Controller synthesis for piecewise affine slab differential inclusions: A duality-based convex optimization approach

Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions

Controller synthesis for piecewise affine slabdifferential inclusions

A duality-based convex optimization approach

Behzad Samadi Luis Rodrigues

Department of Mechanical and Industrial EngineeringConcordia University

CDC 2007, New Orleans

Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 1/ 25


Outline of Topics

1 Introduction

2 Stability Analysis

3 L2 Gain Analysis

4 Controller Synthesis

5 Numerical Example

6 Conclusions



Motivation

Question: What is the dual of a piecewise affine (PWA)system?

It is still an open problem.



Motivation

Question: What is the dual of a piecewise affine (PWA)system?

It is still an open problem.



Piecewise Affine Slab Differential Inclusions

A continuous-time PWA slab differential inclusion is describedas

x ∈ ConvAiκx + aiκ + Buiκu + Bwiκw , κ = 1, 2y ∈ ConvCiκx + ciκ + Duiκu + Dwiκw , κ = 1, 2

for (x ,w) ∈ RX×Wi where Conv stands for the convex hull ofa set.

RX×Wi for i = 1, . . . ,M are M slab regions defined as

Ri = (x ,w) | σi < CRx + DRw < σi+1,

where CR ∈ R1×n, DR ∈ R1×nw and σi for i = 1, . . . ,M + 1are scalars such that

σ1 < σ2 < . . . < σM+1




A continuous-time PWA slab differential inclusion is describedas

x ∈ ConvAiκx + aiκ + Buiκu + Bwiκw , κ = 1, 2y ∈ ConvCiκx + ciκ + Duiκu + Dwiκw , κ = 1, 2

for (x ,w) ∈ RX×Wi where Conv stands for the convex hull ofa set.

RX×Wi for i = 1, . . . ,M are M slab regions defined as

Ri = (x ,w) | σi < CRx + DRw < σi+1,

where CR ∈ R1×n, DR ∈ R1×nw and σi for i = 1, . . . ,M + 1are scalars such that

σ1 < σ2 < . . . < σM+1




Practical examples:

Mechanical systems with hard nonlinearities such assaturation, deadzone, Columb friction

Contact dynamics

Electrical circuits with diodes




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



Objective

To introduce a concept of duality for PWA slab differentialinclusions

To propose a method for PWA controller synthesis for stabilityand L2-gain performance of PWA slab differential inclusionsusing convex optimization

Convex optimization problems are numerically tractable.



Objective

To introduce a concept of duality for PWA slab differentialinclusions

To propose a method for PWA controller synthesis for stabilityand L2-gain performance of PWA slab differential inclusionsusing convex optimization

Convex optimization problems are numerically tractable.



Dual parameter set

PWA slab differential inclusion:

x ∈ ConvAiκx + aiκ, κ = 1, 2, x ∈ Ri

Ri = x | ‖Lix + li‖ < 1

Parameter set:

Ω =

[Aiκ aiκ

Li li

] ∣∣∣∣ i = 1, . . . ,M, κ = 1, 2



Dual parameter set


x ∈ ConvAiκx + aiκ, κ = 1, 2, x ∈ Ri

Ri = x | ‖Lix + li‖ < 1

Parameter set:

Ω =

[Aiκ aiκ

Li li

] ∣∣∣∣ i = 1, . . . ,M, κ = 1, 2



Dual parameter set

Sufficient conditions for stability

P > 0,

ATiκP + PAiκ + αP < 0, ∀i ∈ I(0),

λiκ < 0,[AT

iκP + PAiκ + αP + λiκLTi Li Paiκ + λiκliL

Ti

aTiκP + λiκliLi λiκ(l2

i − 1)

]< 0,

for i /∈ I(0).



Dual parameter set

Dual parameter set

ΩT =

[AT

iκ LTi

aTiκ li

] ∣∣∣∣ i = 1, . . . ,M, κ = 1, 2



Dual parameter set


Q > 0,

AiκQ + QATiκ + αQ < 0, ∀i ∈ I(0), κ = 1, 2

µiκ < 0[AiκQ + QAT

iκ + αQ + µiκaiκaTiκ QLT

i + µiκliaiκ

LiQ + µiκliaTiκ µiκ(l2

i − 1)

]< 0,

for i /∈ I(0).

A new interpretation for the result in Hassibi and Boyd (1998)



L2 gain


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

y ∈ ConvCiκx + ciκ + Dwiκw , κ = 1, 2RX×Wi = (x ,w)| ‖Lix + li + Miw‖ < 1

Parameter set:

Φ =

Aiκ aiκ Bwiκ

Li li Mi

Ciκ ciκ Dwiκ

∣∣∣∣∣∣ i = 1, . . . ,M, κ = 1, 2



L2 gain


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

y ∈ ConvCiκx + ciκ + Dwiκw , κ = 1, 2RX×Wi = (x ,w)| ‖Lix + li + Miw‖ < 1

Parameter set:

Φ =

Aiκ aiκ Bwiκ

Li li Mi

Ciκ ciκ Dwiκ

∣∣∣∣∣∣ i = 1, . . . ,M, κ = 1, 2



L2 gain

Sufficient conditions for L2 gain performance

P > 0,[AT

iκP + PAiκ + CTiκCiκ ∗

BTwiκ

P + DTwiκ

Ciκ −γ2I + DTwiκ

Dwiκ

]< 0, ∀i ∈ I(0, 0),

(AT

iκP + PAiκ

+CTiκCiκ + λiκLT

i Li

)∗ ∗

aTiκP + cT

iκCiκ + λiκ li Li λiκ(l2i − 1) + cTiκciκ ∗

BTwiκ

P + DTwiκ

Ciκ + λiκMTi Li DT

wiκciκ + λiκ li M

Ti

(−γ2I + DT

wiκDwiκ

+λiκMTi Mi

) < 0

and λiκ < 0 for i /∈ I(0, 0).



Dual parameter set

Dual parameter set

ΦT =

AT

iκ LTi CT

iκ

aTiκ li cT

iκ

BTwiκ

MTi DT

wiκ

∣∣∣∣∣∣ i = 1, . . . ,M, κ = 1, 2



Dual parameter set


Q > 0,[AiκQ + QAT

iκ + BwiκBTwiκ

∗CiκQ + DwiκBT

wiκ−γ2I + DwiκDT

wiκ

]< 0, ∀i ∈ I(0, 0)

(AiκQ + QAT

iκ+Bwiκ

BTwiκ

+ µiκaiκaTiκ

)∗ ∗

LiκQ + Mi BTwiκ

+ µiκ li aTiκ µiκ(l2i − 1) + Mi M

Ti ∗

CiκQ + DwiκBT

wiκ+ µiκciκaT

iκ DwiκMT

i + µiκ li ciκ

(−γ2I + Dwiκ

DTwiκ

+µiκciκcTiκ

) < 0

and µiκ < 0 for i /∈ I(0, 0).

A new result that extends the result in Hassibi and Boyd(1998) for ci 6= 0



PWA controller synthesis

Consider the following system:

x ∈ ConvAiκx + aiκ + Buiκu, κ = 1, 2, x ∈ Ri

Ri = x | ‖Lix + li‖ < 1

The stability conditions corresponding to the dual parameterset is used to formulate the synthesis problem.




Controller synthesis problem:

Q > 0,

AiκQ + QATiκ + BuiκYi + Y T

i BTuiκ

+ αQ < 0,

for i ∈ I(0), κ = 1, 2 , and

µi < 0

AiκQ + QAT

iκ

+BuiκYi + Y Ti BT

uiκ

+αQ + µiaiκaTiκ

+aiκZTi BT

uiκ+ BuiκZia

Tiκ

+BuiκWiBTuiκ

∗

(LiQ + µi lia

Tiκ

+liZTi BT

uiκ

)µiκ(l2

i − 1)

≤ 0,

for i /∈ I(0) and κ = 1, 2




New variables:

Yi = KiQ

Zi = µiki

Wi = µikikTi

There is a problem: Wi is not a linear function of theunknown parameters µi , Yi and Zi .




Two 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 = µikikTi ≤ 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 works practically well as a heuristic solution

min Trace Xi , s.t. Xi =

[Wi Zi

ZTi µi

]≤ 0



L2 gain PWA controller synthesis

Consider the following system:

x ∈ ConvAiκx + aiκ + Buiκu + Bwiκw , κ = 1, 2,y ∈ ConvCiκx + ciκ + Duiκu + Dwiκw,for (x ,w) ∈ RX×Wi = (x ,w)| ‖Lix + li + Miw‖ < 1

The L2 conditions corresponding to the dual parameter set isused to formulate the synthesis problem.



L2 gain PWA controller synthesis

L2 gain controller synthesis problem:Q > 0,

AiκQ + BuiκYi

+QATiκ + Y T

i BTuiκ

+BwiκBT

wiκ

∗

(CiκQ + Duiκ

Yi

+DwiκBT

wiκ

)−γ2I + Dwiκ

DTwiκ

< 0

for i ∈ I(0), κ = 1, 2 , andµi < 0

AiκQ + Buiκ

Yi

+QATiκ + Y T

i BTuiκ

+BwiκBT

wiκ+ µi aiκaT

iκ

aiκZTi BT

uiκ+ Buiκ

Zi aTiκ

∗ ∗

(LiκQ + Mi B

Twiκ

+µiκ li aTiκ + li Z

Ti BT

uiκ

)µiκ(l2i − 1) + Mi M

Ti ∗

CiκQ + DuiκYi

+DwiκBT

wiκ+ µiκciκaT

iκ

ciκZTi BT

uiκ+ Duiκ

Zi aTiκ

Dwiκ

MTi

+µiκ li ciκ+li Duiκ

Zi

−γ2I + DwiκDT

wiκ+µiκciκcT

iκ + ciκZTi DT

uiκ+Duiκ

Zi cTiκ

< 0,

for i /∈ I(0) and κ = 1, 2



Surge model of a jet engine

Consider the following model (Kristic et al 1995):x1 = −x2 − 3

2x21 − 1

2x31

x2 = u

A bounding envelope is computed for the nonlinear functionf (x1) = −3

2x21 − 1

2x31



Modeling

By substituting the PWA bounds in the equations of thenonlinear system, we get a differential inclusion

x ∈ ConvAiκx + aiκ + Buu + Bww, x ∈ Ri

y = Cx + Dww + Duu (1)

where i = 1, . . . , 4, κ = 1, 2

The approximation error of the nonlinear function isconsidered as the disturbance input (w) and the objective isto limit the L2-gain from w to x1.



Simulation



Conclusions:

A new concept, dual parameter set, was introduced for PWAdifferential inclusions.

Using the dual parameter set, sufficient conditions for stabilityand L2 gain performance were obtained.

Convex methods were proposed for PWA controller synthesisfor stability and performance.

Note that the dual parameter set does not necessarily define aPWA system. The questions still is:

Does a dual system exist for a PWA system in general?



Conclusions:








Conclusions:








Conclusions:







Technology

Controller synthesis for piecewise affine slab differential inclusions: A duality-based convex optimization approach