Dissipativity-Preserving Model Reduction Based on
Generalized Singular Perturbation

Takayuki Ishizaki (Tokyo Institute of Technology)

Introduction
Classical model reduction problem
Given, findsuch that

Balanced truncation
Hankel norm approximation

Structure-preserving model reduction problem
Given, finds.t.

: a class of realizations that possess particular structure(s)Positivity preservation

Controller reduction

Structure preservation is crucial for practical approximations

2/15

Model Reduction
Based on Singular Perturbation

unitary transform

Imposeon

Singular perturbation approximation✓where3/15

Model Reduction
Based on Generalized Singular Perturbation

unitary transform

ImposeIntroducing parameteronGeneralized singular perturbation approximation✓

wherecoincides withsingular perturbation approximant if
projection-basedif

Problem Formulation: a class of -dimensional systems that possess structure(s)e.g., dissipativity, network structure among subsystems[Problem]Consider a system
perturbation modelin the following. Given, find a generalized singularassociated withsuch that✓ set of projections✓ transfer matrices ofand:

5/15

System Dissipativity[Definition]
A systemis said to be

there exists-dissipative with respect tosuch that

iffor✓ equivalent toalong trajectory of[Lemma]Any-dissipative

has a realization that is✓ Cholesky factorization-dissipative w.r.t.

.

:
6/15

Dissipativity Preservation[Theorem]
Letbe given, and suppose that it isIfandthenis✓ If-dissipative w.r.t.

satisfies-dissipative w.r.t.
,

admits projection-like formula:where-dissipative-dissipative
coordinatetransform-dissipative
generalized
singularperturbation

How much error?
7/15

Error Analysis[Theorem] The error system can be factorized as

wherecoordinatetransform✓ Expect: errorwill be small if
is sufficiently small (whileis chosen so thatis bounded)
8/15

Error Analysis[Theorem] The error system can be factorized aswhere[Theorem]
Let✓such that

DefineIf&forsuch that

andthen
Eigenvalues ofis related to approximation error

9/15

Remarks
Eigenvalue decomposition ofleads tothat achieves: design criterion to regulate approximating quality

for given

Generalization to network structure preservation
allows application to controller reduction problem
block-diagonal

10/15

Application toStructured Passive Controller Reduction

Original Closed-LoopApproximate Model: Passive plant: Passive controller

By virtue of dissipativity & network structure preservation:
Find an approximate modeleachremains passive andsuch thatis small enough in-sense.
11/15

Numerical Example
Vibration suppression control:[Output

Uncontrolled: passive (100-dim.)(40-dim.): passive

passive:](60-dim.)Controlled(supposed to be desirable)

TimeBridge

Limited sensor & actuator allocations
12/15

Numerical Example
Vibration suppression control:[Output

Uncontrolled: passive (100-dim.)(40-dim.)[Dim. of: passive

passive:&[Approximation error]]](60-dim.)decreases asdecreases

Controlled(supposed to be desirable)

Time[Outputof approx. model]

Approx. controllers(passive, 5- & 6-dim.)Original closed-loop
less than 1% error(design criterion)(design criterion)

Time

13/15

Numerical Example
Vibration suppression control:[OutputUncontrolled: passive (100-dim.)(40-dim.)passive:(60-dim.)of approx. models][Output: passive[Output

Our controllers(passive, 5- & 6-dim.)
Balanced truncation(not passive, 5- & 6-dim.)]Controlled(supposed to be desirable)

Timeof approx. models]Our controllers(passive, both 2-dim.)

Balanced truncation(not passive, both 2-dim.)
14/15

Concluding Remarks
Generalized singular perturbation approximation includes:
standard singular perturbation
projection-based model reduction

Structure-preserving model reduction
dissipativity, network structure among subsystems
a priori -error boundApplication to distributed passive controller reduction
vibration suppression for interconnected second-order systems
preservation of passivity and closed-loop performance

Thank you for your attention!
15/15