model reduction of linear time-varying systems using multipoint … · 2020-03-23 · model...

Model Reduction of Linear Time-Varying

Systems using Multipoint Krylov-subspace

Projections

Hossain Mohammad Sahadet

Department of Mathematics

Chemnitz University of Technology.

D-09107 Chemnitz

Germany

Berlin-Braunschweig-Chemnitz Workshop

Recent Advances in Model Reduction

Chemnitz, 27 November 2006

1/22 [email protected] Model reduction of LTV systems

1 Introduction

�What is Model reduction?

�Model reduction based on projection

formulations.

�Why Krylov-subspace projection?

– Process based on ‘moment’ matching.

– Multipoint rational approximations are efficient

for particular cases.

– Relatively faster than other direct factorization

approaches.


� The LTI SystemThe linear time-invariant multi-input, multi-output (MIMO) differential-algebraic system of the form

(1)

where , , .

� Transfer function of the LTI System

Applying the Laplace Transforms on the system and

assuming gives

where denotes the Laplace transform .

Here is called the Transfer function of the LTI system.

( ) nx t ∈�

2 Order Reduction Of LTI Systems

)()()( tuBtxAtxE +−=�

xCty T=)(

0 0x =1( ) ( ) ( )Ty s c sI A B u s−= +� �


1( )Tc sI A B−+

( ) ( )iniu t n n∈ ≤� 0( )

ny t ∈�

.�

Rational approximation and projection

� Suppose that a rational approximation of the LTI system (1) is given by

where and .

The reduced matrices are obtained through the projection matrices

P and Q

� For E = I, the k-th moment of the transfer function of the reduced

model is

� Connect the moments with the projection matrices P and Q.

ˆˆ ˆ( ) ( )E z t Az Bu t= − +�

ˆ( ) Ty t C z=

r n�

ˆ ˆˆ ˆ, , ,T T T TE P EQ A P AQ B P B C Q C= = = =

1 ( 1)T k T kC A B C A B− − − +=


0ˆˆ ˆ, , ( ) , r nr r rE B z t C ××∈ ∈ ∈� � �

Definitions and Theorems

� Definition 1

Krylov subspace is defined as

where and is called the Starting vector.

� The first independent basic vectors can be considered as a basis of a

Krylov subspace.

� Definition 2

Block Krylov subspace is defined as

where and the columns of are the starting vectors.

n nA ×∈�

2 1( , ) { , , , , }m

m A b span b Ab A b A bκ −= ��

2 1( , ) { , , , , }m

m A B span B AB A B A Bκ −= ��


nb∈�

n nA ×∈� n mB ×∈�

Definitions and Theorems

(Joel R. Phillips’1998)

Theorem 1

If the columns of Q form a basis for and

is chosen such that is non-singular, then the first m moments

(about zero) of the original and reduced order system match.

Theorem 2

If the columns of P form a basis for and the matrix

Q is chosen such that is nonsingular, then the first nmoments of

the original and reduced order system match.

Theorem 3

If the columns of P span , and the columns of Q span ,

then matches the first m+nmoments of .

n mP

×∈ℜ

A

A

( , )T

m A Cκ − 1( , )n A Bκ −

1ˆ ˆ ˆ( )TC sI A B−+ 1( )TC sI A B−+


1 1( , )m A E A Bκ − −

( , )T T T

n A E A Cκ − −

Some projection techniques

�The projection techniques

� Lanczos process : (that means, the two

projection bases are bi- orthogonal)

� Arnoldi Process : and .

� Structural properties are inherited.

P Q=

TP Q I=


TP Q I=

Multipoint approximations

� the transfer function and some of its derivatives (moments) are matched at several points in the complex plane.

� P and Qmust contain a basis for the union of the Krylov subspace

constructed at the different expansion points.

� For a complex expansion point, a real model can be efficiently

obtained.

� For a real matrix A, if is in Krylov subspace, then

its conjugate is also in the Krylov subspace.

Multipoint approximation

1( )u I sA p−= −�*u�


� Consider the nonlinear system describing a circuit equation

(2)

� split v(t) and u(t) into two signal parts

, and

� linearize around the large signal part

(3)

where and .

3 LTV signal analysis

( ( ))( ( )) ( ) ( )l

dq v tf v t b t b u t

dt+ = +

( ) ( ).T

lz t c v t=

( ) ( )L su u u= + ( ) ( )L sv v v= +

( )Lv

( ) ( ) ( )( ) ( ( ) ) ( )s s sdG t v C t v bu t

dt+ =

( ) ( )( ) ( )( ) /L sG t f v t v= ∂ ∂ ( ) ( )( ) ( )( ) /L sC t q v t v= ∂ ∂


Frequency-domain matrix of LTV system

� Follow the formalism of L. Zadeh’s(1950) variable transfer functions and then obtain

(4)

� Assume u is a single response and then using the delta function

to obtain

(5)

� write s = iω and substitute in (3) to get

(6)

( )u uω ωδ ω ω′ ′= −

( ) ( , ) ( ) i tv t h i t u e ωω ω=

( ) ( , ) ( ( ) ( , )) ( ) ( , )d

G t h s t C t h s t sC t h s t bdt

+ + =


( ) ( , ) ( ) i tv t h i t u e dωω ω ω∞

′

−∞

′ ′ ′= ∫

Frequency-domain matrix of LTV system

� Now defining

K = and D = ,

equation (6) can be written as

[K + sD] (7)

Equation (7) is known as the Frequency-domain expression of LTI transfer functions.

� h (s, t) is a rational function with infinite number of poles.

For example, in a periodically time-varying system with fundamental frequency , if is a pole of the system,

then , k is an integer, will be a pole of h (s, t) .

( ) ( )d

G t C tdt

+ ( )C t

( , )h s t b=

0ω0kη ω+


η

4 Model reduction of LTV systems� Obtaining Discrete Rational Functions

Let the LTV system be periodic. We discuss only the SISO case.

We seek a representation of transfer functions in terms of finite-dimensional matrices.

� discretization of the operators K and D .

� use backward-Euler discretization method. Then we get

(8)

(9)

(10)

11

1 1

1 22

2 2

1

M

M MM

M M

d dG

h h

d dG

h hK

d dG

h h

−

+ −

− + =

− +

� �

1

2

M

d

dD

d

=

�

1 2( ) [ ( ) ( ) ( )]TMh s h s h s h s= �


Model reduction of LTV systems

� Approximating the Krylov subspace

� for a small-signal steady state system, we need to solve the finite-

difference equations

(11)

where , T is the fundamental period.

� the transfer function then becomes

� factorize K of eqn. (8) into lower and upper triangular parts.

1111

1 1

21 2 22

2 2

1

( )( )( )

( ) ( )

( ) ( )

M

M MM

M M M M

b tv td dG s

h h

v td d b tG

h h

d dG

h h v t b t

α

−

+ − ⋅ − + = − +

��

��

� �

� �

� �

( ) sTs eα −≡

( , ) ( )sTh s t e v t−= �


Model reduction of LTV systems

� Equation (11) then takes the form

(12)

� Defining a small-signal modulation operator Ψ(s) .

(13)

the transfer function has the form

(14)

� Finally the approximation comes to the form

(15)

( ( ) ) ( ) ( )L s U v s b sα+ = ��

1

20( )

0 M

st

st

st

Ie

Ies

Ie

Ψ ≡

� �

( ) ( )Hh s v s= Ψ �

( )[ ( ) ] ( )HK sD s L s U sα+ Ψ + Ψ�


Approximation by Direct approach

� L is lower triangular, L-1 can easily be obtained. Then eqn.(12) takes the form

(16)

� In eqn.(16), L-1 U has nonzero entries. Hence, has the form

where Rj∈ RN×N is the ((j-1)N)+1 through jN rows of the last columns

of L-1 U .

� Advantages : easy to compute, need backsolving technique and simple matrix factorization.

� Disadvantages : cost of computation increases very rapidly with the growth of problem size.

1( ( ) )I s L Uα −+

1 1( ( ) ) ( ) ( )I s L U v s L b sα − −+ = ��

1

2

0 ( )

0 ( )

0 0 ( ) M

I s R

I s R

I s R

αα

α

+

�

� � �

�


� The preconditioned problem can be found in the form

(17)

� Factorization are required only for the diagonal blocks.

� Backward substitution are also needed.

•••• Advantages :

� reduces the cost.

� easy computations.

� convergence may be faster.

•••• Disadvantages :

� reduced model may not preserve the system structure.

Approximation by preconditioned Krylov

subspace method.

1 1( ( ) ) ( ) ( )I s L U v s L b sα − −+ = ��


Projection by using the finite-

difference technique

� Basis for the projection is obtained using finite-difference equations.

� very good approximation to the Krylov-subspaces of spectral

operator can be obtained.

The following algorithm shows the detail study.


Algorithm I (Approximating Multipoint Krylov-subspace Model Reduction


set k =1

for i = 1,…….., nq {

for j = 1,…….., m(i) {

if j = 1 then

w = b

else

w = D vk

for l = 1,…….., k-1 {

}

k = k +1

}

} (continue to next page)

1( )[ ( ) ] ( )Hi i iu s L s U s wα −= Ψ + Ψ

Tlu u v u= −

/kv uu=

Algorithm I (Approximating Multipoint Krylov-subspace Model Reduction

(continue from previous page)


ˆ

ˆ

ˆ

ˆ

T

T

T

T

K V KV

E V EV

C V C

B V B

=

=

=

=

Recycled Krylov-subspace scheme

� Assume E = I and take the expansion point at origin. Then exact Vk is

constructed so that

� Every new ui in is related to for i < m .

� Van der Vorst’87 has shown

can be constructed from nearly the same Krylov

sequence , that is used to construct .

This implies can be constructed from , where

m > k.

1 1 ( 1)( , ) { , , , }k

k kV K K b b K b K b− − − −⊂ ≡ …

qK b−

{ , , , },mb Kb K b m q>…

{ , , , }mb Kb K b…

1( , )iK K b−( , )mK K b

1K b−


kV

• Advantages :

� no extra cost required to obtain the projectors at multi-frequency

points.

� relatively faster convergence.

Recycled Krylov-subspace scheme


� RF circuit simulator

� Switched-Capacitor Synchronous Detector

� Transmitter

Applications


Reference:

The paper is mainly based on the original work of

J. Phillips. Model Reduction of Linear Control systems Using Multipoint

Krylob-subspace Projectors. In Proc. ICCAD, November 1998.

The End

Thanks for your attention

The full- text of this talk can be

provided through mail to

[email protected]

End [email protected] Model reduction of LTV systems

model reduction of linear time-varying systems using multipoint … · 2020-03-23 · model...

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