Sliding mode control and active disturbance rejection control to the stabilization of one-dimensional Schrödinger equation subject to boundary control matched disturbance
SUMMARYIn this paper, we are concerned with the boundary stabilization of a one‐dimensional anti‐stable Schrödinger equation subject to boundary control matched disturbance. We apply both the sliding mode control (SMC) and the active disturbance rejection control (ADRC) to deal with the disturbance....
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| Published in | International journal of robust and nonlinear control Vol. 24; no. 16; pp. 2194 - 2212 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Bognor Regis
Blackwell Publishing Ltd
10.11.2014
Wiley Subscription Services, Inc |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1049-8923 1099-1239 |
| DOI | 10.1002/rnc.2977 |
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| Abstract | SUMMARYIn this paper, we are concerned with the boundary stabilization of a one‐dimensional anti‐stable Schrödinger equation subject to boundary control matched disturbance. We apply both the sliding mode control (SMC) and the active disturbance rejection control (ADRC) to deal with the disturbance. By the SMC approach, the disturbance is supposed to be bounded only. The existence and uniqueness of the solution for the closed‐loop system is proved and the ‘reaching condition’ is obtained. Considering the SMC usually requires the large control gain and may exhibit chattering behavior, we develop the ADRC to attenuate the disturbance for which the derivative is also supposed to be bounded. Compared with the SMC, the advantage of the ADRC is not only using the continuous control but also giving an online estimation of the disturbance. It is shown that the resulting closed‐loop system can reach any arbitrary given vicinity of zero as time goes to infinity and high gain tuning parameter goes to zero. Copyright © 2013 John Wiley & Sons, Ltd. |
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| AbstractList | SUMMARYIn this paper, we are concerned with the boundary stabilization of a one‐dimensional anti‐stable Schrödinger equation subject to boundary control matched disturbance. We apply both the sliding mode control (SMC) and the active disturbance rejection control (ADRC) to deal with the disturbance. By the SMC approach, the disturbance is supposed to be bounded only. The existence and uniqueness of the solution for the closed‐loop system is proved and the ‘reaching condition’ is obtained. Considering the SMC usually requires the large control gain and may exhibit chattering behavior, we develop the ADRC to attenuate the disturbance for which the derivative is also supposed to be bounded. Compared with the SMC, the advantage of the ADRC is not only using the continuous control but also giving an online estimation of the disturbance. It is shown that the resulting closed‐loop system can reach any arbitrary given vicinity of zero as time goes to infinity and high gain tuning parameter goes to zero. Copyright © 2013 John Wiley & Sons, Ltd. In this paper, we are concerned with the boundary stabilization of a one-dimensional anti-stable Schrodinger equation subject to boundary control matched disturbance. We apply both the sliding mode control (SMC) and the active disturbance rejection control (ADRC) to deal with the disturbance. By the SMC approach, the disturbance is supposed to be bounded only. The existence and uniqueness of the solution for the closed-loop system is proved and the 'reaching condition' is obtained. Considering the SMC usually requires the large control gain and may exhibit chattering behavior, we develop the ADRC to attenuate the disturbance for which the derivative is also supposed to be bounded. Compared with the SMC, the advantage of the ADRC is not only using the continuous control but also giving an online estimation of the disturbance. It is shown that the resulting closed-loop system can reach any arbitrary given vicinity of zero as time goes to infinity and high gain tuning parameter goes to zero. Copyright copyright 2013 John Wiley & Sons, Ltd. SUMMARY In this paper, we are concerned with the boundary stabilization of a one-dimensional anti-stable Schrödinger equation subject to boundary control matched disturbance. We apply both the sliding mode control (SMC) and the active disturbance rejection control (ADRC) to deal with the disturbance. By the SMC approach, the disturbance is supposed to be bounded only. The existence and uniqueness of the solution for the closed-loop system is proved and the 'reaching condition' is obtained. Considering the SMC usually requires the large control gain and may exhibit chattering behavior, we develop the ADRC to attenuate the disturbance for which the derivative is also supposed to be bounded. Compared with the SMC, the advantage of the ADRC is not only using the continuous control but also giving an online estimation of the disturbance. It is shown that the resulting closed-loop system can reach any arbitrary given vicinity of zero as time goes to infinity and high gain tuning parameter goes to zero. Copyright © 2013 John Wiley & Sons, Ltd. |
| Author | Liu, Jun-Jun Guo, Bao-Zhu |
| Author_xml | – sequence: 1 givenname: Bao-Zhu surname: Guo fullname: Guo, Bao-Zhu email: Correspondence to: Bao-Zhu Guo, Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, China., bzguo@iss.ac.cn organization: Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, China – sequence: 2 givenname: Jun-Jun surname: Liu fullname: Liu, Jun-Jun organization: School of Mathematical Sciences, Beijing Institute of Technology, Beijing 100081, China |
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| References | Chen G, Delfour MC, Krall AM, Payre G. Modeling, stabilization and control of serially connected beam. SIAM Journal on Control and Optimization 1987; 25:526-546. Guo W, Guo BZ. Stabilization and regulator design for a one-dimensional unstable wave equation with input harmonic disturbance. International Journal of Robust and Nonlinear Control 2013; 23(5):514-533. Machtyngier E. Exact controllability for the Schrödinger equation. SIAM Journal on Control and Optimization 1994; 32:24-34. Luo ZH, Guo BZ, Morgul O. Stability and Stabilization of Infinite Dimensional Systems with Applications. Springer-Verlag: London, 1999. Komornik V, Loreti P. Fourier Series in Control Theory. Springer-Verlag: New York, 2005. Weiss G. Admissibility of unbounded control operators. SIAM Journal on Control and Optimization 1989; 27:527-545. Cheng MB, Radisavljevic V, Su WC. Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties. Automatica 2011; 47:381-387. Krstic M. Adaptive control of an anti-stable wave PDE. Dynamics of Continuous, Discrete & Impulsive Systems. Series A: Mathematical Analysis 2010; 17:853-882. Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag: New York, 1983. Pisano A, Orlov Y. Boundary second-order sliding-mode control of an uncertain heat process with unbounded matched perturbation. Automatica 2012; 48:1768-1775. Krstic M, Guo BZ, Smyshlyaev A. Boundary controllers and observers for the linearized Schrödinger equation. SIAM Journal on Control and Optimization 2011; 49:1479-1497. Guo BZ, Kang W. The Lyapunov approach to boundary stabilization of an anti-stable one-dimensional wave equation with boundary disturbance. International Journal of Robust and Nonlinear Control 2012. DOI: 10.1002/rnc.2874, in press. Pisano A, Orlov Y, Usai E. Tracking control of the uncertain heat and wave equation via power-fractional and sliding-mode techniques. SIAM Journal on Control and Optimization 2011; 49:363-382. Smyshlyaev A, Krstic M. Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations. IEEE Transactions on Automatic Control (2004); 49:2185-2202. Medvedev A, Hillerström G. An external model control system. Control-Theory and Advanced Technology 1995; 10:1643-1665. Krstic M, Smyshlyaev A. Boundary Control of PDEs: A Course on Backstepping Designs. SIAM: Philadelphia, PA, 2008. Orlov YV, Utkin VI. Sliding mode control in infinite-dimensional systems. Automatica 1987; 23:753-757. Guo BZ, Xu CZ. The stabilization of a one-dimensional wave equation by boundary feedback with noncollocated observation. IEEE Transactions on Automatic Control 2007; 52:371-377. Krstic M, Guo BZ, Balogh A, Smyshlyaev A. Output-feedback stabilization of an unstable wave equation. Automatica 2008; 44:63-74. Breger AM, Butkovskii AG, Kubyshkin VA, Utkin VI. Sliding modes for control of distributed parameter entities subjected to a mobile multicycle signal. Automation and Remote Control 1980; 41:346-355. Weiss G. Admissible observation operators for linear semigroups. Israel Journal of Mathematics 1989; 65:17-43. Vincent J, Morris D, Usher N, Gao Z, Zhao S, Nicoletti A, Zheng Q. On active disturbance rejection based control design for superconducting RF cavities. Nuclear Instruments and Methods in Physics Research A 2011; 643:11-16. Guo W, Guo BZ, Shao ZC. Parameter estimation and stabilization for a wave equation with boundary output harmonic disturbance and non-collocated control. International Journal of Robust and Nonlinear Control (2011); 21:1297-1321. Liu WJ. Elementary Feedback Stabilization of the Linear Reaction-Convection-Diffusion Equation and the Wave Equation. Springer-Verlag: New York, 2009. Guo BZ, Jin FF. Sliding mode and active disturbance rejection control to the stabilization of anti-stable one-dimensional wave equation subject to boundary input disturbance. IEEE Transactions on Automatic Control 2012. DOI: 10.1109/TAC.2012.2218669, in press. Guo BZ. Riesz basis approach to the stabilization of a flexible beam with a tip mass. SIAM Journal on Control and Optimization 2001; 39:1736-1747. Han JQ. From PID to active disturbance rejection control. IEEE Transactions on Industrial Electronics 2009; 56:900-906. Guo BZ, Zhao ZL. On the convergence of extended state observer for nonlinear systems with uncertainty. Systems & Control Letters 2011; 60:420-430. 1989; 65 2012 1980; 41 2004; 49 2013; 23 2010; 17 2011; 60 1995; 10 2009 2008 1996 2005 2007; 52 1989; 27 1999 2009; 56 1987; 25 2011; 643 1987; 23 2001 2011; 21 1983 2008; 44 2001; 39 2011; 47 2012; 48 2011; 49 1994; 32 |
| References_xml | – reference: Guo BZ, Kang W. The Lyapunov approach to boundary stabilization of an anti-stable one-dimensional wave equation with boundary disturbance. International Journal of Robust and Nonlinear Control 2012. DOI: 10.1002/rnc.2874, in press. – reference: Guo BZ. Riesz basis approach to the stabilization of a flexible beam with a tip mass. SIAM Journal on Control and Optimization 2001; 39:1736-1747. – reference: Breger AM, Butkovskii AG, Kubyshkin VA, Utkin VI. Sliding modes for control of distributed parameter entities subjected to a mobile multicycle signal. Automation and Remote Control 1980; 41:346-355. – reference: Krstic M, Guo BZ, Balogh A, Smyshlyaev A. Output-feedback stabilization of an unstable wave equation. Automatica 2008; 44:63-74. – reference: Vincent J, Morris D, Usher N, Gao Z, Zhao S, Nicoletti A, Zheng Q. On active disturbance rejection based control design for superconducting RF cavities. Nuclear Instruments and Methods in Physics Research A 2011; 643:11-16. – reference: Luo ZH, Guo BZ, Morgul O. Stability and Stabilization of Infinite Dimensional Systems with Applications. Springer-Verlag: London, 1999. – reference: Weiss G. Admissible observation operators for linear semigroups. Israel Journal of Mathematics 1989; 65:17-43. – reference: Medvedev A, Hillerström G. An external model control system. Control-Theory and Advanced Technology 1995; 10:1643-1665. – reference: Guo W, Guo BZ, Shao ZC. Parameter estimation and stabilization for a wave equation with boundary output harmonic disturbance and non-collocated control. International Journal of Robust and Nonlinear Control (2011); 21:1297-1321. – reference: Han JQ. From PID to active disturbance rejection control. IEEE Transactions on Industrial Electronics 2009; 56:900-906. – reference: Guo BZ, Xu CZ. The stabilization of a one-dimensional wave equation by boundary feedback with noncollocated observation. IEEE Transactions on Automatic Control 2007; 52:371-377. – reference: Guo W, Guo BZ. Stabilization and regulator design for a one-dimensional unstable wave equation with input harmonic disturbance. International Journal of Robust and Nonlinear Control 2013; 23(5):514-533. – reference: Guo BZ, Jin FF. Sliding mode and active disturbance rejection control to the stabilization of anti-stable one-dimensional wave equation subject to boundary input disturbance. IEEE Transactions on Automatic Control 2012. DOI: 10.1109/TAC.2012.2218669, in press. – reference: Pisano A, Orlov Y, Usai E. Tracking control of the uncertain heat and wave equation via power-fractional and sliding-mode techniques. SIAM Journal on Control and Optimization 2011; 49:363-382. – reference: Chen G, Delfour MC, Krall AM, Payre G. Modeling, stabilization and control of serially connected beam. SIAM Journal on Control and Optimization 1987; 25:526-546. – reference: Krstic M. Adaptive control of an anti-stable wave PDE. Dynamics of Continuous, Discrete & Impulsive Systems. Series A: Mathematical Analysis 2010; 17:853-882. – reference: Liu WJ. Elementary Feedback Stabilization of the Linear Reaction-Convection-Diffusion Equation and the Wave Equation. Springer-Verlag: New York, 2009. – reference: Cheng MB, Radisavljevic V, Su WC. Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties. Automatica 2011; 47:381-387. – reference: Orlov YV, Utkin VI. Sliding mode control in infinite-dimensional systems. Automatica 1987; 23:753-757. – reference: Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag: New York, 1983. – reference: Smyshlyaev A, Krstic M. Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations. IEEE Transactions on Automatic Control (2004); 49:2185-2202. – reference: Guo BZ, Zhao ZL. On the convergence of extended state observer for nonlinear systems with uncertainty. Systems & Control Letters 2011; 60:420-430. – reference: Weiss G. Admissibility of unbounded control operators. SIAM Journal on Control and Optimization 1989; 27:527-545. – reference: Krstic M, Smyshlyaev A. Boundary Control of PDEs: A Course on Backstepping Designs. SIAM: Philadelphia, PA, 2008. – reference: Krstic M, Guo BZ, Smyshlyaev A. Boundary controllers and observers for the linearized Schrödinger equation. SIAM Journal on Control and Optimization 2011; 49:1479-1497. – reference: Machtyngier E. Exact controllability for the Schrödinger equation. SIAM Journal on Control and Optimization 1994; 32:24-34. – reference: Komornik V, Loreti P. Fourier Series in Control Theory. Springer-Verlag: New York, 2005. – reference: Pisano A, Orlov Y. Boundary second-order sliding-mode control of an uncertain heat process with unbounded matched perturbation. Automatica 2012; 48:1768-1775. – start-page: 189 year: 2001 end-page: 219 – volume: 27 start-page: 527 year: 1989 end-page: 545 article-title: Admissibility of unbounded control operators publication-title: SIAM Journal on Control and Optimization – year: 2009 – year: 1983 – volume: 17 start-page: 853 year: 2010 end-page: 882 article-title: Adaptive control of an anti‐stable wave PDE publication-title: Dynamics of Continuous, Discrete & Impulsive Systems. Series A: Mathematical Analysis – volume: 643 start-page: 11 year: 2011 end-page: 16 article-title: On active disturbance rejection based control design for superconducting RF cavities publication-title: Nuclear Instruments and Methods in Physics Research A – volume: 56 start-page: 900 year: 2009 end-page: 906 article-title: From PID to active disturbance rejection control publication-title: IEEE Transactions on Industrial Electronics – year: 2005 – volume: 49 start-page: 363 year: 2011 end-page: 382 article-title: Tracking control of the uncertain heat and wave equation via power‐fractional and sliding‐mode techniques publication-title: SIAM Journal on Control and Optimization – volume: 25 start-page: 526 year: 1987 end-page: 546 article-title: Modeling, stabilization and control of serially connected beam publication-title: SIAM Journal on Control and Optimization – volume: 23 start-page: 753 year: 1987 end-page: 757 article-title: Sliding mode control in infinite‐dimensional systems publication-title: Automatica – volume: 10 start-page: 1643 year: 1995 end-page: 1665 article-title: An external model control system publication-title: Control‐Theory and Advanced Technology – start-page: 668 year: 1996 end-page: 672 – volume: 52 start-page: 371 year: 2007 end-page: 377 article-title: The stabilization of a one‐dimensional wave equation by boundary feedback with noncollocated observation publication-title: IEEE Transactions on Automatic Control – volume: 39 start-page: 1736 year: 2001 end-page: 1747 article-title: Riesz basis approach to the stabilization of a flexible beam with a tip mass publication-title: SIAM Journal on Control and Optimization – volume: 32 start-page: 24 year: 1994 end-page: 34 article-title: Exact controllability for the Schrödinger equation publication-title: SIAM Journal on Control and Optimization – volume: 49 start-page: 2185 year: 2004 end-page: 2202 article-title: Closed‐form boundary state feedbacks for a class of 1‐D partial integro‐differential equations publication-title: IEEE Transactions on Automatic Control – volume: 23 start-page: 514 issue: 5 year: 2013 end-page: 533 article-title: Stabilization and regulator design for a one‐dimensional unstable wave equation with input harmonic disturbance publication-title: International Journal of Robust and Nonlinear Control – volume: 48 start-page: 1768 year: 2012 end-page: 1775 article-title: Boundary second‐order sliding‐mode control of an uncertain heat process with unbounded matched perturbation publication-title: Automatica – volume: 65 start-page: 17 year: 1989 end-page: 43 article-title: Admissible observation operators for linear semigroups publication-title: Israel Journal of Mathematics – volume: 44 start-page: 63 year: 2008 end-page: 74 article-title: Output‐feedback stabilization of an unstable wave equation publication-title: Automatica – volume: 49 start-page: 1479 year: 2011 end-page: 1497 article-title: Boundary controllers and observers for the linearized Schrödinger equation publication-title: SIAM Journal on Control and Optimization – year: 2008 – year: 2012 article-title: The Lyapunov approach to boundary stabilization of an anti‐stable one‐dimensional wave equation with boundary disturbance publication-title: International Journal of Robust and Nonlinear Control – year: 2012 article-title: Sliding mode and active disturbance rejection control to the stabilization of anti‐stable one‐dimensional wave equation subject to boundary input disturbance publication-title: IEEE Transactions on Automatic Control – volume: 41 start-page: 346 year: 1980 end-page: 355 article-title: Sliding modes for control of distributed parameter entities subjected to a mobile multicycle signal publication-title: Automation and Remote Control – volume: 47 start-page: 381 year: 2011 end-page: 387 article-title: Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties publication-title: Automatica – volume: 60 start-page: 420 year: 2011 end-page: 430 article-title: On the convergence of extended state observer for nonlinear systems with uncertainty publication-title: Systems & Control Letters – volume: 21 start-page: 1297 year: 2011 end-page: 1321 article-title: Parameter estimation and stabilization for a wave equation with boundary output harmonic disturbance and non‐collocated control publication-title: International Journal of Robust and Nonlinear Control – year: 1999 |
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| Snippet | SUMMARYIn this paper, we are concerned with the boundary stabilization of a one‐dimensional anti‐stable Schrödinger equation subject to boundary control... SUMMARY In this paper, we are concerned with the boundary stabilization of a one-dimensional anti-stable Schrödinger equation subject to boundary control... In this paper, we are concerned with the boundary stabilization of a one-dimensional anti-stable Schrodinger equation subject to boundary control matched... |
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| SubjectTerms | Active control active disturbance rejection control Boundary control disturbance rejection Disturbances Rejection Schroedinger equation Schrödinger equation Sliding mode control stability stability, boundary control Stabilization Tuning |
| Title | Sliding mode control and active disturbance rejection control to the stabilization of one-dimensional Schrödinger equation subject to boundary control matched disturbance |
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