linear matrix inequalities in system and control theory solmaz sajjadi kia adviser: prof. jabbari...
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Linear Matrix Inequalities in System and Control Theory
Solmaz Sajjadi KiaAdviser: Prof. Jabbari
System, Dynamics and Control SeminarUCI, MAE Dept.April 14, 2008
Linear Matrix Inequality (LMI)
miRFFR
FFF
nnTii
m
m
iii
,...,2,1,0,
)0(0)(1
0
x
xx
Set of n polynomial inequalities in x, e.g.,
Convex constraint on x
0)2()3)(1(
01
032
210
11
11
01
11
32
21)(
221221
21
221
212121
xxxxx
xx
xxx
xxxxxxxF
Matrices as Variable
0,0 XXX AAT
010
00
01
10
00
01
.,.
33221132132
21
22
EEE
Rge
xxxxxxxx
xxX
X
0)()()( 333222111 AEEAAEEAAEEA TTT xxx
00
0)(3
1
i i
iiT
iE
AEEAx
Multiple LMIs
)0)(....,,0)(,0)((
,0)(....,,0)(,0)()()2()1(
)()2()1(
xFxFxFDiag
xFxFxFn
nmiRFFR
FFxF
nnTii
m
m
iii
,...,2,1,0,
)0(0)(1
0
x
x
LMI Problems
Feasibility
)0(0)( xF
Minimization Problem
)0(0)(
tosubjectmin
xx
F
cT
How do we cast our control problems in LMI form?
We rely on quadratic function V(x)=x’Px
Three Useful Properties to Cast Problems in Convex LMI From
Congruent Transformation
S-Procedure
Schur Complement
Congruent transformation
invertible and square is where)0(0)(
)0(0)(
PPxTP
xT
T
Stable State Feedback Synthesis Problem
0))()((0x(x)V
0x,xV(x)
Stability Lyapunov
T
T
xBABAxxxx TTTT KPPKPP
PP
xBAxxu
BuAxx)( K
K
0PKKPPP TTT BBAA
1
0
0
FQKFFQ
QTTT BBAQA
TTTT BBAA QQPQQPKKPPPQ , where0)( 1
0TTT BBAA QKKQQQ
KQFFFQQ where0TTT BBAA
S Procedure
12
0)()(such that,..2,1,0:2
,..2,10)(such that for 0)(:1
10
0
SS
TTNkS
NkTVTS
k
N
kkk
k
xx
xxx
12
0)()(such that,..2,1,0:2
,..2,10)(such that for 0)(:1
10
0
SS
TTNkS
NkTVTS
k
N
kkk
k
xx
xxx
Three Useful Properties to Cast Problems in Convex LMI From
Congruent Transformation
S-Procedure
Schur Complement
Reachable Set/Invariant Set for Peak Bound Disturbance
The reachable set (from zero): is the set of points the state vector can reach with zero initial condition, given some limitations on the disturbance.
The invariant set: is the set that the state vector does not leave once it is inside of it, again given some limits on the disturbance.
Reachable Set/Invariant Set for Peak Bound Disturbance
Ellipsoidal Estimate
wBxAx cl 1Peak Bound Disturbance
2max)()( wtwtwT
}:{),( 2cPxxxcP T
0))()()(()(,)()( 2max twtwxVxVwtwtw TT
0*
1
w
x
I
BAAwx cl
TclTT
PPPP
0,)( PPxxxV T
}:{),( 2maxmax wxxxwP T P
0))(()( 2max wxVxV
2max)(for)0)((0)( wxxxVxVxV T P
0))(()( 2max wxVxV
0)( 11 wwxBwwBxxAAx TTTTcl
Tcl
T PPPPP
0*
1
I
BAA clTcl
PPPP
xxSxRxRxQxQxRxS
xSxQ
xSxQxSxRxQ
TTT
T
onaffinely depend)( and),()(),()( where0)()(
)()(
0)()()()(,0)( 1
Linear (thus convex) Verses Nonlinear Convex inequality
Nonlinear (convex) inequalities are converted to LMI form using Schur Complement
Three Useful Properties to Cast Problems in Convex LMI From
Congruent Transformation
S-Procedure
Schur Complement
H∞ or L2 Gain
0)1(*
1
DCD
CBAAT
TT
2γ
PPP
00
21 such that smalles i.e., system, theofgain L2or Energy (t)w(t)wγ(t)z(t)zDwCxz
wBAxx TT
00
2
00
2 0)0())(( (t)w(t)wγ(t)z(t)z(t)w(t)wγ(t)z(t)zVxV TTTT
0(x(t))V
0x,xV(x) :Lyapunov Quadratic2
T
(t)w(t)wγ(t)z(t)z TTPP
0)()()()( 211 (t)w(t)wγDwCxDwCxPxBwwPBxxPAPAx TTTTTTT
0*
0*
11
DD
DCBCCAA
w
x
DD
DCBCCAAwx
T
TTT
T
TTTTT
22 γ
PPP
γ
PPP
0
**
*1
I
D
CBAAT
TT
2γ
PPP
Norm of a vector in an ellipsoid
Find Max of ||u||=||Kx|| for x in {x| xTPx≤c2 }
max2max
22max
2
2max
2
2max
2
2
2max
||||||||)(
0))((0)(,00
uKxuuKxKxcPxxu
cKxKx
xKu
cKPxK
u
cKPP
c
uK
KP
TTTTT
TTT
T
0
**
* 11
1
I
D
CBAAT
Tclcl
Tcl
2γ
PPP
I
I
Q
00
00
00
0
**
* 112
1
I
D
CBAAT
Tcl
Tclcl
QQQ
0*
1
I
BAA clTcl
PPPP
wDuDxCz
wBuBAxx
11121
12
wDxKDCz
wBxKBAx
11121
12
)(
)(Kxu
0 QQ clTcl AA
Q0 cl
Tcl AA PP
I
Q
0
0
0*
1
I
BAA clTcl
QQQ
0*
2
2max
c
u
TQKQ0
2
2max
c
u
T
K
KP
I
Q
0
0
FKQ
A Saturation Problem
11121
112111
21
,
,
DDKDCC
BBBKAAwhere
wDxCz
wBxAx
Kxu
uDwDxCzuBwBAxx
clcl
clcl
clcl
clclcl
Problem: Synthesis/Analysis of a Bounded State Feedback Controller (||u||<umax) exposed wT(t)w(t)≤w2
max
Analysis: What is the largest disturbance this system can tolerate with K
Synthesis: Find a K such that controller never saturates
xT Px<wT
max
Analysis: What is the largest disturbance (e.g. wmax) the system can tolerate ?
umax=Kx-umax=Kx
}:{),( 2maxmax wxxxwP T P
0*
1
I
BAA clTcl
PPPP
0
0*2max
P
w
P
2max
uK T
2maxw
β1
βminimizeβ
wmax
1
xT Px<wT
max
Synthesis: Find a K such that controller never saturates
Kx=umaxKx=-umax
}:{),( 2maxmax wxxxwP T P
0*
1
I
BAA clTcl
QQQ
0
0*
2max
Q
QKQ
2maxw
u
T
FKQ
ilityfeasibfor ckeck
1FQK
0&0
0
min
2max
2max
21222
11
1111
Iw
uI
BAA
I
DI
CBAA
TTclcl
T
Tcl
Tclcl
KQQQQQ
γ
QQQ
2
2
0&0
0
min
2max
2max
21222222
11
1211121211
Iw
uI
BBBAA
I
DI
DCBBBAA
TTT
T
TTTTTT
T2
2
211
KQQQKQKQQQ
γ
KQQKQKQQQ
1
FQK
FKQ,QQ 21
Solution
Good Reference
Prof. Jabbari’s Note on LMIs
S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, “Linear Matrix Inequalities in Systems and Control Theory”
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