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 & 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, (1989). Google Scholar

[2]

S. Balea, A. Halanay, F. Ursu and I. Ursu, Geometric methods in control synthesis for electrohydraulic servoactuators in servoelastic framework,, in, (2009), 25. Google Scholar

[3]

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

[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. Google Scholar

[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. Google Scholar

[6]

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

[7]

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

[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. doi: 10.1080/00207179508921959. Google Scholar

[9]

M. Guillon, "L'asservissement Hydraulique et Electrohydraulique,", Paris, (1972). Google Scholar

[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. Google Scholar

[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. doi: 10.1023/B:ENGI.0000032810.53387.d9. Google Scholar

[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. doi: 10.1016/j.nonrwa.2007.12.009. Google Scholar

[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. doi: 10.1093/imamat/hxp019. Google Scholar

[14]

A. Halanay and I. Ursu, Stabilization of equilibria in switching models for electrohydraulic servoactuators in a servoelastic framework,, in, (2009), 25. Google Scholar

[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, (2009), 25. Google Scholar

[16]

A. Halanay and I. Ursu, Stability analysis of equilibria in a switching nonlinear model of a hydrostatic electrohydraulic actuator,, in, (2010). Google Scholar

[17]

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

[18]

A. Isidori, "Nonlinear Control Systems,", 2nd edition, (1995). Google Scholar

[19]

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

[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. doi: 10.1002/rnc.1462. Google Scholar

[21]

D. Liberzon, "Switching in Systems and Control,", Systems & Control: Foundations & Applications, (2003). doi: 10.1007/978-1-4612-0017-8. Google Scholar

[22]

I. G. Malkin, Theory of Stability of Motion,, (in Russian), (1966). Google Scholar

[23]

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

[24]

T. W. McLain and R. W. Beard, Nonlinear robust control of on electrohydraulic positioning system,, in, (1998). Google Scholar

[25]

H. E. Merritt, "Hydraulic Control Systems,", New York, (1976). doi: 10.1115/1.3601167. Google Scholar

[26]

H. Olsson, "Control Systems with Friction,", Ph.D Thesis, (1996). Google Scholar

[27]

V. Pastrakuljic, "Design and Modeling of a New Electrohydraulic Actuator,", MS Thesis, (1995). Google Scholar

[28]

E. Richard and R. Outbib, Feedback stabilization of an electrohydraulic system,, in, (1995), 330. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/05063516X. Google Scholar

[32]

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

[33]

I. Ursu, G. Tecuceanu, F. Ursu and A. Toader, Nonlinear control synthesis for hydrostatic type flight controls electrohydraulic actuators,, in, (2007), 13. Google Scholar

[34]

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

[35]

I. Ursu and A. Toader, A unitary approach on adaptive control synthesis,, in, (2010), 3. Google Scholar

[36]

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

[37]

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

[38]

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

[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. doi: 10.1016/S0005-1098(96)00222-1. Google Scholar

[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. doi: 10.1109/ACC.1999.783142. Google Scholar

[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. doi: 10.1080/002071700110037515. Google Scholar

show all references

References:
[1]

B. D. O. Anderson and J. B. Moore, "Optimal Control. Linear Quadratic Methods,", Prentice Hall, (1989). Google Scholar

[2]

S. Balea, A. Halanay, F. Ursu and I. Ursu, Geometric methods in control synthesis for electrohydraulic servoactuators in servoelastic framework,, in, (2009), 25. Google Scholar

[3]

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

[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. Google Scholar

[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. Google Scholar

[6]

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

[7]

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

[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. doi: 10.1080/00207179508921959. Google Scholar

[9]

M. Guillon, "L'asservissement Hydraulique et Electrohydraulique,", Paris, (1972). Google Scholar

[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. Google Scholar

[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. doi: 10.1023/B:ENGI.0000032810.53387.d9. Google Scholar

[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. doi: 10.1016/j.nonrwa.2007.12.009. Google Scholar

[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. doi: 10.1093/imamat/hxp019. Google Scholar

[14]

A. Halanay and I. Ursu, Stabilization of equilibria in switching models for electrohydraulic servoactuators in a servoelastic framework,, in, (2009), 25. Google Scholar

[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, (2009), 25. Google Scholar

[16]

A. Halanay and I. Ursu, Stability analysis of equilibria in a switching nonlinear model of a hydrostatic electrohydraulic actuator,, in, (2010). Google Scholar

[17]

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

[18]

A. Isidori, "Nonlinear Control Systems,", 2nd edition, (1995). Google Scholar

[19]

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

[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. doi: 10.1002/rnc.1462. Google Scholar

[21]

D. Liberzon, "Switching in Systems and Control,", Systems & Control: Foundations & Applications, (2003). doi: 10.1007/978-1-4612-0017-8. Google Scholar

[22]

I. G. Malkin, Theory of Stability of Motion,, (in Russian), (1966). Google Scholar

[23]

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

[24]

T. W. McLain and R. W. Beard, Nonlinear robust control of on electrohydraulic positioning system,, in, (1998). Google Scholar

[25]

H. E. Merritt, "Hydraulic Control Systems,", New York, (1976). doi: 10.1115/1.3601167. Google Scholar

[26]

H. Olsson, "Control Systems with Friction,", Ph.D Thesis, (1996). Google Scholar

[27]

V. Pastrakuljic, "Design and Modeling of a New Electrohydraulic Actuator,", MS Thesis, (1995). Google Scholar

[28]

E. Richard and R. Outbib, Feedback stabilization of an electrohydraulic system,, in, (1995), 330. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/05063516X. Google Scholar

[32]

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

[33]

I. Ursu, G. Tecuceanu, F. Ursu and A. Toader, Nonlinear control synthesis for hydrostatic type flight controls electrohydraulic actuators,, in, (2007), 13. Google Scholar

[34]

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

[35]

I. Ursu and A. Toader, A unitary approach on adaptive control synthesis,, in, (2010), 3. Google Scholar

[36]

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

[37]

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

[38]

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

[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. doi: 10.1016/S0005-1098(96)00222-1. Google Scholar

[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. doi: 10.1109/ACC.1999.783142. Google Scholar

[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. doi: 10.1080/002071700110037515. Google Scholar

[1]

Marc Chamberland, Victor H. Moll. Dynamics of the degree six Landen transformation. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 905-919. doi: 10.3934/dcds.2006.15.905

[2]

Ugo Boscain, Grégoire Charlot, Mario Sigalotti. Stability of planar nonlinear switched systems. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 415-432. doi: 10.3934/dcds.2006.15.415

[3]

Philippe Jouan, Said Naciri. Asymptotic stability of uniformly bounded nonlinear switched systems. Mathematical Control & Related Fields, 2013, 3 (3) : 323-345. doi: 10.3934/mcrf.2013.3.323

[4]

Moussa Balde, Ugo Boscain. Stability of planar switched systems: The nondiagonalizable case. Communications on Pure & Applied Analysis, 2008, 7 (1) : 1-21. doi: 10.3934/cpaa.2008.7.1

[5]

Frederic Laurent-Polz, James Montaldi, Mark Roberts. Point vortices on the sphere: Stability of symmetric relative equilibria. Journal of Geometric Mechanics, 2011, 3 (4) : 439-486. doi: 10.3934/jgm.2011.3.439

[6]

Canghua Jiang, Kok Lay Teo, Ryan Loxton, Guang-Ren Duan. A neighboring extremal solution for an optimal switched impulsive control problem. Journal of Industrial & Management Optimization, 2012, 8 (3) : 591-609. doi: 10.3934/jimo.2012.8.591

[7]

Honglei Xu, Kok Lay Teo, Weihua Gui. Necessary and sufficient conditions for stability of impulsive switched linear systems. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1185-1195. doi: 10.3934/dcdsb.2011.16.1185

[8]

Xiang Xie, Honglei Xu, Xinming Cheng, Yilun Yu. Improved results on exponential stability of discrete-time switched delay systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 199-208. doi: 10.3934/dcdsb.2017010

[9]

Jin Feng He, Wei Xu, Zhi Guo Feng, Xinsong Yang. On the global optimal solution for linear quadratic problems of switched system. Journal of Industrial & Management Optimization, 2019, 15 (2) : 817-832. doi: 10.3934/jimo.2018072

[10]

Carlos Arnoldo Morales, M. J. Pacifico. Lyapunov stability of $\omega$-limit sets. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 671-674. doi: 10.3934/dcds.2002.8.671

[11]

Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631

[12]

Jóhann Björnsson, Peter Giesl, Sigurdur F. Hafstein, Christopher M. Kellett. Computation of Lyapunov functions for systems with multiple local attractors. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4019-4039. doi: 10.3934/dcds.2015.35.4019

[13]

Saroj P. Pradhan, Janos Turi. Parameter dependent stability/instability in a human respiratory control system model. Conference Publications, 2013, 2013 (special) : 643-652. doi: 10.3934/proc.2013.2013.643

[14]

Olha P. Kupenko, Rosanna Manzo. Shape stability of optimal control problems in coefficients for coupled system of Hammerstein type. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2967-2992. doi: 10.3934/dcdsb.2015.20.2967

[15]

Zhaohua Gong, Chongyang Liu, Yujing Wang. Optimal control of switched systems with multiple time-delays and a cost on changing control. Journal of Industrial & Management Optimization, 2018, 14 (1) : 183-198. doi: 10.3934/jimo.2017042

[16]

Manuel González-Burgos, Sergio Guerrero, Jean Pierre Puel. Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation. Communications on Pure & Applied Analysis, 2009, 8 (1) : 311-333. doi: 10.3934/cpaa.2009.8.311

[17]

Nicolás Carreño. Local controllability of the $N$-dimensional Boussinesq system with $N-1$ scalar controls in an arbitrary control domain. Mathematical Control & Related Fields, 2012, 2 (4) : 361-382. doi: 10.3934/mcrf.2012.2.361

[18]

Lyudmila Grigoryeva, Juan-Pablo Ortega, Stanislav S. Zub. Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies. Journal of Geometric Mechanics, 2014, 6 (3) : 373-415. doi: 10.3934/jgm.2014.6.373

[19]

Denis Serre, Alexis F. Vasseur. The relative entropy method for the stability of intermediate shock waves; the rich case. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4569-4577. doi: 10.3934/dcds.2016.36.4569

[20]

Hongyan Yan, Yun Sun, Yuanguo Zhu. A linear-quadratic control problem of uncertain discrete-time switched systems. Journal of Industrial & Management Optimization, 2017, 13 (1) : 267-282. doi: 10.3934/jimo.2016016

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]