northeastern university

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


Outlines

Motivation Problem Statement Convex (In)validation Certificates Sparsification Based Approach Moments Based Approach

Extension to Systems with Structural Constraints Examples Conclusions

Motivation --1


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


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


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


Outlines



Problem Statement --1


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


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


Outlines



,

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


, , 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


Sparsification Based Approach --2


, , :

,

,

, : ,

, ,

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



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


Outlines




Moments Based Approach --1


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



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



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


Outlines



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


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


Outlines



Examples --1


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


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


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


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


, 0,1i ts

A Priori information Experimental Data time


Outlines



Conclusions


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

,

northeastern university

Documents

approach moments

approach extension

convex optimization

convex relaxation motivation

hybrid systems

university park

invalidatedthe model

arx system identification