Manifold Construction and Parameterization for ... - ASP-DAC 2018
Manifold Construction and Parameterization for Nonlinear Manifold-Based Model Reduction Chenjie Gu and Jaijeet Roychowdhury {gcj,jr}@eecs.berkeley.edu
University of California, Berkeley
ASPDAC 2010
Slide 1
Outline ●
Background Introduction to MOR and maniMOR ● Manifold construction and parameterization ●
●
Manifold construction using integral curves DC manifold and the normalized integral curve equation ● Ideal and almost-ideal manifold ● Algorithm ●
Experimental results ● Conclusion ●
ASPDAC 2010
Slide 2
Background
ASPDAC 2010
Slide 3
Model Order Reduction Original system (size n)
d~x = f (~x) + B~ u(t) dt
~ u(t)
v : ~x 7! ~ z;
~ x 2 Rn ; ~z 2 Rq ;
y = C~x ~
q¿n
Reduced system (size q)
~ u(t) ASPDAC 2010
d~z = fr (~ z ) + Br ~ u(t) dt
~y = Cr ~z Slide 4
Low-order Linear Subspace 2
3
2
x ¡10 d 4 1 5 4 x2 1 = dt x 1 3
1 ¡1 0
32
3
2
3
1 x1 1 0 5 4 x2 5 + 4 0 5 u(t) ¡1 x3 0
Low-order linear subspace Defined by
ASPDAC 2010
x=Vz
Slide 5
Low-order Nonlinear Manifold 2
3
2
x ¡10 d 4 1 5 4 x2 1 = dt x 1 3
1 ¡1 0
32
3
2
3 2
3
1 x1 0 1 0 5 4 x2 5 ¡ 4 0 5+ 4 0 5 u(t) ¡1 x3 x21 0
Low-order nonlinear manifold
ManiMOR: MOR Based on Nonlinear Projection on Nonlinear Manifolds ASPDAC 2010
Slide 6
Key Steps in ManiMOR ●
“Find” the nonlinear manifold ●
●
Capture important dynamics
“Parameterize” the manifold ●
Build up the coordinate system
ASPDAC 2010
Slide 7
Manifold and Its Parameterization 8 < x = cos(t) y = sin(t) : z=t
ASPDAC 2010
Slide 8
Manifold and Its Parameterization
M
Parameterization of the manifold
Tangent space Tx M ½ Rn
U
x
à v
Rq
~ U
z
System of coordinates Manifold defined by pairs of fx; Tx M g ASPDAC 2010
No explicit mapping may be derived. Instead, use piecewise linear approximation. Slide 9
Manifold and Its Parameterization 1. Identify the manifold that capture important dynamics 2. Compute and store pairs of fx; Tx M g=fz; Tz M g
ASPDAC 2010
Slide 10
DC Manifold DC operating points constitute a DC manifold. d~ x = f (~ x) + B~ u(t) = 0 dt
How to compute and parameterize the DC manifold? ASPDAC 2010
Slide 11
DC Manifold f (~ x) + B~ u(t) = 0
A straight-forward solution:
Computation: Perform DC sweep analysis Parameterization: Define z coordinates using values of
u
Problems: Hard to choose step size in DC sweep analysis Not generalizable to higher dimensions
ASPDAC 2010
Slide 12
Introduction to Integral Curve Given a vector field v(x) , dx = v(x) its integral curve is the curve ° ´ x(t) such that dt
ASPDAC 2010
Slide 13
DC Manifold as an Integral Curve Need to derive the relationship between dx and du f (~ x) + B~ u(t) = 0
@f dx +B =0 @x du
dx = ¡[G(x)]¡1 B du
The first Krylov basis.
Initial condition: x(u = 0) = xDC ju=0 Solutions are DC operating points.
Any numerical integration / transient analysis code can be applied. ASPDAC 2010
Slide 14
Parameterization using Euclidean Distance Parameterization using values of u (x2 ; y2 ) (x1 ; y1 )
(x3 ; y3 ) u0 + 2h
u0 + h
u0
Parameterization using Euclidean Distance (x3 ; y3 ) (x2 ; y2 ) u0 + h
(x1 ; y1 )
u0 + 2h
u0 Sample points equally spaced on the DC manifold ASPDAC 2010
Slide 15
Parameterization using Euclidean Distance
ASPDAC 2010
Slide 16
Normalized Integral Curve Equation Local Euclidean distance is
jjdxjj2 = jduj
dx = ¡[G(x)]¡1 B du
¯¯ ¯¯ ¯¯ dx ¯¯ ¯¯ ¯¯ = jj[G(x)]¡1 Bjj2 = 1 Generally not satisfied ¯¯ du ¯¯ 2 Normalize RHS
dx [G(x)]¡1 B = du jj[G(x)]¡1 Bjj2 ASPDAC 2010
Normalized Integral Curve Equation
Does it define the same integral curve? Slide 17
Validation
ASPDAC 2010
Slide 18
Normalized Integral Curve Equation Theorem: ^(¿ ) satisfy Suppose t = ¾(¿ ); x(t) and x
d d x(t) = g(x(t)) and x ^(¿ ) = ¾0 (¿ )g(^ x(¿ )) , respectively. dt d¿ ^(¿ ) span the same state space. Then x(t) and x Sketch of proof: 0 dt = ¾ (¿ )d¿ . t = ¾(¿ ) Since , we have
^(¿ ) ´ x(t) = x ^(¾(t)) , then Define x
d d^ x(¿ ) dt x ^(¿ ) = = ¾ 0 (¿ )g(x(t)) = ¾0 (¿ )g(^ x(¿ )) d¿ dt d¿
ASPDAC 2010
Slide 19
Normalized Integral Curve Equation dx = ¡[G(x)]¡1 B du
dx [G(x)]¡1 B = du jj[G(x)]¡1 Bjj2
Solution: x(u)
^(^ u) Solution: x
Define u = ¾(^ u) =
Z
0
u ^
1 d¹ jj[G(^ x(¹))]¡1 Bjj2
From the theorem, x(u) and x ^(^ u) define the same integral curve. ASPDAC 2010
Slide 20
Normalized Integral Curve Equation dx [G(x)]¡1 B = du jj[G(x)]¡1 Bjj2
The first normalized Krylov basis.
Directly available from Krylov subspace methods. Generalizable to higher dimensions.
ASPDAC 2010
Slide 21
Ideal Nonlinear Manifold @x = v1 (x); @z1
@x = v2 (x); @z2
¢¢¢ ;
@x = vq (x): @zq
V (x) = [v1 (x); ¢ ¢ ¢ ; vq (x)] is the projection matrix
for the reduced linearized system (at x ).
For example, Arnoldi algorithm generates a basis for Kq ([G(x)]¡1 ; B) = f[G(x)]¡1 B; [G(x)]¡2 B; ¢ ¢ ¢ ; [G(x)]¡q Bg
However, this set of PDEs is over-determined. ASPDAC 2010
Slide 22
Almost-Ideal Manifold Construction @x = v1 (x) @z1
xDC
@x = v3 (x) @z3 ASPDAC 2010
@x = v2 (x) @z2 Slide 23
Almost-Ideal Manifold Construction
ASPDAC 2010
Slide 24
Experimental Results
ASPDAC 2010
Slide 25
A Hand-Calculable Example d x1 = ¡x1 + x2 ¡ u(t) dt d x2 = x21 ¡ x2 dt
f (x) =
G(x) = ASPDAC 2010
·
¡1 2x1
·
¡x1 + x2 x21 ¡ x2
1 ¡1
¸
;
¸
;
[G(x)]
B=
¡1
·
¡1 0
1 = 2x1 ¡ 1
¸ ·
1 2x1
1 1
¸ Slide 26
DC and AC Manifold ¡1
[G(x)]
1 = 2x1 ¡ 1
·
1 2x1
1 1
¸
;
B= ·
·
¡1 0 ¸
¸
1 1 w1 (x) = [G(x)] B = 2x1 ¡ 1 2x1 · ¸ 1 ¡1 ¡ 2x1 ¡2 w2 (x) = [G(x)] B = ¡4x1 (2x1 ¡ 1)2 ¡1
@x w1 (x) = v1 (x) = DC manifold: @z1 jjw1 (x)jj2
AC manifold: ASPDAC 2010
@x w2 ¡ < w2 ; v1 > v1 = v2 (x) = @z1 jjw2 ¡ < w2 ; v1 > v1 jj2 Slide 27
DC and AC Manifold
ASPDAC 2010
Slide 28
Application to MOR 2
3
2
x ¡10 d 4 1 5 4 x2 1 = dt x3 1
1 ¡1 0
32
3
2
3
2
3
1 x1 0 1 0 5 4 x2 5 ¡ 4 0 5 + 4 0 5 u(t) ¡1 x3 x21 0
Trajectory of the full system stays close to the manifold
ASPDAC 2010
Slide 29
Simulation of the Reduced Order Model Response to a step input
Response to a sinusoidal input
ASPDAC 2010
Slide 30
Conclusion ●
Presented a manifold construction and parameterization procedure Based on computing integral curves ● Preserves local distance ● Captures important system responses ● Such as DC and AC responses ●
●
Application to manifold-based MOR ●
ASPDAC 2010
Validated against several examples
Slide 31
University of California, Berkeley
ASPDAC 2010
Slide 1
Outline ●
Background Introduction to MOR and maniMOR ● Manifold construction and parameterization ●
●
Manifold construction using integral curves DC manifold and the normalized integral curve equation ● Ideal and almost-ideal manifold ● Algorithm ●
Experimental results ● Conclusion ●
ASPDAC 2010
Slide 2
Background
ASPDAC 2010
Slide 3
Model Order Reduction Original system (size n)
d~x = f (~x) + B~ u(t) dt
~ u(t)
v : ~x 7! ~ z;
~ x 2 Rn ; ~z 2 Rq ;
y = C~x ~
q¿n
Reduced system (size q)
~ u(t) ASPDAC 2010
d~z = fr (~ z ) + Br ~ u(t) dt
~y = Cr ~z Slide 4
Low-order Linear Subspace 2
3
2
x ¡10 d 4 1 5 4 x2 1 = dt x 1 3
1 ¡1 0
32
3
2
3
1 x1 1 0 5 4 x2 5 + 4 0 5 u(t) ¡1 x3 0
Low-order linear subspace Defined by
ASPDAC 2010
x=Vz
Slide 5
Low-order Nonlinear Manifold 2
3
2
x ¡10 d 4 1 5 4 x2 1 = dt x 1 3
1 ¡1 0
32
3
2
3 2
3
1 x1 0 1 0 5 4 x2 5 ¡ 4 0 5+ 4 0 5 u(t) ¡1 x3 x21 0
Low-order nonlinear manifold
ManiMOR: MOR Based on Nonlinear Projection on Nonlinear Manifolds ASPDAC 2010
Slide 6
Key Steps in ManiMOR ●
“Find” the nonlinear manifold ●
●
Capture important dynamics
“Parameterize” the manifold ●
Build up the coordinate system
ASPDAC 2010
Slide 7
Manifold and Its Parameterization 8 < x = cos(t) y = sin(t) : z=t
ASPDAC 2010
Slide 8
Manifold and Its Parameterization
M
Parameterization of the manifold
Tangent space Tx M ½ Rn
U
x
à v
Rq
~ U
z
System of coordinates Manifold defined by pairs of fx; Tx M g ASPDAC 2010
No explicit mapping may be derived. Instead, use piecewise linear approximation. Slide 9
Manifold and Its Parameterization 1. Identify the manifold that capture important dynamics 2. Compute and store pairs of fx; Tx M g=fz; Tz M g
ASPDAC 2010
Slide 10
DC Manifold DC operating points constitute a DC manifold. d~ x = f (~ x) + B~ u(t) = 0 dt
How to compute and parameterize the DC manifold? ASPDAC 2010
Slide 11
DC Manifold f (~ x) + B~ u(t) = 0
A straight-forward solution:
Computation: Perform DC sweep analysis Parameterization: Define z coordinates using values of
u
Problems: Hard to choose step size in DC sweep analysis Not generalizable to higher dimensions
ASPDAC 2010
Slide 12
Introduction to Integral Curve Given a vector field v(x) , dx = v(x) its integral curve is the curve ° ´ x(t) such that dt
ASPDAC 2010
Slide 13
DC Manifold as an Integral Curve Need to derive the relationship between dx and du f (~ x) + B~ u(t) = 0
@f dx +B =0 @x du
dx = ¡[G(x)]¡1 B du
The first Krylov basis.
Initial condition: x(u = 0) = xDC ju=0 Solutions are DC operating points.
Any numerical integration / transient analysis code can be applied. ASPDAC 2010
Slide 14
Parameterization using Euclidean Distance Parameterization using values of u (x2 ; y2 ) (x1 ; y1 )
(x3 ; y3 ) u0 + 2h
u0 + h
u0
Parameterization using Euclidean Distance (x3 ; y3 ) (x2 ; y2 ) u0 + h
(x1 ; y1 )
u0 + 2h
u0 Sample points equally spaced on the DC manifold ASPDAC 2010
Slide 15
Parameterization using Euclidean Distance
ASPDAC 2010
Slide 16
Normalized Integral Curve Equation Local Euclidean distance is
jjdxjj2 = jduj
dx = ¡[G(x)]¡1 B du
¯¯ ¯¯ ¯¯ dx ¯¯ ¯¯ ¯¯ = jj[G(x)]¡1 Bjj2 = 1 Generally not satisfied ¯¯ du ¯¯ 2 Normalize RHS
dx [G(x)]¡1 B = du jj[G(x)]¡1 Bjj2 ASPDAC 2010
Normalized Integral Curve Equation
Does it define the same integral curve? Slide 17
Validation
ASPDAC 2010
Slide 18
Normalized Integral Curve Equation Theorem: ^(¿ ) satisfy Suppose t = ¾(¿ ); x(t) and x
d d x(t) = g(x(t)) and x ^(¿ ) = ¾0 (¿ )g(^ x(¿ )) , respectively. dt d¿ ^(¿ ) span the same state space. Then x(t) and x Sketch of proof: 0 dt = ¾ (¿ )d¿ . t = ¾(¿ ) Since , we have
^(¿ ) ´ x(t) = x ^(¾(t)) , then Define x
d d^ x(¿ ) dt x ^(¿ ) = = ¾ 0 (¿ )g(x(t)) = ¾0 (¿ )g(^ x(¿ )) d¿ dt d¿
ASPDAC 2010
Slide 19
Normalized Integral Curve Equation dx = ¡[G(x)]¡1 B du
dx [G(x)]¡1 B = du jj[G(x)]¡1 Bjj2
Solution: x(u)
^(^ u) Solution: x
Define u = ¾(^ u) =
Z
0
u ^
1 d¹ jj[G(^ x(¹))]¡1 Bjj2
From the theorem, x(u) and x ^(^ u) define the same integral curve. ASPDAC 2010
Slide 20
Normalized Integral Curve Equation dx [G(x)]¡1 B = du jj[G(x)]¡1 Bjj2
The first normalized Krylov basis.
Directly available from Krylov subspace methods. Generalizable to higher dimensions.
ASPDAC 2010
Slide 21
Ideal Nonlinear Manifold @x = v1 (x); @z1
@x = v2 (x); @z2
¢¢¢ ;
@x = vq (x): @zq
V (x) = [v1 (x); ¢ ¢ ¢ ; vq (x)] is the projection matrix
for the reduced linearized system (at x ).
For example, Arnoldi algorithm generates a basis for Kq ([G(x)]¡1 ; B) = f[G(x)]¡1 B; [G(x)]¡2 B; ¢ ¢ ¢ ; [G(x)]¡q Bg
However, this set of PDEs is over-determined. ASPDAC 2010
Slide 22
Almost-Ideal Manifold Construction @x = v1 (x) @z1
xDC
@x = v3 (x) @z3 ASPDAC 2010
@x = v2 (x) @z2 Slide 23
Almost-Ideal Manifold Construction
ASPDAC 2010
Slide 24
Experimental Results
ASPDAC 2010
Slide 25
A Hand-Calculable Example d x1 = ¡x1 + x2 ¡ u(t) dt d x2 = x21 ¡ x2 dt
f (x) =
G(x) = ASPDAC 2010
·
¡1 2x1
·
¡x1 + x2 x21 ¡ x2
1 ¡1
¸
;
¸
;
[G(x)]
B=
¡1
·
¡1 0
1 = 2x1 ¡ 1
¸ ·
1 2x1
1 1
¸ Slide 26
DC and AC Manifold ¡1
[G(x)]
1 = 2x1 ¡ 1
·
1 2x1
1 1
¸
;
B= ·
·
¡1 0 ¸
¸
1 1 w1 (x) = [G(x)] B = 2x1 ¡ 1 2x1 · ¸ 1 ¡1 ¡ 2x1 ¡2 w2 (x) = [G(x)] B = ¡4x1 (2x1 ¡ 1)2 ¡1
@x w1 (x) = v1 (x) = DC manifold: @z1 jjw1 (x)jj2
AC manifold: ASPDAC 2010
@x w2 ¡ < w2 ; v1 > v1 = v2 (x) = @z1 jjw2 ¡ < w2 ; v1 > v1 jj2 Slide 27
DC and AC Manifold
ASPDAC 2010
Slide 28
Application to MOR 2
3
2
x ¡10 d 4 1 5 4 x2 1 = dt x3 1
1 ¡1 0
32
3
2
3
2
3
1 x1 0 1 0 5 4 x2 5 ¡ 4 0 5 + 4 0 5 u(t) ¡1 x3 x21 0
Trajectory of the full system stays close to the manifold
ASPDAC 2010
Slide 29
Simulation of the Reduced Order Model Response to a step input
Response to a sinusoidal input
ASPDAC 2010
Slide 30
Conclusion ●
Presented a manifold construction and parameterization procedure Based on computing integral curves ● Preserves local distance ● Captures important system responses ● Such as DC and AC responses ●
●
Application to manifold-based MOR ●
ASPDAC 2010
Validated against several examples
Slide 31
