Symbolic Model Reduction for Linear and Nonlinear DAEs

Transcription

1 Symbolic Model Reduction for Linear and Nonlinear DAEs Symposium on Recent Advances in MOR TU Eindhoven, The Netherlands November 23rd, 2007 Thomas Halfmann

2 Overview Introduction to symbolic analysis and approximation motivation of symbolic methods principles of symbolic simplification approximation methods for linear systems nonlinear symbolic model generation EDA tool Analog Insydes platform overview Methodology transfer mechatronics Slide 2

3 Symbolic Methods VDD M8 M5 M7 IB M1 M2 vin M9 CC Vout CL numerical simulation M3 M4 M6 VSS symbolic analysis Slide 3

4 Design Flow f i i Phase comparator Loop filter F(s) A=1 system level validation / redesign (on all levels) VCO block level Ctrl. logic f o IN VDD VSS OUT transistor level Application fields of symbolic methods: error analysis gaining system understanding specification transfer between levels automated behavioral model generation multi-physics modeling and system simulation IN+ IN- VDD OUT VSS Slide 4

5 Symbolic Analysis Folded-Cascode OpAmp Problem What causes resonance at 10 MHz? Classic approach parameter variations and numerical simulation failed because of huge number of parameters Symbolic analysis calculation of the transfer function and derivation of a symbolic formula for the resonance frequency Complexity problem exact symbolic transfer function consists of more than terms printed: paper stack with a height of miles differential order: 19 Slide 5

6 Symbolic Analysis Folded-Cascode OpAmp Problem What causes resonance at 10 MHz? Classic approach parameter variations and numerical simulation failed because of huge number of parameters Symbolic analysis calculation of the transfer function and derivation of a symbolic formula for the resonance frequency Complexity problem symbolic approximation Slide 6

7 Numerical Validation first design design after symbolic analysis Slide 7

8 Complexity Problem Text book: V out RC = R E Exact solution using computer algebra 132 Terms Slide 8

9 Symbolic Approximation: Basic Idea Netlist R 1 = R 2 = 10 Ω R 3 = R 4 = 1000 Ω Symbolic analysis: exact transfer function Numerical evaluation Transfer function after symbolic approximation RR 2 4 RR + RR + RR + R R + R R RR 2 4 ( R + R )( R + R ) relative error: Slide 9

10 Symbolic Approximation: Basic Idea Netlist R 1 = R 2 = 10 Ω R 3 = R 4 = 1000 Ω Symbolic analysis: exact transfer function Numerical evaluation Transfer function after symbolic approximation Problem: RR 2 4 RR + RR + RR + R R + R R RR 2 4 ( R + R )( R + R ) relative error: Exact symbolic transfer function can not be calculated for real world problems. Slide 10

11 Symbolic Approximation: Ranking symbol position relative error netlist -R 2 (1, 2) 2.488E-3 R 2 (2, 2) 5.013E-3 R 3 R 4 -R 2 (2, 2) (2, 2) (2, 1) 9.950E E E+0 equations R1 + R 2 R 2 i1 1 = + + R 2 R 2 R3 R 4 i 2 0 R 1 R 2 (1, 1) (1, 1) 1.000E E+0 numeric result R 1 = R 2 = 10 Ω R 3 = R 4 = 1000 Ω i 2 = A Slide 11

12 Symbolic Approximation: Ranking symbol position relative error netlist -R 2 (1, 2) 2.488E-3 R 2 (2, 2) 5.013E-3 R 3 R 4 -R 2 (2, 2) (2, 2) (2, 1) 9.950E E E+0 equations R1 + R 2 R 2 i1 1 = + + R 2 R 2 R3 R 4 i 2 0 R 1 R 2 (1, 1) (1, 1) 1.000E E+0 numeric result R 1 = R 2 = 10 Ω R 3 = R 4 = 1000 Ω i 2 = A cumulative error Slide 12

13 Symbolic Approximation: Ranking symbol position relative error netlist -R 2 (1, 2) 2.488E-3 R 2 (2, 2) 5.013E-3 R 3 R 4 -R 2 (2, 2) (2, 2) (2, 1) 9.950E E E+0 equations R1 + R 2 R 2 i1 1 = + + R 2 R 2 R3 R 4 i 2 0 R 1 R 2 (1, 1) (1, 1) 1.000E E+0 numeric result R 1 = R 2 = 10 Ω R 3 = R 4 = 1000 Ω i 2 = A cumulative error Slide 13

14 Symbolic Approximation: Ranking symbol position relative error netlist -R 2 R 2 R 3 R 4 -R 2 (1, 2) (2, 2) STOP (2, 2) (2, 2) (2, 1) 2.488E E E E E+0 equations R1 + R 2 R 2 i1 1 = + + R 2 R 2 R3 R 4 i 2 0 R 1 R 2 (1, 1) (1, 1) 1.000E E+0 numeric result R 1 = R 2 = 10 Ω R 3 = R 4 = 1000 Ω i 2 = A cumulative error Slide 14

15 Symbolic Approximation: Ranking symbol position relative error netlist -R 2 (1, 2) 2.488E-3 R 2 (2, 2) 5.013E-3 R 3 R 4 -R 2 (2, 2) (2, 2) (2, 1) 9.950E E E+0 equations R1 + R 2 R 2 i1 1 = + + R 2 R 2 R3 R 4 i 2 0 R 1 R 2 (1, 1) (1, 1) 1.000E E+0 numeric result R 1 = R 2 = 10 Ω R 3 = R 4 = 1000 Ω i 2 = A cumulative error Slide 15

16 Symbolic Approximation: Ranking symbol position relative error netlist -R 2 (1, 2) 2.488E-3 R 2 (2, 2) 5.013E-3 R 3 R 4 -R 2 (2, 2) (2, 2) (2, 1) 9.950E E E+0 equations R1 + R 2 R 2 i1 1 = + + R 2 R 2 R3 R 4 i 2 0 R 1 R 2 (1, 1) (1, 1) 1.000E E+0 symbolic approximated transfer function i 2 = RR 2 4 ( R + R )( R + R ) Slide 16

17 Overview Symbolic Simplification Methods Design task Linear equations Nonlinear equations AC, P/Z, Noise Simplified equations Transfer functions automatic error control AC, DC, Transient, TEMP Simplified equations Insights ABM generation Insights Slide 17

18 Linear Symbolic Analysis Symbolic methods Starting from netlist description Standard analysis techniques (MNA, STA) for setting up analytic system equations Slide 18

19 Linear Symbolic Analysis V out RC = R E Simplification methods SBG (Simplification Before Generation) matrix approximation based on Sherman-Morrison formula for error prediction SAG (Simplification After Generation) approximation of transfer function in canonical SOP form to equal coefficient error Pole/Zero techniques Slide 19

20 Mathematical Formulation Small-signal behavior can mathematically be described by a linear system of equations: With: formulated in Laplace frequency s coefficients of A internal variables parameters output component ( ) ˆ A s; p x= b( s; p) a = a ( p) s ij xˆ R p= ( p, L, p ) y = xˆ k n 1 l N l Slide 20

21 Linear Symbolic Analysis: Benefits and Limits Benefits Limits support for symbolic computation of transfer functions, etc. support for symbolic pole/zero analyses application of industrial-sized circuits (up to 100 transistors) in interactive computation time advanced research area: efficient and sophisticated algorithms available symbolic analysis is a tradeoff between accuracy and complexity symbolic extraction of poles and zeros possible for transfer functions of order 3 only Slide 21

22 Nonlinear Symbolic Analysis netlist description reference simulation original simplified index control error control original DAE system AHDL model simplified DAE system Slide 22

23 Mathematical Formulation System behavior can mathematically be described by a set of differential algebraic equations (DAE system): With: differential part algebraic part inputs internal variables outputs parameters F( f, g) f g ( x( t), x& ( t), y( t), u( t); p) ( x( t), y( t), u( t); p) = 0 u = ( u1, L, um) x= ( x1, L, xn ) y = ( y1, L, yl ) p= ( p, L, p ) 1 N = 0 Slide 23

24 Numerical Analyses Transient analysis dynamic behavior nonlinear DAE system 2 1 t DC/DT analysis static behavior nonlinear equation system AC analysis linear behavior linear equation system in frequency domain 1 2 M agnitude B d P hase g e E0 1.0E2 1.0E4 1.0E6 1.0E8 Frequency d1.0e0 1.0E2 1.0E4 1.0E6 1.0E8 1.0E0 1.0E2 1.0E4 1.0E6 1.0E E0 1.0E2 1.0E4 1.0E6 1.0E8 Frequency v out Vin Slide 24

25 Symbolic Model Reduction DAE F DAE G v F = A( F, u) E( v, F vg ) < ε v G = A( G, u) Specifications inputs u, outputs v numerical analyses A Error control error function E error bound ε Goal find DAE system G with reduced complexity and defined accuracy Slide 25

26 Symbolic Model Reduction DAE F R1 R 2 R k DAE G v F = A( F, u) E( v, F vg ) < ε v G = A( G, u) Specifications inputs u, outputs v numerical analyses A Error control error function E error bound ε Goal find DAE system G with reduced complexity and defined accuracy Simplification process iterative application of reduction techniques R Slide 26

27 Symbolic Reduction Techniques (1) x 0 = = f ( y) g( x, y) 0 = ( f ( y y) g ), Algebraic manipulation elimination of variables removal of independent blocks of equations Slide 27

28 Symbolic Reduction Techniques (2) f (x) = f1( x) f2( x) f ( ) 3 x a x < a x x > b b f ( x ) = f 2( x ) für alle x Branch reduction detection and removal of unused branches of piecewise-defined functions Slide 28

29 Slide 29 Symbolic Reduction Techniques (3) 0 ) ( : 1 = = N i i j x t F 0 ( ~ ) : 1 = = N k i i i j x t G Term reduction replace terms of equations with zero applicable to nested expressions

30 Slide 30 Symbolic Reduction Techniques (4) 0 ) ( : 1 = = N i i j x t F 0 ( ~ ) : 1 = + = κ N k i i i j x t G Term substitution replace terms of equations with constant value applicable to nested expressions

31 Algorithm DAE system reference simulation perform reductions reject error analysis no error ok? yes simplified DAE system Slide 31

32 Ranking accepted arbitrary order: 59 reductions rejected optimized order: 64 reductions Definition Ranking is the estimation of the influence of each reduction on the output behavior and a suitable ordering. Advantages increased number of reductions decreased computation time Implementation accurate estimation simple and efficient computation tradeoff between accuracy and efficiency Slide 32

33 Clustering no clustering: 65 iterations, 26.2 s with clustering: 5 iterations, 3.0 s Definition Clustering is the bundling of reduction steps with similar influence on the output behavior. Advantages many simultaneous reduction steps reduced effort for error analyses highly decreased computation time Implementation ranking data can be used for clustering Slide 33

34 Index Monitor Definition Index k of a DAE system is a measure for its distance to a regular ODE system. For k >1 the numerical solving is an ill-posed problem. Advantages monitoring of the index during model reduction assuring numerical stability of reduced system Implementation many index concepts not appropriate: algebraic/symbolic due to complexity, structural/graph-based due to modified topology numerical computation of tractability index and strangeness index Slide 34

35 Original Algorithm DAE system reference simulation perform reductions reject error analysis no error ok? yes simplified DAE system Slide 35

36 Improved Algorithm DAE system compute ranking (re)compute clustering reference simulation perform reductions reject error analysis, index analysis no error ok? index ok? yes simplified DAE system Slide 36

37 Example Application: Operational Amplifier 11 Q2N2907A Input Q7 Q2N2222 RF1 in 1 1k Vsig 0 Q Q2N2907A Q8 Q2N2222 Q2 I I4 Q2N2907A 5 Q2N2907A Q3 0 Comp 30p Q4 Q5 7 Q2N2222 Output 8 R R2 20 VB 10 6 Q2N2907A Q6 I3 12 VDB VCC 0 VDD 8 bipolar transistors 7 sources 5 resistors 1 capacitor 0 RF2 10k RL 100k Task computation of a behavioral model 0 input: V sig, output: V 9 Slide 37

38 Example Application: Operational Amplifier DT specification error bound error function ε E DT DT = 0.25V = Transient specification error bound error function ε E Tran Tran = 1.0V = Slide 38

39 Example Application: Operational Amplifier Original system 73 equations 350 terms 94 parameters Slide 39

40 Example Application: Operational Amplifier Symbolic reduction 73 6 equations terms parameters Model reduction time 792 s Slide 40

41 Example Application: Operational Amplifier Symbolic reduction 73 6 equations terms parameters Behavioral model original: s model: 2.3 s Slide 41

42 Infineon Design Problem: Folded-cascode OpAmp Application cfcamp original model simplified model equations top-level terms parameters factor % % % % % 94 5 % 137 automatic generation of behavioral models (e.g. VHDL-AMS, Verilog-A) system simulation, optimization, system understanding Slide 42

43 Nonlinear Symbolic Analysis: Benefits and Limits Benefits Limits Synergies support for automated generation of behavioral models for different behavioral modeling languages and simulators automatic and error-controlled complexity reduction supported analysis modes: DC-Transfer, AC, Transient limited circuit size (currently up to 20 transistors) algorithms under development explicit symbolic results for static systems in general not possible due to the general mathematical concept the methods are also applicable to other domains (e.g. system simulation of mechatronical systems) Slide 43

44 EDA Tool Analog Insydes Tool for analysis, modeling, and optimization of analog circuits Commercially distributed since 1998 Current version: Mathematica Add-on Evaluation version available at: Slide 44

45 EDA Tool Analog Insydes Mathematica Kernel High-performance MathLink Binaries Online Help Mathematica Frontend Interfaces: PSpice, Eldo, Saber, Spectre, Titan Symbolic Model Library Slide 45

46 MOR Techniques Current Activities bottom-up behavioral modeling Analog Insydes Device Model Library Circuit Schematic Netlist Import Setup of Circuit Equations Behavioral Model AHDL AHDL Import netlist-based MOR coupling of numeric/symbolic MOR Model Approximation Performance Optimization model optimization device model compilation Device Model SPICE / CMI Model Export Analytic Model AHDL Simulation Involved in public project initiatives: VeronA SymTecO SyreNe Slide 46

47 Transfer of Methodology Enzyme kinetics interface between Analog Insydes and PathwayLab (simulator for enzyme kinetics) Hydraulics modeling of gas pipeline networks Energy distribution networks model reduction for high-voltage energy distribution networks with dispersed power generation Slide 47

48 Transfer of Methodology: Mechatronical Systems Usage of analogies for methodology transfer analogies electronics mechanics through variables current force, torque across variables voltage rotation, displacement equation setup Kirchhoff laws d Alemberts principle Adding support for mechanics Decompose mechanical system in finite elements Implement elements in terms of symbolic component models Support for vector-valued variables Automated mapping to an equivalent netlist with scalar-type nodes and branches Slide 48

49 Application Example: Symbolic Analysis of Mechanical System Parameters: density elasticity shear modulus length A1 length A2 height A1,A2 length B height B width A1, A2, B ρ E G l A1 l A2 h A l B h B w System consists of silicon beams: l A2 = 12 μm l B = 250 μm Goal: Find symbolic formula for displacement as a function of force l A1 = μm Slide 49

50 Application Example: Symbolic Analysis of Mechanical System Step 1: Analyze the system numerically in order to find out a suitable discretisation. Therefore we compare the dynamics with different discretisations. Step 2: Derive the symbolic model equations for the static model from dynamic model. Goal: Find symbolic formula for displacement as a function of force Step 3: Apply symbolic model reduction in order to reduce the static model and solve the reduced symbolic model equations for displacement in C3. Step 4: against the solution of the full model. Validate the derived solution numerical Slide 50

51 Result Validation l l + 2l u = F Ehw l l 3 A1 A1 A2 3 A 2 A1+ A2 Compare reduced against full model: Slide 51

52 Result Validation l l + 2l u = F Ehw l l 3 A1 A1 A2 3 A 2 A1+ A2 Compare reduced against full model: Slide 52

53 Result Validation l l + 2l u = F Ehw l l 3 A1 A1 A2 3 A 2 A1+ A2 Compare reduced against full model: h B l B Slide 53

54 Application Example: Acceleration Sensor Simple mechatronical system Acceleration sensor with electronic circuit attached to a point mass Force F accelerates the system Acceleration sensor component Three metal plates forming series connection of two capacities Movable center plate with mass m dyn and Hooks law In case of acceleration Plate moves Change of capacities Voltage drop at V out Slide 54

55 Application Example: Model Reduction Analog Insydes command CancelTerms approximates the DAE system: Slide 55

56 Application Example: Model Reduction Analog Insydes command CancelTerms approximates the DAE system Slide 56

57 Application Example: Reduced Model Equations Post processing with Analog Insydes command CompressDAE: Reduced equation system 5 equations Comparison of simulation results original model reduced model Slide 57

58 Summary and Outlook Summary Application of symbolic analysis only possible using approximation techniques coupled with numeric simulation Linear symbolic analysis used successfully for industrial circuit analysis tasks Nonlinear symbolic analysis under development with promising results Outlook Extension of nonlinear simplification methods Extension towards support for multi-physics systems Integration of symbolic analysis into industrial design flows Slide 58