Method for determining a switching function for a sliding mode controller, and sliding mode controller

Abstract

The disclosure relates to a method for determining a switching function for a sliding mode controller for controlling a controlled variable of a system, the switching function being selected as a function of a control deviation of the controlled variable and its time derivatives up to at least the second order and on the basis of initial control dynamics of the system, coefficients of the switching function being represented by means of poles of a closed control loop of the system, the poles each being selected as a function of the control deviation, and desired control dynamics of the system being set by shifting at least one first pole of the poles, and to such a sliding mode controller and to a use of such a controller.

Claims

1. A method for operating a hydraulic valve comprising: identifying, with a control device, a plurality of positions of a piston over time for a piston in the hydraulic valve using a displacement transducer in the hydraulic valve, the plurality of positions including a present position of the piston and at least one prior position of the piston; determining, with the control device, a plurality of deviations over time between the plurality of positions of the piston and a predetermined position of the piston that produces a desired volumetric flow through the hydraulic valve, the plurality of deviations including a present deviation e between the present position and the predetermined position of the piston; determining, with the control device, a first derivative e corresponding to a rate of change of deviation of the piston position over time, and a second derivative e corresponding to an acceleration of deviation of the piston position over time based on the plurality of deviations over time; determining, with the control device, a switching function output s for controlling the position of the piston based on e, , , a first pole value .sub.1 that the control device determines based on a function of e: .sub.1(e), and a second pole value .sub.2 that the control device determines based on a function of .sub.1, based on a switching function:
s(e,,)=.sub.1.sub.2e(.sub.1+.sub.2)+; and operating, with the control device, an electromagnet that applies a force to the piston to move the piston from the present position toward the predetermined position based on the switching function output s.

2. The method of claim 1 further comprising: determining, with the control device, the first pole value .sub.1 based on the function of e: .sub.1(e)=|e|+.sub.0 where is a positive linear gradient that is stored in a computer-readable storage medium and .sub.0 is a value that is less than zero that is that is stored in the computer-readable storage medium.

3. The method of claim 2 further comprising: increasing, with the control device, an absolute value of .sub.0 to provide faster dynamics for the switching function.

4. The method of claim 2 further comprising: decreasing, with the control device, an absolute value of .sub.0 to provide slower dynamics for the switching function.

5. The method of claim 2 further comprising: determining, with the control device the second pole value .sub.2 based on the function of .sub.1(e) and a predetermined proportional constant value c.sub.2 that is stored in the computer-readable storage medium based on a function: .sub.2=c.sub.2.sub.1(e)=c.sub.2(|e|+.sub.0).

6. The method of claim 2 further comprising: determining, with the control device, the second pole value .sub.2 based on the function of .sub.1(e) and a predetermined proportional constant value c.sub.2 that is stored in the computer-readable storage medium based on a function: .sub.2=c.sub.2.sub.1(e).

7. The method of claim 1 further comprising: determining, with the control device, the first pole value .sub.1 based on a square root function of e.

8. The method of claim 1 further comprising: determining, with the control device, a value of a manipulated variable u based on a function: u={square root over (|s|)}sign(s) where is a proportional gain factor that is stored in the computer-readable storage medium; and operating, with the control device, the electromagnet based on the value of u to move the piston from the present position toward the predetermined position.

9. The method of claim 8 further comprising: determining, with the control device, the value of the manipulated variable u based on a function: u={square root over (|s|)}sign(s)+{dot over (u)} where {dot over (u)} is a derivative of u with respect to time that the control device determines based on a function: {dot over (u)}= $\{\begin{array}{cc}-\sqrt[3]{.Math.s.Math.}\mathrm{sign}\left(s\right)& \mathrm{for}.Math.u.Math.U_M\\ -u& \mathrm{for}.Math.u.Math.>U_M\end{array}$ where is an integral gain factor that is stored in the computer-readable storage medium and U.sub.M is a predetermined maximum value for the manipulated variable u that is stored in the computer-readable storage medium.

10. A control system for a hydraulic valve comprising: a control device connected to a displacement transducer in the hydraulic valve that identifies a position of a piston in the hydraulic valve, an electromagnet that is configured to apply a force to the piston to move the piston in the hydraulic valve, and a computer-readable storage medium, the control device being configured to: identify a plurality of positions of a piston over time for the piston in the hydraulic valve using the displacement transducer in the hydraulic valve, the plurality of positions including a present position of the piston and at least one prior position of the piston; determine a plurality of deviations over time between the plurality of positions of the piston and a predetermined position of the piston that produces a desired volumetric flow through the hydraulic valve, the plurality of deviations including a present deviation e between the present position and the predetermined position of the piston; determine a first derivative e corresponding to a rate of change of deviation of the piston position over time, and a second derivative e corresponding to an acceleration of deviation of the piston position over time based on the plurality of deviations over time; determine a switching function output s for control of the position of the piston based on e, , , a first pole value .sub.1 that the control device determines based on a function of e: .sub.1(e), and a second pole value .sub.2 that the control device determines based on a function of .sub.1, based on a switching function:
s(e,,)=.sub.1.sub.2e(.sub.1+.sub.2)+; and operate the electromagnet to move the piston from the present position toward the predetermined position based on the switching function output s.

11. The system of claim 10, the control device being further configured to: determine the first pole value .sub.1 based on the function of e: .sub.1(e)=|e|+.sub.0 where is a positive linear gradient that is stored in a computer-readable storage medium and .sub.0 is a value that is less than zero that is that is stored in the computer-readable storage medium.

12. The system of claim 11, the control device being further configured to: increase an absolute value of .sub.0 to provide faster dynamics for the switching function.

13. The system of claim 11, the control device being further configured to: decrease an absolute value of .sub.0 to provide slower dynamics for the switching function.

14. The system of claim 11, the control device being further configured to: determine the second pole value .sub.2 based on the function of .sub.1(e) and a predetermined proportional constant value c.sub.2 that is stored in the computer-readable storage medium based on a function: .sub.2=c.sub.2.sub.1(e)=c.sub.2(|e|+.sub.0).

15. The system of claim 11, the control device being further configured to: determine the second pole value .sub.2 based on the function of .sub.1(e) and a predetermined proportional constant value c.sub.2 that is stored in the computer-readable storage medium based on a function: .sub.2=c.sub.2.sub.1(e).

16. The system of claim 10, the control device being further configured to: determine the first pole value .sub.1 based on a square root function of e.

17. The system of claim 10 further comprising: determine a value of a manipulated variable u based on a function: u={square root over (|s|)}sign(s) where is a proportional gain factor that is stored in the computer-readable storage medium; and operate the electromagnet based on the value of u to move the piston from the present position toward the predetermined position.

18. The system of claim 17 further comprising: determine the value of the manipulated variable u based on a function: u={square root over (|s|)}sign(s)+{dot over (u)} where {dot over (u)} is a derivative of u with respect to time that the control device determines based on a function: $\stackrel{.}{u}=\{\begin{array}{cc}-\sqrt[3]{.Math.s.Math.}\mathrm{sign}\left(s\right)& \mathrm{for}.Math.u.Math.U_M\\ -u& \mathrm{for}.Math.u.Math.>U_M\end{array}$ where is an integral gain factor that is stored in the computer-readable storage medium and U.sub.M is a predetermined maximum value for the manipulated variable u that is stored in the computer-readable storage medium.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Exemplary embodiments of the disclosure are presented in the drawings an are explained in more detail in the description below.

(2) In the drawings:

(3) FIG. 1 schematically shows a hydraulic directional valve, for the control of which a sliding mode controller according to the disclosure can be used.

(4) FIG. 2 schematically shows a control loop having a sliding mode controller according to the disclosure in a preferred embodiment.

(5) FIG. 3 schematically shows a two-dimensional illustration of a switching function which is not according to the disclosure and a switching function according to the disclosure in a preferred embodiment for a sliding mode controller.

(6) FIG. 4 shows, in a graph, illustrations of a pole of a closed control loop in various preferred embodiments, as can be used for a switching function according to the disclosure.

(7) FIG. 5 schematically shows a possible sequence of a method according to the disclosure in a preferred embodiment.

DETAILED DESCRIPTION

(8) FIG. 1 schematically shows, by way of example, a system 100 which is in the form of a hydraulic directional valve and for which a sliding mode controller according to the disclosure in a preferred embodiment can be used for control.

(9) The hydraulic directional valve 100 has a piston 110 which can be moved in a housing in order to connect pressure connections P for a pump, T for a tank and working connections A and B to one another in a suitable manner. A restoring force is applied to the piston 110 at one housing end by means of a spring 120 and a setting force is applied to the piston 110 at another housing end by means of an electromagnet 130. A further spring 121 acts against the spring 120 in order to keep the piston 110 at a zero position without magnetic force.

(10) A voltage can be applied to the electromagnet 130 in order to move the piston 110, depending on the value of the voltage. A displacement transducer 140 is also provided in order to detect a position and possibly a speed and an acceleration of the piston 110 and to forward this signal to a processing unit. In this respect, it is mentioned that the valve shown has, by way of example, only one electromagnet for controlling the piston. However, it is likewise conceivable for a valve having a plurality of electromagnets to be used.

(11) FIG. 2 shows a simple control scheme which can be used to control the position x of the piston 110 of the hydraulic directional valve 100 as a controlled variable, for example. A desired or reference value x.sub.ref for the position can initially be predefined. A control deviation e=xx.sub.ref is formed from an actual value x of the position which is fed back, and is supplied to the sliding mode controller 200. According to the procedure already mentioned above, the sliding mode controller uses the switching function s(e) to determine a value for the manipulated variable u which, in this case, is the voltage to be applied to the electromagnet 130. The position x of the piston 110 is then influenced using a controlled system 210. At this juncture, it is mentioned again that the exact influence of the manipulated variable via the controlled system is not relevant to a sliding mode controller.

(12) In addition to the desired value x.sub.ref, the actual value x and the control deviation e, their respective first and second time derivatives are also concomitantly included in the control, as was explained in detail above. FIG. 2 illustrates only the respective variables which are not derived only for the sake of clarity.

(13) FIG. 3 now shows two switching levels s.sub.1=0 and s.sub.2=0 of two switching functions s.sub.1 and s.sub.2 in a graph. In this case, the first time derivative of the control deviation is plotted against the control deviation e. In this respect, it is noted that only switching levels of first-order switching functions are shown, by way of example, owing to the limited and simpler representability. Strictly speaking, the switching levels are therefore only switching lines.

(14) The switching line s.sub.1=0 belongs to a linear switching function of the form s.sub.1(e, )=r.sub.0e+ or s.sub.1(e,)=.sub.1e with a constant .sub.1. In contrast, the switching line s.sub.2=0 belongs to a non-linear switching function of the form s.sub.2(e,)=.sub.1 with .sub.1=.sub.1(e). Suitably selecting .sub.1 as a function of e therefore makes it possible to achieve a desired curvature of the switching lines, which is indicated only by way of example in FIG. 3. In this respect, it is also noted that such non-linear switching functions can accordingly also be formed for higher orders.

(15) FIG. 4 shows various possible embodiments for .sub.1=.sub.1(e) in a graph. .sub.1,1 is a linear function of the form .sub.1(e)=|e|+.sub.0, where is a gradient and .sub.0 is an associated ordinate intercept, as already explained above. In this manner, a first pole with a smaller absolute value results for control deviations with larger absolute values and a first pole with a larger absolute value results for control deviations with smaller absolute values.

(16) It is therefore possible to take into account the dynamics mentioned at the outset and often desired in practice with very fast compensation for small deflections, but a slow reaction in the case of step changes in the reference variable.

(17) A quadratic and a square root dependence of the first pole are shown, by way of example, with .sub.1,2 and .sub.1,3. These are further possible ways of deliberately influencing the dynamics on the basis of the control deviation, as already explained above.

(18) FIG. 5 schematically shows a possible sequence of a method according to the disclosure in a preferred embodiment for determining a switching function for a sliding mode controller. This is a hardware-in-the-loop experiment in which the system having the controlled variable, the hydraulic directional valve 100 having the position x of the piston in the present case, is connected to a computer 500 and to a real-time system 510 in a suitable manner.

(19) The sliding mode controller 200, for example, is implemented on the real-time system 510 and a desired value x.sub.ref is predefined for said controller. In this case, the sliding mode controller 200 comprises a switching function with initial control dynamics for the hydraulic directional valve 100. A value for the manipulated variable, the voltage u in the present case, is therefore determined using the sliding mode controller 200 and is then set at the electromagnet of the hydraulic directional valve 100.

(20) An actual value x of the position of the piston is determined using the displacement transducer in the hydraulic directional valve 100 and is forwarded both to the real-time system 510 and to the computer 500. Whereas the actual value x in the real-time system is supplied to the sliding mode controller 200 for control, the position x of the piston is evaluated with respect to the dynamics on the computer 500, for example by means of a suitable program, in a step 520.

(21) The first pole .sub.1 of the switching level of the sliding mode controller 200, for example, is then adapted or shifted in a step 530. This makes it possible to quickly and easily find a suitable switching function for desired dynamics by adapting or shifting only a few parameters, for example only the first pole.