Transient Sensitivity Analysis - TU/e · PDF fileTalk Structure Introduction Recap Sensitivity...

Transient Sensitivity AnalysisCASA Day 13th Nov 2007

Zoran Ilievski

Zoran Ilievski

Transient Sensitivity Analysis

Talk Structure

Introduction

Recap Sensitivity

Examples and Results

Further Work

Zoran Ilievski


Introduction

State Sensitivity

∆ L & ∆ W will gives ∆R

Which will change the states/voltages x(t)

Zoran Ilievski


Introduction

Function Sensitivity

More complex functions such as Gain are expressions based onthese states and parameters.

As parameters change, the Gain (or other functions) will alsochange.

Zoran Ilievski


Introduction - Circuit DAE Description

d

dt[q(x(t))] + j(x(t)) = s(t)

x(t) ∈ RN is the state vector.

j & q are current and charge densities

s(t) contains all source values.

Solve the DAE obtained from Kirchhoffs laws, to findbehaviour.

Zoran Ilievski


Introduction - The influence of a parameter P

d

dt[q(x(t,p),p)] + j(x(t,p),p) = s(t,p) (1)

p ∈ RP is the parameter vector.

Optimisation is done by adjusting these parameters.

How sensitive are the states x(t) ∈ RN to these adjustments?

How does a circuits behaviour respond? What is the observedsensitivity of the circuit function?

Zoran Ilievski


Introduction - State sensitivity

x is the sensitivity of the states x to a parameters p, given by.

x(t,p) ≡ ∂x(t,p)/∂p ∈ RNxP

How are these state sensitivities calculated?

What is the operation cost of calculating the observationfunction sensitivities dependant upon these sate sensitivities?

Zoran Ilievski


Introduction - Circuit Observation Functions

Gf (x(p),p) =

∫ T

0F(x(t,p),p)dt. (2)

ddp

Gf (x(p),p) =

∫ T

0

∂F

∂x· x +

∂F

∂pdt. (3)

The Gain of an amplifier circuit can be called the observationfunction of that design

You can see that the state sensitivity is central to the overallfunction sensitivity

The cost of x is the main burden.

Zoran Ilievski


Introduction

How are these state and observation sensitivities calculated?

Zoran Ilievski


Introduction

Recap Sensitivity


Further Work

Zoran Ilievski


Recap - State sensitivity

Backward-Euler (more generally: BDF) is applied:

1

∆t[q(xn+1)− q(xn)] + j(xn+1)− s(tn+1) = 0 (4)

Newton-Raphson solution involves the coefficient matrixY = 1

∆t C + G, in which C = ∂q/∂x and G = ∂j/∂x

We now know q, j and s at each time point t

Zoran Ilievski


Recap - State sensitivity

Differentiation w.r.t. parameter P gives:

0 =1

∆t[∂q

∂p

n+1

− ∂q

∂p

n

] +1

∆t[Cxn+1 − Cxn] + Gxn+1 +

∂j

∂p− ∂s

∂p

After slight re-arrangement:

((1/∆t)C + G)︸︷︷︸Y

xn+1 = − 1

∆t[∂q

∂p

n+1

− ∂q

∂p

n

] + (1/∆t)Cxn − ∂j

∂p+∂s

∂p︸︷︷︸f

Zoran Ilievski


Recap - State sensitivity - Cost

xn+1(p) = Y−1f, in which

f = − 1

∆t[∂q

∂p

n+1

− ∂q

∂p

n

]− ∂j

∂p

n+1

+∂s

∂p

n+1

+1

∆tCxn(p).

The vector f requires O(PN2) operations for the 1∆t Cxn(p)

term

In addition O(PN) evaluations for each other term

Solving the system requires an additional O(PN2) operations.

Zoran Ilievski


Recap - Circuit Observation Functions

We now have an analysis of x

But what part does x play in the sensitivity analysis of anyobservation function.

Gf (x(p),p) =

∫ T

0F(x(t,p),p)dt.

Zoran Ilievski


Recap - Circuit Observation Functions

Gf (x(p),p) =

∫ T

0F(x(t,p),p)dt.

ddp

Gf (x(p),p) =

∫ T

0

∂F

∂x· x +

∂F

∂pdt.

You can see that the state sensitivity is central to the overallfunction sensitivity.

The cost of x is the main burden.

Zoran Ilievski


Recap - Direct Forward Method

It is no supprise that the emphasis in sensitivity analysis is theefficient calculation of x or indeed the inner product.

∂F

∂x· x =

∂F

∂x.Y−1f = [Y−T [

∂F

∂x]T ]T f

This can be calculated in O(min(F ,P)N2 + FPN) operations.

This is for the Direct Forward Method

Circuits often contain many thousands of parameters, need abetter method.

Zoran Ilievski


Recap - Backward Adjoint Method

Small N Large Circuits, Large N

Direct Foward Feasible Expensive

Backward Adjoint Feasible

Elimination of x , and so reducing the complexity is done usinga backward integration technique.

An less expensive expression of the observation sensitivity interms of λ(t)

Zoran Ilievski


Recap - Backward Adjoint Method - general λ expression

[λ?dq

dp]|T0 +

∫ T

0[− (

dλ?

dtC− λ?G)x− dλ?

dt

∂q

∂p+ λ?(

∂j

∂p− ∂s

∂p)]dt = 0

This result holds for any λ

The motivation behind the selectin of λ is to eliminate x in

ddp

Gf (x(p),p) =

∫ T

0

∂F

∂x· x +

∂F

∂pdt.

Zoran Ilievski


Recap - Backward Adjoint Method - general λ expression

ddp

Gf (x(p),p) =

∫ T

0

∂F

∂x· x +

∂F

∂pdt.

ddp

Gf (x(p),p) =

∫ T

0[−dλ?

dtC + λ?G]x +

∂F

∂pdt.

C?dλdt− G?λ = −(

∂F

∂x)?.

The natural conclusion is that this linear ’adjoint’ DAE is theconstraint on λ

Zoran Ilievski


Recap - Backward Adjoint Equation

This linear ’adjoint’ DAE is the constraint on λ


∂F

∂x)?.

Zoran Ilievski



This linear ’adjoint’ DAE is the constraint on λ

The sensitivity can be rewritten in terms of λ if it satisfies theadjoint DAE


∂F

∂x)?.

Zoran Ilievski



Alternative:

Substituting the general λ equation the sensitivity can beexpressed as follows,

ddpG2(x(p),p) = −[λ?(t)dq

dp ]|Tt=0 +∫ T0

(dλ?

dt∂q∂p − λ

?( ∂j∂p −∂s∂p) + ∂F

∂p

)dt.

Zoran Ilievski


Recap - Backward Adjoint Equation - initial conditions

−[λ?(t)(∂q∂x .x + ∂q∂p)]|Tt=0

x(t = T ) can be eliminated, if λ(T ) = 0

With this choice the sensitivity expression simplifies to,

ddp

G2(x(p),p) = λ?(0)[∂q

∂x(0).xDC +

∂q

∂p(0)] +∫ T

0

dλ?

dt

∂q

∂p− λ?(

∂j

∂p− ∂s

∂p) +

∂F

∂pdt. (5)

Zoran Ilievski


Recap - Backward Adjoint Equation - Summary

Standard transient analysis of forward system, initial conditionare set at t=0

d

dt[q(x(t,p),p)] + j(x(t,p),p) = s(t,p)

Solution of Backward Adjoint Equation

Each time integration step requires O(FN2) operations.

Initial condition are set at t=T, λ(T )=0


∂F

∂x)?.

Zoran Ilievski



BAE is always linear, even though original equation isnonlinear.


∂F

∂x)?.

Zoran Ilievski



substitution of λ values to obtain the observation sensitivity

The integrand at each time point requires O(PN + FP)operations and O(FPN) evaluations

ddp

G2(x(p),p) = λ?(0)[∂q

∂x(0).xDC +

∂q

∂p(0)] +∫ T

0

dλ?

dt

∂q

∂p− λ?(

∂j

∂p− ∂s

∂p) +

∂F

∂pdt.

Zoran Ilievski


Method Table

Direct Foward Backward AdjointCost O(min(F ,P),N2) + O(FPN) O(FN + FP) + O(FPN)

Usualy F << N

Zoran Ilievski


Talk Structure

Introduction

Recap Sensitivity


Further Work

Zoran Ilievski


Backward Adjoint Equation - Reduction


∂F

∂x)?.

Each time integration step requires an LU decomposition ofO(Nα)

α=2 for a sparce system and α=3 for a full system

This brings the total cost to O(Nα + FN2)

F << P, in practice

Quite a large cost for a large system, N.

What if we could project this system on to a n << Ndimensional subspace?

Zoran Ilievski


Proper Orthogonal Decomposition - Reduction

C and G are large N x N, matrices

As we have seen, costly system to solve

Would like to reduce this cost

Zoran Ilievski



V TV = Ir

Given a collection of functions x(t1), x(t1), ...x(tM) in X

Find subspace Xr ⊂ X so to minimize the error,

ΣMj=1 =‖ xj − V T xj ‖2

Zoran Ilievski



Zoran Ilievski



For any system, in the state analysis we obtain,

X = [x(t1), x(t2)...x(tnsnapshots))] ∈ RnsnapshotsxN (6)

Next an SVD is computed giving,

X = V ΣU∗ ≈ VnΣnU?nn << N (7)

As long as the singular values decay rapidly the system can bereduced

xr = V ∗k x(t), xr ∈ Rn

Zoran Ilievski



Reduction methods differ in the way V is selected

POD produces data dependant bases, from existing snapshots

We already have snapshot data from the foward analysis.

We can form a POD basis for the forward analysis.

Can we reduce the backward adjoint method with the samebasis?

Zoran Ilievski



Find a projection matrix, V to reduce the system to size n.

n << N

For highly non-linear circuits, POD is a good choice.

V is constructed from the left eigen vectors corresponding tothe most dominant singular values.

As long as the decay in singular values is rapid the system canbe reduced.

Zoran Ilievski



The projection matrix is applied as follows

V′C?Vdλdt− V′G?Vλ = −V′(

∂F

∂x)?.

C?dλdt− G?λ = −

ˆ(∂F

∂x)

?

.

Zoran Ilievski



C?dλdt− G?λ = −

ˆ(∂F

∂x)

?

.

λ can be obtained by projecting back on the the origionalbasis by V λ

This changes the cost contribution of N down to n. Wheren << N, usually.

A very attractive solution.

Zoran Ilievski


Results(1) - Direct Forward

Sensitivity on Obf=Energy(I,R2) on R2, direct fowardmethod, is -3.068e-008

Zoran Ilievski


Results(1) - Direct Forward

Sensitivity Obf=Energy(I,R2) on R2, backward adjointmethod, is -3.068e-008

Previous result was -3.068e-008

Promising result

Zoran Ilievski


Results(2) - Reduction Example

length of R2 dG/dP dG/dP with POD

0.02 −0.235x10−10 −0.235x10−10

0.021 −0.247x10−10 −0.247x10−10

0.022 −0.256x10−10 −0.256x10−10

Zoran Ilievski


Results(3) - Industrial Singular Value Analysis

Use of internal PStar circuit simulation software

Very good at supplying state snapshots, which can berecorded and used in Matlab.

[V S U]=svd(states’); diag(S)

σ Value

σ1 798.8497

σ2 0.2292

σ3 0.0885

σ4 0.0547

σ5 0.0225

Zoran Ilievski


Results(3) - Industrial Example

The rapid decay is singular values is promising, it shows a nicePOD basis can be formed.

Zoran Ilievski


Talk Structure

Introduction

Recap Sensitivity


Further Work

Zoran Ilievski


Conclusions and Further Work

We have written the BAE in forward form,

We use again the x(T-t) data in the state analysis, for theobservation function.

Form a POD bases for the state analysis.

Attempt to apply the same pod bases to the backward step.

Zoran Ilievski


Conclusions and Further Work

Further Work

For a simple circuit, we have had some success.

But were we just lucky.

Can we apply the POD basis on all circuits?

Collaborative work on the thory behind the application ofPOD is being carried out on this at Wuppertal.

We are aiming to submit an abstract for SCEE 2008 and apaper in the proceedings with the subject on POD for BAEmethods.

Zoran Ilievski


The End

THE END

Zoran Ilievski


Problem Description

The task is to reduce the Tline and then plug it back in tothe system.

The important requirement is to keep the nodes V1 and V2

safe.

Figure: Circuit with the Transmission Line

Zoran Ilievski


Modified Nodal Analysis for the Tline

V4 and V5 are the internal nodes and V1 and V2 are the externalpins for transmission line.

0BBBBB@0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0

1CCCCCA·

0BBBBBB@V1

V2

V4

V5q1q2

1CCCCCCA+

0BBBBBBB@

1R

0 −1R

0 0 0

0 2R

0 −1R

0 0−1R

0 3R

−1R

0 0

0 −1R

−1R

3R

0 00 0 C1 0 −1 00 0 0 C2 0 −1

1CCCCCCCA·

0BBBBB@V1V2V4V5q1q2

1CCCCCA =

0BBBBB@I1I20000

1CCCCCA

Zoran Ilievski


AMOR

The AMOR is an Arnoldi based algorithm which is usedPRIMA techniques inside to reduced the dynamical system.

The form which AMOR accepts is like this:{Bu = Ex + Kx ,y = Cx .

Zoran Ilievski


AMOR

bellow figure shows how we can define the AMOR input forthis TLINE.

In block form we have:

(0 00 E

)·(

ux

)+

(D C−B K

)·(

ux

)=

(y0

)Zoran Ilievski


AMOR

What if the T-Line is terminated with a capacitor?

Can this submodel be completely reduced?

Zoran Ilievski


Transient Sensitivity Analysis - TU/e · PDF fileTalk Structure Introduction Recap Sensitivity...

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