This document outlines a presentation on controller synthesis for piecewise affine systems. It introduces piecewise affine slab differential inclusions as a model for systems with hard nonlinearities. The objective is to propose a method for synthesizing piecewise affine controllers for stability and L2-gain performance of such systems using convex optimization. Key points covered include:
- Defining the dual parameter set for piecewise affine slab systems.
- Developing sufficient linear matrix inequality conditions for stability and L2-gain analysis using the dual parameter set.
- Formulating the controller synthesis problem based on the dual stability conditions, though additional work is needed to address non-convex terms.
- Providing examples of practical systems that can
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Controller synthesis for piecewise affine slab differential inclusions: A duality-based convex optimization approach
1. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Controller synthesis for piecewise affine slab
differential inclusions
A duality-based convex optimization approach
Behzad Samadi Luis Rodrigues
Department of Mechanical and Industrial Engineering
Concordia University
CDC 2007, New Orleans
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 1/ 25
2. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Outline of Topics
1 Introduction
2 Stability Analysis
3 L2 Gain Analysis
4 Controller Synthesis
5 Numerical Example
6 Conclusions
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 2/ 25
3. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Motivation
Question: What is the dual of a piecewise affine (PWA)
system?
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 3/ 25
4. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Motivation
Question: What is the dual of a piecewise affine (PWA)
system?
It is still an open problem.
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 3/ 25
5. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Piecewise Affine Slab Differential Inclusions
A continuous-time PWA slab differential inclusion is described
as
˙x ∈ Conv{Aiκx + aiκ + Buiκ u + Bwiκ w, κ = 1, 2}
y ∈ Conv{Ciκx + ciκ + Duiκ u + Dwiκ w, κ = 1, 2}
for (x, w) ∈ RX×W
i where Conv stands for the convex hull of
a set.
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 4/ 25
6. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Piecewise Affine Slab Differential Inclusions
A continuous-time PWA slab differential inclusion is described
as
˙x ∈ Conv{Aiκx + aiκ + Buiκ u + Bwiκ w, κ = 1, 2}
y ∈ Conv{Ciκx + ciκ + Duiκ u + Dwiκ w, κ = 1, 2}
for (x, w) ∈ RX×W
i where Conv stands for the convex hull of
a set.
RX×W
i 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 + 1
are scalars such that
σ1 < σ2 < . . . < σM+1
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 4/ 25
7. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Piecewise Affine Slab Differential Inclusions
Practical examples:
Mechanical systems with hard nonlinearities such as
saturation, deadzone, Columb friction
Contact dynamics
Electrical circuits with diodes
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 5/ 25
8. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Piecewise Affine Slab Differential Inclusions
Hassibi and Boyd (1998) - Quadratic stabilization and control
of piecewise linear systems - Limited to piecewise linear
controllers for PWA slab systems
Johansson and Rantzer (2000) - Piecewise linear quadratic
optimal control - No guarantee for stability
Feng (2002) - Controller design and analysis of uncertain
piecewise linear systems - All local subsystems should be stable
Rodrigues and Boyd (2005) - Piecewise affine state feedback
for piecewise affine slab systems using convex optimization -
Stability analysis and synthesis using parametrized linear
matrix inequalities
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 6/ 25
9. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Objective
To introduce a concept of duality for PWA slab differential
inclusions
To propose a method for PWA controller synthesis for stability
and L2-gain performance of PWA slab differential inclusions
using convex optimization
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 7/ 25
10. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Objective
To introduce a concept of duality for PWA slab differential
inclusions
To propose a method for PWA controller synthesis for stability
and L2-gain performance of PWA slab differential inclusions
using convex optimization
Convex optimization problems are numerically tractable.
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 7/ 25
11. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Dual parameter set
PWA slab differential inclusion:
˙x ∈ Conv{Aiκx + aiκ, κ = 1, 2}, x ∈ Ri
Ri = {x| Li x + li < 1}
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 8/ 25
12. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Dual parameter set
PWA slab differential inclusion:
˙x ∈ Conv{Aiκx + aiκ, κ = 1, 2}, x ∈ Ri
Ri = {x| Li x + li < 1}
Parameter set:
Ω =
Aiκ aiκ
Li li
i = 1, . . . , M, κ = 1, 2
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 8/ 25
13. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Dual parameter set
Sufficient conditions for stability
P > 0,
AT
iκP + PAiκ + αP < 0, ∀i ∈ I(0),
λiκ < 0,
AT
iκP + PAiκ + αP + λiκLT
i Li Paiκ + λiκli LT
i
aT
iκP + λiκli Li λiκ(l2
i − 1)
< 0,
for i /∈ I(0).
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 9/ 25
14. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Dual parameter set
Dual parameter set
ΩT
=
AT
iκ LT
i
aT
iκ li
i = 1, . . . , M, κ = 1, 2
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 10/ 25
15. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Dual parameter set
Sufficient conditions for stability
Q > 0,
AiκQ + QAT
iκ + αQ < 0, ∀i ∈ I(0), κ = 1, 2
µiκ < 0
AiκQ + QAT
iκ + αQ + µiκaiκaT
iκ QLT
i + µiκli aiκ
Li Q + µiκli aT
iκ µiκ(l2
i − 1)
< 0,
for i /∈ I(0).
A new interpretation for the result in Hassibi and Boyd (1998)
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 11/ 25
16. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
L2 gain
PWA slab differential inclusion:
˙x ∈ Conv{Aiκx + aiκ + Bwiκ w, κ = 1, 2}, (x, w) ∈ RX×W
i
y ∈ Conv{Ciκx + ciκ + Dwiκ w, κ = 1, 2}
RX×W
i = {(x, w)| Li x + li + Mi w < 1}
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 12/ 25
17. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
L2 gain
PWA slab differential inclusion:
˙x ∈ Conv{Aiκx + aiκ + Bwiκ w, κ = 1, 2}, (x, w) ∈ RX×W
i
y ∈ Conv{Ciκx + ciκ + Dwiκ w, κ = 1, 2}
RX×W
i = {(x, w)| Li x + li + Mi w < 1}
Parameter set:
Φ =
Aiκ aiκ Bwiκ
Li li Mi
Ciκ ciκ Dwiκ
i = 1, . . . , M, κ = 1, 2
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 12/ 25
18. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
L2 gain
Sufficient conditions for L2 gain performance
P > 0,
AT
iκP + PAiκ + CT
iκCiκ ∗
BT
wiκ
P + DT
wiκ
Ciκ −γ2
I + DT
wiκ
Dwiκ
< 0, ∀i ∈ I(0, 0),
AT
iκP + PAiκ
+CT
iκCiκ + λiκLT
i Li
∗ ∗
aT
iκP + cT
iκCiκ + λiκli Li λiκ(l2
i − 1) + cT
iκciκ ∗
BT
wiκ
P + DT
wiκ
Ciκ + λiκMT
i Li DT
wiκ
ciκ + λiκli MT
i
−γ2
I + DT
wiκ
Dwiκ
+λiκMT
i Mi
< 0
and λiκ < 0 for i /∈ I(0, 0).
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 13/ 25
19. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Dual parameter set
Dual parameter set
ΦT
=
AT
iκ LT
i CT
iκ
aT
iκ li cT
iκ
BT
wiκ
MT
i DT
wiκ
i = 1, . . . , M, κ = 1, 2
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 14/ 25
20. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Dual parameter set
Sufficient conditions for stability
Q > 0,
AiκQ + QAT
iκ + Bwiκ
BT
wiκ
∗
CiκQ + Dwiκ
BT
wiκ
−γ2
I + Dwiκ
DT
wiκ
< 0, ∀i ∈ I(0, 0)
AiκQ + QAT
iκ
+Bwiκ
BT
wiκ
+ µiκaiκaT
iκ
∗ ∗
LiκQ + Mi BT
wiκ
+ µiκli aT
iκ µiκ(l2
i − 1) + Mi MT
i ∗
CiκQ + Dwiκ
BT
wiκ
+ µiκciκaT
iκ Dwiκ
MT
i + µiκli ciκ
−γ2
I + Dwiκ
DT
wiκ
+µiκciκcT
iκ
< 0
and µiκ < 0 for i /∈ I(0, 0).
A new result that extends the result in Hassibi and Boyd
(1998) for ci = 0
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 15/ 25
21. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
PWA controller synthesis
Consider the following system:
˙x ∈ Conv{Aiκx + aiκ + Buiκ u, κ = 1, 2}, x ∈ Ri
Ri = {x| Li x + li < 1}
The stability conditions corresponding to the dual parameter
set is used to formulate the synthesis problem.
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 16/ 25
22. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
PWA controller synthesis
Controller synthesis problem:
Q > 0,
AiκQ + QAT
iκ + Buiκ Yi + Y T
i BT
uiκ
+ αQ < 0,
for i ∈ I(0), κ = 1, 2 , and
µi < 0
AiκQ + QAT
iκ
+Buiκ Yi + Y T
i BT
uiκ
+αQ + µi aiκaT
iκ
+aiκZT
i BT
uiκ
+ Buiκ Zi aT
iκ
+Buiκ Wi BT
uiκ
∗
Li Q + µi li aT
iκ
+li ZT
i BT
uiκ
µiκ(l2
i − 1)
≤ 0,
for i /∈ I(0) and κ = 1, 2
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 17/ 25
23. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
PWA controller synthesis
New variables:
Yi = Ki Q
Zi = µi ki
Wi = µi ki kT
i
There is a problem: Wi is not a linear function of the
unknown parameters µi , Yi and Zi .
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 18/ 25
24. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
PWA controller synthesis
Two solutions:
Convex relaxation: Since Wi = µi ki kT
i ≤ 0, if the synthesis
inequalities are satisfied with Wi = 0, they are satisfied with
any Wi ≤ 0. Therefore, the synthesis problem can be made
convex by omitting Wi .
Rank minimization: Note that Wi = µi ki kT
i ≤ 0 is the
solution of the following rank minimization problem:
min Rank Xi
s.t. Xi =
Wi Zi
ZT
i 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
ZT
i µi
≤ 0
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 19/ 25
25. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
L2 gain PWA controller synthesis
Consider the following system:
˙x ∈ Conv{Aiκx + aiκ + Buiκ u + Bwiκ w, κ = 1, 2},
y ∈ Conv{Ciκx + ciκ + Duiκ u + Dwiκ w},
for (x, w) ∈ RX×W
i = {(x, w)| Li x + li + Mi w < 1}
The L2 conditions corresponding to the dual parameter set is
used to formulate the synthesis problem.
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 20/ 25
26. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
L2 gain PWA controller synthesis
L2 gain controller synthesis problem:
Q > 0,
AiκQ + Buiκ
Yi
+QAT
iκ + Y T
i BT
uiκ
+Bwiκ
BT
wiκ
∗
CiκQ + Duiκ
Yi
+Dwiκ
BT
wiκ
−γ2
I + Dwiκ
DT
wiκ
< 0
for i ∈ I(0), κ = 1, 2 , and
µi < 0
AiκQ + Buiκ
Yi
+QAT
iκ + Y T
i BT
uiκ
+Bwiκ
BT
wiκ
+ µi aiκaT
iκ
aiκZT
i BT
uiκ
+ Buiκ
Zi aT
iκ
∗ ∗
LiκQ + Mi BT
wiκ
+µiκli aT
iκ + li ZT
i BT
uiκ
µiκ(l2
i − 1) + Mi MT
i ∗
CiκQ + Duiκ
Yi
+Dwiκ
BT
wiκ
+ µiκciκaT
iκ
ciκZT
i BT
uiκ
+ Duiκ
Zi aT
iκ
Dwiκ
MT
i
+µiκli ciκ
+li Duiκ
Zi
−γ2
I + Dwiκ
DT
wiκ
+µiκciκcT
iκ + ciκZT
i DT
uiκ
+Duiκ
Zi cT
iκ
< 0,
for i /∈ I(0) and κ = 1, 2
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 21/ 25
27. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Surge model of a jet engine
Consider the following model (Kristic et al 1995):
˙x1 = −x2 − 3
2x2
1 − 1
2x3
1
˙x2 = u
A bounding envelope is computed for the nonlinear function
f (x1) = −3
2x2
1 − 1
2x3
1
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 22/ 25
28. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Modeling
By substituting the PWA bounds in the equations of the
nonlinear system, we get a differential inclusion
˙x ∈ Conv{Aiκx + aiκ + Buu + Bw w}, x ∈ Ri
y = Cx + Dw w + Duu (1)
where i = 1, . . . , 4, κ = 1, 2
The approximation error of the nonlinear function is
considered as the disturbance input (w) and the objective is
to limit the L2-gain from w to x1.
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 23/ 25
29. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Simulation
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 24/ 25
30. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Conclusions:
A new concept, dual parameter set, was introduced for PWA
differential inclusions.
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 25/ 25
31. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Conclusions:
A new concept, dual parameter set, was introduced for PWA
differential inclusions.
Using the dual parameter set, sufficient conditions for stability
and L2 gain performance were obtained.
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 25/ 25
32. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Conclusions:
A new concept, dual parameter set, was introduced for PWA
differential inclusions.
Using the dual parameter set, sufficient conditions for stability
and L2 gain performance were obtained.
Convex methods were proposed for PWA controller synthesis
for stability and performance.
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 25/ 25
33. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Conclusions:
A new concept, dual parameter set, was introduced for PWA
differential inclusions.
Using the dual parameter set, sufficient conditions for stability
and L2 gain performance were obtained.
Convex methods were proposed for PWA controller synthesis
for stability and performance.
Note that the dual parameter set does not necessarily define a
PWA system. The questions still is:
Does a dual system exist for a PWA system in general?
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 25/ 25