Download Analogous Non-Smooth Models of Mechanical and Electrical Systems

Analogous Non-Smooth Models of Mechanical
and Electrical Systems
Michael Möller and Christoph Glocker
IMES - Center of Mechanics, ETH Zurich, 8092 Zurich, Switzerland
[email protected], [email protected]
Summary. The non-smooth modeling of mechanical and electrical systems allows
for ideal unilateral contacts, sprag clutches and dry friction in mechanical systems
and for ideal diodes and switches in electrical systems. The formulation of nonsmooth electrical models is demonstrated by the example of the DC-DC buck converter using the flux approach. The non-smooth electrical elements are described
with set-valued branch relations in analogy with set-valued force laws in mechanics. With the set-valued branch relations, the dynamics of the circuit are described
as measure differential inclusions. The measure differential inclusions obtained for
the DC-DC buck converter are related to an analogous mechanical system. For the
numerical solution, the measure differential inclusions are formulated as a measure
complementarity system and discretised with a difference scheme, known in mechanics as time-stepping. For every time-step a linear complementarity problem is
obtained.
1 Introduction
The well developed formulations and methods used for non-smooth mechanical
systems [5, 8, 9] can be adopted for electrical systems, by extending the classical electromechanical analogy to non-smooth systems. There are basically
three approaches for the description of electrical systems, called the charge approach, the flux approach and mixed approaches [3]. The charge approach for
electrical systems uses the charges and associated currents as variables while
the voltages are balanced. In the flux approach the fluxes with associated
voltages form the variables while balancing the currents. In mechanics usually the positions and their associated velocities are used as variables and the
forces are balanced. The classical analogy links these approaches in mechanics and electronics. In Table 1 the corresponding variables and linear elements
are shown for each approach. The duality between voltage u and current ı
of an electrical system mirror the duality in mechanics between velocity v
and force f . Table 1 is therefore completed with a column for the momentum approach, which is dual to the position approach in the same way as the
Peter Eberhard (ed.), Multiscale Problems in Multibody System Contacts, 45–54.
© 2007 Springer. Printed in the Netherlands.
46
Michael Möller and Christoph Glocker
mechanics
mechanics
electronics
electronics
position app. momentum app. charge approach flux approach
local
variables
position r
velocity v
force f
momentum p
force f
velocity v
charge g
current ı
voltage u
flux ϕ
voltage u
current ı
inertia
mass m
−f = mv̇
stiffness k
−v = k1 f˙
inductivity L
−u = Li
capacity C
−ı = C u̇
dissipation
damping d
−f = dv
damping d
−v = d1 f
resistance R
−u = Rı
resistance R
1
−ı = R
u
energy
storage
stiffness k
−f = kr
mass m
1
−v = m
p
capacity C
−u = C1 g
inductivity L
−ı = L1 ϕ
Table 1. Corresponding variables and elements in mechanics and electronics.
charge approach is dual to the flux approach. In [6], the idealized modeling of
switches and diodes in the charge approach is introduced and linked to nonsmooth mechanics. In this paper the non-smooth modeling of ideal switches
and diodes in the flux approach will be demonstrated by the example of the
DC-DC buck converter.
2 DC-DC Buck Converter
The DC-DC buck converter as shown in Fig. 1 is a circuit that allows for the
Fig. 1. The DC-DC buck converter.
efficient conversion of DC electrical power from the voltage u0 supplied by
Analogous Non-Smooth Models of Mechanical and Electrical Systems
47
the voltage source to a lower voltage uR at the resistive load R. Besides the
classical elements R, C, L and the voltage source u0 , the circuit consists of an
ideal diode D and a unilateral switch S. The part of Fig. 1, which is drawn in
grey, shows the switch control of the DC-DC buck converter, which controls
the voltage at the load R by operating the unilateral switch S.
2.1 The Switch Control of the DC-DC Buck Converter
The switch control of the DC-DC buck converter consist of an amplifier with
gain K, a comparator and a ramp generator with period T , lower voltage ul
and upper voltage uu (cf. Fig. 1). The output voltage a of the switch control
is used to operate the unilateral switch S of the buck converter. With the
output voltage ucomp of the amplifier
ucomp = −K(uR + uref )
(1)
and the explicitly time-dependent output voltage ug (t) of the ramp generator
!
t
− k (uu − ul ) ; kT ≤ t < (k + 1)T ; k = 0, 1, 2, ... (2)
ug (t) = ul +
T
the voltage a at the comparator is set for modeling reasons as
"
0,
ucomp − ug (t) ≤ 0,
a=
+∞, ucomp − ug (t) > 0.
(3)
This relation can be simplified by eliminating ucomp yielding the following
rule for the switch control
"
0,
−uR (t) ≤ h(t),
1
(4)
h(t) := uref + ug (t).
a=
K
+∞, −uR (t) > h(t),
For a = 0 the switch is closed and behaves as an ideal diode, while the switch
is perfectly isolating for a → +∞.
2.2 The Extended DC-DC Buck Converter
The extended DC-DC buck converter has two additional capacitors C ∗ and C ◦
compared to the original DC-DC buck converter, in order to obtain a nonsingular matrix M of capacitances (cf. Fig. 2). This extension of the circuit is
done to ease the analogy to mechanical systems, while the circuit of the original DC-DC buck converter can be obtained by setting the two additional capacities C ∗ and C ◦ to zero. In analogy with the force impulsion measure dF in
mechanics, a current impulsion measure dI is introduced for the flux approach.
The force impulsion measure dF consists of the Lebesgue-measurable forces
48
Michael Möller and Christoph Glocker
Fig. 2. Electrical model of the extended DC-DC buck converter.
f and the purely atomic impulsive forces F , while dI consists of Lebesguemeasurable currents ı and purely atomic impulsive currents I.
dF = f dt + F dη,
dI = ı dt + I dη.
(5)
In the flux approach the voltages u are assumed to be functions of bounded
variation. Higher order discontinuities are out of the scope of this paper. A
general method of classifying such discontinuities may be found in [1].
To describe the dynamics of the circuit, generalized voltages v and associated generalized fluxes q are introduced in analogy with generalized velocities u and generalized coordinates q in mechanics. The circuit of the extended
DC-DC buck converter has four nodes, which have been shaded grey in Fig. 2.
The voltages vi of each node with respect to the ground node are chosen as
generalized voltages. The vector of nodal voltages v and the associated vector
of nodal fluxes q become
T
T
q := q1 , q2 , q3
, v := v1 , v2 , v3
, q̇ = v dt-almost everywhere, (6)
for the extended DC-DC buck converter. All branch voltages ui satisfying
Kirchhoff’s voltage law can be expressed as a linear combination of the nodal
voltages vi , defining the nodal transformation of the circuit
u0 = w T
0 v = v1
⇒ wT
0 = (1, 0, 0)
uC ∗ = w T
⇒ wT
C ∗ v = v1
C ∗ = (1, 0, 0)
T
T
uS = wS v = v2 − v1 ⇒ wS = (−1, 1, 0)
uD = w T
D v = v2
T
uC = wC v = −v3
⇒ wT
D = (0, 1, 0)
⇒ wT
C = (0, 0, −1)
T
uL = w T
L v = v3 − v2 ⇒ w L = (0, −1, 1)
uC ◦ = w T
C ◦ v = v2
uR = w T
R v = −v3
⇒ wT
C ◦ = (0, 1, 0)
⇒ wT
R = (0, 0, −1).
(7)
Analogous Non-Smooth Models of Mechanical and Electrical Systems
49
The nodal transformation (7) holds also in integrated form for the branch
fluxes ϕi = wT
i q, as well as for the associated virtual flux displacements
δϕi = wT
δq.
Kirchhoff’s
current law is evaluated in terms of a virtual work
i
approach by demanding that the virtual action ddWδ has to vanish for arbitrary virtual branch flux displacements δϕadm
that are admissible with the
i
constraints imposed by the topology of the circuit
: 0 = dδW =
dIi δϕadm
.
(8)
∀ δϕadm
i
i
i
The sum is taken over all elements of the system i ∈ {0, C ∗ , S, D, C, L, C ◦ , R}.
With the admissible virtual branch flux displacements δϕadm
= wT
i
i δq, obtained by transforming arbitrary virtual nodal flux displacements δq, the
equation (8) can be further simplified,
T
dIi wT
wi dIi ,
(9)
∀ δq : 0 = dδW =
i δq = δq
i
i
yielding the equilibrium conditions at the nodes
wi dIi = 0.
(10)
i
The branch relations of the capacitors, the resistor and the inductor are singlevalued
dIC ∗ = −C ∗ duC ∗ , dIC = −CduC , dIC ◦ = −C ◦ duC ◦ ,
1
1
ıR dt = − uR dt, ıL dt = − ϕL dt.
R
L
(11)
The branch relations for the capacitors are formulated on the level of measures
to include impulsive currents in analogy with impulsive forces in mechanics.
The branch relations of the diode, the switch and the voltage source are setvalued
(12)
where Upr denotes the unilateral primitive [6]. An ideal diode is an element
through which the current may flow only in the positive direction. To prevent the current from flowing in the negative direction, an ideal diode can
provide an unbounded voltage at zero current. This characteristic can be expressed with the inclusion −ı ∈ Upr(u) and is analogous to an unilateral
kinematic constraint (sprag clutch) in mechanics −f ∈ Upr(v). The relation
given in (12) for the diode and depicted in Fig. 3 is obtained after completion
with an impact law. The unilateral switch is modeled as a series connection of
50
Michael Möller and Christoph Glocker
Fig. 3. Characteristic of sprag clutches and diodes.
a spark gap with break-through voltage a and a diode (cf. [6]). The unilateral
switch is analogous to a series connection of a sprag clutch and a kinematic
excitation with relative velocity a in the position-flux analogy. The unilateral
switch can be operated using the break-through voltage, while it is closed
for a = 0 and open for a → ∞. The characteristic given in (12) for the
unilateral switch contains an impact law as well and is depicted in Fig. 4. The
third set-valued element in (12) is the voltage source, which is analogous to a
bilateral kinematic constraint in mechanics.
Fig. 4. Characteristic of moving sprag clutches and unilateral switches.
After inserting the single-valued branch relations (11) into (10), replacing all branch variables using the nodal transformation (7) and defining the
matrices
∗
◦
T
T
M := wC ∗ wT
C ∗ C + w C ◦ w C ◦ C + w C w C C,
(13)
1
1
, K := wL wT
D := wR w T
R
L ,
R
L
Analogous Non-Smooth Models of Mechanical and Electrical Systems
51
one obtains the measure differential inclusions
describing the dynamics of the DC-DC buck converter. The mechanical
model associated with the extended DC-DC buck converter can be obtained
from (14) using the position-flux analogy. The mechanical model is illustrated
in Fig. 5. The model consists of three masses C ∗ , C ◦ and C corresponding
to the three capacitors of the extended DC-DC buck converter. Since the
position-flux analogy is used, the electrical circuit and the mechanical model
have the same topology. The sprag clutch acting between the environment and
the mass C ◦ is analogous to the diode and allows the mass C ◦ only move to
the right. The masses C ∗ and C ◦ are interacting by the serial connection of a
kinematic excitation with relative velocity −a and a sprag clutch, constituting
the analog to the unilateral switch. The switch control of the DC-DC buck
converter measures the velocity uR and provides the relative velocity −a.
Fig. 5. Mechanical model associated with the extended DC-DC buck converter.
2.3 Numerical Integration
A time-stepping method is used to solve the measure differential inclusions (14) describing the non-smooth dynamics of the circuit for the unknown
nodal voltages v(t) and the associated nodal fluxes q(t). Time-stepping methods discretise directly the inclusion (14) over a time step ∆t. The problem of
solving the measure differential inclusion (14) numerically, is formulated as
follows: For the system (14), with given initial nodal charges q A and initial
nodal voltages v A ,
q A := q(tA ), v A := v(tA ),
(15)
at the initial time tA , find nodal charges q E and nodal voltages v E ,
q E := q(tE ),
v E := v(tE ),
(16)
52
Michael Möller and Christoph Glocker
at the time tE which approximate the exact solution. The time tE is the end
of a chosen time interval [tA , tE ] with length
∆t := tE − tA .
(17)
The resulting algorithms are very robust and easy to implement, but have a
limited accuracy, see e.g. [2, 7] for some versions of time stepping algorithms.
There are different possibilities to treat smooth set valued elements - in this
case the voltage source - for numerical integration. Beside state reduction and
replacement with two unilateral constraints, the third possibility is to append
the current of the voltage source branch to the vector of unknown voltages
after discretisation. This approach, also known in electronics as modified nodal
analysis, is used in the following for the extended DC-DC buck converter. In
order to simplify the expressions, the notations
!
dID
dI :=
, W := wD w S
(18)
dIS
are introduced. Using these notations the equality of measures in (14) may be
written as
M dv + Dvdt + Kqdt − W dI − w 0 dI0 = 0,
(19)
wT
0 v − u0 = 0,
where the constraint of the voltage source has been added as an additional
equation. The notations
!
0
T
γ := W v + ŵ, ŵ :=
(20)
a
are introduced to formulate the measure inclusions of the ideal diode and the
unilateral switch as complementarity conditions
0 γ + ⊥ dI 0,
(21)
allowing to set up the linear complementarity problem after discretisation.
For non-smooth mechanical systems usually Moreau’s midpoint rule is used to
discretise the measure differential inclusions. This rule consists of a trapezoidal
rule for the positions and an Euler step for the velocities. For the discretisation
of the DC-DC buck converter, an implicit Euler scheme is used, which does
not require a regular capacitor matrix M . The integral of the equality of
measures (19) over the time step ∆t is approximated as
M (v E − v A ) + Dv E ∆t + Kq E ∆t − W∆I − w0 ∆I0 = 0,
wT
0 v E − u0 = 0,
(22)
using end-point terms by applying an implicit Euler scheme. The relation
between the nodal voltages v and the nodal fluxes q can be approximated
using one implicit Euler step
Analogous Non-Smooth Models of Mechanical and Electrical Systems
q E = q A + v E ∆t.
53
(23)
The complementarity conditions (21) are expressed in the discretised form
0 γ E ⊥ ∆I 0,
(24)
where the vector of local variables at the end-time tE ,
γ E = W T v E + ŵA ,
(25)
is formed using the vector ŵ A at the beginning tA of the time step. This is
done in order to avoid a nonlinear dependence on the unknown nodal voltages v E , which would lead to a nonlinear complementarity problem. By using
the vector ŵA instead of ŵ E , a small time-delay of ∆t is inserted into the
switch control feedback of the DC-DC buck converter, which seems reasonable
from the modeling point of view as well.
Elimination of the end-point nodal fluxes q E from the equations (22) with
the help of equation (23) yields
(M + D∆t + K∆t2 )v E − w0 ∆I0 − M v A + Kq A ∆t − W∆I = 0,
− wT
0 v E + u0 = 0,
(26)
where the terms have already been regrouped for the unknown variables v E
and ∆I0 . With the definition of the vectors and matrices
!
!
vE
M + D∆t + K∆t2 −w0
ν :=
,
, M̂ :=
∆I0
−wT
0
0
(27)
!
!
M v A − Kq A ∆t
W
ĥ :=
, Ŵ :=
,
−u0
0
the notation can be simplified yielding the mixed linear complementarity problem
T
M̂ ν − ĥ − Ŵ∆I = 0, γ E = Ŵ ν + ŵA , 0 γ E ⊥ ∆I 0.
(28)
If the matrix M̂ is regular then the vector ν can be eliminated from (28)
resulting in the linear complementarity problem
−1
T
γ E = Ŵ
M̂
y
A
T
−1
∆I + Ŵ M̂ ĥ + ŵ A , 0 γ E ⊥ ∆I 0
Ŵ x
b
(29)
0 y ⊥ x 0
in standard form. It has to be noted, that the matrix M̂ is regular not only for
the extended DC-DC buck converter, but for the original version as well. After
solving the linear complementarity problem (29) for the vectors γ E and ∆I,
the vector ν can be calculated from the first equation in (28), yielding the
end-point nodal voltages v E . The nodal fluxes q E can then be calculated with
the help of (23). The numerical results obtained for the original DC-DC buck
converter in a chaotic parameter regime, as published in [4, 6], are shown in
Fig. 6 and agree with those given in the publications.
54
Michael Möller and Christoph Glocker
Fig. 6. Phase space plot and comparator voltages of the DC-DC buck converter.
3 Conclusion
Using the example of the DC-DC buck converter the formulation of the measure differential inclusions, their relation to an analogous mechanical model
and their solution using the time-stepping method has been shown for the
flux approach. Only a small set of non-smooth elements have been described
in this paper to show the basic procedure, but the formulations used are not
limited to this set of elements only.
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