Document Zbl 1129.93497

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Control Lett.</a> <a href="/?q=in%3A185904" title="Articles in this Issue">54, No. 12, 1163-1182 (2005)</a>. </div> <div class="abstract">Summary: This work presents a hybrid nonlinear control methodology for a broad class of switched nonlinear systems with input constraints. The key feature of the proposed methodology is the integrated synthesis, via multiple Lyapunov functions, of “lower-level” bounded nonlinear feedback controllers together with “upper-level” switching laws that orchestrate the transitions between the constituent modes and their respective controllers. Both the state and output feedback control problems are addressed. Under the assumption of availability of full state measurements, a family of bounded nonlinear state feedback controllers are initially designed to enforce asymptotic stability for the individual closed-loop modes and provide an explicit characterization of the corresponding stability region for each mode. A set of switching laws are then designed to track the evolution of the state and orchestrate switching between the stability regions of the constituent modes in a way that guarantees asymptotic stability of the overall switched closed-loop system. When complete state measurements are unavailable, a family of output feedback controllers are synthesized, using a combination of bounded state feedback controllers, high-gain observers and appropriate saturation filters to enforce asymptotic stability for the individual closed-loop modes and provide an explicit characterization of the corresponding output feedback stability regions in terms of the input constraints and the observer gain. A different set of switching rules, based on the evolution of the state estimates generated by the observers, is designed to orchestrate stabilizing transitions between the output feedback stability regions of the constituent modes. The differences between the state and output feedback switching strategies, and their implications for the switching logic, are discussed and a chemical process example is used to demonstrate the proposed approach.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5057058">Cited in <strong>95</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93D15" title="MSC2020">93D15</a> </td> <td class="space"> Stabilization of systems by feedback </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C57" title="MSC2020">93C57</a> </td> <td class="space"> Sampled-data control/observation systems </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aswitched+systems">switched systems</a>; <a href="/?q=ut%3Amultiple+Lyapunov+functions">multiple Lyapunov functions</a>; <a href="/?q=ut%3Ainput+constraints">input constraints</a>; <a href="/?q=ut%3Aoutput+feedback+control">output feedback control</a>; <a href="/?q=ut%3Abounded+nonlinear+control">bounded nonlinear control</a>; <a href="/?q=ut%3Ahybrid+chemical+processes">hybrid chemical processes</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1129.93497" data-ciurl="/ci/05057058" data-biburl="/bibtex/05057058.bib" data-amsurl="/amsrefs/05057058.bib" data-xmlurl="/xml/05057058.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05057058.pdf" title="Zbl 1129.93497 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.sysconle.2005.04.005" aria-label="DOI for “Output feedback control of switched nonlinear systems using multiple Lyapunov functions”" title="10.1016/j.sysconle.2005.04.005">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Artstein, Z., Stabilization with relaxed controls, Nonlinear Anal., 7, 1163-1173 (1983) · <a href="/0525.93053" class="nowrap">Zbl 0525.93053</a></td> </tr><tr> <td>[2]</td> <td class="space">Barton, P. I.; Pantelides, C. C., Modeling of combined discrete/continuous processes, A.I.Ch.E. J., 40, 966-979 (1994)</td> </tr><tr> <td>[3]</td> <td class="space">Bemporad, A.; Morari, M., Control of systems integrating logic, dynamics and constraints, Automatica, 35, 407-427 (1999) · <a href="/1049.93514" class="nowrap">Zbl 1049.93514</a></td> </tr><tr> <td>[4]</td> <td class="space">Branicky, M. S., Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Automat. Control, 43, 475-482 (1998) · <a href="/0904.93036" class="nowrap">Zbl 0904.93036</a></td> </tr><tr> <td>[5]</td> <td class="space">Branicky, M. S.; Borkar, V. S.; Mitter, S. K., A unified framework for hybrid control: model and optimal control theory, IEEE Trans. Automat. Control, 43, 31-45 (1998) · <a href="/0951.93002" class="nowrap">Zbl 0951.93002</a></td> </tr><tr> <td>[6]</td> <td class="space">Christofides, P. D., Robust output feedback control of nonlinear singularly perturbed systems, Automatica, 36, 45-52 (2000) · <a href="/0939.93025" class="nowrap">Zbl 0939.93025</a></td> </tr><tr> <td>[7]</td> <td class="space">Decarlo, R. A.; Branicky, M. S.; Petterson, S.; Lennartson, B., Perspectives and results on the stability and stabilizability of hybrid systems, Proc. IEEE, 88, 1069-1082 (2000)</td> </tr><tr> <td>[8]</td> <td class="space">Demetriou, M.; Kazantzis, N., A new actuator activation policy for performance enhancement of controlled diffusion processes, Automatica, 40, 415-421 (2004) · <a href="/1051.93046" class="nowrap">Zbl 1051.93046</a></td> </tr><tr> <td>[9]</td> <td class="space">El-Farra, N. H.; Christofides, P. D., Integrating robustness, optimality, and constraints in control of nonlinear processes, Chem. Eng. Sci., 56, 1841-1868 (2001)</td> </tr><tr> <td>[10]</td> <td class="space">El-Farra, N. H.; Christofides, P. D., Switching and feedback laws for control of constrained switched nonlinear systems, (Tomlin, C. J.; Greenstreet, M. R., Lecture Notes in Computer Science Series, vol. 2289 (2002), Springer: Springer Berlin), 164-178 · <a href="/1044.93515" class="nowrap">Zbl 1044.93515</a></td> </tr><tr> <td>[11]</td> <td class="space">El-Farra, N. H.; Christofides, P. D., Bounded robust control of constrained multivariable nonlinear processes, Chem. Eng. Sci., 58, 3025-3047 (2003)</td> </tr><tr> <td>[12]</td> <td class="space">El-Farra, N. H.; Christofides, P. D., Coordinating feedback and switching for control of hybrid nonlinear processes, A.I.Ch.E. J., 49, 2079-2098 (2003)</td> </tr><tr> <td>[13]</td> <td class="space">Freeman, R. A.; Kokotovic, P. V., Robust Nonlinear Control Design: State-Space and Lyapunov Techniques (1996), Birkhauser: Birkhauser Boston · <a href="/0863.93075" class="nowrap">Zbl 0863.93075</a></td> </tr><tr> <td>[14]</td> <td class="space">Garcia-Onorio, V.; Ydstie, B. E., Distributed, asynchronous and hybrid simulation of process networks using recording controllers, Internat. J. Robust Nonlinear Control, 14, 227-248 (2004) · <a href="/1033.93003" class="nowrap">Zbl 1033.93003</a></td> </tr><tr> <td>[15]</td> <td class="space">Grossman, R. L.; Nerode, A.; Ravn, A. P.; Rischel, H., Hybrid systems, (Lecture Notes in Computer Science, vol. 736 (1993), Springer: Springer New York) · <a href="/0825.00044" class="nowrap">Zbl 0825.00044</a></td> </tr><tr> <td>[16]</td> <td class="space">Grossmann, I. E.; van den Heever, S. A.; Harjukoski, I., Discrete optimization methods and their role in the integration of planning and scheduling, (Proceedings of Sixth International Conference on Chemical Process Control. Proceedings of Sixth International Conference on Chemical Process Control, Tucson, AZ (2001)), 124-152</td> </tr><tr> <td>[17]</td> <td class="space">Hespanha, J. P.; Morse, A. S., Stability of switched systems with average dwell time, (Proceedings of 38th IEEE Conference on Decision and Control. Proceedings of 38th IEEE Conference on Decision and Control, Phoenix, AZ (1999)), 2655-2660 · <a href="/0108.23602" class="nowrap">Zbl 0108.23602</a></td> </tr><tr> <td>[18]</td> <td class="space">Hu, B.; Xu, X.; Antsaklis, P. J.; Michel, A. N., Robust stabilizing control law for a class of second-order switched systems, Systems Control Lett., 38, 197-207 (1999) · <a href="/0948.93013" class="nowrap">Zbl 0948.93013</a></td> </tr><tr> <td>[19]</td> <td class="space">Isidori, A., Nonlinear Control Systems: An Introduction (1995), Springer: Springer Berlin-Heidelberg · <a href="/0569.93034" class="nowrap">Zbl 0569.93034</a></td> </tr><tr> <td>[20]</td> <td class="space">Khalil, H. K., Robust servomechanism output feedback controller for feedback linearizable systems, Automatica, 30, 1587-1599 (1994) · <a href="/0816.93032" class="nowrap">Zbl 0816.93032</a></td> </tr><tr> <td>[21]</td> <td class="space">Khalil, H. K., Nonlinear Systems (1996), Macmillan Publishing Company: Macmillan Publishing Company New York · <a href="/0626.34052" class="nowrap">Zbl 0626.34052</a></td> </tr><tr> <td>[22]</td> <td class="space">Lemmon, M. D.; Antsaklis, P. J., Timed automata and robust control: can we now control complex dynamical systems?, (Proceedings of 36th IEEE Conference on Decision and Control. Proceedings of 36th IEEE Conference on Decision and Control, San Diego, CA (1997)), 108-113</td> </tr><tr> <td>[23]</td> <td class="space">Liberzon, D., ISS and integral-ISS disturbance attenuation with bounded controls, (Proceedings of 38th IEEE Conference on Decision and Control. Proceedings of 38th IEEE Conference on Decision and Control, Phoenix, AZ (1999)), 2501-2506</td> </tr><tr> <td>[24]</td> <td class="space">Liberzon, D.; Morse, A. S., Basic problems in stability and design of switched systems, IEEE Control Systems Mag., 19, 59-70 (1999) · <a href="/1384.93064" class="nowrap">Zbl 1384.93064</a></td> </tr><tr> <td>[25]</td> <td class="space">Lin, Y.; Sontag, E. D., A universal formula for stabilization with bounded controls, Systems Control Lett., 16, 393-397 (1991) · <a href="/0728.93062" class="nowrap">Zbl 0728.93062</a></td> </tr><tr> <td>[26]</td> <td class="space">Lygeros, J.; Godbole, D. N.; Sastry, S. S., A game theoretic approach to hybrid system design, (Alur, R.; Henzinger, T., Lecture Notes in Computer Science, vol. 1066 (1996), Springer: Springer Berlin), 1-12</td> </tr><tr> <td>[27]</td> <td class="space">Malisoff, M.; Sontag, E. D., Universal formulas for feedback stabilization with respect to Minkowski balls, Systems Control Lett., 40, 247-260 (2000) · <a href="/0985.93044" class="nowrap">Zbl 0985.93044</a></td> </tr><tr> <td>[28]</td> <td class="space">Mhaskar, P.; El-Farra, N. H.; Christofides, P. D., Hybrid predictive control of process systems, A.I.Ch.E. J., 50, 1242-1259 (2004)</td> </tr><tr> <td>[29]</td> <td class="space">Mignone, D.; Bemporad, A.; Morari, M., A framework for control, fault detection, state estimation, and verification of hybrid systems, (Proceedings of American Control Conference. Proceedings of American Control Conference, San Diego, CA (1999)), 134-139</td> </tr><tr> <td>[30]</td> <td class="space">Peleties, P.; DeCarlo, R., Asymptotic stability of $m$-switched systems using Lyapunov-like functions, (Proceedings of American Control Conference. Proceedings of American Control Conference, Boston, MA (1991)), 1679-1684</td> </tr><tr> <td>[31]</td> <td class="space">Pettersson, S.; Lennartson, B., Stability and robustness for hybrid systems, (Proceedings of 35th IEEE Conference on Decision and Control. Proceedings of 35th IEEE Conference on Decision and Control, Kobe, Japan (1996)), 1202-1207 · <a href="/1017.93027" class="nowrap">Zbl 1017.93027</a></td> </tr><tr> <td>[32]</td> <td class="space">Sepulchre, R.; Jankovic, M.; Kokotovic, P., Constructive Nonlinear Control (1997), Springer: Springer Berlin-Heidelberg · <a href="/1067.93500" class="nowrap">Zbl 1067.93500</a></td> </tr><tr> <td>[33]</td> <td class="space">Sontag, E. D., A Lyapunov-like characterization of asymptotic controllability, SIAM J. Control Opt., 21, 462-471 (1983) · <a href="/0513.93047" class="nowrap">Zbl 0513.93047</a></td> </tr><tr> <td>[34]</td> <td class="space">Sontag, E. D., Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34, 435-443 (1989) · <a href="/0682.93045" class="nowrap">Zbl 0682.93045</a></td> </tr><tr> <td>[35]</td> <td class="space">Teel, A.; Praly, L., Global stabilizability and observability imply semi-global stabilizability by output feedback, Systems Control Lett., 22, 313-325 (1994) · <a href="/0820.93054" class="nowrap">Zbl 0820.93054</a></td> </tr><tr> <td>[36]</td> <td class="space">Wicks, M. A.; Peleties, P.; DeCarlo, R. A., Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems, European J. Control, 4, 140-147 (1998) · <a href="/0910.93062" class="nowrap">Zbl 0910.93062</a></td> </tr><tr> <td>[37]</td> <td class="space">Yamalidou, E. C.; Kantor, J., Modeling and optimal control of discrete-event chemical processes using Petri nets, Comp. Chem. Eng., 15, 503-519 (1990)</td> </tr><tr> <td>[38]</td> <td class="space">Ye, H.; Michel, A. N.; Hou, L., Stability theory for hybrid dynamical systems, IEEE Trans. Automat. Control, 43, 461-474 (1998) · <a href="/0905.93024" class="nowrap">Zbl 0905.93024</a></td> </tr><tr> <td>[39]</td> <td class="space">Zefran, M.; Burdick, J. W., Design of switching controllers for systems with changing dynamics, (Proceedings of 37th IEEE Conference on Decision and Control. Proceedings of 37th IEEE Conference on Decision and Control, Tampa, FL (1998)), 2113-2118</td> </tr><tr> <td>[40]</td> <td class="space">Christofides, P. D.; Teel, A. R., Singular perturbations and input-to-state stability, IEEE Trans. Automat. Contr., 41, 1645-1650 (1996) · <a href="/0864.93086" class="nowrap">Zbl 0864.93086</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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