Optimal Noise Rejection in a Tractor Active Suspension System Using Generalized Sampled-Data Hold Functions

Copyright Cl IFAC Control Applications and Ergonomics in Agriculture, Athens, Greece, 1998

OPTIMAL NOISE REJECTION IN A TRACTOR ACTIVE SUSPENSION SYSTEM USING GENERALIZED SAMPLED-DATA HOLD FUNCTIONS

P.N.Paraskevopouios, K.G.Arvanitis and A.A. Vernardos

National Technical University ofAthens. Department ofElectrical and Computer Engineering. Division of Computer Science. Zographou 15773. Athens. GREECE

Abstract: In this paper, a technique for optimal noise rejection, based on generalized sampled-data hold functions is proposed. The technique consists in suitably modulating the sampled output of the system under control by periodically varying functions in order to achieve the control objective. The proposed technique is applied to the active suspension system of a tractor. The effectiveness of the method is demonstrated by various simulation results. Copyright © 1998 IFAC Keywords: Sampled-data control, noise rejection, LQG control, traction control, active vehicle suspension.

have a significant effect upon tracking perfonnance and comfort. It is apparent that operators of off-road vehicles must be isolated to some degree from vehicle vibrations. Passive, active and semi-active suspension systems are used in off-road vehicles. mainly for this important reason (see e.g. Lee and Pradko (1968), Schubert and Ruzicka (1969), Stikeleather and Suggs (1970), Kim et al. (1985), Ulsoy and Hrovat (1990» . In this respect, in the present paper, a technique is presented, for optimal noise rejection. This technique relies on generalized sampled-data hold functions (GSHF) based control, which is among the most interesting periodic feedback control strategies. GSHF based control has been proposed first by Kabamba (1987), and subsequently has successfully been applied in solving a variety of important control problems (see e.g. Arvanitis (1995, 1998), Paraskevopoulos and Arvanitis (1994» . The GSHF based control technique, consists in suitably modulating the sampled output of the system under control by periodically varying functions, in order to achieve the control objective. In particular, the GSHF approach is shown to be very effective, in solving, among other important control problems, the discrete optimal noise rejection control problem

1. INfR.ODUCTION

In recent years, the importance of providing better ride comfort and safety to off-road vehicle operators has been increasingly emphasized (see e.g. Liljedah1 and Strait (1962), Lee and Pradko (1968), Schubert and Ruzicka (1969), Stikeleather and Suggs (1970), Larson et al. (1976), Kim et al. (1985» . As the vehicles increase in size and power for high speed operations, comfort features become important factors in detennining their work perfonnance. Along with recent progress made in comfort features such as roll-over protection, noise reduction, climate control, and more convenient drive, ride comfort associated with vehicle vibrations has become a particular concern of off-road vehicle designers (see e.g. Lee and Pradko (1968), Schubert and Ruzicka (1969), Stikeleather and Suggs (1970), Kim et al. (1985» . Since off-road vehicles are mainly involved in agricultural operations, the operators are frequently exposed to relatively high levels of vibrations for long periods of time. EXl>0sure to excessive vibrations can result in impainnent of human perfonnance, comfort and health. In particular, low frequency vibrations induced from the off-road terrain conditions have been reported to

265

of linear disturbed continuous-time systems. For this reason, it is applied, in this paper, to the active suspension system of a tractor. Several simulations of the proposed GSHF based technique for optimal noise rejection, are performed in the paper, for various values of the sampling period and of the estimator parameters. From these results it is verified that a large gain is expected when the sampling period is chosen to be small. On the other hand, a small gain is expected in cases where the covariance kernel of the noise is large. This reveals that the covariance of the sampled state vector will be large, indicating a poorly regulated system in this case.

where To

= exp(ATo), F.

=f exp[A(To -A)jBF(A)dA. (Sa) o

(k+I)To

f exp{A[(k+l)To -A]}W(A)dA. (Sb)

ro(kTo) =

kTo

Eq. (Sb) implies that ro(kTo) is a stationary zeromean white Gaussian process with covariance kernel E[ro (kTo)ro T(lTo)] =R",o(k -1), k,l eN To R", = exp[ A(To - A) ]Rw exp[ AT (To - A)]dA.

fo

2. OPTIMAL NOISE REGECTION USING GSHF

The optimal noise rejection problem via GSHF control, treated in this paper, is as follows: Given a symmetric positive definite weighting matrix Q eRn xn and a sampling period To, find a GSHF

Consider the controllable and observable linear state space system of the form

.

x(t) = Ax(t) + Bu(t) +w(t) (la) y(t)=Cx(t) , s(kTo)=y(kTo)+v(kTo) (lb)

based control law of the form (3), in order to minimize the following cost function

where x(t) eRn is the state vector, u(t) eR m is the control vector,

y( t) eR P

is the output vector,

wet) eRn is a disturbance vector, To is the samp-

The solution of the above problem has been established by Kabamba (1987), and it can be expressed in terms of a discrete Riccati equation, which resembles the Riccati equations appearing in other well known optimal-estimation problems (e.g. discrete Kalrnan filtering) . More precisely, it has been proven by Kabamba (1987) that if the triplet (A, B, C) is minimal, then for almost all To > 0 , the optimal noise rejection problem is solvable. Its solution is

ling period, s(kTo) eR P is a discrete measurement vector and v(kTo) eR P is a discrete measurement noise vector, and where all matrices have appropriate dimensions. It is supposed that wet) and v (kTo ) are stationary zero-mean white Gaussian process with covariance kernels E[W(t)WT('t)]= R,.8(t-'t), t,'teR

(2a)

E[v(kTo)vT (lTo)] = Rvo(k -1), k,l eN, R .. > 0 (2b) where by abuse of notation, o(e) represents the Dirac function in both the discrete-time and continuous-time case. System (1), will be acted upon by controls of the form u(t)

=F(t)s(kTo),

t e [kTo, (k + l)To) , k ~ 0

where To

W(A,B, To) =

fexp[A(To - A)]BBT exp[A (To - A)]dA. T

o

F.

(3)

= -KC T(CKC T + Rv) - I

(7)

and K eR nxn is the unique positive definite solution of the discrete Riccati equation

F(t+To)=F(t) ,for t e[O, To) where F(t) is a To -periodic integrable and bounded matrix of appropriate dimension representing a hold function. Then, the closed loop system has the form

Clearly, relation (6) provides us a solution to the optimal noise rejection problem via GSHF control, in the case where the hold function F(t) does not have a prespecified structure. Our attention is next focused on the special class of the time-varying To-

x[ (k + l)To] = x(kTo) + F.s(kTo) +ro(kTo) (4a) s(kTo) = Cx(kTo) + v(kTo) (4b)

periodic matrix functions F(t), for which every element of F(t), denoted by fij(t) , is piecewise

or equivalently,

constant over intervals of length Tj

266

=To / N

j ,

with

(9)

for J.1. = 0, ... , N j - l. In this case, as it has been shown by Arvanitis (1995), the following relation holds (10) where, defining by bi the ith column of B, (Ha) T;

A ~ exp(ATj)

,

hj ~ f exp(AA»jdA.

(lIb)

o

Fig.l. Linear tractor active suspension model.

and where the m x p block matrix F has the form

(12)

The state and control matrices of the active suspension system model are In this case, ith row fiT (t) of the matrix F(t) and the ith block row of the matrix ft(t) = [fjl(t)

... f jp(t)]=eN; _,,[ijl

A=

... i jp ] (13)

K", M", 0 0

V J.1.Ti ::::;t«J.1.+I)Tj for i=I, ... ,m and for J.1.=0, ... ,

-1

0

Bs Ms

Ks Ms

0

Bs M", 1

Bs Ms

0 -1 where Ms, Bs and K s' are the sprung mass, the damping coefficient of damper associated with the sprung mass and the stiffness coefficient of the spring associated with the sprung mass, respectively, while M us and K us are the unsprung mass and the stiffness coefficient of the spring associated with the unsprung mass, respectively. A single sensor is available to measure suspension stroke. Thus, in our case

3. TRACTOR ACTIVE SUSPENSION SYSTEM MODEL

c=[o

The particular tractor active suspension system studied in the paper is depicted in Fig. 1. It can be represented by a fourth-order linear model of the form (1). The state vector consists of the tire deflection (X2 - xo), the unsprung mass velocity (x 2 ) , the

1 0 0]'

4. SIMULATION RESULTS In this section, various simulations of the proposed technique for optimal noise rejection will be performed. To this end we next consider a tractor active suspension system, whose parameters have the following values: Ms =4000 kg, M", =600 kg,

x 2 ) and the sprung mass ve-

locity (X I ) ' The disturbance of the suspension system is due to soil roughness

Ks M",

Ms B= Mus

elements are zero except for a unity appearing in the ( N j - J.1. )th position. From the above analysis it becomes clear that in the case where the elements of F(t), are piecewise constant, the admissible hold function can be obtained on the basis of F. To find matrix F one must solve (10). Note that (10) is always solvable if N j ~ n j , i=I,2, ... ,m, where n j comprise a set of locally minimum controllability indices of the pair (A,B) (for details see (Arvanitis, 1995» .

-

0

Bs M",

0

N j -I, where e N;_" eRN; is the row vector, whose

suspension stroke (XI

1

0

F are interrelated as

xo ' and

267

K", =360000 N/m,

Ks =30000 N/m,

The admissible hold function is depicted in Fig. 3. Similarly, selecting N=20, the admissible hold function depicted in Fig. 4, is obtained. Note that in all the three cases presented above J min = tr{KQ} =

B. =2000

Nsec/m. With these particular values we obtain

0

o

1

-600 -3.333

A=

o o

[

50

-1

o

0.5

-75

3.3~33], B=[6.6~63] -05

tr{K} =0.0617. We next study the impact of the sampling period To on the gain F(t). To this end, in what follows, the sampling period is decreased to the value To =0.1 sec and all the other parameters maintain the same values as above. In the case where, F(t) does not have a prespecified structure, we obtain

-1

The covariance kernels ofw(t) and v(kTo) are 4

7 X 10- 0 0 01 0000 -8 Rw= , Rv=1O (14) o 000 [ o 000

-0.0005 0.0244 symmetry

Selecting Q= 14 x 4 , To = 1 sec and applying the proposed technique in the case where F(t) does not have a prespecified structure, we obtain

KfOO01

607 -0.2 F = -7.25051 • [ -0.2580

-0~01]

-0.0003 -0.0001 0.0569 -0.0004 symmetry 0.0004 -0.0006 0.0043

1.0114

The admissible hold function is depicted in Fig. 5. In the sequel, for the same example, a hold function

r00981 ]

F = -0.1902

6

F(t)

0.6459

$

-0.0001 0.0001] -0.0005 0.0003 0.0002 -0.0001 0.0001

0.0251

2

o

The admissible hold function is depicted in Fig. 2. In the sequel, for the same example, a hold function with piecewise constant behavior is designed. To this end, selecting N=5, we obtain

-2

-3.8659

I

-10

5.0701 F = -6.7476 - 10.0618 1.9610

-12

o

0.2 sec

0.4 sec

0.6 sec

O.B sec

1 sec

Fig. 3. F(t) with piecewise constant structure for N=5, To =1 sec and Rw , R v, given by (14). 0 . 5,---~--~--~--~----,

0.5

F(t) 0

F(t) 0

.{).5 -1

-1

-1.5

-1 .5

-2 -2.5 -3

-3.5 -4 -4 .5

o

0 .2 sec

0.4 sec

0.6 sec

O.B sec

0.2 sec

1 sec

0.4 sec

0.6 sec

O.B sec

1 sec

Fig. 4. F(t) with piecewise constant structure for N=20, To=l sec and Rw , R v, given by (14).

Fig. 2. Controller gain F(t) with free structure for To=l sec and R w, R v, given by (14).

268

F(t)

oooo.---~----~----~----~----~

F(t) 4000

2000

o -2000 -4000 -4000 ~OL---~----~----~----~----~

o

0.02 sec

0.04 sec

0.06 sec

0.08 sec

o

0.1 sec

0.02 sec

0.04 sec

0.06 sec

0.08 sec

0.1 sec

Fig. 5. Controller gain F(t) with free structure for To =0.1 sec and Rw , Rv, given by (14).

Fig. 7. F(t) with piecewise constant structure for N=20, To=O.1 sec and Rw , Rv , given by (14).

with piecewise constant behavior is designed. To this end, selecting N=5, we obtain

is increased to the value Rv =0.01, and all the other parameters maintain the same values as above. In the case where, F(t) does not have a prespecified structure, matrix F, is given by.

- 2.7429

F= 10

3

X

4.9450 0.2533 -4.9871 3.0049 -

F •

The admissible hold function is depicted in Fig. 6. Similarly, selecting N=20, the admissible hold function depicted in Fig. 7, is obtained. Note that in all the three cases presented above J nUn = tr{KQ} =0.0248.

=

[~~~~~~51 0.0454 0.0452

The hold function depicted in Fig. 8, is then obtained. Subsequently, for the parameters, a hold function with piecewise constant behavior is designed. Selecting, first N=5, we obtain

Clearly, in all cases the gain of the hold function is larger than the gain obtained in the case where To = I sec. This is due to the fact that, choosing a

-0.4019 0.3477 F = -05274 -0.7077 0.3071

small To yields a small W (A, B, To) or a small :8 . Their inversion produces matrices with large entries, which are involved in the computation of the gain F(t). In the sequel, the impact of the covariance kernels on F(t) is studied. To this end, in what follows, the sampling period is chosen to have the value To = I sec, the measurement noise covariance kernel

and the respective hold function is depicted in Fig. 9. Similarly, for N=20, the hold function depicted in Fig. 10, is obtained. From the above simulation results, it can be easily recognized that when R v is

5000 4000

F(t)

3000 2000 1000 0 -1000 -2000

-3000 -4000 -5000 0

0.02 sec

0.04 sec

0.06 sec

0.08 sec

.0.35 L-__--'____---'-____~_______'______' 0.2 sec 0.4 sec 0.6 sec 0.8 sec 1 sec o

0.1 sec

Fig. 8. Controller gain F(t) with free structure for To=1 sec, Rv=O.OI and Rw given by (14).

Fig. 6. F(t) with piecewise constant structure for N=5, To=O.1 sec and Rw , Rv, given by (14).

269

feedback gains are expected in cases where the covariance kernel of the external disturbance is small or the covariance kernel of the measurement noise is large.

0.4

F(t) 0.2

o

REFERENCES .{l.2

Arvanitis, K.G. (1995). Adaptive decoupling of linear systems using multirate generalized sampled-data hold functions. IMA J. Math. Control Inform., 12, 157-177. Arvanitis, K.G. (1998). On the localization of intersample ripples of linear systems controlled by generalized sampled-data hold functions. A utomatica, in press. Kabamba, P.T. (1987). Control of linear systems using generalized sampled-data hold functions . IEEE Trans. Autom. Control, 32, 772-783. Kim, K.U., D.L.Hoang and D.R.Hunt (1985). Ride simulation of passive, active and semi-active seat suspensions for off-road vehicles. Trans. ASAE, 28, 56-64. Larson, D.L., D.W.Smith and lB.Liljedahl (1976). The dynamics of three-dimensional tractor motion. Trans. ASAE, 19, 195-200. Lee, R.E. and Pradko (1968). Analysis of human vibration. Trans. SAE, 77, 346-360. Liljedahl, L.A. and 1 Strait (1962). Automatic tractor steering. Agric. Eng., 43, 332-333 , 349. Paraskevopoulos, P.N. and K.G.Arvanitis (1994). Exact model matching of linear systems using generalized sampled-data hold functions. Automatica, 30, 503-506. Schubert, D.W. and lE.Ruzicka (1969). Theoretical and experimental investigation of the electrohydraulic vibration isolation system. Trans. ASME J. Eng. Industry, 91,981-990. StikeIeather, L.F. and C.W.Suggs (1970). An active seat suspension system for off-road vehicles. Trans. ASAE, 13, 99-106. UIsoy, A.G. and D. Hrovat (1990). Stability robustness of LQG active suspensions. Proc. 1990 A .C.C., 1347-1356. San Diego, CA, May 1990.

I .{l.80

0.2 sec

0.4 sec

0.6 sec

0.8 sec

1 sec

Fig. 9. F(t) with piecewise constant structure for N=5, To=1 sec, Rv=O.OI and Rw given by (14). .{l . 05;---~--~--~--~---,

F(t)

.{l.350

0.2 sec

0.4 sec

0.6 sec

0.8 sec

1 sec

Fig. 10. F(t) with piecewise constant structure for N=20, To=1 sec, R v=O.OI and Rw given by (14). large a small gain F(t) must be expected, indicating a poorly regulated system. Consider now the case where To = 1 sec, R v = 10-8 , and 10-9 0 0 0

o

R -

w-

[

0

0 0 0 000

o

000

(15)

If the hold function does not have a prespecified

behavior, we obtain the gain depicted in Fig. 11. When, a piecewise constant hold function is to be designed, analogous results can be obtained. From these results, it is easily verified that, when R w is small, a small gain is expected.

F(t)

.{l.25

5. CONCLUSIONS A GSHF based technique for optimal noise rejection, has been applied to a tractor active suspension system. The effectiveness of the method has been illustrated by various simulation results. From these results it has been recognized that large feedback gains are e,,:pected in cases where the sampling period is chosen to be small. Moreover, small

o

0.2 sec

0.4 sec

0.6 sec

0.8 sec

1 sec

Fig. 11. Controller gain F(t) with free structure for To=1 sec, R v=IO-8 and Rw given by (15).

270