Model reduction of port-Hamiltonian systems Rostyslav V. Polyuga [email protected], [email protected]

Arjan van der Schaft [email protected]

Port-Hamiltonian systems

Numerical example

Port-Hamiltonian models arise from network modeling of multi-physics systems, like electrical circuits, 2.2. Port-Hamiltonian systems etc. Some examples of port-Hamiltonian systems are shown 37 in multi-body systems, electrical machines, Figures 1, 2, 3, 4.

Consider the 100-dimensional full order port-Hamiltonian ladder network shown in Fig. 4.

u=I

R1

y = UC 1

L1 , φ1

R(n/2) L(n/2) , φ(n/2)

...

C(n/2)

C2 , q2

C1 , q1

q(n/2)

R(n/2+1)

Figure 4: n-dimensional ladder network

We applied the Arnoldi method to obtain the reduced order port-Hamiltonian models for the orders r = 2 to r = 30 with increments of 2 (see also [3, 4]). Evolution of the relative H2- and H∞-norms is shown in Fig. 5. As expected, both H2 and H∞ relative norms decay as the dimension r of the reduced order models increases. Relative H2 error norm vs r

21

1.5 ||G − Gr||2 / ||G||2

1.8. Kirchhoff’s laws, junctions and the network structure Figure 1: Magnetically levitated ball Figure 2.3: Magnetically levitated ball

L R for some matrices AL , AC , AS . Here IL , IC , IP denote the currents, respectively through the inductors, capacitors and external ports. Likewise, VL , VC , VP denote J K the voltages over the inductors, capacitors and external ports. Kirchhoff’s current and voltage laws define between the flows and efforts Vr a Dirac structure Vl ω V ˙ IP ) ˙ −φ, f = (IC , VL , IP ) = (− VQ, m ∂H τ e = (VC I, IL , VP ) = ( ∂H , , V ) P ∂Q ∂φ

1 0.5 0

2

||G − Gr||∞ / ||G||∞

other bond2.2.2 goes (to. Clearly all the efforts are the same due to the constraint of Example (Electro-mechanical system)). Consider the dynamics of an iron The state-space dimensions of mathematical models arising from port-based network modeling easily ball0-junction. in the magnetic field ofcan a controlled inductor, in Figure 2.3. The the This relation be easily verified by shown just writing the 0-junction become very large. Thus the problem arises of structure preserving model reduction of portport-Hamiltonian description of this system (with q the height of the ball, p the equation: Hamiltonian systems. Structurempreserving methods based on reduced order Dirac structures n X X • Effort-constraint fik = method f( ok ⇒ f1 = f2 + f3 ⇒ f3 = f1 − f2 x˙ 1 = (J11 − R11)(Q11 − Q12Q−1 k=1 k=1 22 Q21)x1 + B1u,

(2)

yec = B1T (Q11 − Q12Q−1 22 Q21)x1.

DC motor example •1.8.5 Flow-constraint method, see (3.22) in [3]. SVD-based structure preserving methods In the schematic model of a DC motor reported in Figure 1.13, we can distinguish •6Positive real balancing for port-Hamiltonian systems (Chapter 4 in [3]). interconnected lumps: Krylov-based structure preserving methods We concentrate on elements the single-input/single-output case.physical The reduced models obtained by • 2 storage with corresponding statesorder (φ, p): idealareinductor projectingL the scaled port-Hamiltonian system (with therefore Q = I) using the variables. orthonormal projection map and rotational inertia J. We have 2 state V ∈ Rn×r , r << n. This results in the r-dimensional reduced order model ( xˆ˙ = resistor V T (J − R R)V xˆ +the V T bu, • 2 dissipative elements: the and friction b. (3) T yˆ = b V xˆ. • 1 gyration effect K

Denote F = J − R. Different methods can be considered: .. F b .. . . . .. F r−1b] then moments at infinity (or Markov parameters) are • If span col V = span col [b • An ideal voltage source V matched; the Arnoldi method is used (for details see [4]). −1b .. . . . .. (F − s I)−r b] then moments at an arbitrary point s •We If span col V = span col [(F − s I) 0 0 can start by writing all the equations characterizing the various ideal elements. 0 are matched; the rational Arnoldi method is used (as discussed in [5]). • If span col V = span col [(s1I − F )−1b .. . . . .. (sr I − F )−1b] then the first moments at a set of distinct points s1, . . . , sr are matched; the points s1, . . . , sr can be chosen to make the reduced order models satisfy first order optimality conditions [2] with respect to an H2 system error metric.

c1

q2

k (n/2)

k2 ... c2

16

18

20

22

24

26

28

30

22

24

26

28

30

0.8

2

4

6

8

10

12

14

16

18

20

m(n/2) q(n/2)

Figure 3: Mass-spring-damper system

c(n/2)

The Amplitude Bode plots of the full, reduced and error systems for r = 20 are shown in Fig. 6. The figure exhibits that the approximation is very good for high frequencies since we match moments at infinity. Amplitutde Bode Plots of the full order and reduced order models for r = 20 Singular Values (dB)

Structure preserving reduction methods

q1

14

Figure 5: Evolution of the relative H2- and H∞-norms

Singular Values (dB)

in matrix general implicit equations not easily amenable matrices to analysis. withClearly, the energy Q =these QT , dissipation matrix R = are RT > 0 and interconnection J = −J T more convenient representations can be Inobtained andHowever, B. If Q is positive semi-definitecoordinate any port-Hamiltonian system is passive. the sequelusing we willthe assume Figure 1.13: DC motor. thattheory Q > 0.exposed in Section 2.4.

m2

12

r

y = B T Qx,

m1

10

0.9

0.7

−φ˙ = AL λ In the linear∂H case, = and A in the absence of algebraic constraints and a feed-through term, port-Hamiltonian λ C systems take∂Qthe following form ([6, 1]) Electrical Mechanical VP = AP λ ( T x T R)Qx + Bu, =A (J − ˙˙ + 0 = AL T ∂H − A IP Q C P ∂φ (1)

u1

8

1

Figure 2: DC motor

k1

6

r Relative H∞ error norm vs r

b with Hamiltonian H(φ, Q) the total energy. This leads to the port-Hamiltonian system in implicit form

u2

4

40

Full order Reduced order

20 0 −20 −4 10

−3

−2

10

−1

0

10 10 10 Frequency (rad/sec) Amplitutde Bode Plots of the error system for r = 20

1

10

20 10 0 −10 −20 −4 10

−3

10

−2

10 Frequency (rad/sec)

−1

10

0

10

Figure 6: Amplitude Bode plots for r = 20

Additionally, the reduced order models are port-Hamiltonian and therefore passive and stable.

References [1] The Geoplex Consortium. Modeling and Control of Complex Physical Systems; The Port-Hamiltonian Approach. Springer Berlin Heidelberg, 2009. [2] S. Gugercin, R.V. Polyuga, C.A. Beattie, and A.J. van der Schaft. Interpolation-based H2 Model Reduction for port-Hamiltonian Systems. In Proceedings of the Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, P.R. China, pages 5362-5369, December 16-18, 2009. [3] R.V. Polyuga. Model Reduction of Port-Hamiltonian Systems. Ph.D. thesis, University of Groningen, 2010. [4] R.V. Polyuga and A.J. van der Schaft. Structure preserving model reduction of port-Hamiltonian systems by moment matching at infinity. Automatica, 46:665-672, 2010. [5] R.V. Polyuga and A.J. van der Schaft. Structure preserving moment matching for port-Hamiltonian systems: Arnoldi and Lanczos. To appear in IEEE Transactions on Automatic Control, 2010. [6] A.J. van der Schaft. L2-Gain and Passivity Techniques in Nonlinear Control. Lect. Notes in Control and Information Sciences, Vol. 218, Springer-Verlag, Berlin, 1996, 2nd revised and enlarged edition, Springer-Verlag, London, 2000 (Springer Communications and Control Engineering series).