The nonlinear control strategy presented in section 8.4 is suitable for the system at hand in so far that it imposes a linear system behavior by feedback compensation of the nonlinearities of the system. Its functionality rests on a sufficiently accurate model of the system. Unmodeled dynamics or modeling errors of different kinds may prevent the nonlinearity compensation inherent to input-output linearization from functioning well. Even though the control strategies discussed so far show robust behavior, inves-tigating alternative approaches to controlling the system may be worthwile. As an al-ternative to input-output-linearizing strategies, it is possible to construct a feedforward control in accordance with the desired output trajectory. The problem with pure feed-forward control, however, is a sensitivity to disturbances and initial condition errors which cannot be compensated and therefore may lead to deviations from the desired trajectory or even destabilize the system. Therefore, control effectiveness is typically enhanced by superposing a possibly linear feedback controller whose purpose it is to stabilize the system about the desired output trajectory, see Figure 8.13. This is known as two-degree-of-freedom control design as the feedforward control is conceived inde-pendently from the feedback control.
The determination of the feedforward control nominally yielding the desired output trajectory is related to system inversion. The problem of determining the control sig-nal(s) required for a certain desired output trajectory is easy to answer for fully actuated systems which can be inverted directly with the computed torque method. Fundamen-tal results treating the question of system inversion have been presented in [35], ad-dressing the question of system inversion for underactuated systems, too.
In the recent past, it has been found that the same differential-geometrical considera-tions allowing for a nonlinearity-compensating input-output-linearizing control of non-linear systems can also be used to construct feedforward control laws. Practically, the
ξ∗= ξ∗(y∗, ˙y∗, ¨y∗)
˙η = q(ξ∗, η)
R
¨ y∗
˙y∗ y∗
−α(ξβ(ξ∗∗,η),η)
ξ∗, η u∗R
˙x = fd(x) + gR(x)uR
uR
y = h(x) x
P
I
R
-Figure 8.13: Feedforward-linearizing control approach with P I output feedback.
nominal control input for a non-flat SISO system to follow a desired output trajectory y∗can be computed from [91]
u∗ = y∗,(r)− α(ξ∗, η)
β(ξ∗, η) . (8.237)
From this it can be seen that a SISO system is invertible only if it has a strict relative degree. The η dynamics can be computed from
η = q(ξ˙ ∗, η) (8.238)
with arbitrary initial conditions. Now in order for the system to perfectly track the desired output trajectory, it has to hold that
ξ(0) = ξ∗(0) . (8.239)
As this will not practically be fulfilled in many cases which is why for exact output tracking this feedforward control will be superposed with an appropriate feedback con-troller. Most popularly, this is the well-known P ID control or one of its phenotypes, i.e.
P I or P D control etc. For differentially flat systems, this approach has become widely treated in the literature as in e.g. [31, 89, 131].
Applying this approach to the vane pump system at hand yields the advantage that in case a simple P I controller works for stabilizing the system about the desired trajectory ξ∗, the problem of state estimation is resolved in that essentially an output-feedback technique is used.
0 0.1 0.2 0.3 0.4 0.5
(b) Main chamber pressure.
0 0.1 0.2 0.3 0.4 0.5
(c) Actuation chamber pressure.
Figure 8.14: Feedforward control: with-out valve dynamics.
(b) Main chamber pressure.
0 0.1 0.2 0.3 0.4 0.5
(c) Actuation chamber pressure.
Figure 8.15: Feedforward control: with valve dynamics.
Simulation results Simulation results for the feedforward-linearizing control featur-ing P I feedback control are shown in Figure 8.14 for a system with an ideal valve and in Figure 8.15 for a system with a non-ideal valve. The results show that stable trajectory tracking with negligible error can be achieved. Transients from valve dynamics show minor influence.
8.8 Intermediate Conclusion
In this chapter, a feedback-linearizing control approach for the nonlinear volume flow control of a variable displacement vane pump system was suggested under the assump-tion of perfect actuator dynamics, i.e. an ideal servo valve.
With the system behavior showing switching behavior in the control input matrix because, it was shown that the internal dynamics of the system feature a switching behavior accordingly. Stability of the internal dynamics was discussed within the available frameworks, indicating stable system behavior as essentially the external coordinates that are asymptotically stabilized about a desired output trajectory do not have switching properties.
Simulation results show excellent tracking performance of the system, even when ac-tuating the system with the inclusion of valve dynamics and the input derived from a system without valve dynamics.
To account for the need of state availability within the control approach suggested, two different observer types were assessed with respect to their applicability within this context. A high-gain nonlinear observer effectively dominating system nonlinearities yielded very good observation and tracking results too. In addition, a nonlinear local tracking observer was investigated. While results here indicate that this observer can in principle be applied to the observation problem at hand, numerical simulations showed severe sensitivity in terms of observer initialization and physical parameters, so that overall, this observer type is likely to remain a theoretical concept.
The investigation of a feedforward-linearizing control law was shown to bear the po-tential to remedy the disadvantages of full state control. With the feedforward control law computable offline, a simple stabilizing P I output feedback was shown to yield very good results for trajectory tracking.
As for the assumption of a static valve in the derivation of the control laws in this section, inclusion of valve dynamics is no promising approach for two reasons: Firstly, as pointed out in [32], inclusion of valve dynamics significantly complicates the control law and secondly and more importantly, inclusion of the valve dynamics in the plant model for control law derivation can easily be shown to result in switching external coordinates. In the control approach suggested here, only the one-dimensional internal
dynamics show switching behavior as a consequence of the switching input matrix gR
while the external coordinate errors are stabilized. Control laws with switching external coordinates can, if at all, not in general be expected to yield stable tracking behavior.
Stability proofs here are expected to be of significant difficulty as the switching system’s stability will have to be analyzed for all six states so that the inclusion of valve dynamics in this problem is advised against for future research.
Featuring a Variable Displacement Vane Pump
9.1 Background
With ever increasing needs to reduce energy consumption in automotive transmissions, a variable displacement vane pump is a means to do so by adapting volume flow to varying needs of the hydraulic consumer(s). In order to investigate the SISO control approach suggested in chapter 8 in the context of a clutch actuation circuit, the pump needs to be integrated in an according hydraulic circuit conceptually so that an ap-propriate model of the system can be derived. Naturally, the interaction of the pump dynamics with potentially a multitude of dynamically responding elements makes non-linear control within automotive transmissions a challenging task. Until now, control concepts for hydraulic control units in automotive transmissions have largely remained linear which is partly due to the use of mainly solenoid valves with pressure feedback.
The concept presented in this chapter demonstrates the possibility of alternative control approaches based on purely servo-valve-based control for pressures and pump volume flow.
9.2 System Description
The clutch actuation system to be controlled with multiple inputs is shown in Figure 9.1. The pump is again to provide the volume flow for the system whose pressures are controled through the inputs to the servo valves with inputs uM and uC. The main pressure valve is to maintain the system pressure whereas the clutch valve is to control the clutch which represents the main hydraulic consumer. The secondary hydraulic consumer is modeled as an ideal consumer with consumer pressure pC. As with uRin the preceding chapter, inputs uM and uC will be taken as nominal valve openings.
Figure 9.1: Clutch actuation system with servo valves.