Conventional and Added-Order Proportional Nonlinear Integral Observers
ABSTRACT
In this paper, a kind of fire new nonlinear integrator and integral action is proposed. Consequently, a conventional Proportional Nonlinear Integral (P_NI) observer and two kinds of added-order P_NI observers are developed to deal with the uncertain nonlinear system. The conditions on the observer gains to ensure the estimated error to be ultimate boundness, which shrinks to zero as the states and control inputs converge to the equilibrium point, are provided. This means that if the observed system is asymptotically stable, the estimated error dynamics is asymptotically stable, too. Moreover, the highlight point of this paper is that the design of nonlinear integral observer is achieved by linear system theory. Simulation results showed that under the normal and perturbed cases, the pure added-order P_NI observer can effectively deal with the uncertain nonlinearities on both the system dynamics and measured outputs.
In this paper, a kind of fire new nonlinear integrator and integral action is proposed. Consequently, a conventional Proportional Nonlinear Integral (P_NI) observer and two kinds of added-order P_NI observers are developed to deal with the uncertain nonlinear system. The conditions on the observer gains to ensure the estimated error to be ultimate boundness, which shrinks to zero as the states and control inputs converge to the equilibrium point, are provided. This means that if the observed system is asymptotically stable, the estimated error dynamics is asymptotically stable, too. Moreover, the highlight point of this paper is that the design of nonlinear integral observer is achieved by linear system theory. Simulation results showed that under the normal and perturbed cases, the pure added-order P_NI observer can effectively deal with the uncertain nonlinearities on both the system dynamics and measured outputs.
KEYWORDS
State Estimation, PI Observer, Nonlinear Integral Observer, Added-Order Observer, Nonlinear Integrator, Nonlinear Integral Action, Model-Based Observer
State Estimation, PI Observer, Nonlinear Integral Observer, Added-Order Observer, Nonlinear Integrator, Nonlinear Integral Action, Model-Based Observer
Cite this paper
Liu, B. , Luo, X. and Li, J. (2014) Conventional and Added-Order Proportional Nonlinear Integral Observers. International Journal of Modern Nonlinear Theory and Application, 3, 210-220. doi: 10.4236/ijmnta.2014.35023.
Liu, B. , Luo, X. and Li, J. (2014) Conventional and Added-Order Proportional Nonlinear Integral Observers. International Journal of Modern Nonlinear Theory and Application, 3, 210-220. doi: 10.4236/ijmnta.2014.35023.
References
[1] Luenberger, D.G. (1971) An Introduction to Observers. IEEE Transactions on Automatic Control, 16, 596-620.
http://dx.doi.org/10.1109/TAC.1971.1099826 [2] Wojciechowski, B. (1978) Analysis and Synthesis of Proportional Integral Observers for Single Input Single Output Time-Invariant Continuous Systems. Ph.D. Thesis, Gliwice. [3] Kaczorek, T. (1979) Proportional Integral Observers for Linear Multivariable Time-Varying Systems. Regelungstechnik, 27, 359-362. [4] Jiang, G.P., Wang, S.P. and Song, W.Z. (2000) Design of Observer with Integrators for Linear Systems with Unknown Input Disturbances. Electronics Letters, 36, 1168-1169.
http://dx.doi.org/10.1049/el:20000799 [5] Busawon, K.K. and Kabore, P. (2001) Disturbance Attenuation Using Proportional Integral Observers. International Journal of Control, 74, 618-627. http://dx.doi.org/10.1080/00207170010025249 [6] Koenig, D. and Mammar, S. (2002) Design of Proportional Integral Observer for Unknown Input Descriptor Systems. IEEE Transactions on Automatic Control, 47, 2057-2062.
http://dx.doi.org/10.1109/TAC.2002.805675 [7] Cao, Z. and Ho, D.W.C. (2004) Proportional Multiple-Integral Observer Design for Descriptor Systems with Measurement Output Disturbances. IEE Proceedings—Control Theory and Applications, 151, 279-288. [8] Ha, Q.P. and Trinh, H. (2004) State and Input Simultaneous Estimation for a Class of Nonlinear Systems. Automatica, 40, 1779-1785. http://dx.doi.org/10.1016/j.automatica.2004.05.012 [9] Saif, M. (1993) Reduced-Order Proportional Integral Observer with Application. Journal of Guidance Control and Dynamics, 16, 985-988. http://dx.doi.org/10.2514/3.21116 [10] Sharifuddin, M., Goutam, C. and Kingshook, B. (2010) LMI Approach to Robust Unknown Input Observer Design for Continuous Systems with Noise and Uncertainties. International Journal of Control, Automation, and Systems, 8, 210-219. http://dx.doi.org/10.1007/s12555-010-0205-9 [11] Kamel, M., Chadli, M. and Chaabane, M. (2012) Unknown Inputs Observer for a Class of Nonlinear Uncertain Systems: An LMI Approach. International Journal of Automation and Computing, 9, 331-336.
http://dx.doi.org/10.1007/s11633-012-0652-2 [12] Liu, B.S. and Tian, B.L. (2012) General Integral Control Design Based on Linear System Theory. Proceedings of the 3rd International Conference on Mechanic Automation and Control Engineering, Baotou, 5, 3174-3177. [13] Liu, B.S., Li, J.H. and Luo, X.Q. (2014) General Integral Control Design via Feedback Linearization. Intelligent Control and Automation, 5, 19-23. http://dx.doi.org/10.4236/ica.2014.51003 [14] Liu, B.S. and Tian, B.L. (2012) General Integral Control Design Based on Sliding Mode Technique. Proceedings of the 3rd International Conference on Mechanic Automation and Control Engineering, Baotou, 5, 3178-3181. [15] Liu, B.S., Luo, X.Q. and Li, J.H. (2013) General Concave Integral Control. Intelligent Control and Automation, 4, 356-361. http://dx.doi.org/10.4236/ica.2013.44042 [16] Liu, B.S., Luo, X.Q. and Li, J.H. (2014) General Convex Integral Control. International Journal of Automation and Computing, 11, 565-570. [17] Liu, B.S. (2014) Constructive General Bounded Integral Control. Intelligent Control and Automation, 5, 146-155.
http://dx.doi.org/10.4236/ica.2014.53017 [18] Liu, B.S. (2014) On the Generalization of Integrator and Integral Control Action. International Journal of Modern Nonlinear Theory and Application, 3, 44-52. http://dx.doi.org/10.4236/ijmnta.2014.32007 [19] Khalil, H.K. (2007) Nonlinear Systems. 3rd Edition, Electronics Industry Publishing, Beijing, 5-6.
[1] Luenberger, D.G. (1971) An Introduction to Observers. IEEE Transactions on Automatic Control, 16, 596-620.
http://dx.doi.org/10.1109/TAC.1971.1099826 [2] Wojciechowski, B. (1978) Analysis and Synthesis of Proportional Integral Observers for Single Input Single Output Time-Invariant Continuous Systems. Ph.D. Thesis, Gliwice. [3] Kaczorek, T. (1979) Proportional Integral Observers for Linear Multivariable Time-Varying Systems. Regelungstechnik, 27, 359-362. [4] Jiang, G.P., Wang, S.P. and Song, W.Z. (2000) Design of Observer with Integrators for Linear Systems with Unknown Input Disturbances. Electronics Letters, 36, 1168-1169.
http://dx.doi.org/10.1049/el:20000799 [5] Busawon, K.K. and Kabore, P. (2001) Disturbance Attenuation Using Proportional Integral Observers. International Journal of Control, 74, 618-627. http://dx.doi.org/10.1080/00207170010025249 [6] Koenig, D. and Mammar, S. (2002) Design of Proportional Integral Observer for Unknown Input Descriptor Systems. IEEE Transactions on Automatic Control, 47, 2057-2062.
http://dx.doi.org/10.1109/TAC.2002.805675 [7] Cao, Z. and Ho, D.W.C. (2004) Proportional Multiple-Integral Observer Design for Descriptor Systems with Measurement Output Disturbances. IEE Proceedings—Control Theory and Applications, 151, 279-288. [8] Ha, Q.P. and Trinh, H. (2004) State and Input Simultaneous Estimation for a Class of Nonlinear Systems. Automatica, 40, 1779-1785. http://dx.doi.org/10.1016/j.automatica.2004.05.012 [9] Saif, M. (1993) Reduced-Order Proportional Integral Observer with Application. Journal of Guidance Control and Dynamics, 16, 985-988. http://dx.doi.org/10.2514/3.21116 [10] Sharifuddin, M., Goutam, C. and Kingshook, B. (2010) LMI Approach to Robust Unknown Input Observer Design for Continuous Systems with Noise and Uncertainties. International Journal of Control, Automation, and Systems, 8, 210-219. http://dx.doi.org/10.1007/s12555-010-0205-9 [11] Kamel, M., Chadli, M. and Chaabane, M. (2012) Unknown Inputs Observer for a Class of Nonlinear Uncertain Systems: An LMI Approach. International Journal of Automation and Computing, 9, 331-336.
http://dx.doi.org/10.1007/s11633-012-0652-2 [12] Liu, B.S. and Tian, B.L. (2012) General Integral Control Design Based on Linear System Theory. Proceedings of the 3rd International Conference on Mechanic Automation and Control Engineering, Baotou, 5, 3174-3177. [13] Liu, B.S., Li, J.H. and Luo, X.Q. (2014) General Integral Control Design via Feedback Linearization. Intelligent Control and Automation, 5, 19-23. http://dx.doi.org/10.4236/ica.2014.51003 [14] Liu, B.S. and Tian, B.L. (2012) General Integral Control Design Based on Sliding Mode Technique. Proceedings of the 3rd International Conference on Mechanic Automation and Control Engineering, Baotou, 5, 3178-3181. [15] Liu, B.S., Luo, X.Q. and Li, J.H. (2013) General Concave Integral Control. Intelligent Control and Automation, 4, 356-361. http://dx.doi.org/10.4236/ica.2013.44042 [16] Liu, B.S., Luo, X.Q. and Li, J.H. (2014) General Convex Integral Control. International Journal of Automation and Computing, 11, 565-570. [17] Liu, B.S. (2014) Constructive General Bounded Integral Control. Intelligent Control and Automation, 5, 146-155.
http://dx.doi.org/10.4236/ica.2014.53017 [18] Liu, B.S. (2014) On the Generalization of Integrator and Integral Control Action. International Journal of Modern Nonlinear Theory and Application, 3, 44-52. http://dx.doi.org/10.4236/ijmnta.2014.32007 [19] Khalil, H.K. (2007) Nonlinear Systems. 3rd Edition, Electronics Industry Publishing, Beijing, 5-6.