Nonlinear Adaptive Control of a CVT Based Parallel Hybrid Passenger Car

Copyright © IFAC Advances i.n Auto~otive Control, Mohican State Park, Loudonvtlle, OhIO, USA, 1998

NONLINEAR ADAPTIVE CONTROL OF A CVT BASED PARALLEL HYBRID PASSENGER CAR T. Mayer, D. Schroder • ·Chair for Electrical Drives, Technische Universitiit Miinchen

Abstract. The driveline of the Autark Hybrid Vehicle, which is developed at the TU Munchen, is based on a CVT (Continuously Variable Transmission). Due to the CVT the model used to design the driveline controller is inherently nonlinear. Moreover several clutches make the model multistructural. Varying vehicle mass and an operating point dependent gearbox efficiency require the application of robust control methods. In this paper a robust controller design based on Sliding Mode Control (SMC) is presented. Since it can be proven that this approach is not robust to a varying gearbox efficiency the controller is extended by an adaptive approach. Copyright © 1998 IFAC

Keywords. Parallel Hybrid Vehicle, Vehicle Control, Sliding Mode Control, Adaptive Control, General Regression Neural Network

1.

INTRODUCTION

Major challenges for future cars are the requirement of low fuel consumption and emissions. Nevertheless vehicle performance must be excellent to make a car acceptable to the public. Internal combustion engines (ICE) are characterised by rather low efficiencies at low loads. Emissions improve when the required engine dynamics is low. A combination of a parallel hybrid driveline with a CVT-based gearbox offers the possibility to operate the combustion engine exclusively in regions of high efficiency at rather low dynamics. Therefore a widespread CVT with a ratio range of about 25 is implemented. The high rate of ratios is realised by a special construction of the gearbox which offers the possibility to switch the power flow in the conventional CVT and therefore use it twice. Figure 1 shows the hybrid drivetrain, a more detailed description can be found in (Mayer and Schroder, 1995a) Control of parallel hybrid vehicles with CVT's has been studied in several projects (Spijker, 1994; Schmid et al., 1995). It was stated that control algorithms based on partial feedback yield results which coincide well with the control objectives. In this paper a robust controller based on Sliding Mode is proposed to control the wheel torque in all cases where the CVT ratio is not fixed. Stability and robustness of the plant will be analysed. For non-robust cases the control algorithm is extended by an adaptive approach.

Fig. 1. The concept of the Autark Hybrid Vehicle

2.

THE DRIVELINE MODEL

Only the eigenfrequencies of the driveline which are within the desired closed loop bandwidth of the controlled vehicle must be considered in the model. As a possible specification which is given in (Spijker, 1994) the desired wheel torque should be available within 0.2 - 0.38 without overshooting. It can be dedicated from the relative degree of the system, which is derived below, that the closed loop behaviour of the driveline can be described by a second order model. A closed loop bandwidth of 2.5Hz and a relative damping of 0.9 are chosen to meet the specifications. Since the lowest eigenfrequency of the driveline is within this range and the second one is above 15Hz just the lowest eigenfrequency is modelled. The dynamic behaviour of the CVT is sufficiently described by a simple integrator (Spijker, 1994),

111

the integrator output R is the gearbox ratio. The efficiency .,., of the gearbox is strongly dependent on the operating point and is included in the model. The dynamics of the controlled engines are modelled by first order lags with the time constants Tice and Temot (Tice » Temot). Additionally disturbing torques due to rolling resistance, gradients of the road and braking are summed in the external torque T ezt . The vehicle mass 8 2 depends on the load, the engine side mass 8 1 is dependent whether the internal combustion engine is coupled to the driveline. Figure 2 shows the reduced order driveline model. The model equations

g(X)

=

0 0. 2X 4 0.2

a a a

0 0 0

a _l_ Tiee

a

0 0 0

a a

(4)

_l_ Tetnot

and the vector of inputs according to figure 2 u

= [Ul

U2

U3] T

(5)

In the hybrid case the control vector must be chosen of third degree to receive a fully determined system. Hence the following output vector is derived which yields on the one hand good vehicle performance by controlling the wheel torque (Tout = CtXl + btX2) and on the other hand good efficiency by setting the engine operation points by controlling the speed of the gearbox entry shaft and the torque of one engine (Mayer and Schr6der, 1996). (6)

Figure 3 shows the signal flow graph of this operating mode. The system is structurally nonlinear Fig. 2. The reduced order driveline model

are described in detail in (Mayer and Schroder, 1995bj Mayer and Schroder, 1996). The nonlinear state space description of the form

x

=

f(x)

y

=

h(x)

+ g(x)u (1)

consists of the states (see nomenclature for details)

x

=

Xl

=

t

X2

i;

R

X3

X4

X5

u.

X6]


]

x.

y,

Fig. 3. The signal flow diagram of the reduced driveline model

(2)

the matrixes

due to the CVT, parameter uncertainties are inherent to the model. This model just describes one important mode of the overall set of operating possibilities. One different special mode of the gearbox is the so called Synchronous Point at a ratio of about 7. In that point, power is transmitted by fixed gear ratios only and not by the CVT. Therefore the overall gearbox efficiency is significantly higher. Since the ratio is fixed, the reduced model of the drlveline is approximately linear. A similar case occurs when the CVT has reached a mechanical limit. Additionally the model must be parametrised dependent on the state of the clutch

(3) Z6+Z6- ='t(C,Zl +b,Z2) 81 _2.A.. race

-~ Temot

112

and following the approach in (Slotine and Li, 1991) the SMC algorithm is

to the ICE. For each case a different model is required, the system is multistructural.

u

3.

= T1 + TZ = 2 + 2 = 4 < 5 = Ta

(12)

and with Pi = 1Ji it is obvious that both controllaws are similar, but the second one has lower switching amplitudes which are only dependent on the parameter uncertainties and not required to ensure the attractivity of S. The final control law is:

Uz

d

= ( dt + Ci

Ters

O.2U1Yl--

T

+ ktxzx3 -ers- + Xs + ...

'TJ

PZszTers81 - kzsat

1J

(:~) Ters 8 1 (14)

The control law for U1 has a singularity at zero speed (X4 = 0). Hence a LQ-optimized controller is applied for low speeds. The next step is to design the parameters Ci, Pi, k i , bi . The parameters Ci determine the behaviour on the sliding surface. The sliding surface can be interpreted as a dynamics. Once on the surface the tracking error tends to zero exponentially with a time constant (Ti - 1)/Ci. The behaviour is that of a filter function. Ci can be derived by designing a filter with the specified characteristics. In our case the bandwidth is chosen to fmaz = 10Hz, so that the lowest eigenfrequency of the drivetrain is within this bandwidth. Maximum damping is required at frequencies> 15Hz, so that higher eigenfrequencies are not stimulated. Therefore an 8.th order butterworth filter is designed and the constants Ci are achieved by mini-

(7)

(8)

)r;-l Yi

=

YZdTers81 - CZ(Y2 - YZd)Ters 8 1 - ...

and the sliding surface S Si

(10)

Since k i is defined in (Slotine and Li, 1991) to

Therefore the system is not input-state linearizable and input-state tracking is not possible. An alternative is to yield the control canonical form by applying the input-output linearisation were B is invertible. Defining the vector of tracking errors

y = Y -Yd

f - ksgn(s))

with y~r) = Yd - CiY for T = 2. To reduce chattering instead of the sgn-function the sat-function with a thin boundary layer b around the sliding surface S is used and the control law is changed to

CONTROLLER DESIGN

In the next sections only the design of the controller for the main case from figure 3 is described. In all other cases either equivalent controller structures with different parameters or LQ optimised linear state space controllers are used. During mechanical braking the CVT is controlled to higher ratios, additionally a steering algorithm for the electric motor is required to charge a maximum amount of energy back to the battery. The above described model is a Multi-InputMulti-Output-System (MIMO). One degree of freedom is whether the torque of the ICE or of the electric motor is added to the control vector. Since the combustion engine is scheduled to run at low dynamics to reduce emissions its torque is defined as a system output and the rate of ratio change Ul and the desired torque of the electric motor Uz are used to control the engine speed X4 and the output torque Tout where higher dynamics are required. For simplicity reasons and without loss of generality the two engine torques are summed up to a virtual engine with the new torque Xs and the time constant Temot. The reduced system has a absolute degree Ta of 5 and two inputs and 2 outputs. The damping of the torsional spring bt is comparatively small and is omitted for controller design. Sliding Mode is a method to control nonlinear systems with structural and nonstructural dynamics (Slotine and Li, 1991). In the considered case the input matrix g(x) (eqn.4) is not quadratic and therefore not invertible which is required for SMC. The relative degree of a system is the number of times the outputs must be differentiated until an explicit relationship between the output y and an input U is derived. Here the total relative degree T r is: Tr

= B- 1 (y~r) -

(9)

113

l00'''-''--~--''''-------~---;(:::''11)

mizing the cost function

(15) The factors PI and P 2 pinpoint the time to reach the sliding surface S. Suitable values can be obtained by the function

1 j

t

Pi = Is(t = 0)1

10

(16)

60

treach

8OOr--~--~-~----~--'"(2)

with the deviation from S at time zero s(t = 0) and the desired time treach to reach S. The factors k i determine the switching amplitude. They can be made operating point dependent to minimize the switching amplitudes. The investigated system can be written in the form

600

'E400 ~

iJ

200

desired_,/' tllIqUe

0

~

·200

y=

f(x)

+ 6.f(x) + B(x)u

(17)

.4001 l r - - - . l r - -.....-_,---"....---.._-..,.,l 3OO.r-_~_ _~_l\.;;,;.m...;:.e..;,.in~I$J

with all parameter uncertainties being summed up in 6.f(x). ki is then determined by ki

> l6.fd

with

i = 1,2

250

I I I I

(18)

The width of the boundary layer is derived by ._ bt -

\

_diCE

2kj

(3-,)

/

~tor

........ 1 1

.

(19)

1

W max - 2 --

'50\;-0-,--1-:::0---;:;;20c--'*30,----::40::---~50",..--...,d60

t reach

Time in [$J

with W max = 21ffmax. The following plots in figure 4 show the system behaviour in the case of exactly known parameters. At low speed, when driving in the Synchronous Point and during mechanical braking different control laws are implemented. The periods are marked in the plots. Additionally the control law is dependent whether the ICE is active. Obviously switching between the control laws is realised without any losses in performance. The controller is implemented discretely with a sample time of lOms. Figure 5 illustrates the differences in the desired rate of ratio change when the sgnfunction or the sat-function is applied.

Fig. 4. Simulation of a driving cycle with perfectly known parameters. Vehicle speed and gearbox ratio (1), wheel torque (2), engine speeds (3)

~(x)

=

L~h2(x) =

X5 -

£I. XIX3 1/

01

(20)

an additional function 5(X) must be determined so that the vector 4.

STABILITY AND ROBUSTNESS

} (x)

Hx)

It is inherent to the design method that the sliding surface is attractive and that therefore the controlled outputs are stable. But due to the fact that controller design was based on IjO-linearisation and that internal dynamics occur, the stability of the internal dynamics must be investigated. To receive the equation for the internal dynamics a local coordinate transformation is required (Isidori, 1989). With

4J} (x)

=

hI (x) =

GtXl

4J~ (x)

=

Ljh 1 (x)

= GtX2

4J~(x)

=

h 2 (x) =

X4

(x) =

et>i(x)

(21)

et>~(x)

et>5(X) has a nonsingular Jacobian matrix and can therefore be used for the local coordinates transformation. The function 5(X) can be derived by far

1 :S j

:S 2

(22)

if the distribution

(23)

114

I

By choosing a function

(1)

OJI

i: j

1 '2

0

i-02

Lyapunov-

-'l/J +k 2

V('l/J) =

ll> 02

positive definite

(30)

we receive

'll-OA

!-Q6

(31)

1!.()Jl .iJ

-l~_~ _ _~ _ ~_ _~_---:._ _~

o

I

OJI

~

02

-6

0

10

20

30 lIm. in [.]

40

SO

60

With a and

(2)

v

i: '0 ·OA

!.Q6

.iJ

=

0

V <

0

constant speed, stillstand else

(32)

Hence the internal dynamics is (asymptotically) stable. In a next step the system behaviour under parameter disturbances is investigated. Figure 6 shows the output torque when the vehicle mass is 200kg higher than the presumed mass for controller design, and with a guessed fixed gearbox efficiency which is lower than the actual efficiency. From

~-02 ] -OJI

'l/J being positive

--2~0----'l1':0--T.;40:---C:50O;--60
-I 0!.---;1"::'0

Tin. inl')

Fig. 5. Desired rate of ratio change with the sgn- (1) and with the sat-function (2)

800...--

- _ -_ _- _ - - - , (1)

600

with

1,

0 (Y 0.2X4

g,(x) =

0

(0?

1

(24)

-200

-400 0,,"--~1O--~20:---~30:---~40"..---!50:.--..".!60 Temot

800 ...-

:-----......:.:Tim:.:.:;.::,.in:::..l:.!I.)-------. (2)

is involutive, that means that the Lie-bracket

600

~

(25)

400

" 200 u

~

is describable as a linear combination of the original vectors gl and g2. Here the Lie-Bracket is zero and therefore the proof is trivially. With

.;j

~

0

desircdtCl'que

·200 -400 0""--1~0--~20:---~30:---~40"...---!50"...--..".!60

Time in I.)

Fig. 6. Wheel torque with increased vehicle mass of 200kg (1) and with a wrongly guessed gearbox efficiency (2)

the following solutions are received:

(27)

these plots it can be stated that the controlled plant is robust to variations of the vehicle mass but not to variations of the gearbox efficiency. A condition for robustness is that variations of parameters do not change the relative degree of the system. Mathematically this can be expressed by

This is exactly the wheel speed for driving forward (positive solution) or backward (negative solution). Therefore in the case of forward driving the internal dynamics is

~f(x, t),~gj(x) E

Ker [V'h i , V'Lf2hi"'" V'L;-2h i ]

(33)

or in compact form •

'l/J =

-a'l/J

2

+b

This requirement is called matching condition (EImall and Olgac, 1992). For the considered system

(29)

115

this condition is satisfied if V'h1~f(x)

=0

and

800...---------------(--,1)

V'h2~r(X)

=0

600

(34)

B

~400 .9

.. 200

After some steps of calculation follows that the matching condition is satisfied for variations of the vehicle mass, but not for variations of the gearbox efficiency. Hence the assumptions taken from figure 6 are proved. To overcome this restriction a robust adaptive approach is presented in the next chapter.

:>

$ ii

~

0

-200 -4OO0~-----:2~0-----40:-::-----60~ lime in Is]

1Ar---------:.-:::::=::::::::--'(?\'12) 13

5.

12

ADAPTIVE SLIDING MODE CONTROL

no adaptatiOll!!!!!

To investigate the adaptive approach the control law for the wheel torque (eqn. 13) is replaced by

(35)

0.7

0.60!:-------,2~0--1i-un-e-in-Is-]-4~0,.-------:-:!60

where bi and f; are replaced by their estimated values and the sat-function is omitted since it yields no additional dynamics. Since the tracking error depends explicitly on ~11, the tracking error dynamics is

Fig. 8. Wheel torque (1) and gearbox efficiency (2) with an adaptive control law

no tracking controller is applied or when the actuators reach their bounds, adaptation is stopped according to figure 10. Figure 8 shows the wheel torque which now is equivalently good as without estimation error and the adapted gearbox efficiency which coincides well with the real efficiency as long as adaptation is active. Since the gearbox efficiency is a three dimensional map (figure 9) the idea arises to learn this map by a neural net-

(36) With b11 > 0 for all operating points when driving forward and P1 > 0 this function is obviously SPR and therefore from Lemma 8.1 in (Slotine and Li, 1991) the adaptation law can be chosen to

;:111 = -'YS l0k--t1 Y1X22

(37)

With the Lyapunov function candidate

v = 1'-1 R(x)

~ 090 c

. ,g

(38)

~o

it can easily be shown that iT < 0 and therefore the tracking error approaches to zero globally. The controller bases on an MRAC (model reference adaptive control) design according to figure 7. Hence adaptation is only allowed during track-

.

~ Cl

085

080 075 070 065 40

200

o

0

Fig. 9. Efficiency map of the gearbox

work approach. According to (Schatfner, 1996) the General Regression Neural Network (GRNN) is a suitable solution were stability can be proven. Due to lack of space the interested reader must be referred to this publication for details. The general form for the adaptation according to figure 10 is similar to the simple integration, just the integrator must be replaced by the NN. The result~ for wheel torque and gearbox efficiency after 6C training runs can be seen in figure 11. The result~ are comparable to the first approach.

Fig. 7. Adaptive Sliding Mode Controller

ing control. In the above mentioned cases where

11 "

NOMENCLATURE

~ ueatCvr Gearbox Ratio

R 1J

Comtanta

Error Si

Gearbox Effici..,cy

1

f


Adaptation Requin>ments

rice

Fig. 10. Adaptation algorithm and exclusions for adaptation 800

(1)

bair Tw

ct, bt m

CVT ratio gearbox efficiency relative angle over torsional spring engine speed, wheel speed moment of inertia of engine side moment of inertia of wheel side torque resulting from rolling resistance, hill climbing, mech. brakes torque of el. motor torque of ICE time constant of controlled el. motor time constant of controlled ICE factor to calculate wind resistance radius of the wheels stiffness, damping of eigenfrequency vehicle mass

600

~

400

5

~

REFERENCES

200

,l!

J

0 -200 -400

20

0

"lime In

[.J

40

lA

60 (2)

13

i 0

i!l

12

1.1

no adaptation!111

~

~

0.7 0.6 0

40

20 "lime In

60

['J

Fig. 11. Wheel torque (1) and gearbox efficiency (2) with the NN approach

6.

CONCLUSION

In this paper a controller for the Autark Hybrid Vehicle based on sliding mode was designed. It was shown that the controller is robust to variations in vehicle mass but it can be proven mathematically that the designed SM-controller is inherently not robust to changes in gearbox efficiency. As a solution an adaptive sliding mode controller either based on a simple adaptation law or based on a neural network approach was presented. The NN approach offers additionally the possibility to learn the efficiency map of the gearbox.

ACKNOWLEDGEMENTS This work is part of a project called Umweltfreundliche Antriebstechnik fUr Fahrzeuge, SFB 365, which is sponsored by the DFG.

Elmali, H. and N. Olgac (1992). Robust output tracking control of nonlinear mimo systems via sliding mode technique. Automatica. Isidori, A. (1989). Nonlinear Control Systems An Introduction. 2. ed.. Springer-Verlag. Berlin Heidelberg. Mayer, T. and D. Schroder (1995a). Operating modes and control aspects for a special hybrid drivetrain. In: Preprints of the l.st IFAC Workshop on Advances in Automotive Control. Ascona, Switzerland. pp. 120-125. Mayer, T. and D. Schroder (1995b). Simulation and hierarchical controller design for a special hybrid drivetrain. In: 6th EPE. Vol. 2.. Sevilla, Spain. pp. 2.407-2.412. Mayer, T. and D. Schroder (1996). Robust control of a parallel hybrid drivetrain with a special cvt. In: SAE, Paper Nr. 960233. Detroit, USA. Schiiffner, C. (1996). Analyse und Synthese neuronaler Regelungsverfahren. PhD thesis. TU Munchen. Herbert Utz Verlag, Munchen. Schmid, A., P. Dietrich, S. Ginsburg and H. P. Geering (1995). Controlling a cvt-equipped hybrid car. SAE, Paper Nr. 950492. Slotine, J.-J. and W. Li (1991). Applied Nonlinear Control. Prentice Hall International Inc.. USA. Spijker, E. (1994). Steering and Control of a CVT based hybrid transmission for a passenger car. PhD thesis. TU Eindhoven. Netherlands.