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Application of second order sliding mode algorithms for output feedback control in hydraulic cylinder drives with profound valve dynamics

Anwendung von Sliding-Mode-Algorithmen zweiter Ordnung für die Ausgabesteuerung in hydraulischen Zylinder-Antrieben mit tiefgreifender Ventildynamik

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Abstract

The application of second order sliding mode algorithms for output feedback control in hydraulic valve-cylinder drives appear attractive due to their simple realization and parametrization, and strong robustness toward bounded parameter variations and uncertainties. However, intrinsic nonlinear dynamic effects of hydraulic valves such as slew rate limitations and time delays arising in the electrical and mechanical amplification stages limits the applicability of such methods, and may lead to partial losses of robustness and limit cycles/oscillations in the outputs, internal states and the valve input signals. The application of some popular second order sliding mode controllers and their smooth counterparts are analyzed and experimentally verified. The controllers are considered for output feedback control and compared with a conventional PI control approach. The controllers under consideration are applied for position tracking control of a hydraulic valve-cylinder drive exhibiting strong variations in inertia- and gravitational loads, and furthermore suffer from profound valve dynamics. Results demonstrate that both the twisting- and super twisting algorithms may be successfully applied for this purpose, when continuous approximations of discontinuous are utilized, and furthermore that excellent performance may be achieved when applying their smooth counterparts directly.

Zusammenfassung

Die Anwendung von Sliding-Mode-Algorithmen zweiter Ordnung für die Ausgabesteuerung in hydraulischen Zylinder-Antrieben mit tiefgreifender Ventildynamik erweist sich aufgrund folgender Eigenschaften als attraktiv: einfache Realisierung und Parametrisierung und eine starke Robustheit gegenüber begrenzten Parametervariationen und Unsicherheiten. Allerdings begrenzen intrinsische nichtlineare dynamische Effekte von Hydraulikventilen wie Einschränkungen der Anstiegsgeschwindigkeit und Zeitverzögerungen in den elektrischen und mechanischen Verstärkungsstufen die Anwendbarkeit solcher Verfahren. Dies kann zu partiellen Verlusten an Robustheit führen und Grenzzyklen/Schwingungen in den Ausgaben, internen Stadien und die Ventileingangssignale begrenzen. Die Anwendung von einigen populären Sliding-Mode-Steuerungen zweiter Ordnung und deren ebenen Pendants sind analysiert und experimentell verifiziert. Die Regler sind für die Regelung der Ausgaben bestimmt und werden mit einem herkömmlichen PI-Regelungsansatz verglichen. Die zu betrachtenden Regler werden zur Kontrolle der Positionsverfolgung eines hydraulischen Ventil-Zylinder-Antriebs verwendet, der starke Schwankungen in Trägheits- und Gravitationslasten aufweist. Außerdem treten Probleme mit profunder Ventildynamik auf. Die Ergebnisse zeigen, dass sowohl die Twisting- als auch Super Twisting-Algorithmen erfolgreich für diesen Zweck eingesetzt werden können, falls Folgendes gewährleistet ist: eine kontinuierliche Annäherung von Diskontinuierlichem. Darüber hinaus besteht die Aussicht, eine ausgezeichnete Leistung zu erzielen, wenn deren ebene Pendants direkt Anwendung finden.

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Abbreviations

\(A_{A}\) :

Piston area on \(A\)-side [m2]

\(A_{B}\) :

Piston area on \(B\)-side [m2]

\(a_{d}\) :

Oscillation amplitude [–]

\(a_{\mathit{sr}}\) :

Valve slew rate limit [m/s]

\(B_{v}\) :

Viscous damping coefficient [N s/m]

\(C_{L}\) :

Leakage coefficient [m3/s/Pa]

\(C_{\mathit{Ll}}\) :

Leakage coefficient [m3/s/Pa]

\(C_{\mathit{Ls}}\) :

Leakage coefficient [m3/s/Pa]

\(e\) :

Position control error [m]

\(F_{\mathit{ad}}\) :

Load force [N]

\(F_{\mathit{ext}}\) :

External disturbance force [N]

\(F_{f}\) :

Friction forces [N]

\(F_{G}\) :

Force due to gravity [N]

\(K_{p}\) :

Pressure gain [Pa/s/m3]

\(K_{Q}\) :

Pres. dependent flow gain [m3/(s V)]

\(K_{vA}\) :

Flow gain of valve port \(A\) [\(\mbox{m}^{3}/(\mbox{s}\sqrt{\mbox{Pa}}\,\mbox{V})\)]

\(K_{vB}\) :

Flow gain of valve port \(B\) [\(\mbox{m}^{3}/(\mbox{s}\sqrt{\mbox{Pa}}\,\mbox{V})\)]

\(K_{v}\) :

Equivalent valve flow gain [\(\mbox{m}^{3}/(\mbox{s}\sqrt{\mbox{Pa}}\,\mbox{V})\)]

\(M_{\mathit{eq}}\) :

Equivalent inertia load [kg]

\(P_{A}\) :

Pressure in \(A\)-chamber [Pa]

\(P_{B}\) :

Pressure in \(B\)-chamber [Pa]

\(P_{L}\) :

Cylinder load pressure [Pa]

\(P_{S}\) :

Supply pressure [Pa]

\(P_{T}\) :

Tank pressure [Pa]

\(\Delta P_{\mathit{AB}}\) :

Pressure difference [Pa]

\(Q_{A}\) :

Flow through flow port \(A\) [\(\mbox{m}^{3}/\mbox{s}\)]

\(Q_{B}\) :

Flow through flow port \(B\) [\(\mbox{m}^{3}/\mbox{s}\)]

\(Q_{L}\) :

Cross port leakage flow [\(\mbox{m}^{3}/\mbox{s}\)]

\(u_{v}\) :

Valve control input [V]

\(v\) :

Control input [V]

\(V_{A0}\) :

Initial volume of \(A\)-chamber [m3]

\(V_{B0}\) :

Initial volume of \(B\)-chamber [m3]

\(\mathbf{x}\) :

State vector

\(x_{P}\) :

Piston position [m]

\(x_{R}\) :

Piston position reference [m]

\(x_{v}\) :

Valve spool position [m]

\(z, \bar{z}\) :

Auxiliary variables [m/s]

\(\alpha\) :

Control gain

\(\beta\) :

Control gain

\(\beta_{e}\) :

Eff. oil/hose bulk modulus [Pa]

\(\delta\) :

Control parameter

\(\sigma\) :

Valve flow gain ratio [–]

\(\mu\) :

Cylinder area ratio [–]

\(\omega_{v}\) :

Valve bandwidth [rad/s]

\(\zeta_{v}\) :

Valve damping ratio [–]

\(\varrho\) :

Cylinder volume ratio [–]

\(\tau_{d}\) :

Oscillation period [s]

\(\tau_{p}\) :

Time constant of pres. dyn. [s]

\(\xi\) :

Model disturbance [m/s]

\(\varXi\) :

Parameter bound [m/s]

\(\bar{\varXi}\) :

Parameter bound [m/s2]

\(\Delta t\) :

Time delay [s]

\(\phi\) :

Homogeneity degree

\(\varLambda\) :

Inv. hydraulic capacitance [Pa/m3]

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Schmidt, L., Andersen, T.O. Application of second order sliding mode algorithms for output feedback control in hydraulic cylinder drives with profound valve dynamics. Elektrotech. Inftech. 133, 238–247 (2016). https://doi.org/10.1007/s00502-016-0425-7

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