robust semidefinite programming and its application to sampled-data control
DESCRIPTION
Workshop on Uncertain Dynamical Systems. Robust Semidefinite Programming and Its Application to Sampled-Data Control. Yasuaki Oishi (Nanzan University) Udine, Italy August 26, 2011. * Joint work with Teodoro Alamo. 1. Introduction. Robust semidefinite programming problems. - PowerPoint PPT PresentationTRANSCRIPT
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