Transcript of "Stability Analysis and Controller Synthesis for a Class of Piecewise Smooth Systems"
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Stability Analysis and Controller Synthesis for a
Class of Piecewise Smooth Systems
The Oral Examination
for the Degree of Doctor of Philosophy
Behzad Samadi
Department of Mechanical and Industrial Engineering
Concordia University
18 April 2008
Montreal, Quebec
Canada
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Practical Motivation
c Quanser
Memoryless Nonlinearities
Saturation Dead Zone Coulomb &
Viscous Friction
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Theoretical Motivation
Richard Murray, California Institute of Technology: “
Rank Top Ten Research Problems in Nonlinear Control
10 Building representative experiments for evaluating controllers
9 Convincing industry to invest in new nonlinear methodologies
8 Recognizing the diﬀerence between regulation and tracking
7 Exploiting special structure to analyze and design controllers
6 Integrating good linear techniques into nonlinear methodologies
5 Recognizing the diﬀerence between performance and operability
4 Finding nonlinear normal systems for control
3 Global robust stabilization and local robust performance
2 Magnitude and rate saturation
1 Writing numerical software for implementing nonlinear theory
...This is more or less a way for me to think online, so I wouldn’t take any of this
too seriously.”
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Special Structure
Piecewise smooth system:
x = f (x) + g (x)u
˙
where f (x) and g (x) are piecewise continuous and bounded
functions of x.
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Objective
To develop a computational tool to design controllers for
piecewise smooth systems using convex optimization
techniques.
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Objective
To develop a computational tool to design controllers for
piecewise smooth systems using convex optimization
techniques.
Why convex optimization?
There are numerically eﬃcient tools to solve convex
optimization problems.
Linear Matrix Inequalities (LMI)
Sum of Squares (SOS) programming
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Literature
Hassibi and Boyd (1998) - Quadratic stabilization and control
of piecewise linear systems - Limited to piecewise linear
controllers for PWA slab systems
Johansson and Rantzer (2000) - Piecewise linear quadratic
optimal control - No guarantee for stability
Feng (2002) - Controller design and analysis of uncertain
piecewise linear systems - All local subsystems should be stable
Rodrigues and How (2003) - Observer-based control of
piecewise aﬃne systems - Bilinear matrix inequality
Rodrigues and Boyd (2005) - Piecewise aﬃne state feedback
for piecewise aﬃne slab systems using convex optimization -
Stability analysis and synthesis using parametrized linear
matrix inequalities
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Major Contributions
To propose a two-step controller synthesis method for a class
1
of uncertain nonlinear systems described by PWA diﬀerential
inclusions.
To introduce for the ﬁrst time a duality-based interpretation of
2
PWA systems. This enables controller synthesis for PWA slab
systems to be formulated as a convex optimization problem.
To propose a nonsmooth backstepping controller synthesis for
3
PWP systems.
To propose a time-delay approach to stability analysis of
4
sampled-data PWA systems.
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Lyapunov Stability for Piecewise Smooth Systems
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Lyapunov Stability for Piecewise Smooth Systems
Theorem (2.1)
For nonlinear system x(t) = f (x(t)), if there exists a continuous
˙
function V (x) such that
V (x ⋆ ) = 0
V (x) > 0 for all x = x ⋆ in X
t1 ≤ t2 ⇒ V (x(t1 )) ≥ V (x(t2 ))
then x = x ⋆ is a stable equilibrium point. Moreover if there exists a
continuous function W (x) such that
W (x ⋆ ) = 0
W (x) > 0 for all x = x ⋆ in X
t2
t1 ≤ t2 ⇒ V (x(t1 )) ≥ V (x(t2 )) + W (x(τ ))dτ
t1
and
x → ∞ ⇒ V (x) → ∞
then all trajectories in X asymptotically converge to x = x ⋆ .
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Lyapunov Stability for Piecewise Smooth Systems
Why nonsmooth analysis?
Discontinuous vector ﬁelds
Piecewise smooth Lyapunov functions
Vector Field
Lyapunov Function
Continuous Discontinuous
Smooth
Piecewise Smooth
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Extension of local linear controllers to global PWA
controllers for uncertain nonlinear systems
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Extension of linear controllers to PWA controllers
Objectives:
Global robust stabilization and local robust performance
Integrating good linear techniques into nonlinear
methodologies
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
PWA Diﬀerential Inclusions
Consider the following uncertain nonlinear system
x = f (x) + g (x)u
˙
Let
x ∈ Conv{σ1 (x, u), . . . , σK (x, u)}
˙
where
σκ (x, u) = Ai κx + ai κ + Bi κ u, x ∈ Ri ,
with
Ri = {x|Ei x + ei ≻ 0}, for i = 1, . . . , M
≻ represents an elementwise inequality.
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Extension of linear controllers to PWA controllers
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Extension of linear controllers to PWA controllers
Theorem (3.2)
¯ ¯ ¯ ¯ ¯
Let there exist matrices Pi = PiT , Ki , Zi , Zi , Λi κ and Λi κ that
verify the conditions for all i = 1, . . . , M, κ = 1, . . . , K and for a
given decay rate α > 0, desired equilibrium point x ⋆ , linear
¯
controller gain Ki ⋆ and ǫ > 0, then all trajectories of the
nonlinear system in X asymptotically converge to x = x ⋆ .
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Extension of linear controllers to PWA controllers
Theorem (3.2)
¯ ¯ ¯ ¯ ¯
Let there exist matrices Pi = PiT , Ki , Zi , Zi , Λi κ and Λi κ that
verify the conditions for all i = 1, . . . , M, κ = 1, . . . , K and for a
given decay rate α > 0, desired equilibrium point x ⋆ , linear
¯
controller gain Ki ⋆ and ǫ > 0, then all trajectories of the
nonlinear system in X asymptotically converge to x = x ⋆ .
If the conditions are feasible, the resulting PWA controller
provides global robust stability and local robust performance.
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Extension of linear controllers to PWA controllers
Theorem (3.2)
¯ ¯ ¯ ¯ ¯
Let there exist matrices Pi = PiT , Ki , Zi , Zi , Λi κ and Λi κ that
verify the conditions for all i = 1, . . . , M, κ = 1, . . . , K and for a
given decay rate α > 0, desired equilibrium point x ⋆ , linear
¯
controller gain Ki ⋆ and ǫ > 0, then all trajectories of the
nonlinear system in X asymptotically converge to x = x ⋆ .
If the conditions are feasible, the resulting PWA controller
provides global robust stability and local robust performance.
The synthesis problem includes a set of Bilinear Matrix
Inequalities (BMI). In general, it is not convex.
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Controller synthesis for PWA slab diﬀerential inclusions: a
duality-based convex optimization approach
Objective:
To formulate controller synthesis for stability and L2 gain
performance of piecewise aﬃne slab diﬀerential inclusions as
a set of LMIs.
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
L2 Gain Analysis for PWA Slab Diﬀerential Inclusions
PWA slab diﬀerential inclusion:
x ∈Conv{Ai κ x + ai κ + Bwi κ w , κ = 1, 2}, (x, w ) ∈ RX ×W
˙ i
y ∈Conv{Ci κ x + ci κ + Dwi κ w , κ = 1, 2}
RX ×W ={(x, w )| Li x + li + Mi w < 1}
i
Parameter set:
Ai κ1 ai κ1 Bwi κ1
Φ = Li Mi i = 1, . . . , M, κ1 = 1, 2, κ2 = 1, 2
li
Ci κ2 ci κ2 Dwi κ2
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
A duality-based convex optimization approach
Dual Parameter Set
T
LT CiT 2
Ai κ1
i κ
ΦT = aiT 1 ciT 2
li i = 1, . . . , M, κ1 = 1, 2, κ2 = 1, 2
κ κ
T MiT T
Bwi κ Dwi κ
1 2
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
LMI Conditions for L2 Gain Analysis
Parameter Set
P > 0,
AT 1 P + PAi κ1 + CiT 2 Ci κ2 ∗
iκ κ
<0
T T −γ 2 I + Dwi κ Dwi κ2
T
Bwi κ P + Dwi κ Ci κ2
1 2 2
for i ∈ I(0, 0), κ1 = 1, 2 and κ2 = 1, 2 and λi κ1 κ2 < 0
AT 1 P + PAi κ1
iκ
+CiT 2 Ci κ2 ∗ ∗
κ
+λi κ1 κ2 LT Li
i
aiT 1 P + ciT 2 Ci κ2 + λi κ1 κ2 li Li λi κ1 κ2 (li2 − 1) + ciT 2 ci κ2 ∗
<0
κ κ κ
T
−γ 2 I
Bwi κ P
1
T
T Dwi κ ci κ2 + λi κ1 κ2 li MiT
T +Dwi κ Dwi κ
+Dwi κ Ci κ2
2
2
2 2
TM
+λi κ1 κ2 MiT Li +λi κ1 κ2 Mi i
for i ∈ I(0, 0), κ1 = 1, 2 and κ2 = 1, 2
/
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
LMI Conditions for L2 Gain Analysis
Dual Parameter Set
Q > 0,
Ai κ1 Q + QAT 1 + Bwi κ1 Bwi κ
T ∗
iκ
<0
1
T −γ 2 I + Dwi κ2 Dwi κ
T
Ci κ2 Q + Dwi κ2 Bwi κ
1 2
for i ∈ I(0, 0), κ1 = 1, 2 and κ2 = 1, 2 and µi κ1κ2 < 0
Ai κ1 Q + QAT 1
iκ
T
+B
wi κ1 Bwi κ ∗ ∗
1
+µi κ1 κ2 ai κ1 aiT 1
κ
Li Q + Mi Bwi κ + µi κ1 κ2 li aiT 1
T µi κ1 κ2 (li2 − 1) + Mi MiT
∗ <0
κ
1
−γ 2 I
T
Ci κ2 Q + Dwi κ2 Bwi κ
+Dw D T
Dwi κ2 MiT + µi κ1 κ2 li ci κ2
1 wi κ2
i κ2
T
+µi κ1 κ2 ci κ2 ai κ1
+µi κ1 κ2 ci κ2 ciT 2
κ
for i ∈ I(0, 0), κ1 = 1, 2 and κ2 = 1, 2
/
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
PWA L2 gain controller synthesis
The goal is to limit the L2 gain from w to y using the
following PWA controller:
u = Ki x + ki , x ∈ Ri
New variables:
= Ki Q
Yi
= µi ki
Zi
= µi ki kiT
Wi
Problem: Wi is not a linear function of the unknown
parameters µi , Yi and Zi .
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
A duality-based convex optimization approach
Proposed solutions:
Convex relaxation: Since Wi = µi ki kiT ≤ 0, if the synthesis
inequalities are satisﬁed with Wi = 0, they are satisﬁed with
any Wi ≤ 0. Therefore, the synthesis problem can be made
convex by omitting Wi .
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
A duality-based convex optimization approach
Proposed solutions:
Convex relaxation: Since Wi = µi ki kiT ≤ 0, if the synthesis
inequalities are satisﬁed with Wi = 0, they are satisﬁed with
any Wi ≤ 0. Therefore, the synthesis problem can be made
convex by omitting Wi .
Rank minimization: Note that Wi = µi ki kiT ≤ 0 is the
solution of the following rank minimization problem:
min Rank Xi
Wi Z i
s.t. Xi = ≤0
ZiT µi
Rank minimization is also not a convex problem. However,
trace minimization (maximization for negative semideﬁnite
matrices) works practically well as a heuristic solution
Wi Z i
max Trace Xi , s.t. Xi = ≤0
ZiT µi
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
A duality-based convex optimization approach
The following problems for PWA slab diﬀerential inclusions with
PWA outputs were formulated as a set of LMIs:
Stability analysis (Propositions 4.1 and 4.2)
L2 gain analysis (Propositions 4.3 and 4.4)
Stabilization using PWA controllers (Propositions 4.5 and 4.6)
PWA L2 gain controller synthesis (Propositions 4.7 and 4.8)
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Backstepping Controller Synthesis for PWP Systems: A
Sum of Squares Approach
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Backstepping Controller Synthesis for PWP Systems: A
Sum of Squares Approach
Objective:
To formulate controller synthesis for a class of piecewise
polynomial systems as a Sum of Squares (SOS) programming.
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Backstepping Controller Synthesis for PWP Systems: A
Sum of Squares Approach
Objective:
To formulate controller synthesis for a class of piecewise
polynomial systems as a Sum of Squares (SOS) programming.
SOSTOOLS, a MATLAB toolbox that handles the general
SOS programming, was developed by S. Prajna, A.
Papachristodoulou and P. Parrilo.
m
fi 2 (x) ≥ 0
p(x) =
i =1
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Sampled-Data PWA Systems: A Time-Delay Approach
Sampled-data PWA controller
u(t) = Ki x(tk ) + ki , x(tk ) ∈ Ri
Continuous−time
Hold
PWA systems
PWA controller
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Sampled-Data PWA Systems: A Time-Delay Approach
The closed-loop system can be rewritten as
x(t) = Ai x(t) + ai + Bi (Ki x(tk ) + ki ) + Bi w ,
˙
for x(t) ∈ Ri and x(tk ) ∈ Rj where
w (t) = (Kj − Ki )x(tk ) + (kj − ki ), x(t) ∈ Ri , x(tk ) ∈ Rj
The input w (t) is a result of the fact that x(t) and x(tk ) are
not necessarily in the same region.
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Sampled-Data PWA Systems: A Time-Delay Approach
Theorem (6.1)
For the sampled-data PWA system, assume there exist symmetric
positive matrices P, R, X and matrices Ni for i = 1, . . . , M such
that the conditions are satisﬁed and let there be constants ∆K and
∆k such that
w ≤ ∆K x(tk ) + ∆k
Then, all the trajectories of the sampled-data PWA system in X
converge to the following invariant set
Ω = {xs | V (xs , ρ) ≤ σa µ2 + σb }
θ
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Sampled-Data PWA Systems: A Time-Delay Approach
Solving an optimization problem to maximize τM subject to the
constraints of the main theorem and η > γ > 1 leads to
⋆
τM = 0.2193
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Summary of Major Contributions
To propose a two-step controller synthesis method for a class
1
of uncertain nonlinear systems described by PWA diﬀerential
inclusions.
To introduce for the ﬁrst time a duality-based interpretation of
2
PWA systems. This enables controller synthesis for PWA slab
systems to be formulated as a convex optimization problem.
To propose a nonsmooth backstepping controller synthesis for
3
PWP systems.
To propose a time-delay approach to stability analysis of
4
sampled-data PWA systems.
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Publications
B. Samadi and L. Rodrigues, “Extension of local linear
1
controllers to global piecewise aﬃne controllers for uncertain
nonlinear systems,” accepted for publication in the
International Journal of Systems Science.
B. Samadi and L. Rodrigues, “Controller synthesis for
2
piecewise aﬃne slab diﬀerential inclusions: a duality-based
convex optimization approach,” under second revision for
publication in Automatica.
B. Samadi and L. Rodrigues, “Backstepping Controller
3
Synthesis for Piecewise Polynomial Systems: A Sum of
Squares Approach,” in preparation
B. Samadi and L. Rodrigues, “Sampled-Data Piecewise Aﬃne
4
Systems: A Time-Delay Approach,” to be submitted.
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Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Publications
1 B. Samadi and L. Rodrigues, “Backstepping Controller Synthesis for Piecewise
Polynomial Systems: A Sum of Squares Approach,” submitted to the 46th
Conference on Decision and Control, cancun, Mexico, Dec. 2008.
2 B. Samadi and L. Rodrigues, “Sampled-Data Piecewise Aﬃne Slab Systems: A
Time-Delay Approach,” in Proc. of the American Control Conference, Seattle,
WA, Jun. 2008.
3 B. Samadi and L. Rodrigues, “Controller synthesis for piecewise aﬃne slab
diﬀerential inclusions: a duality-based convex optimization approach,” in Proc.
of the 46th Conference on Decision and Control, New Orleans, LA, Dec. 2007.
4 B. Samadi and L. Rodrigues, “Backstepping Controller Synthesis for Piecewise
Aﬃne Systems: A Sum of Squares Approach,” in Proc. of the IEEE
International Conference on Systems, Man, and Cybernetics (SMC 2007),
Montreal, Oct. 2007.
5 B. Samadi and L. Rodrigues, “Extension of a local linear controller to a
stabilizing semi-global piecewise-aﬃne controller,” 7th Portuguese Conference
on Automatic Control, Lisbon, Portugal, Sep. 2006.
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