ADVANCED CONTROL STRATEGIES FOR TILTING RAILWAY VEHICLES A C Zolotas1 and R M Goodall2 Department of Electronic and Electrical Engineering, Loughborough University, Leicestershire LE11 3TU, UK tel: +44(0)1509-227015/227007, fax: +44(0)1509-227014 email: 1
[email protected], 2
[email protected] Keywords: railway dynamics; tilt control; Kalman-Bucy filter ABSTRACT This paper presents the findings of the first stage of a fundamental study of the tilt control problem. A tilting railway vehicle is described via appropriate modelling and a frequency domain analysis illustrates the problems associated with straightforward feedback control. A Kalman-Bucy filter is developed based on 3 inertial measurements (body lateral acceleration, roll and yaw rate) to provide a more effective feedback signal. The filter is re-formulated to estimate not only the states but also the cant, cant rate and curve radius of the track on which the vehicle is travelling. Then a controller is developed based on the estimator ‘true’ cant deficiency output. The performance of the estimator and of the controller is assessed via appropriate computer simulations. INTRODUCTION Tilting trains are provided with a facility whereby the body of the vehicle is leaned inwards on curves to reduce the lateral acceleration experienced by the passengers, and thereby to increase the speed through the curve. In practice this has to be achieved actively, and the use of active tilting systems is an area in which control engineering has provided a major contribution to modern rail vehicle technology, with most modern high speed vehicles now incorporating tilt. Initial studies of tilting trains used feedback control from a lateral accelerometer mounted on the body of the vehicle, but it proved difficult to achieve a sufficiently high speed of response on the transitions at the start and end of curves without causing a deterioration of ride quality on straight track. Most tilting train implementations now use a command-driven system in which a signal from an accelerometer on a non-tilting part of the previous vehicle indicates the required tilting angle, with a straightforward tilt angle feedback controller locally ensuring that each vehicle tilts to the commanded angle [2]. This is commonly known as ’precedence’ tilt - the advanced information enables a sufficient level of filtering to be applied to remove the effect of track irregularities on the tilt command signal.
Such a control scheme is quite complex; amongst other things it must reconfigure when the train changes direction, and it is also difficult to provide a satisfactory performance for the leading vehicle of the train. The difficulties arise from the dynamic complexity of the mechanical system being controlled, but surveys show that a fundamental study of the tilt control problem has not been carried out. This paper reports the findings of a fundamental study which is now being undertaken, with the objective of identifying advanced control techniques which can provide simpler, more effective controllers. The paper describes the physical system and its modelling, provides a frequency domain analysis which reveals the problems with straightforward feedback control, and shows how a state estimator can be used to provide a much more effective feedback signal. MATHEMATICAL MODEL OF A TILTING VEHICLE Figure 1 shows an end-view of a railway vehicle. The model possesses both the lateral and roll degrees of freedom for both the body and the bogie. The vertical degrees of freedom are ignored, however the effects of the vertical suspensions upon the roll motion of the vehicle are included. For simplicity wheelset dynamics are not taken into account. Pairs of parallel spring/damper combinations were used to model the primary (bogie-wheelsets) lateral, primary vertical and secondary (body-bogie) Vehicle body d1
y+v
c.o.g
kvr δa kaz krz
h g1
h g2 cpy
h1
ksy
ksz crz
csy
Vehicle bogie kpz kpy
θv+
cpz c.o.g
h2
y+ b
θ+b Wheelset
h3
d2 θ+o
Rail level
y+ o
Figure 1: End-view of a railway tilting vehicle
lateral suspensions. A representation of a pair of airsprings is used to model the roll effect of the secondary vertical suspension as shown in Figure 1. The model also contains the stiffness of an antiroll bar connected between the body and the bogie, and roll damping may be included if necessary. To provide active tilt a rotational displacement actuator (assumed to be ideal) is included in series with the roll stiffness, i.e. the concept of an ‘active anti-roll bar’ [5]. The end-view model can be represented by Equations 1-4, and corresponds to local track references. The translation and rotation of these reference axes associated with curves are allowed for in the equations. For the airspring model see the Appendix. The symbols and the parameters used in this paper are listed in the Appendix. mv y¨v = −2ksy (yv − h1 θv − yb − h2 θb ) . . . − 2csy (y˙ v − h1 θ˙v − y˙ b − h2 θ˙b ) . . . (1)
ivr θ¨v = −kvr (θv − θb − δa ) . . . + 2h1 {ksy (yv − h1 θv − yb − h2 θb ) . . . + csy (y˙ v − h1 θ˙v − y˙ b − h2 θ˙b )} . . . + mv g(yv − yb ) + 2d1 {−kaz (d1 θv . . . − d1 θb ) − ksz (d1 θv − d1 θr )} − ivr θ¨o
u = δa
(2)
yb
and w =
θb £1
R
y˙ v
θ˙v
y˙ b
θo
θ˙o
θ¨o
θ˙b yo
θr
¤T ¤T
y˙ o
To study the vehicle model, both deterministic and stochastic track features were used in the simulation. The deterministic track used was a curved track with a radius of 1000m and a maximum track cant angle of 6o . A nominal vehicle curving was assumed. Note that the speed of 162 km h curved track is canted to reduce the amount of lateral acceleration perceived by the passengers. The resulting acceleration is referred to as cant deficiency. At each end of the curve there are transition sections of around 2secs long during which the curvature and cant increase steadily.
0
−0.5
−1
−1.5
0
2
4
6
8
10
12
14
16
18
20
16
18
20
time.secs Passenger lateral acceleration 1.5
(3)
1
0.5
0
−0.5
ibr θ¨b = kvr (θv − θb − δa ) . . . + 2h2 {ksy (yv − h1 θv − yb − h2 θb ) . . . + csy (y˙ v − h1 θ˙v − y˙ b − h2 θ˙b )} . . .
0
2
4
6
8
10
12
14
time.secs
Figure 2: Passive (non-tilting) vehicle at 162 km h
− 2d1 {−kaz (d1 θv − d1 θb ) − ksz (d1 θv . . . − d1 θr )} + 2d2 (−d2 kpz θb − d2 cpz θ˙b ) . . . + 2h3 {kpy (yb − h3 θb − yo ) . . . + cpy (y˙ b − h3 θ˙b − y˙ o )} − ibr θ¨o
θv
Body roll angle rel to track
− 2kpy (yb − h3 θb − yo ) . . . − 2cpy (y˙ b − h3 θ˙b − y˙ o ) . . . mb v 2 + mb gθo − hg2 mb θ¨o R
£ x = yv
0.5
mb y¨b = 2ksy (yv − h1 θv − yb − h2 θb ) . . . + 2csy (y˙ v − h1 θ˙v − y˙ b − h2 θ˙b ) . . .
−
(5)
where
roll angle.degrees
mv v 2 + mv gθo − hg1 mv θ¨o R
x˙ = Ax + Bu + Γw
accel.m/s2
−
dynamic modes which result are often referred to as the “sway modes”. The next step is to represent the above set of equations in a state space form for the purposes of system analysis and control design [4].
(4)
Equation 2 includes an end moment effect mv g (yv − yb ), which models the roll effect of the body weight due to the lateral displacement of its centre of gravity (this effect is neglected in Equation 4 owing to the high stiffness of the primary suspensions). Clearly, the mathematical model indicates a complex system which is characterised by significant coupling between the lateral and roll directions, and the
Figure 2 shows the body lateral acceleration of the passive model on the curve (11.95%g steady state), together with the corresponding body roll angle. Note that, because the lateral suspension acts significantly lower than the body centre of gravity, the body rolls outwards on curves, increasing the acceleration felt by passengers. The stochastic track inputs represent the irregularities in the track alignment on both straight track and curves, and these were characterised by an ap2 m2 ) proximate spatial spectrum equal to Ωfl v3 ( cycle/m s −8 with a lateral track roughness {Ωl } of 0.33 × 10 m and a nominal forward vehicle speed of 162 km h . The
ride quality provided by the passive vehicle at the nominal speed is 1.4%g(the RMS lateral acceleration). TILT CONTROL OBJECTIVES The advantage of body tilt is the operation of vehicles at higher speeds while traversing curves. Hence, the performance of the tilt control system on the curve transitions is critical. Primarily the passenger ride comfort provided by the tilting vehicle should not be degraded compared to the non-tilting vehicle speeds. The main objective of a tilt control system is to provide an acceptably fast response to changes in track cant and curvature (deterministic features) while not reacting significantly to track irregularities (stochastic features). In any tilt control system there is a fundamental trade-off between the vehicle curve transition response and straight track performance. It should be noted at this point that any control system directly controls the secondary suspension roll angle and not the vehicle lateral acceleration. Incorporating an excessively fast controller may provide high roll rates and also jerk levels which are unacceptable. On the other hand, a slow controller will provide low roll rates and probably jerk levels, giving an unacceptable increase of the lateral acceleration during the curve transition before compensating by tilting the vehicle body. The curving lateral acceleration response of the vehicle consists of the following two components: 1. that due to the body tilt and the deterministic track features (cant and curvature); 2. that due to the suspension dynamic response (lower sway oscillations) to both deterministic and stochastic track features In summary, the main performance requirements for the tilt control system are: 1. to reduce the lateral acceleration perceived by the passengers on curves 2. to provide a comfortable response during curve transitions (tilting trains are designed to operate at higher speeds and the curve transition time therefore decreases) 3. to maintain the straight track performance within acceptable limits (specified as not more than 7.5% deterioration compared to the passive suspension system at the same speed). From a control point of view the objectives of the tilt control system can be translated as: increasing the response of the system at low frequencies (deterministic track features) while reducing the high frequency system response (stochastic track features).
BASIC CONTROL SCHEME The basic tilt control system, which is a classical application of negative feedback, can be seen in Figure 3. It is based on the use of a body mounted lateral accelerometer to provide a measurement of Track input disturbances curvature, cant, lateral track irregularities
-
(measured lateral acceleration)
Vehicle Dynamics (plant model)
Controller
(suspension roll)
-
inferred composite tilt feedback signal
+
..
ym
θs 1 g
θd (equivalent cant deficiency angle)
Figure 3: Basic Tilt control system the equivalent cant deficiency angle, i.e. the true cant deficiency plus the suspension dynamic effects. The control input comprises an angular displacement (δa ) provided by a rotary actuator in series with the anti-roll bar, which in turn provides a torque to the vehicle body. Nulling control is used to provide full compensation for the lateral acceleration on steady curve. The problem is that passengers will feel no acceleration while experiencing the sensation of rotation in the roll direction, consequently a significant portion of them will experience motion sickness. The use of partial tilt [2] can resolve this by using a portion of the secondary suspension roll angle in the feedback path (dotted line in Figure 3) providing a tilt action to compensate only for a proportion of the lateral acceleration. This paper is concentrated upon partial tilt only, because (in practice) this is what will be used. Conclusions about the stability of the closed-loop system may be drawn by investigating the open loop frequency response of the system. Figure 4 shows the uncompensated (nominal) and the designed open loop after the introduction of a P+I controller. Note that while gain reduction was required to stabilise the closed loop system, the opposite applies in the case of fast tilt response. Hence, there must be a compromise between the tilt response and the acceleration. The second part of Figure 4 compares the time response of the non-tilting and tilting vehicles. The tilt controller provides 60% compensation on steady curve at a speed of 209 km h , however the suspension dynamics during the transition has a significant effect on the system (the response is not as desired). The ride quality provided
by the active system at 209 km h is 18.75mg RMS lateral acceleration (6.5% degradation compared to the passive system, 17.61mg, at the same speed). Open−Loop Gain (dB)
Actuator i/p to inferred composite tilt 40
−3 dB −6 dB
0
−12 dB
compensated
−20 dB
−20
−40
−350
−300
−250
−200
−150
−100
−40 dB 0
−50
Open−Loop Phase (deg) Perceived passenger lateral acceleration 3
2
acceleration.m/s
£ x = yv w ˜
tilting at 58m/s 2
(7)
£ ¤T ˜ xk = x w
−1 dB
3 dB 6 dB
nominal
x˙ k = Ak xk + Bk u + Γk wk where
0 dB 0.25 dB 0.5 dB 1 dB
20
as states rather than disturbance inputs [3]. The reformulated state space system is given by:
non−tilting at 45m/s
θv
£ = θo
yb θ˙o
y˙ v
θb
θ˙v
y˙ b
θ˙b
h and wk = θ¨o
¤ 1 T R
θr 1˙ R
¤T
iT
1
· ¸ ˜ A Γ Ak = 0 ∆LP
0 −1 −2
0
2
4
6
8
10
12
14
16
18
03×1
¤T
20
time.secs
Figure 4: Basic partial nulling tilt control scheme The sensor in such a strategy exists within the control loop, which results in a difficult controller design due to the interactions between the suspension and controller dynamics. It was also found that the system is non-minimum phase which results in a stability threshold point for the controller design (i.e. speed of response is limited up to a saturation point set by the non-minimum phase zero). This is actually the primary disadvantage of this strategy. STATE ESTIMATION (KALMAN-BUCY FILTER)
£ ′ Γk = Γ
(0
0)
(1
0)
(0
¤T 1)
Appropriate low-pass filters (∆LP ) were applied to the extra (track) variables in the re-formulated matrix Ak because the high frequency components of the track are in fact track irregularities. The sensor noise levels are represented by vector ν and characterised by a covariance matrix R. The necessary measurements for the design of the estimator were given by three sensors mounted on the body vehicle: an accelerometer and two gyroscopes. The accelerometer was used to measure the lateral acceleration of the vehicle body while the two gyroscopes measured the body roll and yaw speeds respectively. The output equation for the sensors is given by: yk = Ck xk + Dk u + ν
The previous section has shown that the suspension dynamic interactions have a negative effect on the controller design. Therefore a more effective feedback signal is required and this is the ‘true’ cant deficiency angle, i.e. the feedback signal which is largely unaffected by the suspension dynamic interactions. The new feedback signal is given by: 2
v − (θo + θv ) θ˜d = gR
£ Bk = B
(6)
In theory a Kalman-Bucy filter can be designed based upon Equation 5 in connection with the output equation which represents the measurements. However, the cant deficiency feedback is associated with signals of the disturbance vector w as shown in Equation 6. These signals are related to the track, on which the vehicle is travelling, for which there is no a priori knowledge. Measuring such track parameters is not a practical solution. Hence, the system state space should be re-formulated for the design of the Kalman-Bucy filter in order to treat R1 , θo and θ˙o
(8)
Ck and Dk are based upon the relative rows of Ak and Bk . The Kalman-Bucy filter can be now designed using Equations 7 and 8. The state estimates can be calculated by solving the following differential equation: x˙ e = Ak xe + Bk u + Lk (ym − Ck xe − Dk u)
(9)
where xe is the vector of the re-formulated state estimates and Lk is the Kalman-Bucy filter gain matrix which is designed off-line [1]. The performance of the Kalman-Bucy filter can be assessed by tuning the covariance matrix Qk for the track noises where Qk = diag(Qθ¨o , Q 1˙ ). R
Estimation Results For the purposes of tilt control the Kalman-Bucy filter was first designed for the deterministic case and then assessed in the stochastic case. The scheme can be seen in the block diagram of Figure 5. Figure 6 shows the estimated ‘true’ cant deficiency on curved track with the estimation error.
Track input disturbances curvature, cant, lateral track irregularities
u
K(s)
vehicle body roll and yaw gyroscope vehicle body lateral acceleration
Vehicle Dynamics A, B, C, Γ, D, H Sensor Noise R Sensors
estimated outputs
Kalman-Bucy system model (process noise Qk )
-
Kalman Gain
Actuator i/p to inferred composite tilt
+
Open−Loop Gain (dB)
estimated feedback signals
the inferred composite tilt feedback for the two control cases. Both schemes incorporate a P+I classical controller design. It can be easily seen from the figure that the new feedback signal resulted in a much easier controller design compared to the previous basic control scheme. The dynamic complexity is still preserved but the non-minimum phase characteristic has disappeared.
Figure 5: Kalman-Bucy filter estimation scheme
40
scheme 1: basic nulling scheme 2: estimator based
−1 dB
3 dB 6 dB
nominal
−3 dB −6 dB
0
−12 dB
designed scheme 1
−20
−20 dB designed scheme 2
−40
−350
−300
Cant deficiency Estimate
−250
−200
−150
−100
−40 dB 0
−50
Open−Loop Phase (deg) Perceived passenger lateral acceleration
1.4 2
acceleration.m/s2
estimate 1.2
1
cant deficiency.m/s2
0 dB 0.25 dB 0.5 dB 1 dB
20
0.8
1.5
tilting at 58m/s
0.5 0 −0.5 −1
0.6
non−tilting at 45m/s
1
0
2
4
6
8
10
12
14
16
18
20
time.secs
0.4
Figure 7: Estimator based tilt control scheme
0.2
error 0
−0.2
0
5
10
15
20
25
time.secs
Figure 6: ‘True’ Cant deficiency estimate at 162 km h
The errors are mainly due to the sensor noise levels. Note that some extra errors associated with the unknown information of cant acceleration and stochastic lateral track were also expected. In the simulations, the sensors noise levels were set to 3% of the maximum values, determined by adding the peak value of the responses on pure curved track plus 3 times the RMS values on straight track irregularities. The results obtained from the estimator are very close to the true values. The estimator performed well, although the deterministic and stochastic track features are unknown and not included with the vehicle dynamics. ESTIMATOR BASED TILT CONTROLLER DESIGN This section presents the design of a classical controller based upon the estimated ‘true’ cant deficiency to provide partial tilt as in the previous case. For this purpose the performance of the estimator is also assessed. Figure 7, the first part, compares the designed OL frequency responses of the actuator input to
The time domain simulation results shown in Figure 7, second part, shows that the control system responds faster on the curve. Small errors are introduced due to the estimation process as expected (reducing the sensor noise helps to give further improvement). Unfortunately even in this case the oscillations due to the lateral suspension effects still exist. This is mainly due to the coupling through the body roll, which the control system takes into account. These problems can be overcome by using LQR design methods. Noticeably if fewer sensors are included in the Kalman filter design, the estimation process cancels out some of the dynamic oscillations. Moreover, preliminary results have shown that this control scheme improves the straight track performance of the vehicle. The active system provides 17.26mg RMS lateral acceleration at 209 km h , which is 2% improvement compared to the passive system at the same speed. CONCLUSIONS This paper has presented the modelling of a tilting railway vehicle and illustrated the fundamental problem using straightforward control. A state estimator (Kalman-Bucy filter) was designed based on 3 sensors (1 accelerometer and 2 gyroscopes) to provide a more effective feedback signal for control design. The filter was re-formulated in such a way in order to include track variables as states rather than disturbances. Computer simulation has shown that the estimator provides a very good estimate of the ‘true’
cant deficiency. It was also illustrated that the estimator scheme enables a much easier control design compared to the basic control scheme. The next stage is to undertake a full analysis which quantifies the straight and curved track performance, including a comparison with the “precedence tilt” strategies which are currently used in rail vehicles.
h2
APPENDIX
h3
Airspring modelling
hg2 hg1
θv
FZ
FZ2 ksz kaz
krz
FZ1
REFERENCES zr
[1] K. Brammer and G. Siffling. Filters. Artech House, 1989.
crz FZ2
FZ
m Anti-roll bar stiff./bogie, 2,000,000( N rad ) N Primary vertical stiff., 2,000,000( m ) Primary vertical damp., 20,000( Nms ) Primary lateral stiff., 35,000,000( N m) Primary lateral damp., 16,000( Nms ) Airspring semi-spacing, 0.90(m) Prim. vert. suspen. semi-spacing, 1.00(m) 2ndary lateral suspension height(body cog), 0.9(m) 2ndary lateral suspension height(bogie cog), 0.25(m) Primary lateral suspension height(bogie cog), -0.09(m) Bogie cog height(rail level), 0.37(m) Body cog height(rail level), 1.52(m)
zv
FZ1
(+) Sign convention
Kalman-Bucy
[2] R. M. Goodall. Tilting trains and beyond the future for active railway suspensions, part 1 improving passenger comfort. Computing and Control Engineering Journal IEE, pages 153–160, August 1999.
θb zb
Figure 8: Airspring model
Disregarding vertical motions and substituting d1 θr for zr : Fz = −kaz (d1 θv − d1 θb ) − ksz (d1 θv − d1 θr )
(10)
ksz krz (ksz + krz ) θr + θv + θb + θ˙b θ˙r = − crz crz crz
(11)
NOTATION
yv , yb , yo θ v , θ b , δa θo , R θr v mv ivr mb ibr −−− kaz ksz krz crz ksy csy
kvr kpz cpz kpy cpy d1 d2 h1
Lat. displ. of body, bogie and track Roll displ. of body, bogie and actuator Track cant, curve radius Airspring reservoir roll deflection Vehicle forward speed Half body mass, 19,000(kg) Half body roll inertia, 25,000(kgm2 ) Bogie mass, 2,500(kg) Bogie roll inertia, 1,500(kgm2 ) Values per bogie side Airspring area stiff., 210,000( N m) Airspring series stiff., 620,000( N m) Airspring reserv. stiff., 244,000( N m) Airspring reserv. damp., 33,000( Nms ) Secondary lateral stiff., 260,000( N m) Secondary lateral damp., 33,000( Nms )
[3] L. Hong and R. M. Goodall. State estimation for active steering of railway vehicles. IFAC ’99, 1999. [4] T. X. Mei, R. M. Goodall, and L. Hong. Kalman filter for the state estimation of a 2-axle railway vehicle. 5th European Control Conference ’99, Karlsruhe, Germany, Aug-Sept 1999, 1999. [5] J. T. Pearson, R. M. Goodall, and I. Pratt. Control system studies of an active anti-roll bar tilt system for railway vehicles. Proceedings, Institution of Mechanical Engineers, 212(F1):pp43–60, 1998.
