December  2013, 6(6): 1551-1567. doi: 10.3934/dcdss.2013.6.1551

New results on the problem of the stabilization of equilibria for models of electrohydraulic servoactuators

1. 

Department of Mathematics 2, University Politehnica of Bucharest, 313 Splaiul Independenţei, RO-060042, Bucharest, Romania

2. 

Department of Mathematics 1, University Politehnica of Bucharest, 313 Splaiul Independenţei, RO-060042, Bucharest, Romania, Romania

Received  June 2012 Revised  September 2012 Published  April 2013

Control synthesis for electrohydraulic servoactuators (EHSA) is achieved using elements of geometric control theory. Based on a Malkin type theorem for switched systems of ordinary differential equations, the existence of stabilizing feedback controllers is prove to hold in the specific case of EHSAs when the relative degree of the nonlinear control system is one unit less than the order of the system. The proof relies on coordinate transformations that bring the system to some canonical form.
Citation: Silvia Balea, Andrei Halanay, Ioan Ursu. New results on the problem of the stabilization of equilibria for models of electrohydraulic servoactuators. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1551-1567. doi: 10.3934/dcdss.2013.6.1551
References:
[1]

B. D. O. Anderson and J. B. Moore, "Optimal Control. Linear Quadratic Methods," Prentice Hall, Inc., 1989.

[2]

S. Balea, A. Halanay, F. Ursu and I. Ursu, Geometric methods in control synthesis for electrohydraulic servoactuators in servoelastic framework, in "7th International Conference on Mathematical Problems in Engineering and Aerospace Sciences" (ed. Siva Sundaram) (June 25-27, 2008, Genoa, Italy), Cambridge Scientific Publishers, (2009), 51-57.

[3]

S. Balea, A. Halanay and I. Ursu, Stabilization through coordinates transformation for switching systems associated to electrohydraulic servomechanisms, Mathematical Reports, 11 (2009), 279-292.

[4]

S. Balea, A. Halanay and I. Ursu, Coordinates transformation and stabilization for switching models of actuators in servoelastic framework, Applied Mathematical Sciences, 4 (2010), 3625-3642.

[5]

S. Balea, A. Halanay and I. Ursu, Coordinate transformations and stabilization of some switched control systems with application to hydrostatic electrohydraulic servoactuators, Control Engineering and Applied Informatics, 12 (2010), 67-72.

[6]

J. F. Blackburn, G. Reethof and J. L. Shearer, "Fluid Power Control," Technology Press of MIT, 1960.

[7]

L. Dinca, J. Corcau, M. Lungu and A. Tudosie, Mathematical models and numerical simulations for electro-hydrostatic servo-actuators, International Journal of Circuits, Systems and Signal Processing, 2 (2008), 229-238.

[8]

M. Fliess, J. Lévine, P. Martin and P. Rouchon, Flatness and defect of non-linear systems: Introductory theory and examples, International Journal of Control, 61 (1995), 1327-1361. doi: 10.1080/00207179508921959.

[9]

M. Guillon, "L'asservissement Hydraulique et Electrohydraulique," Paris, Dunod, I-II, 1972.

[10]

S. R. Habibi and G. Singh, Derivation of design requirements for optimization of a high performance hydrostatic actuation system, International Journal of Fluid Power, 1 (2000), 11-27.

[11]

A. Halanay, C. A. Safta, I. Ursu and F. Ursu, Stability of equilibria in a four-dimensional nonlinear model of a hydraulic servomechanism, Journal of Engineering Mathematics, 49 (2004), 391-406. doi: 10.1023/B:ENGI.0000032810.53387.d9.

[12]

A. Halanay, C. A. Safta, F. Ursu and I. Ursu, Stability analysis for a nonlinear model of a hydraulic servomechanism in a servoelastic framework, Nonlinear Analysis: Real World Applications, 10 (2009), 1197-1209. doi: 10.1016/j.nonrwa.2007.12.009.

[13]

A. Halanay and I. Ursu, Stability of equilibria of some switched nonlinear systems with applications to control synthesis for electrohydraulic servomechanisms, IMA Journal of Applied Mathematics, 74 (2009), 361-373. doi: 10.1093/imamat/hxp019.

[14]

A. Halanay and I. Ursu, Stabilization of equilibria in switching models for electrohydraulic servoactuators in a servoelastic framework, in "7th International Conference on Mathematical Problems in Engineering and Aerospace Sciences" (ed. Siva Sundaram) (June 25-27, 2008, Genoa, Italy), Cambridge Scientific Publishers, (2009), 73-80.

[15]

A. Halanay, I. Ursu, C. A. Safta and F. Ursu, Control synthesis for electrohydraulic servoactuators in a servoelastic framework, 7th International Conference on Mathematical Problems in Engineering and Aerospace Sciences (June 25-27, 2008, Genoa, Italy), 716-723, Cambridge Scientific Publishers, Ed. Siva Sundaram, (2009).

[16]

A. Halanay and I. Ursu, Stability analysis of equilibria in a switching nonlinear model of a hydrostatic electrohydraulic actuator, in "Mathematical Problems in Engineering Aerospace and Science" (ed. S. Sivasundaram), Vol. 5, Cambridge Scientific Publishers, 2010.

[17]

R. Hermann and A. J. Krener, Nonlinear controllability and observability, IEEE Transactions on Automatic Control, AC-22 (1977), 728-740.

[18]

A. Isidori, "Nonlinear Control Systems," 2nd edition, Springer, 1995.

[19]

M. Jelali and A. Kroll, "Hydraulic Servo-Systems," Advances in Industrial Control, Springer, London, 2003. doi: 10.1007/978-1-4471-0099-7.

[20]

W. Kemmetmüller and A. Kugi, Immersion and invariance-based impedance control for electrohydraulic systems, International Journal of Robust and Nonlinear Control, 20 (2010), 725-744. doi: 10.1002/rnc.1462.

[21]

D. Liberzon, "Switching in Systems and Control," Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-1-4612-0017-8.

[22]

I. G. Malkin, Theory of Stability of Motion, (in Russian), Second revised edition, Izdat. "Nauka", Moscow, 1966; English translation in Atomic Energy Comm., Translation, AEC-TR-3352, 1966.

[23]

M. Margaliot, Stability analysis of switched systems using variational principles: An introduction, Automatica J. IFAC, 42 (2006), 2059-2077. doi: 10.1016/j.automatica.2006.06.020.

[24]

T. W. McLain and R. W. Beard, Nonlinear robust control of on electrohydraulic positioning system, in "Proceedings of the International Mechanical Engineering Congress and Exposition" (IMECE' 98), FPST Division, November, Anaheim, California, 1998.

[25]

H. E. Merritt, "Hydraulic Control Systems," New York, John Wiley & Sons, 1976. doi: 10.1115/1.3601167.

[26]

H. Olsson, "Control Systems with Friction," Ph.D Thesis, Lund Institute of Technology, Lund., 1996.

[27]

V. Pastrakuljic, "Design and Modeling of a New Electrohydraulic Actuator," MS Thesis, University of Toronto, 1995.

[28]

E. Richard and R. Outbib, Feedback stabilization of an electrohydraulic system, in "Proceedings of 3rd European Control Conference," Rome, Italy, September, (1995), 330-334.

[29]

S. Sampson, S. R. Habibi, R. Burton and Y. Chinniah, Effect of controller in reducing steady-state error due to flow and force disturbances in the electrohydraulic actuator system, International Journal of Fluid Power, (2004), 57-66.

[30]

J.-K. Shiau and D.-M. Ma, An autopilot design for the longitudinal dynamics of a low-speed experimental uav using two-time-scale cascade decomposition, Transactions of the Canadian Society for Mechanical Engineering, 33 (2009), 501-521.

[31]

R. N. Shorten, F. Wirth, O. Mason, K. Wulff and C. King, Stability criteria for switched and hybrid systems, SIAM Review, 49 (2007), 545-592. doi: 10.1137/05063516X.

[32]

I. Ursu, F. Ursu and L. Iorga, Neuro-fuzzy synthesis of flight controls electrohydraulic servo, Aircraft Engineering and Aerospace Technology, 73 (2001), 465-472. doi: 10.1108/00022660110403014.

[33]

I. Ursu, G. Tecuceanu, F. Ursu and A. Toader, Nonlinear control synthesis for hydrostatic type flight controls electrohydraulic actuators, in "Proceedings of the International Conference in Aerospace Actuation Systems and Components," Toulouse, June 13-15, (2007), 189-194.

[34]

I. Ursu, F. Ursu and F. Popescu, Backstepping design for controlling electrohydraulic servos, Journal of The Franklin Institute, 343 (2006), 94-110. doi: 10.1016/j.jfranklin.2005.09.003.

[35]

I. Ursu and A. Toader, A unitary approach on adaptive control synthesis, in "Mathematical Methods, Computational Techniques and Intelligent Systems," (12th WSEAS Int. Conf. on Mathematical Methods, Computational Techniques and Intelligent Systems MAMECTIS '10, Kantaoui, Sousse, Tunisia, May 3-6, 2010), Mathematics and Computers in Science and Engineering, A Series of Reference Books and Textbooks, Published by WSEAS Press, (2010), 71-78.

[36]

L. Vu and D. Liberzon, Common Lyapunov functions for families of commuting nonlinear systems, Systems and Control Letters, 54 (2005), 405-416. doi: 10.1016/j.sysconle.2004.09.006.

[37]

S. Waldherr and M. Zeitz, Conditions for the existence of a flat input, International Journal of Control, 81 (2008), 437-441. doi: 10.1080/00207170701561443.

[38]

P. K. C. Wang, Analytical design of electrohydraulic servomechanisms with near time-optimal response, IEEE Trans. Autom Control, 8 (1963), 15-27. doi: 10.1109/TAC.1963.1105512.

[39]

B. Yao and M. Tomizuka, Adaptive robust control of SISO nonlinear systems in a semi-strict feedback form, Automatica J. IFAC, 33 (1997), 893-900. doi: 10.1016/S0005-1098(96)00222-1.

[40]

B. Yao, J. T. Reedy and G. T.-C. Chiu, Adaptive robust motion control of single rod hydraulic actuators: Theory and experiments, IEEE/ASME Transactios on Mechatronics, 5 (2000), 79-91. doi: 10.1109/ACC.1999.783142.

[41]

B. Yao, F. Bu and G. T. Chiu, Non-linear adaptive robust control of electro-hydraulic systems driven by double-rod actuators, International Journal of Control, 74 (2001), 761-775. doi: 10.1080/002071700110037515.

show all references

References:
[1]

B. D. O. Anderson and J. B. Moore, "Optimal Control. Linear Quadratic Methods," Prentice Hall, Inc., 1989.

[2]

S. Balea, A. Halanay, F. Ursu and I. Ursu, Geometric methods in control synthesis for electrohydraulic servoactuators in servoelastic framework, in "7th International Conference on Mathematical Problems in Engineering and Aerospace Sciences" (ed. Siva Sundaram) (June 25-27, 2008, Genoa, Italy), Cambridge Scientific Publishers, (2009), 51-57.

[3]

S. Balea, A. Halanay and I. Ursu, Stabilization through coordinates transformation for switching systems associated to electrohydraulic servomechanisms, Mathematical Reports, 11 (2009), 279-292.

[4]

S. Balea, A. Halanay and I. Ursu, Coordinates transformation and stabilization for switching models of actuators in servoelastic framework, Applied Mathematical Sciences, 4 (2010), 3625-3642.

[5]

S. Balea, A. Halanay and I. Ursu, Coordinate transformations and stabilization of some switched control systems with application to hydrostatic electrohydraulic servoactuators, Control Engineering and Applied Informatics, 12 (2010), 67-72.

[6]

J. F. Blackburn, G. Reethof and J. L. Shearer, "Fluid Power Control," Technology Press of MIT, 1960.

[7]

L. Dinca, J. Corcau, M. Lungu and A. Tudosie, Mathematical models and numerical simulations for electro-hydrostatic servo-actuators, International Journal of Circuits, Systems and Signal Processing, 2 (2008), 229-238.

[8]

M. Fliess, J. Lévine, P. Martin and P. Rouchon, Flatness and defect of non-linear systems: Introductory theory and examples, International Journal of Control, 61 (1995), 1327-1361. doi: 10.1080/00207179508921959.

[9]

M. Guillon, "L'asservissement Hydraulique et Electrohydraulique," Paris, Dunod, I-II, 1972.

[10]

S. R. Habibi and G. Singh, Derivation of design requirements for optimization of a high performance hydrostatic actuation system, International Journal of Fluid Power, 1 (2000), 11-27.

[11]

A. Halanay, C. A. Safta, I. Ursu and F. Ursu, Stability of equilibria in a four-dimensional nonlinear model of a hydraulic servomechanism, Journal of Engineering Mathematics, 49 (2004), 391-406. doi: 10.1023/B:ENGI.0000032810.53387.d9.

[12]

A. Halanay, C. A. Safta, F. Ursu and I. Ursu, Stability analysis for a nonlinear model of a hydraulic servomechanism in a servoelastic framework, Nonlinear Analysis: Real World Applications, 10 (2009), 1197-1209. doi: 10.1016/j.nonrwa.2007.12.009.

[13]

A. Halanay and I. Ursu, Stability of equilibria of some switched nonlinear systems with applications to control synthesis for electrohydraulic servomechanisms, IMA Journal of Applied Mathematics, 74 (2009), 361-373. doi: 10.1093/imamat/hxp019.

[14]

A. Halanay and I. Ursu, Stabilization of equilibria in switching models for electrohydraulic servoactuators in a servoelastic framework, in "7th International Conference on Mathematical Problems in Engineering and Aerospace Sciences" (ed. Siva Sundaram) (June 25-27, 2008, Genoa, Italy), Cambridge Scientific Publishers, (2009), 73-80.

[15]

A. Halanay, I. Ursu, C. A. Safta and F. Ursu, Control synthesis for electrohydraulic servoactuators in a servoelastic framework, 7th International Conference on Mathematical Problems in Engineering and Aerospace Sciences (June 25-27, 2008, Genoa, Italy), 716-723, Cambridge Scientific Publishers, Ed. Siva Sundaram, (2009).

[16]

A. Halanay and I. Ursu, Stability analysis of equilibria in a switching nonlinear model of a hydrostatic electrohydraulic actuator, in "Mathematical Problems in Engineering Aerospace and Science" (ed. S. Sivasundaram), Vol. 5, Cambridge Scientific Publishers, 2010.

[17]

R. Hermann and A. J. Krener, Nonlinear controllability and observability, IEEE Transactions on Automatic Control, AC-22 (1977), 728-740.

[18]

A. Isidori, "Nonlinear Control Systems," 2nd edition, Springer, 1995.

[19]

M. Jelali and A. Kroll, "Hydraulic Servo-Systems," Advances in Industrial Control, Springer, London, 2003. doi: 10.1007/978-1-4471-0099-7.

[20]

W. Kemmetmüller and A. Kugi, Immersion and invariance-based impedance control for electrohydraulic systems, International Journal of Robust and Nonlinear Control, 20 (2010), 725-744. doi: 10.1002/rnc.1462.

[21]

D. Liberzon, "Switching in Systems and Control," Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-1-4612-0017-8.

[22]

I. G. Malkin, Theory of Stability of Motion, (in Russian), Second revised edition, Izdat. "Nauka", Moscow, 1966; English translation in Atomic Energy Comm., Translation, AEC-TR-3352, 1966.

[23]

M. Margaliot, Stability analysis of switched systems using variational principles: An introduction, Automatica J. IFAC, 42 (2006), 2059-2077. doi: 10.1016/j.automatica.2006.06.020.

[24]

T. W. McLain and R. W. Beard, Nonlinear robust control of on electrohydraulic positioning system, in "Proceedings of the International Mechanical Engineering Congress and Exposition" (IMECE' 98), FPST Division, November, Anaheim, California, 1998.

[25]

H. E. Merritt, "Hydraulic Control Systems," New York, John Wiley & Sons, 1976. doi: 10.1115/1.3601167.

[26]

H. Olsson, "Control Systems with Friction," Ph.D Thesis, Lund Institute of Technology, Lund., 1996.

[27]

V. Pastrakuljic, "Design and Modeling of a New Electrohydraulic Actuator," MS Thesis, University of Toronto, 1995.

[28]

E. Richard and R. Outbib, Feedback stabilization of an electrohydraulic system, in "Proceedings of 3rd European Control Conference," Rome, Italy, September, (1995), 330-334.

[29]

S. Sampson, S. R. Habibi, R. Burton and Y. Chinniah, Effect of controller in reducing steady-state error due to flow and force disturbances in the electrohydraulic actuator system, International Journal of Fluid Power, (2004), 57-66.

[30]

J.-K. Shiau and D.-M. Ma, An autopilot design for the longitudinal dynamics of a low-speed experimental uav using two-time-scale cascade decomposition, Transactions of the Canadian Society for Mechanical Engineering, 33 (2009), 501-521.

[31]

R. N. Shorten, F. Wirth, O. Mason, K. Wulff and C. King, Stability criteria for switched and hybrid systems, SIAM Review, 49 (2007), 545-592. doi: 10.1137/05063516X.

[32]

I. Ursu, F. Ursu and L. Iorga, Neuro-fuzzy synthesis of flight controls electrohydraulic servo, Aircraft Engineering and Aerospace Technology, 73 (2001), 465-472. doi: 10.1108/00022660110403014.

[33]

I. Ursu, G. Tecuceanu, F. Ursu and A. Toader, Nonlinear control synthesis for hydrostatic type flight controls electrohydraulic actuators, in "Proceedings of the International Conference in Aerospace Actuation Systems and Components," Toulouse, June 13-15, (2007), 189-194.

[34]

I. Ursu, F. Ursu and F. Popescu, Backstepping design for controlling electrohydraulic servos, Journal of The Franklin Institute, 343 (2006), 94-110. doi: 10.1016/j.jfranklin.2005.09.003.

[35]

I. Ursu and A. Toader, A unitary approach on adaptive control synthesis, in "Mathematical Methods, Computational Techniques and Intelligent Systems," (12th WSEAS Int. Conf. on Mathematical Methods, Computational Techniques and Intelligent Systems MAMECTIS '10, Kantaoui, Sousse, Tunisia, May 3-6, 2010), Mathematics and Computers in Science and Engineering, A Series of Reference Books and Textbooks, Published by WSEAS Press, (2010), 71-78.

[36]

L. Vu and D. Liberzon, Common Lyapunov functions for families of commuting nonlinear systems, Systems and Control Letters, 54 (2005), 405-416. doi: 10.1016/j.sysconle.2004.09.006.

[37]

S. Waldherr and M. Zeitz, Conditions for the existence of a flat input, International Journal of Control, 81 (2008), 437-441. doi: 10.1080/00207170701561443.

[38]

P. K. C. Wang, Analytical design of electrohydraulic servomechanisms with near time-optimal response, IEEE Trans. Autom Control, 8 (1963), 15-27. doi: 10.1109/TAC.1963.1105512.

[39]

B. Yao and M. Tomizuka, Adaptive robust control of SISO nonlinear systems in a semi-strict feedback form, Automatica J. IFAC, 33 (1997), 893-900. doi: 10.1016/S0005-1098(96)00222-1.

[40]

B. Yao, J. T. Reedy and G. T.-C. Chiu, Adaptive robust motion control of single rod hydraulic actuators: Theory and experiments, IEEE/ASME Transactios on Mechatronics, 5 (2000), 79-91. doi: 10.1109/ACC.1999.783142.

[41]

B. Yao, F. Bu and G. T. Chiu, Non-linear adaptive robust control of electro-hydraulic systems driven by double-rod actuators, International Journal of Control, 74 (2001), 761-775. doi: 10.1080/002071700110037515.

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