control of a mls using fast derivative estimation
TRANSCRIPT
7/31/2019 Control of a MLS Using Fast Derivative Estimation
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Control of a magnetic levitation system usingfast derivative estimation
Johann Reger
Institut fur Meß- und Automatisierungstechnik
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Contents
1. Simplified system model
2. Flatness-based tracking controller
3. Trajectory planning
4. An estimate for the velocity
5. Closed-loop dynamics
6. Simulations
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References
We refer to some results exposed in
E. Delaleau, “Differential Flatness in Electrical Drives and Power
Electronics”, Tutorial at the 1st International Conference on
Electrical and Electronics Engineering and 10th
Conference onElectrical Engineering, September, 2004.
J. Levine, J. Lottin, and J.-C. Ponsart, “A Nonlinear Approach to the
Control of Magnetic Bearings”, IEEE Transactions on Control SystemTechnology, Vol. 4., No. 5, September, 1996.
H. Sira Ramırez and M. Fliess, “On the output feedback control of a
synchronous generator”, Proc. 43
rd
IEEE CDC, December, 2004.
J. Reger, H. Sira Ramırez, and M. Fliess, “On non-asymptotic
observation of nonlinear systems”, submitted to 44th IEEE CDC,
December, 2005.
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Simplified system model
Consider the simplified model of a magnetic levitation system
h = v ,
v =k
m„ i
c − h«2
− g
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Simplified system model
Consider the simplified model of a magnetic levitation system
h = v ,
v =k
m„ i
c − h«2
− g
with
h > 0: vertical position of the center of mass — (measured) output,
v: vertical velocity of the center of mass,
i > 0: coil current — input,
g: gravity constant,
c: minimal air gap,
m: mass to be lifted,
k: konstant (air gap permeability, core reluctance, . . . ).
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Flatness-based tracking controller
We design a controller for stabilizing the output reference h = h#(t).
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Flatness-based tracking controller
We design a controller for stabilizing the output reference h = h#(t).
The output h is flat since
v = h ,
i = (c − h)
r m
k(h + g) .
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Flatness-based tracking controller
We design a controller for stabilizing the output reference h = h#(t).
The output h is flat since
v = h ,
i = (c − h)
r m
k(h + g) .
In accordance with Delaleau/Hagenmeyer the reference h = h#(t)
may be tracked by a feedforward linearizing controller
i = (c − h#)
s m
k
„h# + λ2(h# − h) + λ1(h# − h) + λ0
Z t0
(h# − h)dτ + g
«
which is a feedforward of h# enhanced with a PID-controller.
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Flatness-based tracking controller
We design a controller for stabilizing the output reference h = h#(t).
The output h is flat since
v = h ,
i = (c − h)
r m
k(h + g) .
In accordance with Delaleau/Hagenmeyer the reference h = h#(t)
may be tracked by a feedforward linearizing controller
i = (c − h#)
s m
k
„h# + λ2(h# − h) + λ1(h# − h) + λ0
Z t0
(h# − h)dτ + g
«
which is a feedforward of h# enhanced with a PID-controller.
It enforces the error dynamics
h# − h + λ2(h# − h) + λ1(h# − h) + λ0Z t0
(h# − h)dτ = 0 .
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Flatness-based tracking controller
Differentiating once wrt. time we obtain the third order dynamics
...h# −
...h + λ2(h
# − h) + λ1(h# − h) + λ0(h
# − h) = 0 .
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Flatness-based tracking controller
Differentiating once wrt. time we obtain the third order dynamics
...h# −
...h + λ2(h
# − h) + λ1(h# − h) + λ0(h
# − h) = 0 .
Consequently, we may set the error dynamics
...e + λ2 e + λ1 e + λ0 e = 0
of the error e = h# − h
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Flatness-based tracking controller
Differentiating once wrt. time we obtain the third order dynamics
...h# −
...h + λ2(h
# − h) + λ1(h# − h) + λ0(h
# − h) = 0 .
Consequently, we may set the error dynamics
...e + λ2 e + λ1 e + λ0 e = 0
of the error e = h# − h by choosing the coefficients λi such that
s3 + λ2s
2 + λ1s + λ0
is a Hurwitz polynomial.
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Flatness-based tracking controller
Differentiating once wrt. time we obtain the third order dynamics...h# −
...h + λ2(h
# − h) + λ1(h# − h) + λ0(h
# − h) = 0 .
Consequently, we may set the error dynamics
...e + λ2 e + λ1 e + λ0 e = 0
of the error e = h# − h by choosing the coefficients λi such that
s3 + λ2s
2 + λ1s + λ0
is a Hurwitz polynomial.
A possible choice is
λ2 = 400, λ1 = 30000, λ0 = 90000 .
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Trajectory planning
For the reference h# we specify a stationary set point change
h#(t0) = h0 → h
#(t1) = h1 .
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Trajectory planning
For the reference h# we specify a stationary set point change
h#(t0) = h0 → h
#(t1) = h1 .
The reference may be chosen as
h# = h0 + (h1 − h0) B(τ )
with the Bezier polynomial
B(τ ) = τ 5(252− 1050 τ + 1800 τ
2 − 1575 τ 3 + 700 τ
4 − 126 τ 5), τ =
t − t0
t1 − t0
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Trajectory planning
For the reference h# we specify a stationary set point change
h#(t0) = h0 → h
#(t1) = h1 .
The reference may be chosen as
h# = h0 + (h1 − h0) B(τ )
with the Bezier polynomial
B(τ ) = τ 5(252− 1050 τ + 1800 τ
2 − 1575 τ 3 + 700 τ
4 − 126 τ 5), τ =
t − t0
t1 − t0
which guarantees that
B(0) = 0, B(1) = 1 anddi
dτ iB(τ )
˛τ =0
=di
dτ iB(τ )
˛τ =1
= 0, i = 1, 2, 3, 4 .
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An estimate for the velocity
The chosen feedback
i = (c − h#)
s m
k
„h# + λ2(h# − h) + λ1(h# − h) + λ0
Z t0
(h# − h)dτ + g
«
requires knowledge about the velocity h.
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An estimate for the velocity
The chosen feedback
i = (c − h#)
s m
k
„h# + λ2(h# − h) + λ1(h# − h) + λ0
Z t0
(h# − h)dτ + g
«
requires knowledge about the velocity h.
For instants t > tr we obtain a velocity estimate resorting to the formula
h(i)(t) = (k + i − 1)!i! (k − i − 1)! 1(t − tr)i h(t) i = 1, . . . , d ≤ k − 1
+i
Xj=1
k + i − j − 1
i− j
!(k − j − 1)!
(k − i − 1)!
1
(t− tr)k+i−jzj(k, t)
with
zj(k, t) =
k
j + 1
!2(−1)−j( j + 1)! (t − tr)k−j−1
h(t) + zj+1(k, t), j = 1, . . . , k −
zk−1(k, t) = k! (−1)1−k
h(t) and zj(k, tr) = 0, j = 1, . . . , k − 1 .
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An estimate for the velocity
We choose the 4-th order estimate
˙h = 12
1
t − trh +
1
(t − tr)4z1 ,
¨h
= 60
1
(t− tr)2h
+ 8
1
(t − tr)5z1 +
1
(t − tr)4z2
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An estimate for the velocity
We choose the 4-th order estimate
˙h = 12
1
t − trh +
1
(t − tr)4z1 ,
¨h
= 60
1
(t− tr)2h
+ 8
1
(t − tr)5z1 +
1
(t − tr)4z2
with the filter
z1 = −72 (t− tr)2 h + z2 ,
z2 = 96 (t − tr) h + z3 ,
z3 = −24 h
subject to the homogeneous initial values
z1(0) = 0, z2(0) = 0, z3(0) = 0 .
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Closed-loop dynamics
In the closed loop, we may have to deal with a noisy feedback hn(t).
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Closed-loop dynamics
In the closed loop, we may have to deal with a noisy feedback hn(t).
Therefore, the closed-loop dynamics reads
h = v ,
v =
„c− h#
c− h
«2 “
h# + λ2(h
# −˙hn) + λ1(h
# − hn) + λ0ζ + g”− g ,
ζ = h# − hn
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Closed-loop dynamics
In the closed loop, we may have to deal with a noisy feedback hn(t).
Therefore, the closed-loop dynamics reads
h = v ,
v =
„c− h#
c− h
«2 “
h# + λ2(h
# −˙hn) + λ1(h
# − hn) + λ0ζ + g”− g ,
ζ = h# − hn
with the estimate
˙hn = 12
1
t − trhn +
1
(t − tr)4z1(t)
and the filterz1(t) = −72 (t − tr)2 hn + z2(t), z1(0) = 0 ,
z2(t) = 96 (t − tr) hn + z3(t), z2(0) = 0 ,
z3(t) = −24 hn, z3(0) = 0 .
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Simulations
We tackle the estimator singularity at t = tr by using the extrapolation
˙h ≈ h(t
−
r ) + h(t−
r )(t − tr)
¨h ≈ h(t
−
r )
whenever t ∈ [tr, tr + ε).
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Simulations
We tackle the estimator singularity at t = tr by using the extrapolation
˙h ≈ h(t
−
r ) + h(t−
r )(t − tr)
¨h ≈ h(t
−
r )
whenever t ∈ [tr, tr + ε).
We retain the estimation accuracy by reinitializing the estimator either
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Simulations
We tackle the estimator singularity at t = tr by using the extrapolation
˙h ≈ h(t
−
r ) + h(t−
r )(t − tr)
¨h ≈ h(t
−
r )
whenever t ∈ [tr, tr + ε).
We retain the estimation accuracy by reinitializing the estimator either
at equidistant, small time intervals or
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Simulations
We tackle the estimator singularity at t = tr by using the extrapolation
˙h ≈ h(t
−
r ) + h(t−
r )(t − tr)
¨h ≈ h(t
−
r )
whenever t ∈ [tr, tr + ε).
We retain the estimation accuracy by reinitializing the estimator either
at equidistant, small time intervals orby calculation of a next reset time t′r being the first time when
e(t) = |hn(t)− h(t)| > δ ,
i. e., an absolute error bound is passed. The output estimate
h(t) = h(t−
r ) +˙h(t
−
r )(t− tr) +1
2¨h(t
−
r )(t− tr)2 .
accounts for deviations from polynomial evolution.
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Simulations
In the following simulations we assume a set point change from
h#(t0) = h0 = 1 → h
#(t1) = h1 = 5
within the time interval [t0, t1] = [0, 1].
Moreover, we used the normalized systems parameters
k = 58, c = 0.11, g = 981, m = 0.084 .
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Si l ti
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Simulations
In the following simulations we assume a set point change from
h#(t0) = h0 = 1 → h
#(t1) = h1 = 5
within the time interval [t0, t1] = [0, 1].
Moreover, we used the normalized systems parameters
k = 58, c = 0.11, g = 981, m = 0.084 .
We close with some Matlab-Simulations.
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