Control and performance evaluation of a clutch servo system with hydraulic actuation

ARTICLE IN PRESS

Control Engineering Practice 12 (2004) 1369–1379

Control and performance evaluation of a clutch servo system with hydraulic actuation M. Montanari, F. Ronchi, C. Rossi*, A. Tilli, A. Tonielli CASY—Center for Research on Complex Automated Systems, ‘‘G. Evangelisti’’ DEIS—University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy Received 3 April 2003; accepted 11 September 2003

Abstract A hydraulic actuated clutch control system for commercial cars is analyzed. The design of closed-loop controller is presented, based on a simplified system model. A physical full-order model is also described and used to assess through computer simulations the dependence of the closed-loop system performances on some plant and controller key parameters. Selected performance indexes are gear shift timing and position tracking error and it results that they are mostly affected by two key parameters: oil pipeline length and controller sampling time. The resulting dependencies can be used to set performances and cost specifications for both plant configuration and electronic control unit. Experimental tests performed with different plant and controller configurations are reported. They closely match the simulation results, showing the effectiveness of the proposed approach. r 2003 Elsevier Ltd. All rights reserved. Keywords: Performance analysis; Systems design; Modelling; Automotive control; Hydraulic actuators

1. Introduction In recent years, servo actuation in traditional gear shift systems on commercial cars has gained increasing attention, especially on the European market. The system consists in a manual gear-shift device with added actuators, controlled via an Electronic Control Unit (ECU), for clutch and gear control. When compared with automatic gear shift systems, the servo actuated mechanical solution offers some advantages in terms of overall system costs (simpler system) and fuel saving (higher mechanical efficiency). Significant fuel savings of 4–5% on standard driving cycle are reported, in agreement with demands from market and regulations. Servo actuated gear-shift systems can be operated basically into two different modes: *

semi-automatic: gear shift is requested by the driver through a proper interface, and the system executes the shifting, provided it is compatible with engine and vehicle operating conditions;

*Corresponding author. Tel.: +39-051-2093020; fax: +39-0512093073. E-mail address: [email protected] (C. Rossi). 0967-0661/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2003.09.004

*

fully automatic: gear shift is decided by the control system itself, based on operating point (torque requested by the driver, engine speed, current gear).

One drawback associated with this system is the interruption of traction torque occurring during the shifting process, when the clutch has to be disengaged to exit the current gear and entering the new one. To achieve an acceptable comfort, this lack of traction should last as short as possible, typically below 300 ms: The gear-shift operation is managed by a controller, which generates the torque request for the engine and controls both the servoactuated gear-box and the clutch, achieving the proper synchronization. Since the torque transmitted to the driveline during clutch opening and closing heavily depends on the clutch position, the latter must be accurately controlled to prevent unpleasant oscillation due to driveline elasticity. Hence, this paper focuses on the clutch subsystem control and achievable performances. Several papers in the literature address the gear-shift control problem for both the complete driveline system (Fredriksson & Egardt, 2000; Garofalo, Glielmo, Iannelli, & Vasca, 2002; Pettersson & Nielsen, 2000) and the clutch subsystem (Horn, Bamberger, Michau, &

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M. Montanari et al. / Control Engineering Practice 12 (2004) 1369–1379

Pindl, 2002). These papers focus on the control algorithm design for a given plant. Nevertheless, physical and cost constraints on both plant and controller strongly affect the achievable performances for such a complex system. Hence, a system level analysis is needed to define the ‘‘optimal’’ trade-off between plant and controller specifications. Electrohydraulic servo systems as the one used in this paper are widely spread in many industrial applications. The standard control approach for such systems relies on local linearization of the dynamics around an operating point, followed by linear control design (Merritt, 1967). Other significant approaches are: nonlinear modelling (Zavarehi, Lawrence, & Sassani, 1999), feedback linearization (Vossoughi & Donath, 1995), Lyapunov like control (Sohl & Bobrow, 1999), variable structure control (Bonchis, Corke, Rye, & Ha, 2001) and HN control (Tunay, Rodin, & Beck, 2001). In the considered case, however, these standard approaches need to be revised, because a tracking problem in presence of strong nonlinearities is considered and its effects on achievable performances need to be enlightened. According to the above considerations, the main goal of this paper is twofold: *

*

The definition of a proper control architecture for the clutch position tracking problem. The analysis of the dependencies of the overall system performance on both plant and controller configurations, which is fundamental to meet given system level specifications at minimum cost.

Although the latter could be formulated as an optimization problem, its solution is far too complex to be addressed analytically and it needs to be approached through simulations. An accurate system modelling is fundamental for both issues. A physical full-order model of the clutch system is derived and validated, and it is then used in all the simulations through the paper. Due to its complexity, the full-order model is not suitable for the controller design. A reduced-order model is derived for the design of the position controller and it ensures its real-time implementation. To achieve fast dynamic performances the controller is based on feedforward terms obtained by reduced-order model inversion. The closed-loop regulator ensures robustness with respect to uncertainty on system parameters and to neglected dynamics. A pressure control loop is introduced to reduce the effects of non-idealities in the hydraulic circuit. Once the controller has been defined, an analysis can be performed of the dependence of selected performance indexes, namely clutch opening/closing time and tracking error, on plant and controller parameters. In particular, trade-off curves are derived and it is shown how they can be used to set performances and cost

specifications for both plant configuration and electronic control unit. The paper is organized as follows. In Section 2, the clutch control system is described and a full-order physical model is presented. In Section 3, the clutch model is validated and the physical parameters of the considered setup are reported. In Section 4, a simplified model for control purposes is obtained, in particular for the hydraulic servovalve and the oil pipeline. In Section 5, the controller structure and design are presented. In Section 6, the simulation analysis and experimental validation are reported, showing the validity of the proposed approach.

2. System description and model derivation A simplified schematic of the clutch control system is shown in Fig. 1. The clutch is composed by two disks connected to the engine shaft and to the gear-box shaft in the driveline. By means of the hydraulic actuator, it is possible to control the clutch position. In this way, the torque transmitted from the engine to the wheels is modulated and the gear change is made possible during the disengagement phase. The actuator is mainly composed by a hydraulic piston connected to a Belville spring (Almen & Laszlo, 1936) and other preload springs, which keep the clutch closed when no force is applied by the piston. The hydraulic circuit is controlled by a servovalve (a threeway spool flow control valve) which determines the oil flow and pressure in the actuator through a pipeline. The valve is connected to a power supply at highpressure ps ; which is filled with oil by means of a pump, and to a reservoir with oil at atmospheric pressure p0 : In order to disengage the clutch, the servovalve connects the mechanical actuator chamber (see Fig. 1) with the high-pressure source ps : For the clutch engagement, the servovalve connects the chamber with the low pressure p0 and the chamber empties by the action of the spring forces. The spool valve displacement is controlled applying voltage to the windings of the servovalve, which represents the control input of the system. To keep the clutch at a given position, an offset current is needed to keep the spool in its neutral point, that corresponds to no oil flowing in the circuit. For currents greater than this offset value, the actuator is connected to the high pressure power supply, while for currents smaller than the offset value, the actuator is connected to the low-pressure circuit. Due to geometrical layout constraints, the pipeline connecting the servovalve to the hydraulic actuator can be of different lengths and stiffness. It will be shown that the pipeline puts constraints on system performance and controller design. Therefore, different types of pipeline are considered in this work.

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Fig. 1. Clutch system scheme.

For modelling purpose, the overall clutch system is decomposed in three interconnected main parts: servovalve, mechanical actuator and pipeline. The oil pressures ps and p0 of the power supply and of the reservoir are assumed constant, hence the model of accumulator and pump are not taken into account. Relative pressures with respect to the atmospheric one are considered: in particular, it is assumed that p0 ¼ 0: 2.1. Servovalve model The servovalve considered is a three-way spool flow control valve (Merritt, 1967). Its active element consists of a plunger moved by an electromagnetic actuator. The plunger displacement defines the supply and return orifice areas. It is moved by spring forces (which maintain the load connected to the return when no external force is applied) and forces generated by the electric actuator. Detailed servovalve model refers to three subsystems: electromagnetic, mechanical and hydraulic. Mechanical model: The mechanical model of the servovalve is described by x’ v ¼ vv ; 1 v’v ¼ ½F0  kxv  bv ðvv Þ þ fm ðxv ; jÞ mv þ FB ðxv ; ps ; p1 ; p0 Þ;

ð1Þ

where xv ; vv are the plunger position and speed, fm is the magnetic force, j is the magnetic flux, ðF0 þ kxv Þ is the valve spring force, bv ðvv Þ represents the friction forces ðviscous þ static þ CoulombÞ; generated by the oil

flowing in the valve restrictions, FB is the Bernoulli force, acting on the plunger due to fluid flowing through orifices. The expression of the Bernoulli force is 8 > < 2Cd Cv cos yAf ðxv Þjps  p1 j if xv > xvf ; FB ¼ 0 if xvd pxv pxvf ; > : 2Cd Cv cos yAd ðxv Þjp1  p0 j if xv oxvd ; where Cd is the discharge coefficient, Cv is the velocity coefficient, y is the jet angle, Af ðxv Þ; Ad ðxv Þ are the filling and dumping orifice areas, xvf ; xvd define the amplitude of the dead-zone. Note that Bernoulli forces are proportional to the orifice area and the pressure drop and always act in a direction to close the orifice. The reader is referred to (Merritt, 1967) for detailed analytical derivation of the forces acting on the plunger. Electromagnetic model: Electromagnetic force acting on the plunger is generated by a variable reluctance actuator, constituted by a solenoid and a slider, connected to the plunger. The structure of the electromagnetic model is defined as in Filicori, Guarino Lo Bianco, and Tonielli (1993): 1 ðV  riÞ; N Ni ¼ ½Rðxv Þj þ FðjÞ; 1 @R ðxv Þj2 ; fm ðxv ; jÞ ¼  2 @xv j’ ¼

ð2Þ

where N is the number of coil turns, Nj is the linked magnetic flux, i is the winding current, V is the input voltage, r is the winding resistance, Rðxv Þ is the nonlinear air-gap reluctance, FðjÞ is the nonlinear magnetization curve of the iron core.

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1372

2000

1.8

1800

1.6

1600

1.4

1400

1.2

Force [N]

Orifice area [mm2]

2

filling

1

dumping

0.8

1200 1000

600

0.4

400 200

xvd xvf

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

Plunger position [mm]

(a)

fc (x)

800

0.6

0.2

fo (x)

0

1.6

0

(b)

2

4

6

8

10

12

Clutch position [mm]

Fig. 2. (a) Filling and dumping orifice areas Af ðxv Þ; Ad ðxv Þ; (b) clutch spring forces f ðx; vÞ:

Hydraulic model: The valve is an overlapped valve, i.e. the land width is greater than the port width when the spool is neutral (which corresponds to no oil flow) and hence there exists a dead-band in the orifice area vs. spool displacement, as it can be seen from characteristics reported in Fig. 2(a). The servovalve outlet flow can be written according to Bernoulli’s equation:

8 rffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > > sgnðp  p Þ jps  p1 jAf ðxv Þ C > d s 1 > r > < q1 ¼ 0 > rffiffiffi > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 > : Cd sgnðp1  p0 Þ jp1  p0 jAd ðxv Þ r

if xv > xvf ðfillingÞ; if xvd pxv pxvf ðdead-zoneÞ;

ð3Þ

if xv oxvd ðdumpingÞ;

where r is the oil density. 2.2. Mechanical actuator model The clutch actuator is mainly constituted by a mass– spring–damper system, where a mass m is driven by Belville spring forces, friction forces (ArmstrongHe! louvry, Dupont, & Canudas De Wit, 1994) and hydraulic forces. The model is given by the following equations, where the oil flow q2 is the input x’ ¼ v; 1 v’ ¼ ½f ðx; vÞ  bðvÞ þ Ap2 ; m b ðq2  AvÞ; p’ 2 ¼ V0 þ Ax

modulus of the oil. The term Ap2 in (4) represents the actuator hydraulic force. The term f ðx; vÞ collects all the position-dependent nonlinear forces, given by the sum of the pre-load and Belville spring forces. Since the Belville spring has a hysteresis, the analytical expression of its force is given by

ð4Þ

where x; v are the actuator position and speed, p2 is the pressure in the mechanical actuator chamber, m is the actuator mass, V0 is the minimum volume of the chamber, achieved when x ¼ 0 (see Fig. 1), A is the actuator cross-sectional area, b is the bulk

8 > f ðxÞ > > 0 > > > > > > < bðvÞ þ Ap 2 f ðx; vÞ ¼ > > > > > > fc ðxÞ > > > :

if v > 0 or ðv ¼ 0 and bðvÞ þ Ap2 > f0 ðxÞÞ; if v ¼ 0 and fc ðxÞo  bðvÞ þ Ap2 of0 ðxÞ; if vo0 or ðv ¼ 0 and bðvÞ þ Ap2 ofc ðxÞÞ; ð5Þ

where functions f0 ðxÞ and fc ðxÞ are shown in Fig. 2(b). Friction forces bðvÞ take into account viscous forces (bð0Þ ¼ 0 is assumed), which are approximatively linear. 2.3. Pipeline model Due to car lay-out constraints, pipeline can be relatively long and flexible. Delay due to propagation of the flow in the line and pressure/flow oscillations must be taken into account when the pipeline is excited with the large bandwidth signals required for the clutch

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q1(t) Z(s)

1373

q2(t)

exp(-Γ(s))

1/Z(s) -

p1(t)

p2(t)

exp(-Γ(s)) Fig. 3. Pipeline model based on a two-port network.

movement. Moreover, pipeline and actuator chamber introduce complex and quite uncertain dynamics in the system, since oil parameters are temperature dependent and some hydraulics effects cannot be well modelled. In order to guarantee robust stability of the closed-loop system, a careful analysis of the pipeline dynamics is necessary. Moreover, this dynamics strongly affects the achievable system performances. Using a scattering variables approach, it is possible to model the fluid line as a two-port network with inputs and outputs given by the upstream and downstream flows q1 ; q2 and pressures p1 ; p2 : Since the natural output of the valve is the oil flow q1 ; while the input of the actuator is the oil flow q2 ; the two-port configuration depicted in Fig. 3 is considered. From the solution of PDEs describing the fluid line (Goodson & Leonard, 1972; Lozano, Brogliato, Egeland, & Maschke, 2000) and modal approximation technique (Yang & Tobler, 1991), the following reduced-order finite-dimensional LTI model for the fluid line is obtained: 2 3 ð1Þiþ1 ð2=Dn Þlci ð2Z0 =Dn Þðs% þ 8Þ " # 7 n 6 X 6 s%2 þ 8s% þ l2ci P1 ðsÞ s%2 þ 8s% þ l2ci 7 6 7 ¼ 6 7 Q2 ðsÞ ð2=Z0 Dn Þs% ð1Þiþ1 ð2=Dn Þlci 5 i¼1 4  s%2 þ 8s% þ l2ci s%2 þ 8s% þ l2ci " # P2 ðsÞ ; ð6Þ Q1 ðsÞ where ln0 r c0 r2 ; Z0 ¼ 0 2 ; s% ¼ 0 s; 2 n0 c0 r pr0  0  1 p lci ¼ i  ; i ¼ 1; y; n 2 Dn

Dn ¼

and n is the number of modal (normalized) frequencies lci considered in the approximation, l is the axial length of the pipeline, r0 is the radius of the tube, r0 is the oil density, be is the equivalent oil bulk modulus in the pipeline ffiffiffiffiffiffiffiffiffiffiffiffi taking into account pipeline elasticity, c0 ¼ p be =r0 is the sonic velocity of the fluid, n0 is the mean kinematic viscosity. It is worth noting that the hydraulic circuit is provided with a spilling device which ensures that there is no significant amount of air enclosed in the pipe. Hence, the effect of entrained air can be neglected. Since a finite number of modes is used in model (6), it is necessary to correct the steady-state value of the transfer functions, as described in Yang and Tobler

Table 1 Clutch system parameters and methods used for their identification Parameter

Value

Units

Theor.

m A b V0 f ðx; vÞ

0:755 665 106 1800 166 106 (Fig. 2(b))

kg m3 N=ðm=sÞ m3 N



b be l n0 r0 r0

600 106 Variable Variable 94 106 852 3:25 103

Pa Pa m m2 =s kg=m3 m









mv N r L F0 k Kf Rðxv Þ FðjÞ bv ðvv Þ bv FB ðxv ; ps ; p1 ; p0 Þ Af ðxv Þ; Ad ðxv Þ

0:021 410 2:7 42 103 2:02 7570 7:9

kg



53 (Fig. 2(a))

O H N N=m N=A A=Wb A N N=ðm=sÞ N mm2

Exper.

d

s

s d d d d d d





(1991), in order to guarantee that at steady state it holds " # " #" # 1 8Z0 Dn p2 p1 ¼ : ð7Þ q2 0 1 q1 It results that n ¼ 4 modes are sufficient for simulation. Note that the ffi first natural frequency of the pipeline is pffiffiffiffiffiffiffiffiffiffiffi o1 ¼ p be =r0 =ð2lÞ:

3. Model identification and validation System model parameters have been defined either on the basis of theoretical relationships or by identification procedures from experimental data. System model parameters together with the method used for their definition are reported in Table 1: ‘‘s’’ and ‘‘d’’ correspond, respectively, to static and dynamic experiments; nonlinear least-squares error techniques have been used for the parameter identification (Coleman, Branch, & Grace, 1999). Identified valve orifice areas

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1374 2

8 x [mm]

i* [A]

1.5 1 0.5 0

0 6 x 10

0.2

0.4

0

0.6

0 6 x 10

0.2

0.4

0.6

0

0.2

0.4 time (s)

0.6

3 p2 [Pa]

p [Pa]

4 2

3

1

2 1 0

6

2 1

0

0.2

0.4 time (s)

0

0.6

Fig. 4. Validation of the clutch system model ðl ¼ 0:3 m and be ¼ 600 MPaÞ: Position x and pressures p1 ; p2 of the real system (solid) and model (marked), with reference current i : 2 8 6 x [mm]

i [A]

1. 5 *

1 0. 5 0

2 0 6 x 10

0.2

0.4

0

0.6

0 6 x 10

0.2

0.4

0.6

0

0.2

0.4 time (s)

0.6

p 2[Pa]

3

2

1

p [Pa]

3

1 0

4

2 1

0

0.2

0.4 time (s)

0.6

0

Fig. 5. Validation of the clutch system model (l ¼ 1:3 m and be ¼ 75 MPa). Position x and pressures p1 ; p2 of the real system (solid) and model (marked), with reference current i :

and actuator spring forces f0 ðxÞ and fc ðxÞ are shown in Figs. 2(a) and (b). In order to validate the complete model, comparisons between experiments and simulations under the same operating conditions have been performed. Two tests are reported: the same reference current i is imposed by means of a fast current control loop in both the real system and the simulation model. Simulation model is based on considerations reported in Section 2. Both upstream and downstream pressures ðp1 ; p2 Þ are measured in these tests for validation purposes. In Fig. 4, a pipeline with l ¼ 0:3 m and be ¼ 600 MPa; corresponding to rigid pipeline wall, has been adopted. In Fig. 5, a long flexible pipeline with l ¼ 1:3 m; be ¼ 75 MPa is adopted. It is worth noting that these tests are quite tough owing to high sensitivity with respect to the input/

output characteristic between servovalve current and oil flow. Simulated position and pressure match the experimental ones with good accuracy.

4. Control model derivation The model described in Section 2 is fairly complex, since it involves nonlinearities and secondary effects. Moreover, it depends on physical parameters which are function of temperature, operating conditions and aging of components. In addition, some physical parameters are difficult to identify, due to unmeasurable state variables. Hence, a reduced model for control design purposes is developed, neglecting fast dynamics and second-order phenomena.

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4.1. Approximate model of the servovalve

goal is achieved satisfying two conditions. Firstly, the closed-loop pressure/position controllers must be designed with a sufficiently small bandwidth with respect to the first resonance frequency o1 introduced by the pipeline. Besides, the position reference trajectory itself must have harmonic content below the o1 frequency. This condition imposes a limitation on the fastest achievable clutch movement (and consequently on the duration of the gear shift), that depends on the properties of the pipeline (namely pipeline length and elasticity).

Under usual operating conditions, the magnetic force acting on the servovalve plunger is considered proportional to the valve current fm ðxv ; jÞ ¼ Kf i;

1375

ð8Þ

where Kf is the force constant. This assumption holds because the servovalve is purposely designed to have pffi @R=@xðxÞCconst and jp i; i.e. FðjÞpj2 and FðjÞbRðxÞj in the saturation region, which is the normal operating condition. Designing a high gain current controller, the servovalve can be considered a current-driven actuator. Analyzing the time constants of the resulting system and assuming that the bandwidth of the outer loops is sufficiently slow with respect to the bandwidth of the servovalve, the following algebraic relation between the imposed servovalve current and the spool displacement is obtained: F0 þ Kf i xv ¼ : ð9Þ k

4.3. Simplified actuator model Hysteresis due to the Belville spring is neglected for the controller design, hence the mean spring force f ðx; vÞ ¼ f ðxÞ ¼ 12 ðf0 ðxÞ þ fc ðxÞÞ is considered. Friction forces are approximated with a linear friction model bðvÞ ¼ bv: It is worth to note that the obtained reduced-order model is almost linear. The only nonlinearities are introduced by the actuator spring and the hydraulic servovalve model (10).

This equation is obtained from (1) neglecting the Bernoulli force and considering relation (8). Hence, the model of the valve used for the control design, described by Bernoulli equation (3), is expressed by:

5. Controller design Based on the reduced model, a position tracking controller for the clutch system is designed. Main

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8  if i > if ðfillingÞ; > < sgnðps  p1 Þ jps  p1 j ff ði Þ if id pi pif ðdead-zoneÞ; q1 ¼ 0 > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : sgnðp1  p0 Þ jp1  p0 j fd ði Þ if i oid ðdumpingÞ; where functions ff ðÞ and fd ðÞ are obtained from orifice area profiles and relation (9), while if ; id are the threshold current values corresponding to spool displacements on the dead-zone limits xvf ; xvd ; respectively.

ð10Þ

objectives of the controller are: (A) good tracking of clutch position trajectories with large harmonic content, always considering the bandwidth limitation introduced by the pipeline resonances, (B) robustness with respect to parameter uncertainties and unmodelled dynamics. The conceptual control scheme is depicted in Fig. 6. The controller is based on a cascade structure with nested position, pressure and current loops. The valve current i is measured with Hall effect current sensor, oil pressure ps (supply pressure) and p2 (downstream pressure) are measured with semiconductor pressure

4.2. Approximate model of the pipeline For control purpose, a simplified model of the pipeline, which is given by the steady state behaviour of the hydraulic line (7), is utilized. This model can be adopted if the pipeline resonances are not significantly excited. This x

Hydraulic model inv. x

Actuator inversion

v*,a*,j*

p0, ps

Reference Generator

x

x*

x

position regulator

+

+

p2*

-

pressure regulator

+

q*

valve inversion

p2

Fig. 6. Position tracking controller structure.

p2

i*

current drive i

V

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1376

sensors, the clutch position x with an LVDT sensor. With respect to commercial standard solutions, a feedback pressure control is introduced in order to minimize the effects of valve uncertainties on the hydraulic circuit. Position and pressure controllers are composed by feedforward and feedback actions. The first ones provide nonlinearities compensation and allow for fast response, whereas the second ones are designed to assure robustness with respect to parameter variations and uncertainties. The fast current controller is not presented here. As already described, it can be thought as a fast inner loop whose dynamics is neglected. In the controller block diagram, the nonlinear filter proposed in Zanasi, Guarino Lo Bianco, and Tonielli (2000) is used for the generation of the smooth position reference x and its time derivatives v ¼ x’  ; a ¼ v’ and j  ¼ v. : This filter also ensures bandwidth limitation of the reference signals. Hence, pipeline dynamics can be neglected according to (7) in the computation of the feedforward actions, reducing the computational burden without impairing their effectiveness. Moreover, uncertainties on the oil parameters may impair the improvements theoretically achievable with more complex feedforward actions obtained by model inversion of the pipeline. On the other hand, effects of the pipeline reference frequency o1 must be considered for the tuning of the closed-loop controller parameters. Referring to the servovalve model (10) and the pipeline model (7), it is possible to consider the oil flow q ¼ q1 ¼ q2 as the input of the system, thanks to model inversion. In fact, recalling that ff ðÞ and fd ðÞ are invertible functions, the reference current i are defined as

i ¼

2

q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   sgnððp2 þ 8Z0 Dn q Þ  p0 Þ jðp2 þ 8Z0 Dn q Þ  p0 j if q o0 ðdumpingÞ;

where kp is the constant positive pressure gain and x is an auxiliary signal to be defined. Recalling that q2 ¼ q1 ¼ q ; the error model of the pressure dynamics is Ab v*  kp p* þ x  p’ 2 : ð14Þ p’* ¼  V0 þ Ax Note that reference derivative p’  is not completely 2

known, since its expression depends on v as follows:   1 @f     ðxÞv þ b’v þ mv.  kx ðv  v Þ : p’ 2 ¼ A @x In order to compensate for the time derivative of p’ 2 ; after defining x as   1 @f    ðxÞv þ b’v þ mv. x¼ A @x

1 v’* ¼ ðkx x*  b*v þ Ap* 2 Þ; m Ab kx 1 @f ðxÞ*v  kp p* 2 : v* þ v*  p’* 2 ¼  V0 þ Ax A @x A

ðid þ if Þ

q ¼ 0 ðdead-zoneÞ;

1 v’* ¼ ðb*v  kx x* þ Ap* 2 Þ; ð13Þ m where p* 2 ¼ p2  p2 is the downstream pressure tracking error. Since pressure p2 is not the available control input in the clutch system, a pressure controller is designed by dynamic inversion of the hydraulic actuator model. Let us define the following reference flow q : ðV0 þ AxÞ ðV0 þ AxÞ p* 2 þ x; q ¼ Av  kp b b

the overall error model is x’* ¼ v*;

! 8  q > 1 > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ff > > sgnðps  ðp2 þ 8Z0 Dn q ÞÞ jps  ðp2 þ 8Z0 Dn q Þj > > > > > if q > 0 ðfillingÞ; > > > >1 <

> > if > > > > > > > fd1 > > > > :

control input p2 : 1 * ð12Þ p2 ¼ ðf ðxÞ þ bv þ m’v  kx xÞ: A Feedforward actions and compensating term for the spring force are introduced and a proportional position regulator, with constant positive gain kx ; is designed. The error model for the actuator dynamics is x’* ¼ v*;

!

ð11Þ where q is the reference oil flow to be defined later. Note that upstream pressure p1 has been computed from the measured pressure p2 and the control input q : The position and speed tracking errors are defined as x* ¼ x  x and v* ¼ v  v ; respectively. The control design is based on a backstepping approach (Kristic, Kanellakopoulos, & Kokotovich, 1995). Referring to the mechanical actuator dynamics, the following position tracking controller is defined, with

ð15Þ

With a proper tuning of control parameters kp ; kx ; the controller guarantees exponential tracking of the position reference. Details of the stability proof are reported in Appendix A.1. The resulting controller is then discretized by using Euler algorithm to derive the digital implementation, whose performances are analyzed in next section.

6. Performance analysis In this section, position tracking performances are analyzed with respect to some key factors of the global system. The goal is twofold: identification of the system key parameters and choice of their ‘‘optimal’’

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configuration for the clutch actuation system. Owing to strong nonlinearity and complexity of the controlled device, this investigation cannot be carried out in an analytical way. Hence, activity has been performed through simulations. A benchmark position reference relative to a gear shift is used. The reference trajectory is parameterized in time in order to simulate different gear shift times (see Figs. 9(a), (b)). The selected performance index is the rms value of the position tracking error. Simulations have shown that electrical and mechanical servovalve parameter variations up to 50% of the nominal values do not significantly affect system performances, owing to the presence of current and pressure feedback controllers. Also the servovalve dynamics does not significantly influence the system performance for the considered gear shift times. In addition, the servovalve orifice areas and the power supply pressure guarantee that the required oil flow can be provided by the servovalve for all the considered reference trajectories. On the other side, it results that length and elasticity of the pipeline are key parameters for the overall system performance. It is worth noting that in the following considerations, the controller structure is not modified, while the control parameters are suitably tuned for each configuration to match the bandwidth limitation imposed by the pipeline. In Fig. 7, the effects of different pipelines are shown: short ð0:3 mÞ and long ð1:3 mÞ pipeline are considered, while rigid or flexible pipeline refers to equivalent bulk modulus be ; respectively, equal to 600 MPa (equal to oil bulk) or 75 MPa: In order to minimize the influence of sampling time, comparisons of Fig. 7 are obtained by using a sampling time Ts ¼ 0:2 ms for controller

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implementation. Fig. 7 allows to find the minimum achievable gear shift time for each pipeline configuration, given a maximum allowable rms value of the tracking error. It is worth observing that the length of the pipeline influences the maximum achievable performance, while pipeline stiffness is less relevant. Another key parameter is the sampling time of the digital controller, since it strongly affects the sizing of the ECU due to the induced computational load. In Fig. 8, the effect of sampling time is analyzed for two different pipelines: long-flexible and short-rigid. In the diagram, the rms tracking error value has been represented versus different sampling times (0:2; 2; 5 and 10 ms), considering different gear shift times ð0:25; 0:7; 1:3 sÞ: As expected, for each gear shift time there exists a minimum sampling time under which no performance improvement is obtained. In this situation, the dynamics limits are imposed by the first pipeline resonance frequency, hence it is of no help to choose smaller sampling times. This fact is relevant mostly for long flexible pipelines. On the other hand, once a pipeline has been selected, the selection of the greatest sampling time compatible with the performances can be performed based on the results of Fig. 8, where it is shown that different system configurations meet the same desired performance expressed by a tracking rms error. For example, a tracking error of 0:55 mm; for the 0:7 s position reference corresponds to two system configurations: the first one is characterized by the long flexible pipeline and sampling time equal to 0:2 ms; the second one is characterized by the short rigid pipeline and sampling time equal to 10 ms: Hence, a trade-off between the pipeline and the ECU features can be

long−flexible long−rigid short−flexible short−rigid

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Fig. 7. Performance analysis for different pipeline configurations.

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lf−0.25s lf−0.7 s lf−1.3 s sr−0.25 s sr−0.7 s sr−1.3 s experimental

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Fig. 8. Tracking error analysis for different sampling times and pipeline configurations: solid lines correspond to long-flexible (lf) and dashed lines to short-rigid (sr) pipelines.

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6 4

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Fig. 9. (a) Experimental test: short-rigid pipeline, Ts ¼ 5 ms: Position x and reference position x (marked). Downstream pressure p2 and reference pressure p2 (marked); (b) experimental test: long-flexible pipeline, Ts ¼ 0:2 ms: Position x and reference position x (marked). Downstream pressure p2 and reference pressure p2 (marked).

performed using Fig. 8, leading to an ‘‘optimized’’ system configuration. Since all the activities are carried out by simulation, it is important to verify how reliable the prediction is. In order to validate the developed model, experiments have been performed on a prototyping station. The experimental setup is composed by a DSP board equipped with a DSP TMS320C32, installed on a standard PC. It is linked with the clutch control system by means of an interface board used to acquire and filter sensor signals and to impose the servovalve voltage. Experiments with two different configurations, selected via the proposed procedure, are reported in Figs. 9(a) and (b). The

resulting rms values of the tracking error are also reported in Fig. 8, where it is shown that they closely match the values predicted by simulations.

7. Conclusions It has been shown how proper modelling and control design for a hydraulic clutch control system can lead to different system configurations capable to achieve the desired performances. A proper model is essential for studying system behaviour via simulation: the approach followed in the paper showed valuable results. On the

ARTICLE IN PRESS M. Montanari et al. / Control Engineering Practice 12 (2004) 1369–1379

other hand, the closed-loop controller cannot be neglected. Again, deriving controller parameters directly from system model, through the use of a simplified model, allows to carry out the performance analysis directly in simulation. This is instrumental in fixing system specifications before actual implementation, avoiding expensive trial and error design.

Appendix A A.1. Stability proof of the proposed controller By means of Lyapunov-like technique (Khalil, 1996), it can be shown that the position reference trajectory is exponentially tracked by means of the controller of Section 6. Consider the Lyapunov function candidate     1 b2 2 V¼ 2mv*2 þ 2kx þ x* þ 2b*vx* þ Zp* 22 ; 2 m where Z is a constant positive parameter to be defined. The time derivative of V along the trajectories of (15) is bkx 2 Ab x* p* 2 þ 2A*vp* 2 x*  Zkp p* 22 þ V’ ¼  b*v2  m  m  kx Ab 1 @f ðxÞ v*p* 2 :   þZ A V0 þ Ax A @x

ðA:1Þ

Considering bounded clutch position xA½0; xM ; there exist a constant K > 0 such that    Ab  K 1 @f   V þ Ax þ A @x ðxÞo A : 0 Applying Young’s inequality to the fourth term of (A.1), it holds   Ab Ab kx 2 A 2 x* p* 2 p x* þ p* 2 ; m 2m A kx hence it follows that   2 ’  bkx x* 2  b*v2  Zkp  bA p* 22 Vp 2m 2mkx    kx þ K þ 2A þ Z v*p* 2 : A

ðA:2Þ

It can be shown that with proper choice of constant Z and control parameters kx ; kp the time derivative V’ is negative definite. Hence, tracking errors tend exponentially to zero.

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References Almen, J., & Laszlo, A. (1936). The uniform-section disk spring. American Society of Mechanical Engineers, 58, 305–314. Armstrong-H!elouvry, B., Dupont, P., & Canudas De Wit, C. (1994). A survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica, 30(7), 1083–1138. Bonchis, A., Corke, P., Rye, D., & Ha, Q. (2001). Variable structure methods in hydraulic servo systems control. Automatica, 37, 589–595. Coleman, T., Branch, M., & Grace, A. (1999). Matlab optimization toolbox user’s guide. Natick, MA: The MathWorks Inc. Filicori, F., Guarino Lo Bianco, C., & Tonielli, A. (1993). Modelling and control strategies for a variable reluctance direct-drive motor. IEEE Transactions on Industrial Electronics, 40(1), 105–115. Fredriksson, J., & Egardt, B., 2000. Nonlinear control applied to gearshifting in automated manual transmissions. Proceedings of the 39th IEEE conference on decision control, December 2000, Sydney, Australia (pp. 444–449). Garofalo, F., Glielmo, L., Iannelli, L., & Vasca, F. (2002). Optimal tracking for automotive dry clutch engagement. Fifteenth triennial world congress of IFAC, 21–26 July 2002, Barcelona, Spain. Goodson, R., & Leonard, R. (1972). A survey of modeling techniques for fluid line transients. ASME Journal of Basic Engineering, 94, 474–482. Horn, J., Bamberger, J., Michau, P., & Pindl, S. (2002). Flatness-based clutch control for automated manual transmissions. Fifteenth triennial world congress of IFAC, 21–26 July 2002, Barcelona, Spain. Khalil, H. (1996). Nonlinear systems (2nd ed.). Upper Saddle River, NJ: Prentice-Hall. Kristic, M., Kanellakopoulos, I., & Kokotovich, P. (1995). Nonlinear and adaptive control design. New York: Wiley. Lozano, R., Brogliato, B., Egeland, O., & Maschke, B. (2000). Dissipative systems analysis and control. London: Springer. Merritt, H. (1967). Hydraulic control systems. New York: Wiley. Pettersson, M., & Nielsen, L. (2000). Gear shifting by engine control. IEEE Transactions on Control System Technology, 8(3), 495–507. Sohl, G., & Bobrow, J. (1999). Experiments and simulations on the nonlinear control of a hydraulic servosystem. IEEE Transactions on Control System Technology, 7, 238–247. Tunay, I., Rodin, E., & Beck, A. (2001). Modeling and robust control design for aircraft brake hydraulics. IEEE Transactions on Control System Technology, 9, 319–329. Vossoughi, G., & Donath, M. (1995). Dynamic feedback linearization for electrohydraulically actuated control systems. Transactions of the ASME, 117, 468–477. Yang, W., & Tobler, W. (1991). Dissipative modal approximation of fluid transmission lines using linear friction model. ASME Journal of Dynamic Systems of Measurement and Control, 113, 152–162. Zanasi, R., Guarino Lo Bianco, C., & Tonielli, A. (2000). Nonlinear filter for the generation of smooth trajectories. Automatica, 36(3), 439–448. Zavarehi, M., Lawrence, P., & Sassani, F. (1999). Nonlinear modeling and validation of solenoid-controlled pilot-operated servovalves. IEEE/ASME Transactions on Mechatronics, 4, 324–333.