Chattering in the reach control problem. (English) Zbl 1388.93019
Summary: The Reach Control Problem (RCP) is a fundamental problem in hybrid control theory. The goal of the RCP is to find a feedback control that drives the state trajectories of an affine system to leave a polytope through a predetermined exit facet. In the current literature, the notion of leaving a polytope through a facet has an ambiguous definition. There are two different notions. In one, at the last time instance when the trajectory is inside the polytope, it must also be inside the exit facet. In the other, the trajectory is required to cross from the polytope into the outer open half-space bounded by the exit facet. In this paper, we provide a counterexample showing that these definitions are not equivalent for general continuous or smooth state feedback. On the other hand, we prove that analyticity of the feedback control is a sufficient condition for equivalence of these definitions. We generalize this result to several other classes of feedback control previously investigated in the RCP literature, most notably piecewise affine feedback. Additionally, we clarify or complete a number of previous results on the exit behaviour of trajectories in the RCP.
MSC:
| 93B03 | Attainable sets, reachability |
| 93B52 | Feedback control |
| 93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |
References:
| [1] | Ames, A., & Sastry, S. S. (2005). Characterization of Zeno behavior in hybrid systems using homological methods. In 2005 American control conference; Ames, A., & Sastry, S. S. (2005). Characterization of Zeno behavior in hybrid systems using homological methods. In 2005 American control conference |
| [2] | Ashford, G.; Broucke, M. E., Time-varying affine feedback for reach control on simplices, Automatica, 49, 5, 1365-1369, (2013) · Zbl 1319.93027 |
| [3] | Belta, C.; Habets, L. C.G. J.M., Controlling a class of nonlinear systems on rectangles, IEEE Transactions on Automatic Control, 51, 11, 1749-1759, (2006) · Zbl 1366.93278 |
| [4] | Belta, C., Habets, L.C.G.J.M., & Kumar, V. (2002). Control of multi-affine systems on rectangles with applications to biomolecular networks. In 41st IEEE conference on decision and control; Belta, C., Habets, L.C.G.J.M., & Kumar, V. (2002). Control of multi-affine systems on rectangles with applications to biomolecular networks. In 41st IEEE conference on decision and control |
| [5] | Blanchini, F., Nonquadratic Lyapunov functions for robust control, Automatica, 31, 3, 451-461, (1995) · Zbl 0825.93653 |
| [6] | Blanchini, F.; Pellegrino, F. A., Relatively optimal control: a static piecewise-affine solution, SIAM Journal on Control and Optimization, 46, 2, 585-603, (2007) · Zbl 1141.93028 |
| [7] | Branicky, M. S., Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Transactions on Automatic Control, 43, 4, 475-482, (1998) · Zbl 0904.93036 |
| [8] | Brezis, H., On a characterization of flow-invariant sets, Communications on Pure and Applied Mathematics, 23, 2, 261-263, (1970) · Zbl 0191.38703 |
| [9] | Broucke, M. E., Reach control on simplices by continuous state feedback, SIAM Journal on Control and Optimization, 48, 5, 3482-3500, (2010) · Zbl 1202.93036 |
| [10] | Broucke, M. E.; Ganness, M., Reach control on simplices by piecewise affine feedback, SIAM Journal on Control and Optimization, 52, 5, 3261-3286, (2014) · Zbl 1304.93034 |
| [11] | Çamlibel, M. K. (2008). Well-posed bimodal piecewise linear systems do not exhibit Zeno behavior. In 17th IFAC world congress; Çamlibel, M. K. (2008). Well-posed bimodal piecewise linear systems do not exhibit Zeno behavior. In 17th IFAC world congress |
| [12] | Clarke, F. H.; Ledyaev, Y. S.; Stern, R. J.; Wolenski, P. R., Nonsmooth analysis and control theory, (1998), Springer · Zbl 1047.49500 |
| [13] | Cortés, J., Discontinuous dynamical systems -a tutorial on solutions, nonsmooth analysis, and stability, IEEE Control Systems Magazine, 28, 3, 36-73, (2008) · Zbl 1395.34023 |
| [14] | Courant, R.; Hilbert, D., Methods of mathematical physics, Volume 2, (1962), Interscience (Wiley) · Zbl 0729.00007 |
| [15] | Ezzine, J., & Haddad, A. H. (1988). On the controllability and observability of hybrid systems. In 1988 American control conference; Ezzine, J., & Haddad, A. H. (1988). On the controllability and observability of hybrid systems. In 1988 American control conference |
| [16] | Garabedian, P. R., Partial differential equations, (1964), Wiley · Zbl 0124.30501 |
| [17] | Goebel, R., & Teel, A. R. (2008). Lyapunov characterization of Zeno behavior in hybrid systems. In 47th IEEE conference on decision and control; Goebel, R., & Teel, A. R. (2008). Lyapunov characterization of Zeno behavior in hybrid systems. In 47th IEEE conference on decision and control |
| [18] | Habets, L. C.; Collins, P. J.; van Schuppen, J. H., Reachability and control synthesis for piecewise-affine hybrid systems on simplices, IEEE Transactions on Automatic Control, 51, 6, 938-948, (2006) · Zbl 1366.93348 |
| [19] | Habets, L. C.G. J.M.; van Schuppen, J. H., Control of piecewise-linear hybrid systems on simplices and rectangles, (Benedetto, M. D.D.; Sangiovanni-Vincentelli, A. L., Hybrid systems: Computation and control, (2001), Springer), 261-274 · Zbl 0991.93075 |
| [20] | Habets, L. C.G. J.M.; van Schuppen, J. H., A control problem for affine dynamical systems on a full-dimensional polytope, Automatica, 40, 1, 21-35, (2004) · Zbl 1035.93008 |
| [21] | Hale, J. K., Ordinary differential equations, (1980), Krieger · Zbl 0186.40901 |
| [22] | Haugwitz, S.; Hagander, P., Analysis and design of startup control of a chemical plate reactor with uncertainties: a hybrid approach, (IEEE conference on control applications, (2007)), 1426-1431 |
| [23] | Heemels, W. P.M. H.; Çamlibel, M. K.; van der Schaft, A. J.; Schumacher, J. M., On the existence and uniqueness of solution trajectories to hybrid dynamical systems, (Johannson, R.; Rantzer, A., Nonlinear and hybrid systems in automotive control, (2002), Springer), 391-422 |
| [24] | Helwa, M. K.; Broucke, M. E., Monotonic reach control on polytopes, IEEE Transactions on Automatic Control, 58, 10, 2704-2709, (2013) · Zbl 1369.93399 |
| [25] | Helwa, M. K., & Broucke, M. E. (2014). Reach control of single-input systems on simplices using multi-affine feedback. In International symposium on mathematical theory of networks and systems; Helwa, M. K., & Broucke, M. E. (2014). Reach control of single-input systems on simplices using multi-affine feedback. In International symposium on mathematical theory of networks and systems |
| [26] | Helwa, M. K.; Broucke, M. E., Flow functions, control flow functions, and the reach control problem, Automatica, 55, 108-115, (2015) · Zbl 1378.93020 |
| [27] | Helwa, M. K.; Lin, Z.; Broucke, M. E., On the necessity of the invariance conditions for reach control on polytopes, Systems & Control Letters, 90, 16-19, (2016) · Zbl 1335.93023 |
| [28] | Heymann, M.; Lin, F.; Meyer, G.; Resmerita, S., Analysis of Zeno behaviors in a class of hybrid systems, IEEE Transactions on Automatic Control, 50, 3, 376-383, (2005) · Zbl 1365.93293 |
| [29] | Imura, J.-I.; van der Schaft, A., Characterization of well-posedness of piecewise-linear systems, IEEE Transactions on Automatic Control, 45, 9, 1600-1619, (2000) · Zbl 0978.34012 |
| [30] | Krantz, S. G.; Parks, H. R., A primer of real analytic functions, (2002), Birkhäuser · Zbl 1015.26030 |
| [31] | Kroeze, Z., & Broucke, M. E. (2016). A viability approach to the output reach control problem. In 2016 American control conference; Kroeze, Z., & Broucke, M. E. (2016). A viability approach to the output reach control problem. In 2016 American control conference |
| [32] | Lee, C. W., Subdivisions and triangulations of polytopes, (Goodman, J. E.; O’Rourke, J., Handbook of discrete and computational geometry, (2004), Chapman and Hall/CRC), 383-406 · Zbl 1056.52001 |
| [33] | Liberzon, D., Switching in systems and control, (2003), Springer · Zbl 1036.93001 |
| [34] | Lygeros, J.; Johansson, K. H.; Simić, S.; Zhang, J.; Sastry, S. S., Dynamical properties of hybrid automata, IEEE Transactions on Automatic Control, 48, 1, 2-17, (2003) · Zbl 1364.93503 |
| [35] | Moarref, M.; Ornik, M.; Broucke, M. E., An obstruction to solvability of the reach control problem using affine feedback, Automatica, 71, 229-236, (2016) · Zbl 1343.93013 |
| [36] | Ornik, M.; Broucke, M. E., Characterization of a topological obstruction to reach control by continuous state feedback, Mathematics of Control, Signals, and Systems, 29, 2, (2017) · Zbl 1366.93040 |
| [37] | Ornik, M., Moarref, M., & Broucke, M. E. (2017). An automated parallel parking strategy using reach control theory. In 20th IFAC world congress; Ornik, M., Moarref, M., & Broucke, M. E. (2017). An automated parallel parking strategy using reach control theory. In 20th IFAC world congress |
| [38] | Petterson, S., & Lennartson, B. (1996). Stability and robustness for hybrid systems. In 35th IEEE conference on decision and control; Petterson, S., & Lennartson, B. (1996). Stability and robustness for hybrid systems. In 35th IEEE conference on decision and control |
| [39] | Roszak, B., & Broucke, M. E. (2005). Necessary and sufficient conditions for reachability on a simplex. In 44th IEEE conference on decision and control and 2005 european control conference; Roszak, B., & Broucke, M. E. (2005). Necessary and sufficient conditions for reachability on a simplex. In 44th IEEE conference on decision and control and 2005 european control conference |
| [40] | Roszak, B.; Broucke, M. E., Necessary and sufficient conditions for reachability on a simplex, Automatica, 42, 11, 1913-1918, (2006) · Zbl 1261.93015 |
| [41] | Semsar-Kazerooni, E.; Broucke, M. E., Reach controllability of single input affine systems on a simplex, IEEE Transactions on Automatic Control, 59, 3, 738-744, (2014) · Zbl 1360.93105 |
| [42] | Sussmann, H. J., Bounds on the number of switchings for trajectories of piecewise analytic vector fields, Journal of Differential Equations, 43, 3, 399-418, (1982) · Zbl 0488.34033 |
| [43] | Vukosavljev, M., & Broucke, M. E. (2014). Control of a gantry crane: a reach control approach. In 53rd IEEE conference on decision and control; Vukosavljev, M., & Broucke, M. E. (2014). Control of a gantry crane: a reach control approach. In 53rd IEEE conference on decision and control |
| [44] | Vukosavljev, M., Jansen, I., Broucke, M. E., & Schoellig, A. P. (2016). Safe and robust robot maneuvers based on reach control. In 2016 IEEE international conference on robotics and automation; Vukosavljev, M., Jansen, I., Broucke, M. E., & Schoellig, A. P. (2016). Safe and robust robot maneuvers based on reach control. In 2016 IEEE international conference on robotics and automation |
| [45] | Wu, Y.; Shen, T., Reach control problem for linear differential inclusion systems on simplices, IEEE Transactions on Automatic Control, 61, 5, 1403-1408, (2016) · Zbl 1359.93161 |
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