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Design of an active suspension for a heavy duty truck using optimal control theory
0045.7949/89 S3.00+0.00 Q 1989PergamonPress plc
Computers d Strucrures Vol.31.No. 3.pp.38M93, 1989
Printedin GreatBritain.
DESIGN OF AN ACTIVE SUSPENSION FOR A HEAVY DUTY TRUCK USING OPTIMAL CONTROL THEORY M. M. ELMADANY Mechanical Engineering Department, College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia (Received 11 Jmwry
1988)
Abstract-The optimal active suspension for the cab in a tractor-semitrailer vehicle model is obtained using the linear stochastic optimal control theory. Two state controllers, full state and limited state feedback, are determined. The control law minimizes a performance function representing operator discomfort and cab suspension travel. The randomly profiled road is modelled as a filtered white noise excitation and time delays due to the multiaxle excitation inputs are included. The performance of the optimally active controlled system is compared with the performance of an optimal passive system.
NOTATION
vehicle-road system dynamic matrix road representation dynamic matrix vehicle dynamic matrix road disturbance input distribution matrix excitation matrix of road shape filter excitation distribution matrix vehicle dimensions gain matrix damping matrix suspension damping rates transformation matrix expectation operator forcing function distribution matrix control force distribution matrices measurement transformation matrix unit matrix moments of inertias of the cab, semitrailer and tractor, respectively.
v W X
-G&X, Y Y Y,l Ys, YI
u* rv 4*(w) 0 e,, e,,e,
&irmance index gain matrix stiffness matrix suspension spring rates tire spring rates Lagrangian function mass matrix masses of the cab, semitrailer and tractor, respectively masses of the wheel-axle assemblies generalized external force solution of Riccati equation vector of generalized coordinates weighting matrices covariance matrix of forcing function identity matrix matrix of zeros except for #h element which is unity weighting factors weighting matrix output vector solution of Riccati equation state spectral density matrix time trace Rayleigh dissipation function control force vector total potential energy
vehicle speed vector of coloured noise inputs state vector fore-aft displacements of the cab, semitrailer and tractor, respectively covariance matrix state vector vertical displacements of the cab, semitrailer and tractor, respectively. vertical displacements for axles measurement vector weighting factors coefficient Dirac delta function road excitation white noise process vector of white noise process variance of road irregularities time delay between ith and jth tires single-sided power spectral density temporal angular frequency pitch angles of the cab, semitrailer and tractor, respectively INTRODUCTION
One important facet of the work currently being carried out at research institutes and industrial organizations is aimed at providing better ride qualities for tractor-semitrailer vehicles. Research efforts have included measurement of ride sensation, subjective evaluation of human response to vibrations, development of mathematical models of vehicle, road and human systems, development of innovative passive
and active suspension systems and application of analytical and optimization techniques in the design of vehicle suspensions. The optimum design of vehicle suspensions subject to random road excitations has generally been based on one of two approaches. The first uses analysis techniques and formal optimization procedures for the optimum parameters selection of suspension systems consisting of passive elements. The other approach utilizes synthesis techniques in designing 385
386
M. M. ELMADANY
the optimal configuration of the transfer function of the suspension with the assumption that the transfer function can be realized with passive or active elements. The synthesis of the actively suspended cab in a tractor-semitrailer vehicle is the subject of this paper, Several papers have been published during the past two decades on the application of synthesis techniques to the design of suspensions of passenger cars, trucks and high speed rail vehicles. The vehicle suspensions have been synthesized using two techniques: the Wiener-Hiipf technique and optimal control theory. Bender and Paul [l] and Bender et al. [2] have employed the Wiener-Hopf technique to minimize a weighted sum of mean-squared vehicle acceleration and suspension stroke for a two-degreeof-freedom (Z-DOP) vehicle model. Kamopp and Trikha [3], Hullender et al. [4] and Young and Wormley[5] obtained the optimal configuration of the vibration isolator of a I-DOF vehicle model utilizing the Wiener-HBpf technique, with the meansquare vehicle acceleration being used as a ride comfort criterion. Applications of the optimal control theory to the design of vehicle suspensions have drawn much attention in recent years. Thompson [6] applied the optimal control theory to a vehicle model of 2-DOF and optimized a performance index comprising a weighted sum of mean-squared values for the body forces, tire dynamic deflections and relative wheel travel. Thompson and Pearce (1 extended the result in [6] to include the effects of front and rear suspension parameters on a two-dimensional linear vehicle model. Guenther and Leondes [8] presented results of optimal sus~nsion structure of a vehicle subject to both random roadway roughness and random external disturbance force. Hrovat and Hubbard [9] addressed the problem of optimum suspension structure for a I-DOF vehicle model using the Linear Quadratic Regulator theory. The performance index included RMS suspension stroke, acceleration and jerk. Using stochastic optimal control theory Ha6 analyzed a lumped Z-DOF vehicle model [lo] and a discrete-continuous vibrating system representing a vehicle with a long wheelbase [ 111. The active suspension system was optimized with respect to ride comfort measured in mean-squared acceleration, road holding and suspension travel with the limitation on the expenditure of control forces. It is evident from the number of studies cited that the possible application of the optimal control theory to remedy the inherent problems [ 121of the passively suspended tractor-semitrailer trucks has not been addressed. The present work is undertaken in order to gain an understanding of the capabilities of advanced suspension designs as they affect operator comfort through the application of the optimal control theory. In this paper, a 9-DOF lumped parameter vehicle model which includes heave and pitch motions is
considered. The primary emphasis is on the use of active suspension to improve vehicle dynamic performance. The input disturbance is the apparent road motion, caused by the vehicle’s forward speed along a road having an irregular profile. The road surface is described by a filtered white noise with the excitation time delays in the multi-axled vehicle being included in the analysis. Linear stochastic optimal control theory is applied to design the optimally active controlled suspension system for cab ride. Two state controllers aiming at improving cab performance are considered. The first controller relies on the availability of full state information including that of road surface height. In this case, a Riccati controller is used to implement the active suspension. In the second controller, a limited feedback, based on absolute velocity signals, is assumed and the optimal gains are obtained using a gradient search technique.
PASSIVE VERSUS ACTIVE SUSPENSION
The main functions of the road vehicle suspension systems (passive or active) include the following. (a) It must provide basic support. (b) It must isolate passenger and cargo compartments from road surface irregularities and external forces disturbances. (c) It must transmit braking and accelerating forces to the road. The requirements to fulfil these various functions conflict to varying degrees. The severity of the conflicts is likely to grow in view of increased demand for upgraded ride quality and improving operator working conditions. Practically all suspensions in use today on tractor-semitrailer trucks utilize passive elements, and typically include tires, linkages, coil and leaf springs, rubber donuts, bump stops and hydraulic dampers. Passive suspensions do not require a supply of power and thus their capabilities are fundamentally limited. On the other hand, passive elements are simple, inexpensive and reliable. Active suspension systems may use continuous or intermittent power from an external source. Typically an active feedback control suspension system consists of a power source (e.g. electric, hydraulic or pneumatic), an actuator (e.g. hydraulic, pneumatic, magnetic or electro-mechanical), sensing instruments (e.g. accelerometers, relative displacement transducers), signal processing, feedback and amplifying elements. Actively controlled suspensions offer several advantages compared with passive suspensions. They permit the use of more general control forces than are available from passive suspensions comprised of energy storage and dissipative elements. They permit selectivity in design with only selected adverse characteristics of vehicle behavior being modified. In contrast to passive suspensions whose
Active suspension for a heavy duty truck properties are fixed, active suspensions are capable of being tuned or adjusted and can also be made self adaptive to varying conditions of speed and loadings. These features must, of course, be balanced against the increased complexity, cost, power requirement and decreased reliability of the active controller. OPTIMAL
Characterization
CONTROL
FORMULATION
of road irregularities
As the vehicle travels along the road, it experiences disturbance forces due to road surface roughness. The suspension system is designed to account for this disturbance spectrum since the acceleration and control motion will largely depend upon the traversed type of road surface. Measurements on road profiles have shown that road irregularities can be described by stationary ergodic Gaussian random variables with zero mean values. A road roughness model for a vehicle moving at a constant forward speed, v, is given by the single-sided power spectral density [13, 141.
cW=~&l&*
(1)
where u2 is the variance of road irregularities, c1is a coefficient depending on the shape of road irregularities, and w is the temporal frequency. The vehicle excitation model A(t), given by the stationary Gaussian colored noise (l), can be obtained from a white noise process I;(t) by means of a shape filter i(t) = -ad(t)
+ 5(t),
(2)
where El5
011= 0
a??[r(t){(t -
7)] = ~cwu%(~).
is the Dirac delta function. Since the vehicle has multiple contacts with the roadway, the time delay between successive contact points has to be considered. The road excitations to the vehicle model take the form
387
+ 0148
(4 -
12 + 714) + QSl6
+ Qd(4
-
t2 +
713 -
712)
+
0326
(t2 -
tl +
713 -
712)
+
0248
(6
-
t2 +
714 -
712)
+
Q426tt2
-
tl +
714 -
712)
+
0346
-
t2 +
714 -
713)
+
Q43602
-
6 +
714 -
21311
0,
(t2 -
t, +
714)
(4)
where Qll is a (4 x 4) identity matrix and Qii, i # j, is a (4 x 4) matrix of zeros except for the ijth element which is unity. Vehicle mathematical
model
In the design and evaluation of the statistical performance characteristics of a specific suspension concept, an abstract mathematical model is developed to represent the actual road vehicle. It is desirable to develop the simplest credible model which is able to describe the motions of the vehicle system. The simplicity of the model is determined to a large extent by the number of degrees of freedom. The credibility of the model is determined in part by whether the degrees of freedom used will describe the system to the desired accuracy. The process of developing the vehicle model is strongly influenced by the available information about the characteristics of the actual system and the purpose for which it will be used. In general, the fidelity of the model is limited by the assumptions, approximations and restrictions that are made in the development of the mathematical model. The design of the model always includes a trade-off between fidelity of the model and the ease, cost, interpretation and understanding of the solution of the governing equations which describe the behavior of the model. In order to make the mathematics tractable in developing the tractor-semitrailer model used in this study, the following assumptions are made.
d(7)
+=A,,,w+B,t
(3)
where A,=
-avI
B,=I
cT=[~I,1;2,ts,t41T.
I is an identity matrix of dimension 4 x 4 and { is a zero mean correlated white noise process with the following covariance matrix: EK(4W(t2)l = 2ava2[Qll~(t,- t2) +Qd(tl - 4 + %)I + + Q,J(t,
-
t2 +
Q2lW2
-
tl +
712)
7.13) + Q3l602
-
t1 +
713)
(a) The tractor chassis, semitrailer, cab and axles are rigid, and their stiffnesses are lumped into the suspension elements. (b) All linear and angular displacements are small. (c) The springing and damping elements in the suspension system are linear. (d) Tire nonuniformity, wheel unbalance effects and engine and drivetrain excitations are ignored. (e) Nonlinearities resulting from wheel-hop, suspension stop and dry friction in the suspension elements are neglected. (f) Gyroscopic effects are neglected. The model of the tractor-semitrailer vehicle considered is presented in Fig. 1. Each unit representing the cab, tractor or semitrailer has rigid body modes of heave and pitch except the semitrailer which is constrained in the vertical direction due to the artic-
M. M. E&%UMNY
kt3
$
kt4
f
Fig. 1. Tractor-semitrailer model.
ovation at the I%& wheel. Each WheeI-axle assembP$ Vanstates in the veti dire&ion. Tire stiffnesses are represented by &near spring e&nents, The tractor and semitrailer bodies are suspended using passive spring and damping elements. The cab is actively suspended on the tractor chassis with t(r and uz r~pr~~~ting the controlled active suspension forces. ~~~~~~s
of
motim
and state space ~5r~~~~ti5~
The equations of motion of the vehicle system are derived using Lagrange’s equation which is given by
~ornb~n~n~ eqns (3) and (7) yiefds where the Lagrangian
function L, is given by
L=T-V,
T Y U N;:(r) Pi
is the tota kinetic energy of the system is the total potential energy of the system is the Rayleigh dissipation function is the generalized external force is the generalized coordinate.
where the augmented state vector y and the matrices A, G and B are given by
The resulting equations of motion take the form ~~~~~~~~=~u~~
(63
where ICT, T3 and R are the mass, damping and stiffness matrices, respectively. F is the distribution matrix of the forcing function and G is the distributioa matrix of the active forces. u = [a,, uz]’ is the active forces. The matrices R, C, P, F and G are given in the Appendix. In order to perform Linear Quadratic Regulator (LQR) design, it is necessary to model the system in state space form.
The basic measures of the controller performance considered in this study are the cab ride comfort and the cab s~~ns~~~ travels. The equations governing the motions of the cab reveals that cab vertical acceleration is proportional to the control forces. Thus limiting the values of the control forces will limit the Ievei of cab acceleration which is the measunz of ride (~s~omfo~. The cab front and rear sus~ns~~~~ trave$s are &tear ~orn~~~ons of the
Active suspension for a heavy duty truck
389
where P satisfies the steady-state matrix Riczati equa-
elements of the state and are given by
tion
PA + ATP - PGR-‘G’P
where matrix D of dimension 4 x 22 has all elements equal to zero except the following:
+ Q = 0.
(13)
P is a unique non-negative
definite matrix. Eigenveetor technique [16] in which P is derived from the eigenvectors of the canonical form of the equation is used to solve for P. The closed-loop state matrix equation takes the form $=(A-GC)y+Bw.
Since the design of vehicle suspension systems is a function of several criteria, the probtem is to assign some means for comparing the effects of each in order to determine which design has the optimum combination of all the specified criteria. The problem is one of assigning a weight to each criterion in order to assist in making an engineering decision. The quadratic performance index used takes the form
(14)
The solution of eqn (14) is the steady-state variance matrix governed by (A - GC)Y + Y(A - GC)‘+
0 = 0
co-
(W
which is the Lyapunov matrix equation, where 0 is the intensity of the forcing function and is given by
f = Efr’Q,r + aTRu] = E[ yTIFQ,Dy f u=Ru] = Efy’Qy
+ n%]
where
Q and R are positive definite, symmetric matrices. E is the expectation operator. qI, gtl p1 and p2 are weighting factors. The absolute value of each term of the weighting factors is not important. More important is the relative value. By selecting the weighting factors in different ways, various optimal control laws can he determined. They are selected to achieve appropriate relative weights between the responses and control efforts.
The optimal regulator requires all the states of the system, including the road inputs, to be available for feedback. Given a system described by eqn (8) and an output given by (9), it is well known [lS] that the optimal control law which minimizes the performance index described by eqn (11) is given by II=-cy
C=R.-“GrPV
(12)
J-
ftfP@*
Wf
The power spectral densities of the state variables are obtained from
S,(o) = [id
-(A - GC)]-‘B
2WO2/rt
Were, it is assumed that not all the system state variables can be measured. The controller configuration chosen has the absolute cab and tractor velocities at the front and rear cab suspension locations as feedback signals. These elements of the measurement vector z are linear combinations of the elements of the state vector y. The measurement
390
M. M. ELMADANY
equation therefore can be written in the form z=Hy.
Table 1. Values of baseline vehicle oarameters (19)
950 kg 24,800 kg 3450 kg 803 kg 1503 kg 1503 kg 1503 kg 800 kg m2 200,000 kg m* 9500 kg m2
The matrix H of dimension (4 x 22) has all elements equal to zero except the following: H IO,1 -1 - ?
HI,,, = -be,
HI,,, = 1,
H,,,r = bs,
HI,,, = 1,
H,,c = - @, + bs 1
H 13,3 -1 -
HI,,,=
9
A linear time invariant
-@,-&I.
(20)
k, = 400 kN/m (21)
is to be determined to minimize the performance index (11). Using eqns (8), (19) and (21), the closed loop system is given by $=(A-GKH)y+Bw.
The performance mined from
(22)
= tr [So]
(23)
where (A - GKH)Y + Y(A - GKH)r + 0 = 0
(24)
and + S(A - GKH) -HTKTRKH
+ Q = 0.
(25)
A gradient search technique is used to determine the optimum value of K. The gradient of the performance index (11) with respect to any element Ku of K can be found from ;
= 2[RKHYHr - GrSYHT].
Hence, a cyclic iterative procedure follows.
(26) is devised as
(1) Assume an initial value for K. (2) Obtain Y and S by solving eqns (24) and (25). (3) Determine the value of the performance index J from eqn (23) and %I/dK from eqn (26). Update K using a gradient search routine. (4) Go to step (2) until results from cycle to cycle are similar. RESULTS
baseline vehicle parameters used in the numerical calculations are given in Table 1. The cab The
b, = b, = b, = b, = b, = b,= h, = h, = h, =
5.62 m 2.2 m 0.7 m 0.9 m 0.85 m 1.15m 0.3 m l.Om l.Om
k, k, k, k,
= = = =
700 kN/m 1000 kN/m 1000 kN/m 100 kN/m k6 = 60 kN/m
c, = cy = c) = c, = c5 = c, =
33 kNsec/m 58.3 kNsec/m 83.3 kNsec/m 83.3 kNsec/m 8.3 kNsec/m 5.0 kNsec/m
Tire characteristics k,3 = 4000 kN/m
k,, = 2000 kN/m kr2 = 4000 kN/m
k,4 = 4000 kN/m
index (11) may then be deter-
J = tr Ty(Q + (HTK*KH)]
(A - GKH)‘S
2.8 m b; = 3.68m
Suspension characteristics
control law of the form
u= -Kz
b, = 0.65 m b, =
suspension parameters k,, k,, c5 and c, correspond to an optimized design. The optimization is based on minimization of a performance index comprising of the weighted sum of the mean-squared cab vertical acceleration and suspension deflections. The passively optimized vehicle is assumed to travel at 80 km/hr over a concrete road. The road characteristics are a = 0.03 m-i and u = 0.0087 m. The weighted RMS vertical and longitudinal accelerations for the optimal passive suspension are 0.084 and O.O5g, respectively, while the suspension deflection is approximately 0.36 x 10m2m. A speed of 80 km/hr for the vehicle operating on the concrete road is used for designing the optimal controllers. The weighting factors in the performance index are taken as p, = p2 =0.8 x lO-9 and q, = q2= 1, unless otherwise stated. Table 2 displays the eigenvalues corresponding to the heave and pitch modes of the cab for the vehicle with full state feedback controller and for the optimally passively suspended cab. Compared to the passive vehicle, the actively controlled vehicle shows a marked reduction in the natural frequencies. The damping ratios are lower than for the passive vehicle. Figures 2 and 3 show a comparison of the IS0 ride quality evaluation for the full state feedback conTable 2. Eigenvalues for the cab with passive and active suspensions Passive suspension Damping Eiaenvalues rat’0 _ (SC-‘) -9.28 kjl4.66 -7.18 kjll.10
0.535 0.543
Active suspension Damping Eigenvalues ratio (WC’) -3.21 fj8.72 -4.73 f j9.89
0.345 0.432
Active suspension for a heavy duty truck passive Full state feedback Limited state feedback
Optimally
*a .
Frequency (Hz) Fig. 2. Ride quality evaluation for the optimally passive cab and two controlled cab configurations-vertical acceleration.
trolled, limited state feedback controlled and baseline vehicle for vertical and lon~tudinal accelerations, respectively. The one-third octave band acceleration is used for the ride quality evaluation. It can be seen that the optimally baseline vehicle response exceeds the IS0 fatigue decreased proficiency in the 2-5 Hz frequency band for trips longer than 8 hr, and hardly meets the 4 hr fatigue decreased proficiency contours. The configuration with full state feedback gives the best performance improvement in both the vertical and longitudinal directions, and the ride is improved
n Optimally passive o Full state feedback A Limited state feedback
-
in the most sensitive frequency region for human beings. The limited feedback controller satisfies the IS0 criterion for 8 hr fatigue decreased proficiency and is very effective in solving the riding problem of the cab. Thus, a significant improvement over the passive vehicle is obtained using either full or limited state feedback, with the weighted RMS acceleration levels being markedly reduced. For the full state feedback controller, the weighted RMS vertical and longitudinal accelerations are 0.024 and O.O12g, respectively, while the corresponding values using the limited state feedback gives 0.037 and O.OlSg, respectively. Figure 4 shows the weighted RMS vertical and longitudinal accelerations vary with the weighting parameters p, and p2. Variation of the RMS cab front suspension deflection with p, and pz is also shown in the figure. Full state feedback is considered. For acts as passive p, = p* = 10-u the suspension elements (with very stiff suspension) with weighted RMS values of 0.104 and 0.044g for vertical and longitudinal accelerations, respectively. A thorough utilization of the active system possibilities is indicated for pi = p2 = IO-' with a significant reduction in both the vertical and lon~tudinal accelerations which are 0.~7 and 0.01 g, respectively. Note that the weighted RMS longitudinal acceleration cannot be reduced below 0.01 g because of the contributions from the pitching motions of both the tractor and semitrailer units, while the active suspension is principally affe@ing the heave and pitch motions of the cab. It is of interest to examine the case of having an active suspension at either end of the cab. This is done by varying the weighting factors p, and p2.For p, = 0.8 x IO-“ and p2 = IO-i0 which represents the case of the cab front suspension optimally controlled while the rear suspension is passive the weighted RMS vertical and lon~tudinal accelerations are 0.028 and O.O17g, respectively. For the other case in which the cab is actively suspended at the rear and
4
i
lo-;o”
Cab
front deflection
IO’
2
E
391
Frequency (Hz)
Fig. 3. Ride quality evaluation for the optimally passive cab and two controlled cab configurations-longitudinal acceleration.
Fig. 4. Effect of weighting factors p, and p2 on cab performance.
M. M. ELMADANY
332
passively suspended at the front (p, = lo-lo and p2 = 0.8 x 10M9)a large increase in the acceleration levels is noticed with the weighted RMS acceleration reaching values of 0.052 and 0.03 g in the vertical and longitudinal directions respectively. This indicates that the control loses its effectiveness as the cab front susDension becomes stiff. In the extreme case where the-cab front is pinned to the tractor chassis while the cab rear is actively suspended (p, = 10-13, p2 = 0.8 x 10-9) the acceleration levels go up to 0.1 I and 0.051 g for the vertical and longitudinal directions, respectively. This result is due to the high level of vibration transmitted through the front suspension to the cab which shows the ~nt~bution of the tractor pitching mode. It may be concluded that of the two cab suspensions, stiffening the front suspension has a much stronger negative effect on cab riding quality than the rear suspension, and if only one active suspension system is to be used, it should be installed at the cab front location. The sensitivity of the statistical active suspension performance to the choice of the weighting factors p, and p2 is shown in Table 3. The weighted RMS vertical acceleration has been reduced by a factor of three by changing p, = p2 = 0.02 x 10m9to p, = p2 = 1.4 x 10m9.This reduction is achieved with an almost 100% increase in cab suspension deflection. Figures 5 and 6 show, respectively, the weighted RMS vertical and longitudinal accelerations as functions of vehicle forward speed for the controlled cab with full and limited state feedback and the passively suspended cab. The optimal control gain matrix obtained for the vehicle travelling at 80 km/hr is used for operation at the other velocities. It is clear that the actively controlled suspensions have a significant iniluence on cab riding quality at all speeds considered. Compared with the passive baseline model, the weighted RMS accelerations are reduced by approximately 70% for the full state feedback in the speed range. The control forces required vary between 135 N at 20 km/hr and 230 N at 80 km/hr actuator RMS, or less than 405 N and 690 peak assuming Gaussian disturbance. It is worth mentioning that for full state feedback using the optimal control gain matrix obtained for 80 km/hr for the other velocities
^o0.19 3 0)
PI=P2
0.2 x 0.4 x 0.6 x 0.8 x 1.0 x 1.2 x 1.4 x
1o-9 1o-q 1O-9 1O-9 1o-9 1o-9 1o-9
cp)
0.046 0.034 0.028 0.024 0.022 0.019 0.016
Gp)
0.019 0.014 0.012 0.011 0.011 0.010 0.010
0.36 0.48 0.56 0.61 0.65 0.68 0.70
0.19 0.28 0.33 0.37 0.41 0.43 0.45
Limited
feedback
state
feedback
(km/h)
degrades the cab acceleration performance by just 8% compared with that computed using the actual optimal control gains obtained at the different speeds. The roadway state variables feedbacks do not influence the natural dynamics of the vehicle but act to reduce the disturbance seen by the vehicle. CONCLUSIONS
The ride quality improvements achievable with an actively suspended cab in a typical tractor-semitrailer vehicle are evaluated. The actively controlled suspensions with full state feedback provide superior riding qualities, while the active control force required is kept within a practical range. Using absolute velocities as feedback signals degrades the performance to some extent, but the performance is still considerably better than the passively suspended cab. The optimal control theory is a viable method for designing active suspensions and could better direct future design efforts towards solution of the cab ride problem. It also yields high performance active vibration controller. Covariance analysis is a very useful tool in determining the stochastic response characteristics of road vehicles.
^o 0.15
s
Cab deflection Front Rear x 1O-2m x 10e2 m
Full state
b
Fig. 5. RMS vertical acceleration as a function of vehicle speed for the optimally passive cab and two controlled cab configurations.
0
Cab acceleration Vertical Longitudinal
Optimally pasave
0
Speed
2
‘fable 3. Effect of weighting factors, pi and &, on active suspension performance
0
t
-I
2 ?j 3 *I
n
Optimally
0
Futt stota
A Limited
0.10
passive feedback
state
feedback
t
Speed (km/h)
Fig. 6. RMS longitudinal acceleration as a function of vehicle speed for the optimally passive cab and two controlled cab configurations.
Active suspension for a heavy duty truck Acknowledgetnenr-The author would like to thank the Research Center, King Saud University for supporting this research. REFERENCES
1. E. K. Bender and 1. L. Paul, Active vibration isolation and active vehicle suspension. Depi. Mech. Engng, Massachusetts Institute of Technology, DSR 76109-L PB 173 648 (1966). 2. E. K. Bender, D. C. Karnopp and I. L. Paul, On the optimi~tion of vehicle suspensions using random process theory. ASME Publication 6%Tran-12 (1967). 3. D. C. Karnopp and A. K. Trikha, Comparative study of optimization techniques for shock and vibration isolation. ASME J. Engng Ind. 91, 41-49. 4. D. A. Hullender, D. N. Wormley and H. H. Richardson, Active control of vehicle air cushion susoensions. ASME J. Dvnum. Svsf Meas. Control 94, 41-49 (1972). 5. J. W. Youna and D. N. Wormlev, Optimization Of linear vehiclg suspensions subjected to simultaneous guideway and external force disturbances. ASME J. Dynom. Syst. Meas. Control 95, 213-219 (1973). 6. A. G. Thompson, An active suspension with optimal linear state feedback. Veiricie Syst. 5, 187-203 . Dynam. . (1976). 7. A. G. Thompson and C. E. M. Pearce, An optimal susoension for an automobile on a random road. SAE Paper No. 790478 (1979). 8. D. R. Guenther and C. T. Leondes. Svnthesis of a high-speed tracked vehicle suspension system. Part I: problem statement, suspension structure, and decomposition, and Part II: definition and solution of the control problem. IEEE Trans. Axiom Control AC-22, 158-172 (1977). 9. D. Hrovat and M. Hubbard, Optimum vehicle suspensions minimizing RMS rattlespace, sprung-mass acceleration and jerk. ASME J. Dynam. Syst. Meus. Control 103, 228-236 (1981). 10. A. Ha&, Suspension optimization of a 2-DGF vehicle model using stochastic optimal control technique. J. Sound Vibr. 100, 343-357 (1985). 11. A. Hat, Stochastic optimal control of vehicies with elastic body and active suspension. ASME J. Dynam. Syst. Meas. Confrol 108, 106110 (1986). 12. M. M. ElMadany, Active damping and load levelling for ride improvement. Compur. Strucr. 29,8997 (1988). 13. L. Ilosvai and B. S&es, Random vehicle vibrations as effected by dry friction in wheel suspensions. Vehicle Syst. Dynam. i, 197-209 (1972). Vehicle Suspension (in Russian) 14. R. W. Rote&erg, Mashinostroenie, Moscow (1972). 15. A. E. Bryson and Y. C. Ho, Applied optimal control. In Ootimization. Estimation and Control. John Wiley, _ New-York (1975). 16. J. E. Potter, Matrix quadratic solutions. SIAM J. appl. Math. 14 (1966).
393
where m =(m,+m,+m,+m,)(m,+m,+m,) 5 m=+m,+m~+m,+m~+m~+m,. All other elements of R are equal to zero. All elements of the stiffness Matrix K(9 x 9) are equal to zero except the following: ?&=k,+k2+k3+k4 &,,=&,=
-b,k,~b~k~+b~(k~+k~)
R,,=R,,=(b,+b,-b,)k, + (b4 + b, + be)kd &,,=G,,=
-k,,
R,,, = R,,, = -kz
R,,=&.,=
-k,,
-ke
L=$,= &,b = b:k, + b:k, + b:(k, + k,,)
gq,~=Rs.4=65tb4+b3-66)k3 + b,(b, + b, + bdkd K,, = R,, = b,k,,
Rd.,= R,,, = -bzk2
R4.s = &,, = -bsk,,
I&,, = %.a = -bsk,
&. 5 = (64 + bs - bd2k, + (be + b, + b,)2k, &,,=&,,=
-(be+bx-bdks
&., = %, 5 = -(b,
+ b, + bdk,
&,,=k,+k,,,
F,=kz+krz
&,,=k,+k,,
h,, = k4 + k,4
All elements of the damping matrix c(9 x 9) are zero except the following: C,, = c, + c2 + C)+ c,, C3,r=&=
-b,c,+b,c,+b,(c,+c,)
C,,=~,,=(b,+b,-b,)c,+(b,+b,+b,)c, c,,,=
q,
Q,,, = -cl,
qs
= q)
c,,
=
=
-c3,
= c;,, = --cz
%=G.,=
-c4
b:c, + b:c2 + b:(c, + c.,)
G,s = G,, = bdb, + b, - b,h
+ bdb, + b, + bdc,
L=G,,,=b,c,
C,, = G,
= -b,cz
G4.8 = G.4 = -b,c,
C.9 = CA = -b,c,
C5,s = (be + b, - b,)2c, + (b4 + b3 + b,)*c,
Matrix
C,, = G,
5= - @4+ b, - bdc,
C,,=C,,=
-@,+b,+b&,
q,,= c,,
CT,,= %
G,, =
G,9
R%
c39
4
=
c4.
all zero elements except the following:
has
APPENDIX
The mass matrix M(9 x 9) is symmetric with elements given as follows:
Fe.1 = k,,,
F,.2= km
h
Fw=k,.
3 =
k3r
Matrix G(9,2) has all zero elements except the following: G,., = 1,
Cl.2 = 1
$.I = -4,
%=bs
G,,,
=
G,r=b,+ba
-1,
G.2
=
-1,
G,,=b,-b,.
Computers d Strucrures Vol.31.No. 3.pp.38M93, 1989
Printedin GreatBritain.
DESIGN OF AN ACTIVE SUSPENSION FOR A HEAVY DUTY TRUCK USING OPTIMAL CONTROL THEORY M. M. ELMADANY Mechanical Engineering Department, College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia (Received 11 Jmwry
1988)
Abstract-The optimal active suspension for the cab in a tractor-semitrailer vehicle model is obtained using the linear stochastic optimal control theory. Two state controllers, full state and limited state feedback, are determined. The control law minimizes a performance function representing operator discomfort and cab suspension travel. The randomly profiled road is modelled as a filtered white noise excitation and time delays due to the multiaxle excitation inputs are included. The performance of the optimally active controlled system is compared with the performance of an optimal passive system.
NOTATION
vehicle-road system dynamic matrix road representation dynamic matrix vehicle dynamic matrix road disturbance input distribution matrix excitation matrix of road shape filter excitation distribution matrix vehicle dimensions gain matrix damping matrix suspension damping rates transformation matrix expectation operator forcing function distribution matrix control force distribution matrices measurement transformation matrix unit matrix moments of inertias of the cab, semitrailer and tractor, respectively.
v W X
-G&X, Y Y Y,l Ys, YI
u* rv 4*(w) 0 e,, e,,e,
&irmance index gain matrix stiffness matrix suspension spring rates tire spring rates Lagrangian function mass matrix masses of the cab, semitrailer and tractor, respectively masses of the wheel-axle assemblies generalized external force solution of Riccati equation vector of generalized coordinates weighting matrices covariance matrix of forcing function identity matrix matrix of zeros except for #h element which is unity weighting factors weighting matrix output vector solution of Riccati equation state spectral density matrix time trace Rayleigh dissipation function control force vector total potential energy
vehicle speed vector of coloured noise inputs state vector fore-aft displacements of the cab, semitrailer and tractor, respectively covariance matrix state vector vertical displacements of the cab, semitrailer and tractor, respectively. vertical displacements for axles measurement vector weighting factors coefficient Dirac delta function road excitation white noise process vector of white noise process variance of road irregularities time delay between ith and jth tires single-sided power spectral density temporal angular frequency pitch angles of the cab, semitrailer and tractor, respectively INTRODUCTION
One important facet of the work currently being carried out at research institutes and industrial organizations is aimed at providing better ride qualities for tractor-semitrailer vehicles. Research efforts have included measurement of ride sensation, subjective evaluation of human response to vibrations, development of mathematical models of vehicle, road and human systems, development of innovative passive
and active suspension systems and application of analytical and optimization techniques in the design of vehicle suspensions. The optimum design of vehicle suspensions subject to random road excitations has generally been based on one of two approaches. The first uses analysis techniques and formal optimization procedures for the optimum parameters selection of suspension systems consisting of passive elements. The other approach utilizes synthesis techniques in designing 385
386
M. M. ELMADANY
the optimal configuration of the transfer function of the suspension with the assumption that the transfer function can be realized with passive or active elements. The synthesis of the actively suspended cab in a tractor-semitrailer vehicle is the subject of this paper, Several papers have been published during the past two decades on the application of synthesis techniques to the design of suspensions of passenger cars, trucks and high speed rail vehicles. The vehicle suspensions have been synthesized using two techniques: the Wiener-Hiipf technique and optimal control theory. Bender and Paul [l] and Bender et al. [2] have employed the Wiener-Hopf technique to minimize a weighted sum of mean-squared vehicle acceleration and suspension stroke for a two-degreeof-freedom (Z-DOP) vehicle model. Kamopp and Trikha [3], Hullender et al. [4] and Young and Wormley[5] obtained the optimal configuration of the vibration isolator of a I-DOF vehicle model utilizing the Wiener-HBpf technique, with the meansquare vehicle acceleration being used as a ride comfort criterion. Applications of the optimal control theory to the design of vehicle suspensions have drawn much attention in recent years. Thompson [6] applied the optimal control theory to a vehicle model of 2-DOF and optimized a performance index comprising a weighted sum of mean-squared values for the body forces, tire dynamic deflections and relative wheel travel. Thompson and Pearce (1 extended the result in [6] to include the effects of front and rear suspension parameters on a two-dimensional linear vehicle model. Guenther and Leondes [8] presented results of optimal sus~nsion structure of a vehicle subject to both random roadway roughness and random external disturbance force. Hrovat and Hubbard [9] addressed the problem of optimum suspension structure for a I-DOF vehicle model using the Linear Quadratic Regulator theory. The performance index included RMS suspension stroke, acceleration and jerk. Using stochastic optimal control theory Ha6 analyzed a lumped Z-DOF vehicle model [lo] and a discrete-continuous vibrating system representing a vehicle with a long wheelbase [ 111. The active suspension system was optimized with respect to ride comfort measured in mean-squared acceleration, road holding and suspension travel with the limitation on the expenditure of control forces. It is evident from the number of studies cited that the possible application of the optimal control theory to remedy the inherent problems [ 121of the passively suspended tractor-semitrailer trucks has not been addressed. The present work is undertaken in order to gain an understanding of the capabilities of advanced suspension designs as they affect operator comfort through the application of the optimal control theory. In this paper, a 9-DOF lumped parameter vehicle model which includes heave and pitch motions is
considered. The primary emphasis is on the use of active suspension to improve vehicle dynamic performance. The input disturbance is the apparent road motion, caused by the vehicle’s forward speed along a road having an irregular profile. The road surface is described by a filtered white noise with the excitation time delays in the multi-axled vehicle being included in the analysis. Linear stochastic optimal control theory is applied to design the optimally active controlled suspension system for cab ride. Two state controllers aiming at improving cab performance are considered. The first controller relies on the availability of full state information including that of road surface height. In this case, a Riccati controller is used to implement the active suspension. In the second controller, a limited feedback, based on absolute velocity signals, is assumed and the optimal gains are obtained using a gradient search technique.
PASSIVE VERSUS ACTIVE SUSPENSION
The main functions of the road vehicle suspension systems (passive or active) include the following. (a) It must provide basic support. (b) It must isolate passenger and cargo compartments from road surface irregularities and external forces disturbances. (c) It must transmit braking and accelerating forces to the road. The requirements to fulfil these various functions conflict to varying degrees. The severity of the conflicts is likely to grow in view of increased demand for upgraded ride quality and improving operator working conditions. Practically all suspensions in use today on tractor-semitrailer trucks utilize passive elements, and typically include tires, linkages, coil and leaf springs, rubber donuts, bump stops and hydraulic dampers. Passive suspensions do not require a supply of power and thus their capabilities are fundamentally limited. On the other hand, passive elements are simple, inexpensive and reliable. Active suspension systems may use continuous or intermittent power from an external source. Typically an active feedback control suspension system consists of a power source (e.g. electric, hydraulic or pneumatic), an actuator (e.g. hydraulic, pneumatic, magnetic or electro-mechanical), sensing instruments (e.g. accelerometers, relative displacement transducers), signal processing, feedback and amplifying elements. Actively controlled suspensions offer several advantages compared with passive suspensions. They permit the use of more general control forces than are available from passive suspensions comprised of energy storage and dissipative elements. They permit selectivity in design with only selected adverse characteristics of vehicle behavior being modified. In contrast to passive suspensions whose
Active suspension for a heavy duty truck properties are fixed, active suspensions are capable of being tuned or adjusted and can also be made self adaptive to varying conditions of speed and loadings. These features must, of course, be balanced against the increased complexity, cost, power requirement and decreased reliability of the active controller. OPTIMAL
Characterization
CONTROL
FORMULATION
of road irregularities
As the vehicle travels along the road, it experiences disturbance forces due to road surface roughness. The suspension system is designed to account for this disturbance spectrum since the acceleration and control motion will largely depend upon the traversed type of road surface. Measurements on road profiles have shown that road irregularities can be described by stationary ergodic Gaussian random variables with zero mean values. A road roughness model for a vehicle moving at a constant forward speed, v, is given by the single-sided power spectral density [13, 141.
cW=~&l&*
(1)
where u2 is the variance of road irregularities, c1is a coefficient depending on the shape of road irregularities, and w is the temporal frequency. The vehicle excitation model A(t), given by the stationary Gaussian colored noise (l), can be obtained from a white noise process I;(t) by means of a shape filter i(t) = -ad(t)
+ 5(t),
(2)
where El5
011= 0
a??[r(t){(t -
7)] = ~cwu%(~).
is the Dirac delta function. Since the vehicle has multiple contacts with the roadway, the time delay between successive contact points has to be considered. The road excitations to the vehicle model take the form
387
+ 0148
(4 -
12 + 714) + QSl6
+ Qd(4
-
t2 +
713 -
712)
+
0326
(t2 -
tl +
713 -
712)
+
0248
(6
-
t2 +
714 -
712)
+
Q426tt2
-
tl +
714 -
712)
+
0346
-
t2 +
714 -
713)
+
Q43602
-
6 +
714 -
21311
0,
(t2 -
t, +
714)
(4)
where Qll is a (4 x 4) identity matrix and Qii, i # j, is a (4 x 4) matrix of zeros except for the ijth element which is unity. Vehicle mathematical
model
In the design and evaluation of the statistical performance characteristics of a specific suspension concept, an abstract mathematical model is developed to represent the actual road vehicle. It is desirable to develop the simplest credible model which is able to describe the motions of the vehicle system. The simplicity of the model is determined to a large extent by the number of degrees of freedom. The credibility of the model is determined in part by whether the degrees of freedom used will describe the system to the desired accuracy. The process of developing the vehicle model is strongly influenced by the available information about the characteristics of the actual system and the purpose for which it will be used. In general, the fidelity of the model is limited by the assumptions, approximations and restrictions that are made in the development of the mathematical model. The design of the model always includes a trade-off between fidelity of the model and the ease, cost, interpretation and understanding of the solution of the governing equations which describe the behavior of the model. In order to make the mathematics tractable in developing the tractor-semitrailer model used in this study, the following assumptions are made.
d(7)
+=A,,,w+B,t
(3)
where A,=
-avI
B,=I
cT=[~I,1;2,ts,t41T.
I is an identity matrix of dimension 4 x 4 and { is a zero mean correlated white noise process with the following covariance matrix: EK(4W(t2)l = 2ava2[Qll~(t,- t2) +Qd(tl - 4 + %)I + + Q,J(t,
-
t2 +
Q2lW2
-
tl +
712)
7.13) + Q3l602
-
t1 +
713)
(a) The tractor chassis, semitrailer, cab and axles are rigid, and their stiffnesses are lumped into the suspension elements. (b) All linear and angular displacements are small. (c) The springing and damping elements in the suspension system are linear. (d) Tire nonuniformity, wheel unbalance effects and engine and drivetrain excitations are ignored. (e) Nonlinearities resulting from wheel-hop, suspension stop and dry friction in the suspension elements are neglected. (f) Gyroscopic effects are neglected. The model of the tractor-semitrailer vehicle considered is presented in Fig. 1. Each unit representing the cab, tractor or semitrailer has rigid body modes of heave and pitch except the semitrailer which is constrained in the vertical direction due to the artic-
M. M. E&%UMNY
kt3
$
kt4
f
Fig. 1. Tractor-semitrailer model.
ovation at the I%& wheel. Each WheeI-axle assembP$ Vanstates in the veti dire&ion. Tire stiffnesses are represented by &near spring e&nents, The tractor and semitrailer bodies are suspended using passive spring and damping elements. The cab is actively suspended on the tractor chassis with t(r and uz r~pr~~~ting the controlled active suspension forces. ~~~~~~s
of
motim
and state space ~5r~~~~ti5~
The equations of motion of the vehicle system are derived using Lagrange’s equation which is given by
~ornb~n~n~ eqns (3) and (7) yiefds where the Lagrangian
function L, is given by
L=T-V,
T Y U N;:(r) Pi
is the tota kinetic energy of the system is the total potential energy of the system is the Rayleigh dissipation function is the generalized external force is the generalized coordinate.
where the augmented state vector y and the matrices A, G and B are given by
The resulting equations of motion take the form ~~~~~~~~=~u~~
(63
where ICT, T3 and R are the mass, damping and stiffness matrices, respectively. F is the distribution matrix of the forcing function and G is the distributioa matrix of the active forces. u = [a,, uz]’ is the active forces. The matrices R, C, P, F and G are given in the Appendix. In order to perform Linear Quadratic Regulator (LQR) design, it is necessary to model the system in state space form.
The basic measures of the controller performance considered in this study are the cab ride comfort and the cab s~~ns~~~ travels. The equations governing the motions of the cab reveals that cab vertical acceleration is proportional to the control forces. Thus limiting the values of the control forces will limit the Ievei of cab acceleration which is the measunz of ride (~s~omfo~. The cab front and rear sus~ns~~~~ trave$s are &tear ~orn~~~ons of the
Active suspension for a heavy duty truck
389
where P satisfies the steady-state matrix Riczati equa-
elements of the state and are given by
tion
PA + ATP - PGR-‘G’P
where matrix D of dimension 4 x 22 has all elements equal to zero except the following:
+ Q = 0.
(13)
P is a unique non-negative
definite matrix. Eigenveetor technique [16] in which P is derived from the eigenvectors of the canonical form of the equation is used to solve for P. The closed-loop state matrix equation takes the form $=(A-GC)y+Bw.
Since the design of vehicle suspension systems is a function of several criteria, the probtem is to assign some means for comparing the effects of each in order to determine which design has the optimum combination of all the specified criteria. The problem is one of assigning a weight to each criterion in order to assist in making an engineering decision. The quadratic performance index used takes the form
(14)
The solution of eqn (14) is the steady-state variance matrix governed by (A - GC)Y + Y(A - GC)‘+
0 = 0
co-
(W
which is the Lyapunov matrix equation, where 0 is the intensity of the forcing function and is given by
f = Efr’Q,r + aTRu] = E[ yTIFQ,Dy f u=Ru] = Efy’Qy
+ n%]
where
Q and R are positive definite, symmetric matrices. E is the expectation operator. qI, gtl p1 and p2 are weighting factors. The absolute value of each term of the weighting factors is not important. More important is the relative value. By selecting the weighting factors in different ways, various optimal control laws can he determined. They are selected to achieve appropriate relative weights between the responses and control efforts.
The optimal regulator requires all the states of the system, including the road inputs, to be available for feedback. Given a system described by eqn (8) and an output given by (9), it is well known [lS] that the optimal control law which minimizes the performance index described by eqn (11) is given by II=-cy
C=R.-“GrPV
(12)
J-
ftfP@*
Wf
The power spectral densities of the state variables are obtained from
S,(o) = [id
-(A - GC)]-‘B
2WO2/rt
Were, it is assumed that not all the system state variables can be measured. The controller configuration chosen has the absolute cab and tractor velocities at the front and rear cab suspension locations as feedback signals. These elements of the measurement vector z are linear combinations of the elements of the state vector y. The measurement
390
M. M. ELMADANY
equation therefore can be written in the form z=Hy.
Table 1. Values of baseline vehicle oarameters (19)
950 kg 24,800 kg 3450 kg 803 kg 1503 kg 1503 kg 1503 kg 800 kg m2 200,000 kg m* 9500 kg m2
The matrix H of dimension (4 x 22) has all elements equal to zero except the following: H IO,1 -1 - ?
HI,,, = -be,
HI,,, = 1,
H,,,r = bs,
HI,,, = 1,
H,,c = - @, + bs 1
H 13,3 -1 -
HI,,,=
9
A linear time invariant
-@,-&I.
(20)
k, = 400 kN/m (21)
is to be determined to minimize the performance index (11). Using eqns (8), (19) and (21), the closed loop system is given by $=(A-GKH)y+Bw.
The performance mined from
(22)
= tr [So]
(23)
where (A - GKH)Y + Y(A - GKH)r + 0 = 0
(24)
and + S(A - GKH) -HTKTRKH
+ Q = 0.
(25)
A gradient search technique is used to determine the optimum value of K. The gradient of the performance index (11) with respect to any element Ku of K can be found from ;
= 2[RKHYHr - GrSYHT].
Hence, a cyclic iterative procedure follows.
(26) is devised as
(1) Assume an initial value for K. (2) Obtain Y and S by solving eqns (24) and (25). (3) Determine the value of the performance index J from eqn (23) and %I/dK from eqn (26). Update K using a gradient search routine. (4) Go to step (2) until results from cycle to cycle are similar. RESULTS
baseline vehicle parameters used in the numerical calculations are given in Table 1. The cab The
b, = b, = b, = b, = b, = b,= h, = h, = h, =
5.62 m 2.2 m 0.7 m 0.9 m 0.85 m 1.15m 0.3 m l.Om l.Om
k, k, k, k,
= = = =
700 kN/m 1000 kN/m 1000 kN/m 100 kN/m k6 = 60 kN/m
c, = cy = c) = c, = c5 = c, =
33 kNsec/m 58.3 kNsec/m 83.3 kNsec/m 83.3 kNsec/m 8.3 kNsec/m 5.0 kNsec/m
Tire characteristics k,3 = 4000 kN/m
k,, = 2000 kN/m kr2 = 4000 kN/m
k,4 = 4000 kN/m
index (11) may then be deter-
J = tr Ty(Q + (HTK*KH)]
(A - GKH)‘S
2.8 m b; = 3.68m
Suspension characteristics
control law of the form
u= -Kz
b, = 0.65 m b, =
suspension parameters k,, k,, c5 and c, correspond to an optimized design. The optimization is based on minimization of a performance index comprising of the weighted sum of the mean-squared cab vertical acceleration and suspension deflections. The passively optimized vehicle is assumed to travel at 80 km/hr over a concrete road. The road characteristics are a = 0.03 m-i and u = 0.0087 m. The weighted RMS vertical and longitudinal accelerations for the optimal passive suspension are 0.084 and O.O5g, respectively, while the suspension deflection is approximately 0.36 x 10m2m. A speed of 80 km/hr for the vehicle operating on the concrete road is used for designing the optimal controllers. The weighting factors in the performance index are taken as p, = p2 =0.8 x lO-9 and q, = q2= 1, unless otherwise stated. Table 2 displays the eigenvalues corresponding to the heave and pitch modes of the cab for the vehicle with full state feedback controller and for the optimally passively suspended cab. Compared to the passive vehicle, the actively controlled vehicle shows a marked reduction in the natural frequencies. The damping ratios are lower than for the passive vehicle. Figures 2 and 3 show a comparison of the IS0 ride quality evaluation for the full state feedback conTable 2. Eigenvalues for the cab with passive and active suspensions Passive suspension Damping Eiaenvalues rat’0 _ (SC-‘) -9.28 kjl4.66 -7.18 kjll.10
0.535 0.543
Active suspension Damping Eigenvalues ratio (WC’) -3.21 fj8.72 -4.73 f j9.89
0.345 0.432
Active suspension for a heavy duty truck passive Full state feedback Limited state feedback
Optimally
*a .
Frequency (Hz) Fig. 2. Ride quality evaluation for the optimally passive cab and two controlled cab configurations-vertical acceleration.
trolled, limited state feedback controlled and baseline vehicle for vertical and lon~tudinal accelerations, respectively. The one-third octave band acceleration is used for the ride quality evaluation. It can be seen that the optimally baseline vehicle response exceeds the IS0 fatigue decreased proficiency in the 2-5 Hz frequency band for trips longer than 8 hr, and hardly meets the 4 hr fatigue decreased proficiency contours. The configuration with full state feedback gives the best performance improvement in both the vertical and longitudinal directions, and the ride is improved
n Optimally passive o Full state feedback A Limited state feedback
-
in the most sensitive frequency region for human beings. The limited feedback controller satisfies the IS0 criterion for 8 hr fatigue decreased proficiency and is very effective in solving the riding problem of the cab. Thus, a significant improvement over the passive vehicle is obtained using either full or limited state feedback, with the weighted RMS acceleration levels being markedly reduced. For the full state feedback controller, the weighted RMS vertical and longitudinal accelerations are 0.024 and O.O12g, respectively, while the corresponding values using the limited state feedback gives 0.037 and O.OlSg, respectively. Figure 4 shows the weighted RMS vertical and longitudinal accelerations vary with the weighting parameters p, and p2. Variation of the RMS cab front suspension deflection with p, and pz is also shown in the figure. Full state feedback is considered. For acts as passive p, = p* = 10-u the suspension elements (with very stiff suspension) with weighted RMS values of 0.104 and 0.044g for vertical and longitudinal accelerations, respectively. A thorough utilization of the active system possibilities is indicated for pi = p2 = IO-' with a significant reduction in both the vertical and lon~tudinal accelerations which are 0.~7 and 0.01 g, respectively. Note that the weighted RMS longitudinal acceleration cannot be reduced below 0.01 g because of the contributions from the pitching motions of both the tractor and semitrailer units, while the active suspension is principally affe@ing the heave and pitch motions of the cab. It is of interest to examine the case of having an active suspension at either end of the cab. This is done by varying the weighting factors p, and p2.For p, = 0.8 x IO-“ and p2 = IO-i0 which represents the case of the cab front suspension optimally controlled while the rear suspension is passive the weighted RMS vertical and lon~tudinal accelerations are 0.028 and O.O17g, respectively. For the other case in which the cab is actively suspended at the rear and
4
i
lo-;o”
Cab
front deflection
IO’
2
E
391
Frequency (Hz)
Fig. 3. Ride quality evaluation for the optimally passive cab and two controlled cab configurations-longitudinal acceleration.
Fig. 4. Effect of weighting factors p, and p2 on cab performance.
M. M. ELMADANY
332
passively suspended at the front (p, = lo-lo and p2 = 0.8 x 10M9)a large increase in the acceleration levels is noticed with the weighted RMS acceleration reaching values of 0.052 and 0.03 g in the vertical and longitudinal directions respectively. This indicates that the control loses its effectiveness as the cab front susDension becomes stiff. In the extreme case where the-cab front is pinned to the tractor chassis while the cab rear is actively suspended (p, = 10-13, p2 = 0.8 x 10-9) the acceleration levels go up to 0.1 I and 0.051 g for the vertical and longitudinal directions, respectively. This result is due to the high level of vibration transmitted through the front suspension to the cab which shows the ~nt~bution of the tractor pitching mode. It may be concluded that of the two cab suspensions, stiffening the front suspension has a much stronger negative effect on cab riding quality than the rear suspension, and if only one active suspension system is to be used, it should be installed at the cab front location. The sensitivity of the statistical active suspension performance to the choice of the weighting factors p, and p2 is shown in Table 3. The weighted RMS vertical acceleration has been reduced by a factor of three by changing p, = p2 = 0.02 x 10m9to p, = p2 = 1.4 x 10m9.This reduction is achieved with an almost 100% increase in cab suspension deflection. Figures 5 and 6 show, respectively, the weighted RMS vertical and longitudinal accelerations as functions of vehicle forward speed for the controlled cab with full and limited state feedback and the passively suspended cab. The optimal control gain matrix obtained for the vehicle travelling at 80 km/hr is used for operation at the other velocities. It is clear that the actively controlled suspensions have a significant iniluence on cab riding quality at all speeds considered. Compared with the passive baseline model, the weighted RMS accelerations are reduced by approximately 70% for the full state feedback in the speed range. The control forces required vary between 135 N at 20 km/hr and 230 N at 80 km/hr actuator RMS, or less than 405 N and 690 peak assuming Gaussian disturbance. It is worth mentioning that for full state feedback using the optimal control gain matrix obtained for 80 km/hr for the other velocities
^o0.19 3 0)
PI=P2
0.2 x 0.4 x 0.6 x 0.8 x 1.0 x 1.2 x 1.4 x
1o-9 1o-q 1O-9 1O-9 1o-9 1o-9 1o-9
cp)
0.046 0.034 0.028 0.024 0.022 0.019 0.016
Gp)
0.019 0.014 0.012 0.011 0.011 0.010 0.010
0.36 0.48 0.56 0.61 0.65 0.68 0.70
0.19 0.28 0.33 0.37 0.41 0.43 0.45
Limited
feedback
state
feedback
(km/h)
degrades the cab acceleration performance by just 8% compared with that computed using the actual optimal control gains obtained at the different speeds. The roadway state variables feedbacks do not influence the natural dynamics of the vehicle but act to reduce the disturbance seen by the vehicle. CONCLUSIONS
The ride quality improvements achievable with an actively suspended cab in a typical tractor-semitrailer vehicle are evaluated. The actively controlled suspensions with full state feedback provide superior riding qualities, while the active control force required is kept within a practical range. Using absolute velocities as feedback signals degrades the performance to some extent, but the performance is still considerably better than the passively suspended cab. The optimal control theory is a viable method for designing active suspensions and could better direct future design efforts towards solution of the cab ride problem. It also yields high performance active vibration controller. Covariance analysis is a very useful tool in determining the stochastic response characteristics of road vehicles.
^o 0.15
s
Cab deflection Front Rear x 1O-2m x 10e2 m
Full state
b
Fig. 5. RMS vertical acceleration as a function of vehicle speed for the optimally passive cab and two controlled cab configurations.
0
Cab acceleration Vertical Longitudinal
Optimally pasave
0
Speed
2
‘fable 3. Effect of weighting factors, pi and &, on active suspension performance
0
t
-I
2 ?j 3 *I
n
Optimally
0
Futt stota
A Limited
0.10
passive feedback
state
feedback
t
Speed (km/h)
Fig. 6. RMS longitudinal acceleration as a function of vehicle speed for the optimally passive cab and two controlled cab configurations.
Active suspension for a heavy duty truck Acknowledgetnenr-The author would like to thank the Research Center, King Saud University for supporting this research. REFERENCES
1. E. K. Bender and 1. L. Paul, Active vibration isolation and active vehicle suspension. Depi. Mech. Engng, Massachusetts Institute of Technology, DSR 76109-L PB 173 648 (1966). 2. E. K. Bender, D. C. Karnopp and I. L. Paul, On the optimi~tion of vehicle suspensions using random process theory. ASME Publication 6%Tran-12 (1967). 3. D. C. Karnopp and A. K. Trikha, Comparative study of optimization techniques for shock and vibration isolation. ASME J. Engng Ind. 91, 41-49. 4. D. A. Hullender, D. N. Wormley and H. H. Richardson, Active control of vehicle air cushion susoensions. ASME J. Dvnum. Svsf Meas. Control 94, 41-49 (1972). 5. J. W. Youna and D. N. Wormlev, Optimization Of linear vehiclg suspensions subjected to simultaneous guideway and external force disturbances. ASME J. Dynom. Syst. Meas. Control 95, 213-219 (1973). 6. A. G. Thompson, An active suspension with optimal linear state feedback. Veiricie Syst. 5, 187-203 . Dynam. . (1976). 7. A. G. Thompson and C. E. M. Pearce, An optimal susoension for an automobile on a random road. SAE Paper No. 790478 (1979). 8. D. R. Guenther and C. T. Leondes. Svnthesis of a high-speed tracked vehicle suspension system. Part I: problem statement, suspension structure, and decomposition, and Part II: definition and solution of the control problem. IEEE Trans. Axiom Control AC-22, 158-172 (1977). 9. D. Hrovat and M. Hubbard, Optimum vehicle suspensions minimizing RMS rattlespace, sprung-mass acceleration and jerk. ASME J. Dynam. Syst. Meus. Control 103, 228-236 (1981). 10. A. Ha&, Suspension optimization of a 2-DGF vehicle model using stochastic optimal control technique. J. Sound Vibr. 100, 343-357 (1985). 11. A. Hat, Stochastic optimal control of vehicies with elastic body and active suspension. ASME J. Dynam. Syst. Meas. Confrol 108, 106110 (1986). 12. M. M. ElMadany, Active damping and load levelling for ride improvement. Compur. Strucr. 29,8997 (1988). 13. L. Ilosvai and B. S&es, Random vehicle vibrations as effected by dry friction in wheel suspensions. Vehicle Syst. Dynam. i, 197-209 (1972). Vehicle Suspension (in Russian) 14. R. W. Rote&erg, Mashinostroenie, Moscow (1972). 15. A. E. Bryson and Y. C. Ho, Applied optimal control. In Ootimization. Estimation and Control. John Wiley, _ New-York (1975). 16. J. E. Potter, Matrix quadratic solutions. SIAM J. appl. Math. 14 (1966).
393
where m =(m,+m,+m,+m,)(m,+m,+m,) 5 m=+m,+m~+m,+m~+m~+m,. All other elements of R are equal to zero. All elements of the stiffness Matrix K(9 x 9) are equal to zero except the following: ?&=k,+k2+k3+k4 &,,=&,=
-b,k,~b~k~+b~(k~+k~)
R,,=R,,=(b,+b,-b,)k, + (b4 + b, + be)kd &,,=G,,=
-k,,
R,,, = R,,, = -kz
R,,=&.,=
-k,,
-ke
L=$,= &,b = b:k, + b:k, + b:(k, + k,,)
gq,~=Rs.4=65tb4+b3-66)k3 + b,(b, + b, + bdkd K,, = R,, = b,k,,
Rd.,= R,,, = -bzk2
R4.s = &,, = -bsk,,
I&,, = %.a = -bsk,
&. 5 = (64 + bs - bd2k, + (be + b, + b,)2k, &,,=&,,=
-(be+bx-bdks
&., = %, 5 = -(b,
+ b, + bdk,
&,,=k,+k,,,
F,=kz+krz
&,,=k,+k,,
h,, = k4 + k,4
All elements of the damping matrix c(9 x 9) are zero except the following: C,, = c, + c2 + C)+ c,, C3,r=&=
-b,c,+b,c,+b,(c,+c,)
C,,=~,,=(b,+b,-b,)c,+(b,+b,+b,)c, c,,,=
q,
Q,,, = -cl,
qs
= q)
c,,
=
=
-c3,
= c;,, = --cz
%=G.,=
-c4
b:c, + b:c2 + b:(c, + c.,)
G,s = G,, = bdb, + b, - b,h
+ bdb, + b, + bdc,
L=G,,,=b,c,
C,, = G,
= -b,cz
G4.8 = G.4 = -b,c,
C.9 = CA = -b,c,
C5,s = (be + b, - b,)2c, + (b4 + b3 + b,)*c,
Matrix
C,, = G,
5= - @4+ b, - bdc,
C,,=C,,=
-@,+b,+b&,
q,,= c,,
CT,,= %
G,, =
G,9
R%
c39
4
=
c4.
all zero elements except the following:
has
APPENDIX
The mass matrix M(9 x 9) is symmetric with elements given as follows:
Fe.1 = k,,,
F,.2= km
h
Fw=k,.
3 =
k3r
Matrix G(9,2) has all zero elements except the following: G,., = 1,
Cl.2 = 1
$.I = -4,
%=bs
G,,,
=
G,r=b,+ba
-1,
G.2
=
-1,
G,,=b,-b,.