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Integral and state variable feedback controllers for improved performance in automotive vehicles
Com~rerr % Strucrr~es Vol. 42, No. 2, pp. 237-244, Printed in Gnat Britain.
1992 0
~~~9192 $5.00 + 0.00 1991 pergamon m?ss plc
INTEGRAL AND STATE VARIABLE FEEDBACK CONTROLLERS FOR IMPROVED PERFORMANCE IN AUTOMOTIVE VEHICLES M. M. ELMADANY Mechanical Engineering Department, College of Engineering, King Saud University, Biyadh, Saudi Arabia (Received 12 October 1990)
Abstract-An analytical investigation of a half-ear model using an integral plus state variable feedback controller for tracking and regulation is performed. The potential benefits of incorporating, actively or semi-actively, such a controller to remedy the inherent problems associated with conventional passive suspensions and active suspensions based on state variable feedback controllers are examined. Both random and dete~inistic roadway inputs as well as de~~inistic body force and moment dist~~n~ are used. The results demonstrate that an optimal suspension using an integral plus state variable feedback controller retains both excellent ride and attitude control characteristics.
1. ~RODU~ION The demand for increased comfort and controllability in road vehicles has resulted in many automotive industries seriously considering incorporating microprocessor driven active suspensions [la]. These intelligent electronically controlled suspension systems can ~tentially increase overall performance. The objectives of improving comfort and controllability simultaneously, however, cannot be achieved using optimally controlled suspensions based on state variable feedback controllers. The trade-off between vibration isolation and overall system stiffness is the main reason. In order to maximize isolation quality for greater ride comfort, active suspension has to have a lower body natural frequency. As a consequence, a large clearance space is required both for relative excursions due to road excitations, as well as for displacements resulting from externally applied loads. In other words, the co~esponding overall static stiffness will not be sufficiently high to control body motion due to changes in payload, forces from steering, braking and traction, and aerodynamic forces. This problem can be overcome if the active sus~nsion control strategy is envisioned as shown in Fig. 1 [S-7]. In this case, the controller is composed of two parts: (a) a ride controller to provide improved ride comfort and safety, and (b) an attitude controller to perform load leveling action, control and roll response of the vehicle during cornering and control the pitch response during braking and accelerating. Karnopp [8,9] and ElMadany [7], using an active high gain load leveling device in parallel with an active damper, demonstrated that such as active suspension system has the potential to provide excellent ride and handling characteristics as well as
zero static deflection for body force disturbances. Thompson and Davis [6] and ElMadany 171, by including an integrator in the forward path (ahead of the control input), have obtained an active suspension system with infinite stiffness towards static loading, but which is soft with respect to road inputs. The resulting system, however, does not have zero steady-state axle-to-body deflection in response to ramp road inputs. Using multivariable integral optimal control, an suspension system is derived [lo, 111. The optimal control law consists of two controllers. The first is an integral controller acting on suspension deflection (rattle space) to ensure zero steady-state offset due to body and maneuvering forces as well as road rump inputs. The second is a state variable feedback controller for vibration control and performance improvement, The derived optimal suspension system retains the excellent ride comfort associated with lower body natural frequency with greatly enhanced controllability. In the previous work, a quarter-car model was selected. Despite the simplicity of the quarter-car model, it contains many important features of automotive suspensions, leads to a basic understanding of the limitations to suspension performance, and allows thorough investigation of the vehicle dynamic problems. However, when detailed study of the motion of the vehicle is required, two- as well as three-dimensional models should be considered. In this study, a two-dimensional linear vehicle model with coupled bounce and pitch modes, subjected to both random and dete~inistic disturbances, is considered. Optimal control theory is applied to the model to design two control schemes: an integral-plus-state variable feedback controller (IPSVF) and state variable feedback controller 237
M.M. External force
Road disturbances
Attitude controller I
ELMADANY
about the static equilibrium are assumed and a linear discrete mathematical model is derived. A set of eight variables is needed to describe the system. The eight variables are: the suspension deflections at the front and rear (x, and x5); the tire deflections at the front and rear (x2 and x,); the vertical velocities of the two ends of the sprung mass (x3 and x,); and the vertical velocity of each of the two unsprung masses (x4 and x8). The following state equations can be derived using Newton’s second law:
I
Fig. 1. Ride and attitude
controllers.
f,=
_ul+k,,
1, =
X8 -
x,
1, =
v2 -
xg
is=
_!L+k,x,.
ml
(SVFB). The controllers are based on measurements of all states. These measurements are used to drive the control signal required to generate the active forces or for modulating the activated dampers (semiactive dampers). The performance levels that result are compared to identify superior control strategies. 2. VEHICLE
SYSTEM
The model of the vehicle system is shown in Fig. 2. The sprung mass is considered as a rigid body, with mass, m, and pitch moment of inertia, J, undergoing heave and pitch motions. The unsprung masses, m, and m2, represent front and rear wheel-axle assemblies, respectively. The linear spring elements, k,, and kt2, represent front and rear tire stiffnesses. Continuous tire contact with the road is assumed. U, and u2 are active or semi-active suspension forces, which are control variables. The road disturbances at the front and rear tires are represented by velocity inputs, D, and v2, respectively. The disturbances also include an external applied force, F, due to a change in load and a moment, M, that may result from accelerating/or braking the vehicle. Small deviations
ml
m2
m2
These equations can be expressed in matrix form as: k=Ax+Bu+Ew,
(2)
where x is an 8-state vector, II is a 2-control force vector, w is a 4-disturbance vector, A is an 8 x 8 system matrix, B is an 8 x 2 control force distribution matrix, and E is an 8 x 2 disturbance distribution matrix. To achieve zero steady-state response for the suspension deflections, the system state equations (2) are appended with two state variables, x,(t) and x,,(t), such that: i, = x,
(3)
i 10= x5
(4)
or in matrix form:
p=cx,
(5)
where C is a 2 x 8 matrix with C,, , = 1, C,, 5= 1 and all other elements are zero. The augmented system equations then become: i=Az+Bu+Ew, where z = [x’ pq,
Fig. 2. Vehicle model.
(6)
Integral and state variable feedback controllers The performance characteristics that are of the utmost interest in vehicle suspension design are passenger ride comfort, suspension travel and road-holding ability. Sprung mass acceleration has been used as an indicator of passenger ride comfort. Suspension deflection should be kept small for good rigid body control. It is also important from a packaging point of view. The tire-road contact force (or equivalently dynamic tire deflection) should be kept as small as possible for safety reasons.
3. DESIGN
OF OPTIMAL
CONTROLLERS
Consider the linear time-invariant system described by eqn (2), i.e.
controlled
The optimal follows:
239
closed-loop
system is expressed
3r=(A-BG)x+Ew =&x+Ew.
(12)
The LQR design procedure is guaranteed to produce closed-loop stable designs if the system is “controllable” and “observable”, which it is for the system in hand. If the disturbance w(t) = 0, the regulator ensures that the initial state disturbances, x,, decay to zero at a rate prescribed by the eigenvalues of 4. Consider eqn (12) again, but now with x, = 0, and w(t) a step input of magnitude w, . Then the response is: x(t) = A;’ [exp(A, t) - Z]Ew,.
%=Ax+Bu+Ew.
(13)
(2)
For controller design it is assumed that all the states are available and exact measurements are possible. First, consider a state variable feedback regulator: u = -Gx,
(8)
where G is the state variable feedback gain matrix. The optimization consists of determining the control, u, which minimizes the following performance index:
Since all the modes of A, = (A - BG) are damped, then the maximum value of x(t) occurs at steadystate [ 131 and gives: x, = -(A - BG)-‘Ew,
(14)
u,=G(A-BG)-‘Ew,.
(15)
and
Consider next an integral plus state feedback control algorithm for the system in eqn (6): u= -Gz=
-Gx-Hp,
(16)
where H is an integral action gain matrix. The optimal control input, II, is obtained minimizing the following performance index: j=; whereQ,.,=y,,Q,,,=yZ,Qs.s=Y~,Q6.6=Y4andall other elements of Q are zero, R,~, = p, , R,, 2 = pz, and all other elements of R are zero. In this performance index, j U: dr and s a: dt are related to ride comfort requirements, since they are proportional to car body acceleration, j xi dt and j x: dt reflect body-to-axle space requirements, and j x2 dt and s xz dt indicate the road following requirements. The desired relative levels of these quantities are determined by the choice of the weighting factors pi 9p2 and yI-y4. Optimal control theory provides the solution of eqn (9) in the form of eqn (8). The gain matrix G is computed from [ 121: G = R-‘BTP
(IO)
and the matrix P is obtained as the solution matrix to the steady-state algebraic Riccati equation: PA + ATP - PBR- ‘BTP +.X0 = 0.
as
(11) \--I
by
m(P,t(:+P211i+Y,X:+yZX:+y?X:+y4Xa s0 +YSX; + Y~X:O)dt
= ;
m (z’Qz + uTRu) dt,
(17)
I0 where
Q=
QO oy,o. I o o
0 76
I
The optimal control law that minimizes the performance index [eqn (17)] is given by: u = -Gz = -R-%A, where P satisfies the steady-state equation:
(18)
matrix
Riccati
Pii + dTP - PBR-‘BTP + (5 = 0.
(19)
240
M. M.
ELMADANY
The closed-loop system equations are:
(A-W
e,
--H-I
C
0
E
I II z+
OW
=$z+Ew.
(20)
It is of interest to examine the case where the control law given by eqn (16) is invoked by a semi-active controller (activated damper). The semiactive controller behaves like an active controller as long as the control action requires energy dissipation. If the control action calls for imparting energy to the system the activated damper is off, delivering zero force [14,15]. Two activated dampers are incorporated into the passive suspension systems at the front and rear. The forces in the dampers are adjusted by continuously modulating the fluid-flow orifices, based on the integral and state variable feedback control law [eqn (16)]. This leads to the control of energy dissipation without delivering energy to the system. The logical function that satisfies the dissipative phase of the control action is: uSal= -,g,
dljzj9
uSBI- 0, ha2 = -jg,
G2,zj,
usa2 --0
when
a,,, (xj - x4) > 0
(21)
when
u,] (xj - x4) < 0
(22)
when
u,,~(x, - x8) > 0
(23)
when
u,r(x, - xe) ,< 0. (24)
4. EVALUATION
OF VEHICLE
RIDE
4.1. Eigenvalue analysis The following parameter values for a baseline vehicle are assumed in the calculations [16]: m, = 28.6 kg,
- 15.74 fj76.35
(5, = 0.2)
- 11.15 k5’55.33 (r, = 0.2) -2.37 fj10.02
(5 = 0.23),
- 1.3 kj8.04 (l = 0.16). The eigenvalues for a number of optimal active suspension systems obtained by varying the weighting factors in the performance index are shown in Table 1. The eigenvalues presented are for state variable feedback controllers and integral plus state variable feedback controllers. The wheel-hop damping ratio takes the value of 0.2 or 0.3 in all cases considered. By comparing the eigenvalues of the actively suspended vehicle with the corresponding eigenvalues of the passively suspended vehicle, it may be noted that the body (sprung mass) modes for the actively suspended vehicle are better damped than for the passively suspended vehicle. The integral plus state variable feedback controller introduces two extra real roots. With increasing the weighting factors ys and y6, the real roots become more negative. 4.2. Response to random road disturbances The vehicle is assumed to traverse a random road surface at a constant forward speed, V. The road roughness is described by its power spectra1 density [17, 181:
k,, = k,* = 155.9 kN/m
mz = 54.4 kg,
a= 1.1 m
m = 505 kg,
6= 1Sm
J=651
on the damping ratio of the wheel-hop mode limits the extent to which the r.m.s. sprung mass acceleration can be reduced. On the other hand, for very lightly damped wheel motions, lower road-holding qualities will result. A damping ratio of 0.2 was suggested in ref. [5] as the lower limit on the damping of the wheel-hop mode. The eigenvalues of the vehicle incorporating passive suspensions with damping ratio of 0.2 for the wheel-hop modes are:
kgm’.
The following additional passive suspension: k, = 19.96 kN/m,
data values pertain to a
k, = 22.59 kN/m.
The vehicle equipped with the semi-active suspension is assumed to be identical to the passively suspended vehicle. It has been recognized [5,7,8], for both active and passive suspension systems, that the constraint placed
where R, is the reference spatial angular frequency (rad/m), 4, (m2/rad/m) is the single-sided PSD of the random variable x0, &, = 4,.,(Q) is a constant describing reference waveness, and o is the temporal angular frequency (rad/sec). The road input velocity sequences, therefore, can be treated as white noise Gaussian random inputs. Thus, the road velocity disturbance v (vI and v2) is specified by:
Eb(t,h&)l = ~Vdd%W, - tA
(26)
where E denotes the expectation, S(e) represents the Dirac delta function, and rrVe$&, is the intensity of the white noise process.
Integral and state variable feedback controllers
241
Table 1. Eigenvalues of optimal active suspension systems Control type
Case no. 1
PI =P2 8x10-‘@
2
Integral plus state variable feedback (IPSVFB)
State variable feedback (SVFB)
Y2
Y3
Y4
1
2.5
1
2.5
Ys -
Y6 -
8x10-”
1
8
1
8
-
-
3
8x10-”
1
2.5
1
2.5
1
4
8x10-”
1
8
1
8
5
8x lo-lo
1
2.5
1
6
8x10-lo
1
8
1
YI
- 11.31 * j54.33 (l = 0.2) -6.88 f j 8.54 (< = 0.63)
-24.02 + j77.52 r = 0.3), -7.97 + j 9.81 (5 = 0.63),
- 17.56 + j56.02 (r = 0.3) -6.59 k j 8.02 (l = 0.64)
1
-15.44kj75.28(l =0.2), -8.34 f j10.45 (5 = 0.62), -1.0, - 1.0
-11.31 kj54.33 (l =0.2) -6.86 k j 8.55 (5 = 0.63)
1
1
-24.03 + j77.52 (r = 0.3), -7.95&j 9.82(< =0.63), -l.O,-1.0
- 17.56 f j56.02 (l = 0.3) -6.57kj 8.04(( =0.63)
2.5
10
10
- 15.45 k j75.28 (5 = 0.2), -8.21 f j10.56 (< = 0.61), -3.16, - 3.15
- 11.31 f j54.33 (5 = 0.2) -6.71 f j8.69 (5 = 0.61)
8
10
10
-24.03 + j77.53 (5 = 0.3), -7.82fj 9.94(5 =0.62), -3.16, - 3.15
- 17.56 f j56.03 (c = 0.3) -6.42&j 8.18(5 =0.62)
In computing the covariance matrix of the state vectors, x and z, the time delay between the white noise inputs, a, and u2, is taken into consideration. For example, the steady-state covariance matrix, Z, for the system described by eqn (20) is [19]: f,z+zAf+Q=o,
(27)
where 0 is the intensity of the forcing function and is given by:
where E = [E, : EJ, and ‘512is the time delay between the front and rear tires. The effect of vehicle speed on the r.m.s. acceleration of the center of gravity, r.m.s. front suspension deflection, and r.m.s. front tire deflection is shown in Figs 3-5, respectively. Both actively (based on the 005
------ Passwe,
---9
-
0.04
'-.-
&,=
Closed-loop eigenvalues, I/set - 15.45 f j75.27 (r = 0.2), -8.35 f j10.44 ({ = 0.63),
two schemes of controllers) and passively suspended vehicles exhibit increases in r.m.s. vehicle response variables with vehicle speed. The actively controlled suspensions exhibit marked improvements in ride comfort (vibration isolation) and suspension travel, without compromising the road-holding ability. The ride qualities offered by integral plus state variable feedback controller with yS= y6 = 1 are hardly different from the ones offered by the state variable feedback controller. When ys = y6 = 10, improvements in suspension travel and tire deflection are achieved with an increase in the r.m.s. sprung mass acceleration. Based on the vehicle parameters used in this study and in particular having a damping ratio of 0.2 for the wheel-hop mode, Figs 3-5 demonstrate the beneficial effects of active controlled suspensions. They also demonstrate that it is possible to improve vehicle performance by simultaneously reducing r.m.s. sprung mass acceleration, and r.m.s. suspension 005
0.2
SVFB, case I IPSVF, case 3 IPSVF, case 5
,A ,/
----- Passive,
_-
I’
E
---
SVFB,
<,= 0 2 case
-
IPSVFB,
case 3
004
.-.-
IPSVFB,
case 5,)” I’ ,’ I’ ,’
003-
/.//
5 I= s
5 D
,’ .’
I
.’
,,e*’
,’
,' .A."
I'8' ,' I'<' / 8' /' /'I' ./‘
5 E % 002Lz
./ ./ ./
z k 2
OOI-
L
0
I IO
I 20
I 30
I 40
Speed (m/set)
Fig. 3. Effect of vehicle speed on the r.m.s. acecleration at the center of gravity.
0
e
40 Speed (m/set)
Fig. 4. Effect of vehicle speed on the r.m.s. front suspension deflection.
242
M.M. ELMADANY 0020 ------Passwe, -z 2
5
0.015
c
----
SVFB,
-
IPSVFB
case
I IO
0
I
, case 3 and case 5
I 20 Speed
--------
&, =0.2
I 30
- - --
Passwe, &,= 0.2 SVFB, case I IPSVFB, case 3
-
IPSVFB,
-
case 5
I 40
(m/set)
Fig. 5. Effect of vehicle speed on the r.m.s. front tire deflection.
Frequency
(Hz)
Fig. 7. Front suspension deflection power spectra for the actively and passively suspended vehicles.
sion deflection without negatively influencing r.m.s. tire deflection. The sprung mass acceleration spectra for the actively controlled and passive suspensions are shown in Fig. 6. The response spectra for the actively suspended vehicle are attenuated well in the 0.8-2 Hz range, with negligible compromise of isolation at the tire frequency, compared with passive suspensions. The integral plus variable feedback controller shows a less damped peak associated with sprung mass modes. Only a marginal effect on ride comfort results if yS and y6 have small values (around 1). Increasing the weighting factors yS and y6, which leads to faster roots, accentuates frequencies from 0.8 to 3 Hz. The power spectra of the suspension deflection at the front end are illustrated in Fig. 7. The figure shows compromising results as active suspension is used. Increases in low frequency stroke response are noticed, accompanied by decreasing amplification near body frequencies for the controlled vehicle, compared with the uncontrolled vehicle. The introduction of the integral controller limits the low frequency stroke response, but it is still higher than 16'
e
-Passive, --------
xi 5
-2 IO
:g
IO-3
4.3. Response to discrete disturbances Three types of discrete inputs to the vehicle system are considered. These external inputs are: (1) An applied step load of 2000 N at the center of gravity of the vehicle sprung mass. (2) A ramp input having a unit slope with a vehicle speed of 10 mjsec. 10-4
&,= 0.2
SVFB,
case I
- - - - IPSVFB,
case 3
-
case 5
-
the passive suspension. The low frequency suspension travel for the actively suspended vehicle is due to the different levels of inertial damping for the sprung and unsprung masses [20]. The front tire deflection variation response spectra are shown in Fig. 8. The low frequency asymptotes of the spectral profiles are not altered. Control action takes effect in the intermediate frequency range starting at 0.6 Hz, showing a trade-off of attenuation and amplification over the frequency range. The active suspension attenuates tire deflection in the 0.6-2 Hz region, but amplifies the response in the 2-10 Hz region.
IPSVFB,
iG I 10-s N’ F
-
Passwe,
--------
SVFB,
(;
0.2
case I
- - - - IPSVFE,
case 3
-
case 5
-
IPSVFB,
z :! 2 10-4 i! I
IO-5
E al -6 C IO 2 :: cl lo-7 B 10-e10.'
IO Frequency
IO2
(Hz)
Fig. 6. Sprung mass acceleration power spectra for the actively and passively suspended vehicles.
Fig. 8. Front tire deflection power spectra for the actively and passively suspended vehicles.
243
Integral and state variable feedback controllers -
Passive, E,= 0.2
Q-*-Z SVFB, - e -0 IPSVFB,
case
I
case 3
B-o--OIPSVFB,
case 5
c-t -+ IPSVFB,
case 3 -semi-active
I:::;_ 0
3
2
4
Time kecf
Fig. 9. Front suspension deflections of the actively, semiactively and passively suspended vehicles when subjected to a step external force.
(3) A moment applied at the body center of gravity to simulate the vehicle behavior during accelerating. The moment increases linearly to 2000 Nm in 1 set, remains at 2000 Nm for the next 4 set, and then falls off to zero in 1 sec. The simulation is performed for each case. A set of plots comparing the responses of the actively, semiactively and passively suspended vehicles are presented and discussed next. The suspension deflection responses for the different systems due to the step external force are shown in Fig. 9. The active suspension with state variable feedback controller exhibits a smaller and better damped response than the passive suspension. Both of them, however, exhibit a permanent defiection. The steady-state deflection for the controlled 030
E G z
020
.$,=0.2 I
suspension is 0.03 m compared with 0.06 m for the passive system. This is due to the fact that the active suspension has an overall stiffness of 71 kN/m compared with 36 kN/m for the passive suspension [16]. The integral plus state variable feedback controller reacts fast to the external disturbance and reaches a zero steady-state deflection in 4 set for the case with ys = y6 = 1. With faster real roots, ys = y6 = 10, the system reaches zero steady-state body-to-axle deflection just after 1.6 sec. The semi-active controller is based on the integral plus state variable feedback with the gains obtained using yS= ys = 1, with retaining the passive spring rates. The effect of the semi-active controller is clearly demonstrated in the figure where the deflection of the suspension is eliminated. Although the maximum suspension deflection for the semi-active controller is smaller than its active counterpart, the reaction to the external force is much slower for the semi-active control. The suspension deflection responses to ramp inputs are shown in Fig. 10, which illustrates the fact that the state variable feedback controller exhibits steadystate deflection. A zero steady-state offset for a ramp input will result with the use of the integral of the rattle spaces. The active suspension with higher values of ys and ys undergoes less deflection and damps more quickly. The semi-active controller tracks its active counterpart demonstrating the effectiveness of the semi-active control in generating a damping force that could annihilate the steady-state position offset. The suspension deflection and pitch angle response time histories due to the application of a moment at the sprung mass center of gravity are shown in Figs 11 and 12, respectively. The results obtained are clearly different for each of the sus~nsion systems considered. The suspension deflection and pitch angle do not take zero values during the accelerating of the vehicle for the passive and the state variable feedback
-
Passive,
Q-*Q
SVFB,
**4
IPSVFB,
case 3
+.----+
Posswe,
+-G--V IPSVFB,
case 5
w----f)
SVFB,
++ -+ IPSVFB,
case
case 3-
-ig
semi-actwe
-6
3
-.,,h 0
2
3
004
i
&=0.2 case
I
p - - * IPSVFB,
case 3
v-
case 5
-0
IPSVFB,
4
Time fsec)
Fig. 10. Front suspension deflections of the actively, semiactively and passively suspended vehicles when subjected to ramp inputs (step velocity inputs).
Fig. II. Front suspension deflections of the actively and passively suspended vehicles when subjected to a moment applied at the sprung mass center of gravity.
244
M. M. ELMADANY
006 E
-
Passive,
Q-------•
SVFB,
D - - -Q
IPSVFB,
0.2
(,= case
I
case 3
Acknowledgement-The
author would like to thank the Research Center, King Saud University for supporting this research. REFERENCES
A. Baker, Lotus active suspension. Automat. Engnr 9, I (1984). Y. Yokoya,
Time kec)
Fig. 12. Pitch angle responses of the actively and passively suspended vehicles to a moment applied at the sprung mass center of gravity.
systems. These offsets are eliminated by the introduction of the integral of the rattle spaces.
controller
5. SUMMARY AND
CONCLUSIONS
The objective of this paper has been to examine the use of integral plus state variable feedback controllers for improving the vertical dynamic performance of road vehicles. The potential for improved vehicle ride comfort and ride attitude control, resulting from the use of controllable forces generated actively or semiactively, is examined. The performance characteristics of such suspension systems are evaluated and compared with those of the pertinent active suspension systems based on state variable feedback and conventional suspension systems. The results of the comparison, presented and discussed in this paper, lead to the conclusion that the optimal control theory provides a useful mathematical tool for the design of active suspension systems. The suspension designs which may have emerged from the use of optimal control theory prove to be effective in controlling vehicle vibrations and achieve better performance than the conventional passive suspension. The integral plus state variable controllers are essential and viable means for achieving excellent vehicle attitude control without sacrificing the superior ride comfort offered by the state variable feedback controllers. Such controllers eliminate low frequency suspension deflections due to external loads or maneuvering forces. The semi-active systems based on integral plus state variable feedback controller tracks its active counterpart. The low power consumption capabilities render this class of semi-active systems to be an attractive candidate for controlling vehicle vibrations, with its real time implementation being promising.
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1992 0
~~~9192 $5.00 + 0.00 1991 pergamon m?ss plc
INTEGRAL AND STATE VARIABLE FEEDBACK CONTROLLERS FOR IMPROVED PERFORMANCE IN AUTOMOTIVE VEHICLES M. M. ELMADANY Mechanical Engineering Department, College of Engineering, King Saud University, Biyadh, Saudi Arabia (Received 12 October 1990)
Abstract-An analytical investigation of a half-ear model using an integral plus state variable feedback controller for tracking and regulation is performed. The potential benefits of incorporating, actively or semi-actively, such a controller to remedy the inherent problems associated with conventional passive suspensions and active suspensions based on state variable feedback controllers are examined. Both random and dete~inistic roadway inputs as well as de~~inistic body force and moment dist~~n~ are used. The results demonstrate that an optimal suspension using an integral plus state variable feedback controller retains both excellent ride and attitude control characteristics.
1. ~RODU~ION The demand for increased comfort and controllability in road vehicles has resulted in many automotive industries seriously considering incorporating microprocessor driven active suspensions [la]. These intelligent electronically controlled suspension systems can ~tentially increase overall performance. The objectives of improving comfort and controllability simultaneously, however, cannot be achieved using optimally controlled suspensions based on state variable feedback controllers. The trade-off between vibration isolation and overall system stiffness is the main reason. In order to maximize isolation quality for greater ride comfort, active suspension has to have a lower body natural frequency. As a consequence, a large clearance space is required both for relative excursions due to road excitations, as well as for displacements resulting from externally applied loads. In other words, the co~esponding overall static stiffness will not be sufficiently high to control body motion due to changes in payload, forces from steering, braking and traction, and aerodynamic forces. This problem can be overcome if the active sus~nsion control strategy is envisioned as shown in Fig. 1 [S-7]. In this case, the controller is composed of two parts: (a) a ride controller to provide improved ride comfort and safety, and (b) an attitude controller to perform load leveling action, control and roll response of the vehicle during cornering and control the pitch response during braking and accelerating. Karnopp [8,9] and ElMadany [7], using an active high gain load leveling device in parallel with an active damper, demonstrated that such as active suspension system has the potential to provide excellent ride and handling characteristics as well as
zero static deflection for body force disturbances. Thompson and Davis [6] and ElMadany 171, by including an integrator in the forward path (ahead of the control input), have obtained an active suspension system with infinite stiffness towards static loading, but which is soft with respect to road inputs. The resulting system, however, does not have zero steady-state axle-to-body deflection in response to ramp road inputs. Using multivariable integral optimal control, an suspension system is derived [lo, 111. The optimal control law consists of two controllers. The first is an integral controller acting on suspension deflection (rattle space) to ensure zero steady-state offset due to body and maneuvering forces as well as road rump inputs. The second is a state variable feedback controller for vibration control and performance improvement, The derived optimal suspension system retains the excellent ride comfort associated with lower body natural frequency with greatly enhanced controllability. In the previous work, a quarter-car model was selected. Despite the simplicity of the quarter-car model, it contains many important features of automotive suspensions, leads to a basic understanding of the limitations to suspension performance, and allows thorough investigation of the vehicle dynamic problems. However, when detailed study of the motion of the vehicle is required, two- as well as three-dimensional models should be considered. In this study, a two-dimensional linear vehicle model with coupled bounce and pitch modes, subjected to both random and dete~inistic disturbances, is considered. Optimal control theory is applied to the model to design two control schemes: an integral-plus-state variable feedback controller (IPSVF) and state variable feedback controller 237
M.M. External force
Road disturbances
Attitude controller I
ELMADANY
about the static equilibrium are assumed and a linear discrete mathematical model is derived. A set of eight variables is needed to describe the system. The eight variables are: the suspension deflections at the front and rear (x, and x5); the tire deflections at the front and rear (x2 and x,); the vertical velocities of the two ends of the sprung mass (x3 and x,); and the vertical velocity of each of the two unsprung masses (x4 and x8). The following state equations can be derived using Newton’s second law:
I
Fig. 1. Ride and attitude
controllers.
f,=
_ul+k,,
1, =
X8 -
x,
1, =
v2 -
xg
is=
_!L+k,x,.
ml
(SVFB). The controllers are based on measurements of all states. These measurements are used to drive the control signal required to generate the active forces or for modulating the activated dampers (semiactive dampers). The performance levels that result are compared to identify superior control strategies. 2. VEHICLE
SYSTEM
The model of the vehicle system is shown in Fig. 2. The sprung mass is considered as a rigid body, with mass, m, and pitch moment of inertia, J, undergoing heave and pitch motions. The unsprung masses, m, and m2, represent front and rear wheel-axle assemblies, respectively. The linear spring elements, k,, and kt2, represent front and rear tire stiffnesses. Continuous tire contact with the road is assumed. U, and u2 are active or semi-active suspension forces, which are control variables. The road disturbances at the front and rear tires are represented by velocity inputs, D, and v2, respectively. The disturbances also include an external applied force, F, due to a change in load and a moment, M, that may result from accelerating/or braking the vehicle. Small deviations
ml
m2
m2
These equations can be expressed in matrix form as: k=Ax+Bu+Ew,
(2)
where x is an 8-state vector, II is a 2-control force vector, w is a 4-disturbance vector, A is an 8 x 8 system matrix, B is an 8 x 2 control force distribution matrix, and E is an 8 x 2 disturbance distribution matrix. To achieve zero steady-state response for the suspension deflections, the system state equations (2) are appended with two state variables, x,(t) and x,,(t), such that: i, = x,
(3)
i 10= x5
(4)
or in matrix form:
p=cx,
(5)
where C is a 2 x 8 matrix with C,, , = 1, C,, 5= 1 and all other elements are zero. The augmented system equations then become: i=Az+Bu+Ew, where z = [x’ pq,
Fig. 2. Vehicle model.
(6)
Integral and state variable feedback controllers The performance characteristics that are of the utmost interest in vehicle suspension design are passenger ride comfort, suspension travel and road-holding ability. Sprung mass acceleration has been used as an indicator of passenger ride comfort. Suspension deflection should be kept small for good rigid body control. It is also important from a packaging point of view. The tire-road contact force (or equivalently dynamic tire deflection) should be kept as small as possible for safety reasons.
3. DESIGN
OF OPTIMAL
CONTROLLERS
Consider the linear time-invariant system described by eqn (2), i.e.
controlled
The optimal follows:
239
closed-loop
system is expressed
3r=(A-BG)x+Ew =&x+Ew.
(12)
The LQR design procedure is guaranteed to produce closed-loop stable designs if the system is “controllable” and “observable”, which it is for the system in hand. If the disturbance w(t) = 0, the regulator ensures that the initial state disturbances, x,, decay to zero at a rate prescribed by the eigenvalues of 4. Consider eqn (12) again, but now with x, = 0, and w(t) a step input of magnitude w, . Then the response is: x(t) = A;’ [exp(A, t) - Z]Ew,.
%=Ax+Bu+Ew.
(13)
(2)
For controller design it is assumed that all the states are available and exact measurements are possible. First, consider a state variable feedback regulator: u = -Gx,
(8)
where G is the state variable feedback gain matrix. The optimization consists of determining the control, u, which minimizes the following performance index:
Since all the modes of A, = (A - BG) are damped, then the maximum value of x(t) occurs at steadystate [ 131 and gives: x, = -(A - BG)-‘Ew,
(14)
u,=G(A-BG)-‘Ew,.
(15)
and
Consider next an integral plus state feedback control algorithm for the system in eqn (6): u= -Gz=
-Gx-Hp,
(16)
where H is an integral action gain matrix. The optimal control input, II, is obtained minimizing the following performance index: j=; whereQ,.,=y,,Q,,,=yZ,Qs.s=Y~,Q6.6=Y4andall other elements of Q are zero, R,~, = p, , R,, 2 = pz, and all other elements of R are zero. In this performance index, j U: dr and s a: dt are related to ride comfort requirements, since they are proportional to car body acceleration, j xi dt and j x: dt reflect body-to-axle space requirements, and j x2 dt and s xz dt indicate the road following requirements. The desired relative levels of these quantities are determined by the choice of the weighting factors pi 9p2 and yI-y4. Optimal control theory provides the solution of eqn (9) in the form of eqn (8). The gain matrix G is computed from [ 121: G = R-‘BTP
(IO)
and the matrix P is obtained as the solution matrix to the steady-state algebraic Riccati equation: PA + ATP - PBR- ‘BTP +.X0 = 0.
as
(11) \--I
by
m(P,t(:+P211i+Y,X:+yZX:+y?X:+y4Xa s0 +YSX; + Y~X:O)dt
= ;
m (z’Qz + uTRu) dt,
(17)
I0 where
Q=
QO oy,o. I o o
0 76
I
The optimal control law that minimizes the performance index [eqn (17)] is given by: u = -Gz = -R-%A, where P satisfies the steady-state equation:
(18)
matrix
Riccati
Pii + dTP - PBR-‘BTP + (5 = 0.
(19)
240
M. M.
ELMADANY
The closed-loop system equations are:
(A-W
e,
--H-I
C
0
E
I II z+
OW
=$z+Ew.
(20)
It is of interest to examine the case where the control law given by eqn (16) is invoked by a semi-active controller (activated damper). The semiactive controller behaves like an active controller as long as the control action requires energy dissipation. If the control action calls for imparting energy to the system the activated damper is off, delivering zero force [14,15]. Two activated dampers are incorporated into the passive suspension systems at the front and rear. The forces in the dampers are adjusted by continuously modulating the fluid-flow orifices, based on the integral and state variable feedback control law [eqn (16)]. This leads to the control of energy dissipation without delivering energy to the system. The logical function that satisfies the dissipative phase of the control action is: uSal= -,g,
dljzj9
uSBI- 0, ha2 = -jg,
G2,zj,
usa2 --0
when
a,,, (xj - x4) > 0
(21)
when
u,] (xj - x4) < 0
(22)
when
u,,~(x, - x8) > 0
(23)
when
u,r(x, - xe) ,< 0. (24)
4. EVALUATION
OF VEHICLE
RIDE
4.1. Eigenvalue analysis The following parameter values for a baseline vehicle are assumed in the calculations [16]: m, = 28.6 kg,
- 15.74 fj76.35
(5, = 0.2)
- 11.15 k5’55.33 (r, = 0.2) -2.37 fj10.02
(5 = 0.23),
- 1.3 kj8.04 (l = 0.16). The eigenvalues for a number of optimal active suspension systems obtained by varying the weighting factors in the performance index are shown in Table 1. The eigenvalues presented are for state variable feedback controllers and integral plus state variable feedback controllers. The wheel-hop damping ratio takes the value of 0.2 or 0.3 in all cases considered. By comparing the eigenvalues of the actively suspended vehicle with the corresponding eigenvalues of the passively suspended vehicle, it may be noted that the body (sprung mass) modes for the actively suspended vehicle are better damped than for the passively suspended vehicle. The integral plus state variable feedback controller introduces two extra real roots. With increasing the weighting factors ys and y6, the real roots become more negative. 4.2. Response to random road disturbances The vehicle is assumed to traverse a random road surface at a constant forward speed, V. The road roughness is described by its power spectra1 density [17, 181:
k,, = k,* = 155.9 kN/m
mz = 54.4 kg,
a= 1.1 m
m = 505 kg,
6= 1Sm
J=651
on the damping ratio of the wheel-hop mode limits the extent to which the r.m.s. sprung mass acceleration can be reduced. On the other hand, for very lightly damped wheel motions, lower road-holding qualities will result. A damping ratio of 0.2 was suggested in ref. [5] as the lower limit on the damping of the wheel-hop mode. The eigenvalues of the vehicle incorporating passive suspensions with damping ratio of 0.2 for the wheel-hop modes are:
kgm’.
The following additional passive suspension: k, = 19.96 kN/m,
data values pertain to a
k, = 22.59 kN/m.
The vehicle equipped with the semi-active suspension is assumed to be identical to the passively suspended vehicle. It has been recognized [5,7,8], for both active and passive suspension systems, that the constraint placed
where R, is the reference spatial angular frequency (rad/m), 4, (m2/rad/m) is the single-sided PSD of the random variable x0, &, = 4,.,(Q) is a constant describing reference waveness, and o is the temporal angular frequency (rad/sec). The road input velocity sequences, therefore, can be treated as white noise Gaussian random inputs. Thus, the road velocity disturbance v (vI and v2) is specified by:
Eb(t,h&)l = ~Vdd%W, - tA
(26)
where E denotes the expectation, S(e) represents the Dirac delta function, and rrVe$&, is the intensity of the white noise process.
Integral and state variable feedback controllers
241
Table 1. Eigenvalues of optimal active suspension systems Control type
Case no. 1
PI =P2 8x10-‘@
2
Integral plus state variable feedback (IPSVFB)
State variable feedback (SVFB)
Y2
Y3
Y4
1
2.5
1
2.5
Ys -
Y6 -
8x10-”
1
8
1
8
-
-
3
8x10-”
1
2.5
1
2.5
1
4
8x10-”
1
8
1
8
5
8x lo-lo
1
2.5
1
6
8x10-lo
1
8
1
YI
- 11.31 * j54.33 (l = 0.2) -6.88 f j 8.54 (< = 0.63)
-24.02 + j77.52 r = 0.3), -7.97 + j 9.81 (5 = 0.63),
- 17.56 + j56.02 (r = 0.3) -6.59 k j 8.02 (l = 0.64)
1
-15.44kj75.28(l =0.2), -8.34 f j10.45 (5 = 0.62), -1.0, - 1.0
-11.31 kj54.33 (l =0.2) -6.86 k j 8.55 (5 = 0.63)
1
1
-24.03 + j77.52 (r = 0.3), -7.95&j 9.82(< =0.63), -l.O,-1.0
- 17.56 f j56.02 (l = 0.3) -6.57kj 8.04(( =0.63)
2.5
10
10
- 15.45 k j75.28 (5 = 0.2), -8.21 f j10.56 (< = 0.61), -3.16, - 3.15
- 11.31 f j54.33 (5 = 0.2) -6.71 f j8.69 (5 = 0.61)
8
10
10
-24.03 + j77.53 (5 = 0.3), -7.82fj 9.94(5 =0.62), -3.16, - 3.15
- 17.56 f j56.03 (c = 0.3) -6.42&j 8.18(5 =0.62)
In computing the covariance matrix of the state vectors, x and z, the time delay between the white noise inputs, a, and u2, is taken into consideration. For example, the steady-state covariance matrix, Z, for the system described by eqn (20) is [19]: f,z+zAf+Q=o,
(27)
where 0 is the intensity of the forcing function and is given by:
where E = [E, : EJ, and ‘512is the time delay between the front and rear tires. The effect of vehicle speed on the r.m.s. acceleration of the center of gravity, r.m.s. front suspension deflection, and r.m.s. front tire deflection is shown in Figs 3-5, respectively. Both actively (based on the 005
------ Passwe,
---9
-
0.04
'-.-
&,=
Closed-loop eigenvalues, I/set - 15.45 f j75.27 (r = 0.2), -8.35 f j10.44 ({ = 0.63),
two schemes of controllers) and passively suspended vehicles exhibit increases in r.m.s. vehicle response variables with vehicle speed. The actively controlled suspensions exhibit marked improvements in ride comfort (vibration isolation) and suspension travel, without compromising the road-holding ability. The ride qualities offered by integral plus state variable feedback controller with yS= y6 = 1 are hardly different from the ones offered by the state variable feedback controller. When ys = y6 = 10, improvements in suspension travel and tire deflection are achieved with an increase in the r.m.s. sprung mass acceleration. Based on the vehicle parameters used in this study and in particular having a damping ratio of 0.2 for the wheel-hop mode, Figs 3-5 demonstrate the beneficial effects of active controlled suspensions. They also demonstrate that it is possible to improve vehicle performance by simultaneously reducing r.m.s. sprung mass acceleration, and r.m.s. suspension 005
0.2
SVFB, case I IPSVF, case 3 IPSVF, case 5
,A ,/
----- Passive,
_-
I’
E
---
SVFB,
<,= 0 2 case
-
IPSVFB,
case 3
004
.-.-
IPSVFB,
case 5,)” I’ ,’ I’ ,’
003-
/.//
5 I= s
5 D
,’ .’
I
.’
,,e*’
,’
,' .A."
I'8' ,' I'<' / 8' /' /'I' ./‘
5 E % 002Lz
./ ./ ./
z k 2
OOI-
L
0
I IO
I 20
I 30
I 40
Speed (m/set)
Fig. 3. Effect of vehicle speed on the r.m.s. acecleration at the center of gravity.
0
e
40 Speed (m/set)
Fig. 4. Effect of vehicle speed on the r.m.s. front suspension deflection.
242
M.M. ELMADANY 0020 ------Passwe, -z 2
5
0.015
c
----
SVFB,
-
IPSVFB
case
I IO
0
I
, case 3 and case 5
I 20 Speed
--------
&, =0.2
I 30
- - --
Passwe, &,= 0.2 SVFB, case I IPSVFB, case 3
-
IPSVFB,
-
case 5
I 40
(m/set)
Fig. 5. Effect of vehicle speed on the r.m.s. front tire deflection.
Frequency
(Hz)
Fig. 7. Front suspension deflection power spectra for the actively and passively suspended vehicles.
sion deflection without negatively influencing r.m.s. tire deflection. The sprung mass acceleration spectra for the actively controlled and passive suspensions are shown in Fig. 6. The response spectra for the actively suspended vehicle are attenuated well in the 0.8-2 Hz range, with negligible compromise of isolation at the tire frequency, compared with passive suspensions. The integral plus variable feedback controller shows a less damped peak associated with sprung mass modes. Only a marginal effect on ride comfort results if yS and y6 have small values (around 1). Increasing the weighting factors yS and y6, which leads to faster roots, accentuates frequencies from 0.8 to 3 Hz. The power spectra of the suspension deflection at the front end are illustrated in Fig. 7. The figure shows compromising results as active suspension is used. Increases in low frequency stroke response are noticed, accompanied by decreasing amplification near body frequencies for the controlled vehicle, compared with the uncontrolled vehicle. The introduction of the integral controller limits the low frequency stroke response, but it is still higher than 16'
e
-Passive, --------
xi 5
-2 IO
:g
IO-3
4.3. Response to discrete disturbances Three types of discrete inputs to the vehicle system are considered. These external inputs are: (1) An applied step load of 2000 N at the center of gravity of the vehicle sprung mass. (2) A ramp input having a unit slope with a vehicle speed of 10 mjsec. 10-4
&,= 0.2
SVFB,
case I
- - - - IPSVFB,
case 3
-
case 5
-
the passive suspension. The low frequency suspension travel for the actively suspended vehicle is due to the different levels of inertial damping for the sprung and unsprung masses [20]. The front tire deflection variation response spectra are shown in Fig. 8. The low frequency asymptotes of the spectral profiles are not altered. Control action takes effect in the intermediate frequency range starting at 0.6 Hz, showing a trade-off of attenuation and amplification over the frequency range. The active suspension attenuates tire deflection in the 0.6-2 Hz region, but amplifies the response in the 2-10 Hz region.
IPSVFB,
iG I 10-s N’ F
-
Passwe,
--------
SVFB,
(;
0.2
case I
- - - - IPSVFE,
case 3
-
case 5
-
IPSVFB,
z :! 2 10-4 i! I
IO-5
E al -6 C IO 2 :: cl lo-7 B 10-e10.'
IO Frequency
IO2
(Hz)
Fig. 6. Sprung mass acceleration power spectra for the actively and passively suspended vehicles.
Fig. 8. Front tire deflection power spectra for the actively and passively suspended vehicles.
243
Integral and state variable feedback controllers -
Passive, E,= 0.2
Q-*-Z SVFB, - e -0 IPSVFB,
case
I
case 3
B-o--OIPSVFB,
case 5
c-t -+ IPSVFB,
case 3 -semi-active
I:::;_ 0
3
2
4
Time kecf
Fig. 9. Front suspension deflections of the actively, semiactively and passively suspended vehicles when subjected to a step external force.
(3) A moment applied at the body center of gravity to simulate the vehicle behavior during accelerating. The moment increases linearly to 2000 Nm in 1 set, remains at 2000 Nm for the next 4 set, and then falls off to zero in 1 sec. The simulation is performed for each case. A set of plots comparing the responses of the actively, semiactively and passively suspended vehicles are presented and discussed next. The suspension deflection responses for the different systems due to the step external force are shown in Fig. 9. The active suspension with state variable feedback controller exhibits a smaller and better damped response than the passive suspension. Both of them, however, exhibit a permanent defiection. The steady-state deflection for the controlled 030
E G z
020
.$,=0.2 I
suspension is 0.03 m compared with 0.06 m for the passive system. This is due to the fact that the active suspension has an overall stiffness of 71 kN/m compared with 36 kN/m for the passive suspension [16]. The integral plus state variable feedback controller reacts fast to the external disturbance and reaches a zero steady-state deflection in 4 set for the case with ys = y6 = 1. With faster real roots, ys = y6 = 10, the system reaches zero steady-state body-to-axle deflection just after 1.6 sec. The semi-active controller is based on the integral plus state variable feedback with the gains obtained using yS= ys = 1, with retaining the passive spring rates. The effect of the semi-active controller is clearly demonstrated in the figure where the deflection of the suspension is eliminated. Although the maximum suspension deflection for the semi-active controller is smaller than its active counterpart, the reaction to the external force is much slower for the semi-active control. The suspension deflection responses to ramp inputs are shown in Fig. 10, which illustrates the fact that the state variable feedback controller exhibits steadystate deflection. A zero steady-state offset for a ramp input will result with the use of the integral of the rattle spaces. The active suspension with higher values of ys and ys undergoes less deflection and damps more quickly. The semi-active controller tracks its active counterpart demonstrating the effectiveness of the semi-active control in generating a damping force that could annihilate the steady-state position offset. The suspension deflection and pitch angle response time histories due to the application of a moment at the sprung mass center of gravity are shown in Figs 11 and 12, respectively. The results obtained are clearly different for each of the sus~nsion systems considered. The suspension deflection and pitch angle do not take zero values during the accelerating of the vehicle for the passive and the state variable feedback
-
Passive,
Q-*Q
SVFB,
**4
IPSVFB,
case 3
+.----+
Posswe,
+-G--V IPSVFB,
case 5
w----f)
SVFB,
++ -+ IPSVFB,
case
case 3-
-ig
semi-actwe
-6
3
-.,,h 0
2
3
004
i
&=0.2 case
I
p - - * IPSVFB,
case 3
v-
case 5
-0
IPSVFB,
4
Time fsec)
Fig. 10. Front suspension deflections of the actively, semiactively and passively suspended vehicles when subjected to ramp inputs (step velocity inputs).
Fig. II. Front suspension deflections of the actively and passively suspended vehicles when subjected to a moment applied at the sprung mass center of gravity.
244
M. M. ELMADANY
006 E
-
Passive,
Q-------•
SVFB,
D - - -Q
IPSVFB,
0.2
(,= case
I
case 3
Acknowledgement-The
author would like to thank the Research Center, King Saud University for supporting this research. REFERENCES
A. Baker, Lotus active suspension. Automat. Engnr 9, I (1984). Y. Yokoya,
Time kec)
Fig. 12. Pitch angle responses of the actively and passively suspended vehicles to a moment applied at the sprung mass center of gravity.
systems. These offsets are eliminated by the introduction of the integral of the rattle spaces.
controller
5. SUMMARY AND
CONCLUSIONS
The objective of this paper has been to examine the use of integral plus state variable feedback controllers for improving the vertical dynamic performance of road vehicles. The potential for improved vehicle ride comfort and ride attitude control, resulting from the use of controllable forces generated actively or semiactively, is examined. The performance characteristics of such suspension systems are evaluated and compared with those of the pertinent active suspension systems based on state variable feedback and conventional suspension systems. The results of the comparison, presented and discussed in this paper, lead to the conclusion that the optimal control theory provides a useful mathematical tool for the design of active suspension systems. The suspension designs which may have emerged from the use of optimal control theory prove to be effective in controlling vehicle vibrations and achieve better performance than the conventional passive suspension. The integral plus state variable controllers are essential and viable means for achieving excellent vehicle attitude control without sacrificing the superior ride comfort offered by the state variable feedback controllers. Such controllers eliminate low frequency suspension deflections due to external loads or maneuvering forces. The semi-active systems based on integral plus state variable feedback controller tracks its active counterpart. The low power consumption capabilities render this class of semi-active systems to be an attractive candidate for controlling vehicle vibrations, with its real time implementation being promising.
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