A convex optimization approach to model in validation of switched arx systems with unknown switches
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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

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A convex optimization approach to model in validation of switched arx systems with unknown switches

51st IEEE Conference on Decision and Control

A Convex Optimization Approach to Model (In)validation of Switched ARX Systems with Unknown Switches

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 Engineering

Penn State University, University Park, PA, USA

Northeastern University


Northeastern university

Outlines

  • Motivation

  • Problem Statement

  • Convex (In)validation Certificates

  • Sparsification Based Approach

  • Moments Based Approach

  • Extension to Systems with Structural Constraints

  • Examples

  • Conclusions

Northeastern University


Motivation 1

Cooling

Air Conditioner

Motivation --1

Powersaving

Traffic

Heating

Can be modeled by

hybrid systems

with

both continuous dynamics and discrete dynamics

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Motivation 2

Switched ARX System

Motivation --2

are mode signal which can switch arbitrarily amongst subsystems, .

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Motivation 3

Motivation --3

NP-hard

Switched ARX System Identification

Given:

Order of Submodels ; Number of submodels ;

A priori bound on noise ; Experimental data

Find:

A piecewise linear affine model such that

Solutions based on:

Heuristics, Optimization, Probabilistic Priors, Convex Relaxation…

Model Invalidation Problem

Is the model identified valid for other data?

Northeastern University


Northeastern university

Outlines

  • Motivation

  • Problem Statement

  • Convex (In)validation Certificates

  • Sparsification Based Approach

  • Moments Based Approach

  • Extension to Systems with Structural Constraints

  • Examples

  • Conclusions

Northeastern University


Problem statement 1

Model Invalidation of Switched ARX System

Problem Statement --1

Given

Parameters of submodels of a hybrid system:

A priori bound on noise

Experimental data

Determine

Whether the consistency set is nonempty, where

Northeastern University


Northeastern university

Problem Statement --2

not invalidated at t

Hybrid decoupling constraint

is feasible.

Invalidated

Not Invalidated

NONCONVEX PROBLEM, HOW TO SOLVE?

Northeastern University


Northeastern university

Outlines

  • Motivation

  • Problem Statement

  • Convex (In)validation Certificates

  • Sparsification Based Approach

  • Moments Based Approach

  • Extension to Systems with Structural Constraints

  • Examples

  • Conclusions

Northeastern University


Convex in validation certificates 1 sparsification based approach 1

is feasible.

is feasible.

Convex (In)validation Certificates 1Sparsification Based Approach --1

is feasible.

Relaxation of cardinality

Re-weighted heuristic

Sparsification Based (In)validation Certificates

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Convex in validation certificates 1 sparsification based approach 2

Sparsification Based (In)validation Certificates

Solve iteratively

Convex (In)validation Certificates 1Sparsification Based Approach --2

Update the weight by

Northeastern University


Northeastern university

Convex (In)validation Certificates 1Sparsification Based Approach --3

Northeastern University


Northeastern university

Outlines

  • Motivation

  • Problem Statement

  • Convex (In)validation Certificates

  • Sparsification Based Approach

  • Moments Based Approach

  • Extension to Systems with Structural Constraints

  • Examples

  • Conclusions

Northeastern University


Convex in validation certificates 2 moments based approach 1

Convex (In)validation Certificates 2Moments Based Approach --1

is feasible.

  • The model is invalidated

Possible to use Positivstellensatz to get invalidation certificates.

However, easier to utilize problem structure via moment-based

polynomial optimization.

Northeastern University


Convex in validation certificates 2 moments based approach 2

A Moments-based Relaxation

Convex (In)validation Certificates 2Moments-Based Approach --2

  • The model is invalidated if and only if

There exists an N such that

  • The set is nonempty if and only if

There exists an N such that and

(the moment matrix hits the flat extension)

Specifically, in our problem, T+1 is a choice for N.

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Convex in validation certificates 2 moments based approach 3

A Moments-based Relaxation

Problem has a sparse structure (running intersection property holds)

Convex (In)validation Certificates 2Moments-Based Approach --3

Northeastern University


Northeastern university

Outlines

  • Motivation

  • Problem Statement

  • Convex (In)validation Certificates

  • Sparsification Based Approach

  • Moments Based Approach

  • Extension to Systems with Structural Constraints

  • Examples

  • Conclusions

Northeastern University


Extension to systems with structural constraints

Time instant

Extension to Systems with Structural Constraints

Northeastern University


Northeastern university

Outlines

  • Motivation

  • Problem Statement

  • Convex (In)validation Certificates

  • Sparsification Based Approach

  • Moments Based Approach

  • Extension to Systems with Structural Constraints

  • Examples

  • Conclusions

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Examples 1

Given submodels without structural constraints

and the measurement equation

Examples --1

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Examples 11

Given submodels without structural constraints

Examples --1

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Examples 2

Examples --2

run

walk

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Examples 21

A Priori information

Experimental Data

time

Examples --2

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Examples 22

A Priori information

Experimental Data

time

Examples --2

Northeastern University


Northeastern university

Outlines

  • Motivation

  • Problem Statement

  • Convex (In)validation Certificates

  • Sparsification Based Approach

  • Moments Based Approach

  • Extension to Systems with Structural Constraints

  • Examples

  • Conclusions

Northeastern University


Conclusions

The model (in)validation problem for switched ARX systems with unknown switches:

Given

Is there any consistency set ?

Conclusions

  • 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

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