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## Model Order Reduction

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**Model Order Reduction**Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Joel Phillips, Cadence Berkeley Labs Jacob White, Massachusetts Instit. of Technology**Funct. Spec**RTL Behav. Simul. Stat. Wire Model Logic Synth. Gate-level Net. Front-end Gate-Lev. Sim. Floorplanning Back-end Parasitic Extrac. Place & Route Layout Conventional Design Flow**Layout parasitics**• Wires are not ideal. Parasitics: • Resistance • Capacitance • Inductance • Why do we care? • Impact on delay • noise • energy consumption • power distribution Picture from “Digital Integrated Circuits”, Rabaey, Chandrakasan, Nikolic**Parasitic Extraction**thousands of wires e.g. critical path e.g. gnd/vdd grid Parasitic Extraction • identify some ports • produce equivalent circuit that models response of wires at those ports tens of circuit elements for gate level spice simulation**Electromagnetic**Analysis (Tuesday) small surface panels with constant charge thin volume filaments with constant current million of elements Model Order Reduction (Today) tens of elements Parasitic Extraction (the two steps)**Why build reduced models?**• Compression for Efficiency • It is possible to represent the system under study “precisely” with millions of elements • But the simulation is too slow with the complicated representation • Abstraction • I do not care at all about the precise representation • In fact I would rather those details were not even there. I may not be able to create or manipulate the precise representation at all.**Challenges for reduction algorithms**• Accuracy • Must be controllable and predictable • Efficiency • Algorithms should be scalable to handle large systems • Numerical robustness • Algorithms should work reliably for all reasonable inputs & accuracy requests • Models must work in simulation • Composability : Combination of two good models is a good model**Overview**• Introduction and Motivations • State-space models • Reduction via eigenmode analysis • Reduction via rational function fitting (point matching) • Reduction via moment matching (Pade, AWE) • Reduction via moment matching: (Projection Framework) • general Krylov Subspace methods • case 1: Arnoldi • case 2: PVL • case 3: multipoint moment matching • Importance of preserving passivity • PRIMA**State-Space Models**• Linear system of ordinary differential equations (ABCD form) Input State Output**State-Space Model Example:Interconnect Segment**• Step 1: Identify internal state variables • Example : MNA uses node voltages & inductor current**+**+ - - State-Space Model Example:Interconnect Segment • Step 2: Identify inputs & outputs • Example : For Z-parameter representation, choose port currents inputs and port voltage outputs**+**+ - - State-Space Model Example:Interconnect Segment • Step 3: Write state-space & I/O equations • Example : KCL + inductor equation**State-Space Model Example:Interconnect Segment**• Step 4: Identify state variables & matrices**LARGE!**A linear circuit can be expressed as a state space model • So in general….**A canonical form for model order reduction**Assuming A is non-singular we can cast the dynamical linear system into one canonical form for model order reduction Note: not necessarily always the best, but the simplest for educational purposes**Construct a linear system model with:**• smaller complexity • same fidelity • small reduction cost 10 x 10 Our goal: smaller model, still accurate • Given a large linear system model: 500,000 x 500,000**Frequency Domain Representation**Bilateral Laplace Transform: Key Transform Property:**Transfer Function:**System Transfer Function Express y(s) as a function of u(s)**Connection Between the Transfer FunctionTime Domain Impulse**Response Frequency domain representation u(s) y(s) = H(s) u(s) H(s) Linear system Time domain representation u(t) h(t) Linear system The transfer function H(s) is the Laplace Transform of the impulse response h(t)**Overview**• Introduction and Motivations • State-space models • Reduction via eigenmode analysis • Reduction via rational function fitting (point matching) • Reduction via moment matching (Pade, AWE) • Reduction via moment matching: (Projection Framework) • general Krylov Subspace methods • case 1: Arnoldi • case 2: PVL • case 3: multipoint moment matching • Importance of preserving passivity • PRIMA**Model Order Reduction via Eigenmode Analysis**Pole-Residue Form Pole-Zero Form (SISO) • Ideas for reducing order: • Drop terms with small residues • Drop terms with large (“fast” modes) • Remove pole/zero near-cancellations • Cluster poles that are “together” • How to compute poles and residues?**Diagonalize E**Computing Poles & Residues • Poles are eigenvalues of E-1 residues poles**Eigenvalue Based Reduction**• Advantages • Conceptually familiar • Simple physical interpretation : retains dominant system modes/poles • Drawbacks • Relatively expensive : have to find the poles first • Relatively inefficient. For a given model size, many other approaches can provide better accuracy • Rule of thumb • Anything that can be done by manipulating pole/eigenvalues/eigenvectors can probably be done better with more sophisticated analysis, at the same or smaller cost.**Defining Accuracy**• Time-domain response should be “close” • For which possible inputs? • Frequency response should match • At what frequencies?**Matching Frequency Response**• Ensure accuracy for only some inputs? • Example: • low frequency inputs, • or some band, • or some points in the frequency response matching some part of the frequency response Original**Overview**• Introduction and Motivations • State-space models • Reduction via eigenmode analysis • Reduction via rational function fitting (point matching) • Reduction via moment matching (Pade, AWE) • Reduction via moment matching: (Projection Framework) • general Krylov Subspace methods • case 1: Arnoldi • case 2: PVL • case 3: multipoint moment matching • Importance of preserving passivity • PRIMA**Model Order Reduction via Rational Transfer Function Fitting**Original System Transfer Function: rational function Model Reduction = Find a low order (q << N) rational function matching reduced order rational function**Rational Transfer Function Fitting: Degrees of Freedom**Reduced Model Dynamical System coefficients Reduced Model Transfer Function coefficients**Rational Transfer Function Fitting: Degrees of Freedom**(cont.) Reduced Model Transfer Function Apply any invertible change of variables to the state I I Many Dynamical Systems have the same transfer function!!**Can match 2q points**• cross multiplying generates a linear system For i = 1 to 2q Rational Transfer Function Fitting: via Point Matching**Rational Transfer Function Fitting: Point Matching matrix**can be ill-conditioned • Columns contain progressively higher powers of the test frequencies: problem is numerically ill-conditioned • also... missing data can cause severe accuracy problems**Overview**• Introduction and Motivations • State-space models • Reduction via eigenmode analysis • Reduction via rational function fitting (point matching) • Reduction via moment matching (Pade, AWE) • Reduction via moment matching: (Projection Framework) • general Krylov Subspace methods • case 1: Arnoldi • case 2: PVL • case 3: multipoint moment matching • Importance of preserving passivity • PRIMA**Point matching vs. Moment Matching**Point matching: can be very inaccurate in between points Moment (derivatives) matching: accurate around expansion point, but inaccurate on wide frequency band**Frequency Domain "Moments" (or Taylor coefficients) of the**transfer function Taylor Series Expansion of the original transfer function around s=0 The Taylor coef. = frequency domain moments = = derivatives of the transfer function (up to a constant)**Time domain moments of the impulse response**Definition:**Compare:**Hence the the Taylor coeff. are, up to a constant, the time-domain moments of the circuit response. Connection to the time-domain moments of the circuit response Time-domain moments**Rational function fitting via moment matching: Pade**Approximation (AWE)**Rational function fitting via moment matching: Pade**Approximation (AWE) • Step 1: calculate the first 2q moments of H(s) • Step 2: calculate the 2q coeff. of the Pade’ approx, matching the first 2q moments of H(s)**Step 2: Calculation of Pade’ coeff. (AWE)**For coeff. a’s solve the following linear system: For coeff. b’s simply calculate:**Pade matrix can be very ill-conditioned**• matrix powers converge to the eigenvector corresponding to the largest eigenvalue. Columns become linearly dependent for large q the problem is numerically very ill-conditioned!**Overview**• Introduction and Motivations • State-space models • Reduction via eigenmode analysis • Reduction via rational function fitting (point matching) • Reduction via moment matching (Pade, AWE) • Reduction via moment matching: (Projection Framework) • general Krylov Subspace methods • case 1: Arnoldi • case 2: PVL • case 3: multipoint moment matching • Importance of preserving passivity • PRIMA**Projection Framework:Change of variables**reduced state Note: q << N original state**Projection Framework**• Original System • Substitute • Note: now few variables (q<<N) in the state, but still thousands of equations (N)**Projection Framework (cont.)**• Reduction of number of equations: test multiplying by VqT • If V and U biorthogonal**Projection Framework (cont.)**qxn qxq nxn nxq**Projection Framework**Change of variables Equation Testing**Approaches for picking V and U**• Use Eigenvectors • Use Time Series Data • Compute • Use the SVD to pick q < k important vectors • Use Frequency Domain Data • Compute • Use the SVD to pick q < k important vectors • Use Singular Vectors of System Grammians? • Use Krylov Subspace Vectors?**Overview**• Introduction and Motivations • State-space models • Reduction via eigenmode analysis • Reduction via rational function fitting (point matching) • Reduction via moment matching (Pade, AWE) • Reduction via moment matching: (Projection Framework) • general Krylov Subspace methods • case 1: Arnoldi • case 2: PVL • case 3: multipoint moment matching • Importance of preserving passivity • PRIMA**Intuitive view of Krylov subspace choice for change of base**projection matrix Taylor series expansion: • change base and use only the first few vectors of the Taylor series expansion: equivalent to match first derivatives around expansion point U**Combine point and moment matching: multipoint moment**matching • Multipole expansion points give larger band • Moment (derivates) matching gives more • accurate behavior in between expansion points