Real-Time-Capable Trajectory Planning for Pivoting-Plate-Type Axial Piston Pumps with Systematic Consideration of System Limitations

Abstract

A method is for producing, for a hydraulic machine having an actuator, a setpoint-value trajectory satisfying predefined limitations in order to influence an output variable of the hydraulic machine. A trajectory of unlimited setpoint values is fed to a trajectory planning function, which produces the setpoint-value trajectory from the trajectory of unlimited setpoint values. In the trajectory planning function, the trajectory of unlimited setpoint values is differentiated at least twice in order to obtain a trajectory of unlimited setpoint values that is differentiated n times. In the trajectory planning function, at least one limitation is applied to the differentiated trajectory of unlimited setpoint values in order to obtain a differentiated trajectory of limited setpoint values. The differentiated trajectory of limited setpoint values is fed to a filter integrator chain in order to obtain the setpoint-value trajectory.

Claims

1. A method for generating a setpoint trajectory which satisfies predetermined limitations for a hydraulic machine with an actuator for influencing an output variable of the hydraulic machine, the method comprising: supplying a trajectory of unlimited setpoints to a trajectory planning function, which generates the setpoint trajectory from the trajectory of unlimited setpoints; differentiating, in the trajectory planning function, the trajectory of unlimited setpoints at least n times in order to obtain an n-times differentiated trajectory of unlimited setpoints with n≥2; applying, in the trajectory planning function, at least one limitation to the n-times differentiated trajectory of unlimited setpoints in order to obtain an n-times differentiated trajectory of limited setpoints; and supplying the n-times differentiated trajectory of limited setpoints to a filter integrator chain to obtain the setpoint trajectory.

2. The method as claimed in claim 1, wherein the at least one limitation includes a control variable limitation.

3. The method as claimed in claim 1, wherein the at least one limitation includes a limitation as a function of a control range of the actuator.

4. The method as claimed in claim 3, wherein: the hydraulic machine is a pivoting plate or bent axis machine, and the limitation further includes a limitation of a pivot angle to a value between a minimum value and a maximum value depending on a control range of the actuator.

5. The method as claimed in claim 3, wherein the limitation further includes a limitation to a value between a minimum value and a maximum value identical to the minimum value depending on a control range of the actuator.

6. The method as claimed in claim 5, wherein a setpoint is specified as the minimum value and a maximum value identical with the minimum value.

7. The method as claimed in claim 1, wherein the at least one limitation includes a limitation of the setpoint.

8. The method as claimed in claim 1, wherein the actuator comprises a hydraulic adjustment cylinder with a movable piston.

9. The method as claimed in claim 2, wherein the actuator comprises an electrically controllable valve.

10. The method as claimed in claim 9, wherein the control variable limitation is a magnetic force limitation or a control current limitation of the electrically controllable valve to a value between a minimum value and a maximum value.

11. The method as claimed in claim 9, wherein the limitation further includes a limitation of a valve slider position to a value between a minimum value and a maximum value depending on a control range of the actuator.

12. The method as claimed in claim 1, wherein: the at least one limitation includes a lower limit and/or an upper limit, and the upper limit is preferably different from the lower limit.

13. The method as claimed in claim 1, further comprising: supplying the setpoint trajectory to a flatness-based pilot control, which produces therefrom a control variable trajectory for the actuator.

14. The method as claimed in claim 1, wherein the hydraulic machine is a pump and the setpoint is a delivery pressure of the pump.

15. The method as claimed in claim 1, wherein a computing unit is configured to carry out the method as.

16. The method as claimed in claim 15, wherein a computer program causes the computing unit to carry out the method when the computer program is executed on the computing unit.

17. The method as claimed in claim 16, wherein the computer program is stored on a non-transitory machine-readable memory medium.

Description

DESCRIPTION OF FIGURES

[0018] FIG. 1 shows schematically an axial piston machine with which a method according to the invention can be carried out.

[0019] FIG. 2 shows a principle structure of the flatness-based pilot control in combination with the advanced trajectory filter for real-time control of hydraulic systems using the example of the control of the delivery pressure of an axial piston machine of a pivoting plate design.

[0020] FIG. 3 shows a qualitative comparison of different trajectories for the variable to be controlled, an internal limited system state and the control variable.

DETAILED DESCRIPTION OF THE DRAWING

[0021] The invention describes a general approach to real-time trajectory planning taking into account control variable and state variable limitations for nonlinear single-variable systems of any order. This new approach is particularly suitable for hydraulic systems and is used as an example for the control of a pivoting plate axial piston machine (AKM) with pressure control of the pivot angle.

[0022] In FIG. 1, an axial piston machine 100, for example of a pivoting plate or bent axis design, with an adjustment device 130 is shown schematically. A pivot angle of the axial piston machine is adjustable by means of the adjustment device 130, wherein the conveying volume or volumetric displacement can be adjusted by adjusting the pivot angle. The axial piston machine 100 can be operated both as a motor and as a pump with a revolution rate n and a torque M. The axial piston machine is connected to a high pressure side p.sub.Hi and a low pressure side p.sub.Lo and is subjected to a pressure difference Δp.sub.p. In an implementation as a pump, the pressure difference Δp.sub.p represents the delivery pressure and an output variable of the pump.

[0023] The pivot angle can be adjusted by means of the adjustment device 130. The adjustment device 130 here comprises an adjustment cylinder 131, which engages with its piston 133 at a swivel cradle 101 of the axial piston machine 100, for example. A position of the piston is denoted with x.sub.K. The piston is subjected to a pressure difference Δp.sub.x, which can be adjusted by means of two electroproportional valves 132, 134. The position x.sub.K of the piston 133 in the adjustment cylinder 131 represents the output variable of the adjustment device 130.

[0024] The electroproportional valves 132, 134 each have a coil or an electromagnet 132a, 134a, which is supplied with a current i, and a restoring spring 132b, 134b, which are used to change a valve slider position.

[0025] The two pressure control valves 132, 134, which determine the adjustment cylinder position x.sub.k, are conveniently considered as one valve. For this purpose, one valve is energized in such a way that no flow is generated, while the calculated control current I is switched to the other. A necessary conversion from one control current I to the two valves takes place outside the advanced state variable filter. In the following, therefore, a valve with a control current I is assumed. If the pressure control valves 132, 134 have a significantly faster dynamic than the adjustment cylinder, the valve slider position x.sub.v and the control current I are algebraically related as a function of the pressure difference Δp.sub.x via the valves by means of a static force balance (1).

F.sub.m(I)+F.sub.f(x.sub.v)+F.sub.jet(x.sub.v;Δp.sub.v)+F.sub.p(Δp.sub.v)=0;  (1)

where F.sub.m(I) is a magnetic force, F.sub.f(x.sub.v) is a restoring spring force, F.sub.jet (x.sub.v;Δp.sub.v) is a flow force, and Fp(Δp.sub.v) is a pressure force. These forces can be applied, for example, by means of nonlinear functional relationships or characteristic fields as a function of the input variables. Δp.sub.v describes the pressure difference across the valves and is defined as follows:

[00001]$\begin{array}{cc}\Delta p_v=\{\begin{array}{cc}{p}_s-p_x,& x_v\ge 0\\ p_x-p_t,& x_v<0\end{array},& \left(2\right)\end{array}$

wherein the control pressure p.sub.s≥p.sub.x and the tank pressure p.sub.t≤p.sub.x are constant pressures.

[0026] The control current I or the valve slider position x, determines the piston position x.sub.k via the following dynamic:

[00002]$\begin{array}{cc}{\stackrel{.}{x}}_k=\frac{1}{A_k}{q}_l\left(x_v,\Delta p_v\right)=\frac{1}{A_k}\sqrt{\frac{2}{{\rho }_c}}{\alpha }_v{A}_v\left(x_v\right)\sqrt{\Delta p_v},& \left(3\right)\end{array}$

[0027] This dynamic depends on the cross-sectional area of the adjustment cylinder, A.sub.k, as well as other constants α.sub.v, ρ.sub.v. In addition, the pressure difference across the valves, Δp.sub.v, has a direct influence on the first derivative of the adjustment cylinder position.

[0028] Due to the mechanical coupling, there is a bijective, algebraic relationship between the piston position x.sub.k and the pivot angle α.sub.p, which can be given, for example, by means of a nonlinear functional relationship or a characteristic field:

α.sub.p=f(x.sub.k)bzw.Math.x.sub.k=f.sup.−1(α.sub.p)  (4)

[0029] Finally, the differential pressure across the pump results from the following dynamic:

[00003]$\begin{array}{cc}\Delta {\stackrel{.}{p}}_p=\frac{K}{V}\left(q_p\left({\alpha }_p\right)-q_l\left(\Delta p_p\right)\right)=\frac{K}{V}\left(V_p.Math.{\alpha }_p.Math.{\omega }_p-q_l\left(\Delta p_p,{\alpha }_m,n_m\right)\right)& \left(5\right)\end{array}$

wherein K, V and V, are constants, the angular velocity of the pump ω.sub.p can be measured via the pump revolution rate or estimated by means of observers and q.sub.1(Δp.sub.p) represents a load volumetric flow, which is a function, for example, of the differential pressure app. If the pump is used in a hydrostatic transmission, the load volumetric flow can also be a function of the pivot angle α.sub.m or the revolution rate n.sub.m of a hydraulic motor.

[0030] In the case of axial piston pumps of a pivoting plate design, as for example represented in FIG. 1, the supply flow rate is adjusted by adjusting the pivot plate angle, in short the pivot angle. In the closed hydraulic system, the supply flow rate influences the differential pressure or delivery pressure Δp.sub.p via the pump. The pivot angle adjustment is carried out by means of a mechanical coupling of the pivot plate to a differential cylinder. Both chambers of the differential cylinder can be pressurized in this case. The chamber pressure is controlled by means of proportional directional valves with pressure recirculation.

[0031] Depending on the mechanical adjustment device and, if appropriate, the sensor configuration, various functions can be implemented for these pumps. An example is a speed control with mechanical pivot angle control (load-resistant operation). For this purpose, the pivot angle of the pivoting plate is mechanically returned to the proportional directional control valve by means of a spring and is thus kept within a control range. Another example is an electronic output pressure control (load-sensitive operation) without mechanical return of the pivot angle to the control valve. With the invention, even load-sensitive pumps can be used for load-resistant operation without the need for mechanical adaptations.

[0032] In one embodiment of the invention, for an output pressure-controlled axial piston pump of a pivoting plate design, an electronic four-quadrant control of the differential pressure taking into account a pivot angle limitation or a pivot angle control taking into account a differential pressure limitation can be realized here. In addition, a current limitation for the proportional directional valves can also be taken into account.

[0033] In the embodiment of the invention in accordance with FIG. 2 this is achieved by means of an advanced trajectory filter or a state variable filter (eZVF) 210 for the differential pressure Δp.sub.p.sup.des across the pump as a setpoint, the output of which contains a trajectory Δp.sub.p.sup.plan, which maintains the existing limitation, and the time derivatives of which are d/dt . . . (d/dt).sup.n. Compliance with the Limitations is carried out here by cascaded, model-based limitations 214, 213, 212 of the highest (here the second) derivative z″.sub.ref of the trajectory filter.

[0034] If limitations of the current I (212), pivot angle or piston position x.sub.k (213) or differential pressure Δp.sub.p (214) are reached, a differential pressure trajectory is obtained which complies with the limitations. The order of the limitations determines the priority of the corresponding limitations. If no limitation is reached, the differential pressure reference trajectory is filtered via an unlimited filter dynamic 211 which is adjustable by the user.

[0035] If the upper and lower limits follow the reference trajectory identically, this reference trajectory is implemented (as long as higher prioritizations do not override this). For example, a pivot angle control can be implemented taking into account a differential pressure limitation. The pivot angle reference trajectory is then specified as upper and lower limits for the pivot angle. These limitations will override the unlimited differential pressure planning and thus the specified pivot angle reference trajectory will be precisely implemented as long as the differential pressure limitation is not violated.

[0036] The valve slider position x.sub.v determines the adjustment cylinder position x.sub.k. A nonlinear dynamic of the first order is assumed below here. In addition to the valve slider position, the pressure difference Δp.sub.x has a direct influence via the valves on the first derivative of the adjustment cylinder position.

[0037] Due to the mechanical coupling, there is a bijective, algebraic relationship between the adjustment cylinder position x.sub.k and the pivot angle α.sub.p, which can be given, for example, by a nonlinear function relationship or a characteristic field. Due to the mechanical coupling, it also applies to this relationship that a limitation of the adjustment cylinder position x.sub.k leads to a limitation of the pivot angle α.sub.p.

[0038] The pivot angle, the angular velocity of the pump, which can be measured by means of the pump revolution rate or estimated by observers, and a possible load volumetric flow determine, among other things, the differential pressure across the pump. Here, a nonlinear dynamic of the first order is assumed below. The load volumetric flow can depend, for example, on the differential pressure, but when used in a hydrostatic transmission can also depend on the pivot angle or the revolution rate of a hydraulic motor.

[0039] Based on this dynamic model, the eZVF 210 can be calculated. The model is flat in the real output Δp.sub.p. Therefore, in the following, the real output y=Δp.sub.p is selected identically to the flat output. Thus, there are no zero dynamics in the model.

[0040] However, it is also possible to use the invention if a stable zero dynamic is available. In this case, a stable differential equation must be solved to replan the setpoint from y to the flat output, see also Joos, S., Bitzer, M., Karrelmeyer, R., & Graichen, K. (2017). Online trajectory planning for state- and input-constrained linear SISO systems using a switched state variable filter. IFAC-PapersOnLine, 50(1), 2639-2644.

[0041] The simplified, non-linear model of the AKP is obtained using equations (3), (4) and (5) in flat coordinates:

[00004]$\begin{array}{cc}& \left(6\right)\end{array}$$\Delta {\stackrel{.}{p}}_p=\frac{K}{V}\left(V_p.Math.{\alpha }_p.Math.{\omega }_p-q_l\left(\Delta \text{?}\right)\right)=\frac{K}{V}\left(V_p.Math.f\left(x_k\right).Math.{\omega }_p-\text{?}\left(\Delta p_p\right)\right)$$\begin{array}{c}\Delta \text{?}=\frac{K}{V}\left(V_p.Math.\left({\stackrel{.}{\alpha }}_p.Math.{\omega }_p+{\alpha }_p.Math.{\stackrel{.}{\omega }}_p\right)-\text{?}\left(\Delta p_p\right)\right)\\ =\frac{K}{V}\left(V_p.Math.\left(\frac{\partial }{\partial x_k}f\left(x_k\right){\stackrel{.}{x}}_k.Math.{\omega }_p+{\alpha }_p.Math.{\stackrel{.}{\omega }}_p\right)-\text{?}\left(\Delta p_p\right)\right)\\ =\frac{K}{V}(V_p.Math.\left(\frac{\partial }{\partial x_k}f\left(x_k\right).Math.\frac{1}{A_k}\sqrt{\frac{2}{p_v}}\text{?}{A}_v\left(x_v\right)\sqrt{\Delta \text{?}}.Math.{\omega }_p+{\alpha }_p.Math.{\omega }_p\right)-\\ \text{?}\left(\Delta p_p\right)).\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

[0042] The partial derivative

[00005]$\frac{\partial }{\partial x_k}f\left(x_k\right)$

can be calculated either analytically or, in the presence of a characteristic field, numerically. The states of the system in flat coordinates are given by the vector Δp.sub.p=[Δp.sub.p; Δ{dot over (p)}.sub.p]. The model of the AKM in flat coordinates then consists of the dynamics of the differential pressure and the first time derivate of the differential pressure dynamics. This also requires the dynamics for the adjustment cylinder position.

[0043] A dynamic pilot control for the AKM can be specified directly from inversion of the model equations (6), (7) in flat coordinates. The trajectories Δp.sub.ref, Δ{dot over (p)}.sub.ref, Δp″.sub.ref (=z.sub.ref, ż.sub.ref, z″.sub.ref) can be generated with the help of a state variable filter (ZVF) 211, which can be implemented, for example, as a second-order delay element. The dynamics of the ZVF can be specified, for example, by means of a filter time constant.

[0044] In order to be able to comply with state variable and control variable limitations when controlling the AKM, these limitations must be taken into account in the planned trajectories. For this purpose, the ZVF is extended so that the input Δp″.sub.ref of the filter integrator chain 215 is dynamically limited. For each k-th limitation, a limit γ.sub.k is calculated in 216 and is applied to Δp″.sub.ref via a dynamic limitation element 212, 213, 214.

[0045] The integrator chain 215 has a number n of integrators to obtain the flat output from the limited (n) derivative Y.sub.Lim(t) (n is the order of the system) by n-times integration. This is the setpoint for the flat output Δp.sub.p.sup.plan (and the time derivatives thereof), which are required, among other things, to realize the flat pilot control.

[0046] If there are limitations in the form of so-called box constraints for a control variable or a state, the limits γ.sub.k for the upper and lower limitations can be combined into a pair of limits γ.sub.⊕,k;γ.sub.⊖,k). A box constraint exists when there is an upper and lower, possibly time-variant, limitation for each variable x.sub.i, i.e. x.sub.i,min≤x.sub.i≤x.sub.i,max applies for all x.sub.i. For the AKM, all limitations are in the form of box constraints, so that they can always be calculated in pairs in 216.

[0047] The limit for a flat system with system order n and state z and a box constraint limitation f.sup.Tz−z.sub.i,Lim≤0, wherein f is only for an element other than 0, is then:

[00006]${\gamma }_{\lim }=\frac{r\left(\xi \right)}{f\left(\xi \right)}.Math.z_{\xi ,\lim }-{\left[0,.Math.,0,r\left(\xi \right),.Math.,r\left(n\right)\right]}^{T}z,$

[0048] Wherein r(ξ), . . . ,r(n) represent tuning parameters. In order to determine the variable ξ from all states on which the limitation f.sup.Tz−z.sub.i,Lim≤0 depends directly, the one with a minimum relative degree must be determined.

[0049] The variable ξ is then the index associated with this state. For a more detailed description, see Joos, S.; Bitzer, M.; Karrelmeyer, R.; Graichen, K.: Prioritization-based switched feedback control for linear SISO systems with time-varying state and input limitations. Proc. European Control Conference, p. 2935-2940, 2018, referred.

[0050] The order of the limitation elements determines the priority of the individual limitation. Typically, control variable limitations are prioritized the highest, i.e. the corresponding limitation element is placed last. Thus, the feasibility of the pilot control signal is ensured by the actuator.

[0051] To take into account (212) the limitations of the control current I.sub.min≤I≤I.sub.max, these are first represented with the help of the force balance (1) as limitations of the valve slider position, i.e. I.sub.min leads to an x.sub.v;min and I.sub.max to an x.sub.v;max. If no analytical resolution of the force balance (1) according to x.sub.v is possible, this can be done for example by means of a zero point search. The dynamic limits γ.sub.⊕,I and γ.sub.⊖,I for taking into account the control variable limitation result from the insertion of x.sub.v=x.sub.v;min/max into the highest derivative.

[00007]$\begin{array}{cc}{\gamma }_{\oplus ,I}\left(\Delta p_p,\text{?}\right))=\frac{K}{V}\left(V_p.Math.\left(\frac{\partial }{\partial x_k}f\left(x_k\right).Math.\frac{1}{A_k}\surd \frac{2}{{\rho }_v}{\alpha }_v{A}_x\left(\text{?}\right)\sqrt{\Delta p_r}.Math.{\omega }_p+f\left(x_k\right).Math.{\stackrel{.}{\omega }}_p\right)-{\stackrel{.}{q}}_l\left(\Delta p_p\right)\right)& \left(8\right)\end{array}$$\begin{array}{cc}{\gamma }_{\oplus ,I}\left(\Delta p_{p,}\text{?}\right))=\frac{K}{V}\left(V_p.Math.\left(\frac{\partial }{\partial x_k}f\left(x_k\right).Math.\frac{1}{A_k}\sqrt{\frac{2}{{\rho }_v}}{\alpha }_v{A}_v\left(x_{v,\max }\right)\sqrt{\Delta \text{?}}.Math.{\omega }_p+f\left(x_k\right).Math.{\stackrel{.}{\omega }}_p\right)-{\stackrel{.}{q}}_l\left(\Delta p_p\right)\right).& \left(9\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

[0052] The required piston position X.sub.k is obtained by resolving from (6) to α.sub.p and using (4) to

[00008]$\begin{array}{cc}{x}_k=f^{-1}\left({\alpha }_p\right)=f^{-1}(\frac{1}{V_p.Math.{\omega }_p}.Math.\left(q_l\left(p_{p)}+\frac{V}{K}\Delta {\stackrel{.}{p}}_p\right)\right).& \left(10\right)\end{array}$

[0053] The dynamic limits γ.sub.⊕,xk; γ.sub.⊖,xk for taking into account (213) the geometric limitation of the adjustment cylinder position x.sub.k;min≤x.sub.k≤x.sub.k;max result from the first derivative of the flat output, i.e. ξ=2 and (6) to

[00009]$\begin{array}{cc}{\stackrel{.}{p}}_{p,\min }\left(x_{k,\min }\right)=\frac{K}{V}\left(V_p.Math.f\left(x_{k,\min }\right).Math.{\omega }_p-q_l\left(\Delta p_p\right)\right)& \left(11\right)\end{array}$$\begin{array}{cc}{p}_{p,\max }\left(x_{k,\max }\right)=\frac{K}{V}\left(V_p.Math.f\left(x_{k,\max }\right).Math.{\omega }_p-q_l\left(\Delta p_p\right)\right).& \left(12\right)\end{array}$

[0054] The functions for limiting the highest derivative, in this case z″, are then:

γ.sub.⊖,xk(Δp.sub.p,x.sub.k,min)=r.sub.⊖,x.sub.k.Math.(Δ{dot over (p)}.sub.p,min(x.sub.k,min)−Δ{dot over (p)}.sub.p)  (13)

γ.sub.⊕,xk(Δp.sub.p,x.sub.k,max)=r.sub.⊕,x.sub.k.Math.(Δ{dot over (p)}.sub.p,max(x.sub.k,max)−Δ{dot over (p)}.sub.p)  (14)

Thus, there is one tuning parameter each, r.sub.⊕,xk and r.sub.⊖,xk.

[0055] These influence how hard the trajectory is planned in the stop and are to be adjusted pump-specifically depending on the available control variable reserve.

[0056] In addition, the differential pressure across the pump should be limited (214) to the interval Δ.sub.p,min≤Δp.sub.p≤ΔP.sub.p;max. Here ξ=1 and thus γ.sub.⊖,,Δpp and γ.sub.⊕,,Δpp each depend on a parameter vector r.sub.⊖,Δp=[r.sub.⊖,Δpp(1);r.sub.⊖,Δpp(2)] or r.sub.⊕,Δp=[r.sub.⊕,Δpp(1);r.sub.⊕,Δpp(2)] The functions for complying with this limitation result from:

γ.sub.⊖,Δp.sub.y(Δp.sub.p,Δp.sub.p,min)=r.sub.⊖,Δp.sub.p(1).Math.(Δp.sub.p,min−Δp.sub.p)−r.sub.⊖,Δp.sub.p(2).Math.Δ{dot over (p)}.sub.p  (15)

γ.sub.⊕,Δp.sub.y(Δp.sub.p,Δp.sub.p,max)=r.sub.⊕,Δp.sub.p(1).Math.(Δp.sub.p,max−Δp.sub.p)−r.sub.⊕,Δp.sub.p(2).Math.Δ{dot over (p)}.sub.p  (15)

[0057] For this please refer to Joos, S.; Bitzer, M.; Karrelmeyer, R.; Graichen, K.: Prioritization-based switched feedback control for linear SISO systems with time-varying state and input limitations. In: Proceedings European Control Conference, Limassol, Cyprus, 2018, p. 2935-2940, referred.

[0058] The resulting limits thus correspond to a second-order integrator chain, the state of which [Δp.sub.p,max−Δp.sub.p, Δ{dot over (p)}.sub.p] or [Δp.sub.p,min−Δp.sub.p, Δ{dot over (p)}.sub.p] is stabilized by means of the tuning parameters. The faster/slower the poles and thus the tuning parameters [r,.sub.Δpp(l); r,.sub.Δpp(2)] are selected for calculation in 216, the later/earlier and the stronger/weaker the system is pulled to the differential pressure limitation. The tuning parameters are to be set pump-specifically depending on the available control variable reserve.

[0059] A flatness-based pilot control 220 determines a setpoint I.sup.FF for the valve current from the setpoint Δp.sub.p for the feed pressure or pump pressure. This is based on a dynamic system model with a control current as input and a differential pressure as output.

[0060] The qualitative profile of the limited control and the associated behavior of the controlled AKM compared to the behavior thereof with unlimited control are shown in FIG. 3. FIG. 3 shows a qualitative comparison of different trajectories for the variable differential pressure Δp.sub.p to be controlled, an internal limited system state pivot angle or piston position x.sub.k as well as the control variable current I.

[0061] If an AKM on the basis of the unlimited reference trajectory 303 is controlled by the prior art according to a setpoint trajectory 302 generated or planned in a non-inventive manner, see left figure, then for each differential pressure, Δpp and piston position, x.sub.k the profile 301 is measured on the system. The system cannot follow the planned trajectory 302 precisely due to the limitation 304 of the piston position which is not taken into account.

[0062] In comparison, the control trajectory 302′ replanned by the invention, see right picture, can be realized by the system, i.e. the planned trajectory 302′ and the actual trajectory 301′ are almost exactly coincident.