Ajoint Transient Sensitivity Analysis in Circuit Simulation Z. Ilievski 1, H. Xu 1, A. Verhoeven 1, E.J.W. ter Maten 1,2, W.H.A. Schilers 1,2 an R.M.M. Mattheij 1 1 Technische Universiteit Einhoven; e-mail: Z.Ilievski@tue.nl 2 NXP Semiconuctors Einhoven Summary. Sensitivity analysis is an important tool that can be use to assess an improve the esign an accuracy of a moel escribing an electronic circuit. Given a moel escription in the form of a set of ifferential-algebraic equations it is possible to observe how a circuit s output reacts to varying input parameters, which are introuce at the requirements stage of esign. In this paper we consier the ajoint metho more closely. This metho is efficient when the number of parameters is large. We exten the transient sensitivity work of Petzol et al., in particular we take into account the parameter epenency of the ynamic term. We also compare the complexity of the irect an ajoint sensitivity an erive some error estimates. Finally we sketch out how Moel Orer Reuction techniques coul be use to improve the efficiency of ajoint sensitivity analysis. Keywors Sensitivity Analysis; Transient Analysis; Ajoint Metho; Moel Orer Reuction 1 Introuction A typical quantity in circuit analysis is the prouct of voltage ifference times the current through an electronic component (power) an, when integrate of time, this reflects the total power that is issipate. Another time omain problem is the etermination of the time moment when a certain unknown, or an expression, crosses a particular value. Such a moment can be the moment at which synchronization is require in co-simulation between a circuit simulator an another simulation tool. In transient analysis, the ajoint metho can be formulate as a convolution of the circuit equations with a carefully constructe function, that, by its nature, requires a backwar integration in time of a relate DAE (an for which a proper initial value has to be etermine). The metho has been popularize in 8, 12] for linear problems. For more general DAEs the metho has been stuie in 7] in a more mathematical way.
2 Z. Ilievski et al In 16] the application to the nonlinear DAEs of circuit equations was stuie more closely. Nice applications can be erive for the problem of fining optimal sources in etecting faults in analog circuits 5]. However, in stuying sizing problems (in which for instance the physical area of a capacitor has to be taken into account), it appears that especially parameters of capacitors give rise to terms that require aitional investigation. Here the effect of the inex of the relate DAE shows up. Apart from purposes of optimization, ajoint systems are of interest in etermining optimal reuce orer moels 3, 1], in which case a large number of parameters occurs. Because the ajoint systems are linear the equations themselves can be mae subject to a reuce orer moeling process. We will escribe ways how to calculate sensitivities in a stable an an efficient way. 2 Transient sensitivity analysis Equation (1) is a general Differential-Algebraic Equation (DAE) that can be use to escribe how any circuit behaves over a perio of time. In Moifie Noal Analysis 12], x(t) R N is the state vector an represents the noe voltages an the currents through voltage sources an inuctors, j an q are vector functions that escribe the current an charge (capacitors) or flux (inuctors) behavior. All source values are comprise in s(t) q(x(t))] + j(x(t)) = s(t). (1) The initial solution at t =, the DC-solution x DC, satisfies j(x DC ) = s(). (2) Applying Euler-Backwar time integration between time points t n an t n+1 = t n + t enables to calculate x n+1 as approximation at t n+1 : 1 t q(xn+1 ) q(x n )] + j(x n+1 ) s(t n+1 ) = (3) A Newton-Raphson proceure involves the coefficient matrix Y = 1 t C + G, in which C = q/ x an G = j/ x. Making explicit that the equations an its solution epen on a parameter p R P we will write q(x(t, p), p)] + j(x(t, p), p) = s(t, p). (4) By ajusting these parameters it is possible to optimize the behavior of a require functionality. The sensitivity of x(t, p) with respect to p is enote by
Ajoint Transient Sensitivity Analysis 3 ˆx(t, p) x(t, p)/ = ( x i (t, p)/ j ) R N P, an similarly for ˆx DC (p). After solving (3), an saving of the LU-ecomposition of the matrix Y = LU, the sensitivity ˆx n+1 (p) ˆx(t n+1, p) may be calculate by recursion 9, 11] ˆx n+1 (p) = Y 1 f, in which (5) f = 1 q n+1 q n] j n+1 + s n+1 + 1 t t Cˆxn (p). (6) The vector f requires O(P N 2 ) operations for the last term in aition to O(P N) evaluations for a term like q etc... For simplicity we assume full matrices. Solving the system requires an aitional O(P N 2 ) operations. A more general basic observation function is enote by F(x(t, p), p) R F from which other observation functions can be obtaine, like From (7) we erive If F x G(x(p), p) = F(x(t, p), p). (7) T ( ) F F G(x(p), p) = ˆx +. (8) x can be etermine rather cheaply in (8), the main emphasis in sensitivity analysis is in the efficient calculation of ˆx, or even in efficiently calculating the inner-prouct F F F x ˆx. Note that, from (5), we erive x ˆx = x Y 1 f = Y T F x ]T ] T f, which can be calculate in O(min(F, P )N 2 +F P N) operations (in aition to those alreay mentione above: the overall leaing N 2 -term still has coefficient P ). This is a irect, forwar, analysis. When, aitionally, some library for evaluating q, j, or s, oes not allow symbolic ifferentiation, here also a symmetric finite ifference will be mae (at the cost of two aitional evaluations for each quantity q q(p+ p) q(p p) 2 p ). This means that at each interior time point of (8) the integran will have an error O( p 2 ) (assuming this iscretization error is ominant). A quarature rule like the Trapezoial Rule as up to O( p 2 / t) leaing to p = o( t) if t an no persistent errors in sensitivities are wante. In the sequel, we now consier an approach base on (backwar) ajoint integration 7]. We ifferentiate (4) w.r.t. p an multiply the result with a function λ (t) R F N (in which the means transpose), yieling = = = λ (t) λ (t) q λ (t) q ] T + ] (9) q + j s ( λ q j + λ s ( λ q x x + q ] T + )] ) + λ ( j x x + j s )]
4 Z. Ilievski et al = = λ (t) q λ (t) q ] T + ] T + λ ( λ ( Cˆx + q ) + λ ( Gˆx + j C λ G )ˆx λ q + λ s ( j This result hols for any λ. We now consier some choices. )] s )] (1) 2.1 Backwar, ajoint sensitivity for G(x(p), p) In (8) we encounter the prouct F x ˆx. Equation (1) will enable us to get ri of the ˆx, which oes not nee to be calculate explicitly. We choose λ(t) R N F appropriately an require that λ(t) satisfies the linear ajoint DAE (we assume the inex 1 case, which is not trivial 4]). C λ ( F ). G λ = (11) x This oes not yet make λ(t) unique, because we i not specify the initial value yet. But we are now able to express F λ x ˆx in terms of C G λ] ˆx, after which we can apply (1). T ( ) F F G(x(p), p) = ˆx + x ( ] = λ C + λ G ˆx + F ) = λ (t) q ] ( ( T λ q j t= + λ s ) + F ). The first term involves ˆx() = ˆx DC, which we know. However, if λ (T ) q x (T ), one still nees ˆx(T ), which we i not want to etermine explicitly. Hence goo choices to efine the initial conitions for λ(t) are C(T ) = q (T ) = λ(t ) =, or if (13) x C(t) = q ( F ) x (t) = λ(t ) λ DC, with G λ DC = (14) x t= Note that (13) is only allowe for DAEs of inex up to 1. With this choice (12) simplifies to (12) q G(x(p), p) = λ () x () ˆx DC + q ] () + ( λ q λ ( j s ) + F ). (15)
Ajoint Transient Sensitivity Analysis 5 The circuit that escribes the ajoint system (11) is escribe in more etails in 8, 12]. There the above expression is erive by conservation properties base on the Kirchhoff Laws which imply Tellegen s Theorem. Note that (9) is just the time integral of inner-proucts of branch-currents AjointCircuit * branch-voltages OriginalCircuit. The formulation use above follows 7], but reveals more closely the effect of a non-trivial function q in (12) that explicitly epens on p. By this, also the velocity λ will be neee. Alreay a linear function q(x, p) = C(p)x requires this. Because of the DAE-nature of the problem (1), in 16], λ was estimate by symmetric finite ifferences in the interior of the interval an with one-sie approximations at the bounaries. When, as before, a library for evaluating q, j, or s, oes not allow symbolic ifferentiation, here also a symmetric finite ifference will be mae (at the cost of two aitional evaluations for each quantity). This means that at each interior time point the integran will have an error O( t 2 + p 2 ). Any quarature rule (like the Trapezoial Rule) may a these errors up to an error O( t + p 2 / t) which is aitional to the error of the quarature rule, which means that one will require p = O( t) (note that a one-sie ifference will even nee p = O( t 2 )). It also shows that the Trapezoial Rule may not give better results than simple, first orer, Euler integration. System (11) can be etermine by integrating backwars in time after x(t, p) has been etermine in the nominal analysis. This backwar time integration 1 of (11) requires Jacobian matrices tc + G (assuming Euler backwars), similar as in the forwar analysis. When the same step sizes are use an the LU-ecompositions of the Jacobian matrices at the converge values have been save from the nominal analysis, the transpose ecompositions can be reuse for the Jacobians when integrating (11) (one may also save approximative inverse matrices, or preconitioning matrices). However we will assume that one will re-ecompose them uring the backwar integration. Note that in 6] it is remarke that if the same time step is use in the forwar (for x) an backwar analysis (for λ) this may give rise to very inaccurate solutions for λ. This general step size approach introuces effects ue to interpolation (effects which we have not yet stuie). Let W = O(N α ) represent the number of operations for the LU-ecompositions with 1 α 2 for sparse systems an α = 3 for full systems. Note that λ () q x () ˆx DC = C λ()] ˆx DC, which can be hanle similarly as in ajoint sensitivity analysis in the DC-problem, without explicitly calculating ˆx DC, in W + O(min(F, P )N 2 + F P N) operations (assuming that G has been ecompose again at t = ). Each time integration step of (11) requires W + O(F N 2 ) operations, after which the integran in (15) at each time point requires O(P N + F P ) evaluations an O(F P N) aitional operations. In practise F P, which (apart from the W term) makes (15) more efficient as the irect, forwar metho for (5)-(6). Reuction of the W term is iscusse in Section 2.2.
6 Z. Ilievski et al 2.2 MOR applie to the global ajoint sensitivity equations The main buren of the backwar ajoint sensitivity equations still is the W = O(N α ) work neee for the LU-ecompositions in the case α = 3 when integrating backwars in time for the ajoint equations for λ(t). In orer to reuce this we observe that in the interior we only nee to know λ(t) for coorinates where q, j s, an are non trivial. However, at t = also for the nontrivial rows of q x () coorinates of λ(t) shoul be known (which may significantly increase the number of neee coorinates). More precisely, in (15) a term like λ () q x () ˆx DC R F P shows that the total number of output is r = F P, an when r N, one may think to apply MOR. We observe that there is a tentative opportunity to apply Proper Orthogonal Decomposition (POD) 1, 13], since the forwar time integration to etermine x(t) elivere a nice series of snapshots {x(t ),..., x(t N )} (an, even cheaply, also of F x (t k)). With POD a matrix V R N r is foun such that x V x, x R r, r N, in which V is time inepenent an similar snapshots coul have been obtaine from a moifie problem VT q(v x(t))] + V T j(v x(t)) = V T s(t). (16) For (16) the matrices are of size r r, which inicates a nice option for MOR. In this paper we apply the V T, V matrices from the POD irectly to the ajoint problem (11). Note that in this approach we can neglect the epenency of V on p. For each p one integrates (1) an calculates the snapshots, resulting in a POD Moel Orer reuction projection matrix V. With V we can reuce the system of DAEs for λ (in which only the error ue to POD matters) resulting in λ P OD (t) λ(t) (please see 13] for error estimates). Note that POD can also be use to reuce (1) itself. In this case the epenency of V on p introuces an aitional error in the proceure that can not be neglecte. This last approach coul further reuce costs an has to be stuie further. In 2] it was shown that (in general) POD not irectly applies to DAEs. Here a Least-Square POD remey was introuce that can be applie to the linear DAE (11). Alternatively to POD, uring the forwar integration one coul aitionally etermine projection matrices for the Trajectory Piecewise-Linear Metho 14, 15]. Next, similarly as to the POD case, a global matrix V is etermine that allows for Piecewise-Linear MOR. In this case the reuce DAE problems can irectly be solve 15].
Ajoint Transient Sensitivity Analysis 7 3 Results p=l R (m) G/ G/ withp OD.2 -.235 1 1 -.235 1 1.21 -.247 1 1 -.247 1 1.22 -.256 1 1 -.256 1 1 Table 1. Sensitivities, where p is the parameter, in this case the parameter is the length of the secon resistor. Units are (W tm 1 ) Fig. 1. Voltage ifferences for successive sensitivity values, noe 2 Equation (12) was implemente in Matlab for a simple circuit an the sensitivities of the energy G issipate for a resistor R = R(l R ) were observe while changing the length parameter l R G = I(R) V(R). (17) To see the epenency of V R for ifferent values of p, in Fig.1. we plotte, A: (V R (p =.21) V R (p =.2))(t) B: (V R (p =.22) V R (p =.21))(t). In Table 1 the sensitivity G/, calculate by the backwar ajoint metho, are shown for 3 ifferent values of p. To test POD we create a larger, equivalent system by splitting one resistor in to a number of smaller resistors. Doing this enable us to compare the sensitivity of our observe resistor in the reuce large system with the sensitivity in the original system. We generate
8 Z. Ilievski et al our projection matrices by applying the singular value ecomposition on the new state snapshot matrix for the enlarge problem. Inee the number of nontrivial singular values for both systems was the same. We applie the projection matrices to reuce only the backwar ajoint calculation steps. The size of the matrices was consierably reuce an the calculate sensitivities from the reuce system were exact to at least 3 significant figures, please see Table 1 for sensitivity results for the POD an for the non POD approach. 4 Conclusions The backwar, ajoint sensitivity methos are immeiately attractive when the original DAE (1) is linear an when the number of parameters P 1. Direct forwar an backwar ajoint approaches impose ifferent accuracy conitions to finite ifference approximations. The irect forwar metho exploits the re-use of LU-ecompositions. The backwar ajoint methos becomes more of interest when MOR can be applie, or when otherwise approximate LU-ecompositions coul have been save uring the forwar time integration. In these cases they can outperform the irect forwar metho when the number of parameters P is large (but still smaller than N, usually F P ). We have shown that applying POD MOR to the backwar ajoint step is possible an works very well. MOR techniques can also be use to reuce the effort in sensitivity calculation in the forwar analysis. In future we want to stuy more closely the application of our sensitivity calculations to larger circuits an with more inustrial relevance. We also want to consier the effect of ifferent time stepping in forwar an backwar analysis. An, finally, we want to stuy the sensitivity of the reuce DAE (16) in which V = V(p) epens on p as well. References 1. P. Astri: Reuction of process simulation moels: a proper orthogonal ecomposition approach, PhD-Thesis, Department of Electrical Engineering, Einhoven University of Technology, 24. 2. P. Astri, A. Verhoeven: Application of least squares MPE technique in the reuce orer moeling of electrical circuits, TU Einhoven, Center for Analysis, Scientific Computing an Applications, CASA Report 11, 26. 3. O. Balima, Y. Favennec, M. Girault, D. Petit: Comparison between the moal ientification metho an the POD-Galerkin metho for moel reuction in nonlinear iffusion systems, Int. J.Numer. Meth. Engng., Vol. 67, pp. 895 915, 26. 4. K. Balla, R. März : Linear ifferential-algebraic equations of inex-1 an their ajoint equations, Results in Maths, Vol.37, pp.12-35, 2. 5. B. Buriek: Generation of optimal test stimuli for nonlinear analog circuits using nonlinear programming an time-omain sensitivities, Proc. DATE 21, Munchen, pp. 63-68. 6. B. Buriek: Zur Berechnung von Testsignalen für nichtlineare analoge Schaltungen unter Verwenung von Methoen er Optimalsteurungstheorie, Shaker, Achen, 25 (PhD Thesis Univ. of Haannover, 25). 7. Y. Cao, S. Li, L. Petzol, R. Serban: Ajoint sensitivity for ifferential-algebraic equations: the ajoint DAE system an its numerical solution, SIAM J. Sci. Comput., Vol. 24-3, pp. 176 189, 22.
Ajoint Transient Sensitivity Analysis 9 8. A.R. Conn, P.K. Coulman, R.A. Haring, G.L. Morrill, C. Visweswariah, C.W. Wu: JiffyTune: circuit optimization using time-omain sensitivities, IEEE Trans. on CAD of ICs an Systems, Vol. 17-12, pp. 1292 139, 1998. 9. L. Daloss, P. Gubian, M. Quarantelli: Multiparameter time-omain sensitivity computation, IEEE Trans. on Circuits an Systems - I: Fun. Theory an Applics, Vol. 48-11, pp. 1296 137, 21. 1. Y. Favennec, M. Girault, D. Petit: The ajoint metho couple with the moal ientification metho for nonlinear moel reuction, Inverse Probl. in Science an Engng., Vol. 14, No. 3, 153 17, 26. 11. D.E. Hocevar, P. Yang, T.N. Trick, B.D. Epler: Transient sensitivity computation for MOS- FET circuits, IEEE Trans. on CAD of Integr. Circuits an Systems, Vol. CAD-4, Nr. 4, pp. 69 62, 1985. 12. L.T. Pillage, R.A. Rohrer, C. Visweswariah: Electronic circuit an system simulation methos, McGraw-Hill, Inc, New York, USA, ISBN 751696, 1994. 13. M. Rathinam, L.R. Petzol: A new look at proper orthogonal ecomposition, SIAM J. Numer. Analysis, Vol. 41-5, pp. 1893 1925, 23. 14. M.J. Rewienski: A trajectory piecewise-linear approach to moel orer reuction of nonlinear ynamical systems, PhD-Thesis, Massachutes Institute of Technology, 23. 15. T. Voss: Moel reuction for nonlinear ifferential algebraic equations, MSc. Thesis University of Wuppertal, 25; Unclassifie Report PR-TN-25/919, Philips Research Laboratories, 25. 16. H. Xu: Transient Sensitivity Analysis in Circuit Simulation, MSc-Thesis, Department of Mathematics an Computing Science, Einhoven University of Technology, 24. Online via author search at Library at http://w3.win.tue.nl/en/]