Robust Optimization in Robust
Control
Carsten Scherer
Delft Center for Systems and Control (DCSC)
Delft University of Technology
The Netherlands
Supported by Dutch Technology Foundation
Outline
• Dissipativity and Linear Matrix Inequalities
• Linear parameter-varying system analysis and synthesis
• Robust LMI problems
• Matrix sum-of-squares relaxations
• Verification of relaxation exactness
• Examples
2/48
Carsten Scherer
Biking = Optimized Control
feedback
Riding a bicycle means
dynamical system
3/48
Carsten Scherer
Biking = Optimized Control
feedback
Riding a bicycle means
• Stabilization
keep bike upright
dynamical system
3/48
Carsten Scherer
Biking = Optimized Control
feedback
Riding a bicycle means
• Stabilization
keep bike upright
• Tracking
follow a given path
dynamical system
3/48
Carsten Scherer
Biking = Optimized Control
feedback
Riding a bicycle means
• Stabilization
keep bike upright
• Tracking
follow a given path
• Disturbance Suppression
counteract wind gusts
dynamical system
3/48
Carsten Scherer
Biking = Optimized Control
feedback
Riding a bicycle means
• Stabilization
keep bike upright
• Tracking
follow a given path
• Disturbance Suppression
counteract wind gusts
A biker is a feedback controller!
dynamical system
3/48
Carsten Scherer
Simple Model of Bike Dynamics
Forces
steering
angle u
velocity
v
Newton’s law:
"
#
" #
0 1
1
ẋ =
x+
u
γ 0
0
h
i
2
y =
αv βv x.
tilt angle y
Dynamical system u → y.
4/48
Carsten Scherer
Simple Model of Bike Dynamics
Forces
steering
angle u
velocity
Newton’s law:
"
#
" #
0 1
1
ẋ =
x+
u
γ 0
0
h
i
2
y =
αv βv x.
v
tilt angle y
Dynamical system u → y.
Model neglects many aspects:
non-linearities, resonances in frame, ...
Mismatch between real system and model: Uncertainty.
4/48
Carsten Scherer
Controlling a Bike
Keep Small
Wind Gust
Bike Model
Tilt angle
Steering angle
Controller
Automatic Control
• Given a model design controller which performs multiple tasks
• Develop algorithms for optimal design of controllers
• Include effects of uncertainties: Robust optimal design
5/48
Carsten Scherer
How to Handle Neglected Effects?
Consider static nonlinearity:
∆
ẋ = Ax + B∆(Cx)
w
z
Pull out nonlinearity:
ẋ = Ax + Bw, w = ∆(z)
Nominal
System
z = Cx
6/48
Carsten Scherer
How to Handle Neglected Effects?
Consider static nonlinearity:
∆
ẋ = Ax + B∆(Cx)
w
z
Pull out nonlinearity:
ẋ = Ax + Bw, w = ∆(z)
Nominal
System
z = Cx
Linear Fractional Representation (LFR)
• Naturally related to modeling of system interconnections
• Very flexible: Easy extension to dynamic uncertainties
6/48
Carsten Scherer
Nominal Stability
ẋ = Ax asymptotically stable iff exists solution X of LMI
X 0 and AT X + XA ≺ 0.
7/48
Carsten Scherer
Nominal Stability
ẋ = Ax asymptotically stable iff exists solution X of LMI
X 0 and AT X + XA ≺ 0.
Proof with Lyapunov function V (x) := xT Xx since
d
V (x(t)) = x(t)T AT X + XA x(t) < 0.
dt
7/48
Carsten Scherer
Nominal Stability
ẋ = Ax asymptotically stable iff exists solution X of LMI
X 0 and AT X + XA ≺ 0.
Proof with Lyapunov function V (x) := xT Xx since
d
V (x(t)) = x(t)T AT X + XA x(t) < 0.
dt
All control systems must at least be stabilized ...
... LMI stability characterization for LTI systems ...
... first reason for success of SDP’s in control.
7/48
Carsten Scherer
Robust Stability
System with linear fractional representation
ẋ = Ax + Bw, z = Cx + Dw, w = ∆(z).
is exponentially stable if ...
8/48
Carsten Scherer
Robust Stability
System with linear fractional representation
ẋ = Ax + Bw, z = Cx + Dw, w = ∆(z).
is exponentially stable if ...
... exists X 0 and Hermitian multiplier P with
"
# "
#T "
#
AT X + XA XB
0 I
0 I
+
P
≺0
BT X
0
C D
C D
and such that
"
∆(z)
z
#T "
P
∆(z)
z
#
≥ 0 for all z.
8/48
Carsten Scherer
Proof
Fully elementary Lyapunov arguments:
"
#T "
# "
#T "
# "
#

T
x(t)
A X + XA XB
0 I
0 I  x(t)
+
P
≤0

w(t)
BT X
0
C D
C D  w(t)
⇐⇒
d
x(t)T Xx(t) +
dt
"
w(t)
z(t)
#T "
P
w(t)
z(t)
#
≤0
9/48
Carsten Scherer
Proof
Fully elementary Lyapunov arguments:
"
#T "
# "
#T "
# "
#

T
x(t)
A X + XA XB
0 I
0 I  x(t)
+
P
≤0

w(t)
BT X
0
C D
C D  w(t)
⇐⇒
d
x(t)T Xx(t) +
dt
"
w(t)
z(t)
#T "
P
w(t)
z(t)
#
≤0
⇐⇒
d
x(t)T Xx(t) +
dt
"
|
∆(z(t))
z(t)
#T "
P
∆(z(t))
z(t)
#
≤0
{z
}
can be dropped since ≥ 0
9/48
Carsten Scherer
Dissipativity (Willems 1972)
"
∃ X:
AT X + XA XB
BT X
0
#
"
+
0 I
C D
#T "
P
0 I
C D
#
≺0
⇐⇒
ẋ = Ax + Bw
z = Cx + Dw
)
"
w
dissipative for supply s(w, z) = −
z
#T "
P
#
w
.
z
No loss of generality: Quadratic storage function V (x) = xT Xx
Z
t2
s(w(t), z(t)) dt ≤ V (x(t1 ))
V (x(t2 )) −
t1
Initial energy at time t1
Final energy at time t2
Supplied energy
10/48
Carsten Scherer
Dissipativity
System Movement
Initial Energy
Final Energy
Supplied Energy
• Is generalization of physical law of conservation of energy
• Extends Lyapunov theory to open dynamical systems
• Fundamental concept behind LMI’s in control
11/48
Carsten Scherer
Energy-Gain = Dissipation
Energy gain is minimal γ with
Z ∞
Z ∞
2
kz(t)k dt ≤ γ
kw(t)k2 dt
0
w
z
System
y
0
u
Controller
Closed-loop system description:
ẋ = Ax + Bw
z = Cx + Dw
z
Closed-Loop
System
w
12/48
Carsten Scherer
Energy-Gain = Dissipation
Energy gain is minimal γ with
Z ∞
Z ∞
2
kz(t)k dt ≤ γ
kw(t)k2 dt
0
w
z
System
y
0
u
Controller
Closed-loop system description:
ẋ = Ax + Bw
z = Cx + Dw
z
Closed-Loop
System
w
Minimize γ such that exists X with
"
# "
#T"
#"
#
AT X + XA XB
0 I
−γI 0
0 I
+
≺ 0.
BT X
0
C D
0
I
C D
12/48
Carsten Scherer
Robust Stability = Separation by Dissipation
Uncertainty anti-dissipative


T
∆(z) 
P
z



∆(z) 
0
z
∆
z
w
Nominal
System
Separation with
same Multiplier P
Nominal system dissipative




T
AT X + XA XB   0 I 
+
P
BT X
0
C D



0 I 
≺0
C D
13/48
Carsten Scherer
Dissipativity – A colorful concept
Lyapunov
stability
backstepping
integral quadratic
constraints
theory of MPC
small gain
theorem
Hamilton-Jacobi-Bellman
structured
singular value
circle criterion
Outline
• Dissipativity and Linear Matrix Inequalities
• Linear parameter-varying system analysis and synthesis
• Robust LMI problems
• Matrix sum-of-squares relaxations
• Verification of relaxation exactness
• Examples
14/48
Carsten Scherer
Linear Parameter Varying Systems
Bike dynamics model:
"
#
" #
0 1
1
ẋ =
x+
u
γ 0
0
i
h
2
y =
αv βv x.
feedback
Depends on time-varying velocity.
dynamical system
15/48
Carsten Scherer
Linear Parameter Varying Systems
Bike dynamics model:
"
#
" #
0 1
1
ẋ =
x+
u
γ 0
0
i
h
2
y =
αv βv x.
feedback
Depends on time-varying velocity.
Feedback controller ... adapted to
on-line measured value of parameter.
dynamical system
15/48
Carsten Scherer
Wafer-Stage: The System
Waferscanners
Lens system
Silicon wafer
on wafer-stage
Positioning
16/48
Carsten Scherer
Wafer-Stage: Performance Limitations
wafer
Moving wafer-stage …
… has to accurately track surface
Movements … oscillations … performance limitations
17/48
Carsten Scherer
Wafer-Stage: Illustration of Dimensions
Accuracy
Layer with 17 silicon atoms
18/48
Carsten Scherer
Wafer-Stage: Parameter Dependence of Model
Position-dependent
resonances
... long-term collaboration DCSC (TU-Delft) and CFT (Philips)
19/48
Carsten Scherer
Linear Parameter Varying Systems
parameter
curve ¼(t)
LPV system
¼2
¼3
ẋ(t) = A(π(t))x(t)
with value & velocity bounds
¼1
π(t) ∈ Π and π̇(t) ∈ Φ.
¼5
¼4
20/48
Carsten Scherer
Linear Parameter Varying Systems
parameter
curve ¼(t)
LPV system
¼2
¼3
ẋ(t) = A(π(t))x(t)
¼1
with value & velocity bounds
π(t) ∈ Π and π̇(t) ∈ Φ.
¼5
¼4
Exponentially stable if exists smooth X(π) with
X
X(π) 0,
∂j X(π)φj + A(π)T X(π) + X(π)A(π) ≺ 0
j
for all δ = (π, φ) ∈ Π × Φ.
20/48
Carsten Scherer
Proof
For arbitrary parameter trajectory π(t) and system trajectory x(t):
d
x(t)T X(π(t))x(t) =
dt
= 2x(t)T X(π(t))ẋ(t) +
X
x(t)T [∂j X(π(t))π̇j (t)] x(t)
j
Algebraic inequality guarantees strict negativity ...
... standard Lyapunov arguments finish proof.
21/48
Carsten Scherer
Performance
Input-output LPV system
ẋ(t) = A(π(t))x(t) + B(π(t))w(t)
z(t) = C(π(t))x(t) + D(π(t))w(t)
L2 -gain smaller than γ if exists smooth X(π) 0 with
T P



X(π)φ
X(π)
0
0
I
0
∂
I
0
j
j
 j



 A(π) B(π) 
 A(π) B(π)  
X(π)
0
0
0


≺ 0


 0

 0
2
I
0
0
−γ
I
0
I




C(π) D(π)
0
0
0 I
C(π) D(π)
for all δ = (π, φ) ∈ Π × Φ.
22/48
Carsten Scherer
Synthesis of LPV Controller
System and Controller ...
z¢
¢(±(t))
w¢
... Linear Fractional Representations
z
Design Goals
• exponential stability
• minimal energy gain
w
System
y
u
Controller
Convex Optimization ...
... gain-scheduled controller
zc
¢c(±(t))
wc
23/48
Carsten Scherer
Reduction to Finite Dimensions
Stability inequality
X
∂j X(π)φj + A(π)T X(π) + X(π)A(π) ≺ 0
j
24/48
Carsten Scherer
Reduction to Finite Dimensions
Stability inequality
X
∂j X(π)φj + A(π)T X(π) + X(π)A(π) ≺ 0
j
Search X(π) in span of X1 (π), . . . , Xn (π).
Results in semi-infinite LMI problem
!
X
k
xk
X
∂j Xk (π)φj + A(π)T Xk (π) + Xk (π)A(π)
≺ 0.
j
24/48
Carsten Scherer
Reduction to Finite Dimensions
Stability inequality
X
∂j X(π)φj + A(π)T X(π) + X(π)A(π) ≺ 0
j
Search X(π) in span of X1 (π), . . . , Xn (π).
Results in semi-infinite LMI problem
!
X
k
xk
X
∂j Xk (π)φj + A(π)T Xk (π) + Xk (π)A(π)
≺ 0.
j
With δ = (π, φ) ∈ Π × Φ = δ results in generic formulation
F0 (δ) + x1 F1 (δ) + · · · + xn Fn (δ) ≺ 0.
24/48
Carsten Scherer
Outline
• Dissipativity and Linear Matrix Inequalities
• Linear parameter-varying system analysis and synthesis
• Robust LMI problems
• Matrix sum-of-squares relaxations
• Verification of relaxation exactness
• Examples
24/48
Carsten Scherer
Robust LMI Problems
Infimize cT x over LMI-constraint
F0 (δ) + x1 F1 (δ) + · · · + xn Fn (δ) ≺ 0 for all δ ∈ δ.
• Have seen concrete engineering examples ... there are many more!
• Is convex problem. Semi-infinite constraints render it NP-hard.
• Computationally tractable approximations: Relaxations
Polytope δ and affine dependence on parameter δ ...
... easy for scalar constraints
... tough for matrix constraints !
25/48
Carsten Scherer
Linear Fractional Representation
Suppose F (δ) is rational in δ = (δ 1 , . . . , δ k ) without pole at zero.
Can construct A, B, C, D such that
"
#
h
i ∆(δ) (I − A∆(δ))−1 B
F (δ) = C D
I
with diagonal and affine
"
∆(δ) =
δ 1 I1 .
0
..
0
δ k Ik
#
.
State-space realization of multi-dimensional system.
26/48
Carsten Scherer
Example: Velocity in Bike Model
With v = v0 + δ consider
"
#
" #
0 1
1
ẋ =
x+
u
γ 0
0
h
i
y =
α(v0 + δ) β(v0 + δ)2 x
27/48
Carsten Scherer
Example: Velocity in Bike Model
With v = v0 + δ consider
"
#
" #
0 1
1
ẋ =
x+
u
γ 0
0
h
i
y =
α(v0 + δ) β(v0 + δ)2 x
∆(δ)
w
z
y
Has Linear Fractional Representation
"
#
" #

0 1
1



ẋ =
x+
u


γ 0
0



h
i
h
i 
2
y =
v0 α v0 β x + α 2v0 α w


"
#
"
#




1 0
0 1


z =
x+
w

0 β/α
0 0
Nominal
System
"
w=
δ 0
0 δ
u
#
z
27/48
Carsten Scherer
Alternative Description of Uncertain LMI
Construct linear fractional representation



C0
 .. 

−1
 .  ∆(δ)(I − A∆(δ)) B + 
Cn
and define affine
"
W (x) =
C0 +
0
Pn
j=1
xj Cj



D0
F0 (δ)
..  1  .. 
. =  . 
2
Dn
Fn (δ)
#
P
C0T + nj=1 xj CjT
P
(D0 + D0T ) + nj=1 xj (Dj + DjT )
Alternative description of LMI:
" #T
"
#
n
X
∆(δ)(I − A∆(δ))−1 B
∗
≺0
F0 (δ) +
xj Fj (δ) =
W (x)
∗
I
j=1
28/48
Carsten Scherer
Infimize cT x such that for all δ ∈ δ:
"
"
#T
#
∆(δ)(I −A∆(δ))−1 B
∆(δ)(I −A∆(δ))−1 B
W (x)
≺ 0.
I
I
ROB
Robust LMI Problem
• Huge range of applications in robust optimization and control:
Uncertain LP’s, robust performance, Lyapunov design, ...
• Captures affine dependence on parameters: A = 0.
• Only few problem classes computationally tractable.
Goals: Compute upper bounds by efficiently solvable relaxation.
Numerically check quality for specific problem instance.
29/48
Carsten Scherer
How to Construct Relaxations?
Choose linear mappings G(y) and H(y) such that
"
#T
"
#
∆(δ)
∆(δ)
G(y) 4 0 =⇒
H(y)
< 0 for all δ ∈ δ.
I
I
30/48
Carsten Scherer
How to Construct Relaxations?
Infimum of cT x such that there exists y with
"
#T
"
#
I 0
I 0
H(y)
+ W (x) ≺ 0
G(y) ≺ 0,
A B
A B
REL
Choose linear mappings G(y) and H(y) such that
"
#T
"
#
∆(δ)
∆(δ)
G(y) 4 0 =⇒
H(y)
< 0 for all δ ∈ δ.
I
I
is upper bound on value of ROB.
Separation by dissipation ... for LMI-parametrized multipliers!
30/48
Carsten Scherer
Proof
Feasibility of REL
"
G(y) ≺ 0,
I 0
A B
#T
"
H(y)
I 0
A B
#
+ W (x) ≺ 0
31/48
Carsten Scherer
Proof
Feasibility of REL
"
G(y) ≺ 0,
I 0
A B
#T
"
H(y)
I 0
A B
#
+ W (x) ≺ 0
implies feasibility of ROB since
"
#T
"
#
∆(δ) (I − A∆(δ))−1 B
∆(δ) (I − A∆(δ))−1 B
W (x)
+
I
I
"
#T
"
#
∆(δ)
∆(δ)
+[(I − A∆(δ))−1 B]T
H(y)
(I − A∆(δ))−1 B ≺ 0.
I
I
|
{z
}
<0
31/48
Carsten Scherer
Example: Convex Hull Relaxation
Suppose parameter set is finitely generated: δ = co{δ 1 , . . . , δ N }.
"
∆(δ)
I
#T
"
H(y)
∆(δ)
I
#
< 0 for δ ∈ co{δ 1 , . . . , δ N }
⇑
32/48
Carsten Scherer
Example: Convex Hull Relaxation
Suppose parameter set is finitely generated: δ = co{δ 1 , . . . , δ N }.
"
∆(δ)
I
#T
"
H(y)
∆(δ)
I
#
< 0 for δ ∈ co{δ 1 , . . . , δ N }
⇑
#T "
#
" #T " #
"
∆(δ j )
∆(δ j )
I
I
< 0, j = 1, . . . , N
H(y) = y,
y
4 0,
y
I
I
0
0
|
{z
}
G(y) 4 0
32/48
Carsten Scherer
Example: Convex Hull Relaxation
Suppose parameter set is finitely generated: δ = co{δ 1 , . . . , δ N }.
"
∆(δ)
I
#T
"
H(y)
∆(δ)
I
#
< 0 for δ ∈ co{δ 1 , . . . , δ N }
⇑
#T "
#
" #T " #
"
∆(δ j )
∆(δ j )
I
I
< 0, j = 1, . . . , N
H(y) = y,
y
4 0,
y
I
I
0
0
|
{z
}
G(y) 4 0
Value of ROB = Value of REL in case of δ ⊂ R (one parameter).
32/48
Carsten Scherer
Outline
• Dissipativity and Linear Matrix Inequalities
• Linear parameter-varying system analysis and synthesis
• Robust LMI problems
• Matrix sum-of-squares relaxations
• Verification of relaxation exactness
• Examples
32/48
Carsten Scherer
Key Point
Uncertainty set δ = {δ ∈ Rk : g1 (δ) ≥ 0, . . . , gm (δ) ≥ 0}
33/48
Carsten Scherer
Key Point
Uncertainty set δ = {δ ∈ Rk : g1 (δ) ≥ 0, . . . , gm (δ) ≥ 0}.
δ region
Compact polytope ... gj (δ) affine
33/48
Carsten Scherer
Key Point
Uncertainty set δ = {δ ∈ Rk : g1 (δ) ≥ 0, . . . , gm (δ) ≥ 0}.
δ region
Compact polytope ... gj (δ) affine
δ region
Semi-algebraic ... gj (δ) polynomial
33/48
Carsten Scherer
Key Point
Uncertainty set δ = {δ ∈ Rk : g1 (δ) ≥ 0, . . . , gm (δ) ≥ 0}.
δ region
Compact polytope ... gj (δ) affine
δ region
Semi-algebraic ... gj (δ) polynomial
Positivity of matrix-valued polynomial for scalar constraints:
"
#T
"
#
∆(δ)
∆(δ)
H(y)
< 0 if g1 (δ) ≥ 0, . . . , gm (δ) ≥ 0.
I
I
33/48
Carsten Scherer
Matrix Sum-of-Squares (SOS)
The polynomial matrix S(δ) is SOS if there exists a rectangular
polynomial matrix T (δ) such that
S(δ) = T (δ)T T (δ).
• S(δ) is scalar-valued and tj (δ) denote elements of column T (δ):
X
S(δ) =
tj (δ)2 .
j
Our definition is generalization of standard scalar one.
• If S(δ) is SOS then S(δ) < 0 for all δ ∈ Rn .
Converse not true. (Hilbert 1888, Motzkin 1967)
34/48
Carsten Scherer
Checking SOS Property is Linear SDP
Fix scalar monomial basis u1 (δ), . . . , ut (δ) and search T1 , . . . , Tt in
T (δ) =
t
X
i=1
Ti ui (δ) =
t
X
Ti [ui (δ) ⊗ I].
i=1
Collect u1 (δ), . . . , ut (δ) into column vector u(δ).
35/48
Carsten Scherer
Checking SOS Property is Linear SDP
Fix scalar monomial basis u1 (δ), . . . , ut (δ) and search T1 , . . . , Tt in
T (δ) =
t
X
i=1
Ti ui (δ) =
t
X
Ti [ui (δ) ⊗ I].
i=1
Collect u1 (δ), . . . , ut (δ) into column vector u(δ).
S(δ) is SOS with respect to monomial basis u(δ) if and only if
S(δ) = [I ⊗ u(δ)]T W [I ⊗ u(δ)] for some W < 0.
• Linear affine and semi-definite constraints: Standard LMI problem.
• Full flexibility in choice of monomials. Can be made explicit.
35/48
Carsten Scherer
Constrained Positivity
It is trivial that
S(δ) < 0 for all δ ∈ Rk with g1 (δ) ≥ 0, . . . , gm (δ) ≥ 0
36/48
Carsten Scherer
Constrained Positivity
It is trivial that
S(δ) < 0 for all δ ∈ Rk with g1 (δ) ≥ 0, . . . , gm (δ) ≥ 0
⇑
S(δ) −
m
X
Sj (δ)gj (δ) is SOS
for
Sj (δ) which are SOS
j=1
36/48
Carsten Scherer
Constrained Positivity
It is trivial that
S(δ) < 0 for all δ ∈ Rk with g1 (δ) ≥ 0, . . . , gm (δ) ≥ 0
⇑
S(δ) −
m
X
Sj (δ)gj (δ) is SOS
for
Sj (δ) which are SOS
j=1
Latter is LMI problem for SOS matrices with respect to finite basis.
36/48
Carsten Scherer
Constraint Qualification Implies Equivalence
Suppose g1 (δ), . . . , gm (δ) satisfy constraint qualification. Then
S(δ) 0 for all δ ∈ Rk with g1 (δ) ≥ 0, . . . , gm (δ) ≥ 0
⇓
S(δ) −
m
X
Sj (δ)gj (δ) is SOS for Sj (δ) which are SOS
j=1
• Extends scalar results of Schmüdgen (1991) and Putinar (1993)
Scherer and Hol (2004)
• Complementary version: S(δ) scalar-valued and gj (δ) matrix-valued
Kojima (2003), Hol and Scherer (2003,2004)
37/48
Carsten Scherer
Constraint Qualification
2
r − kδk −
m
X
sj (δ)gj (δ) is SOS for some r > 0 and SOS sj (δ).
j=1
38/48
Carsten Scherer
Constraint Qualification
2
r − kδk −
m
X
sj (δ)gj (δ) is SOS for some r > 0 and SOS sj (δ).
j=1
If affine g1 (δ), . . . , gm (δ) define a compact polytope δ then the
SOS polynomials s1 (δ),..., sm (δ) can be taken of at most degree 2.
• Obtained by Jacobi & Prestel (2001) using deep results from real
algebraic geometry without degree information.
• Our proof: Elementary but tricky application of semi-definite duality.
Scherer and Hol (2004)
38/48
Carsten Scherer
New Matrix SOS Relaxation
"
∆(δ)
I
#T
"
H(y)
∆(δ)
I
#
< 0 if g1 (δ) ≥ 0, . . . , gm (δ) ≥ 0
⇑
"
∆(δ)
I
#T
H(y)
"
∆(δ)
I
#
−
m
X
yj (δ)gj (δ) is SOS ...
j=1
... for SOS polynomial matrices y1 (δ), . . . , ym (δ)
Search for SOS matrices w.r.t. finite basis: Fits in our framework!
Value of REL converges to value of ROB: Exhaustive basis sequence!
39/48
Carsten Scherer
Infimize cT x such that for all δ ∈ δ:
"
"
#T
#
−1
−1
∆(δ)(I −A∆(δ)) B
∆(δ)(I −A∆(δ)) B
≺ 0.
W (x)
I
I
ROB
Infimize cT x such that there exists y with
"
#T
"
#
I 0
I 0
G(y) ≺ 0,
H(y)
+ W (x) ≺ 0.
A B
A B
REL
Summary
• Can construct relaxations with suitable G(.) and H(.) such that
value of REL is an upper bound of value of ROB.
• Can systematically construct sequence to guarantee convergence.
40/48
Carsten Scherer
Discussion
• Undeniable Problems
Growth in computational complexity to reduce gap
Unavoidable for worst-case scenarios
• Practical Observations
Good approximation with small-sized relaxations ...
... for concrete problem instances in control
How can we determine approximation quality? Detect exactness?
41/48
Carsten Scherer
Outline
• Dissipativity and Linear Matrix Inequalities
• Linear parameter-varying system analysis and synthesis
• Robust LMI problems
• Matrix sum-of-squares relaxations
• Verification of relaxation exactness
• Examples
41/48
Carsten Scherer
Infimize cT x such that there exists y with
"
#T
"
#
I 0
I 0
G(y) ≺ 0,
H(y)
+ W (x) ≺ 0.
A B
A B
REL
Dualization
42/48
Carsten Scherer
Infimize cT x such that there exists y with
"
#T
"
#
I 0
I 0
G(y) ≺ 0,
H(y)
+ W (x) ≺ 0.
A B
A B
REL
Maximize hW0 , M i such that there exist M < 0, N < 0 with
hWj , M i + cj = 0, j = 1, . . . , n, and
"
# "
#T 
I 0
I 0
 + G∗ (N ) = 0.
H∗ 
M
A B
A B
DUAL
Dualization
G∗ (.) and H ∗ (.) are usual Hilbert adjoints.
42/48
Carsten Scherer
Characterizing Exactness
The following statements are equivalent:
• Exists a worst δ0 ∈ δ and the relaxation is exact.
• Exists an optimal solution M of DUAL with
"
#
i I 0
h
M = 0 for some δ ∈ δ.
I −∆(δ)
A B
43/48
Carsten Scherer
Characterizing Exactness
The following statements are equivalent:
• Exists a worst δ0 ∈ δ and the relaxation is exact.
• Exists an optimal solution M of DUAL with
"
#
i I 0
h
M = 0 for some δ ∈ δ.
I −∆(δ)
A B
If δ has LMI description ...
• Solve REL and DUAL to compute optimal value and optimal M .
• Try to determine solution δ ∈ δ (LMI problem).
• If solvable then the values of ROB and REL are equal.
43/48
Carsten Scherer
Outline
• Dissipativity and Linear Matrix Inequalities
• Linear parameter-varying system analysis and synthesis
• Robust LMI problems
• Matrix sum-of-squares relaxations
• Verification of relaxation exactness
• Examples
43/48
Carsten Scherer
Illustrative Academic Example
δ unit box ... choose g1,2 (δ) = ±δ1 + 1 and g3,4 (δ) = ±δ2 + 1.
3.5
3
Degrees of SOS factor:
2.5
y1 (δ1 )
y2 (δ1 )
y3 (δ2 )
y4 (δ2 )
2
1.5
1
0
0
1
1
1
1
1
1
0.5
0
0.5
0.6
0.7
0.8
0.9
1
44/48
Carsten Scherer
High-Performance Aircraft System
nz
®
q
u
u: Control input
α: Measurable parameter
nz : Tracked output
Nonlinear system description with aerodynamic effects:
α̇ = KM an α2 +bn α+cn (2−M /3) α+dn u +q
q̇ = M 2 am α2 +bm α−cm (7−8M /3) α+dm u
nz = M 2 an α2 +bn α+cn (2−M /3) α+dn u
45/48
Carsten Scherer
Main Idea
Rewrite as linear parameter-varying system
α̇ = Kδ1 an δ2 2 +bn δ2 +cn (2−δ1 /3) α+dn u +q
q̇ = δ1 2 am δ2 2 +bm δ2 −cm (7−8δ1 /3) α+dm u
nz = δ1 2 an δ2 2 +bn δ2 +cn (2−δ1 /3) α+dn u
with bounds 2 ≤ δ1 (t) ≤ 4 and −20 ≤ δ2 (t) ≤ 20.
Design good controller scheduled with δ1 (t), δ2 (t)
→ Is good controller for nonlinear system
46/48
Carsten Scherer
Application to Aircraft Model
Synthesis with Convex Hull Relaxation
M
decreases
5 seconds
M(t)
(t) decreases
in 5inseconds
from 4 from
to 2. 4 to 2.
40
Normal
Reference
Response
Reference
Response
30
Normal acceleration
Normal
acceleration
acceleration
20
10
0
−10
−20
0
1
2
3
4
5
Time
47/48
42/
Carsten Scherer
Conclusions
• Dissipativity ... Linear Matrix Inequalities in Control
• Range of applications: stability, performance, robustness
• Linear Parameter Varying systems ... robust LMI’s
• Novel matrix SOS relaxations with convergence
• Numerical verification of exactness
How to exploit system theoretic structure ...
... to improve reliability of LMI solvers ...
... to reduce computational complexity
?
48/48
Carsten Scherer