Optimal ride height and pitch control for championship race cars

0005 1098/86 $3.00+ 0.00 PergamonJournals Ltd. © 1986 International Federation of Automatic Control

Automatica, Vol.22, No. 5, pp. 509 -520, 1986 Printed in Great Britain.

Optimal Ride Height and Pitch Control for Championship Race Cars* D. METZ? and J. MADDOCK:~

Racing cars which enhance performance through the use of unstable ground effects aerodynamics can be height-regulated using linear optimal regulator theory and an active suspension system, producing stable, controllable vehicles capable of braking and cornering at the extreme limits of performance. Key Words--Attitude control; automobiles; computer control; control system synthesis; feedback control; linear optimal regulator; vehicles; vehicle dynamics; ride height control. Abstract--Car performance in championship auto racing is strongly influenced by aerodynamics, particularly ground effects aerodynamics. Vehicles employing this technology require careful control of ride height and unsprung mass pitch in order to be stable, perform properly and achieve maximum success on the racetrack. In the past, attempts for controlling these vehicle variables centred around employment of very stiff "suspension systems" so as to prevent aerodynamic load buildup, which occurs at high race velocities, from causing excessive suspension travel. Control of these variables, however, may require use of an active suspension system, coupled with a controller which can adjust the suspension system in such a way as to mask road surface disturbance and aerodynamic noise. In the present work, a controller is developed for such a vehicle/suspension combination using linear quadratic stochastic regulator theory. Vehicle road inputs are modelled as Gaussian white noise, and are nearly completely rejected by the controller. In addition, representative aerodynamic inputs are also well controlled so that vehicle outputs of unsprung mass pitch angle and centre of gravity height are held constant.

mf~ front unsprung mass (suspension, tyres etc.) mr ~= rear unsprung mass (suspension, tyres etc.) m 2 _~ chassis sprung mass I & chassis mass moment of inertia in pitch mode about centre of gravity lf_A distance from centre of gravity to front wheel axis of rotation lr_~ distance from centre of gravity to rear wheel axis of rotation pf ~ distance from sprung mass centre of pressure to front wheel axis of rotation p, ~_ distance from sprung mass centre of pressure to rear wheel axis of rotation kf_~ spring constant of both front tyres (chassis spring constant ~ or) kr _~ spring constant of both rear tyres (chassis spring constant --,

oo)

Ff_~ front active suspension force on tyres and unsprung mass Fr~ rear active suspension force on tyres and unsprung mass A, B, C, E = usual system state coupling, input coupling, output and noise matrices J = Index of Performance to be extremized F = penalty matrix of final state values (symmetric and positive semidefinite) Q = penalty matrix on outputs (symmetric and positive semidefinite) R = penalty matrix on controls (symmetric and positive definite) x(t) = system state vector y(t) = system output vector u(t) = system control vector P11, PI 2 = solution matrices from algebraic Riccati equations F~, F 2 = feedback and feedforward matrices, respectively.

NOMENCLATURE X l ~ displacement of front unsprung mass (suspension, tyres etc.) x2 =- ~tt __4velocity of front unsprung mass x3& vertical displacement of chassis sprung mass centre of gravity x 4 _= ~3_~ vertical velocity of chassis sprung mass centre of gravity x5 ~ displacement of rear unsprung mass (suspension, tyres etc.) x6 ~- ~5 & velocity of rear unsprung mass xT_~ angular displacement (pitch of chassis sprung mass about centre of gravity x8 =- :~7 & angular velocity of chassis sprung mass about centre of gravity wf ~ road motion applied to (both) front wheels w,~_ road motion applied to (both) rear wheels

I. I N T R O D U C T I O N AND SURVEY OF THE LITERATURE

AN ACTIVEsuspension system can be defined as a means of isolating the unsprung mass of a vehicle from externally imposed disturbances through the use of hardware which consumes external power. By contrast a passive suspension system attempts to achieve the same goal without the consumption of power. Conventional vehicle suspension systems are nearly always of the passive design. Passive suspension systems inherently involve trade-offs between several competing dynamic specifications. These may include passenger comfort, small suspension deflections and roadway tracking ability,

* Received 28 April 1985; revised 27 October 1985; revised 12 February 1986. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor B. Friedland under the direction of Editor H. Austin Spang III. I Professor, Department of General Engineering, 113A Transportation Building, 104 S. Mathews Avenue, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A. :~Research Engineer, General Dynamics Corporation, Space Systems Division, Kearny Mesa Plant, Mail Stop ~ 22-6880, San Diego, CA 92138, U.S.A. A~TO

22:5-A

509

510

D. METZ and J. MADDOCK

among other considerations. The trade-offs are usually made specific in the form of a performance index (Hrovat and Hubbard, 1981) with passenger comfort being measured in terms of sprung and unsprung mass displacements. The conventional passive suspension system has always been limited by its inability to deal with all of the objectives that a suspension system must satisfy.. Without extensive modifications, the passive suspension system cannot handle the demands of high speed ground transportation vehicles. Semiactive designs have been pursued, and have realized improvements over totally passive designs. In addition, the semiactive approach often requires very little external power (Hrovat and Margolis, 1981; Margolis et al. 1975), whereas a fully active suspension system may require considerable external power in order to operate (Margolis et al., 1975; Leatherwood, 1969). The synthesis of fully active suspension systems for high speed ground transportation vehicles is an active research area. Optimal suspension design to minimize RMS rattlespace, sprung mass acceleration and the time derivative of acceleration was investigated in an early work involving rolling vehicles (Hrovat and Hubbard, 1981). Tracked vehicles and magnetic levitation suspended vehicles were also investigated from a similar point of view (Guenther and Leondes, 1977; Gottzein and Lange, 1975). As speeds increase, disturbances caused by ground or track irregularities and wind noise inputs cause degradation of ride quality and passenger comfort (Hrovat and Margolis, 1981). Optimal control methods and the well-developed theory of quadratic synthesis and regulator theory have also been used (Hedrick, 1973; Hedrick et al., 1974) to synthesize linear state feedback structures for active suspension systems. Optimal suspension systems must accommodate three separate regimes of disturbances: external force and moment inputs, vehicle body vibration isolation and guideway tracking irregularities, when these are present (Guenther and Leondes, 1977). in automobile racing, vehicle performance during turning manoeuvres is often the difference between victory and a lower place finish. Cornering performance is obviously influenced by a number of interactive engineering considerations, one of which is aerodynamic performance. Through the use of airfoils, winglets, wings and finally, the shape of the racing car itself, it is possible to create aerodynamic downloading of the vehicle. Performance improvements result because of the characteristics of tyre behaviour. As shown in Fig. 1, an increase in the normal loading applied to a tyre produces a higher lateral cornering force for a given slip angle setting. Thus, it is possible to generate more external torque on the vehicle when the tyres are operating under

high normal loading conditions. If the higher normal loads are obtained by simply increasing the weight of the vehicle, then no net improvement will be realized. Aerodynamic downforce simply provides a way to increase the tyre normal loading without increasing the mass of the racing car itself. The vehicle of interest in the present work is a racing car of the single seat, open wheel configuration, similar to the vehicles raced in worldwide Formula I competition; see Fig. 2. A crucial difference between this style of racing car, hereafter called a Championship car, and those used in current Formula I racing is that the use of ground effects aerodynamics is presently permitted in Championship tsar racing. Ground effects aerodynamics consists of the development of the underbody shape ot the car so that, in conjunction with the surface ot the track itself, the car has a strong negative lift coefficient and is actually forced down against the track surface by aerodynamic fo~zes. A schematic of this racing design philosophy is shown in Fig. 3. Considerable downforce can be generated through such designs; as of this date (June, 1985) lift coefficients approaching - 3 . 0 are well within the realm of the practical. A photograph of the underside of a typical ground effects racing car is shown in Fig. 4. The chassis is a 1983 March 83C. It is easy to recognize the venturi shape present in each sidepod of the car. The striped lines in the photograph are the result of carbon fibre reinforced composite material construction techniques. Wind tunnel testing done on various modifications of simplified venturi models (idealizations of present ground effects racing cars) verify that

"I 8000 z c

2000

Indianapol,s_ 2 7 0 x l 4 5 IS racing tyre

j .% ~L 600' ~ 51500

2

2

-¢c 400C

~000

~- 200C

" 500

I Tyre J Loading -4-~ 9786N 2200 Ibf

~

~

c

~ 1 2 0 0

~

Ibf

i 3114 N tO0 Ibf

i 0--

0 C

t 4

2 Slip Angle in Degrees

Nondlmenslonol Maximum Forces Tyre Loading

Laterot Force/Tyre Loud

3114 N

700 Ibf

5336 N

1200 Ibf

I 13 I OI

9786N

2 2 0 0 Ibf

098

FIG. I. Tyre lateral force vs slip angle for Indianapolis 27.0 x 14.5-15 racing tyre.

Championship race cars

511

FIG. 2. Side view photograph of 1983 March/Cosworth DFX Championship racing car.

there exists an optimum ride height or ground clearance which maximizes the negative lift coefficient (George and Donis, 1983). Figure 5 shows the results of one series of tests as a graph of lift coefficient vs skirt clearance or ride height of the vehicle for various models. Similar data for actual Formula I racing cars utilizing ground effects, before the technology became illegal under Formula I rules, are available (Wright, 1982). Aerodynamic leakage from underneath the vehicle causes the downforce to decrease at heights below the optimum ground clearance; at heights greater than the optimum height, the curve of lift coefficient vs ride height is nearly linear.

approximate venturi

The synthesis of active suspensions in racing and passenger cars has passed the hypothetical stage. A hydraulically powered active suspension system controlled by a microcomputer is under development at Lotus Cars Ltd, Norfolk, England (Wright, 1985). A production road car, the Lotus Turbo Esprit, has been equipped with the system and has already shown enhanced control and performance characteristics. Present power consumption is about 10 HP (7500W) but this may be reduced in future versions of the system. Tests with this vehicle, operated by experienced drivers, have shown that it is approximately 10% faster through a given corner than a standard Esprit (itself a high

airflow pattern

pre-1981 location of continuous contact skirts FIG. 3. Schematicof groundeffectsaerodynamicsand air flow.

512

D. METZ and J. MADDOCK

FIG. 4. Photograph of underbody of 1983 March/Cosworth DFX showing ground effects venturi sidepod shape.

performance automobile), even allowing for its power deficit and the fact that it carries 75 801bf (334 N) extra weight. As of this writing, no attempts have yet been made to optimize the suspension system. 2. DEVELOPMENT OF RIDE HEIGHT MODEL

Figure 6 shows a schematic ride height model for ground effects Championship cars. The model

requires some explanation. Firstly, there is the question of tyre mechanics. Pneumatic tyres essentially function to support the weight of the vehicle and cushion its motion over irregular surfaces. Applying a normal (vertical) load to a tyre causes it to progressively deflect as the load is increased. Except at very low values of vertical loading the load/deflection characteristics of both radial and bias-ply racing tyres are nearly linear for a given

Base

" ~ +

~ +

v.

(

....

4

....

-t

Side pods

Bose

P o d s and ~.ub

kirts CL

Inner L-skirts _ ~

o Wheel bumps

OuterL-skirts Labyrinth skirts

0

5, Skirt

]0, clearance

, 15

WheeL' bumps"

(

~

(mm)

FIG. 5. Lift coefficient vs skirt clearance for some idealized chassis ground effects configurations (taken from George and Donis (1983) by kind permission of ASME).

Championship race cars

m2'T

/~ko

x7

Ff

front rear octuator octuotor

mf 1 wf

gr -I

Fr

mr

wr

FIG. 6. Schematicdrawing of active suspension car for ride height and pitch anglecontrol.

inflation pressure. Tyre stiffness, therefore, can be modelled independently of load, leading to a linear model for tyre mechanics. For studies involving vehicle ride and vibration simulation, the most commonly employed pneumatic tyre model consists of a mass element attached to a linear spring in parallel with a damping element (Wong, 1972a). For preliminary studies of ride dynamics and stabilizability the tyre can be modelled without the damping element (Hrovat, 1982). Suspension systems of Championship cars have become enormously stiff over the past several years in order to minimize ride height variation with aerodynamic downforce build-up. The same design strategy was also employed in Formula I racing when aerodynamic ground effects were allowed. Suspension stiffness coefficients on today's Championship cars approach 10 0001bfin- 1 (1.75 x 10 4 N cm- 1)! Such incredibly stiff suspension systems in essence are not a part of the vehicle ride dynamics, except at high frequencies; in effect the tyres themselves constitute the primary suspension of the vehicle. The effect of distributed aerodynamic downforce, generated by ground effects a n d external wings, is modelled as acting at a centre of aerodynamic pressure of the vehicle sprung mass. Ground effects aerodynamics are modelled as a negative linear spring, which is consistent with wind tunnel results (George and Donis, 1983). The negative sign simply implies that as the "spring" is compressed, the force generated by the spring acts to further compress it, and vice-versa. The resulting

513

vehicle model is thus height-unstable. Driver interviews by the senior author at the 1983, 1984 and 1985 Indianapolis 500 Mile Race and Time Trials confirm the presence of this ride height instability phenomenon. Aerodynamic pitch damping and/or other non-zero stability derivatives are not included in the model for purposes of simplification of the original analysis. Additionally, the aerodynamic stability derivatives, if modelled, only serve to reduce vehicle instability and active suspension requirements, at least as long as only "reasonably bounded" vehicle transgressions are permitted. The forward velocity of the car, assumed to be constant at 200mph (89.4ms -1) determines the nominal value ofdownforce on the car (Metz, 1985). Forward velocity also determines the nominal vertical displacement Xo of the vehicle's centre of gravity (c.g.) from the race track, and would, of course, affect any aerodynamic damping and stabilization tendencies, as well as the negative aerodynamic "spring constant", were these to be included in the model formulation. Mathematically, then, the vehicle operates at an equilibrium position characterized by Xo; all states are measured as perturbations away from this equilibrium point. Equations of motion in state variable form are derived using small angle approximations and are as follows: (1)

X1 = X2

Yc2 = for(wf - x l ) + F d m f

(2) (3)

X3 = X4 2~4 = 0)al[X3 - - (Pf - - /f)X7 - - 0)3]

-- [(Ff + Fr)/m2]

(4)

2£5 = X 6 X6 = 0)r(Wr - - XS) +

(5)

Fr/mr

x7 = Xs :x8 =

-Wa2(Pf

-- /f)[X3

+ [(FfIr -- F f l f ) / l ]

(6)

(7)

-- (Pr -- lf)x7 -- 0)33

(8)

fof~ kf/mf, foal~ ka/m2, 0)3~ [-(prWf "31"for~ kr/mr, and forE~ kJl, we and wf represent disturbances away from the mean value of the track height and this mean value is zero (Athans and Falb, 1966). At any time then, some portions of the vehicle will be closer to the track than Xo, while others will be farther away. If the local p. dA downforce is integrated over the active ground effects surfaces of the vehicle, the total downforce present leads to a centre of pressure

where

PfWr)/(Pr + Pf)],

514

D. METZ and J. MADDOCK

and downforce location that is assumed equivalent to the situation when no road disturbances are present. Therefore, when stochastic wf and Wr are considered, the term (prWf -1- lgfWr)/(pf q- Pr), which is a measure of the average displacement of the centre of pressure with respect to a non-smooth track, can be eliminated from (4) and (8). The simplified equations of motion become: 2l = x2

A=

0 -27000 0 0 0 0 0 0

1 0 0 0 0 0 0 0

0 0 0 83 0 0 0 - 12

0 0 1 0 0 0 0 0

0 0 0 0 0 -27000 0 0

0 0 0 0 1 0 0 0

0 0 0 - 15 0 0 0 0

00 0 0 0 0 1 0

(9) B

"~2 = (,Of(Wf -- X1) -[- F f / m f 23 .~_ X4 J¢4 = COal[X3 -- (Pf -- lf)XT] -- (Ff + F r ) / m 2

(lo)

(12)

(13)

5:0 = OOr(Wr-- XS) + F~/mr

(14)

27 = x8

(15)

2~8 = --COa2(Pf -- If)[X3 -- (pf -- /f)X7] (16)

For a 1983 March Championship car chassis and racing tyres of that era, realistic values for the above parameters are: kf~ spring constant of front tyres = 35001bf in -x (6130Ncm 11 kr&spring constant of rear t y r e s = 35001bf in 1 ( 6 1 3 0 N c m - 1 ) k. ~ spring constant modelling downf o r c e = 3 0 0 1 b f i n 1 (525Ncm -1) (from wind tunnel data) mf & front unsprung mass = 49.9 lbm (23 kg) mr & rear unsprung mass = 49.9 lbm (23 kg) m2& chassis sprung mass = 1400.7 lbm (635 kg) l& chassis mass moment of inertia w.r.t. c.g. = 2.7E61bm in 2 (790kgm 2) p f ~ distance from c.p. to front wheels = 65 in (1.65 m) @ 200 mph (89.4 m s- 1) Pr~ distance from c.p. to rear wheels = 31 in (0.78m) @ 200mph (89.4ms -~) lf~ distance from c.g. to front wheels = 58in (1.74m) IrA distance from c.g. to rear wheels = 38in (0.96 m). Using these values, the system can be put in the standard form:

.';;(t) = Ax(t) + Bu(t) + Ew(t), where u = [ F f

Fr] T, w = [We wr]r and

0

000

6

0

0

0

-0.0019] 1

0

0

toO.0016

0

0.0435

o

0.00121

(1[ 1)

2s = x6

+ [(Fflr - F f l f ) / l ] .

00435

(17)

and

E=

E:

27000 0

0 0

0 0

0 0

0 27000

0 0

:1r.

Equations (12) and (16) have the common term, x 3 - ( p f - l f ) x T . This term is a measure of the displacement of the spring, ka, and determines any change in nominal downforce. If x 3 - ( P f - If)x7 is held to zero at all times, the aerodynamic downforce will remain at its nominal value. Because vehicle stability depends strongly on downforce, it is desirable to maintain downforce at its nominal value. This makes vehicle cornering performance both predictable and repeatable. Requiring that downforce remain constant leads to a natural definition of the system output equation in the form, y = Cx, where

C=

E0o0,000 0

0

0

0

0

0,8 00j 1

and p ~ - l~-~ = 7in (0.18m) for a 1983 vintage Championship car. The second output equation is necessary because the location of the centre of pressure is a strong and highly non-linear function of the pitch angle, xT. Because all values are deviations from nominal values, y is required to remain at zero at all times (or as close as possible) to assure stable vehicle performance. This leads to a definition of a quadratic performance index of the form I r J = 5y (T)Fy(T) + 5l f l r [yrQy + urRu]dt,

(18)

where y is now tracking an input which is zero for all t >1 0, i.e. a regulator problem. The next issue is to select elements for Q, R, F, and 7". In R, the assumption that neither control is more or less important than the other is made. The same assumption is made about both outputs,

Championship race cars which leads to diagonal matrices for both Q and R. The value for T is set to infinity because (1) this guarantees that the state will stay near zero after an initial transient period, and (2) it avoids the arbitrary specification of a large terminal time. The matrix F is then set to zero because a terminal cost at infinity is not realistic (Athans and Falb, t966). The actual diagonal elements of Q and R are discussed below. The road disturbances wf and Wr are represented as stochastic processes. The disturbances can be modelled as the output of a linear system driven by Gaussian white noise (Hrovat and Hubbard, 1981; Kwakernaak and Sivan, 1972). To accomplish this, the state vector x is augmented with two additional states, Xdl and Xd2, equivalent to wf and wr, respectively. Mathematically, w(t) = xd(t),

(19)

Yea(t) = Aaxd(t) + r(t)

(20)

where

and r(t) is Gaussian white noise. Substituting (19) and (20) into (17) yields the following augmented system: E

B

y(t) = [C

0]~(t),

possible to solve for the optimal control u which will drive the output y toward zero (thus keeping downforce at its nominal value) and simultaneously minimize J. It is obviously not possible to minimize (23) and simultaneously reduce y to zero, except in the sense that all quantities approach zero in an asymptotically stable system. Minimization of (23) in the presence of non-steady disturbance inputs is a compromise between the employment of control energy and the achievement of the control objective. The compromise chosen is reflected in the relative scaling of matrices Q and R, and is discussed in greater detail below.

3. SOLUTION TECHNIQUE The system specified in (21) and (22) with the performance index given in (23) is referred to as the time-invariant stochastic linear optimal regulator problem and has a well-known solution (Kwakernaak and Sivan, 1972). For the solution to exist, it is sufficient for the augmented system to be stabilizable. If the pair [A, B] is controllable, then system stability ultimately depends on the matrix A d. TO insure stabilizability, Ad must be asymptotically stable (Kwakernaak and Sivan, 1972). The choice for A d is quite arbitrary and is made to quickly damp the response of Xd due to the white noise. With this in mind, Ad is chosen as:

,00

(22)

where)~(t)=[x, xa] T. Alternatively, the relative tyre displacements could have been defined as states, obviating the need for the above system augmentation. However, existing road surface roughness measurement techniques are well suited to characterization of road profiles as stochastic processes. Because white noise inputs only allow a stochastic characterization of system behaviour, the control, u(t), cannot be optimally determined over the whole period (O,T], a priori. Rather, the situation is reconsidered at each intermediate time r on the basis of available information. This procedure demands certain observability requirements (Kwakernaak and Sivan, 1972), which are easily shown to be satisfied here. The performance index is then averaged over all possible realizations of the disturbances and becomes

515

01'

-10

The solution, then, to the optimal controller is as follows: x u(t) = -- IF 1 F2] I Z I , .

(24)

where F 1 = R - 1BTp 1 l

(25)

F2 = R - XBTp12.

(26)

and

The matrices Pal and P12 are found by solving the following algebraic equations: 0 = CTQC - P 1 1 B R - 1 B T p t 1 + A T p l l + P l l A

(27) and

{';o

J = E ~

[yrQy + urRu]dt

}

,

(23)

where E represents the expected or average value. The problem is now properly posed so that it is

0 = P12 E + [A -- BF1]Tp12 + P12Ad.

(28)

F1, the feedback link, and F2, the feedforward link, characterize the controller. This linear feedback

516

D. METZ and J. MADDOCK

guarantees optimality (Kwakernaak and Sivan, 1972; Kushner, 1967, 1971;/~str6m, 1970). Substituting (24) into (21) leads to the optimal system: xtj-'t'= A - BF~ 0

E

F2

(29)

y(t) = [C ] 0])?(t).

(30)

4. RESULTS

The feedback from the states x to the inputs u is independent of the disturbances, but the state of the disturbances is fed forward to the input through F2. Figure 7 shows the structure of the resulting optimal system including feedback and disturbances. This linear feedback structure is optimal with respect to the Gaussian white noise, r(t) (Kwakernaak and Sivan, 1972; Kushner, 1967, 1971; ~str6m, 1970). The gain matrices F1 and F2 model the controller needed to implement the state feedback, and the gains present in F1 and F2 depend on the choices made for Q and R (see (25)-(28)). As discussed previously, from a physical standpoint the structures of Q and R are such that both are diagonal matrices. The diagonal elements within each matrix may or may not be assumed identical. Selection of the elements for Q and R globally reflects the compromise between control energy and achievement of control objectives discussed above, and locally reflects the relative weighting assigned to each control and/or output. These

white noise

r(t)

feedforword link

disturbance dynamics

xd 'r

ii 4-

disturbance

control input

input w(t)

plant dynamics

feedback link

FIG. 7. Block diagram of active suspension system.

choices represent design parameters or design degrees of freedom, to be chosen at the option of the vehicle engineer. Obviously some experimentation is necessary. The selection process can begin by choosing R to be the identity matrix. This is done to use the penalty on the controls as a reference point and because there is no natural or physical reason to indicate that each control should not be penalized equally. The outputs are then penalized relative to the controls by selecting the diagonal elements of Q to be a constant, p. In doing this, the outputs are also weighted equally and, therefore, penalized p times that of the controls. In the remaining text, the constant p is referred to as the QR ratio. It is important to recognize that the selection of equal weighting for all elements for Q places equal importance on vertical centre of pressure diaplacement Yl = x 3 - ( p f - 1 0 x 7 and pitch angle ./'2= x7 - q~ even though these quantities may not actually be equally critical (indeed, they do not even have the same units). Unequal Q element weighting is obviously possible and can be effected by the designer without loss of generality. Physically, equal weighting of control penalties in R is likely, because (nearly) identical actuators are expected to be used at all four wheels. Conversely, unequal weighting of Q elements is likely, and would require further experimentation. Controller gains, system performance and energy requirements may be studied as a function of a varying QR ratio and under the above choices. QR ratios are selected to demonstrate their effects on the above parameter, and certain trends in these parameters can be identified. The following QR ratios were studied assuming that each output is of equal importance: (1) 10°, (2) 103, (3) 106 and (4) 109. The solution technique outlined in (24)-(30) was repeated for each QR ratio. The effects of the QR ratio on the gain matrices are not substantial. As expected, the significant gains (those >>0) in F~ are greatest for the QR ratio 109. It is not true, however, that these gains monotonically increase with the QR ratio, i.e. some significant gains in F1 for QR = 10° are greater than the corresponding gains in F 1 for QR = 103 a n d 10 6. Regardless, the gains in F1 do not vary substantially over the wide range of QR ratios used. The QR ratio has little effect on the matrix F2; in fact, for all cases the gains are negligible. Substantial increases in the QR ratio produce only a comparatively slight increase in the controller gains. This is important from a hardware construction point of view. Using (29) and (30), the system can be simulated to study its performance and energy requirements. The vehicle's performance is gauged by its ability to remain at (or restore) its equilibrium position.

517

Championship race cars This translates mathematically into requiring output vector y to be zero. Control requirements are measured in terms of the suspension forces needed to maintain (or restore) the vehicle's equilibrium position• Initially, the car is assumed to be at equilibrium and travelling 200 mph (89.4 m s-~); at t = 0, the road motion begins to disturb the vehicle• Road motion amplitudes of + 1.0 in (+2.54cm), realistic at Indianapolis Motor Speedway, were supplied at various frequencies. Results for all QR ratios are essentially the same. The road disturbances only affect the equilibrium position of the vehicle slightly, and for every case, the effect is negligible; front and rear suspension forces are also negligible for each case. Because the system contains no practical hardware constraints, sensor or computer delays, or any other time delays (only inertial resistance), it can react nearly instantaneously to any road disturbance, and deviations from equilibrium are (almost) only a function of numerical integration step size. Therefore, the system is able to maintain the vehicle's equilibrium position with .2O

negligible variations, hence, the negligible suspension forces. A more interesting result is obtained by providing a non-zero initial angle of attack, or pitch angle, and then examining output performance and energy requirements• This is analogous to the racing situation arising when an aerodynamically influenced vehicle pulls out into clear, undisturbed air from the draught of another vehicle. In this situation, an unsprung mass pitch angle ~0.1 rad (6 °) is not uncommon. The output performance and suspension force requirements for the system with the non-zero initial conditions Y~o~= centre of pressure displacement (CPDP) att = 0 :

X30 - - (pf - - lf)xT0 = 0 - - 0 . 1 8 ( 0 . 1

= -0.018 m

(31)

Y2o A pitch angle = Xvo = 0.1 rad

1400 ~

/ i~ ~ ...... ~

12

I

/

.i008 •

T 5 !

~¢

__ __

\ \

\

....

• 06 • 04

"\-\

QR ratio i0o 103

\

QR ratio

.....

800

Jo °

t_/~!'11 ~ "~ ]

6oo 400 ,

106 109

--

N

~

J

[ ~" -.02

2

FIG. 8. Pitch

I

I

3

I

i

4

I

i

5

l

t

6

I

J

I

J

I

J

200 ,'

't

a n g l e vs t i m e for n o n - z e r o

initial pitch angle.

I

I

I

I

I

i

I

I

I

I

I

I

I

I

I

I

I

400

1.0

QR ratio

006 ,i

008

s

014016 /

.\

LV

SECONDS

TIME IN

./

~---"

~ --

I00

....

103

_ _

106

.....

109

2OO R 0 E A -200 B -400 5

I

I

i

/i

-~oo

-800

~1'

-1000

010 012

/"

FIG. 10. Front suspension force Ff vs time for non-zero initial pitch angle.

I

0°2

R

- .

/~

-400 ~//i

I

? 8 9 i0 TIME IN 5ECON05

-.oo

i E

I06 io

400 I

O

.....

o

• 002

P NI

iOs

i~

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-.

-----

i

-200

.02

(32)

are shown in Figs 8-11.

I0001200~-j ,~

• 14

rad)

,//

,/

018 FIG. 9. Centre of pressure displacement vs time for non-zero initial pitch angle.

N E H T 0 N S

i

-1200

1400 -1600 -1800 -2000 -2200

i i

/ t i i

I ,/i

I

~

i

I

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~o3 ~o6 109

:7

FIG. 11. Rear suspension force F, vs time for non-zero initial pitch angle.

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D. METZ and J. MADDOCK

System performance improves unconditionally with increasing QR ratios as shown in Figs 8 and 9. These results indicate that weighting both outputs equally may be inappropriate. Increasing the QR ratio significantly improves the pitch angle response, but improvement in the C P D P response is insignificant. Therefore, the penalty on the C P D P has only a slight effect on system performance, but the penalty on the pitch angle should be more severe in order to obtain an improved response. Because the gains in F1 increase only slightly over a wide range of QR ratios, however, the penalty on the C P D P can be increased at a relatively low cost. Maintaining the penalty on the C P D P equal to that of the pitch angle apparently poses few problems in terms of hardware cost. Force requirements of the system are shown in Figs 10 and 11, which plot the front suspension force (S.F.) and rear S.F., respectively vs time. Again, the disturbance is a non-zero initial pitch angle of ~0.1 rad. The forces at both the front and rear suspensions are nearly identical for the QR ratios of 10° and 103, At QR = 10 6, there is a very slight increase in the forces necessary at both suspensions, but a significant improvement in system performance is obtained. Not until QR = 10 9, however, do the forces at both the front and rear suspensions show a dramatic increase, but again system performance in terms of the pitch angle response is significantly improved. Note that the early front suspension force peak at ~0.05s becomes increasingly less negative to positive as QR increases. In every case, both forces settle to approximately zero after about 1 s. Therefore, regardless of the QR ratio, the system responds, in terms of suspension forces, to the initial condition given within 1 s, and the suspension force response after that initial period is negligible. It is important to put such settling times into perspective. Recall that this vehicle is travelling at approximately 90m s-1; a very large distance on the race track can therefore be consumed very quickly! Generally speaking, system performance improves and control force requirements increase as the QR ratio increases. The pitch angle Y2 response is unacceptable at lower QR ratios but dramatically improves with increasing QR ratios. On the other hand, the C P D P yl response improves only slightly from QR = 10° to QR = 10 9, but because it does improve and the gain changes in the controller are relatively small, the penalty on the CPDP can be increased at a relatively low cost. The energy requirements of the system are nearly equal up to a QR ratio of 106 but increase significantly at QR = 10 9. The forces needed to drive the system quickly die out after 1 s, regardless of the QR ratio. Therefore, when the disturbance is a 0.1 rad initial pitch angle, QR ratios below 10 6 a r e unsatisfactory

because of the pitch angle response. A trade-off between increasing initial suspension forces and an improving system response exists as QR is increased from 106 to 109. A closed loop pole constellation as a function of the QR ratio and developed from (29) is shown in Fig. 12. 5, C L O S I N G C O M M E N T S : C O N C L U S I O N S

5.1. Hardware realization The question of hardware realization of active suspension systems is non-trivial. State measurement(s) and/or the question of the necessity for an observer clearly impact the design of such a system. The robustness of any resultant hardware realization particularly in the presence of unstable aerodynamics, is an important consideration. A presently-operating experimental vehicle, the Lotus Turbo Elite active suspension road sports car, actually measures all states (Wright, 1985). However, this vehicle does not employ ground effects aerodynamics and hence is not unstable w.r.t, ride height. It is precisely the presence ot conditional stability which requires robust controller and sensor performance. A second hardware question of particular significance in racing applications concerns the ability of the controller/processor to effect control calculations quickly enough. When the time constants of the plant are short and/or the system order is high, time delays due to the computation time of the processor may not be negligible. Additionally, the suspension actuators are likely to be hydraulic servovalves, and thus inherently analogue devices with bandwidth response limitations; a D/A converter will therefore be required, introducing additional (though small) delays. Fortunately, a well developed theory for accommodating such delays was recently given (Mita, 1985). Serious and interesting research remains to be done regarding physical implementation of active suspension systems in the presence of robustness requirements, computation time requirements and unstable ride height dynamics. 5.2. Accommodation of other vehicle dynamical modes In addition to the pitch and heave modes (x3 and xT) modelled above, racing cars are subject to roll, lateral motion (sideslip), yaw and forward + acceleration. Fortunately, most of these modes can be decoupled from one another (Ellis, 1969; Wong, 1972b). ride studies handling studies acceleration studies

pitch, heave yaw, sideslip (roll m extreme situations) forward velocity (pitch in extreme situations)

Championship race cars

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-4,274E-17 -~1,643E+02J

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-5,548E-12 -*1,643E+02J -i, 748E-01 -~6,524E-03J -9,229E+00 -+1,209E-04J -10,0O,O (REPEATED 2X)

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FIG. 12. Closed loop pole locations vs

The decoupling of modes is especially justified in view of the application at hand. A very constant ride height is desired regardless of the vehicle motion (turning manoeuvres, braking, acceleration, etc.) being executed, because of ground effects aerodynamics. A further simplification resulting from such a performance objective is that, even if other modes are included, a geometrically linear analysis will suffice. Clearly, a practical controller must prevent lateral movement of the aerodynamic centre of pressure in addition to vertical and foreand-aft motion. No additional actuators will be necessary, but roll motion would have to be included in such a model. Interestingly, second order system motions due to chassis and suspension elasticity, which are commonly ignored in vehicle dynamics analyses, are actually negligible in this application because of the tremendously stiff aluminum honeycomb/carbon fibre reinforced monocoque chassis construction techniques used on championship racing cars.

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QR ratio.

5.3. Future research There are many directions that future research into this subject may head. One may do a more exhaustive study of the QR ratio. For example, the weighting of the CPDP need not be identical to that of the pitch angle for reasons discussed previously. The QR ratio can also be increased above 10 9 while examining system performance and suspension force requirements to determine if the trends discussed in this work continue. A more logical direction to head, however, is to include realistic hardware constraints and time delays into the system as described in Mita (1985). This will give improved results in the situation when the vehicle encounters road disturbances only (no initial angle of attack). From this point, other additions may be incorporated into the model. These may include such items as a damping element in the pneumatic tyre model and the introduction of roll motion. To introduce roll motion, the vehicle model must include all four wheels which requires six additional state equations and an obvious increase in model complexity.

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D. M E T Z a n d J. M A D D O C K

REFERENCES Astr6m, K. J. (1970). Introduction to Stochastic" Control Theory. Academic Press, New York. Athans, M. and P. Falb (1966). Optimal Control An Introduction to the Theory and Its Applications, pp. 750-814. McGrawHill, New York. Ellis, J. R. (1969). Vehicle Dynamics, p. 60. London Business Books, London. George, A. R. and J. E. Donis (1983). Flow patterns, pressures and forces on the underside of the idealized ground effect vehicles. In Aerodynamics of Transportation-II, Fluids Engineering Division, ASME, FED-Vol. 7, pp. 69-79. Gottzein, E. and B. Lange (1975). Magnetic suspension control system for the MBB high speed train. Automatica, II, 271284. Guenther, D. R. and C. T. Leondes (1977). Synthesis of a highspeed tracked vehicle suspension system. Part I: Problem statement, suspension structure and decomposition and Part II: Definition and solution of the control problem. IEEE Trans. Aut. Control, AC-22, 158 172. Hedrick, J. K. (1973). Some optimal control techniques applicable to suspension system design. ASME Paper No. 73-ICT55. Hedrick, J. K., G. E. Billington and D. A. Dreesbach (1974). Analysis, design and optimization of high speed vehicle suspensions using state variable techniques. ASME J. Dyn. Syst., Meas. Control, 96, 193-203. Hrovat, D. and D. L. Margolis (1981). An experimental comparison between semi-active and passive suspensions for aircushion vehicles. Int. J. Vehicle Des., 2, 308-321. Hrovat, D. and M. Hubbard (1981). Optimum vehicle suspensions minimizing RMS rattlespace, sprung-mass accel-

eration and jerk. Automatic Control Division, ASME Paper No. 81-WA/DSC-23. Hrovat, D. (1982). A class of active LQG optimal actuators. Automatica, 18, 117 119. Kushner, H. J. (1976). Stochastic Stability and Control. Academic Press, New York. Kushner, H. J. (1971). Introduction to Stochastic Control. Reinhart and Winston, New York. Kwakernaak, H., and R. Sivan (1972). Linear Optimal Control Systems, pp. 237 263. John Wiley, New York. Leatherwood, J. D., G. V. Dixon and D. G. Stephens (1969). Heave dynamics of a plenum air cushion including passive and active control concepts. NASA Report, TND-5202. Margolis, D. L., J. L. Tylee and D. Hrovat (1975). Heave mode dynamics of a tracked air cushion vehicle with semiactive airbag secondary suspension. Automatic Control Division, ASME Paper No. 75-WA/Aug-3. Metz, L. D. (1985). Aerodynamic requirements at the Indianapolis motor speedway. AIAA J. Guid., Control Dyn., 8, 530 531. Mira, Tsutomu (1985). Optimal digital feedback control systems counting computation time of control laws. IEEE Trans. Aut. Control, AC-30, 542 548. Wong, J. Y. (1972a). Theory of Ground Vehicles, pp. 43-52, 268 272. John Wiley, New York. Wong, J. Y. (1972b). Theory of Ground Vehicles, pp. 210 240, 274 278. John Wiley, New York. Wright, P. G. (1982). The influence of aerodynamics on the design of Formula One racing cars. Int. J. Vehicle Des., 3, 383 397. Wright, P. G. (1985). Director, Research and Development, Team Lotus International Ltd, Norwich, England. Private communication.