Stability Analysis and Controller Synthesis for a Class of Piecewise Smooth Systems

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


Outline


Introduction


Practical Motivation

c Quanser

Memoryless Nonlinearities

Saturation Dead Zone Coulomb &
Viscous Friction


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.”


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.


Objective

To develop a computational tool to design controllers for
piecewise smooth systems using convex optimization
techniques.


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


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


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.


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 ⋆ .



Why nonsmooth analysis?
Discontinuous vector ﬁelds
Piecewise smooth Lyapunov functions

Vector Field
Lyapunov Function
Continuous Discontinuous
Smooth
Piecewise Smooth


Extension of local linear controllers to global PWA
controllers for uncertain nonlinear systems


Extension of linear controllers to PWA controllers

Objectives:
Global robust stabilization and local robust performance
Integrating good linear techniques into nonlinear
methodologies


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.


PWA Diﬀerential Inclusions

PWA bounding envelope for a nonlinear function
10

0

-10

-20

-30

-40

f (x)
-50 δ1 (x)
δ2 (x)
-60
-4 -3 4
-2 -1 0 1 2 3



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 ⋆ .



Theorem (3.2)
¯ ¯ ¯ ¯ ¯
¯

If the conditions are feasible, the resulting PWA controller
provides global robust stability and local robust performance.



Theorem (3.2)
¯ ¯ ¯ ¯ ¯
¯

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.


Controller synthesis for PWA slab diﬀerential inclusions: a
duality-based convex optimization approach


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.


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
 


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


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
/


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
/


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 .


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 .


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



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)


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.



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



PWP system in strict feedback from

 x1 = f1i1 (x1 ) + g1i1 (x1 )x2 ,
˙ for x1 ∈ P1i1
x1


˙
 x2 = f2i (x1 , x2 ) + g2i (x1 , x2 )x3 , for ∈ P2i2

2 2
x2


.

.
.

 
x1


x2

  

˙
 xk = fki (x1 , x2 , . . . , xk ) + gki (x1 , x2 , . . . , xk )u, for   ∈ Pkik

.
 
.
k k
.

  


xk


where  

x1
 
.  E (x , . . . , x ) ≻ 0
 
.  jij 1
Pjij = 

. j
 
xj
 


Sampled-Data PWA Systems: A Time-Delay Approach
Motivation: Toycopter, a 2 DOF helicopter model



Example:
Pitch model of the experimental helicopter:

x1 =x2
˙
1
x2 = (−mheli lcgx g cos(x1 ) − mheli lcgz g sin(x1 ) − FkM sgn(x2 )
˙
Iyy
− FvM x2 + u)

where x1 is the pitch angle and x2 is the pitch rate.
Nonlinear part:

f (x1 ) = −mheli lcgx g cos(x1 ) − mheli lcgz g sin(x1 )

PWA part:
f (x2 ) = −FkM sgn(x2 )



0.4
f (x1 )
ˆ
0.3 f (x1 )

0.2

0.1
f (x1 )

0

-0.1

-0.2

-0.3

-0.4
3.1416
-3.1416 -1.885 -0.6283 0.6283 1.885
x1

PWA approximation - Helicopter model



2

1.5

1

0.5

0
x2

-0.5

-1

-1.5

-2
2 3
-3 -2 -1 0 1
x1

Continuous time PWA controller



Sampled-data PWA controller

u(t) = Ki x(tk ) + ki , x(tk ) ∈ Ri

Continuous−time
Hold
PWA systems

PWA controller



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.



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



Solving an optimization problem to maximize τM subject to the
constraints of the main theorem and η > γ > 1 leads to
⋆
τM = 0.2193



2.5

2

1.5

1

0.5

0
x2

-0.5

-1

-1.5

-2

-2.5
3
-3 -2 -1 0 1 2
x1

Sampled data PWA controller for Ts = 0.2193


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.


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.


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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