Decentralized Model Order Reduction of Linear Networks with Massive Ports Boyuan Yan, Lingfei Zhou, Sheldon X.-D. Tan, Jie Chen University of California, Riverside Bruce McGaughy Cadence Design Systems Inc.

Outline Model order reduction Measure of interaction Decentralized model order reduction Examples Conclusion

RLC circuit model RL C State-space model (MNA) Transfer function CG B LTLT C B LTLT G s+Y (s) = ( ) -1 U (s) H (s) Frequency |H(s)|

Model order reduction (MOR) C B LTLT G s+ H (s) = ( ) -1 BrBr LrTLrT GrGr s+ H r (s) = ( ) -1 CrCr G V VTVT GrGr How to pick a projection matrix V C V VTVT CrCr BrBr B VTVT LTLT V LrTLrT such that H (s) ≈ H r (s)To find

V Moment-matching Taylor expansion: M0M0 LTLT s + H (s) = ( ) M1M1 + M2M2 s2s2 … M0M0 M1M1 M2M2 Frequency Magnitude Hr(s) H(s) Projection matrix: V=

Krylov subspace method M0M0 LTLT s + H (s) = ( ) M1M1 + M2M2 s2s2 … K m ( A, R ) = span( A, A 2 R,…A m-1 R) = K m (G -1 C,G -1 B) M0M0 M1M1 M2M2 span ( )… M m-1 Lanzos, Arnoldi: [Feldmann, TCAD 95, Silveira, ICCAD 96, Odabasioglu, TCAD 98] M 0 =G -1 BM 1 =(G -1 C)G -1 BM k =(G -1 C) k G -1 B … Moments formula: Krylov subspace: To find an orthogonal basis V for Krylov subspace:

A fundamental problem M0M0 M1M1 M2M2 V = M0M0 M1M1 M2M2 BB EKS [Wang, DAC 00] Dependent on input SVDMOR [Feldmann, DATE 04, ICCAD 04] Still not compact enough Problem: Centrality Each input-output pair is implicitly assumed to be equally interacted! Explore input information as well as system information: Explore system information only: Existing solutions MOR degrades as the number of inputs increases!

Motivation u1u1 u2u2 y1y1 y2y2 g 22 g 21 g 12 y 1 = g 11 u 1 + g 21 u 2 g 11 G Solution: 1.Introduce some tools to measure interaction 2.A decentralized framework is needed

Outline Model order reduction Measure of interaction Decentralized model order reduction Examples Conclusion

Relative gains u1u1 u2u2 y1y1 y2y2 g 22 g 21 g 12 g 11 C r Open loop gain: Relative gain: If λ 11 =0, y 1 is NOT influenced by u 1 at all Δu1Δu1 Δy 11 Δy 21 Δu2Δu2 g 11 |u 2 = |u 2 Δy 11 Δu1Δu1 Closed loop gain: g 11 |y 2 = |y 2 Δy 11 + Δy 21 Δu1Δu1 g 11 |u 2 g 11 |y 2 λ 11 = + Δy 21 If λ 11 =1, y 1 is influenced by u 1 Only.

Relative gain array (RGA) Choose pairings corresponding to RGA elements close to 1 Columns and rows always sum to 1 u1u1 u2u2 u3u3 y1y1 y2y2 y3y3 Relative gain array: Pairing in decentralized control: u1u1 u2u2 u3u3 y1y1 y2y2 y3y3 [Bristol, IEEE Trans. Automatic Control, 1966]

Scaled RGA Scaled to [0,1]: The closer λ ij is to 1, the more important u j is in terms of y i RGA Output Input Relative gain RGA in terms of one output (one row) Input Relative gain

Computation of RGA A function of frequency Typically evaluated at zero frequency Higher frequency components tend to be more localized! Pseudoinverse is used for non-square system s = 0 Output Input Relative gain s = 1 GHz Output Input Relative gain RGA at DC is conservative and valid at higher frequencies

Outline Model order reduction Measure of interaction Decentralized model order reduction Examples Conclusion

C B G s+ H (s) = ( ) -1 Decentralize BrBr LrTLrT GrGr s+H ir (s) = ( ) -1 CrCr MOR Decentralized MOR framework Partition the output matrix and decentralize the system into a set of subsystems corresponding to each output of interests. LiTLiT LpTLpT L1TL1T : : C B G s+ H i (s) = ( ) -1 LiTLiT Spatial dominant Krylov subspace method LTLT : : : :

C B G s+ H i (s) = ( ) -1 LiTLiT BrBr L ir T GrGr s+H ir (s) = ( ) -1 CrCr K m (G -1 C,G -1 B) Spatial dominant Krylov subspace M0M0 M1M1 M2M2 V = B M0M0 V i = BiBi M1M1 M2M2 BiBi Spatial dominant: K m (G -1 C,G -1 B i ) The moments of dominant inputs are exactly preserved. The energy transfers from other inputs are also coarsely preserved.

Electrical distance and locality RC network can be viewed as a cascaded low-pass RC filter. Far away nodes have little electrical impact on each other because of the attenuation. The voltage response at a node is only dominated by a small number of inputs nearby. RC network Observed node Dominant input Minor input

Principle components in terms of both frequency and space Existing MOR preserves the principle components in terms of frequency (time) only and ignores the other degree of freedom. In decentralized MOR, the principle components are in terms of both frequency (time) and space. As a result, the reduced model can be made much more compact ! Observed node Dominant input Minor input Frequency Magnitude Frequency (time) Space Projection subspace

Outline Model order reduction Measure of interaction Decentralized model order reduction Examples Conclusion

Frequency domain evaluation Original transfer function matrix: Reduced transfer function matrix corresponding to 2 nd output: Instead of comparing each element in the transfer function matrix, we compare the sum of the corresponding row. We hope

A simple RC mesh Node #: 1600 Port #: 33 Reduced order : 7 Dominant input #: 1 Expansion point: 0 Hz

A larger RC mesh Nodes: Ports: 100 Reduced order : 8 Dominant inputs: 2 Expansion point: 0 Hz

An RLC mesh Node #: Port #: 250 Reduced order : 400 Dominant input #: 10 Expansion point: 10 GHz Since RLC circuit is less localized and much more complicated, it is really hard to match the wide- band frequency response!

Outline Model order reduction Measure of interaction Decentralized model order reduction Examples Conclusion

Propose a decentralized model order reduction framework. Introduce relative gain array to evaluate the relative importance of ports. Propose the concept of spatial dominant subspace to create one more degree of freedom in addition to frequency. The proposed method can take advantage of parallel computation in modeling and simulation. The proposed method is more efficient if only a small number of nodes are to be observed.

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