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51 st IEEE Conference on Decision and Control. A Convex Optimization Approach to Model (In)validation of Switched ARX Systems with Unknown Switches. Yongfang Cheng 1 , Yin Wang 1 , Mario Sznaier 1 , Necmiye Ozay 2 , Constantino M. Lagoa 3 - PowerPoint PPT Presentation
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A Convex Optimization Approach to Model (In)validation of Switched ARX Systems with
Unknown Switches
Northeastern University
Yongfang Cheng1, Yin Wang1, Mario Sznaier1, Necmiye Ozay2, Constantino M. Lagoa3
1Department of Electrical and Computer Engineering Northeastern University, Boston, MA, USA
2Department of Computing and Mathematical Sciences Caltech, Pasadena, CA, USA
3Department of Electrical EngineeringPenn State University, University Park, PA, USA
51st IEEE Conference on Decision and Control
Northeastern University
Outlines
Motivation Problem Statement Convex (In)validation Certificates Sparsification Based Approach Moments Based Approach
Extension to Systems with Structural Constraints Examples Conclusions
Motivation --1
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Air ConditionerCooling
Heating
Power savingTraffic
Can be modeled by hybrid systems
with both continuous dynamics and discrete dynamics
are mode signal which can switch arbitrarily amongst subsystems, .
Motivation --2
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1 1a cn n
t k t t k k t t k tk k
t t t
ξ A ξ C u f
y ξ η
Switched ARX System
tsn iG
1, ,t s sn N
Switched ARX System Identification
Given:Order of Submodels ; Number of submodels ; A priori bound on noise ; Experimental dataFind:A piecewise linear affine model such that
Solutions based on:Heuristics, Optimization, Probabilistic Priors, Convex Relaxation…
Motivation --3
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an sn
1
0, T
t t t
u y
1 1a cn n
t k t t k k t t k tk k
t t t
ξ A ξ C u f
y ξ η
NP-hard
Is the model identified valid for other data?Model
Invalidation Problem
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Outlines
Motivation Problem Statement Convex (In)validation Certificates Sparsification Based Approach Moments Based Approach
Extension to Systems with Structural Constraints Examples Conclusions
Problem Statement --1
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Parameters of submodels of a hybrid system:
A priori bound on noise Experimental data
Whether the consistency set is nonempty, where
1
0, T
t t t
u y
sn
1 1, , , 1, ,a cn n
i k k sk kG i i i i n
A C f
Given
Determine
1 1
,
, , 0, 1a c
sn n
t k t k k t
t t
t t t
t
kk k
t t
t T
N
A C u f
y
,
1
:
1
,
,
, , 0, 1
0t a
a c
t
sn n
k t k t k t k
t t
t t k t t t
t t
k k
t n
t T
N
A y y C u f
g h
t
Model Invalidation of Switched ARX System
: :,1, , , , such that 0a t t ast t n tt ts t nn N g h
Northeastern University
Problem Statement --2 not invalidated at t 1, ,
snG G
is feasible.
, :1
: , such that 0s
t t
t
a at t n t t n
n
t
g h
, :
, ,1
:
, 0
0,1 , 1
a
s
a
i t t t n
i t i
i t i
t
t t n
n
i
s
s s
g h
Hybrid decoupling constraint
NONCONVEX PROBLEM, HOW TO SOLVE?
1, 11, : 0at ntt ts g h
2, 22, : 0at ntt ts g h
3, 33, : 0at ntt ts g h
Invalidated
Not Invalidated
, :,ai t t t ns
, :,ai t t t ns
Northeastern University
Outlines
Motivation Problem Statement Convex (In)validation Certificates Sparsification Based Approach Moments Based Approach
Extension to Systems with Structural Constraints Examples Conclusions
,
1
, , :
, ,
, : , 1 , : :
0
0,1 , and 1
, and
a
a a a
s
s
i t i
n
i t i t t n
i t i t
i t t n i t
i
n
i i t t n t t n
s
s s
s
g h
, , :
, ,
, : , ,
,
0
:
1
:1
0
0 1, 1, min
, and
a
a a
s
s
a
i t ii t i t t n
i t i t
i t t n i t i t t
n
i
n
i n t t n
s
s s
s
T
g h
s is feasible.
is feasible.
Convex (In)validation Certificates 1
Sparsification Based Approach --1
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, , 010 1, 1, mins
i t i tn
is s T
s
0 Relaxation of cardinality
Re-weighted heuristic
Sparsification Based (In)validation Certificates
1
, :
, ,1
:
, 0
0,1 , 1
a
s
a
i t t t n
i t i
i t i
t
t t n
n
i
s
s s
g h
is feasible.
1,0 1,1 1, 1
2,0 2,2 2, 1
,0 ,2 , 1s s s
T
T
n n n T
s s ss s s
s s s
s
0 1 1T
Convex (In)validation Certificates 1
Sparsification Based Approach --2
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, , :
,
,
, : ,
, ,
1
: :
,
,
,
,
1
min
subject to0 ,
0 1 ,
1
,
s
s
a
a
a a
i t i
ki t i t
t t n
i t
i t
i t t n i t
i t t
i ts
i
n t t
t i
n
i
n
i n
s
s
w s
i t
i t
t
ts i
s
t
g h
Sparsification Based (In)validation Certificates
Solve iteratively
Update the weight by
11 0
, , ,, 1,1, ,1 Tk ki t i t i tw s w
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Convex (In)validation Certificates 1
Sparsification Based Approach --3Results of the Sparsification Algorithm Interpretation
Infeasible Invalidated
Feasible, Not invalidated
Feasible, ?
*, 0,1i ts
*, 0,1i ts
1 0 0 0 0
0 1 1 0 1
0 0 0 1 0
s
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Outlines
Motivation Problem Statement Convex (In)validation Certificates Sparsification Based Approach Moments Based Approach
Extension to Systems with Structural Constraints Examples Conclusions
Convex (In)validation Certificates 2
Moments Based Approach --1
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is feasible.
, :
,
, :
,
1
0 ,
0,1 ,
1,s
a
a
i t i
n
i t t t n
i t
i t t t ni
s
s
s
i t
i t
t
g h
, :
1 2* 2,
0 1
2
, :,
, , ,
:
1
min
subject to
, 1
ai na
a
s
t t t
sn
i t t t ns
i
T
i t it i
t i t i in
t
t t n
s
s s
p
t
s t
g h
* 0p
The model is not invalidated
The model is invalidated* 0p
Possible to use Positivstellensatz to get invalidation certificates.However, easier to utilize problem structure via moment-basedpolynomial optimization.
There exists an N such that and (the moment matrix hits the flat extension)
A Moments-based Relaxation
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Convex (In)validation Certificates 2
Moments-Based Approach --2
The model is invalidated if and only if
The set is nonempty if and only if
There exists an N such that
1*
,0 1
min
subject to
anT
N i tt i
N
N
l
L
p
M
m
0
0
m
m
m
±±
, :
1 2* 2,
0 1
2
, :,
, , ,
:
1
min
subject to
, 1
ai na
a
s
t t t
sn
i t t t ns
i
T
i t it i
t i t i in
t
t t n
s
s s
p
t
s t
g h
* 0Np
* *Np p
* 0Np
Specifically, in our problem, T+1 is a choice for N.
,
1rank rankN N M m M m
1 1a an n T Tp p p p
1 2 1a an n T T
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Convex (In)validation Certificates 2
Moments-Based Approach --3
A Moments-based Relaxation
1*
,0 1
min
subject to
anT
N i tt i
N
N
l
L
p
M
m
0
0
m
m
m
±±
Problem has a sparse structure (running intersection property holds)
1*
, :0 1
:
:
min
subject to
10,
0, 1
a
a
a
a
nT
N i t t t nt i
N t n t
N t n t
p l
M Tt
L Tt
mm
m
m
0
0
±±
0 1
1
1
1 1
1, 1, 1 1, 1 1,
, , 1 , 1 ,
1
1
a a
s a s a
a a
a
a
a
s s
a
T n T n
T n
n T
T T
n n T T
n n n
n
n
n
n T n T
s s s s
s s s s
Northeastern University
Outlines
Motivation Problem Statement Convex (In)validation Certificates Sparsification Based Approach Moments Based Approach
Extension to Systems with Structural Constraints Examples Conclusions
1,0 1,1 1,2 1, 2 1, 1
2,0 2,1 2,2 2, 2 2, 1
3,0 3,1 3,2 3, 2 3, 1
0 1 2 2 1
T T
T T
T T
s s s s ss s s s ss s s s
T
s
T
Extension to Systems with Structural Constraints
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Time instant
2,0 3,1 1s s
2,1 3,2 1s s
2, 2 3, 1 1T Ts s
, , 1 1, ,i t j ts s i I j J
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Outlines
Motivation Problem Statement Convex (In)validation Certificates Sparsification Based Approach Moments Based Approach
Extension to Systems with Structural Constraints Examples Conclusions
Examples --1
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Given submodels without structural constraints
and the measurement equation
ActualA Priori Information
Results using sparsification feasible, feasible,
Interpretation Not invalidated no decision
Time (sec.) 3.6808 4.4611
Results using Moments -3.0399e-07 -2.4735e-07
Interpretation not invalidated not invalidated
Times (sec.) 6.8146 6.4891
1 2 3, ,G G G
1 2 3, ,G G G
1 2 3, ,G G G
1 2 3, ,G G G
*, 0,1i ts *
, 0,1i ts
1 2 1
1 2 1 2
1 2 3
1
1
0.53
0.7
0.2 0.24 2
1.4
1.7 02 .5
t t t t
t t t t
t t t t
G
G
G
u
u
u
t t ty
Examples --1
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ActualA Priori Information
Results using sparsification infeasible infeasible
Interpretation invalidated invalidated
Time (sec.) 0.1949 0.1980
Results using Moments 7.3123 14.2226
Interpretation invalidated invalidated
Times (sec.) 2.7642 2.4306
1 2 3, ,G G G
1 2,G G 1 2,G G
2 3,G G
Given submodels without structural constraints
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Examples --2
1 2
1 2
0.6058 0.0003 0.3608 0.18530.2597 0.8589 0.2381 0.1006
t t t
t t t
x x xy y y
run
walk 1 2
1 2
0.4747 0.0628 0.5230 0.11440.3424 1.2250 0.3574 0.2513
t t t
t t t
x x xy y y
Examples --2
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Algorithms results
Results using Sparsification with Constraints Infeasible
Interpretation Invalidated
Time (sec.) 2.1836
Results using Moments with Constraints 0.1751
Interpretation Invalidated
Times (sec.) 1.2968e03
A Priori information Experimental Data time
Algorithms results
Results using Sparsification with Constraints Feasible,
Interpretation Not Invalidated
Time (sec.) 1.0577
Results using Moments with Constraints -3.3581e-08
Interpretation Not Invalidated
Times (sec.) 0.8407e03
Examples --2
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, 0,1i ts
A Priori information Experimental Data time
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Outlines
Motivation Problem Statement Convex (In)validation Certificates Sparsification Based Approach Moments Based Approach
Extension to Systems with Structural Constraints Examples Conclusions
Conclusions
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The model (in)validation problem for switched ARX systems with unknown switches:Given
Is there any consistency set ?
Sparsification Based (In)validation Certificates Sufficient Conditions Moments Based (In)validation Certificates Necessary and Sufficient Conditions A finite order of moment matrix, N=T+1, is found to hit the flat extension Extension to systems with constraints on switch sequences
1
1 0, , ,sn T
i t t ti tG
u y
,