Seminar at KTH Royal Institute of Technology

Multiresolved Control of Discrete-Time Linear Systems

Takayuki Ishizaki (Tokyo Institute of Technology)

Motivation
Biological systems

Power system control

Organs, cells, proteins, …
are interacting hierarchically

Few minutes: governor free control
Few dozen minutes: LFC, EDC

Efficient control of systems on different spatiotemporal scales?
Multiresolved control: Utilize global/local properties in time and space
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A Multiresolved Control Systemex) Power system control

conventional observer-based controller

identical input to all:: average offrequency variation: generatorseasy to synchronize
in each area

freq. variation

Area 1

Area 2Global (uniform) variation
Area 1

Area 2

Appropriately stabilized
Time

Good for uniformvariation but…?
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A Multiresolved Control Systemex) Power system control

conventional observer-based controller

identical input to all:: average offrequency variation: generators
easy to synchronize
in each area

freq. variation

Area 1
Global (uniform) variation
Area 1

Area 2

Appropriately stabilized
Time

Area 2not suitable for local(nonuniform) variation

Local variation in Area 2
Area 2
Area 1Time

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A Multiresolved Control Systemex) Power system control

conventional observer-based controller

identical input to all:: average of
frequency variation: input to Area 2: generators

low-dimcontroller

easy to synchronize
in each area

Area 1

Design a low-dim local controller

Area 2: average of Area 2

cooperating with a given

Plug-in when and where necessary
Local action for a specific time interval

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A Multiresolved Control Systemex) Power system control

conventional observer-based controller

identical input to all:: average of
frequency variation

anotherone

low-dimcontrollerArea 1

Design a low-dim local controller

Area 2cooperating with a given

Plug-in when and where necessary
Local action for a specific time interval

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Outline
Motivation
Key idea: Superposition by state-space expansion
Main result: Resultant multiresolved control system
Technical details
Numerical example

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How to systematically design
Based on superposition?

stabilize

cooperation

stabilize

Superposition by state-space expansion
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Main Result【Theorem】
Each ofis stabilized bycooperation

n-dim controller+ k-dim memory

k-dim local controller

after time kemergency
actiontime9/18

Outline
Motivation
Key idea: Superposition by state-space expansion
Main result: Resultant multiresolved control system
Technical details
Numerical example

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Superposition by State-Space Expansion
block triangularState-space expansion

n-dim

2n-dimfor any【Lemma】

if

Decentralized state FB:

stabilize
stabilize

are not directly measurable…
Decentralized observation of
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Decentralized Observation ofSuperposition States?
Redundant realization (2n-dim)

unknown local variation

Luenberger-type observer

decentralized output feedback

Error dynamics

block triangular

Can we make?
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Block-Triangulation of Error Dynamics【Lemma】
If

andafter time k , thenk-dim Krylov projection modelk-dimn-dimn-dim…?

Some with “lower complexity”
such that?
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Wedderburn Rank Reduction【Wedderburn Rank Reduction】
For given[Chu et al. 1995]

there exist
such thatrank-onedecomposition

andwhere: generalization of Gram-Schmidt basis orthogonalization
if

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Wedderburn Rank Reduction【Wedderburn Rank Reduction】
For given[Chu et al. 1995]

there exist
such thatandwhere【Lemma】

Ifandare controllability and observability matrices,

Markov parameter matching
the low-rank

is equivalent to a k-dim Krylov projection model

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Main Result (reshown)【Theorem】
Each ofis stabilized bycooperation

n-dim controller+ k-dim memory

k-dim local controller

after time kemergency
action

time

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Numerical Example
25 generators (50-dim): identical input to all

average variation:
Area 2: generator(swing eq. model)
Area 3
Area 4

Area 1
Area 5

with

Freq. variation

Only
Area 1(7-dim)Area 1
suppressed

local variation

withTimeTime

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Concluding RemarksMultiresolved control
state-space expansion (global model + local model)
cooperation of global and local controllers
global control at all times
local control when and where necessary

one possible application of model reduction techniques

Future works
multi-layer case (local-middle-global)
multi-rate sampling
make an appropriate criterion to plug in local controllers

Thank you for your attention!

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