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 Examinationfor the Degree of Doctor of Philosophy

Behzad Samadi

Department of Mechanical and Industrial EngineeringConcordia University

18 April 2008Montreal, 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.”

http://www.cds.caltech.edu/~murray/archive/topten/


Special Structure

Piecewise smooth system:

x = f (x) + g(x)u

where f (x) and g(x) are piecewise continuous and boundedfunctions of x .


Objective

To develop a computational tool to design controllers forpiecewise smooth systems using convex optimization

techniques.


Objective

To develop a computational tool to design controllers forpiecewise smooth systems using convex optimization

techniques.

Why convex optimization?

There are numerically efficient tools to solve convexoptimization problems.Linear Matrix Inequalities (LMI)Sum of Squares (SOS) programming


Literature

Hassibi and Boyd (1998) - Quadratic stabilization and controlof piecewise linear systems - Limited to piecewise linearcontrollers for PWA slab systems

Johansson and Rantzer (2000) - Piecewise linear quadraticoptimal control - No guarantee for stability

Feng (2002) - Controller design and analysis of uncertainpiecewise linear systems - All local subsystems should be stable

Rodrigues and How (2003) - Observer-based control ofpiecewise affine systems - Bilinear matrix inequality

Rodrigues and Boyd (2005) - Piecewise affine state feedbackfor piecewise affine slab systems using convex optimization -Stability analysis and synthesis using parametrized linearmatrix inequalities


Major Contributions

1 To propose a two-step controller synthesis method for a classof uncertain nonlinear systems described by PWA differentialinclusions.

2 To introduce for the first time a duality-based interpretation ofPWA systems. This enables controller synthesis for PWA slabsystems to be formulated as a convex optimization problem.

3 To propose a nonsmooth backstepping controller synthesis forPWP systems.

4 To propose a time-delay approach to stability analysis ofsampled-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 6= 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 6= 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 FunctionVector 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

Letx ∈ Convσ1(x , u), . . . , σK(x , u)

whereσκ(x , u) = Aiκx + aiκ + Biκu, x ∈ Ri ,

withRi = x |Eix + 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 = PTi , 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 controllerprovides global robust stability and local robust performance.



Theorem (3.2)






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

The synthesis problem includes a set of Bilinear MatrixInequalities (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 asa set of LMIs.


L2 Gain Analysis for PWA Slab Differential Inclusions

PWA slab differential inclusion:

x ∈ConvAiκx + aiκ + Bwiκw , κ = 1, 2, (x ,w) ∈ RX×W

i

y ∈ConvCiκx + ciκ + Dwiκw , κ = 1, 2

RX×Wi =(x ,w)| ‖Lix + li + Miw‖ < 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 =

ATiκ1

LTi CT

iκ2

aTiκ1

li cTiκ2

BTwiκ1

MTi DT

wiκ2

∣

∣

∣

∣

∣

∣

i = 1, . . . ,M, κ1 = 1, 2, κ2 = 1, 2


LMI Conditions for L2 Gain Analysis

Parameter SetP > 0,

[

ATiκ1

P + PAiκ1 + CTiκ2

Ciκ2 ∗BT

wiκ1P + DT

wiκ2Ciκ2 −γ2I + DT

wiκ2Dwiκ2

]

< 0

for i ∈ I(0, 0), κ1 = 1, 2 and κ2 = 1, 2 and λiκ1κ2 < 0

ATiκ1

P + PAiκ1

+CTiκ2

Ciκ2

+λiκ1κ2LT

i Li

∗ ∗

aTiκ1

P + cTiκ2

Ciκ2+ λiκ1κ2

liLi λiκ1κ2(l2i − 1) + cT

iκ2ciκ2

∗

BTwiκ1

P

+DTwiκ2

Ciκ2

+λiκ1κ2MT

i Li

DT

wiκ2ciκ2

+ λiκ1κ2liM

Ti

−γ2I+DT

wiκ2Dwiκ2

+λiκ1κ2MT

i Mi

< 0

for i /∈ I(0, 0), κ1 = 1, 2 and κ2 = 1, 2


LMI Conditions for L2 Gain Analysis

Dual Parameter SetQ > 0,

[

Aiκ1Q + QATiκ1

+ Bwiκ1BT

wiκ1∗

Ciκ2Q + Dwiκ2BT

wiκ1−γ2I + Dwiκ2

DTwiκ2

]

< 0

for i ∈ I(0, 0), κ1 = 1, 2 and κ2 = 1, 2 and µiκ1κ2 < 0

Aiκ1Q + QAT

iκ1

+Bwiκ1BT

wiκ1

+µiκ1κ2aiκ1

aTiκ1

∗ ∗

LiQ + MiBTwiκ1

+ µiκ1κ2lia

Tiκ1

µiκ1κ2(l2i − 1) + MiM

Ti ∗

(

Ciκ2Q + Dwiκ2

BTwiκ1

+µiκ1κ2ciκ2

aTiκ1

)

Dwiκ2MT

i + µiκ1κ2liciκ2

−γ2I+Dwiκ2

DTwiκ2

+µiκ1κ2ciκ2

cTiκ2

< 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 thefollowing PWA controller:

u = Kix + ki , x ∈ Ri

New variables:

Yi = KiQ

Zi = µiki

Wi = µikikTi

Problem: Wi is not a linear function of the unknownparameters µi , Yi and Zi .



Proposed solutions:

Convex relaxation: Since Wi = µikikTi ≤ 0, if the synthesis

inequalities are satisfied with Wi = 0, they are satisfied withany Wi ≤ 0. Therefore, the synthesis problem can be madeconvex by omitting Wi .



Proposed solutions:

Convex relaxation: Since Wi = µikikTi ≤ 0, if the synthesis

inequalities are satisfied with Wi = 0, they are satisfied withany Wi ≤ 0. Therefore, the synthesis problem can be madeconvex by omitting Wi .Rank minimization: Note that Wi = µikik

Ti ≤ 0 is the

solution of the following rank minimization problem:

min Rank Xi

s.t. Xi =

[

Wi Zi

ZTi µi

]

≤ 0

Rank minimization is also not a convex problem. However,trace minimization (maximization for negative semidefinitematrices) works practically well as a heuristic solution

maxTrace Xi , s.t. Xi =

[

Wi Zi

ZTi µi

]

≤ 0



The following problems for PWA slab differential inclusions withPWA 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 piecewisepolynomial systems as a Sum of Squares (SOS) programming.




Objective:

To formulate controller synthesis for a class of piecewisepolynomial systems as a Sum of Squares (SOS) programming.

SOSTOOLS, a MATLAB toolbox that handles the generalSOS programming, was developed by S. Prajna, A.Papachristodoulou and P. Parrilo.

p(x) =m∑

i=1

f 2i (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 lcgxg cos(x1) − mheli lcgzg sin(x1) − FkM sgn(x2)

− FvMx2 + u)

where x1 is the pitch angle and x2 is the pitch rate.

Nonlinear part:

f (x1) = −mheli lcgxg cos(x1) − mheli lcgzg sin(x1)

PWA part:f (x2) = −FkM sgn(x2)



x1

f(x

1)

f (x1)

f (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) = Kix(tk) + ki , x(tk) ∈ Ri

Continuous−time

PWA systems

PWA controller

Hold



The closed-loop system can be rewritten as

x(t) = Aix(t) + ai + Bi(Kix(tk) + ki ) + Biw ,

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) arenot 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 Xconverge to the following invariant set

Ω = xs | V (xs , ρ) ≤ σaµ2θ + σb



Solving an optimization problem to maximize τM subject to theconstraints 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



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


Conclusions


Summary of Major Contributions

1 To propose a two-step controller synthesis method for a classof uncertain nonlinear systems described by PWA differentialinclusions.

2 To introduce for the first time a duality-based interpretation ofPWA systems. This enables controller synthesis for PWA slabsystems to be formulated as a convex optimization problem.

3 To propose a nonsmooth backstepping controller synthesis forPWP systems.

4 To propose a time-delay approach to stability analysis ofsampled-data PWA systems.


Publications

1 B. Samadi and L. Rodrigues, “Extension of local linearcontrollers to global piecewise affine controllers for uncertainnonlinear systems,” accepted for publication in theInternational Journal of Systems Science.

2 B. Samadi and L. Rodrigues, “Controller synthesis forpiecewise affine slab differential inclusions: a duality-basedconvex optimization approach,” under second revision forpublication in Automatica.

3 B. Samadi and L. Rodrigues, “Backstepping ControllerSynthesis for Piecewise Polynomial Systems: A Sum ofSquares Approach,” in preparation

4 B. Samadi and L. Rodrigues, “Sampled-Data Piecewise AffineSystems: 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 aconvex 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 betweenlinearity and nonlinearity, but convexity andnonconvexity.” (Rockafellar, SIAM review, 1993)

The hard part is to find out if a problem can be formulated asa convex optimization problem.



Piecewise Quadratic Lyapunov function

V (x) = xTPix + 2qTi x + ri , for x ∈ Ri



Conditions on the PWA controller:

Ki = Ki⋆ , if x⋆ ∈ Ri

(Aiκ + BiκKi )x⋆ = 0, if x⋆ ∈ Ri

(Aiκ + BiκKi )Fij = (Ajκ + BjκKj )Fij , if Ri

⋂

Rj 6= ∅







⋂

Rj 6= ∅

Continuity of the Lyapunov function:

FTij (Pi − Pj)Fij = 0, if Ri

⋂

Rj 6= ∅







⋂

Rj 6= ∅

Continuity of the Lyapunov function:

FTij (Pi − Pj)Fij = 0, if Ri

⋂

Rj 6= ∅

Positive definiteness of the Lyapunov function:

Pi x⋆ = 0, if x⋆ ∈ Ri

Pi > ǫI , if x⋆ ∈ Ri ,Eix⋆ + ei 6= 0

Zi ∈ Rn×n,Zi 0

Pi − ETi ZiEi > ǫI

, if x⋆ ∈ Ri ,Eix⋆ + ei = 0

Zi ∈ R(n+1)×(n+1), Zi 0

Pi − ETi Zi Ei > ǫI

, if x⋆ /∈ Ri



Monotonicity of the Lyapunov function:

for i such that x⋆ ∈ Ri ,Eix⋆ + ei 6= 0,

Pi (Aiκ + BiκKi) + (Aiκ + BiκKi )TPi < −αPi

for i such that x⋆ ∈ Ri ,Eix⋆ + ei = 0,

Λiκ ∈ Rn×n, Λiκ 0

Pi(Aiκ + BiκKi) + (Aiκ + BiκKi)TPi + ET

i ΛiκEi < −αPi

for i such that x⋆ /∈ Ri ,

Λiκ ∈ R(n+1)×(n+1), Λiκ 0

Pi(Aiκ + BiκKi) + (Aiκ + BiκKi)T Pi + ET

i ΛiκEi < −αPi



Consider the following second order system

x1 = x2

x2 = −x1 + 0.5x2 − 0.5x21 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 asa set of Bilinear Matrix Inequalities (BMI).

Linear systems analogy: Consider the linear systemx = Ax + Bu with a candidate Lyapunov functionV (x) = xTPx and a state feedback controller u = Kx .Sufficient conditions for the stability of this system are:

Positivity: V (x) = xTPx > 0 for x 6= 0Monotonicity: V (x) = xTPx + xTPx < 0

P > 0

(AT + KTBT)P + P(A + BK) < 0

BMIs are nonconvex problems in general.



Contributions:

To propose a two-step controller synthesis method for a classof uncertain nonlinear systems described by PWA differentialinclusions.

The proposed method has two objectives: global stability andlocal performance. It thus enables to use well knowntechniques in linear control design for local stability andperformance while delivering a global PWA controller that isguaranteed to stabilize the nonlinear system.Differential inclusions are considered. Therefore the controlleris robust in the sense that it can stabilize any piecewisesmooth 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)Tz

with a candidate Lyapunov function

V (x) = xTQx

Sufficient conditions for the stability of this system are:

Positive definiteness: Q > 0Monotonicity: (A + BK )Q + Q(AT + KTBT) < 0

Define Y = KQ.

Positive definiteness: Q > 0Monotonicity: AQ + BY + QAT + Y TBT < 0

Then K = YQ−1.



PWA slab system:

x =Aix + ai , x ∈ Ri

Ri =x | ‖Lix + li‖ < 1

Literature

Hassibi and Boyd (1998)Rodrigues and Boyd (2005)



Sufficient conditions for stability

P > 0,

ATi P + PAi + αP < 0, for 0 ∈ Ri ,

λi < 0,[

ATi P + PAi + αP + λiL

Ti Li Pai + λi liL

Ti

aTi P + λi liLi λi(l

2i − 1)

]

< 0, for 0 /∈ Ri

Hassibi and Boyd (1998)



Sufficient conditions for stability

Q > 0,

AiQ + QATi + αQ < 0, for 0 ∈ Ri ,

µi < 0,[

AiQ + QATi + αQ + µiaia

Ti QLT

i + µi liai

LiQ + µi liaTi µi(l

2i − 1)

]

< 0, for 0 /∈ Ri




Parameter set:

Ω =

[

Ai ai

Li li

] ∣

∣

∣

∣

i = 1, . . . ,M

Dual parameter set

ΩT =

[

ATi LT

i

aTi li

] ∣

∣

∣

∣

i = 1, . . . ,M



PWA slab system:

x =Aix + ai + Bwiw ,

x ∈Ri = x | ‖Lix + li‖ < 1,

y =Cix + Dwiw ,

Parameter set:

Φ =

Ai ai Bwi

Li li 0Ci 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 slabdifferential inclusions with PWA outputs

Extending the definition of the regions of a PWA slabdifferential inclusion

Proposing two methods to formulate the PWA controllersynthesis for PWA slab differential inclusions as a convexproblem


Sum of Squares Programming

A sum of squares program is a convex optimization program of thefollowing form:

Minimize

J∑

j=1

wjαj

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 aresome given real numbers, and the fi ,j are some given multivariatepolynomials.


Sum of Squares Programming

A sum of squares program is a convex optimization program of thefollowing form:

Minimize

J∑

j=1

wjαj

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 aresome given real numbers, and the fi ,j are some given multivariatepolynomials.

SOSTOOLS, a MATLAB toolbox that handles the generalSOS 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

wherePi = 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) andV (x) is an SOS Lyapunov function for the closed loop systemverifying

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 positivedefinite SOS polynomial, Γi(x) is an SOS vector function






Assume that there exists a polynomial control z = γ(x) andV (x) is an SOS Lyapunov function for the closed loop systemverifying

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 positivedefinite 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 formulatedas 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)Tu

−Γ(x , z)TEi (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 functionof x1 and x2.




Example: Single link flexible joint robot:




Example: Single link flexible joint robot:

x1 = x2

x2 = −MgL

Isin(x1) −

K

I(x1 − x3)

x3 = x4

x4 = −f2(x4)

J+

K

J(x1 − x3) +

1

Ju

where x1 = θ1, x2 = θ1, x3 = θ2 and x4 = θ2. u is the motortorque and f2(x4) denotes the motor friction which is described by

f2(x4) = bmx4 + sgn(x4)

(

Fcm + (Fsm − Fcm) exp(−x24

c2m

)

)




PWP approximation:

x1

f 1(x

1)

f1(x1)

f1(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

f 2(x

4)

f2(x4)

f2(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

!!CC

CCCC

CCP33

!!CC

CCCC

CC

(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 PWPsystems as a convex optimization problem

Polynomial Lyapunov functions for PWP systems withdiscontinuous vector fieldsPWP Lyapunov functions for PWP systems with continuousvector fieldsNumerical tools such as SOSTOOLS and Yalmip/SeDuMi



PWA system

x = Aix + ai + Biu, for x ∈ Ri

with the region Ri defined as

Ri = x |Eix + ei ≻ 0,



PWA system

x = Aix + ai + Biu, for x ∈ Ri

with the region Ri defined as

Ri = x |Eix + ei ≻ 0,

Continuous-time PWA controller

u(t) = Kix(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) := xTPx

V2(xs , ρ) :=

∫ 0

−τM

∫ t

t+rxT(s)Rx(s)dsdr

V3(xs , ρ) := (τM − ρ)(x(t) − x(tk))TX (x(t) − x(tk))

and P , R and X are positive definite matrices.



for all i such that 0 ∈ Ri ,

Ψi + τMM1i

[

P0

]

Bi + τM

[

I−I

]

XBi

BTi

[

P 0]

+ τMBTi X

[

I −I]

−γI

< 0

Ψi + τMM2i τM

[

ATi

KTi BT

i

]

RBi +

[

P0

]

Bi τMNi

τMBTi R

[

Ai BiKi

]

+ BTi

[

P 0]

τMBTi RBi − γI 0

τMNTi 0 −

τM2

R



for all i such that 0 /∈ Ri , Λi ≻ 0,

Ψi + τMM1 i

P00

Bi + τM

I−I0

XBi

BTi

[

P 0 0]

+ τMBTi X

[

I −I 0]

−γI

< 0

Ψi + τMM2i

τM

ATi

KTi BT

ikTi BT

i + aTi

RBi

+

P00

Bi

τM

[

Ni

0

]

τMBTi R

[

Ai BiKi Biki + ai

]

+BTi

[

P 0 0] τMBT

i RBi − γI 0

τM

[

NTi 0

]

0 −τM2

R

< 0



Summary of the contributions

Formulating stability analysis of sampled-data PWA systemsas 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)

Dam

ping

For

ce (

N)

NonlinearPWA Approximation