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
1 To propose a two-step controller synthesis method for a class
of uncertain nonlinear systems described by PWA diﬀerential
inclusions.
2 To introduce for the ﬁrst time a duality-based interpretation of
PWA systems. This enables controller synthesis for PWA slab
systems to be formulated as a convex optimization problem.
3 To propose a nonsmooth backstepping controller synthesis for
PWP systems.
4 To propose a time-delay approach to stability analysis of
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
t1 ≤ t2 ⇒ V (x(t1)) ≥ V (x(t2)) +
t2
t1
W (x(τ))dτ
and
x → ∞ ⇒ V (x) → ∞
then all trajectories in X asymptotically converge to x = x⋆.

Why nonsmooth analysis?
Discontinuous vector ﬁelds
Piecewise smooth Lyapunov functions
Lyapunov Function
Vector Field
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) = Aiκx + aiκ + Biκu, 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
f (x)
δ1(x)
δ2(x)
-4 -3 -2 -1 0 1 2 3 4
-60
-50
-40
-30
-20
-10
0
10

Theorem (3.2)
Let there exist matrices ¯Pi = ¯PT
i , ¯Ki , Zi , ¯Zi , Λiκ and ¯Λiκ 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{Aiκx + aiκ + Bwiκ
w, κ = 1, 2}, (x, w) ∈ RX×W
i
y ∈Conv{Ciκx + ciκ + Dwiκ
w, κ = 1, 2}
RX×W
i ={(x, w)| Li x + li + Mi w < 1}
Parameter set:
Φ =





Aiκ1 aiκ1 Bwiκ1
Li li Mi
Ciκ2 ciκ2 Dwiκ2

 i = 1, . . . , M, κ1 = 1, 2, κ2 = 1, 2




A duality-based convex optimization approach
Dual Parameter Set
ΦT
=





AT
iκ1
LT
i CT
iκ2
aT
iκ1
li cT
iκ2
BT
wiκ1
MT
i DT
wiκ2

 i = 1, . . . , M, κ1 = 1, 2, κ2 = 1, 2




LMI Conditions for L2 Gain Analysis
Parameter Set
P > 0,
AT
iκ1
P + PAiκ1 + CT
iκ2
Ciκ2 ∗
BT
wiκ1
P + DT
wiκ2
Ciκ2 −γ2I + DT
wiκ2
Dwiκ2
< 0
for i ∈ I(0, 0), κ1 = 1, 2 and κ2 = 1, 2 and λiκ1κ2 < 0














AT
iκ1
P + PAiκ1
+CT
iκ2
Ciκ2
+λiκ1κ2
LT
i Li

 ∗ ∗
aT
iκ1
P + cT
iκ2
Ciκ2
+ λiκ1κ2
li Li λiκ1κ2
(l2
i − 1) + cT
iκ2
ciκ2
∗



BT
wiκ1
P
+DT
wiκ2
Ciκ2
+λiκ1κ2
MT
i Li


 DT
wiκ2
ciκ2
+ λiκ1κ2
li MT
i


−γ2I
+DT
wiκ2
Dwiκ2
+λiκ1κ2
MT
i Mi














< 0
for i /∈ I(0, 0), κ1 = 1, 2 and κ2 = 1, 2

LMI Conditions for L2 Gain Analysis
Dual Parameter Set
Q > 0,
Aiκ1Q + QAT
iκ1
+ Bwiκ1
BT
wiκ1
∗
Ciκ2 Q + Dwiκ2
BT
wiκ1
−γ2I + Dwiκ2
DT
wiκ2
< 0
for i ∈ I(0, 0), κ1 = 1, 2 and κ2 = 1, 2 and µiκ1κ2 < 0















Aiκ1
Q + QAT
iκ1
+Bwiκ1
BT
wiκ1
+µiκ1κ2
aiκ1
aT
iκ1


 ∗ ∗
Li Q + Mi BT
wiκ1
+ µiκ1κ2
li aT
iκ1
µiκ1κ2
(l2
i − 1) + Mi MT
i ∗
Ciκ2
Q + Dwiκ2
BT
wiκ1
+µiκ1κ2
ciκ2
aT
iκ1
Dwiκ2
MT
i + µiκ1κ2
li ciκ2



−γ2I
+Dwiκ2
DT
wiκ2
+µiκ1κ2
ciκ2
cT
iκ2















< 0
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:
Yi = Ki Q
Zi = µi ki
Wi = µi ki kT
i
Problem: Wi is not a linear function of the unknown
parameters µi , Yi and Zi .

Proposed solutions:
Convex relaxation: Since Wi = µi ki kT
i ≤ 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 kT
i ≤ 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 kT
i ≤ 0 is the
solution of the following rank minimization problem:
min Rank Xi
s.t. Xi =
Wi Zi
ZT
i µi
≤ 0
Rank minimization is also not a convex problem. However,
trace minimization (maximization for negative semidefinite
matrices) works practically well as a heuristic solution
max Trace Xi , s.t. Xi =
Wi Zi
ZT
i µi
≤ 0

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.
p(x) =
m
i=1
f 2
i (x) ≥ 0

PWP system in strict feedback from



˙x1 = f1i1 (x1) + g1i1 (x1)x2, for x1 ∈ P1i1
˙x2 = f2i2 (x1, x2) + g2i2 (x1, x2)x3, for
x1
x2
∈ P2i2
...
˙xk = fkik
(x1, x2, . . . , xk ) + gkik
(x1, x2, . . . , xk )u, for





x1
x2
...
xk





∈ Pkik
where
Pjij
=






x1
...
xj


 Ejij
(x1, . . . , xj ) ≻ 0




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

Example:
Pitch model of the experimental helicopter:
˙x1 =x2
˙x2 =
1
Iyy
(−mheli lcgx g cos(x1) − mheli lcgz g sin(x1) − FkM sgn(x2)
− FvMx2 + 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)

x1
f(x1) f (x1)
ˆf (x1)
-3.1416 -1.885 -0.6283 0.6283 1.885 3.1416
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
PWA approximation - Helicopter model

x1
x2
-3 -2 -1 0 1 2 3
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Continuous time PWA controller

Sampled-data PWA controller
u(t) = Ki x(tk ) + ki , x(tk ) ∈ Ri
Continuous−time
PWA systems
PWA controller
Hold

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) ≤ σaµ2
θ + σb}

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

x1
x2
-3 -2 -1 0 1 2 3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Sampled data PWA controller for Ts = 0.2193

Conclusions

Summary of Major Contributions
1 To propose a two-step controller synthesis method for a class
inclusions.
2 To introduce for the ﬁrst time a duality-based interpretation of
PWA systems. This enables controller synthesis for PWA slab
systems to be formulated as a convex optimization problem.
3 To propose a nonsmooth backstepping controller synthesis for
PWP systems.
4 To propose a time-delay approach to stability analysis of
sampled-data PWA systems.

Publications
1 B. Samadi and L. Rodrigues, “Extension of local linear
controllers to global piecewise affine controllers for uncertain
nonlinear systems,” accepted for publication in the
International Journal of Systems Science.
2 B. Samadi and L. Rodrigues, “Controller synthesis for
piecewise affine slab differential inclusions: a duality-based
convex optimization approach,” under second revision for
publication in Automatica.
3 B. Samadi and L. Rodrigues, “Backstepping Controller
Synthesis for Piecewise Polynomial Systems: A Sum of
Squares Approach,” in preparation
4 B. Samadi and L. Rodrigues, “Sampled-Data Piecewise Affine
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.

Questions

Open Problems
Can general PWP/PWA controller synthesis be converted to a
convex problem?
What is the dual of a PWA system?

Special Structure
Type f (x) g(x)
Piecewise Smooth Piecewise Continuous Piecewise Continuous
Piecewise Polynomial Piecewise Polynomial Piecewise Polynomial
Piecewise Aﬃne Piecewise Aﬃne Piecewise Constant
Piecewise Linear Piecewise Linear Piecewise Constant
Linear Linear Constant

Convex Optimization
“In fact the great watershed in optimization is not between
linearity and nonlinearity, but convexity and
nonconvexity.” (Rockafellar, SIAM review, 1993)
The hard part is to ﬁnd out if a problem can be formulated as
a convex optimization problem.

Piecewise Quadratic Lyapunov function
V (x) = xT
Pi x + 2qT
i x + ri , for x ∈ Ri

Conditions on the PWA controller:
¯Ki = ¯Ki⋆ , if x⋆
∈ Ri
(¯Aiκ + ¯Biκ
¯Ki )¯x⋆
= 0, if x⋆
∈ Ri
(¯Aiκ + ¯Biκ
¯Ki )¯Fij = (¯Ajκ + ¯Bjκ
¯Kj )¯Fij , if Ri Rj = ∅

¯Ki = ¯Ki⋆ , if x⋆
∈ Ri
(¯Aiκ + ¯Biκ
¯Ki )¯x⋆
= 0, if x⋆
∈ Ri
(¯Aiκ + ¯Biκ
Continuity of the Lyapunov function:
¯FT
ij (¯Pi − ¯Pj )¯Fij = 0, if Ri Rj = ∅

¯Ki = ¯Ki⋆ , if x⋆
∈ Ri
(¯Aiκ + ¯Biκ
¯Ki )¯x⋆
= 0, if x⋆
∈ Ri
(¯Aiκ + ¯Biκ
Continuity of the Lyapunov function:
¯FT
ij (¯Pi − ¯Pj )¯Fij = 0, if Ri Rj = ∅
Positive deﬁniteness of the Lyapunov function:
¯Pi ¯x⋆
= 0, if x⋆
∈ Ri
Pi > ǫI, if x⋆
∈ Ri , Ei x⋆
+ ei = 0
Zi ∈ Rn×n, Zi 0
Pi − ET
i Zi Ei > ǫI
, if x⋆
∈ Ri , Ei x⋆
+ ei = 0
¯Zi ∈ R(n+1)×(n+1), ¯Zi 0
¯Pi − ¯ET
i
¯Zi
¯Ei > ǫ¯I
, if x⋆
/∈ Ri

Monotonicity of the Lyapunov function:
for i such that x⋆
∈ Ri , Ei x⋆
+ ei = 0,
Pi (Aiκ + BiκKi ) + (Aiκ + BiκKi )T
Pi < −αPi
∈ Ri , Ei x⋆
+ ei = 0,
Λiκ ∈ Rn×n, Λiκ 0
Pi (Aiκ + BiκKi ) + (Aiκ + BiκKi )TPi + ET
i ΛiκEi < −αPi
/∈ Ri ,
¯Λiκ ∈ R(n+1)×(n+1), ¯Λiκ 0
¯Pi (¯Aiκ + ¯Biκ
¯Ki ) + (¯Aiκ + ¯Biκ
¯Ki )T ¯Pi + ¯ET
i
¯Λiκ
¯Ei < −α¯Pi

Consider the following second order system
˙x1 = x2
˙x2 = −x1 + 0.5x2 − 0.5x2
1 x2 + u
with the following domain:
X =
x1
x2
− 30 < x1 < 30, −60 < x2 < 60

x1
x2
-6 -4 -2 0 2 4 6
-10
-5
0
5
10
Trajectories of the open loop system

x1
x2
-30 -20 -10 0 10 20 30
-60
-40
-20
0
20
40
60
Trajectories of the closed-loop system for the linear controller

x1
x2
-30 -20 -10 0 10 20 30
-60
-40
-20
0
20
40
60
Trajectories of the closed-loop system for the PWA controller

The main limitation: The controller synthesis is formulated as
a set of Bilinear Matrix Inequalities (BMI).
Linear systems analogy: Consider the linear system
˙x = Ax + Bu with a candidate Lyapunov function
V (x) = xTPx and a state feedback controller u = Kx.
Suﬃcient conditions for the stability of this system are:
Positivity: V (x) = xT
Px > 0 for x = 0
Monotonicity: ˙V (x) = ˙xT
Px + xT
P ˙x < 0
P > 0
(AT
+ KT
BT
)P + P(A + BK) < 0
BMIs are nonconvex problems in general.

Contributions:
To propose a two-step controller synthesis method for a class
inclusions.
The proposed method has two objectives: global stability and
local performance. It thus enables to use well known
techniques in linear control design for local stability and
performance while delivering a global PWA controller that is
guaranteed to stabilize the nonlinear system.
Diﬀerential inclusions are considered. Therefore the controller
is robust in the sense that it can stabilize any piecewise
smooth nonlinear system bounded by the diﬀerential inclusion.
Stability is guaranteed even if sliding modes exist.

Linear systems analogy:
Consider the dual system
˙z = (A + BK)T
z
with a candidate Lyapunov function
V (x) = xT
Qx
Sufficient conditions for the stability of this system are:
Positive definiteness: Q > 0
Monotonicity: (A + BK)Q + Q(AT
+ KT
BT
) < 0
Define Y = KQ.
Positive definiteness: Q > 0
Monotonicity: AQ + BY + QAT
+ Y T
BT
< 0
Then K = YQ−1.

PWA slab system:
˙x =Ai x + ai , x ∈ Ri
Ri ={x| Li x + li < 1}
Literature
Hassibi and Boyd (1998)
Rodrigues and Boyd (2005)

Suﬃcient conditions for stability
P > 0,
AT
i P + PAi + αP < 0, for 0 ∈ Ri ,



λi < 0,
AT
i P + PAi + αP + λi LT
i Li Pai + λi li LT
i
aT
i P + λi li Li λi (l2
i − 1)
< 0, for 0 /∈ Ri

Suﬃcient conditions for stability
Q > 0,
Ai Q + QAT
i + αQ < 0, for 0 ∈ Ri ,



µi < 0,
Ai Q + QAT
i + αQ + µi ai aT
i QLT
i + µi li ai
Li Q + µi li aT
i µi (l2
i − 1)
< 0, for 0 /∈ Ri

Parameter set:
Ω =
Ai ai
Li li
i = 1, . . . , M
Dual parameter set
ΩT
=
AT
i LT
i
aT
i li
i = 1, . . . , M

PWA slab system:
˙x =Ai x + ai + Bwi
w,
x ∈Ri = {x| Li x + li < 1},
y =Ci x + Dwi
w,
Parameter set:
Φ =





Ai ai Bwi
Li li 0
Ci 0 Dwi

 i = 1, . . . , M




Summary:
Introducing PWA slab differential inclusions
Introducing the dual parameter set
Extending the L2 gain analysis and synthesis to PWA slab
differential inclusions with PWA outputs
Extending the definition of the regions of a PWA slab
differential inclusion
Proposing two methods to formulate the PWA controller
synthesis for PWA slab differential inclusions as a convex
problem

Sum of Squares Programming
A sum of squares program is a convex optimization program of the
following form:
Minimize
J
j=1
wj αj
subject to fi,0 +
J
j=1
αj fi,j(x) is SOS, for i = 1, . . . , I
where the αj ’s are the scalar real decision variables, the wj ’s are
some given real numbers, and the fi,j are some given multivariate
polynomials.

Sum of Squares Programming
A sum of squares program is a convex optimization program of the
following form:
Minimize
J
j=1
wj αj
subject to fi,0 +
J
j=1
αj fi,j(x) is SOS, for i = 1, . . . , I
where the αj ’s are the scalar real decision variables, the wj ’s are
some given real numbers, and the fi,j are some given multivariate
polynomials.
SOSTOOLS, a MATLAB toolbox that handles the general
SOS programming, was developed by S. Prajna, A.
Papachristodoulou and P. Parrilo.

Consider the following PWP system:
˙x =fi (x) + gi (x)z, x ∈ Pi
˙z =u
where
Pi = {x|Ei (x) ≻ 0}
where Ei (x) ∈ Rpi is a vector polynomial function of x and ≻
represents an elementwise inequality.

Backstepping as a Lyapunov function construction method:
Consider ˙x = fi (x) + gi (x)z, x ∈ Pi

Assume that there exists a polynomial control z = γ(x) and
V (x) is an SOS Lyapunov function for the closed loop system
verifying
V (x) − λ(x) is SOS
−∇V (x)T(fi (x) + gi (x)γ(x)) − Γi (x)TEi (x) − αV (x) is SOS
for i = 1, . . . , M and any α > 0, where λ(x) is a positive
deﬁnite SOS polynomial, Γi (x) is an SOS vector function

Assume that there exists a polynomial control z = γ(x) and
V (x) is an SOS Lyapunov function for the closed loop system
verifying
V (x) − λ(x) is SOS
−∇V (x)T(fi (x) + gi (x)γ(x)) − Γi (x)TEi (x) − αV (x) is SOS
for i = 1, . . . , M and any α > 0, where λ(x) is a positive
deﬁnite SOS polynomial, Γi (x) is an SOS vector function
Consider now the following candidate Lyapunov function
Vγ(x, z) = V (x) +
1
2
(z − γ(x))T
(z − γ(x))
Note that Vγ(x, z) is a positive deﬁnite function.

Controller synthesis: The synthesis problem can be formulated
as the following SOS program.
Find u = γ2(x, z) and Γ(x, z)
s.t. −∇xVγ(x, z)T
(fi (x) + gi (x)z)
−∇zVγ(x, z)T
u
−Γ(x, z)T
Ei (x, z) − αVγ(x, z) is SOS,
Γ(x, z) is SOS
γ2(0, 0) = 0
where i2 = 1, . . . , M2 and γ2(x1, x2) is a polynomial function
of x1 and x2.

Example: Single link ﬂexible joint robot:

Example: Single link ﬂexible joint robot:
˙x1 = x2
˙x2 = −
MgL
I
sin(x1) −
K
I
(x1 − x3)
˙x3 = x4
˙x4 = −
f2(x4)
J
+
K
J
(x1 − x3) +
1
J
u
where x1 = θ1, x2 = ˙θ1, x3 = θ2 and x4 = ˙θ2. u is the motor
torque and f2(x4) denotes the motor friction which is described by
f2(x4) = bmx4 + sgn(x4) Fcm + (Fsm − Fcm) exp(−
x2
4
c2
m
)

PWP approximation:
x1
f1(x1)
f1(x1)
ˆf1(x1)
-3.1416 -0.8976 0.8976 3.1416
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
x4
f2(x4)
f2(x4)
ˆf2(x4)
-8 -6 -4 -2 0 2 4 6 8
-3
-2
-1
0
1
2
3

Typical structure of the regions of a PWP system in strict feedback form
x1 P11
vvmmmmmmmmmmmmmmmm
((QQQQQQQQQQQQQQQQ
(x1, x2) P21

P22

P23

(x1, x2, x3) P31
}}{{{{{{{{

P32
}}{{{{{{{{
!!CCCCCCCC P33
!!CCCCCCCC
(x1, x2, x3, x4) P41 P44 P42 P45 P43 P46

State variables of the nonlinear model - PWA controller:
Time
x1
Time
x2
Time
x3
Time
x4
0 2 4 60 2 4 6
0 2 4 60 2 4 6
-400
-200
0
200
-1
0
1
2
3
4
-8
-6
-4
-2
0
0
1
2
3
4

Control input - PWA controller:
Time
u
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-7
-6
-5
-4
-3
-2
-1
0
1
×104

State variables of the nonlinear model - PWP controller:
Time
x1
Time
x2
Time
x3
Time
x4
0 2 4 60 2 4 6
0 2 4 60 2 4 6
-10
-5
0
5
10
0
1
2
3
-5
-4
-3
-2
-1
0
0
1
2
3
4

Control input - PWP controller:
Time
u
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-300
-250
-200
-150
-100
-50
0
50
100
150
200

Summary of the contributions:
Introducing PWP systems in strict feedback form
Formulating backstepping controller synthesis for PWP
systems as a convex optimization problem
Polynomial Lyapunov functions for PWP systems with
discontinuous vector ﬁelds
PWP Lyapunov functions for PWP systems with continuous
vector ﬁelds
Numerical tools such as SOSTOOLS and Yalmip/SeDuMi

PWA system
˙x = Ai x + ai + Bi u, for x ∈ Ri
with the region Ri deﬁned as
Ri = {x|Ei x + ei ≻ 0},

PWA system
˙x = Ai x + ai + Bi u, for x ∈ Ri
with the region Ri deﬁned as
Ri = {x|Ei x + ei ≻ 0},
Continuous-time PWA controller
u(t) = Ki x(t) + ki , x(t) ∈ Ri

Lyapunov-Krasovskii functional:
V (xs, ρ) := V1(x) + V2(xs , ρ) + V3(xs , ρ)
where
xs(t) :=
x(t)
x(tk )
, tk ≤ t tk+1
V1(x) := xT
Px
V2(xs , ρ) :=
0
−τM
t
t+r
˙xT
(s)R ˙x(s)dsdr
V3(xs , ρ) := (τM − ρ)(x(t) − x(tk))T
X(x(t) − x(tk ))
and P, R and X are positive deﬁnite matrices.

for all i such that 0 ∈ Ri ,

 Ψi + τM M1i
P
0
Bi + τM
I
−I
XBi
BT
i P 0 + τM BT
i X I −I −γI

 0




Ψi + τM M2i τM
AT
i
KT
i BT
i
RBi +
P
0
Bi τM Ni
τM BT
i R Ai Bi Ki + BT
i P 0 τM BT
i RBi − γI 0
τM NT
i 0 − τM
2
R

for all i such that 0 /∈ Ri , ¯Λi ≻ 0,




Ψi + τM M1i


P
0
0

 Bi + τM


I
−I
0

 XBi
BT
i P 0 0 + τM BT
i X I −I 0 −γI



 0
















Ψi + τM M2i
τM


AT
i
KT
i BT
i
kT
i BT
i + aT
i

 RBi
+


P
0
0

 Bi
τM
Ni
0
τM BT
i R Ai Bi Ki Bi ki + ai
+BT
i P 0 0
τM BT
i RBi − γI 0
τM NT
i 0 0 − τM
2
R
















0

Summary of the contributions
Formulating stability analysis of sampled-data PWA systems
as a convex optimization problem

Active Suspension System
Example: Active suspension with a nonlinear damper
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
−30
−20
−10
0
10
20
30
40
Suspension deflection speed (m/s)
DampingForce(N)
Nonlinear
PWA Approximation

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

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