H_infinity-based Control System and Its Digital Implementation for

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H∞-based control system and its digital implementation for the
integrated tilt with active lateral secondary suspensions in high
speed trains
Ronghui Zhou1 , Argyrios Zolotas2 , Roger Goodall3
1. Real-Time Controls and Instrumentation Lab., Global Research Center, GE, Shanghai, China, 201203,
E-mail: ronghui.zhou@ge.com
2. School of Engineering and Informatics, University of Sussex, Sussex House, Brighton, BN1 9QT, UK,
E-mail: a.zolotas@sussex.ac.uk
3. Control systems group, Department of Electronic and Electrical Engineering, Loughborough University,
Loughborough, LE11 3TU, UK,
E-mail: r.m.goodall@lboro.ac.uk
Abstract: In this paper, H∞ -based control system and its digital implementation for the integrated tilt with active lateral secondary suspensions in high speed railway vehicles are discussed, in which mixed-sensitivity H∞ control is designed for the tilting
suspension, while skyhook damping control is employed for the active lateral secondary suspensions. Compared with classical
decentralized control, the proposed control system can well attenuate the strong coupling between the roll and lateral dynamic
modes of the vehicle body. Particular emphasis is also on the digital implementation of the reduced order H∞ tilt controller in
an embedded control unit. Proposed digital controllers are validated via a FPGA-based Hardware-In-the-Loop system.
Key Words: H∞ Control, Active Suspensions, Railway System Control, Integrated Tilt With Active Lateral Suspensions, HIL
Digital Implementation
1
Introduction
High speed trains which are able to operate at 200km/h
and faster are nowadays widely spread in the world, i.e.
France, Germany and United Kingdom in Europe, Japan
and China in Asia. The recent world rail speed record is
578km/h achieved by French V150 TGV [1]. However, in
order to develop the TGV, new rail ifrastructures are needed,
i.e. new rail tracks. One type of high speed train is tilting
train, which can operate at increased speeds without the need
to upgrade the rail infrastructure. The idea is to tilt the vehicle body inwards on the curved sections of the track to compensate the large lateral acceleration perceived by passengers
at higher speeds. Early passive tilting trains completely relied upon the natural pendulum motion laws which caused
safety issues, i.e. vehicle body over turning [2], and a tilt
mechanism (tilting bolster in most cases) in conjunction with
an actuator to tilt the vehicle body was introduced, which has
become a standard technology used in trains worldwide.
The active Anti-Roll Bar (ARB) is one of the tilting system mechanical configuration, as shown in Fig. 1. It is configured by a transversely-mounted torsion tube on the bogie
with vertical links to the vehicle body, except that one of the
links is replaced by a tilt actuator, and thereby applies tilt via
the torsion tube.
Control systems for the tilt actuator can be designed based
on either the local vehicle body measurements or the bogiemounted sensors from the front vehicle. The control system
based on the local measurements (named as “Nulling Control”) however cannot well address the strong interaction between vehicle roll and lateral dynamic modes, the industrial
sector nowadays adopts a control structure called “precedence control”, in which, the bogie-mounted lateral accelerometer from the front vehicle is used to provide “precedence” information which minimises the lateral and roll dy-
Vehicle body
Lateral
actuator
Anti-Roll Bar
Tilt actuator
Bogie frame
Fig. 1: Tilting train with tilt and lateral actuator
namic interaction problem. Appropriate low pass filters are
employed to attenuate the high frequency signal caused by
the track irregularity response of the bogie. The delay introduced by the filter is compensated by the carefully designed
precedence control strategy [3]. Although precedence control is an accepted commercial solution for the tilting train,
research on local tilt control still has practical benefits which
make the system simpler and more straightforward in terms
of detecting sensor failure.
Tilt control based on local vehicle body signals with H∞
and Fuzzy logic controllers were studied in [4][5], but due
to the dynamic interaction between roll and lateral modes of
the railway vehicle body, there is further research potential of
improving the overall transient performance. In addition, the
high speeds associated with tilting trains result in worse ride
quality on straight track. In the work of [6], a lateral actuator
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was proposed to be installed between the vehicle body and
bogie in parallel with (or to replace) the original secondary
damper, as shown in Fig. 1. The control system design for
this dual-actuator system (tilt and lateral) was carried out in
both decentralised and centralised way, in which, the Classical Decentralised (CD) control, and LQG centralised control were investigated. Genetic Algorithm was employed to
optimize the controller parameters due to the multiple design requirements. In this paper, H∞ -based Decentralised
(HD) control is investigated, mixed-sensitivity H∞ control
is designed for the tilting suspension, while skyhook damping control [7] is employed for the active lateral secondary
suspension. Compared with CD control, the proposed control system can further attenuate the strong coupling between
the roll and lateral dynamic modes of the vehicle body.
The proposed HD control strategy is validated via a
FPGA-based Hardware-In-the-Loop (HIL) system. Controller reduction technique is also employed for the H∞ tilt
control before its digital implementation. The remainder of
this paper is organized as follows: Part II presents railway
vehicle model and controller performance assessment approaches, Part III refers to the basics of CD control, while
Part IV gives the details of the HD control system design. It
is followed by the HIL implementation and validation. Conclusions are discussed in the last part.
2
Body lateral dynamics:
mv ÿv
Vehicle body
d1
y+v
c.o.g
kvr
δa
kaz
krz
h g1
ksz
θv+
csy
crz
Fa
Vehicle bogie
c.o.g
kpz
cpz
kpy
h g2
h1
ksy
h2
y+
−2ksy (yv − h1 θv −yb − h2 θb )
−2csy (ẏv − h1 θ̇v − ẏb − h2 θ̇b )
mv v 2
+ mv gθ0 −hg1 mv θ̈0 + Fa
−
R
(1)
Body roll dynamics:
ivr θ̈v
=
2h1 ksy (yv − h1 θv −yb − h2 θb ) + 2h1 csy (ẏv
−h1 θ̇v − ẏb − h2 θ̇b ) − kvr (θv − θb − δa )
+mv g(yv − yb ) − 2d1 2 kaz (θv − θb )
−2d1 2 ksz (θv − θr ) − ivr θ̈0 − Fa h1
(2)
Bogie lateral dynamics:
mv ÿb
=
2ksy (yv − h1 θv −yb − h2 θb ) + 2csy (ẏv − h1 θ̇v
−ẏb − h2 θ̇b ) + 2kpy (yb − h3 θb − y0 ) − 2cpy (ẏb
−h3 θ̇b ẏ0 ) −
mv v 2
+ mv gθ0 −hg1 mv θ̈0 − Fa(3)
R
Bogie roll dynamics:
ibr θ̈b
=
2h2 ksy (yv − h1 θv −yb − h2 θb ) + 2h2 csy (ẏv
−h1 θ̇v − ẏb − h2 θ̇b ) − 2h3 (kpy (yb − h3 θb − y0 )
+cpy (ẏb − h3 θ̇b ẏ0 )) + kvr (θv − θb − δa )
+2d1 2 (kaz (θv − θb ) + ksz (θv − θr ))
−2d2 2 (kpz θb + cp z θ̇b ) − ibr θ̈0 − Fa h2
Railway Vehicle Model And Controller Performance Assessment
2.1 Railway Vehicle Model
The simplified mechanical configuration of the integrated
tilt and lateral system is shown in Fig. 1.
=
(4)
for the additional air-spring state:
θ̇r = −
ksz + krz
ksz
krz
θr +
θv +
θb + θ̇b
crz
crz
crz
(5)
The vehicle model and control system are tested with
specific track inputs including both deterministic (low frequency signals) and stochastic (high frequency signals) features. The deterministic track input was a curved track with
a radius of 1000m and a maximum track cant angle (θ0max )
of 60 , with a transition (150(m)) at the start and end of the
steady curve. The stochastic track inputs represent the irregularities in the track alignment on both straight track and
curves, and these were characterised by an approximate spatial spectrum equal to (2π)2 Ωl v 2 /ft (m2 /(cycle/m)) with a
lateral track roughness (Ωl ) of 0.33x10−8 (m) [4].
b
θ+b
h3
Wheelset
d2
cpy
θ+o
y+
o
Fig. 2: Integrated ARB with lateral actuator
The end-view model consists of a four Degree-OfFreedom (DOF) dynamic system, illustrated in Fig. 2. The
lateral and roll degrees of freedom for both the body and
bogie systems are included. A rotational displacement actuator shown by δa is included in series with the roll stiffness.
Moreover, a lateral actuator shown by Fa is installed in parallel with the original lateral damper between the bogie and
the body. Further details about the model can be found in
[6]. The equations of motion are:
2.2 Controller Performance Assessment
Two main design criterion for the dual-actuator system
controller are summarized below, which needs to meet both
tilt performance and lateral suspension requirements [8].
(i) Provide a fast response on curved track (deterministic
criterion) which is divided into two aspects:
• Pct value for the curve transitions: this is a criterion on
quasi-static lateral acceleration and lateral jerk perceived by
the passengers and was suggested by a British Rail research
study, see [9]. It indicates the percentage of passengers who
will feel uncomfortable as a result of the transition onto the
curve, calculated via a non-linear formula.
• Investigation of the transitional dynamic suspension effects based upon the “ideal tilting” approach [10], a technique which essentially quantifies how closely a particular
control solution fits to the ideal response.
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(ii) Maintain good ride quality in response to track irregularities on straight track (stochastic criterion). The root Mean
Square (R.M.S.) value of the body lateral acceleration on
straight track in response to the track irregularities is traditionally utilized to assess the straight track performance.
More information about tilting train control assessment can
be found in [10]. Associated with ride quality improvement is the constraint on lateral suspension deflection, which
should not exceed the maximum available before bump stops
are reached, i.e. ±60 (mm) is used in this study.
3
Classical Decentralized Control
Details of the CD control can be found in [6], refreshing
here to provide a comparison object for the HD control. Fig.
3 shows the overall system configuration.
provides a faster response compared to PID tilting when the
train starts to negotiate the curve transition, hence reducing
the interaction between the tilting response and lateral suspension. Centring control loop is still used.
4.1
Intuitive Skyhook Damping Lateral Actuator Control
The configuration of intuitive skyhook damping control
with centering loop for lateral actuators is illustrated in Fig.
5. The actuation force is proportional to the absolute body
velocity. A High Pass filter (HP) is used to eliminate the integrator drifting due to zero-offset and also to reduce the low
frequency velocity signal, which in turn reduces the suspension deflection for the deterministic inputs.
Intuitive Skyhook damping control
HP*1/s
Body lateral
acceleration
Active tilting
+
Effective cant deficiency
k2
_
0 (zero) +
Tilting
controller
+
-
Centring control
Suspension roll
Lateral secondary
suspension deflection
kdf
s
Track inputs
Tilt angle
command
Force command to
the lateral actuator
cs
Skyhook damper
1/g
k1
_
s2
1
*
s 2 2[ wi s wi2 s
Lateral acc.
Railway vehicle
system
Fig. 5: Intuitive skyhook damping control with centring loop
Lateral actuator force
The parameters for the lateral actuator control in this design are listed as follows:
Suspension deflection
Lateral actuator
controller
Active lateral suspension
Fig. 3: Classical decentralized control system configuration
Complementary filters
HP*1/s
Body lateral
acceleration
Lateral secondary
suspension deflection
Skyhook damping control
s2
1
*
s 2 2[ wi s wi2 s
Force command to
the lateral actuator
+
cs
+
2[ wi s wi2
*s
s 2 2[ wi s wi2
Skyhook damper
+
-
LP*s
Centring control
kdf
s
Fig. 4: Skyhook damping lateral actuator control with body
lateral centering control
Sequential design process is adopted because the lateral
actuator control loop is a high bandwidth strategy (to attenuate high frequency lateral irregularities) that is intended to
respond faster than the tilting action. The complementary filter skyhook damping control [7] with centring loop is used
for the lateral actuator control driven by the measured body
lateral acceleration and lateral secondary suspension deflection (shown in Fig. 4). Effective cant deficiency (e.c.d.) [6]
is used to drive the tilt actuator with approximate PID control.
4 H∞ -based Decentralised Control
H∞ based Decentralised (HD) control is introduced in
this section. It was found that the H∞ tilting control combined with the intuitive skyhook damping lateral actuator
control can meet all the design requirements, which simplifies the controller design. This is because the H∞ tilting
cs : 59000N ·s/m;
wi : 0.7rad/s;
kdf : 590000N/m;
4.2
Mixed-Sensitivity H∞ Tilting Control
Research on H∞ control started in early 1980s with the
objective to compensate the weakness of LQG control to
deal with good robustness properties [11]. The design process involves the minimization of the H∞ norm of the transfer function from exogenous signals (such as disturbances
and input commands) to the signals which are to be minimized to meet the control objectives. Mixed-sensitivity H∞
control, signal-based H∞ control and H∞ loop-shaping are
three basic types of H∞ control [12].
Mixed-sensitivity is studied in this paper. It addresses the
transfer function shaping problems in which the sensitivity
function S = (I + GK)−1 is shaped along with one or more
other closed-loop transfer functions such as R = KS or
the complementary sensitivity function T = I − S. The
objective of Mixed-sensitivity design is to minimize the H∞
norm of the closed-loop transfer function:
⎡
⎤
W1 S ⎣ W2 R ⎦
W3 T ∞
The norm is usually required to be below a level γ, where
W1 , W2 and W3 are weighting filters for sensitivity transfer function (S), complementary sensitivity transfer function
(T) and control inputs sensitivity (R) respectively. The returned values of S, R and T should satisfy the following loop
shaping inequalities:
σ(S(jw)) ≤ γσ(W1 −1 (jw))
σ(R(jw)) ≤ γσ(W2 −1 (jw))
σ(T (jw)) ≤ γσ(W3 −1 (jw))
5556
(6)
Singular Values
z1 : e
80
W1
z2 : u
S
y1
W2
T
60
L
z3 : y
W3
GAM/w1
40
GAM*G/ss(w2)
K
G
20
y2
Singular Values (dB)
r1 : 0 0
−20
−40
Fig. 6: Mixed-Sensitivity control formulation
−60
s/30 + 1
W1 = 1100
s/0.001 + 1
s/0.1 + 1
W2 = 0.0032
s/30 + 1
s/0.008 + 1
W3 = 0.00032
s/300 + 1
−80
−100
−4
−2
10
0
10
4
10
10
Fig. 7: S and T to conform to GAM/W1 and GAM ∗G/W2 ,
respectively (GAM := γ)
Nichols Chart
40
0 dB
30
0.25 dB
0.5 dB
1 dB
20
−3 dB
0
−6 dB
−10
−12 dB
−20 dB
−20
GM = 5.6 dB
PM = 58.9 deg
−40
−360
(7)
−1 dB
3 dB
6 dB
10
−30
−315
−270
−225
−180
−135
Open−Loop Phase (deg)
−90
−45
−40 dB
0
Fig. 8: Nichols plot for e.c.d
The design can be done in a straightforward way, i.e. using Matlab’s Robust Control Toolbox capabilities[13]. In
fact, using function mixsyn(G,W1 ,W2 ,W3 ) to shape sigma
plots of S and T to conform to GAM/W1 and GAM*G/W2
respectively, as shown in Fig. 7. G is the plant transfer function.
1.4
CD control
HD control
PT
Ideal tilt
1.2
1
Lateral acceleration (m/s2)
4.3 Simulation Results
The proposed HD control system is tested with the 4 DOF
vehicle model and track data presented in section 2.1. The
assessment values are presented in Table 1. The Nichols
chart for e.c.d. is illustrated in Fig. 8, time domain simulation results are illustrated in Fig. 9 and Fig. 10. The
simulation results of HD control show the improvement of
the performance and system robustness compared to the CD
control. The Pct value for seated passengers is reduced to
14%, which is very close to the value for Precedence Tilt
(PT) control (13.5%) (Refering to [4] for the PT and Nulling
Tilt (NT) control assessment). The R.M.S. value of the lateral acceleration on straight track is less than 3.778%, which
illustrates the good ride quality can be guaranteed on the
straight track. The Gain Margin for the tilting control system (with closed lateral actuator control loop) now is 5.6dB
and Phase Margin is 58.9deg.
2
10
Frequency (rad/sec)
Open−Loop Gain (dB)
Fig. 6 illustrates the general control problem configuration for tilting control, r represents a set-point zero reference
command, and the regulated outputs are z1 (the weighted
e.c.d. error signal), z2 (the weighted control signal u) and
z3 (the weighted e.c.d. output signal). Note that regulating
z1 to zero will provide the required 60% tilt compensation,
the regulation of z2 will satisfy control limitation and noise
attenuation at high frequencies, while regulation of z3 is for
system robustness and modelling uncertainty. The usual difficulty in H∞ control design is choosing the weighting filters, normally based on rule of thumb choice or designer’s
experience. Examples of weighting filter choice can be seen
in [12]. The filters employed in this paper: W1 was chosen
to be a low-pass filter with a very low cut-off frequency essentially to enforce integral action on z1 . In contrast, W2 and
W3 were chosen as high-pass filters with pole and zero cutoff frequencies. The weighting filters for the tilting control
are chosen as:
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
0
200
400
600
track (m)
800
1000
1200
Fig. 9: Measured body lateral acceleration
5
Digital Implementation of H∞ -based Decentralised Control
The proposed HD control is further investigated with the
consideration of the controller practical implementation. A
HIL simulation system is setup.
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a 58(m/s)
Table 1: Control system assessment for HD control Lateral acceleration.
Roll gyroscope
Pct (P-factor)
passenger comfort
Deterministic(CURVED TRACK)
HD
-Steady-state(%g)
9.530
-R.M.S. deviation error(%g)
1.800
-Peak value(%g)
12.144
- R.M.S. deviation(rad/s)
0.020
-Peak value(rad/s)
0.111
-Peak jerk level(%g/s)
7.349
-standing(% of pasengers)
50.548
-seated(% of pasengers)
14.214
Stochastic(STRAIGHT TRACK)
- R.M.S. passive(%g)
3.778
- R.M.S. active(%g)
3.569
- degradation (%g)
-5.553
9
CD control
HD control
PT
Ideal tilt
8
7
Body roll angle (deg)
0
−1
0
200
400
600
track (m)
800
1000
1200
Fig. 10: Actual body roll angle
Host PC
Target PC
3.778
3.998
5.802
3.778
3.31
-12.12
(8)
H∞ controller design usually results to rather higher order controller structures (relative to the original size of the
design model). Model reduction based on Schur method is
applied to reduce the controller order down to five order for
further efficient embedded implementation. The frequency
responses of the original controller and reduced order controllers (9th order controller is also studied here) are illustrated in Fig. 12. The equation for H∞ tilt controller (5th
order):
Tt =
TCP/IP
0.1626z 4 − 0.6422z 3 + 0.952z 2 − 0.6277z + 0.1553
z 5 − 4.771z 4 + 9.124z 3 − 8.745z 2 + 4.201z − 0.8099
B
Controller Bode Plot
ta g
–J
RS
23
2/
CA
N
US
FPGA-based
Controller
3.778
3.568
-5.558
−726.4z 2 + 726.4
;
z 2 − 1.974z + 0.9742
−7247z 2 + 7247
Centring loop = 2
z − 2z + 0.9998
4
1
PT [4]
9.530
1.54
12.18
0.018
0.104
6.80
47.62
13.455
HP ∗ 1/s =
5
2
NT [4]
n/a
5.555
19.510
0.032
0.086
10.286
71.411
22.640
are the intuitive skyhook damping control with centring loop
for the lateral actuator:
6
3
CD [6]
9.530
4.576
13.714
0.021
0.104
7.687
53.846
15.674
50
Original Controller (13th order)
5th order Controller
9th order Controller
40
30
Magnitude (dB)
20
Fig. 11: HIL system configuration
5.1 HIL System Configuration
The MATLAB/xPC-Target [14] is employed to provide
the real-time environment for the 4 DOF tilting railway vehicle model. As shown in Fig. 11, the railway vehicle model
is developed in MATLAB/SIMULINK in the host PC. It is
downloaded into the Target PC via the TCP/IP link. High
speed RS232 serial communication (Baud rate is configured as 115200bit/s) is adopted for the data transmission
between xPC-Target and FPGA-based controller. Digital
controllers designed in MATLAB m code are compiled to
C code via Embedded MATLAB, then downloaded into the
Microblaze soft processor in FPGA.
5.2
Digital Controller Design
The controllers designed in s domain are converted to z
domain using Tustin transformation. The equations below
10
0
−10
−20
−30
−40
−50
−60
−2
10
−1
10
0
1
2
10
10
10
Frequency (rad/sec)
3
10
4
10
Fig. 12: Controller frequency response
5.3 HIL Simulation Result
Fig. 13 shows the HIL validation result for the proposed
digital HD controller for integrated tilt and active lateral secondary suspension in high speed railway vehicle (Measured
body lateral acceleration) on curved track. The performance
of the digital controllers with reduced order H∞ controller
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Full order controller
Ideal tilting
LateralAcc (m/s^2)
Reduced order FPGAbased controller
Curved track (m)
Fig. 13: Measured body lateral acceleration
(5th order, in FPGA) is similar to the full order continuous
controller (in simulation). Further work will focus on the
upgrade of the HIL system and investigation of the digital
controller performance in the straight track case (high frequency excitation).
6
[7] H. Li, R. M. Goodall, “Linear and non-linear skyhook damping control laws for active railway suspensions”, Control Engineering Practice, vol7, no7, 1999, pp 843-850.
[8] R. M. Goodall, T.X. Mei, Active suspensions, Chapter 11 in
Handbook of Railway Vehicle Dynamics, (Taylor and Francis). 2006.
[9] CEN: (2007), Railway applications-Ride comfort for passengers Measurements and evaluation, enquiry version pr
EN12299.CEN. Brussels.
[10] R. M. Goodall, A. C. Zolotas, and J. Evans, “Assessment of
the Performance of Tilt System Controllers”, The Railway
Conference a Railtex 2000, NEC Birmingham, UK, November 2000.
[11] G. Zames. “Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms and approximate inverse”, IEEE Transactions on Automatic Control,
vol26, no2, 1981,pp 301-320.
[12] S. Skogestad, I. Postlethwaite, Multivariable feedback control: Analysis and design, Wiley, 2000, Reprinted Version.
[13] The MathWorks. http://www.mathworks.co.uk/products/robust, 2013.
[14] The MathWorks. http://www.mathworks.com/products/xpctarget, 2013.
Summary
In this paper, the integrated mixed-sensitivity H∞ tilt control with skyhook damping lateral actuator control in high
speed railway vehicle is firstly discussed. It aims to further
overcome the control loop interactions in the decentralized
control and improve the performance of using the local integrated suspension control. The simulation results show that
the proposed HD control system can meet both the tilt and
lateral active suspension design requirements. The performance of the HD control is closer to the precedence control
compared to the CD control.
The HIL digital implementation of the proposed HD control is also investigated. The digital controllers are discreted based on Tustin transformation and implemented into
a FPGA-based electronic control unit (with reduced order
H∞ tilting control), which are validated in a xPC-Targetbased HIL system.
References
[1] Alstom. http://www.transport.alstom.com, 2007.
[2] R. Persson, Tilting trains Technology, benefits and motion
sickness, Licentiate Thesis, Railway Technology, Stockholm,
Sweden 2008.
[3] R. M. Goodall, “Tilting trains and beyond - the future for
active railwaysuspensions: Part 1 Improving passenger comfort”, Computing and Control Engineering Journal, vol.10,
no.4, 1999, pp 153-160.
[4] A. C. Zolotas, R. M. Goodall, “Modelling and control of railway vehiclesuspensions”, Lecture notes in control and information sciences, vol.367, 2007, pp 373-412.
[5] H. Zamzuri, A. C. Zolotas, and R. M. Goodall, “Tilt control design for high-speed trains: a study on multi-objective
tuning approaches”, Vehicle System Dynamics, vol.46, no.1,
2008, pp 535-547.
[6] R. Zhou, A. C. Zolotas, R. M. Goodall, “LQG control for
the integration of tilt and active lateral secondary suspension
in high speed railway vehicles”, proceedings of the 8th IEEE
international conference on control automation (ICCA’10),
Xiamen, China, 9-11 June 2010.
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