Structure preserving reduced order models based on ... · expansions, empirical balancing (e.g.,...

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University of Groningen

Structure preserving reduced ordermodels based on balancing fornonlinear systems

Jacquelien M.A. ScherpenUniversity of Groningen

Autumn school Terschelling, 24 September 2009

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Autumn school Terschelling, 24 September 2009 2/53


Purpose of this presentation

To give a overview of our contributions to positive real andbounded real balancing for nonlinear systems. It can beincluded in the previous nonlinear balancing framework,and structure preservation is obtained for model reduction.Finally, reduction to minimal port-Hamiltonian systems isdiscussed.

Positive and bounded real results are joint work with TudorIonescu, and Kenji Fujimoto. Port-Hamiltonian results arejoint work with Arjan van der Schaft.



Problem setting

Consider an input-output system Σ:u −→ Σ −→ y

and a time-invariant state space realization

x = f(x) + g(x)u

y = h(x)(∗)

Assume that (∗) is a valid state space realization of Σ

about x0.



Problem setting (continued)

Questions treated here:

• If (∗) is minimal and dissipative, can we reformulatethe problem in standard balancing form withobservability and controllability?

• Does model reduction based on the above preservethe dissipativity structure?

• If (∗) is non-minimal and port-Hamiltonian, how canwe obtain a minimal port-Hamiltonian realization?

• Do we need duality between controllability andobservability notions?




• For linear control systems, rather complete picture,though structure preserving methods still developed.

• Complexity increase: need for nonlinear tools.




• For linear control systems, rather complete picture,though structure preserving methods still developed.

• Complexity increase: need for nonlinear tools.

Some examples:

• Nonlinear circuits, power systems, fluid systems,MEMS, NEMS .....

• Inflatable space structures



Background

Linear realization theory more or less complete, since 70’s.

• Balanced realizations and Gramians (Moore 1981, . . .).• Balance between past minimal control energy and

generated future output energy .• Hankel operator, Hankel singular values, similarity

invariants, input-output view, tool for model reduction.



Background

Linear realization theory more or less complete, since 70’s.

• Balanced realizations and Gramians (Moore 1981, . . .).• Balance between past minimal control energy and

generated future output energy .• Hankel operator, Hankel singular values, similarity

invariants, input-output view, tool for model reduction.

• Transparent structure, Hankel singular values aremeasure for controllability and observability of state⇒ i.e., in balanced form: if xi is badly controllable and

badly observable then xi is almost “non-minimal”.



Background (continued)

• Unstable systems: normalized right and left coprimefactorizations, (e.g., Meijer, 1990, Ober and McFarlane, 1989)





• For nonlinear control systems model reduction oftendone on “ad hoc” basis, i.e., dependent on applicationor by singular perturbation techniques.





• For nonlinear control systems model reduction oftendone on “ad hoc” basis, i.e., dependent on applicationor by singular perturbation techniques.

• Proper Orthogonal Decomposition, Karhoenen-Loèveexpansions, empirical balancing (e.g., Lall et. al., 2002)

⇒ data-based linear projection methods.

• “Analytical” methods based on nonlinear extension ofbalancing theory: • Verriest and Gray, 2000

• Scherpen, Fujimoto, Gray, since 1993



Structure preservation

• Both Krylov and balancing method recent interest in(dissi)passivity structure preservation: bounded realand positive real structure, e.g., Antoulas, Sorensen,Brenner, Ha Bin Minh/Trentelman, Meyer, etc. .

• Also interest in physical interpretation, i.e.,port-Hamiltonian structure preservation, RLC structureinterpretation, e.g., Oelof/van der Schaft, Meyer,Polyuga/van der Schaft, etc.

• Control structure preservation from optimizationproblem, e.g., Weiland, ...



Outline

• Review stable linear systems balancing

• Review stable nonlinear systems balancing

• Storage functions in terms of observability andcontrollability functions

• Storage functions and coprime factorizations

• Dissipativity preserving model reduction

• Simple example

• Port-Hamiltonian structure preserving reduction to aminimal system



Stable linear systems

Continuous-time, causal linear input-output systemS : u → y with impulse response H(t).If S is also BIBO stable then the system Hankel operator :

H : Lm2 [0, +∞) → L

p2[0, +∞)

: u → y(t) =

∫ ∞

0

H(t + τ)u(τ) dτ.





H : Lm2 [0, +∞) → L

p2[0, +∞)

: u → y(t) =

∫ ∞

0

H(t + τ)u(τ) dτ.

Time flipping operator F : Lm2 [0, +∞) → Lm

2 (−∞, 0]u

t t

u

⇒





H : Lm2 [0, +∞) → L

p2[0, +∞)

: u → y(t) =

∫ ∞

0

H(t + τ)u(τ) dτ.

Time flipping operator F : Lm2 [0, +∞) → Lm

2 (−∞, 0]u

t t

u

⇒H(u) = S ◦ F(u)



Stable linear systems (continued)

H = OC, with the controllability and observabilityoperators C and O.

H∗H is a self-adjoint compact operator with σi are Hankelsingular values , i.e., σ2

i are eigenvalues of H∗H.




H = OC, with the controllability and observabilityoperators C and O.

H∗H is a self-adjoint compact operator with σi are Hankelsingular values , i.e., σ2

i are eigenvalues of H∗H.

(A,B,C) as. stable state space realization of S of order n.

• σ2i are eigenvalues of MW , where W ≥ 0 and M ≥ 0

are the usual controllability and observabilityGramians fulfilling

AW + WAT = −BBT

AT M + MA = −CT C




(A,B,C) is minimal ⇔ M > 0 and W > 0.

If (A,B,C) is minimal and as. stable, then there exists astate space representation where

Σ := M = W =

σ1 0. . .

0 σn

σ1 ≥ σ2 ≥ ... ≥ σn > 0 Hankel singular values. Thensystem is in balanced form .



Outline






• Simple example




Review stable nonlinear systems balancing

Smooth system

x = f(x) + g(x)u

y = h(x)

where u ∈ Rm, y ∈ R

p, and x ∈ M (manifold of dim n).

Assumptions:

• f(0) = 0, 0 as. stable eq. point for u = 0, x ∈ X.

• h(0) = 0.

• Controllability function Lc and observability function Lo

smooth and exist.



Energy functions: Gramian extensions

Lc(x0) = min

u ∈ L2(−∞, 0)

x(−∞) = 0, x(0) = x0

1

2

∫ 0

−∞

‖ u(t) ‖2 dt

Minimum amount of control energy necessary to reachstate x0. Lc is the so-called controllability function .




Lc(x0) = min

u ∈ L2(−∞, 0)

x(−∞) = 0, x(0) = x0

1

2

∫ 0

−∞

‖ u(t) ‖2 dt


Lo(x0) =1

2

∫ ∞

0

‖ y(t) ‖2 dt,x(0)= x0

u(τ)=0, 0≤τ<∞

Output energy generated by state x0.Lo is the so-called observability function.



The controllability and observability function

• In linear case: Lo(x) = 12xT Mx and Lc(x) = 1

2xT W−1x.

• Lyapunov and Hamilton-Jacobi-Bellmann equationscharacterize Lo and Lc.

• Role of observability and controllability for linearsystems is replaced by zero-state observabilityand asymptotic reachability (or anti-stabilizability).



The controllability and observability function

• In linear case: Lo(x) = 12xT Mx and Lc(x) = 1

2xT W−1x.

• Lyapunov and Hamilton-Jacobi-Bellmann equationscharacterize Lo and Lc.

• Role of observability and controllability for linearsystems is replaced by zero-state observabilityand asymptotic reachability (or anti-stabilizability).

• Minimality can be expressed in terms of Fliessexpansions and rank conditions, e.g., Isidori 1995.

• If 0 < Lc(x) < ∞ and 0 < Lo(x) < ∞ for x ∈ X, x 6= 0,then, under appropriate additional assumptions, thesystem is minimal , Scherpen, Gray 2000.



Example

l1

l2m1

m2

θ1

θ2

Frictionless two-link robotarm with actuator (torque)at first link.

System not as. stable. However, associated gradientsystem is as. stable!

Therefore, study of gradient system.Other advantage: reduction based on gradient system isstructure preserving when translated to original system.



Example

Approximately (series approximation, method of Lukes)solving eqs for Lo and Lc (m1 = m2 = l1 = l2 = 1) yields

-0.2-0.1

00.1

0.2-0.2

-0.1

0

0.1

0.2

0

0.001

0.002

0.003

-0.2-0.1

00.1

0.2

-0.2-0.1

0

0.1

0.2-0.2

-0.1

0

0.1

0.2

0

20

40

-0.2-0.1

0

0.1

0.2

Everywhere Lo > 0 and Lc > 0, thus minimal!

However, where Lo and Lc are both close to 0 correspondsto “weakly” obs./contr. subspaces ⇒ almost non-minimal .



Hankel norm

• Hankel norm for linear systems

‖Σ‖2H = max

u∈L2+

‖H(u)‖2

‖u‖2

=

• Hankel norm for nonlinear systems

‖Σ‖2H = max

u∈L2+

‖H(u)‖2

‖u‖2

=



Hankel norm


‖Σ‖2H = max

u∈L2+

‖H(u)‖2

‖u‖2= max

u∈L2+

〈u,H∗H(u)〉

〈u, u〉

=


‖Σ‖2H = max

u∈L2+

‖H(u)‖2

‖u‖2= max

u∈L2+

〈u,H∗(H(u), u)〉

〈u, u〉

=



Hankel norm


‖Σ‖2H = max

u∈L2+

‖H(u)‖2

‖u‖2= max

u∈L2+

〈u,H∗H(u)〉

〈u, u〉

= maxx

xT M x

xT W−1x


‖Σ‖2H = max

u∈L2+

‖H(u)‖2

‖u‖2= max

u∈L2+

〈u,H∗(H(u), u)〉

〈u, u〉

= maxx

Lo(x)

Lc(x)



Hankel norm


‖Σ‖2H = max

u∈L2+

‖H(u)‖2

‖u‖2= max

u∈L2+

〈u,H∗H(u)〉

〈u, u〉

= maxx

xT M x

xT W−1x= λmax(H

∗H) = λmax(MW ) = σ21


‖Σ‖2H = max

u∈L2+

‖H(u)‖2

‖u‖2= max

u∈L2+

〈u,H∗(H(u), u)〉

〈u, u〉

= maxx

Lo(x)

Lc(x)= ???



Balanced realizations (continued)

How to determine ???• For relation with Hankel operator and Hankel norm,

state dependent balanced form does not suffice.

• By considering both eigenstructure of

⋆ differential adjoint (dH(·))∗ (H(·)) and

⋆ full nonlinear Hilbert adjoint H∗(H(u), u),

characterization based on sort of parametrization thatis related to the input value yields form that fill inthe ??? , i.e., give explicit expression for Hankel norm.




• Appropriate assumptions, then there existsneighborhood X of 0 and x = Φ(z) s.t.

Lc(Φ(z)) =1

2

n∑

i=1

z2i

σi(zi)Lo(Φ(z)) =

1

2

n∑

i=1

z2i σi(zi).

In particular, on X, ‖Σ‖H = supz1

Φ(z1,0,...,0)∈X

σ1(z1).

• Singular value functions unique at coordinate axes.

• Tool for balanced structure preserving model reduction.




• Appropriate assumptions, then there existsneighborhood X of 0 and x = Φ(z) s.t.

Lc(Φ(z)) =1

2

n∑

i=1

z2i

σi(zi)Lo(Φ(z)) =

1

2

n∑

i=1

z2i σi(zi).

In particular, on X, ‖Σ‖H = supz1

Φ(z1,0,...,0)∈X

σ1(z1).

• Singular value functions unique at coordinate axes.

• Tool for balanced structure preserving model reduction.

• Discrete time version similar! Fujimoto, Scherpen 2007.



Example (continued)

Gradient system of frictionless two-link robot arm, withm1 = 1, m2 = 10, l1 = 1, l2 = 10

ρ1(s) = 4.2543 × 10−4 + 3.7070 × 10−11s2 − 1.4619 × 10−17s4 + o(s4)

ρ2(s) = 3.7915 × 10−5 − 4.5718 × 10−10s2 − 5.4584 × 10−13s4 + o(s4).

Hankel norm in small neighborhood U of the origin, e.g,U = { z | ‖z‖ ≤ 10 }, (based on 4-th order Taylor seriesapproximation)

‖Σ‖H ≈ sups∈[−10,10]

ρ1(s) = 4.2543 × 10−4.



Outline






• Simple example




Dissipativity

Consider smooth nonlinear system Σ

x = f(x) + g(x)u

y = h(x) + d(x)u,

x ∈ Rn, u ∈ R

m, y ∈ Rp.

Assumptions:• asymptotically reachable from 0, zero-state detectable.• f(0) = 0, 0 as. stable eq. point for u = 0.• h(0) = 0.



Dissipativity

Definition: Σ is dissipative with respect to supply rates(u, y), if ∃ storage function S : R

n → R, S(x) ≥ 0, suchthat dissipation inequality holds:

S(x0) +

∫ t1

t0

s(u, y)dt ≥ S(x1),

for all x, u and t1 ≥ t0, with x0 = x(t0) and x1 = x(t1).

Differential version:∂S(x)

∂x(f(x) + g(x)u) ≤ s(u, y).

Assumption:There exists ϕ(·), such that s(ϕ(y), y) < 0, ϕ(0) = 0.



Dissipativity

Available storage function of Σ is:

Sa(x0) = sup

u ∈ L2(0,∞)

x(∞) = 0, x(0) = x0

−

∫ ∞

0

s(u(t), y(t)) dt.

Required supply function of Σ is

Sr(x0) = inf

u ∈ L2(−∞, 0)

x(−∞) = 0, x(0) = x0

∫ 0

−∞

s(u(t), y(t)) dt.



Dissipativity

Available storage function of Σ is:

Sa(x0) = sup

u ∈ L2(0,∞)

x(∞) = 0, x(0) = x0

−

∫ ∞

0

s(u(t), y(t)) dt.

Required supply function of Σ is

Sr(x0) = inf

u ∈ L2(−∞, 0)

x(−∞) = 0, x(0) = x0

∫ 0

−∞

s(u(t), y(t)) dt.

Lemma : If Σ dissipative w.r.t. s(u, y), then 0 ≤ Sa ≤ Sr



Dissipativity

• Dissipativity of Σ w.r.t. s(u, y) = 12[uT yT ]J

u

y

.

• Define r(x) = [I dT (x)]J

I

d(x)

.

Assumption: r(x) > 0.



Dissipativity

• Dissipativity of Σ w.r.t. s(u, y) = 12[uT yT ]J

u

y

.

• Define r(x) = [I dT (x)]J

I

d(x)

.

Assumption: r(x) > 0.

• Sa is the stabilizing (minimal and Sr is theanti-stabilizing (maximal) solution of aHamilton-Jacobi-Bellman equation.



Dissipativity

Minimal linear system (A,B,C,D). ThenSa(x) = 1

2xT Kminx and Sr = 1

2xT Kmaxx, with Kmin and Kmax

stabilizing and antistabilizing sol. of

AT K + KA +

KB − [0 CT ]J

I

D

[I DT ]J

I

D

−1

·

BT K − [I DT ]J

0

C

−[0 CT ]J

0

C

= 0.



Dissipativity

Minimal linear system (A,B,C,D). ThenSa(x) = 1

2xT Kminx and Sr = 1

2xT Kmaxx, with Kmin and Kmax

stabilizing and antistabilizing sol. of

AT K + KA +

KB − [0 CT ]J

I

D

[I DT ]J

I

D

−1

·

BT K − [I DT ]J

0

C

−[0 CT ]J

0

C

= 0.Specific J : positive real,

bounded real Riccati equa-

tion.



Passivity/positive real

Formulation of positive real balancing (i.e., s(u, y) = uT y) interms of observability and controllability functions:

p = m, J =

0 Im×m

Im×m 0

, r(x) = d(x) + dT (x)



Passivity/positive real

Formulation of positive real balancing (i.e., s(u, y) = uT y) interms of observability and controllability functions:

p = m, J =

0 Im×m

Im×m 0

, r(x) = d(x) + dT (x)

For linear system (A,B,C,D) Riccati equation becomes:

KA + AT K + (KB − CT )R−1(BT K − C) = 0.

Positive real balancing: transformation to equalize anddiagonalize Kmin and Kmax.



Controllability and observability formulation

Consider system Σextended

x = f(x) + g(x)r−1(x)h(x) − g(x)r−1/2(x)u1 + K(x)r−1/2(x)u2

y1 = −r−12 (x)gT (x)

∂T Sa(x)

∂x

y2 = r−12 (x)h(x)

.

Theorem: Under the assumptions mentioned, consider Sa

and Sr of strictly passive Σ = (f(x), g(x), h(x), d(x)), thenSa = Lo, Sr = Lc, Lo observability function and Lc iscontrollability function of Σextended.



Bounded real

Formulation of bounded real balancing (i.e.,s(u, y) = 1

2(||u||2 − ||y||2)) in terms of observability and

controllability functions:

J =

I 0

0 −I

, r(x) = I − dT (x)d(x), c(x) = −dT (x)h(x)



Bounded real

Formulation of bounded real balancing (i.e.,s(u, y) = 1

2(||u||2 − ||y||2)) in terms of observability and

controllability functions:

J =

I 0

0 −I

, r(x) = I − dT (x)d(x), c(x) = −dT (x)h(x)

For linear system (A,B,C,D) Riccati equation becomes:

AK+KA+(KB+CT D)(I−DT D)−1(BT K+DT C)+CT C = 0.

Bounded real balancing on Kmin and Kmax.



Controllability and observability formulation

Consider system Σboundedreal

x = f(x) + g(x)r−1(x)dT (x)h(x) − g(x)r−1/2(x)u1 + K(x)l−1/2(x)u2

y1 = −r−12 (x)gT (x)

∂Sa(x)

∂x

y2 = l−12 (x)h(x), with

∂Sr(x)

∂xK(x) = hT (x).

Theorem: Under the assumptions mentioned, consider Sa

and Sr of strictly bounded real Σ = (f(x), g(x), h(x), d(x)),then Sa = Lo, Sr = Lc, Lo observability function and Lc iscontrollability function of Σboundedreal.



Background

• Linear version of results found in Ober, Jonckheere,recently thesis by Ha Binh Min, Trentelman.

• Linear analysis further developed for model reductionwith error bounds.

• Normalized coprime factorizations for nonlinearsystems by Scherpen, van der Schaft, Paice, Ball



Outline






• Simple example




A factorization approach

Assumptions and dissipative system, supply rate s(u, y).Consider Σfact:

x = f(x) + g(x)r−1(x)

(gT (x)

∂T Sa(x)

∂x− c(x)

)+ g(x)r−1/2(x)v

y1 = r−1(x)

(gT (x)

∂T Sa(x)

∂x− c(x)

)

y2 = h(x) + d(x)r−1(x)

(gT (x)

∂T Sa(x)

∂x− c(x)

).

Generalization of normalized coprime factorization!



A factorization approach

Theorem:Σfact has controllability and modified observability functionsLc(x) = Sr(x) − Sa(x) > 0 and LJ

o (x) = Sa(x) > 0, with

LMo (x) =

∫ ∞

0

1

2yT My dt, x(0) = x, x(∞) = 0 for M = MT .

Note that minimal energy required to reach a state is givenby difference in required and available storage. Energyobserved at the output is the maximal storage available atthat state.



Outline






• Simple example




Model reduction using factorization approach

Assumptions:• 0 < Sa < Sr, exist, as. stab. requirement,• Hessians Lo and Lc positive definite.

For system Σ similar to Hankel approach

π2i (s) =

Sa(ξi(s))

Sr(ξi(s))

Define for Σfact:

ρ2i (s) =

LJo (ξi(s))

Lc(ξi(s))

parametrized in s.




Theorem:Assume appropriate assumptions are fulfilled, so that ρi(s)

exist. Then if πi(s) are the axis singular values frombalancing Sa and Sr, then:

πi(s) =ρi(s)√

1 + ρ2i (s)

.




Theorem:Assume appropriate assumptions are fulfilled, so that ρi(s)

exist. Then if πi(s) are the axis singular values frombalancing Sa and Sr, then:

πi(s) =ρi(s)√

1 + ρ2i (s)

.

Theorem:Appropriate assumptions, then there exists coordinatetransformation x = Φ(z) such that:

Sr(Φ(z)) =1

2

n∑

i=1

z2

i

πi(zi)and Sa(Φ(z)) =

1

2

∑

i

z2

i πi(zi), πi(zi) = πi(Φi(z)).




• If πk > πk+1, then split accordingly in Σ1 and Σ2 fortruncation.

• Available storage and required supply preserved!




• If πk > πk+1, then split accordingly in Σ1 and Σ2 fortruncation.

• Available storage and required supply preserved!

Σ1 and Σ2 :

S1a(z

1) = Sa(z1, 0), S1

r (z1) = Sr(z

1, 0) and

S2a(z

2) = Sa(0, z2), S2

r (z2) = Sr(0, z

2).




• Singular value functions of subsystem Σ1 areπi(zi, 0), i = 1, . . . , k and the singular value functions ofsubsystem Σ2 are πj(0, zj), j = k + 1, . . . , n.

• Σ1,2 are dissipative with respect to the supply rates(u, y1,2).

• Similar result for non factorization case.

• Obstacle: s < 0. Not useful for e.g., LQG (HJB)balancing? Still dissipation preserved in "half line"dissipativeness manner, i.e., via one storage function,see Ha Binh Min/Trentelman.



Outline






• Simple example




Electrical circuit

L2L1 R1

V

R

x1 = −x1 + x2 + u

x2 = x1 − x2 − x32

y = −x1 + u = x1 − x2 + 2u.

xi

current in inductor i, i = 1, 2. System strictly positive real,strictly passive from input voltage to voltage over resistors.

Taylor approximations of Sa and Sr yield extended system.



Electrical circuit

Σextended:

x1 = −5

4x1 +

5

4x2 +

1

2u1 +

1

2K(x)u2

x2 = −x1 − x2 − x3

2

y1 = −0.08675x1 + 0.008485x2 − 0.5571926155x3

1+ 1.38155909x2

1x2−

0.950623382x1x2

2+ 0.04010539255x3

2

y2 =1

2x1 −

1

2x2,

with ∂Sr

∂xK(x) = x1 − x2 and

ρ1(s) = 2.506079510 + 69.19812137s2 + o(s4)

ρ2(s) = 0.4508902128 + 0.8176340704s2 + o(s4).



Intermediate conclusions

• Unifying framework for balancing, bounded and positive real

balancing, etc. via Hankel operator approach.

• Both extension and factorization system relations.

• Dissipativity structure preserving methods.

• Application to SMIB system of positive real balancing (8

states), raises new questions important for multi-physics

systems, i.e., what if mechanical part is important from

physics point of view, but not from passivity point of view!

• Missing: good numerical tools for larger scale systems. (NB:

for nonlinear systems 20 states is sometimes already large!)



Outline






• Simple example

• Port-Hamiltonian structure preserving reduction toa minimal system



Port-Hamiltonian structure preserving reduction

Consider port-Hamiltonian (PH) system of the form

x = (J(x) − R(x))∂H

∂x(x) + g(x)u

y = g(x)T ∂H

∂x(x)

J(x) = −J(x)T interconnection matrixR(x) = R(x)T ≥ 0 dissipation matrix.

H: Hamiltonian, internal energy of system, u and y ports:

H = uT y −∂T H

∂x(x)R(x)

∂H

∂x(x)




Properties

• Power preserving interconnection

• Passivity

• H is Lyapunov function, i.e., stability

• Interconnection two pH systems preserves structure,passivity and stability

• Etc.




Properties


• Passivity



• Etc.

How to reduce a non-minimal pH system to a minimal pHsystem?




• For linear PH systems, see e.g., Polyuga, van derSchaft, 2008.

• Recall: a nonlinear system is minimal (Isidori 95) if thesystem is strongly accessible (controllable for linearsystems) and observable.

First strong accessibility:

• Use nonlinear version of Kalman decomposition, i.e.,under appropriate conditions there exists coordinatessuch that x1 is strongly accessible and x2 is not.




Suppose that F = J − R, then pH dynamics restricted tostrongly accessible subspace can be written as

x1 =(F11(x

1) − F12(x1)F−1

22 (x1)F21(x1)

)︸︷︷︸

F (x1)

∂H

∂x1(x1, 0) + g1(x

1)u

y = g1(x1)

∂H

∂x1(x1, 0)

where Fij(x1) = Fij(x

1, 0) for i, j = 1, 2, g1(x1) = g1(x

1, 0),which is again a PH system.

Hamiltonian H(x1, 0). Interconnection and damping F .




Observability is more complicated. In linear case J11 − R11

stays, and Hamiltonian changes via Schur complement.

In nonlinear case Kalman decomposition can be done, butadditional assumptions are needed:

• If J − R and g are constant matrices, then similar tolinear case.

• If not, then with zero-state observability, via dualitynotions and observability and controllability functions,result can be obtained.

Scherpen and Van der Schaft 2008



General concluding remarks

• Framework for nonlinear balanced realizations inrelation with input-output interpretation from Hankelnorm. Rather complete picture. Extension of linearconcepts were introduced.





• Minimality and PH systems use similar concepts.






• In principle, analytical framework. However,computation of Lo and Lc is “not easy”, (e.g., Krener)and balancing step requires heavy computations,though proofs are constructive. Work in progress.






• In principle, analytical framework. However,computation of Lo and Lc is “not easy”, (e.g., Krener)and balancing step requires heavy computations,though proofs are constructive. Work in progress.

• Many open issues.

1/30

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A structure preserving minimalrepresentation of a nonlinearport-Hamiltonian systems

Jacquelien Scherpen

Arjan van der Schaft


Autumn school Terschelling, 24 September 2009


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Contribution

Exact model reduction method for a non-observable andnon-strongly accessible port-Hamiltonian system to anobservable and strongly accessible port-Hamiltoniansystem.

The nonlinear version of the Kalman decomposition isinstrumental for the approach. Both descriptions in energyand in co-energy variables are considered, depending onwhich description yields better properties for the reductionstep.


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Background

• Well-known balanced trunction methods for model order

reduction approximate system by removing the “almost

non-minimal parts” of the state space, e.g. ......


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Background




• Order reduction while preserving some energy/power

structure, passivity properties, and/or observability and

controllability properties studied for linear systems in many

papers, e.g., . . . . . .


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Background







papers, e.g., . . . . . .

• First: exact reduction, i.e., from non-minimal to minimal.


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Background







papers, e.g., . . . . . .

• First: exact reduction, i.e., from non-minimal to minimal.

• For linear pH systems: from non-controllable /

non-observable to controllable/observable pH system,

Polyuga, Van der Schaft, 2008.


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Problem setting

Consider port-Hamiltonian (PH) system of the form

x = (J(x) − R(x))∂H

∂x(x) + g(x)u

y = g(x)T∂H

∂x(x)

J(x) = −J(x)T interconnection matrixR(x) = R(x)T ≥ 0 dissipation matrix.

H: Hamiltonian, internal energy of system, u and y ports:

H = uT y −∂T H

∂x(x)R(x)

∂H

∂x(x)


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Problem setting

Properties


• Passivity



• Etc.


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Problem setting

Properties


• Passivity



• Etc.

How to reduce a non-minimal pH system to a minimalpH system?


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Outline

• Nonlinear systems and minimality

• Reduction to a strongly-accessible pH system

• Reduction to an observable pH system

• Approximate model reduction

• Concluding remarks


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Nonlinear systems and minimality

Smooth system

x = f(x) + g(x)u

y = h(x)



• Analytic realization (f, g, h) about x0 of formal powerseries is minimal if and only if realization is locallyaccessible and locally observable about x0.


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Smooth system

x = f(x) + g(x)u

y = h(x)



• Analytic realization (f, g, h) about x0 of formal powerseries is minimal if and only if realization is locallyaccessible and locally observable about x0.

• Under constant dim. assumption: “nonlinear” Kalmandecomposition for loc. strongly acc./loc. obs. case.


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Lc(x0) = min

u ∈ L2(−∞, 0)

x(−∞) = 0, x(0) = x0

1

2

∫ 0

−∞

‖ u(t) ‖2 dt



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Lc(x0) = min

u ∈ L2(−∞, 0)

x(−∞) = 0, x(0) = x0

1

2

∫ 0

−∞

‖ u(t) ‖2 dt


Lo(x0) =1

2

∫ ∞

0

‖ y(t) ‖2 dt,x(0)= x0

u(τ)=0, 0≤τ<∞

Output energy generated by state x0.Lo is the so-called observability function.


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Assumptions: f(0) = 0, 0 as. stable eq. point for u = 0,x ∈ X, h(0) = 0. Controllability function Lc andobservability function Lo smooth and exist.


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Assumptions: f(0) = 0, 0 as. stable eq. point for u = 0,x ∈ X, h(0) = 0. Controllability function Lc andobservability function Lo smooth and exist.

• If 0 < Lc(x) < ∞ and 0 < Lo(x) < ∞ for x ∈ X, x 6= 0,then, under appropriate additional assumptions, thesystem is minimal , Scherpen, Gray 2000.


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Outline







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Reduction to a strongly-accessible pH system

Assume strong acc. distribution C0 const. dim.. Then thereexist local coordinates such that C0 =span{ ∂

∂x1}

x1

x2

=

F11(x) F12(x)

F21(x) F22(x)

∂H

∂x1(x)

∂H

∂x2(x)

+

g1(x)

0

u

y =(g1(x)T 0

)

∂H

∂x1(x)

∂H

∂x2(x)

with F (x) = J(x) − R(x), x1 strongly acc., x2 not.


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Reduction to a strongly accessible pH system

Assume F22(x1, 0) invertible. Then pH dynamics restricted

to C0 can be written as

x1 =(F11(x

1) − F12(x1)F−1

22 (x1)F21(x1))

︸︷︷︸F (x1)

∂H

∂x1(x1, 0) + g1(x

1)u

y = g1(x1)

∂H

∂x1(x1, 0)

where Fij(x1) = Fij(x

1, 0) for i, j = 1, 2, g1(x1) = g1(x

1, 0),which is again a pH system.

Hamiltonian H(x1, 0). Interconnection and damping F (x1).


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Reduction to a stronly accessible pH system

Idea: Since C0 =span{ ∂∂x1},

F21(x)∂H

∂x1(x) + F22(x)

∂H

∂x2(x) = f(x2).

For any value of x2, the x1 sub-system is stronglyaccessible. Plug in x2 = 0, and result is obtained.


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Outline







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Reduction to an observable pH system

Observability is more complicated. In linear caseF11 = J11 − R11 remains, and Hamiltonian changes viaSchur complement.

In nonlinear case Kalman decomposition can be done, butadditional assumptions are needed:

• If J − R and g are in special form, then similar to linearcase.

• If not, then with zero-state observability, via dualitynotions and observability and controllability functions,result can be obtained.


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Assumptions:

• Observability space O. Observability co-distribution dO

is constant dimensional( ⇒ there exist local coordinates (x1, x2) such thatker dO =span{ ∂

∂x2} ).

• Assume F and g are such that F11, F12, g1, and g2 onlydepend on x1.

•∂2H

(∂x2)2(x) is invertible for all x.

Then, the pH system takes the form


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x1

x2

=

F11(x1) F12(x

1)

F21(x) F22(x)

∂H

∂x1(x)

∂H

∂x2(x)

+

g1(x1)

g2(x1)

u

y =(gT1 (x1) gT

2 (x1))(

∂H

∂x1(x)

∂H

∂x2(x)

)T

where

F11(x1)

∂H

∂x1(x) + F12(x

1)∂H

∂x2(x) = f(x1)

gT1 (x1)

∂H

∂x1(x) + gT

2 (x1)∂H

∂x2(x) = h(x1)


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Differentiating to x2 yields that the pH system restricted toobservable part can be written as

x1 = F11(x1)

∂H

∂x1(x1) + g1(x

1)u

y = g1(x1)T ∂H

∂x1(x1)

where∂H

∂x2(x1, x2) = 0

can be solved (at least locally) for x2 as a function x2(x1),determining H(x1) := H(x1, x2(x1)).


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What if assumption on g and F is not fulfilled? Thenconsider pH system in “physically dual” co-energycoordinates, i.e.,

z =∂H

∂x(x) =: γ(x),

under the assumption that transformation is non-singular.Take H(z) as the full Legendre transform of H(x), i.e.,

H(z) = xT z − H(x),


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• For system in z coordinates observability functionLo(z) is considered. Then duality (see Fujimoto,

Scherpen, Gray, 2002) via Legendre transform of Lo canbe considered, resulting in characterizations via Lc.


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• Then strong accessibility result can be applied,resulting in restriction to strongly accessible system.


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• Dual system via Legendre transform of Lc then resultsin a zero-state observable system that has the form ofa co-energy variable presentation.


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• Dual system via Legendre transform of Lc then resultsin a zero-state observable system that has the form ofa co-energy variable presentation.

• Open issue: link co-energy to Hamiltonian.


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System in co-energy coordinates:

z =

(∂2H

∂z2(z)

)−1 (J(z) − R(z)

)z

+

(∂2H

∂z2(z)

)−1

g(z)u

y = g(z)T z

with J(z) := J(γ−1(z)), R(z) := R(γ−1(z)), andg(z) := g(γ−1(z)).

Split in z1 (zero-observable) and z2 (not zero-observable).


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Lo(0, z2) = 0. Assume F22(z) invertible and Lo(z

1).Then ∃ coordinates z = ξ(z) s.t. co-energy variabledynamics restricted zero observable part

˙z = Q11(z)(F11(z) − F12(z)

(F22(z)

)−1F21(z)

)

︸︷︷︸F (z)

z+Q11(z)g1(z)u

y = gT1 (z)z

with Q11(z) = Q11(ξ(z)) =∂2H

(∂x1)2(γ−1(ξ(z))

Fij(z) = Fij(ξ(z)). F (z) + F T (z) ≤ 0.


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Outline







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Approximate model reduction

Can these “Kalman like” decompositions be used forfurther reduction of the model, similar to balancedtruncation, but preserving the pH structure?


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Can these “Kalman like” decompositions be used forfurther reduction of the model, similar to balancedtruncation, but preserving the pH structure?

• Note that the observability and strong accessibilityreduction methods to a minimal pH systems both yielddifferent minimal pH models.

⇓

Hence, immediate generalization of balancedtruncation in this setting is not clear.


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The observability reduction for special form F and g ismost easily applied for approximation, i.e.,

x1 = F11(x1, x2(x1)) ∂ eH

∂x1 (x1) + g1(x

1, x2(x1))u

y = g1(x1, x2(x1))T ∂ eH

∂x1 (x1)

H = H(x1, x2(x1)), with x2(x1) solution of ∂H∂x2 (x

1, x2) = 0.


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The observability reduction for special form F and g ismost easily applied for approximation, i.e.,

x1 = F11(x1, x2(x1)) ∂ eH

∂x1 (x1) + g1(x

1, x2(x1))u

y = g1(x1, x2(x1))T ∂ eH

∂x1 (x1)

H = H(x1, x2(x1)), with x2(x1) solution of ∂H∂x2 (x

1, x2) = 0.

Effort constraint reduction, i.e., setting the “effort”

∂H

∂x2= 0.


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Reduction via strongly accessible method also possible, but

technically more involved.x1

x2

=

F11 F12

F21 F22

e1

e2

+

g1

g2

u

y =(gT1 gT

2

)e1

e2

where e1 = ∂H∂x1 (x), e2 = ∂H

∂x2 (x). Set x2 equal to zero (flowconstraint ). If F22 is invertible, this yields

0 = F21e1 + F22e

2 + g2u ⇒ e2 = −F−122 F21e

1 − F−122 g2u


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Substitution flow constraint and x2 = 0 yield

x1 =(F11(x

1) − F12(x1)F−1

22 (x1)F21(x1)) ∂H

∂x1(x1, 0)

+(g1(x1, 0) − F12(x

1)F−122 (x1)g2(x

1, 0))u

y = (gT1 (x1, 0) − gT

2 (x1, 0)F−122 (x1)F21(x

1)

−gT2 (x1, 0)F−1

22 (x1)g2(x1, 0)u

with Fij(x1) = Fij(x

1, 0), i, j = 1, 2.

Again pH system (with through-term) provided that

(F12F−122 )T = F−1

22 F21.


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Outline







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Concluding remarks

• Reduction of non-strongly acc. or non-observable pHsystem to strongly acc. or observable pH system,respectively.

• Observability least straightforward, additionalassumptions made, and zero-observability considered.


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Concluding remarks

• Reduction of non-strongly acc. or non-observable pHsystem to strongly acc. or observable pH system,respectively.

• Observability least straightforward, additionalassumptions made, and zero-observability considered.

Open issues

• Zero-observability pH structure from co-energyrespresentation.

• Which method to use for approximate model reduction.Balance possible?


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Structure preserving reduced order models based on ... · expansions, empirical balancing (e.g.,...

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