Transient Sensitivity Analysis CASA Day 13th Nov 2007 Zoran Ilievski. Zoran Ilievski Transient Sensitivity Analysis

Transcription

1 CASA Day 13th Nov 2007

2 Talk Structure

3 Talk Structure Introduction

4 Talk Structure Introduction Recap Sensitivity

5 Talk Structure Introduction Recap Sensitivity Examples and Results

6 Talk Structure Introduction Recap Sensitivity Examples and Results Further Work

7 Talk Structure Introduction Recap Sensitivity Examples and Results Further Work

8 Introduction State Sensitivity L & W will gives R Which will change the states/voltages x(t)

9 Introduction Function Sensitivity More complex functions such as Gain are expressions based on these states and parameters. As parameters change, the Gain (or other functions) will also change.

10 Introduction - Circuit DAE Description d [q(x(t))] + j(x(t)) = s(t) dt x(t) R N 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 find behaviour.

11 Introduction - The influence of a parameter P d [q(x(t, p), p)] + j(x(t, p), p) = s(t, p) (1) dt p R P is the parameter vector. Optimisation is done by adjusting these parameters. How sensitive are the states x(t) R N to these adjustments? How does a circuits behaviour respond? What is the observed sensitivity of the circuit function?

12 Introduction - State sensitivity ˆx is the sensitivity of the states x to a parameters p, given by. ˆx(t, p) x(t, p)/ p R NxP How are these state sensitivities calculated? What is the operation cost of calculating the observation function sensitivities dependant upon these sate sensitivities?

13 Introduction - Circuit Observation Functions G f (x(p), p) = T 0 F(x(t, p), p)dt. (2) d dp G f (x(p), p) = T 0 F x F ˆx + dt. (3) p The Gain of an amplifier circuit can be called the observation function of that design You can see that the state sensitivity is central to the overall function sensitivity The cost of ˆx is the main burden.

14 Introduction How are these state and observation sensitivities calculated?

15 Introduction Recap Sensitivity Examples and Results Further Work

16 Recap - State sensitivity Backward-Euler (more generally: BDF) is applied: 1 t [q(xn+1 ) q(x n )] + j(x n+1 ) s(t n+1 ) = 0 (4) Newton-Raphson solution involves the coefficient matrix Y = 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

17 Recap - State sensitivity Differentiation w.r.t. parameter P gives: 0 = 1 n+1 t [ q p After slight re-arrangement: q n ] + 1 p t [Cˆxn+1 Cˆx n ] + Gˆx n+1 + j p s p n+1 ((1/ t)c + G) ˆx n+1 = 1 }{{} t [ q q ] + (1/ t)cˆx n j p p p + s p Y }{{} f n

18 Recap - State sensitivity - Cost ˆx n+1 (p) = Y 1 f, in which f = 1 n+1 t [ q p q n ] j n+1 p p + s n p t Cˆxn (p). The vector f requires O(PN 2 ) operations for the 1 t Cˆxn (p) term In addition O(PN) evaluations for each other term Solving the system requires an additional O(PN 2 ) operations.

19 Recap - Circuit Observation Functions We now have an analysis of ˆx But what part does ˆx play in the sensitivity analysis of any observation function. G f (x(p), p) = T 0 F(x(t, p), p)dt.

20 Recap - Circuit Observation Functions G f (x(p), p) = T 0 F(x(t, p), p)dt. d dp G f (x(p), p) = T 0 F F ˆx + x p dt. You can see that the state sensitivity is central to the overall function sensitivity. The cost of ˆx is the main burden.

21 Recap - Direct Forward Method It is no supprise that the emphasis in sensitivity analysis is the efficient calculation of ˆx or indeed the inner product. F F ˆx = x x.y 1 f = [Y T [ F x ]T ] T f This can be calculated in O(min(F, P)N 2 + FPN) operations. This is for the Direct Forward Method Circuits often contain many thousands of parameters, need a better method.

22 Recap - Backward Adjoint Method Small N Large Circuits, Large N Direct Foward Feasible Expensive Backward Adjoint Feasible Elimination of ˆx, and so reducing the complexity is done using a backward integration technique. An less expensive expression of the observation sensitivity in terms of λ(t)

23 Recap - Backward Adjoint Method - general λ expression [λ dq dp ] T 0 + T 0 [ ( dλ dt C λ G)ˆx dλ q dt p + λ ( j p s p )]dt = 0

24 Recap - Backward Adjoint Method - general λ expression [λ dq dp ] T 0 + T 0 [ ( dλ dt C λ G)ˆx dλ q dt p + λ ( j p s p )]dt = 0 This result holds for any λ

25 Recap - Backward Adjoint Method - general λ expression [λ dq dp ] T 0 + T 0 [ ( dλ dt C λ G)ˆx dλ q dt p + λ ( j p s p )]dt = 0 This result holds for any λ The motivation behind the selectin of λ is to eliminate ˆx in d dp G f (x(p), p) = T 0 F F ˆx + x p dt.

26 Recap - Backward Adjoint Method - general λ expression [λ dq dp ] T 0 + T 0 [ ( dλ dt C λ G)ˆx dλ q dt p + λ ( j p s p )]dt = 0 This result holds for any λ The motivation behind the selectin of λ is to eliminate ˆx in d dp G f (x(p), p) = T 0 F F ˆx + x p dt.

27 Recap - Backward Adjoint Method - general λ expression d dp G f (x(p), p) = T 0 F F ˆx + x p dt.

28 Recap - Backward Adjoint Method - general λ expression d dp G f (x(p), p) = T 0 F F ˆx + x p dt. d dp G f (x(p), p) = T 0 [ dλ dt C + λ G]ˆx + F p dt.

29 Recap - Backward Adjoint Method - general λ expression d dp G f (x(p), p) = T 0 F F ˆx + x p dt. d dp G f (x(p), p) = T 0 [ dλ dt C + λ G]ˆx + F p dt. C dλ dt G λ = ( F x ). The natural conclusion is that this linear adjoint DAE is the constraint on λ

30 Recap - Backward Adjoint Equation This linear adjoint DAE is the constraint on λ C dλ dt G λ = ( F x ).

31 Recap - Backward Adjoint Equation This linear adjoint DAE is the constraint on λ The sensitivity can be rewritten in terms of λ if it satisfies the adjoint DAE C dλ dt G λ = ( F x ).

32 Recap - Backward Adjoint Equation Alternative: Substituting the general λ equation the sensitivity can be expressed as follows, d dp G 2(x(p), p) = [λ (t) dq ( T dλ 0 dt dp ] T t=0 + q p λ ( j p s p ) + F p ) dt.

33 Recap - Backward Adjoint Equation - initial conditions [λ (t)( q q x.ˆx + p )] T t=0

34 Recap - Backward Adjoint Equation - initial conditions [λ (t)( q q x.ˆx + p )] T t=0 ˆx(t = T ) can be eliminated, if λ(t ) = 0

35 Recap - Backward Adjoint Equation - initial conditions [λ (t)( q q x.ˆx + p )] T t=0 ˆx(t = T ) can be eliminated, if λ(t ) = 0 With this choice the sensitivity expression simplifies to, d dp G 2(x(p), p) = λ (0)[ q x (0).ˆx DC + q p (0)] + T 0 dλ dt q p λ ( j p s p ) + F dt. (5) p

36 Recap - Backward Adjoint Equation - Summary Standard transient analysis of forward system, initial condition are set at t=0 d [q(x(t, p), p)] + j(x(t, p), p) = s(t, p) dt Solution of Backward Adjoint Equation Each time integration step requires O(FN 2 ) operations. Initial condition are set at t=t, λ(t )=0 C dλ dt G λ = ( F x ).

37 Recap - Backward Adjoint Equation - Summary BAE is always linear, even though original equation is nonlinear. C dλ dt G λ = ( F x ).

38 Recap - Backward Adjoint Equation - Summary substitution of λ values to obtain the observation sensitivity The integrand at each time point requires O(PN + FP) operations and O(FPN) evaluations d dp G 2(x(p), p) = λ (0)[ q x (0).ˆx DC + q p (0)] + T 0 dλ dt q p λ ( j p s p ) + F p dt.

39 Method Table Direct Foward Backward Adjoint Cost O(min(F, P), N 2 ) + O(FPN) O(FN + FP) + O(FPN) Usualy F << N

40 Talk Structure Introduction Recap Sensitivity Examples and Results Further Work

41 Backward Adjoint Equation - Reduction C dλ dt G λ = ( F x ). Each time integration step requires an LU decomposition of O(N α ) α=2 for a sparce system and α=3 for a full system This brings the total cost to O(N α + FN 2 ) F << P, in practice Quite a large cost for a large system, N. What if we could project this system on to a n << N dimensional subspace?

42 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

43 Proper Orthogonal Decomposition - Reduction V T V = I r Given a collection of functions x(t 1 ), x(t 1 ),...x(t M ) in X Find subspace X r X so to minimize the error, Σ M j=1 = x j V T ˆx j 2

44 Proper Orthogonal Decomposition - Reduction

45 Proper Orthogonal Decomposition - Reduction For any system, in the state analysis we obtain, X = [x(t 1 ), x(t 2 )...x(t nsnapshots ))] R nsnapshotsxn (6) Next an SVD is computed giving, X = V ΣU V n Σ n U nn << N (7) As long as the singular values decay rapidly the system can be reduced x r = V k x(t), x r R n

46 Proper Orthogonal Decomposition - Reduction Reduction methods differ in the way V is selected

47 Proper Orthogonal Decomposition - Reduction Reduction methods differ in the way V is selected POD produces data dependant bases, from existing snapshots

48 Proper Orthogonal Decomposition - Reduction 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.

49 Proper Orthogonal Decomposition - Reduction 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.

50 Proper Orthogonal Decomposition - Reduction 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 same basis?

51 Backward Adjoint Equation - Reduction 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 to the most dominant singular values. As long as the decay in singular values is rapid the system can be reduced.

52 Backward Adjoint Equation - Reduction The projection matrix is applied as follows V C V dλ dt V G Vλ = V ( F x ). Ĉ dˆλ dt Ĝ ˆλ = ( F ˆ x ).

53 Backward Adjoint Equation - Reduction Ĉ dˆλ dt Ĝ ˆλ ˆ = ( F x ). λ can be obtained by projecting back on the the origional basis by V ˆλ This changes the cost contribution of N down to n. Where n << N, usually. A very attractive solution.

54 Results(1) - Direct Forward Sensitivity on Obf=Energy(I,R2) on R2, direct foward method, is e-008

55 Results(1) - Direct Forward Sensitivity Obf=Energy(I,R2) on R2, backward adjoint method, is e-008 Previous result was e-008 Promising result

56 Results(2) - Reduction Example length of R2 dg/dp dg/dp with POD x x x x x x10 10

57 Results(3) - Industrial Singular Value Analysis σ Use of internal PStar circuit simulation software Very good at supplying state snapshots, which can be recorded and used in Matlab. Value σ σ σ σ σ [V S U]=svd(states ); diag(s)

58 Results(3) - Industrial Example The rapid decay is singular values is promising, it shows a nice POD basis can be formed.

59 Talk Structure Introduction Recap Sensitivity Examples and Results Further Work

60 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 the observation function. Form a POD bases for the state analysis. Attempt to apply the same pod bases to the backward step.

61 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 of POD is being carried out on this at Wuppertal. We are aiming to submit an abstract for SCEE 2008 and a paper in the proceedings with the subject on POD for BAE methods.

62 The End THE END

63 Problem Description The task is to reduce the Tline and then plug it back in to the system. The important requirement is to keep the nodes V 1 and V 2 safe. Figure: Circuit with the Transmission Line

64 Modified Nodal Analysis for the Tline V 4 and V 5 are the internal nodes and V 1 and V 2 are the external pins for transmission line C A V 1 V 2 V 4 V 5 q1 q C A R R R R R R 0 0 R B R R R 0 0 C A C V 1 V 2 V 4 V 5 q 1 q = C B I 1 I C A

65 AMOR The AMOR is an Arnoldi based algorithm which is used PRIMA techniques inside to reduced the dynamical system. The form which AMOR accepts is like this: { Bu = Eẋ + Kx, y = Cx.

66 AMOR bellow figure shows how we can define the AMOR input for this TLINE. In block form we have: ( E ) ( u ẋ ) ( D C + B K ) ( u x ) = ( y 0 )

67 AMOR What if the T-Line is terminated with a capacitor? Can this submodel be completely reduced?