Robust Semidefinite Programming andIts Application to Sampled-Data Control

Yasuaki Oishi (Nanzan University)

Udine, Italy

August 26, 2011

Workshop on Uncertain Dynamical Systems

* Joint work with Teodoro Alamo

1. Introduction

2

Robust semidefinite programming problems

Tminimize subject to ( , ) ( )

c xF x Oq q Q" Îf

-dim.p polytopeaffine pos. semidef.

Optimization problems constrained by uncertain linear matrix inequalities

Many applications in robust control

Robust SDP problem

Affine parameter dependence

Polynomial or rational par. dep.

equivalent cond.®

sufficient cond.®

3

This talk: general nonlinear parameter dependenceTminimize

subject to ( , ) ( )c xF x Oq q Q" Îf

1( ) , , ( )

mF x O F x Of K fÝ

How to obtain the sufficient condition?

How to make the condition less conservative?

Key idea: DC-representations“difference of two convex functions”

[Tuan--Apkarian--Hosoe--Tuy 00][Bravo--Alamo--Fiacchini--Camacho 07]

2. Preparations

4

Tminimize c x

subject to 0

1 1

( ) ( ) ( ( )p q

i i j p ji j

F x F x a F x Oq q+

= =

+ + )å å f

( )q Q" Î

nonlinear fn.

Problem

AssumptionEach of has a DC-representation( )

ja q

DC-representation

5

A fn. is said to have a DC-representation if( )a q(( )) ( ) cba qq q= -

convex convex

Examplee e

sinh( )2 2

aq q

q q-

= = -

1]Q =[- 1, ( )bq ( )c q

DC-representation®

( )bq

( )c q-

q

( )a q

Example

6

sin( )a q q= 2, 2Q =[- ]

sin2 2

2 2q q

q= + -

d

d

2

2qsinq- +1³ 0

DC-rep.®

Lower bound: ( )a q d¢¢ ³ -

DC-representation: 2 2( )2 2

ad d

q q q® + -

cf. [Adjiman--Floudas 96]

Mild enough to assume

q( )a q

( )bq

( )c q-

3. Proposed approach

7

Tminimize c x

subject to 0

1 1

( ) ( ) ( ( )p q

i i j p ji j

F x F x a F x Oq q+

= =

+ + )å å f

( )q Q" Î Assumption: DC-representation is available

( ) ( ) ( )j j ja b cq q q= -

convex convex

Key step: obtaining bounds ) )) ((( ( )

jj jrar Qqq q q£ £ Î

sufficient condition for the constraint®approximate solution®

concave convex

Obtaining bounds

8

q( )bq

( )c q-(( ) : ( ))cb r llqq q=-

q

(( )) ( ) cba q qq -=

:concave

(( )) ) : (cb rm mqq q- =

q:convex

( )blq

grad.: l

( )cmq-

q( )bq

( )c q-

grad.: m-

( )a q

q

( )ja q , ( )( )j jr l q

, ( )( )

j jr

mq

9

Ý

0

1 1

( ) ( ) ( ) ( )p q

i i j p ji j

F x F x F x Oq r q+

= =

+ +å å f

ver , ( ) , ( )( { , }, 1, 2, , )j jj j j

r r j ql mq Q r" Î ; " Î = K

For each nonlinear fn. ( )ja q

Choose any .Then,

( ) ( ) pj jl m, Î ¡

) ), (( ,(( )) ()j jj j j

rr aml qq q£ £

concave convex

0

1 1

( ) ( ) ( ) ( ) ( )p q

i i j p ji j

F x F x a F x Oq q q Q+

= =

+ + " Îå å f

10

Approximate problem

subject to 0

1 1

( ) ( ) ( ) ( )p q

i i j p ji j

F x F x F x Oq r q+

= =

+ +å å f

ver , ( ) , ( )( { , }, 1, 2, , )j jj j j

r r j ql mq Q r" Î ; " Î = K

Tminimize c x

Number of LMIs ver2q QL

Approximate solution

q

( )ja q , ( )( )j jr l q

, ( )( )

j jr

mq

cf. NP-hardness

Conservative

Choice of and ( ) ( )j jl m

Reduction of conservatism

11

suff. cond.x

approx. sol.Tc x

true sol.[1]Q[2]Q[3]Q

Division of Q

Adaptive division

12

Choice of and l m Quality of the approximation depends on the choice q

( )ja q , ( )jr l q

,( )

jr

mq

Measure of conservatism

d,,( ) : ( ) ( )jj j

V r r lmQl m q q q, = -ò q

( )j

V l m, , ( )jr l q

,( )

jr

mq ( )

ja q

optimal choice w.r.t.®some measure

13

Measure of conservatism

d,,( ) : ( ) ( )jj j

V r r lmQl m q q q, = -ò q

( )j

V l m, , ( )jr l q

,( )

jr

mq ( )

ja q

Theorem is minimized( )

jV l m,Û c

c

is a subgradient of at and is a subgradient of at

( )( ) ,

j

j

bc

l q qm q q

cwhere d d (gravity center of ):Q Q

q q q q Q=ò ò

Example

14

e esinh( )

2 2a

q q

q q-

= = -

1, 1Q =[- ]

c 0q =

optsubgradient of 1

(0) : :2

b l=

optsubgradient of 1

(0) : :2

c m- =

( )bq ( )c q

( )a q

( )r l q

( )rmq

sin sin2 2

( )2 2

aq q

q q q= = + -

Example

15

2, 2Q =[- ] ( )bq ( )c q

( )a q

( )r l q

( )rmq

c 0q =

optsubgradient of (0) :1 :b l=optsubgradient of (0) : 0 :c m=

4. Application to sampled-data control

16

tt

1

0 1 0, , 0 2000 0.1 0.1 k k

A B t t+

æ ö æ ö÷ ÷ç ç= = < - £÷ ÷ç ç÷ ÷-ç ç÷ ÷è ø è ø

Analysis and design of such sampled-data systems

holdsamplerdiscrete discrete

x Ax Bu= +&K

[Fridman et al. 04][Hetel et al. 06][Mirkin 07][Naghshtabrizi et al. 08][Suh 08][Fujioka 09][Skaf--Boyd 09][O.--Fujioka 10][Seuret 11]...

17

(no division) failed0, 200Q =[ ] L

succee ded 0, 100 [100, 200]Q =[ ]È L comparable with the specialized methodL

[O.--Fujioka 10]

holdsamplerdiscrete discrete

x Ax Bu= +&K

Design of a stabilizing K

200]q QÎ [0, =e

0.1

1 2

1( ) , ( )

0.1a a

q

q q qq

- -= =

-

Formulation into a robust SDP Avoiding a numerical problem for a small sampling

[O.--Fujioka 10]

interval

6. Summary

18

Robust SDP problems with nonlinear param. dep. Conservative approach using DC-representations

Concave and convex bounds Approximate problem Reduction of conservatism

Combination with the polynomial-based methods[Chesi--Hung 08][Peaucelle--Sato 09][O. 09]

Optimization of the bounds w.r.t. some measure

Application to sampled-data control