EP2933502B1 - Digital hydraulic drive system - Google Patents

Description

The invention relates to a digital hydraulic drive system having the features in the preamble of claim 1.

Although the widespread use of digital hydraulic systems in industrial applications remains the subject of controversial discussions ( Scheidl, R .; Linjama, M .; Schmidt, S .: Is the future of fluid power digital? In: Proc. IME J. Syst. Contr. Eng., Vol. 226 (2012), No. 6, pp. 721-723 ) within the professional world, there are already some successful implementations of such systems. Their implementation, however, often requires an increased effort in the field of control and regulation. Recent work in this area makes use of non-linear model-based control methods. Hießl et al. used a slip regime approach for the control of a synchronous cylinder with fast switching 3/2-way valves ( Hießl, A .; Plöckinger, A .; Winkler, B .; Scheidl, R .: Sliding mode control for digital hydraulic applications. In: Laamanen, A .; Linjama, M. (ed.): Proc. 5th Workshop Digital Fluid Power (DFP12), October 2012, pp. 15-26 ). Valve control takes place by means of PWM signals, a characteristic feature for a subgroup of digital hydraulic systems ( Linjama, M .: Digital fluid power - State of the art. In: Proc. 12th Scandinavian Int'l Conf. Fluid Power (SICFP'11), May 2011, pp. 331-353 ).

Advanced control methods are also used in another important subgroup of digital hydraulic systems. Linjama et al. used an optimal control approach for a system of digital flow control units (DFCU) in Linjama, M .; Huova, M .; Boström, P .; Laamanen, A .; Siivonen, L .; Morel, L .; Waiden, M .; Vilenius, M .: Design and Implementation of an Energy Saving Digital Hydraulic Control System. In: Vilenius, J .; Koskimies, KT; Uusi-Heikkilä, J. (ed.): Proc. 10th Scandinavian Int'l Conf. Fluid Power (SICFP'07), Vol. 2 (2007), pp. 341-359 , A DFCU is a group of switching valves in parallel, which allows a quantized adjustment of the volumetric flow by selectively switching the individual valves. An in-depth look at this technology will be made in Linjama, M .; Laamanen, A .; Vilenius, M .: Is it time for digital hydraulics? In: Proc. 8th Scandinavian Int'l Conf. Fluid Power (SICFP'03), Tampere University of Technology, 2003, pp. 347-366 , The aforementioned optimal control approach was used for a differential cylinder which is driven by DFCUs based on the principle of the resolved control edge.

The principle of "dissolved control edges" is a concept in which the volume flows at the connections of a hydraulic actuator (such as a cylinder or motor) can be adjusted independently of each other. Compared with conventional servo-hydraulic systems, they offer potential for saving energy by reducing the backpressure. An overview of this principle will be found in Eriksson, B .; Palmberg, J.-O .: Individual metering fluid power systems: Challenges and opportunities. In: Proc. IME J. Syst. Contr. Eng., Vol. 225 (2011), No. 2, pp. 196-211 given.

As far as the quality of the respectively used control is concerned, however, the known solutions still leave something to be desired, and frequently the expenditure required for computation is too high for a timely control of actuator systems.

Based on this prior art, the present invention seeks to further improve the known solutions while maintaining their advantages, a functionally reliable control for a digital hydraulic drive system, that a high control quality is achieved with low computational complexity, so that too insofar as the costs of the desired regulation are reduced.

The item Kock, F., et al .: Flatness-based high frequency control of a hydraulic actuator. In: J. Dynamic Systems, Vol. 134 (2012), No. 2, p. 021003 (7 p en) describes a digital hydraulic drive system consisting of an actuator and at least one independently operable valve device for controlling the volume flows in the supply and / or Abströmanschlüssen the actuator, a flatness-based follow-up control is used and a lower-level control is used, which is used by the Configuration of the valve device depends.

This object is achieved by a digital hydraulic drive system according to the feature configuration of claim 1 in its entirety.

According to the characterizing part of claim 1, it is provided that the flatness-based sequence control uses the volume flows as a manipulated variable and that as the valve device to be configured a digital hydraulic full bridge circuit using pulse width modulated valve units (PWM) and / or pulse-code-modulated valve units (PCM) and / or or digital volume flow units (DFCU) is used.

The fact that a flatness-based follow-up control is used, which uses the volume flows as a manipulated variable, and a subordinate control is used, which depends on the configuration of the valve device, a control method is provided which is particularly suitable for Use by using quick-change valves (pulse width modulation) and / or parallel valves (digital flow control unit).

According to the present inventive solution, the additional degree of freedom inherent in the principle of the dissolved control edge is used to control the pressure drop across the respective valve or valve group in the return flow and thus prevent cavitation and emptying of the reservoirs. Further criteria for the use of this additional degree of freedom are described by Bindel et al. ( Bindel, R .; Nitsche, R .; Rothfuss, R .; Zeitz, M .: Flatness-based control of a hydraulic drive with two valves for a large manipulator. In: at-Automatisierungstechnik, Vol. 48 (2000), No. 3, pp. 124-131 ), which use this additional degree of freedom to control a manipulator joint with 3/2-Wegeservoventilen.

Further advantageous embodiments of the digital hydraulic drive system are the subject of the dependent claims. In a particularly preferred embodiment of the solution according to the invention, an observer-based load estimation for the respective actuator used is carried out within the controller design.

Fig. 1a, 1b, 1c

mit üblichen hydraulischen Schaltsymbolen versehen verschiedene Antriebssystemkonzepte, einmal in der Art einer Vollbrücke (Fig. 1a) und einmal eine Ansteuerung im Zweiquadrantenbetrieb über den Zu- und Ablauf des Aktuators (Fig. 1 b) sowie gemäß der Darstellung nach der Fig. 1c verschiedene digital ansteuerbare hydraulische Schalt- und Steuerventile, die an die Stelle der einstellbaren Drosseln in den Fig. 1a, 1b treten;

Fig. 2

vergleichbar den Darstellungen nach den Fig. 1a und 1b die wesentlichen Komponenten eines digitalhydraulischen Antriebssystems mit vorgeschalteter Ventileinrichtung;

Fig. 3

den grundsätzlichen Aufbau einer Regelungsstruktur zum Regeln des digitalhydraulischen Antriebssystems;

Fig. 4a, 4b, 4c

in der Art von Graphen Angaben über das Regelungsverhalten unter Einsatz von DFCU-Ventilen;

Fig. 5a, 5b, 5c

in der Art von Graphen Angaben über das Regelungsverhalten unter Einsatz von PWM-Ventilen;

Fig. 6a, 6b, 6c, 6d

Auswertegraphen betreffend einen Systemvergleich, einmal unter Einsatz eines Lastschätzers und einmal ohne Lastschätzer; und

Fig. 7

in der Art eines hydraulischen Schaltplanes eine digitalhydraulische Ansteuerungseinrichtung als 6 BitVollbrücke konzipiert.

In the following, the solution according to the invention is explained in more detail with reference to the drawing. This show in principle and not to scale representation of the

Fig. 1a, 1b, 1c

with conventional hydraulic symbols provided different drive system concepts, once in the manner of a full bridge ( Fig. 1a ) and once a two-quadrant drive via the inlet and outlet of the actuator ( Fig. 1 b) and as shown in the Fig. 1c various digitally controlled hydraulic Switching and control valves, which replace the adjustable chokes in the Fig. 1a, 1b to step;

Fig. 2

comparable to the representations after the Fig. 1a and 1b the essential components of a digital hydraulic drive system with upstream valve device;

Fig. 3

the basic structure of a control structure for controlling the digital hydraulic drive system;

Fig. 4a, 4b, 4c

in the manner of graphs information on the control behavior using DFCU valves;

Fig. 5a, 5b, 5c

in the manner of graphs information about the control behavior using PWM valves;

Fig. 6a, 6b, 6c, 6d

Evaluation graphs relating to a system comparison, once using a load estimator and once without load estimator; and

Fig. 7

designed in the manner of a hydraulic circuit diagram, a digital hydraulic control device as a 6 bit full bridge.

The considered digital hydraulic drive system consists of a hydrostatic constant motor 10 with hydropneumatic damping accumulators 12 at both terminals 14, 16. The control is effected by separate valve units or valve groups 18 of a valve device 20 at the inlet and outlet ports 14, 16 of the engine 10. Die Fig. 1a, 1b show two possible embodiments of such a drive solution with dissolved control edge. By "resolved control edges" one understands technical language, in that each control edge of a conventional proportional directional control valve is released via at least one valve with at least one basic and / or one switching position. A valve with, for example, five control edges is thus replaceable over at least five switching valves. Preferably, very small, temporally very fast-switching switching valves are used in the manner of 2/2-way switching valves (see. Fig. 1c ). The motor 10 is connected to a pressure supply source with the supply pressure p _S and to a tank or return to the tank pressure p _T.

In the Fig. 1a a full bridge is shown, which allows a four-quadrant operation. The system off Fig. 1b however, can only be operated in two quadrants, since the volume flow at both ports 14, 16 can only flow in one direction. Nevertheless, both circuits are suitable for control with resolved control edge, since in both cases, the volume flows at the terminals 14, 16 can be specified independently. However, the focus of the present invention is on the full bridge circuit Fig. 1 a and the Fig. 2 , However, in order to take the flexibility of possible circuit concepts into account, the presented design method is divided into two parts: a flatness-based follow-up control, which uses the volume flows as control variables and a lower-level control of the volume flow, which depends on the valve configuration. According to this division, the mathematical models for the drive 10 and the valve units 18, 20 are given below in detail.

First, the actuator model will be presented in principle.

modelliert, wobei J das Rotorträgheitsmoment, dd den Koeffizienten der viskosen Reibung, τ das Lastmoment, p₁ und p₂ die Drücke an den Motoranschlüssen 14, 16 und V_M das Schluckvolumen des Motors bezeichnen. Es sei angemerkt, dass das Lastmoment τ nicht als Systemgröße, sondern als zeitvarianter Parameter aufgefasst wird, d.h. es wird beim Reglerentwurf als bekannt vorausgesetzt. In Ermangelung der Kenntnis des Lastmoments kann ein Lastbeobachter in der Reglerimplementierung eingesetzt werden.The hydrostatic motor 10 is called a first order system $$\mathrm{J}\dot{\mathrm{\omega }}+\mathrm{dw}+\mathrm{\tau }-{\mathrm{V}}_{\mathrm{M}}\left({\mathrm{p}}_{\mathrm{1}}-{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{2}}\right)=\mathrm{0}$$

where J is the rotor inertia, dd is the coefficient of viscous friction, τ is the load torque, p ₁ and p _{2 are} the pressures at the engine ports 14, 16 and V _{M is} the displacement of the engine. It should be noted that the load torque τ is not understood as a system variable, but as a time-variant parameter, ie it is assumed that the controller design is known. In the absence of knowledge of the load torque, a load observer may be employed in the controller implementation.

$$-{\mathrm{q}}_{\mathrm{2}}-{\mathrm{q}}_{\mathrm{A},\mathrm{2}}+{\mathrm{V}}_{\mathrm{M}}\mathrm{\omega }+\mathrm{G}\left({\mathrm{p}}_{\mathrm{1}}-{\mathrm{p}}_{\mathrm{2}}\right)=\mathrm{0}$$

The balancing of the volume flows at the motor connections 14, 16 supplies $${\mathrm{q}}_{\mathrm{1}}-{\mathrm{q}}_{\mathrm{A}.\mathrm{1}}-{\mathrm{V}}_{\mathrm{M}}\mathrm{\omega }-\mathrm{G}\left({\mathrm{p}}_{\mathrm{1}}-{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{2}}\right)=\mathrm{0}$$

$$-{\mathrm{q}}_{\mathrm{2}}-{\mathrm{q}}_{\mathrm{A}.\mathrm{2}}+{\mathrm{V}}_{\mathrm{M}}\mathrm{\omega }+\mathrm{G}\left({\mathrm{p}}_{\mathrm{1}}-{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{2}}\right)=\mathrm{0}$$

basierend auf der Polytropengleichung $${\mathrm{p}}_{\mathrm{0},\mathrm{i}}{\mathrm{V}}_{\mathrm{0},\mathrm{i}}^{\mathrm{n}}-{\mathrm{p}}_{\mathrm{i}}{\mathrm{V}}_{\mathrm{i}}^{\mathrm{n}}=\mathrm{0}\mathrm{mit\; i}=\mathrm{1},\mathrm{2}$$

verwendet, wobei mit V_i die Gasvolumina der Speicher 12, mit p_0,i die Vorspanndrücke, mit V_0,i die Gesamtvolumina und mit n der Polytropenexponent bezeichnet sind.The volume flows entering the attenuation memories 12 are denoted by q _{A, 1} and q _{A, 2} , the leakage coefficient of the motor 10 by G. For the two memories 12, simple first-order non-linear models are designated $${\dot{\mathrm{p}}}_{\mathrm{i}}{\mathrm{V}}_{\mathrm{i}}-.\mathrm{n}{\mathrm{p}}_{\mathrm{i}}{\mathrm{q}}_{\mathrm{A}.\mathrm{i}}=\mathrm{0}\mathrm{with\; i}=\mathrm{1}.\mathrm{2}$$

based on the polytope equation $${\mathrm{p}}_{}$$$${\mathrm{}}_{\mathrm{0}.\mathrm{i}}{\mathrm{V}}_{\mathrm{0}.\mathrm{i}}^{\mathrm{n}}-{\mathrm{p}}_{\mathrm{i}}{\mathrm{V}}_{\mathrm{i}}^{\mathrm{n}}=\mathrm{0}\mathrm{with\; i}=\mathrm{1}.\mathrm{2}$$

where V _{i is} the gas volumes of the memories 12, p _{0, i} the bias pressures, V _{0, i} the total volumes and n the polytropic exponent.

$${\mathrm{V}}_{\mathrm{0},\mathrm{1}}{\mathrm{p}}_{\mathrm{0},\mathrm{1}}^{\frac{\mathrm{1}}{\mathrm{n}}}{\dot{\mathrm{p}}}_{\mathrm{1}}-\mathrm{n}{\mathrm{p}}_{\mathrm{1}}^{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}}}\left({\mathrm{q}}_{\mathrm{1}}-{\mathrm{V}}_{\mathrm{M}}\mathrm{\omega }-\mathrm{G}\left({\mathrm{p}}_{\mathrm{1}}-{\mathrm{p}}_{\mathrm{2}}\right)\right)=\mathrm{0}$$

$${\mathrm{V}}_{\mathrm{0},\mathrm{2}}{\mathrm{p}}_{\mathrm{0},\mathrm{2}}^{\frac{\mathrm{1}}{\mathrm{n}}}{\dot{\mathrm{p}}}_{\mathrm{2}}-\mathrm{n}{\mathrm{p}}_{\mathrm{2}}^{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}}}\left(-\mathrm{q}\mathrm{2}+{\mathrm{V}}_{\mathrm{M}}\mathrm{\omega }+\mathrm{G}\left({\mathrm{p}}_{\mathrm{1}}-{\mathrm{p}}_{\mathrm{2}}\right)\right)=\mathrm{0}$$

schreiben.Consequently, the overall model of the drive ( Fig. 5 ) in the shape $$\mathrm{J}\dot{\mathrm{\omega }}+\mathrm{dw}+\mathrm{\tau }-{\mathrm{V}}_{\mathrm{M}}\left({\mathrm{p}}_{\mathrm{1}}-{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{2}}\right)=\mathrm{0}$$

$${\mathrm{<figure-callout\; id="0"\; label="V"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">V</figure-callout>}}_{\mathrm{0}.\mathrm{1}}{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{0}.\mathrm{1}}^{\frac{\mathrm{1}}{\mathrm{n}}}{\dot{\mathrm{p}}}_{\mathrm{1}}-\mathrm{<figure-callout\; id="1"\; label="n\; p"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">n\; </figure-callout>}{\mathrm{p}}_{}^{}{\mathrm{}}_{\mathrm{1}}^{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}}}\left({\mathrm{q}}_{\mathrm{1}}-{\mathrm{V}}_{\mathrm{M}}\mathrm{\omega }-\mathrm{G}\left({\mathrm{p}}_{\mathrm{1}}-{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{2}}\right)\right)=\mathrm{0}$$

$${\mathrm{<figure-callout\; id="0"\; label="V"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">V</figure-callout>}}_{\mathrm{0}.\mathrm{2}}{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{0}.\mathrm{2}}^{\frac{\mathrm{1}}{\mathrm{n}}}{\dot{\mathrm{p}}}_{\mathrm{2}}-\mathrm{<figure-callout\; id="2"\; label="n\; p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">n\; </figure-callout>}{\mathrm{p}}_{}^{}{\mathrm{}}_{\mathrm{2}}^{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}}}\left(-\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">q</figure-callout>}\mathrm{2}+{\mathrm{V}}_{\mathrm{M}}\mathrm{\omega }+\mathrm{G}\left({\mathrm{p}}_{\mathrm{1}}-{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{2}}\right)\right)=\mathrm{0}$$

write.

The valve units 18 of the valve device 20 are described below from a control point of view closer. Since, as already mentioned at the beginning, the proposed approach to the design of a sequence control for different valve configurations is valid, two types of digital hydraulic full bridge circuit ( Fig. 1c, 2nd ) discussed by way of example. In both cases, the dynamics of valves 18 and valve solenoids are neglected. A full-bridge DFCU can then be modeled as follows: $$\begin{array}{l}{\mathrm{q}}_{\mathrm{i}}=\frac{{\mathrm{\sigma }}_{\mathrm{i}.\mathrm{s}}}{{\mathrm{2}}^{\mathrm{m}}-\mathrm{1}}{\mathrm{K}}_{\mathrm{DFCU}}\sqrt{\left|{\mathrm{p}}_{\mathrm{s}}-{\mathrm{p}}_{\mathrm{i}}\right|}\mathrm{sgn}\left({\mathrm{p}}_{\mathrm{s}}-{\mathrm{p}}_{\mathrm{i}}\right)-\frac{{\mathrm{\sigma }}_{\mathrm{i}.\mathrm{t}}}{{\mathrm{2}}^{\mathrm{m}}-\mathrm{1}}{\mathrm{K}}_{\mathrm{DFCU}}\sqrt{\left|{\mathrm{p}}_{\mathrm{i}}-{\mathrm{p}}_{\mathrm{t}}\right|}\mathrm{sgn}\left({\mathrm{p}}_{\mathrm{i}}-{\mathrm{p}}_{\mathrm{t}}\right)\\ \mathrm{with\; i}=1.2\end{array}$$

The supply pressure and the tank pressure are respectively denoted by p _s and p _t . The pressure-volume flow characteristics of the DFCUs are represented by the coefficient K _DFCU . The switching indices σ _{i, s} , σ _{i, t} ∈ {0,1,2, ..., 2 ^m -1} determine the switching state of the m-bit DFCUs.

The full bridge with PWM controlled valves 18 is modeled in a similar way: $$\begin{array}{l}{\mathrm{q}}_{\mathrm{i}}={\mathrm{\kappa }}_{\mathrm{i}.\mathrm{s}}{\mathrm{K}}_{\mathrm{PWM}}\sqrt{\left|{\mathrm{p}}_{\mathrm{s}}-{\mathrm{p}}_{\mathrm{i}}\right|}\mathrm{sgn}\left({\mathrm{p}}_{\mathrm{s}}-{\mathrm{p}}_{\mathrm{i}}\right)-{\mathrm{\kappa }}_{\mathrm{i}.\mathrm{s}}{\mathrm{K}}_{\mathrm{PWM}}\sqrt{\left|{\mathrm{p}}_{\mathrm{i}}-{\mathrm{p}}_{\mathrm{t}}\right|}\mathrm{sgn}\left({\mathrm{p}}_{\mathrm{i}}-{\mathrm{p}}_{\mathrm{t}}\right)\\ \mathrm{with\; i}=\mathrm{1}.\mathrm{2}\end{array}$$

In this case, κ _{i, s} and κ _{i, t designate} the duty cycle of the respective valves 18 connected to the pressure or fluid supply and tank. The coefficient K _PWM determines a linear approximation of the relationship between volume flow and duty cycle.

$${\mathrm{\sigma }}_{\mathrm{i},\mathrm{t}}=\left(\begin{array}{c}0,\\ -\left({\mathrm{2}}^{\mathrm{m}}-\mathrm{1}\right)+\frac{{\mathrm{q}}_{\mathrm{i}}}{{\mathrm{K}}_{\mathrm{DFCU}\sqrt{\left|{\mathrm{p}}_{\mathrm{i}}-{\mathrm{p}}_{\mathrm{t}}\right|}}},\end{array}\right)\begin{array}{c}{\mathrm{q}}_{\mathrm{i}}>\mathrm{0}\\ {\mathrm{q}}_{\mathrm{i}}\le \mathrm{0}\end{array},$$

und $${\mathrm{\kappa }}_{\mathrm{i},\mathrm{s}}=\left(\begin{array}{c}\frac{{\mathrm{q}}_{\mathrm{i}}}{{\mathrm{K}}_{\mathrm{PWM}}\sqrt{\left|{\mathrm{p}}_{\mathrm{s}}-{\mathrm{p}}_{\mathrm{i}}\right|}}\\ \mathrm{0},\end{array}\right),\begin{array}{c}{\mathrm{q}}_{\mathrm{i}}>\mathrm{0}\\ {\mathrm{q}}_{\mathrm{i}}\le \mathrm{0}\end{array},{\mathrm{\kappa }}_{\mathrm{i},\mathrm{t}}=\left(\begin{array}{c}\mathrm{0}\\ \frac{{\mathrm{q}}_{\mathrm{i}}}{{\mathrm{K}}_{\mathrm{PWM}}\sqrt{\left|{\mathrm{p}}_{\mathrm{s}}-{\mathrm{p}}_{\mathrm{i}}\right|}}\end{array}\right),\begin{array}{c}{\mathrm{q}}_{\mathrm{i}}>\mathrm{0}\\ {\mathrm{q}}_{\mathrm{i}}\le \mathrm{0}\end{array}\mathrm{.}$$

In order to avoid short-circuit currents, only one volumetric current path is active in each bridge branch. A distinction based on the sign of the requested volume flow q _{i is provided} by the control equations $${\mathrm{\sigma }}_{\mathrm{i}.\mathrm{s}}=\left(\begin{array}{c}\left({\mathrm{2}}^{\mathrm{m}}-\mathrm{1}\right)+\frac{{\mathrm{q}}_{\mathrm{i}}}{{\mathrm{K}}_{\mathrm{DFCU}\sqrt{\left|{\mathrm{p}}_{\mathrm{s}}-{\mathrm{p}}_{\mathrm{i}}\right|}}}.\\ \mathrm{0}.\end{array}\right)\begin{array}{c}{\mathrm{q}}_{\mathrm{i}}>\mathrm{0}\\ {\mathrm{q}}_{\mathrm{i}}\le \mathrm{0}\end{array}.$$

$${\mathrm{\sigma }}_{\mathrm{i}.\mathrm{t}}=\left(\begin{array}{c}0.\\ -\left({\mathrm{2}}^{\mathrm{m}}-\mathrm{1}\right)+\frac{{\mathrm{q}}_{\mathrm{i}}}{{\mathrm{K}}_{\mathrm{DFCU}\sqrt{\left|{\mathrm{p}}_{\mathrm{i}}-{\mathrm{p}}_{\mathrm{t}}\right|}}}.\end{array}\right)\begin{array}{c}{\mathrm{q}}_{\mathrm{i}}>\mathrm{0}\\ {\mathrm{q}}_{\mathrm{i}}\le \mathrm{0}\end{array}.$$

and $${\mathrm{\kappa }}_{\mathrm{i}.\mathrm{s}}=\left(\begin{array}{c}\frac{{\mathrm{q}}_{\mathrm{i}}}{{\mathrm{K}}_{\mathrm{PWM}}\sqrt{\left|{\mathrm{p}}_{\mathrm{s}}-{\mathrm{<figure-callout\; id="0"\; label="p\; i"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p\; </figure-callout>}}_{\mathrm{i}}\right|}}\\ \mathrm{0}.\end{array}\right).\begin{array}{c}{\mathrm{q}}_{\mathrm{i}}>\mathrm{0}\\ {\mathrm{q}}_{\mathrm{i}}\le \mathrm{0}\end{array}.{\mathrm{\kappa }}_{\mathrm{i}.\mathrm{t}}=\left(\begin{array}{c}\mathrm{0}\\ \frac{{\mathrm{q}}_{\mathrm{i}}}{{\mathrm{K}}_{\mathrm{PWM}}\sqrt{\left|{\mathrm{p}}_{\mathrm{s}}-{\mathrm{p}}_{\mathrm{i}}\right|}}\end{array}\right).\begin{array}{c}{\mathrm{q}}_{\mathrm{i}}>\mathrm{0}\\ {\mathrm{q}}_{\mathrm{i}}\le \mathrm{0}\end{array}\mathrm{,}$$

The model of the drive presented above represents a non-linear multi-variable system. The control of such systems often exceeds the possibilities of simple PID controller. This applies in particular to the follow-up regulation. The so-called differential flatness is a system feature that facilitates not only the design of the controller but also the analysis and sizing of a system as well as the planning of suitable reference trajectories.

The property of differential flatness implies the existence of a so-called flat output. This (virtual) output is generally a function of system sizes and their time derivatives. A central feature of the flatness is that the trajectories of all system quantities, including the manipulated variables, are uniquely determined by the trajectories of the flat output, while these can in turn be freely specified. This implies that the desired system behavior can be given in the form of trajectories for the components of a flat output. The resulting control task is then limited to ensuring the trajectory sequence of the flat output, which in turn is facilitated by the fact that the manipulated variables can be calculated directly from the components of the flat output.

$${\mathrm{p}}_{\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{2}{\mathrm{V}}_{\mathrm{M}}}\left({\mathrm{V}}_{\mathrm{M}}{\mathrm{y}}_{\mathrm{2}}+\mathrm{J}{\dot{\mathrm{y}}}_{\mathrm{1}}+\mathrm{d}{\mathrm{y}}_{\mathrm{1}}+\mathrm{\tau }\right)$$

$${\mathrm{p}}_2=\frac{\mathrm{1}}{\mathrm{2}{\mathrm{V}}_{\mathrm{M}}}\left({\mathrm{V}}_{\mathrm{M}}{\mathrm{y}}_{\mathrm{2}}-\mathrm{J}{\dot{\mathrm{y}}}_{\mathrm{1}}-\mathrm{d}{\mathrm{y}}_{\mathrm{1}}-\mathrm{\tau }\right)$$

$${\mathrm{q}}_{\mathrm{1}}=\frac{{\mathrm{V}}_{\mathrm{0},\mathrm{1}}{\mathrm{p}}_{\mathrm{0},\mathrm{1}}^{\frac{\mathrm{1}}{\mathrm{n}}}}{\mathrm{n}{\left(\mathrm{2}{\mathrm{V}}_{\mathrm{M}}\right)}^{-\frac{\mathrm{1}}{\mathrm{n}}}}\frac{{\mathrm{V}}_{\mathrm{M}}{\dot{\mathrm{y}}}_{\mathrm{2}}+\mathrm{J}{\ddot{\mathrm{y}}}_{\mathrm{1}}+\mathrm{d}{\dot{\mathrm{y}}}_{\mathrm{1}}+\dot{\mathrm{\tau }}}{{\left({\mathrm{V}}_{\mathrm{M}}{\mathrm{y}}_{\mathrm{2}}+\mathrm{J}{\dot{\mathrm{y}}}_{\mathrm{1}}+\mathrm{d}{\mathrm{y}}_{\mathrm{1}}+\mathrm{\tau }\right)}^{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}}}}+{\mathrm{V}}_{\mathrm{M}}{\mathrm{y}}_{\mathrm{1}}+\frac{\mathrm{G}}{{\mathrm{V}}_{\mathrm{M}}}\left(\mathrm{J}{\dot{\mathrm{y}}}_{\mathrm{1}}+\mathrm{d}{\mathrm{y}}_{\mathrm{1}}+\mathrm{\tau }\right)$$

$${\mathrm{q}}_2=-\frac{{\mathrm{V}}_{\mathrm{0},2}{\mathrm{p}}_{\mathrm{0},2}^{\frac{\mathrm{1}}{\mathrm{n}}}}{\mathrm{n}{\left(\mathrm{2}{\mathrm{V}}_{\mathrm{M}}\right)}^{-\frac{\mathrm{1}}{\mathrm{n}}}}\frac{{\mathrm{V}}_{\mathrm{M}}{\dot{\mathrm{y}}}_{\mathrm{2}}-\mathrm{J}{\ddot{\mathrm{y}}}_{\mathrm{1}}-\mathrm{d}{\dot{\mathrm{y}}}_{\mathrm{1}}-\dot{\mathrm{\tau }}}{{\left({\mathrm{V}}_{\mathrm{M}}{\mathrm{y}}_{\mathrm{2}}-\mathrm{J}{\dot{\mathrm{y}}}_{\mathrm{1}}-\mathrm{d}{\mathrm{y}}_{\mathrm{1}}-\mathrm{\tau }\right)}^{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}}}}+{\mathrm{V}}_{\mathrm{M}}{\mathrm{y}}_{\mathrm{1}}+\frac{\mathrm{G}}{{\mathrm{V}}_{\mathrm{M}}}\left(\mathrm{J}{\dot{\mathrm{y}}}_{\mathrm{1}}+\mathrm{d}{\mathrm{y}}_{\mathrm{1}}+\mathrm{\tau }\right)$$

The considered model of the drive has the property of differential flatness. A flat output y consists of the angular velocity y ₁ = ω and the sum pressure y ₂ = p ₁ + p ₂ at the motor terminals 14, 16. Using the already presented model equations concerning the actuator model, any system size can be represented by the flat output y and its time derivatives be expressed: $$\mathrm{\omega }={\mathrm{y}}_{\mathrm{1}}$$

$${\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{2}{\mathrm{V}}_{\mathrm{M}}}\left({\mathrm{<figure-callout\; id="2"\; label="V\; M\; y"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">V\; </figure-callout>}}_{\mathrm{M}}{\mathrm{y}}_{}{\mathrm{}}_{\mathrm{2}}+\mathrm{J}{\dot{\mathrm{y}}}_{\mathrm{1}}+\mathrm{<figure-callout\; id="1"\; label="d\; y"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">d\; </figure-callout>}{\mathrm{y}}_{}{\mathrm{}}_{\mathrm{1}}+\mathrm{\tau }\right)$$

$${\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_2=\frac{\mathrm{1}}{\mathrm{2}{\mathrm{V}}_{\mathrm{M}}}\left({\mathrm{V}}_{\mathrm{M}}{\mathrm{y}}_{\mathrm{2}}-\mathrm{J}{\dot{\mathrm{y}}}_{\mathrm{1}}-\mathrm{d}{\mathrm{y}}_{\mathrm{1}}-\mathrm{<figure-callout\; id="1"\; label="\tau \; q"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">\tau \; </figure-callout>}\right)$$

$${\mathrm{q}}_{}$$$${\mathrm{}}_{\mathrm{1}}=\frac{{\mathrm{<figure-callout\; id="0"\; label="V"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">V</figure-callout>}}_{\mathrm{0}.\mathrm{1}}{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{0}.\mathrm{1}}^{\frac{\mathrm{1}}{\mathrm{n}}}}{\mathrm{n}{\left(\mathrm{2}{\mathrm{V}}_{\mathrm{M}}\right)}^{-\frac{\mathrm{1}}{\mathrm{n}}}}\frac{{\mathrm{V}}_{\mathrm{M}}{\dot{\mathrm{y}}}_{\mathrm{2}}+\mathrm{J}{\ddot{\mathrm{y}}}_{\mathrm{1}}+\mathrm{d}{\dot{\mathrm{y}}}_{\mathrm{1}}+\dot{\mathrm{\tau }}}{{\left({\mathrm{<figure-callout\; id="2"\; label="V\; M\; y"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">V\; </figure-callout>}}_{\mathrm{M}}{\mathrm{y}}_{}{\mathrm{}}_{\mathrm{2}}+\mathrm{J}{\dot{\mathrm{y}}}_{\mathrm{1}}+\mathrm{<figure-callout\; id="1"\; label="d\; y"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">d\; </figure-callout>}{\mathrm{y}}_{}{\mathrm{}}_{\mathrm{1}}+\mathrm{<figure-callout\; id="1"\; label="\tau "\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">\tau </figure-callout>}\right)}^{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}}}}+{\mathrm{<figure-callout\; id="1"\; label="V\; M\; y"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">V\; </figure-callout>}}_{\mathrm{M}}{\mathrm{y}}_{}{\mathrm{}}_{\mathrm{1}}+\frac{\mathrm{G}}{{\mathrm{V}}_{\mathrm{M}}}\left(\mathrm{J}{\dot{\mathrm{y}}}_{\mathrm{1}}+\mathrm{<figure-callout\; id="1"\; label="d\; y"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">d\; </figure-callout>}{\mathrm{y}}_{}{\mathrm{}}_{\mathrm{1}}+\mathrm{\tau }\right)$$

$${\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">q</figure-callout>}}_2=-\frac{{\mathrm{<figure-callout\; id="0"\; label="V"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">V</figure-callout>}}_{\mathrm{0}.2}{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{0}.2}^{\frac{\mathrm{1}}{\mathrm{n}}}}{\mathrm{n}{\left(\mathrm{2}{\mathrm{V}}_{\mathrm{M}}\right)}^{-\frac{\mathrm{1}}{\mathrm{n}}}}\frac{{\mathrm{V}}_{\mathrm{M}}{\dot{\mathrm{y}}}_{\mathrm{2}}-\mathrm{J}{\ddot{\mathrm{y}}}_{\mathrm{1}}-\mathrm{d}{\dot{\mathrm{y}}}_{\mathrm{1}}-\dot{\mathrm{\tau }}}{{\left({\mathrm{V}}_{\mathrm{M}}{\mathrm{y}}_{\mathrm{2}}-\mathrm{J}{\dot{\mathrm{y}}}_{\mathrm{1}}-\mathrm{d}{\mathrm{y}}_{\mathrm{1}}-\mathrm{<figure-callout\; id="1"\; label="\tau "\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">\tau </figure-callout>}\right)}^{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}}}}+{\mathrm{<figure-callout\; id="1"\; label="V\; M\; y"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">V\; </figure-callout>}}_{\mathrm{M}}{\mathrm{y}}_{}{\mathrm{}}_{\mathrm{1}}+\frac{\mathrm{G}}{{\mathrm{V}}_{\mathrm{M}}}\left(\mathrm{J}{\dot{\mathrm{y}}}_{\mathrm{1}}+\mathrm{<figure-callout\; id="1"\; label="d\; y"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">d\; </figure-callout>}{\mathrm{y}}_{}{\mathrm{}}_{\mathrm{1}}+\mathrm{\tau }\right)$$

It should be noted that the flatness property is also retained when the valve models according to the formulas (7) and (8) are taken into account, since the manipulated variables σ _{i, s} / t and κ _{i, s / t} directly from the volume flows q _i and the pressures p _{i are} calculated, which in turn can be calculated from the flat output y by means of formula (11). To maintain the flexibility and the clarity of the controller design is still performed on the basis of the actuator model according to formula (6). The flatness property is not limited exclusively to digital hydraulic drives, but can be transferred to all systems having the structure (6). This also applies to hydraulic linear drives such as differential cylinders, provided that the first component of the flat output y is replaced by the cylinder position.

Hereinafter, without limiting general principles, it is assumed that no leakage occurs at the motor 10, ie, G = 0. Moreover, the bias conditions of both memories 12 are assumed to be equal: V _0.1 = V _0.2 = V ₀ and p _{0 , 1} = p _0.2 = p ₀ .

In the following, the flatness-based follow-up control is explained in more detail. The design of the flatness-based sequence control is based on three steps. First, suitable reference trajectories must be set for the flat output y. Subsequently, the control laws for the follow-up control are determined. Finally, the setpoint flows calculated by the slave controller are used as input for valve control. The pertinent valve control is as a functional block in the Fig. 3 represented there and (9), (10), since this function block is associated with the formulas (9) and (10) described above.

An essential advantage of the flatness-based design is that it is not necessary to distinguish between different modes of operation and switch between them. Since the nominal volume flows can be calculated analytically from the reference trajectories and the measured variables, the only necessary distinction is that of the sign of these nominal volume flows, as already explained in the laws (9) and (10). In Vollbrückensytem after the Fig. 1a is used in each case a connecting valve of a group 18 for supplying positive volume flows and the respective further connecting valve of another group 18 to the tank for negative volume flows. In the second example after the Fig. 1b This results in a manipulated variable restriction due to the fact that only positive volume flows can be realized. This can be taken into account in the generation of suitable target trajectories, such as in Löwis, J .; Rudolph, J .: Real-time trajectory generation for flat systems with constraints. In: Nonlinear and Adaptive Control, Springer: Berlin, Heidelberg, 2003, pp. 385-394 described.

The use of the reference trajectories will be described in more detail below. The first step in the design of a sequence control is the specification of the desired system behavior in the form of reference trajectories for the flat output y. Apart from limitations of a technological nature, such as manipulated variable constraints, these trajectories can be free and be defined independently by definition. As demonstrated by the present example, it may nevertheless be advantageous to introduce an artificial dependence of these trajectories.

realisieren. Schließlich bleibt mit der Festlegung der Trajektorie des Summendrucks ein zweiter Freiheitsgrad. Durch die Wahl $$\mathrm{t}\mapsto {\mathrm{y}}_{\mathrm{2},\mathrm{r}}\left(\mathrm{t}\right)=\mathrm{2}{\mathrm{p}}_{\min }+\left|\mathrm{J}{\dot{\mathrm{y}}}_{\mathrm{1},\mathrm{t}}\left(\mathrm{t}\right)+\mathrm{d}{\mathrm{y}}_{\mathrm{1},\mathrm{r}}\left(\mathrm{t}\right)+\mathrm{\tau }\left(\mathrm{t}\right)\right|,$$

werden die Trajektorien der Drücke $$\begin{array}{c}\mathrm{t}\mapsto {\mathrm{p}}_{\mathrm{1},\mathrm{r}}\left(\mathrm{t}\right)={\mathrm{p}}_{\min }+\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\mathrm{sgn}\left(\mathrm{J}{\dot{\mathrm{y}}}_{\mathrm{1},\mathrm{r}}\left(\mathrm{t}\right)+\mathrm{d}{\mathrm{y}}_{\mathrm{1},\mathrm{r}}\left(\mathrm{t}\right)+\mathrm{\tau }\left(\mathrm{t}\right)\right)\right)\left(\mathrm{J}{\dot{\mathrm{y}}}_{\mathrm{1},\mathrm{r}}\left(\mathrm{t}\right)+\mathrm{d}{\mathrm{y}}_{\mathrm{1},\mathrm{r}}\left(\mathrm{t}\right)+\mathrm{\tau }\left(\mathrm{t}\right)\right)\\ t\mapsto p_{2,r}\left(t\right)=p_{\min }-\frac{1}{2}\left(1-\mathrm{sgn}\left(J{\dot{y}}_{1,r}\left(t\right)+d{y}_{1,r}\left(t\right)+\tau \left(t\right)\right)\right)\left(J{\dot{y}}_{1,r}\left(t\right)+d{y}_{1,r}\left(t\right)+\tau \left(t\right)\right)\end{array}$$

durch p_min, nach unten beschränkt während gleichzeitig die benötigte Antriebsdruckdifferenz p₁-p₂ bereitgestellt wird, d.h. die Gleichung (6a) wird durch die Referenztrajektorien erfüllt. Folglich kann die Schranke p_min für den Druck dazu verwendet werden, Kavitation (insbesondere bei Lastwechseln) oder auch den Abfall des Speicherdrucks unter den Vorspanndruck p₀ zu verhindern. Darüberhinaus entspricht die Referenz des niedrigeren der beiden Drücke p₁(t) und p₂(t) jederzeit p_min. Folglich können durch einen geeigneten Kompromiss zwischen Druckabfall und Vorspanndruck die Drosselverluste verringert werden.The trajectory for the first component of the flat output y, in the form of the angular velocity ω, results directly from the control task. An operating point change from ω ₀ to ω _f in the transition time t _f could be, for example, a polynomial reference trajectory of the form $$\mathrm{t}\mapsto {\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">y</figure-callout>}}_{\mathrm{1}.\mathrm{r}}\left(\mathrm{t}\right)={\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}+\left({\mathrm{\omega }}_{\mathrm{f}}-{\mathrm{\omega }}_{\mathrm{0}}\right)\frac{{\mathrm{t}}^{\mathrm{3}}}{{\mathrm{<figure-callout\; id="3"\; label="t\; f"\; filenames="imgf0005.png"\; state="\{\{state\}\}">t\; </figure-callout>}}_{\mathrm{f}}^{\mathrm{}}}\left(\mathrm{10}-\mathrm{15}\frac{\mathrm{t}}{{\mathrm{t}}_{\mathrm{f}}}+\mathrm{6}\frac{{\mathrm{t}}^{\mathrm{2}}}{{\mathrm{<figure-callout\; id="2"\; label="t\; f"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">t\; </figure-callout>}}_{\mathrm{f}}^{\mathrm{}}}\right)$$

realize. Finally, with the determination of the trajectory of the total pressure remains a second degree of freedom. By choice $$\mathrm{t}\mapsto {\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">y</figure-callout>}}_{\mathrm{2}.\mathrm{r}}\left(\mathrm{t}\right)=\mathrm{2}{\mathrm{p}}_{\min }+\left|\mathrm{J}{\dot{\mathrm{y}}}_{\mathrm{1}.\mathrm{t}}\left(\mathrm{t}\right)+\mathrm{<figure-callout\; id="1"\; label="d\; y"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">d\; </figure-callout>}{\mathrm{y}}_{}{\mathrm{}}_{\mathrm{1}.\mathrm{r}}\left(\mathrm{t}\right)+\mathrm{\tau }\left(\mathrm{t}\right)\right|.$$

become the trajectories of the pressures $$\begin{array}{c}\mathrm{t}\mapsto {\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{1}.\mathrm{r}}\left(\mathrm{t}\right)={\mathrm{p}}_{\min }+\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\mathrm{sgn}\left(\mathrm{J}{\dot{\mathrm{y}}}_{\mathrm{1}.\mathrm{r}}\left(\mathrm{t}\right)+\mathrm{<figure-callout\; id="1"\; label="d\; y"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">d\; </figure-callout>}{\mathrm{y}}_{}{\mathrm{}}_{\mathrm{1}.\mathrm{r}}\left(\mathrm{t}\right)+\mathrm{\tau }\left(\mathrm{t}\right)\right)\right)\left(\mathrm{J}{\dot{\mathrm{y}}}_{\mathrm{1}.\mathrm{r}}\left(\mathrm{t}\right)+\mathrm{<figure-callout\; id="1"\; label="d\; y"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">d\; </figure-callout>}{\mathrm{y}}_{}{\mathrm{}}_{\mathrm{1}.\mathrm{r}}\left(\mathrm{t}\right)+\mathrm{\tau }\left(\mathrm{t}\right)\right)\\ t\mapsto {\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{2.r}\left(t\right)=p_{\min }-\frac{1}{2}\left(1-\mathrm{sgn}\left(J{\dot{y}}_{1.r}\left(t\right)+\mathrm{<figure-callout\; id="1"\; label="d\; y"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">d\; </figure-callout>}{y}_{}{}_{1.r}\left(t\right)+\tau \left(t\right)\right)\right)\left(J{\dot{y}}_{1.r}\left(t\right)+\mathrm{<figure-callout\; id="1"\; label="d\; y"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">d\; </figure-callout>}{y}_{}{}_{1.r}\left(t\right)+\tau \left(t\right)\right)\end{array}$$

limited by p _min , while at the same time the required driving pressure difference p ₁ -p _{2 is} provided, ie the equation (6a) is satisfied by the reference trajectories. Consequently, the barrier p _{min can be used} for the pressure cavitation (especially at Load changes) or to prevent the drop of the accumulator pressure below the biasing pressure p ₀ . In addition, the reference of the lower of the two pressures p ₁ (t) and p ₂ (t) at any time p _min . Consequently, by a suitable compromise between pressure drop and bias pressure, the throttle losses can be reduced.

zugewiesen, mit K_i,j > 0 und k₁₀ < k_1,2k_1,1. Diese Aufgabe umfasst zwei Schritte: Zunächst wird das System durch eine statische Rückführung exakt linearisiert. Dieser Schritt profitiert erneut von der Flachheitseigenschaft insofern, dass es immer möglich ist, ein flaches System durch eine quasistatische Rückführung exakt zu linearisieren (vgl. Delaleau, E.; Rudolph, J.: Control of flat systems by quasi-static feedback of generalized states. In: Int'l J. Control, Bd. 71 (1998), Nr. 5, S. 745-765 ). Es sei betont, dass die Linearisierung durch Rückführung in keiner Weise eine Approximation darstellt, sondern lediglich eine Kompensation der Nichtlinearitäten. Da das resultierende System linear bezüglich eines neuen (virtuellen) Eingangs ist, genügt ein linearer Regler zur Sicherstellung der Fehlerdynamik.In the next step, the individual control laws for the follow-up regulation are derived. The goal here is that the following errors e ₁ = y ₁ -y _{1, r} and e ₂ = y ₂ -y _{2, r} converge asymptotically to zero. This is a linear error dynamics $$\begin{array}{l}{\ddot{\mathrm{e}}}_{\mathrm{1}}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}}_{\mathrm{1}.\mathrm{2}}{\dot{\mathrm{e}}}_{\mathrm{1}}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}}_{\mathrm{1}.\mathrm{1}}{\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">e</figure-callout>}}_{\mathrm{1}}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}}_{\mathrm{1}.\mathrm{0}}{\int }_{{\mathrm{t}}_{\mathrm{0}}}^{\mathrm{t}}{\mathrm{e}}_{\mathrm{1}}\mathrm{d}\tilde{\mathrm{t}}=\mathrm{0}\\ {\dot{\mathrm{e}}}_2+{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}}_{2.1}{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}}_{2.0}{\int }_{{\mathrm{t}}_{\mathrm{0}}}^{\mathrm{t}}{\mathrm{e}}_{\mathrm{2}}\mathrm{d}\tilde{\mathrm{t}}=\mathrm{0}\end{array}$$

assigned, with K _{i, j} > 0 and k ₁₀ <k _1.2 k _1.1 . This task involves two steps: First, the system is exactly linearized by a static feedback. This step again benefits from the flatness property in that it is always possible to exactly linearize a flat system by quasistatic feedback (cf. Delaleau, E .; Rudolph, J .: Control of flat systems by quasi-static feedback of generalized states. In: Int'l J. Control, Vol. 71 (1998), No. 5, pp. 745-765 ). It should be emphasized that the linearization by feedback in no way represents an approximation, but only a compensation of the nonlinearities. Since the resulting system is linear with respect to a new (virtual) input, a linear controller is sufficient to ensure the error dynamics.

$${\dot{\mathrm{x}}}_{\mathrm{2}}=-\frac{\mathrm{d}}{\mathrm{j}}{\mathrm{x}}_{\mathrm{2}}-\frac{\dot{\mathrm{\tau }}}{\mathrm{J}}+\frac{\mathrm{n}{\mathrm{V}}_{\mathrm{M}}}{\mathrm{J}{\mathrm{V}}_{\mathrm{0}}{\mathrm{p}}_{\mathrm{0}}^{\frac{\mathrm{1}}{\mathrm{n}}}}\left({\mathrm{g}}_{\mathrm{1}}\left(\mathrm{x}\right)\left({\mathrm{q}}_{\mathrm{1}}-{\mathrm{V}}_{\mathrm{M}}{\mathrm{x}}_{\mathrm{1}}\right)+{\mathrm{g}}_{\mathrm{2}}\left(\mathrm{x}\right)\left({\mathrm{q}}_{\mathrm{2}}-{\mathrm{V}}_{\mathrm{M}}{\mathrm{x}}_{\mathrm{1}}\right)\right)$$

$${\dot{\mathrm{x}}}_3=\frac{\mathrm{n}}{{\mathrm{V}}_{\mathrm{0}}{\mathrm{p}}_{\mathrm{0}}^{\frac{\mathrm{1}}{\mathrm{n}}}}\left({\mathrm{g}}_{\mathrm{1}}\left(\mathrm{x}\right)\left({\mathrm{q}}_{\mathrm{1}}-{\mathrm{V}}_{\mathrm{M}}{\mathrm{x}}_{\mathrm{1}}\right)-{\mathrm{g}}_{\mathrm{2}}\left(\mathrm{x}\right)\left({\mathrm{q}}_{\mathrm{2}}-{\mathrm{V}}_{\mathrm{M}}{\mathrm{x}}_{\mathrm{1}}\right)\right)$$

mit dem Zustand x = (x₁, x₂, x₃)^T = (y₁, y₁, y₂)^T und $${\mathrm{g}}_{\mathrm{1}}\left(\mathrm{x}\right)={\left(\frac{\mathrm{1}}{\mathrm{2}{\mathrm{V}}_{\mathrm{M}}}\left({\mathrm{V}}_{\mathrm{M}}{\mathrm{x}}_{\mathrm{3}}+\mathrm{J}{\mathrm{x}}_{\mathrm{2}}+\mathrm{d}{\mathrm{x}}_{\mathrm{1}}+\mathrm{\tau }\right)\right)}^{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}}},$$

$${\mathrm{g}}_2\left(\mathrm{x}\right)={\left(\frac{\mathrm{1}}{\mathrm{2}{\mathrm{V}}_{\mathrm{M}}}\left({\mathrm{V}}_{\mathrm{M}}{\mathrm{x}}_{\mathrm{3}}-\mathrm{J}{\mathrm{x}}_{\mathrm{2}}-\mathrm{d}{\mathrm{x}}_{\mathrm{1}}-\mathrm{\tau }\right)\right)}^{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}}}\mathrm{.}$$

A state representation of the system (6) with respect to the input (q ₁ , q ₂₎ reads $${\dot{\mathrm{x}}}_{\mathrm{1}}={\mathrm{x}}_{\mathrm{2}}$$

$${\dot{\mathrm{x}}}_{\mathrm{2}}=-\frac{\mathrm{d}}{\mathrm{j}}{\mathrm{x}}_{\mathrm{2}}-\frac{\dot{\mathrm{\tau }}}{\mathrm{J}}+\frac{\mathrm{n}{\mathrm{V}}_{\mathrm{M}}}{\mathrm{J}{\mathrm{V}}_{\mathrm{0}}{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{0}}^{\frac{\mathrm{1}}{\mathrm{n}}}}\left({\mathrm{G}}_{\mathrm{1}}\left(\mathrm{x}\right)\left({\mathrm{q}}_{\mathrm{1}}-{\mathrm{V}}_{\mathrm{M}}{\mathrm{x}}_{\mathrm{1}}\right)+{\mathrm{G}}_{\mathrm{2}}\left(\mathrm{x}\right)\left({\mathrm{q}}_{\mathrm{2}}-{\mathrm{V}}_{\mathrm{M}}{\mathrm{x}}_{\mathrm{1}}\right)\right)$$

$${\dot{\mathrm{x}}}_3=\frac{\mathrm{n}}{{\mathrm{V}}_{\mathrm{0}}{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{0}}^{\frac{\mathrm{1}}{\mathrm{n}}}}\left({\mathrm{G}}_{\mathrm{1}}\left(\mathrm{x}\right)\left({\mathrm{q}}_{\mathrm{1}}-{\mathrm{V}}_{\mathrm{M}}{\mathrm{x}}_{\mathrm{1}}\right)-{\mathrm{G}}_{\mathrm{2}}\left(\mathrm{x}\right)\left({\mathrm{q}}_{\mathrm{2}}-{\mathrm{V}}_{\mathrm{M}}{\mathrm{x}}_{\mathrm{1}}\right)\right)$$

with the state x = (x ₁ , x ₂ , x ₃ ) ^T = (y ₁ , y ₁ , y ₂ ) ^T and $${\mathrm{G}}_{\mathrm{1}}\left(\mathrm{x}\right)={\left(\frac{\mathrm{1}}{\mathrm{2}{\mathrm{V}}_{\mathrm{M}}}\left({\mathrm{V}}_{\mathrm{M}}{\mathrm{x}}_{\mathrm{3}}+\mathrm{J}{\mathrm{x}}_{\mathrm{2}}+\mathrm{d}{\mathrm{x}}_{\mathrm{1}}+\mathrm{<figure-callout\; id="1"\; label="\tau "\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">\tau </figure-callout>}\right)\right)}^{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}}}.$$

$${\mathrm{G}}_2\left(\mathrm{x}\right)={\left(\frac{\mathrm{1}}{\mathrm{2}{\mathrm{V}}_{\mathrm{M}}}\left({\mathrm{V}}_{\mathrm{M}}{\mathrm{x}}_{\mathrm{3}}-\mathrm{J}{\mathrm{x}}_{\mathrm{2}}-\mathrm{d}{\mathrm{x}}_{\mathrm{1}}-\mathrm{<figure-callout\; id="1"\; label="\tau "\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">\tau </figure-callout>}\right)\right)}^{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}}}\mathrm{,}$$

$$\mathrm{q}2=\frac{\mathrm{J}{\mathrm{V}}_{\mathrm{0}}{\mathrm{p}}_{\mathrm{0}}^{\frac{\mathrm{1}}{\mathrm{n}}}}{\mathrm{n}{\mathrm{V}}_{\mathrm{M}}}\frac{{\mathrm{v}}_{\mathrm{1}}-\frac{{\mathrm{V}}_{\mathrm{M}}}{\mathrm{J}}{\mathrm{v}}_{\mathrm{2}}+\frac{\mathrm{d}}{\mathrm{J}}{\mathrm{x}}_{\mathrm{2}}+\frac{\dot{\mathrm{\tau }}}{\mathrm{J}}}{\mathrm{2}{\mathrm{g}}_2\left(\mathrm{x}\right)}+{\mathrm{V}}_{\mathrm{M}}{\mathrm{x}}_{\mathrm{1}}$$

The return $${\mathrm{q}}_{}$$$${\mathrm{}}_{\mathrm{1}}=\frac{\mathrm{J}{\mathrm{V}}_{\mathrm{0}}{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{0}}^{\frac{\mathrm{1}}{\mathrm{n}}}}{\mathrm{n}{\mathrm{<figure-callout\; id="1"\; label="V\; M\; v"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">V\; </figure-callout>}}_{\mathrm{M}}}\frac{{\mathrm{v}}_{}}{}\frac{{\mathrm{}}_{\mathrm{1}}+\frac{{\mathrm{V}}_{\mathrm{<figure-callout\; id="2"\; label="M\; J\; v"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">M\; </figure-callout>}}}{\mathrm{J}}{\mathrm{v}}_{}{\mathrm{}}_{\mathrm{2}}+\frac{\mathrm{d}}{\mathrm{J}}{\mathrm{x}}_{\mathrm{2}}+\frac{\dot{\mathrm{\tau }}}{\mathrm{J}}}{\mathrm{2}{\mathrm{G}}_{\mathrm{1}}\left(\mathrm{x}\right)}+{\mathrm{V}}_{\mathrm{M}}{\mathrm{x}}_{\mathrm{1}}$$

$$\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">q</figure-callout>}2=\frac{\mathrm{J}{\mathrm{V}}_{\mathrm{0}}{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{0}}^{\frac{\mathrm{1}}{\mathrm{n}}}}{\mathrm{n}{\mathrm{V}}_{\mathrm{M}}}\frac{{\mathrm{v}}_{\mathrm{1}}-\frac{{\mathrm{V}}_{\mathrm{<figure-callout\; id="2"\; label="M\; J\; v"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">M\; </figure-callout>}}}{\mathrm{J}}{\mathrm{v}}_{}{\mathrm{}}_{\mathrm{2}}+\frac{\mathrm{d}}{\mathrm{J}}{\mathrm{x}}_{\mathrm{2}}+\frac{\dot{\mathrm{\tau }}}{\mathrm{J}}}{\mathrm{2}{\mathrm{G}}_2\left(\mathrm{x}\right)}+{\mathrm{V}}_{\mathrm{M}}{\mathrm{x}}_{\mathrm{1}}$$

the system (17) linearizes with respect to the new (virtual) input v = (v _1, v ₂ ): $${\dot{\mathrm{x}}}_{\mathrm{1}}={\mathrm{x}}_{\mathrm{2}}.{\dot{\mathrm{x}}}_{\mathrm{2}}={\mathrm{<figure-callout\; id="1"\; label="v"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">v</figure-callout>}}_{\mathrm{1}}.{\dot{\mathrm{x}}}_{\mathrm{3}}={\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">v</figure-callout>}}_{\mathrm{2}}\mathrm{,}$$

$${\mathrm{v}}_2={\dot{\mathrm{y}}}_{\mathrm{2},\mathrm{r}}-{\mathrm{k}}_{2,\mathrm{1}}\left({\mathrm{y}}_{\mathrm{2}}-{\mathrm{y}}_{\mathrm{2},\mathrm{r}}\right)-{\mathrm{k}}_{\mathrm{2},\mathrm{0}}{\int }_{{\mathrm{t}}_{\mathrm{0}}}^{\mathrm{t}}\left({\mathrm{y}}_{\mathrm{2}}-{\mathrm{y}}_{\mathrm{2},\mathrm{r}}\right)\mathrm{d}\bar{\mathrm{t}}$$

auf die Fehlerdynamik. In Summe wird der Folgeregler durch die Gleichungen (19) und (21) beschrieben (vgl. Fig. 3). In der Fig. 3 sind die einschlägigen Formeln für die jeweiligen Funktionsblöcke in Zahlen ausgedrückt und in Klammern gesetzt. Dabei betrifft der erste Funktionsblock 30 die Generierung von Trajektorien. Der zweite Funktionsblock 32 symbolisiert den Controller oder Regler. Der dritte Funktionsblock 34 bezieht sich auf die linearisierende Rückführung, und der Funktionsblock 36 soll den Schätzer betreffen. Ansonsten werden die bisher eingeführten Bezugsgrößen und Bezugszeichen auch für die Fig. 3 eingesetzt.Finally, the application of the rule laws $${\mathrm{v}}_{\mathrm{1}}={\ddot{\mathrm{y}}}_{\mathrm{1}.\mathrm{r}}-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}}_{\mathrm{1}.\mathrm{2}}\left({\dot{\mathrm{y}}}_{\mathrm{1}}-{\dot{\mathrm{y}}}_{\mathrm{1}.\mathrm{r}}\right)-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}}_{\mathrm{1}.\mathrm{1}}\left({\mathrm{y}}_{\mathrm{1}}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">y</figure-callout>}}_{\mathrm{1}.\mathrm{r}}\right)-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}}_{\mathrm{1}.\mathrm{0}}{\int }_{{\mathrm{t}}_{\mathrm{0}}}^{\mathrm{t}}\left({\mathrm{y}}_{\mathrm{1}}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">y</figure-callout>}}_{\mathrm{1}.\mathrm{r}}\right)\mathrm{d}\tilde{\mathrm{t}}$$

$${\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">v</figure-callout>}}_2={\dot{\mathrm{y}}}_{\mathrm{2}.\mathrm{r}}-{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}}_{2.\mathrm{1}}\left({\mathrm{y}}_{\mathrm{2}}-{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">y</figure-callout>}}_{\mathrm{2}.\mathrm{r}}\right)-{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}}_{\mathrm{2}.\mathrm{0}}{\int }_{{\mathrm{t}}_{\mathrm{0}}}^{\mathrm{t}}\left({\mathrm{y}}_{\mathrm{2}}-{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">y</figure-callout>}}_{\mathrm{2}.\mathrm{r}}\right)\mathrm{d}\tilde{\mathrm{t}}$$

on the error dynamics. In sum, the slave controller is described by equations (19) and (21) (cf. Fig. 3 ). In the Fig. 3 the pertinent formulas for the respective function blocks are expressed in numbers and in brackets. The first function block 30 relates to the generation of trajectories. The second function block 32 symbolizes the controller or controller. The third functional block 34 refers to the linearizing feedback and the function block 36 is intended to relate to the estimator. Otherwise, the previously introduced reference quantities and reference numerals for the Fig. 3 used.

In addition, an observer 36 can be used, which will be explained in more detail below.

The controller design from the previous section is based on the knowledge of the load torque τ. Such knowledge can be based either on a measurement or a very accurate knowledge of the underlying process. However, if these conditions are not met, an observer-based load estimate can be used.

$$\dot{\hat{\mathrm{V}}}={\hat{\mathrm{q}}}_{\mathrm{1}}-{\mathrm{V}}_{\mathrm{M}}\hat{\mathrm{\omega }}+{\mathrm{I}}_{\mathrm{2},\mathrm{1}}{\tilde{\mathrm{V}}}_{\mathrm{1}}+{\mathrm{I}}_{\mathrm{2},\mathrm{2}}{\tilde{\mathrm{V}}}_{\mathrm{2}}$$

$${\dot{\hat{\mathrm{V}}}}_2=-{\hat{\mathrm{q}}}_2+{\mathrm{V}}_{\mathrm{M}}\hat{\mathrm{\omega }}+{\mathrm{I}}_{\mathrm{3},\mathrm{1}}{\tilde{\mathrm{V}}}_{\mathrm{1}}+{\mathrm{I}}_{\mathrm{3},\mathrm{2}}{\tilde{\mathrm{V}}}_{\mathrm{2}}$$

$$\dot{\hat{\mathrm{\tau }}}={\mathrm{I}}_{\mathrm{4},\mathrm{1}}{\tilde{\mathrm{V}}}_{\mathrm{1}}+{\mathrm{I}}_{\mathrm{4},\mathrm{2}}{\tilde{\mathrm{V}}}_{\mathrm{2}}$$

dazu verwendet werden, sowohl die Winkelgeschwindigkeit ω als auch das Lastmoment τ zu schätzen. Im Folgenden werden die geschätzten Größen durch ein "Dach" und Schätzfehler durch eine "Tilde" gekennzeichnet (e.g. ω̃ = (ω - ω̂). Zwar bringt die Verwendung der Speichervolumina V₁ und V₂ als Zustandsvariablen die nichtlineare Aufschaltung der Schätzfehler $${\tilde{\mathrm{V}}}_{\mathrm{1}}=\left({\mathrm{V}}_{\mathrm{0}}{\mathrm{p}}_{\mathrm{0}}^{\frac{\mathrm{1}}{\mathrm{n}}}{\mathrm{p}}_{\mathrm{1}}^{-\frac{\mathrm{1}}{\mathrm{n}}}-{\hat{\mathrm{V}}}_{\mathrm{1}}\right),{\tilde{\mathrm{V}}}_{\mathrm{2}}=\left({\mathrm{V}}_{\mathrm{0}}{\mathrm{p}}_{\mathrm{0}}^{\frac{\mathrm{1}}{\mathrm{n}}}{\mathrm{p}}_{\mathrm{2}}^{-\frac{\mathrm{1}}{\mathrm{n}}}-{\hat{\mathrm{V}}}_{\mathrm{2}}\right)$$

basierend auf den Druckmessungen mit sich, allerdings führt sie auf eine lineare Schätzfehlerdynamik $$\left(\begin{array}{c}\dot{\tilde{\mathrm{\omega }}}\\ {\dot{\tilde{\mathrm{V}}}}_{\mathrm{1}}\\ {\dot{\tilde{\mathrm{V}}}}_{\mathrm{2}}\\ \dot{\tilde{\mathrm{\tau }}}\end{array}\right)=\left(\begin{array}{cccc}-\frac{\mathrm{d}}{\mathrm{J}}& -{\mathrm{I}}_{\mathrm{1},\mathrm{1}}& -{\mathrm{I}}_{\mathrm{1},\mathrm{2}}& -\frac{\mathrm{1}}{\mathrm{J}}\\ -{\mathrm{V}}_{\mathrm{M}}& -{\mathrm{I}}_{\mathrm{2},\mathrm{1}}& -{\mathrm{I}}_{\mathrm{2},\mathrm{2}}& \mathrm{0}\\ {\mathrm{V}}_{\mathrm{M}}& -{\mathrm{I}}_{\mathrm{3},\mathrm{1}}& -{\mathrm{I}}_{\mathrm{3},\mathrm{2}}& \mathrm{0}\\ \mathrm{0}& -{\mathrm{I}}_{\mathrm{4},\mathrm{1}}& -{\mathrm{I}}_{\mathrm{4},\mathrm{2}}& \mathrm{0}\end{array}\right)\left(\begin{array}{c}\tilde{\mathrm{\omega }}\\ {\tilde{\mathrm{V}}}_{\mathrm{1}}\\ {\tilde{\mathrm{V}}}_{\mathrm{2}}\\ \tilde{\mathrm{\tau }}\end{array}\right)+\left(\begin{array}{c}\mathrm{0}\\ {\tilde{\mathrm{q}}}_{\mathrm{1}}\\ -{\tilde{\mathrm{q}}}_{\mathrm{2}}\\ \mathrm{0}\end{array}\right)$$

If only the pressures p ₁ and p _{2 are} measured, an observer of the form $$\dot{\hat{\mathrm{\omega }}}=-\frac{\mathrm{d}}{\mathrm{J}}\hat{\mathrm{\omega }}-\frac{\mathrm{1}}{\mathrm{J}}\hat{\mathrm{\tau }}+\frac{{\mathrm{V}}_{\mathrm{M}}}{\mathrm{J}}\left({\mathrm{p}}_{\mathrm{1}}-{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{2}}\right)+{\mathrm{<figure-callout\; id="1"\; label="I"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\mathrm{1}.\mathrm{1}}{\tilde{\mathrm{V}}}_{\mathrm{1}}+{\mathrm{<figure-callout\; id="1"\; label="I"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\mathrm{1}.\mathrm{2}}{\tilde{\mathrm{V}}}_{\mathrm{2}}$$

$$\dot{\hat{\mathrm{V}}}={\hat{\mathrm{q}}}_{\mathrm{1}}-{\mathrm{V}}_{\mathrm{M}}\hat{\mathrm{\omega }}+{\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\mathrm{2}.\mathrm{1}}{\tilde{\mathrm{V}}}_{\mathrm{1}}+{\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\mathrm{2}.\mathrm{2}}{\tilde{\mathrm{V}}}_{\mathrm{2}}$$

$${\dot{\hat{\mathrm{V}}}}_2=-{\hat{\mathrm{q}}}_2+{\mathrm{V}}_{\mathrm{M}}\hat{\mathrm{\omega }}+{\mathrm{<figure-callout\; id="3"\; label="I"\; filenames="imgf0005.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\mathrm{3}.\mathrm{1}}{\tilde{\mathrm{V}}}_{\mathrm{1}}+{\mathrm{<figure-callout\; id="3"\; label="I"\; filenames="imgf0005.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\mathrm{3}.\mathrm{2}}{\tilde{\mathrm{V}}}_{\mathrm{2}}$$

$$\dot{\hat{\mathrm{\tau }}}={\mathrm{<figure-callout\; id="4"\; label="I"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\mathrm{4}.\mathrm{1}}{\tilde{\mathrm{V}}}_{\mathrm{1}}+{\mathrm{<figure-callout\; id="4"\; label="I"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\mathrm{4}.\mathrm{2}}{\tilde{\mathrm{V}}}_{\mathrm{2}}$$

be used to estimate both the angular velocity ω and the load torque τ. In the following, the estimated quantities are indicated by a "roof" and estimation errors by a "tilde" (eg ω = (ω-ω).) Although the use of the storage volumes V ₁ and V ₂ as state variables brings about the non-linear switching of the estimation errors $${\tilde{\mathrm{V}}}_{\mathrm{1}}=\left({\mathrm{V}}_{\mathrm{0}}{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{0}}^{\frac{\mathrm{1}}{\mathrm{n}}}{\mathrm{p}}_{\mathrm{1}}^{-\frac{\mathrm{1}}{\mathrm{n}}}-{\hat{\mathrm{V}}}_{\mathrm{1}}\right).{\tilde{\mathrm{V}}}_{\mathrm{2}}=\left({\mathrm{V}}_{\mathrm{0}}{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{0}}^{\frac{\mathrm{1}}{\mathrm{n}}}{\mathrm{p}}_{\mathrm{2}}^{-\frac{\mathrm{1}}{\mathrm{n}}}-{\hat{\mathrm{V}}}_{\mathrm{2}}\right)$$

based on the pressure measurements, but it leads to a linear estimation error dynamics $$\left(\begin{array}{c}\dot{\tilde{\mathrm{\omega }}}\\ {\dot{\tilde{\mathrm{V}}}}_{\mathrm{1}}\\ {\dot{\tilde{\mathrm{V}}}}_{\mathrm{2}}\\ \dot{\tilde{\mathrm{\tau }}}\end{array}\right)=\left(\begin{array}{cccc}-\frac{\mathrm{d}}{\mathrm{J}}& -{\mathrm{<figure-callout\; id="1"\; label="I"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\mathrm{1}.\mathrm{1}}& -{\mathrm{<figure-callout\; id="1"\; label="I"\; filenames="imgf0001.png,imgf0004.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\mathrm{1}.\mathrm{2}}& -\frac{\mathrm{1}}{\mathrm{J}}\\ -{\mathrm{V}}_{\mathrm{M}}& -{\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\mathrm{2}.\mathrm{1}}& -{\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\mathrm{2}.\mathrm{2}}& \mathrm{0}\\ {\mathrm{V}}_{\mathrm{M}}& -{\mathrm{<figure-callout\; id="3"\; label="I"\; filenames="imgf0005.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\mathrm{3}.\mathrm{1}}& -{\mathrm{<figure-callout\; id="3"\; label="I"\; filenames="imgf0005.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\mathrm{3}.\mathrm{2}}& \mathrm{0}\\ \mathrm{0}& -{\mathrm{<figure-callout\; id="4"\; label="I"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\mathrm{4}.\mathrm{1}}& -{\mathrm{<figure-callout\; id="4"\; label="I"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\mathrm{4}.\mathrm{2}}& \mathrm{0}\end{array}\right)\left(\begin{array}{c}\tilde{\mathrm{\omega }}\\ {\tilde{\mathrm{V}}}_{\mathrm{1}}\\ {\tilde{\mathrm{V}}}_{\mathrm{2}}\\ \tilde{\mathrm{\tau }}\end{array}\right)+\left(\begin{array}{c}\mathrm{0}\\ {\tilde{\mathrm{q}}}_{\mathrm{1}}\\ -{\tilde{\mathrm{q}}}_{\mathrm{2}}\\ \mathrm{0}\end{array}\right)$$

$$\dot{\hat{\mathrm{\tau }}}={\mathrm{I}}_{\mathrm{2}}\tilde{\mathrm{\omega }}$$

erfolgen. Es ergibt sich die Schätzfehlerdynamik $$\left(\begin{array}{c}\dot{\tilde{\mathrm{\omega }}}\\ \dot{\tilde{\mathrm{\tau }}}\end{array}\right)=\left(\begin{array}{cc}-\frac{\mathrm{d}}{\mathrm{J}}-{\mathrm{I}}_{\mathrm{1}}& -\frac{\mathrm{1}}{\mathrm{J}}\\ -{\mathrm{I}}_{\mathrm{2}}& \mathrm{0}\end{array}\right)\left(\begin{array}{c}\tilde{\mathrm{\omega }}\\ \tilde{\mathrm{\tau }}\end{array}\right),$$

die für jede Wahl ${\mathrm{l}}_{\mathrm{1}}>-\frac{\mathrm{d}}{\mathrm{J}},{\mathrm{l}}_{\mathrm{2}}<\mathrm{0}$

asymptotisch stabil ist. Fig. 3 verdeutlicht die Struktur des insoweit geschlossenen Regelkreises.For q ₁ , q 2 = 0, this error dynamics can easily be made asymptotically stable by choosing suitable observer gains l _{i, j} . If the volume flows q ₁ and q _{2 are} not known exactly, which is often the case in the application, the error dynamics are carried out non-autonomously with the errors q ₁ and q ₂ as excitation. This affects the usability of the estimator, especially in the case of digital hydraulic systems, in which the deviations by the switching operations of the valves 18 represent a highly dynamic excitation. Remedy can be provided by taking an additional measurement of the angular velocity w. In this case, the load estimation by means of the linear observer $$\dot{\hat{\mathrm{\omega }}}=-\frac{\mathrm{d}}{\mathrm{J}}\hat{\mathrm{\omega }}-\frac{\mathrm{1}}{\mathrm{J}}\hat{\mathrm{\tau }}+\frac{{\mathrm{V}}_{\mathrm{M}}}{\mathrm{J}}\left({\mathrm{p}}_{\mathrm{1}}-{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">p</figure-callout>}}_{\mathrm{2}}\right)+{\mathrm{I}}_{\mathrm{1}}\tilde{\mathrm{\omega }}$$

$$\dot{\hat{\mathrm{\tau }}}={\mathrm{I}}_{\mathrm{2}}\tilde{\mathrm{\omega }}$$

respectively. The result is the estimation error dynamics $$\left(\begin{array}{c}\dot{\tilde{\mathrm{\omega }}}\\ \dot{\tilde{\mathrm{\tau }}}\end{array}\right)=\left(\begin{array}{cc}-\frac{\mathrm{d}}{\mathrm{J}}-{\mathrm{I}}_{\mathrm{1}}& -\frac{\mathrm{1}}{\mathrm{J}}\\ -{\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">I</figure-callout>}}_{\mathrm{2}}& \mathrm{0}\end{array}\right)\left(\begin{array}{c}\tilde{\mathrm{\omega }}\\ \tilde{\mathrm{\tau }}\end{array}\right).$$

the for every choice ${\mathrm{l}}_{}$${\mathrm{}}_{\mathrm{1}}>-\frac{\mathrm{d}}{\mathrm{J}}.{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">l</figure-callout>}}_{\mathrm{2}}<\mathrm{0}$

asymptotically stable. Fig. 3 illustrates the structure of the extent closed loop.

Two variants of the considered digital hydraulic system were simulated with the simulation program AMESim to illustrate the proposed sequence control. In the first case, a full bridge with 6-bit DFCUs ( Fig. 7 ) used as bridge resistors for driving. The DFCUs consist of modified HYDAC WS08W valves with switching times of 5 ms and downstream apertures with diameters of 0.45 mm, 0.62 mm, 0.9 mm, 1.28 mm, 1, 83 mm and 3 mm. The simulation models of the valves 18 form the mechanical valve piston dynamics, a simple magnetic model of the first order with saturation and a subordinate current control. In the second example, the bridge resistors consist of valve groups 18 of the same type, driven by a 50 Hz PWM signal. The Redlich Kwong Soave provided by AMESim Gas model ( Soave, G .: Equilibrium constants from a modified Redlich-Kwong equation of state. In: Chem. Eng. Sci., Vol. 27 (1972), No. 6, pp. 1197-1203 ) was used to simulate the damping memory 12. The applied motor model 10 again corresponds to equation (1). The physical parameters used can be found in the following table. The controller parameters became k _1.0 = 8000 s ^-3 , k _1.1 = 1280 s ^-2 , k _1.2 = 64 s ^-1 , k _2.0 = 400 s ^-2 and k _2.1 = 40 s ^{-1 is} selected. Table: Physical system parameters tcA parameters symbol value unit Rotor inertia J 2.35 · 10 ^-2 Kgm ² Motor damping coefficient d 3.18 · 10 ^-2 nms displacement V _M 2.4 · 10 ^-6 m ³ rad storage volume V ₀ 1 · 10 ^-4 m ³ Memory bias pressure p ₀ 1.5 · 10 ⁶ Pa

The reference trajectory of the angular velocity w comprises three operating point changes. First, the engine 10 is accelerated from standstill to 900 min ^-1 , then braked to 100 min ^-1 and finally reversed to -600 min ^-1 . The results of the simulation of the DFCU bridge are in the Fig. 4a, 4b, 4c represented, wherein in the x-direction, the time is plotted in seconds and in the Fig. 4a in y-direction, the angular velocity w with the unit 1 / min. In the Fig. 4b and 4c the pressure in the unit bar is indicated in the y-direction. In the direction of the Fig. 4a Seen on the top right is a detail shown, from the graph after the Fig. 4a , If reference trajectories are used in the control model, the curves are smoothed and, in particular, the jagged courses in the Fig. 4b and 4c are then smoothed out accordingly.

It can also be seen that the system of reference trajectory can follow the angular velocity very well. In the detailed view smaller oscillations can be seen, which are caused by the non-ideal switching operations of the valves 18. As can be seen from the representation of the pressure curves, deviations from the reference trajectories lead to a slight violation of the lower barrier p _min . Consequently, it is advisable to consider a safety footprint when defining this barrier. These deviations have their origin in the simplified modeling of the valves and in the limited bandwidth of the controllers. Fig. 5a, 5b, 5c show the simulation results for the PWM controlled system. While significantly larger oscillations can be seen on the pressure signals, the follow-up behavior of the angular velocity ω is similar to that of the DFCU system. As far as the designation of the x- and y-coordinates is concerned, as well as the further statements correspond to the Fig. 5a of the Fig. 4a and the Fig. 5b and 5c the Fig. 4b or 4c.

The influence of the load estimator is through Fig. 6 clarified. The DFCU system was simulated with a sudden load change within 100 ms from 0 Nm to 20 Nm at t = 2s and from 20 Nm to 0 Nm at t = 3s. The observer parameters were chosen to be l ₁ = 2000 s ^-1 and l ₂ = -2.35 10 ⁴ Nm s ^-1 . It can be seen that the deviations of the angular velocity from its reference trajectory from 18% to 2% can be significantly reduced by using the load observer. It also shows that the reference trajectory for the total pressure can not be calculated correctly without the load estimate (see equation (13)). For this reason, the pressure p ₂ permanently violates the lower barrier at p _min = 20 bar. In contrast, the observer-based system avoids such violations of limitation other than spikes due to the dynamic limitations of the system. The strongly fluctuating graphs refer to simulation values without load observers.

KpWM PWM Ventil-Koeffizient $$\left[{\mathrm{m}}^{\mathrm{4}}/\sqrt{\mathrm{kg}\cdot \mathrm{m}}\right]$$

K_i,j Reglerparameter $$\left[{\mathrm{s}}^{\mathrm{i}+\mathrm{j}-\mathrm{4}}\right]$$

K_i,j Beobachterverstärkungen misc. nn Polytropenexponent [1] m Anzahl der DFCU-Ventile [1] p₁, p₂ Drücke [Pa] p_0,1 p_0,2 Speicherladedrücke [Pa] p_s, p_t Versorgungs-, Tankdruck [Pa] q₁, q₂ Volumenströme an den Aktuatoranschlüssen [m³/s] q_1,A, q_2,A Speichervol umenströem [m³/s] V₁, V₂ Speichervolumen [m³] V_0,1, V_0,2 Gesamt-Speichervolumesn [m³] V_M Schluckvolumen des Konstantmotors [m³] v₁, v₂ Rückführgrößen misc. x = (x₁, x₂, x₃)^T Zustandsvariablen misc. y = (y₁, y₂)^T Flacher Ausgang misc. κ_i,s/t PWM Tastgrad [1] σ_i,s/t DFCU Schaltindizes [1] τ Lastmoment [Nm] ω Winkelgeschwindigkeit [rad/s] The present invention relates to a flatness-based follow-up control for a digital hydraulic drive, based on the principle of the resolved control edge. The presented control strategies avoid the distinction of operating modes and the resulting switching between such modes. The additional degree of freedom associated with the second manipulated variable is used to set the minimum pressure at the motor terminals 14, 16. In this way, the emptying of the damping memory 12 and cavitation can be prevented. In addition, the pressure losses can be limited to the necessary minimum when using a variable supply. A load estimator is used as shown to determine the load torque τ on the motor shaft of the constant velocity motor 10. nomenclature variable meaning contents unit d Coefficient of viscous friction [N / m ² ] G Leakage factor of the motor [M4 / kg · s] J Rotor inertia [kgm ² ] KDFCU DFCU coefficient $$\left[{\mathrm{m}}^{}\right]$$$$\left[{\mathrm{}}^{\mathrm{4}}/\sqrt{\mathrm{kg}\cdot \mathrm{m}}\right]$$

KpWM PWM valve coefficient $$\left[{\mathrm{m}}^{}\right]$$$$\left[{\mathrm{}}^{\mathrm{4}}/\sqrt{\mathrm{kg}\cdot \mathrm{m}}\right]$$

K _{i, j} controller parameters $$\left[{\mathrm{s}}^{\mathrm{i}+\mathrm{j}-\mathrm{4}}\right]$$

K _{i, j} observers reinforcements misc. n n polytropic [1] m Number of DFCU valves [1] p ₁ , p ₂ pressures [Pa] p _0.1 p _0.2 Memory boost pressures [Pa] p _s , p _t Supply, tank pressure [Pa] q ₁ , q ₂ Volume flows at the actuator connections [m ³ / s] q _{1, A} , q _{2, A} Storage volume flow [m ³ / s] V ₁ , V ₂ storage volume [m ³ ] V _0.1 , V _0.2 Total Speichervolumesn [m ³ ] V _M Suction volume of the constant motor [m ³ ] v ₁ , v ₂ Feedback variables misc. x = (x ₁ , x ₂ , x ₃ ) ^T state variables misc. y = (y ₁ , y ₂ ) ^T Flat outlet misc. κ _{i, s /} t PWM duty cycle [1] σ _{i, s / t} DFCU switching indexes [1] τ load torque [Nm] ω angular velocity [Rad / s]

Claims (8)

1. A digital hydraulic drive system consisting of

   - an actuator and

   - at least one independently operable valve device (20) for the control of the volumetric flows in the inflow and/or outflow connections (14, 16) of the actuator,

   flatness-based follow-up regulation being used and a secondary control being employed which is dependent upon the configuration of the valve device (20), characterised in that the flatness-based follow-up regulation uses the volumetric flows as the control variable and that a digital hydraulic full bridge circuit using pulse width-modulated valve units (PWM) and/or pulse code-modulated valve units (PCM) and/or digital volumetric flow units (DFCU) is used as the valve device (20) to be configured.
2. The drive system according to Claim 1, characterised in that for the implementation of the flatness-based follow-up regulation reference trajectories for a flat output (y) are first of all established, then the regulating laws for the follow-up regulation are determined and then the target volumetric flows calculated by a follow-up regulator are used as the input for the valve control.
3. The drive system according to Claim 1 or 2, characterised in that the flatness-based follow-up regulation is based on the principle of the resolution control edge.
4. The drive system according to any of the preceding claims, characterised in that the actuator is a fixed displacement motor (10) and that the control of the fixed displacement motor (10) takes place in four quadrant operation.
5. The drive system according to any of the preceding claims, characterised in that the additional degree of freedom inherent to the principle of the resolution control edge is used to control the decrease in pressure at the respective valve of the valve device (20) in the return flow.
6. The drive system according to any of the preceding claims, characterised in that the actuator is a fixed displacement motor (10) and that the flat output (y) that forms part of the design is determined from the angular velocity (co) of the fixed displacement motor (10) as the actuator and the total pressure (p₁ + p₂) at its fluid-conveying motor connections (14, 16).
7. The drive system according to any of the preceding claims, characterised in that the regulator is designed with knowledge of a load reference value and that a monitor-supported load assessment is used for this purpose.
8. The drive system according to any of the preceding claims, characterised in that the load assessment takes place by means of a linear monitor.

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