A more reliable reduction algorithm for behavioral model extraction
Dmitry Vasilyev, Jacob White
Massachusetts Institute of Technology
Outline
Background Projection framework for model reduction Balanced Truncation algorithm and
approximations AISIAD algorithm
Description of the proposed algorithm
Modified AISIAD and a low-rank square root algorithm
Efficiency and accuracy
Conclusions
Model reduction problem
• Reduction should be automatic • Must preserve input-output properties
Many (> 104) internal states
inputs outputs
few (<100) internal states
inputs outputs
Differential Equation Model
Model can represent: Finite-difference spatial discretization of PDEs Circuits with linear elements
A – stable, n x n (large)E – SPD, n x n
- state
- vector of inputs
- vector of outputs
Model reduction problem
n – large(thousands)!
Need the reduction to be automatic and preserve input-output properties (transfer function)
q – small (tens)
Approximation error Wide-band applications: model should have
small worst-case error
ω
=> maximal difference over all frequencies
Projection framework for model reduction
Pick biorthogonal projection matrices W and V
Projection basis are columns of V and W
Vxr x
x
n x xrV q
WTAVxr
Ax
Most reduction methods are based on projection
LTI SYSTEM
X (state)
tu
t
y
input output
P (controllability)Which modes are easier to reach?
Q (observability)Which modes produce more output? Reduced model retains
most controllable and most observable modes
Mode must be both very controllable and very observable
Projection should preserve important modes
Reduced system: (WTAV, WTB, CV, D)
Compute controllability and observability
gramians P and Q :
(~n3)AP + PAT + BBT =0 ATQ + QA + CTC = 0
Reduced model keeps
the dominant eigenspaces of PQ : (~n3)
PQvi = λivi wiPQ = λiwi
Balanced truncation reduction (TBR)
Very expensive. P and Q are dense even for sparse models
• Arnoldi [Grimme ‘97]:V = colsp{A-1B, A-2B, …}, W=VT , approx. Pdom only
• Padé via Lanczos [Feldman and Freund ‘95]colsp(V) = {A-1B, A-2B, …}, - approx. Pdom colsp(W) = {A-TCT, (A-T )2CT, …}, - approx. Qdom
• Frequency domain POD [Willcox ‘02], Poor Man’s TBR [Phillips ‘04]
Most reduction algorithms effectively separately approximate dominant eigenspaces of P and Q :
However, what matters is the product PQ
colsp(V) = {(jω1I-A)-1B, (jω2I-A)-1B, …}, - approx. Pdom
colsp(W) = {(jω1I-A)-TCT, (jω2I-A)-TCT, …}, - approx. Qdom
RC line (symmetric circuit)
Symmetric, P=Q all controllable states are observable and vice
versa
V(t) – inputi(t) - output
RLC line (nonsymmetric circuit)
P and Q are no longer equal! By keeping only mostly controllable
and/or only mostly observable states, we may not find dominant eigenvectors of PQ
Vector of states:
Lightly damped RLC circuit
Exact low-rank approximations of P and Q of order < 50 leads to PQ ≈ 0!!
R = 0.008, L = 10-5
C = 10-6
N=100
Lightly damped RLC circuit
Union of eigenspaces of P and Qdoes not necessarily approximate
dominant eigenspace of PQ .
Top 5 eigenvectors of P Top 5 eigenvectors of Q
AISIAD model reduction algorithm
Idea of AISIAD approximation:Approximate eigenvectors using power iterations:
Vi converges to dominant eigenvectors of PQ
Need to find the product (PQ)Vi
Xi = (PQ)Vi => Vi+1
= qr(Xi)
“iterate”
How?
Approximation of the product Vi+1 =qr(PQVi), AISIAD algorithm
Wi ≈ qr(QVi) Vi+1
≈ qr(PWi)
Approximate using solution of Sylvester equation
Approximate using solution of Sylvester equation
More detailed view of AISIAD approximation
Right-multiply by Wi
X X H, qxq (original AISIAD)
M, nxq
X X H, qxq
Modified AISIAD approximation
Right-multiply by Vi
Approximate!
M, nxq
^
Modified AISIAD approximation
Right-multiply by Vi
We can take advantage of numerous methods, which approximate P and Q!
X X H, qxqApproximate!
M, nxq
^
n x qn x n
Specialized Sylvester equation
A X + X H =-M
q x q
Need only column span of X
Solving Sylvester equation
Schur decomposition of H :
A X + X =-M~ ~
Solve for columns of X~
~
X
Solving Sylvester equation
Applicable to any stable A
Requires solving q times
Schur decomposition of H :
Solution can be accelerated via fast MVPAnother methods exists, based on IRA, needs A>0 [Zhou ‘02]
Solving Sylvester equation
Applicable to any stable A
Requires solving q times
Schur decomposition of H :
For SISO systems and P=0 equivalent to matching at frequency points –Λ(WTAW)
^
Modified AISIAD algorithm
1.Obtain low-rank approximations of P and Q2.Solve AXi +XiH + M = 0, => Xi≈ PWi
where H=WiTATWi, M = P(I - WiWi
T)ATWi + BBTWi
3. Perform QR decomposition of Xi =ViR
4. Solve ATYi +YiF + N = 0, => Yi≈ QVi
where F=ViTAVi, N = Q(I - ViVi
T)AV + CTCVi
5.Perform QR decomposition of Yi =Wi+1 R to get new
iterate. 6.Go to step 2 and iterate.7.Bi-orthogonalize W and V and construct reduced
model:(WTAV, WTB, CV, D)
LR-sqrt^ ^
^
^
For systems in the descriptor form
Generalized Lyapunov equations:
Lead to similar approximate power iterations
mAISIAD and low-rank square root
Low-rank gramians
LR-square root
mAISIAD
(inexpensive step) (more expensive)
For the majority of non-symmetric cases, mAISIAD works better than low-rank square root
(cost varies)
RLC line example results
H-infinity norm of reduction error (worst-case discrepancy over all frequencies)
N = 1000,1 input
2 outputs
Steel rail coolling profile benchmark
Taken from Oberwolfach benchmark collection, N=1357 7 inputs, 6 outputs
mAISIAD is useless for symmetric models
For symmetric systems (A = AT, B = CT) P=Q, therefore mAISIAD is equivalent to LRSQRT for P,Q of order q
RC line example
^ ^
Cost of the algorithm
Cost of the algorithm is directly proportional to the cost of solving a linear system:
(where sjj is a complex number)
Cost does not depend on the number of inputs and outputs
(non-descriptor case)
(descriptor case)
Conclusions The algorithm has a superior accuracy and
extended applicability with respect to the original AISIAD method
Very promising low-cost approximation to TBR
Applicable to any dynamical system, will work (though, usually worse) even without low-rank gramians
Passivity and stability preservation possible via post-processing
Not beneficial if the model is symmetric