-semisimple,
consist of eigenvectors
and generalized eigenvectors of .
• can be decomposed as
where
Second-Order Series: Semisimple Case
Second-Order Analysis
Non-Semisimplicity Index
The Stability Problem
q
k
sk
k
s eAsIesa0
det) ,(
• Step 1—Continuity: To determine the critical delay values, and the imaginary zeros.
• Two-Variable Criterion:
q
k
sk
k
s esaesa0
)() ,(
Schur-Cohen Criterion
• Schur-Cohen matrix:
• Schur-Cohen Theorem: The complex polynomial is Schur stable
if and only if the Schur-Cohen matrix is positive definite.
kq
k
k zpzp
0
)(
• A complex polynomial:
H
qqq
q
H
q
q
ppp
pp
p
ppp
pp
p
p
pp
ppp
p
pp
ppp
C
011
01
0
011
01
0
2
11
2
11
......
...
...
...
...
)(zp
Step 1: The Matrix Pencil Solution
• Schur-Cohen Criterion: Consider the frequency-dependent polynomial
and construct the Schur-Cohen matrix, which is a matrix polynomial in . The
matrix is positive definite iff the polynomial in z has no root outside the unit disc.
• Matrix Pencil Solution: Set the determinant of the Schur-Cohen matrix to be zero
(a polynomial equation), whose positive roots correspond to unitary solutions
Critical Delays Critical Zeros
),( zja
k
kj
k ez
k
kk
All critical delays and imaginary zeros/eigenvalues can be computed efficiently by solving a matrix pencil problem.
q
k
k
k zsazsa0
)() ,(
• Bivariate Polynomial:
Step 2: Compute Eigenvalue Series
q
k
jk
keAT0
(
A General Semisimple Result
The imaginary
eigenvalue enters
RHP (LHP).
Inconclusive!
The imaginary
eigenvalue enters
RHP (LHP).
Simple Imaginary Eigenvalue
The imaginary
eigenvalue enters
RHP (LHP).
Inconclusive!
The imaginary
eigenvalue enters
RHP (LHP).
A Non-Semisimple ResultThe critical
eigenvalue radiates
at equi-distant
angles
The critical
eigenvalue radiates
at equi-distant
angles.
Example 1
• First-order condition: A crossing zero, from left to right, at
• Fourth-order, 3 delays
Example 1: Zero Locus
Example 2: First-Order Condition Fails!
Example 3: Non-Semisimple Eigenvalue
Comparisons
• To conventional algebraic methods:
Numerical vs Symbolic computation
• Delay Stabilization Margin: For a delay system
or in state space form
Feedback Stabilization
)()(
)()()(
tCxty
tButAxtx
yu
The Delay Stabilization Margin Problem
Delay Stabilization Margin:
Gain Margin:
Phase Margin:
•The delay stabilization problem is fundamentally more difficult. No viable necessary and sufficient condition exists. The problem remains open.•Suggested as an unresolved challenge in Unsolved Problem in Mathematical Systems and Control Theory.
• First-order condition: A crossing zero, from left to right, at
Bounds and Sufficient Conditions
), ,0[ min
Rational Approximation
Guaranteeing Stabilization: Compute Eigenvalues
Analytical Bounds
Generalizations:
• The approach can be generalized to systems with time-varying delays.
• It can also be generalized to MIMO systems, to find estimates on the so-called delay radius.
• Similar results can be obtained using PID controllers.• Extensions to networked delay systems, multi-agent delayed
communications.
Summary
• Main Results:1. Eigenvalue Perturbation Approach:: A general mathematical result
providing a computationally efficient stability analysis approach,
requiring only the solution of a matrix pencil problem.
2. Linear Delay Systems: A full characterization of stability for all
possible delay parameter values.
3. Stabilization: Bounds on the fundamental delay stabilization margin.
• Highlights:1. Full Leverage on Computation: Joint and judicious use of
eigenvalues and eigenvectors.
2. Analytical vs Continuity Properties: Much like continuity, the
analyticity of eigenvalues plays an important role.
3. A unified operator-theoretic approach for stability and stabilization
of linear time-delay systems, generalizable to networked control and
multi-agent robustness problems.
The Amazing Dujiangyan … for Real!
Thanks to
Dalian University of Technology!
Thanks to Students and Collaborators—My Third Eye!
C. Méndez-Barrios
V. Kharitonov D. Ma
S.-I. Niculescu
P. Fu K. Gu
T. Qi
J. Zhu
Thank you all!
The radiance of the star that leans on me
Was shining years ago. The light that now
Glitters up there my eyes may never see
And so the time lag teases me with how
--Delay by Elizabeth Jennings