vehicle system using stochastic optimal control

Journal ofSound

and

Vibration (1987) 115(3), 473-482

AN ACTIVE LATERAL SUSPENSION TO A TRACK/VEHICLE SYSTEM USING STOCHASTIC OPTIMAL CONTROL T. YOSHIMURA Department of Mechanical Engineering, The University of Tokushima, Minamijosanjima 2- 1, Tokushima 770, Japan AND

N. ANANTHANARAYANA

AND

D. DEEPAK

Research Designs and Standards Organisation, Ministry of Railways, Manak Nagar, Lucknow 226011, India (Received 26 April 1986, and in revised form 21 August 1986)

An active suspension design to improve lateral ride quality and stability in a track/vehicle system subject to lateral track irregularity is presented. The measurement of the state variables is performed in a noisy environment, and unknown state variables of the system are estimated from the measurement data by using a Kalman filter. It is assumed that the lateral track irregularity and measurement noise are Gaussian random processes, respectively. The optimal control for the active suspension is determined by minimizing the quadratic performance index composed of the state variables and control efforts, and then the active suspension structure has a cascade feedback loop composed of the Kalman filter and the optimal controller. The numerical results indicate that the proposed active suspension provides much improved lateral ride quality and stability. 1. INTRODUCTION

Attention to active suspension designs of track/vehicle systems for high speed transportation has increased in recent years, and a number of investigations concerning vertical, longitudinal and lateral motions subject to random irregularity of tracks have been proposed [l-8]. Passive suspensions [9], which are composed of a function of relative quantities for the state variables of systems, have frequently been used for the improvement of ride quality and stability, but performance limits with passive suspensions are wellknown for high speed transportation, occurring because the suspensions are determined by selection of primary and secondary stiffness and damping parameters. Unlike passive suspensions, active suspensions require an external force, sensors and actuators, and are composed of a function of absolute quantities for the state variables of systems. The suspension structure is determined by a trade-off between the controlled state variables and control efforts, and the control policy for the active suspension is found by minimizing the given performance index. Therefore, active suspensions are more costly in respect to their structure, but can provide greater improvements for high speed transportation than passive suspensions. In the reports already published, the state variables to be selected as the performance index were frequently the rattlespaces and the carbody accelerations, and some state variables of systems were measured directly [l-7]. This paper is concerned with active suspension design from the view-point of stochastic optimal control [8,10, 1l] to improve lateral ride quality and stability in the track/vehicle system subject to lateral irregularity of the tracks [4]. The measurement of the state 473 0022-460X/87/120473+10$03.00/0

0 1987AcademicPress

Inc. (London)

Limited

474

T.

YOSHIMURA,

N. ANANTHANARAYANA

AND

D. DEEPAK

variables is performed in a noisy environment [S], and the noise contained in the measurement data is assumed to be a Gaussian random process with known statistical properties. A Kalman filter is used in the suspension system in connection with estimating all the state variables of the system from the measurement data. The control policy for the active suspension is determined by the linear combination of the estimated state variables. 2. DESCRIPTION OF THE PROBLEM Consider the model shown in Figure 1 [4] for the active suspension design of a track/vehicle system with tangent tracks and constant vehicle speed. The equations of motion for the carbody and trucks are described by a set of linear differential equations subject to lateral track irregularity, and the effects of conicity and creep at each wheel, the creep characteristic being assumed to be linear in the operation range. The equations of motion for a reduced half carbody lateral mass roll inertia (LMRIO) model [4], where the stiffness between axles and bogie frame is assumed very large and the yaw motion of the carbody is neglected [4], are as follows: M,ci;,= -(I$

+=$,)q,

-cc,,

+4fJ

V)ti, +4&I*+ &,q,+ C,& - &rfVI4

-Cyth244+&(Y,+Y,);

(1)

truck yaw equation: I,&= -~f,~~~l~,~q~-~~,,+~~J,+~~,,~2~q2-~~~f,~2+f,~2~l~+~,,l~2 +GLb~l~o+

carbody lateral equation: M&=2&q,

K,,a)y,

+ W&~/r0

+2C,,Q, -2K,,q,

- &,a)~,;

(2)

-2Cyrgs+2K,,h,q4+2C.“rh294;

(3)

carbody roll equation: I&, = -2K,&I,

-2C,,U,

+ 2K,,hzq, + X,,h,&

-2(K,,h:+2K,b:)q,-2(Cy,h:+2C,b:)&. The definitions of variables, ql, q2, q3 and q4, used in equations (l)-(4)

Figure 2. (A list of nomenclature

is given in the Appendix.)

0

Ib Mb

I

Figure 1. Geometry of lateral dynamics model.

(4)

are indicated in

TRACK/VEHICLE

Figure

ACTIVE

2. Definitions

LATERAL

q2,q3and

of q,,

475

SUSPENSION

q4.

The lateral track irregularity is described by a center-line misalignment error y, from the reference center-line. The irregularity has the spectral density function [3] @,(L’) = A/O2

(9

and y is generated by processing n(t) as (6)

i = 77(t), where the covariance of q(t) is given by E[~(f)+)]=~AVS(t-+q+(t-T).

(7)

The assumption expressed in equation (5) is the most popular one for spectral densities of irregularities of tracks; other forms have been presented [2,4,12]. 3. DYNAMIC SYSTEM WITH NOISY MEASUREMENT The active suspension efforts, which take place internal to the vehicle, work on individual trucks and react against the carbody as lateral force and steering torque. Since the ride quality mainly depends upon the effects of lateral and roll accelerations of carbody in the system, an integral-type controller composed of lateral force and steering torque is needed to control these accelerations. When the terms corresponding to the controller are substituted into the equations of motion (l)-(4), the resultant equations differentiated with respect to time become ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ii;

=

Fz,,

41+

F&2

+

F&2

+

(8)

G2u2

+

r,,

TI

+

(9)

r427),,

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

(10)

& = F,,& +

(11)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

F,,ci’,

where the coefficients of these equations are given by 6, =-(&,+2&,)/M,, 66 = C,,,/ M,, Fa

=

F22 F27

=

-J&h,/

=

-CC,,,

F2,

M,

= Fd4

-(K,,+2K,,+2K,,a2)/I,,

F.zs, = 2&u/Mbr Ft,,= 2&M Mb,

V)l M,,

+4&l

-CA,/ =

F4,

M,

-[4(_Lb2+~~a2V

Fe5= --2&t/M,, Fa = 2C,t/Mb, Fa = 2Cy,h,lMb, 5, = -2~y,h,/~b, 615

=

2Kyrh2l

lb,

66

=

2 Cyth2l

6, = &,I M,,

Fz = 4fi.I M,,

lb,

=

--WN(cJ,),

&I/k FM= -2Cytl Mb, V-C

432

=

-2C,,h,l

Ib,

476

T. YOSHIMURA, Fx7 =

N. ANANTHANARAYANA

G, = -21 M,n

Ga = l/I,,

r,, = KwI M,,

D. DEEPAK

F,, = -2( C,.,hf + 2C,,b:)/ I,,,

-2(K,.,h:+2K,b:)/I,,

G, = 11 M,,

AND

G, = 2hJ I,,,

r4, = (21;bh I r. + K,,,a )I I,,

I;> = Ku, I M,,

&+)I&. r4, = ( 2fxbhlraFrom equations can be derived:

(8)-( 1 l), the following

state variable

expression

in a vector-matrix

form

where

0

1

0

0

0

0

0

0

6, 0

62 0

53

0

65

56

6,

68

0

1

0

0

0

0

FJ, 0

0 0

6, 0

F44 0

0

0

0

0

0

1

0

0’

FU 0

F.Q 0

0

0

Fm

0

FM 0

Fti,

0

F65 0

0

1

F,,

62

0

0

6,

66

EST

0

0

0

0

G 0

0 0

rz, 0

r,, 0

0 0

G2 0

r,, 0

r,, 0

G, 0

0 0

0

0

0

0

Gs,

0

0

0

From equation

,

yP

(7), the covariance

E[qWqT(dl= The measurement

device,

designed

of q is given as

’ 6(t-~)~Q~8(t-~). [4f 0 4/ 1 for a noisy environment, z=&h2ij4+&

where the covariance

(13) is expressed

by (14)

of .!j is given by

E[tYtM~)l= rf 6 (t- 7). From equation

68 _

(14), the state variable

expression

(15)

becomes

z=hTx+& where hA [O,O, O,O, 0, l,O, -hZ-jT.

(16)

TRACK/VEHICLE

ACTIVE

LATERAL

SUSPENSION

The system described by equations (12) and (16) includes the controllability

477

and observa-

bility conditions [ 10, 111.

4. DETERMINATION

The performance given by

OF THE ACTIVE

SUSPENSION

index to determine the optimal control for the active suspension is J = E[x’Q,x + uTR,u]

(17)

and this expression can be equivalently replaced by I=lim$ T+a

I

T E[xTQ,x+uTRcu]

dt.

(18)

0

The steady state optimal control u” to minimize expression (18) is determined as [ 10, 1l] u” = -K ci = _R-‘eGTP*; c 7

(19)

where PF is the steady state solution of the equation P, = P,F+ FTP, - P,GR;‘GTP,

+ Qc

(20)

and i is the estimate of x obtained as the output of the Kalman filter, i=Fi+Gu’+kJ-hTx),

k, = PFh/ r,

(21,22)

In equation (22), PT is the steady state solution of the equation P, = PfFT + FP, - qfhhTPf/ rf + rQfrT.

(23)

When equations (19) and (16) are substituted into equations (12) and (21), respectively, the latter become ii = (F - GK, - k,h’)f

i=Fx-GK,i+rq,

+ k,hTx + k&.

(24,25)

The dynamic system combined with x and i then has the form e=ke+&,

(26)

where

and the covariance of u is given by E[u( t)uT( r)] =

[

1

Qf o r0 s(t-7). f

(27)

$ has stable eigenvalues even if F has unstable eigenvalues, since equations (12) and (16) include the controllability and observability conditions. The elements of QE and R, are selected so as to contain the lateral and roll accelerations of the carbody, and the lateral force and steering torque. Therefore, other elements of QC except for QCb6and Q ,288, and those of R, except for R,,, and I& may be set to zero, respectively. The stable property of equations (20) and (23), PC = P, = 0 at t + co, is guaranteed since the system includes the controllability and observability conditions. The suspension system is designed as shown in Figure 3 where the Kalman filter and optimal controller are combined in cascade form.

478

T. YOSHIMURA,

N. ANANTHANARAYANA

AND

D. DEEPAK I

K 7

I

Dynamic system

r

I------’

+-J+ L------l

Optimal controller

Kolman filter

L

Figure 3. Suspension

J

system designed by stochastic optimal control.

TABLE

1

Control gains with Rc,, and Rcz2 as parameters

10-* 1O-4 1o-6 lo-” 1o-4

1o-4 1o-6 1o-4 1o-8

10-l 1o-4 1o-6 lo+ 1o-4

lo@ 1o-4 1o-6 1o-4 lo-*

IO-*

0.324x 0.324 x 0.324 x 0.597 x 0.374 x

10’ 10’ lo5 10” 10

-0.120~ 10’ -0.119~ 10’ -o~119x105 -0.300x lo* -0,134x 10’

0.346 x 10’ 0.343 x 10’ 0.342 x 10’ 0.616X lo4 0.357 x 10-l

-0.208 -0.208 -0.208 -0.405 -0.255

x x x x x

lo6 10’ 10’ IO’ IO’

-0,305 -0.303 -0,301 -0.586 -0.368

0.589x 10’ 0.598~10’ 0~608x10’ 0.102x 10 0.713x 10’

0.134x 10’ 0~171X105 0.134~10’ 0.171 x 10’ 0,134~ 10’ 0.171 x 10’ 0.252x lo4 0.325 x 10’ 0.155x 10’ 0.199x lo5

TABLE

x x x x

lo4 lo4 lo4 IO’

0.837 x 0.872x 0.877 x 0,289~ 0,117~

10’ 10’ 10’ 10 IO-’

-0.364~ -0.364 x -0.364x -0.774x -0.106

0.111 x 10’ 0.111 x 10’ 0.111 x 10’ 0.204x 10’ 0,129~ lo5

0.278x 0.278x 0.278x 0.527x 0.353~

10’ 10’ 10’ lo4

lo4 lo4 lo4 10 lo4

-0.455 -0.455 -0.455 -0.825 -0.401

x x x x

0,161 x 0.162x 0.163x 0.414x 0.228x

lo4 lo4 lo4 10’

0.200~ 10’ 0.200~ 10’ 0.200~ 10’ 0.405x 10” 0.793x lo-’

lo4 -0,346x 104 lo4 -0.346x 10’ lo4 -0.346~ lo4 10 -0640 x 10 lo4 -0.387~ lo4

2

Filter gains with rf as a parameter !i lo-* lo-’ IO-*

k II 0.555 0.141 x 10 0.316~ 10

42

k13

0.570 0.192x 10’ 0.652x lo*

0.646 x 10-l 0.238 0.885

b

kIS

-0.982 -0.882x 10-l -0.250 x 10 -0.207 -0.517 x 10 -0.289

sb 0.360x 10 0.908x 10 0.195 x lo*

4,

47 -0.%5x -0.227 -0.316

10-l

-0.271 x 10 -0.539 x 10 -0.103 x lo2

TRACK/VEHICLE

ACTIVE

LATERAL

SUSPENSION

479

5. SIMULATION STUDY The parameter values used for the simulation are all the same as in reference [4] except for V = 83.33 m/s, and Qf, R, and 9 are taken as parameters. The steady state solutions, Pr and P/*, of equations (20) and (23) are obtained by iterative computation in which the Adams method [13] is used, the initial guesses for the solutions being simply taken as P,(O) = I and PI(O) = I, respectively. The computed results for the control and filter gains are, respectively, listed in Tables 1 and 2, where QCh6= Qcs8= 1. From the tables, it is seen that the control gains seem insensitive to variation of R,,i (&), but sensitive to variation of R,, ,/ Rcz2, and that the filter gains increase more in absolute values as the values of rf decrease. Applying the Adams method to the discrete-time evolutions of 0, the covariance of u at the discrete time id t (i and At denote a positive integer and time increment, respectively) being given by

0 s,, 1w v/At

(28)

which is derived from the continuous expression given in equation (27). Typical results of time evolutions for the carbody lateral acceleration & and its estimate &, the carbody roll acceleration C&and its estimate &, and the controls u, and uz are,

T E z f ? 2 s 5 P E s

6 4 2 0

-7

If

-2 -4 -6

Time f(s)

Time 1(s)

Figure 4. Time evolutions of (a) ij3 (-) and & (- - - -) in the suspension system, (b) & (-) in the suspension system, (c) u, (-) and u2 (- - - -) in the suspension system and (d) ii3 (-) under no control.

and & (- - - -) and C&(- - - -)

480

l-. Y0SHIMURA.N.

ANANTHANARAYANA

AND

D. DEEPAK

respectively, shown in Figures 4(a), (b) and (c) where R,, , = Rczz = 1 and rf = 10e3. Figure 4(d) shows the time evolutions for & and ijQunder no control. It is assumed that all the initial conditions of 8 are set to zero, and the time increment At for the Adams method is taken as 0.01 second, as determined by considerations of numerical accuracy and computational load. It is seen from Figures 4(a)-(c) that the time evolutions of &, ij4, U, and u2 reach steady state after a transient state, and that the estimates ij3 and $, from the Kalman filter show significant performance of the system. Figure 4(d) shows that the time evolutions of & and & diverge with time, while those when the active suspension is operative remain stable.

6. CONCLUSION

An active suspension design to improve ride quality and stability in a track/vehicle system has been proposed from the viewpoint of stochastic optimal control. Estimates of state variables in the system have been obtained as output of a Kalman filter. The active suspension system structure is a cascade combination of the Kalman filter and optimal controller. The numerical results have indicated the effectiveness of the proposed active suspension system in improving ride quality and stability.

REFERENCES 1. G. N. SARMA and F. KOZIN 1971 American Society of Mechanical Engineers Journal of Dynamic Systems, Measurements, and Control 93, 233-241. An active suspension system design for the

lateral dynamics of a high-speed wheel-rail system. 2. J. K. HEDRICK, G. F. BILLINGTON and D. A. DREESBACH

1974. American Society of Mechanical Engineers Journal of Dynamic Systems, Measurements, and Control 96, 192-203.

Analysis, design and optimization of high speed vehicle suspensions using state variables techniques. 3. M. ‘fOMrzUKA 1976 American Society of Mechanical Engineers Journal of Dynamic Systems, Measurements, and Control 98, 309-315. Optimum linear preview control with application to vehicle suspension-revisited. 4. P. K. SINHA, D. N. WORMLEY and J. K. HEDRICK 1978 American Society of Mechanical Engineers Journal of Dynamic Systems, Measurements, and Control 100,270-282. Rail passenger vehicle lateral dynamic performance improvement through active control. 5. D. HROVAT and M. HUBBARD 1981 American Society of Mechanical Engineers Journal of Dynamic Systems, Measurements and Control 103, 228-236. Optimum vehicle suspensions minimizing RMS rattlespace, sprung-mass acceleration and jerk. 6. G. M. CELNIKER and J. K. HEDRICK 1982 American Society of Mechanical Engineers Journal of Dynamic Systems, Measurements, and Control 104, 100-106. Rail vehicle active suspensions for lateral ride and stability improvement. 7. R. J. CAUDILL and L. M. SWEET 1982 American Society of Mechanical Engineers Journal of Dynamic Systems, Measurments, and Control 104,238-246. Magnetic guidance of conventional railroad vehicles. 8. T.YOSHIMURA,N.ANANTHANARAYANA~~~D.DEEPAK~~~~ JournalofSoundandVibration 106, 217-225. An active vertical suspension to track/vehicle systems with noisy measurement. 9. T. MATSUDAIRA, N. MATSUI, S. ARAI and K.YOKOSE 1969 American Society ofMechanical Engineers Journal of Engineering for Industry 91, 879-890. Problems on hunting of railway vehicle on test stand. 10. J.S.MEDITCH 1969Stochastic Optimal Linear Estimation and Control. New York: McGraw-Hill. 11. K. J. ASTROM 1970 Introduction to Stochastic Control Theory. New York: Academic Press. 12. D. S. GARIVALTIS, V. K. GARG and A. F. D'SOUZA 1981 American Society of Mechanical Engineers Journal of Mechanical Design 103, 871-880. Fatigue damage of the locomotive suspension elements under random loading. 13. A. RALSTON and H. S.WILF 1967 Mathematical Methods for Digital Computers I. New York: John Wiley & Sons.

TRACK/VEHICLE

ACTIVE

APPENDIX:

EC1 F

fY f, G

h

h, I

Ih I, J

KC

kf

Kh K,.,

f&l 4x

4k

Mts M PC

PT Pf

PT 91

q2 93

q4 Qf

5‘ K ‘/ fb

LATERAL

SUSPENSION

NOMENCLATURE

track irregularity parameter half truck wheel base half track gauge half lateral distance between vertical springs secondary vertical damping per truck side secondary lateral damping parameter secondary yaw damping parameter expectation operator system dynamics matrix longitudinal creep coefficient lateral creep coefficient control distribution matrix observation matrix distance of carbody from lateral suspension plane identity matrix carbody roll moment of inertia truck yaw moment of inertia performance index control gain matrix filter gain matrix secondary vertical stiffness secondary lateral stiffness secondary yaw stiffness lateral gravitational stiffness yaw gravitational stiffness carbody mass truck mass solution of Ricatti equation for control steady state solution of Ricatti equation for control solution of Ricatti equation for filter steady state solution of Ricatti equation for filter truck lateral displacement truck yaw angle carbody lateral displacement carbody roll angle intensity of q intensity of 7j positive semidefinite matrix in performance index positive definite matrix in performance index intensity of 5 nominal wheel rolling radius time time in performance index superscript, transpose of a vector or matrix control vector lateral force steering torque vehicle forward velocity state vector estimate of x track misalignment at front wheelset of truck track misalignment at rear wheelset of truck measurement noise disturbance matrix Kronecker’s delta function Dirac’s delta function zero-mean Gaussian white noise vector zero-mean Gaussian white noise

481

482

@lx) R

At

T. YOSHIMURA,

N. ANANTHANARAYANA

zero-mean Gaussian white noise combined state vector wheel conicity zero-mean Gaussian white noise vector zero-mean Gaussian white noise time power spectral density of track irregularity wave number time difference differentiation with respect to time

AND

D. DEEPAK