Input-output-to-state stability of systems related through simulation relations. (English) Zbl 1458.93216
Summary: There have been efforts to simplify analysis of a complex system by relating it to a simpler system through certain system relations. In this paper, we consider continuous-time systems related by a graph simulation relation and focus on preservation of input-output-to-state stability (IOSS) and its integral variant, integral IOSS (iIOSS). We establish that, under mild continuity and boundedness assumptions on some appropriately defined set-valued functions, the (i)IOSS property for the simulating system can lead to the (i)IOSS property for the original simulated system. The results thus demonstrate the possibility of inferring the (i)IOSS of a system by analyzing a potentially simpler system.
MSC:
| 93D25 | Input-output approaches in control theory |
| 93C10 | Nonlinear systems in control theory |
| 93B17 | Transformations |
Keywords:
simulation relations; nonlinear control systems; input-output-to-state stability; related systems; property preservationReferences:
| [1] | D. Angeli, Asymptotic characterizations of integral input-output to state stability, in Proc. European Control Conf., IEEE, Piscataway, NJ, 2003, pp. 1322-1326. |
| [2] | D. Angeli, B. Ingalls, E. D. Sontag, and Y. Wang, Separation principles for input-output and integral-input-to-state stability, SIAM J. Control Optim., 43 (2004), pp. 256-276. · Zbl 1101.93070 |
| [3] | D. Angeli and E. D. Sontag, Forward completeness, unboundedness observability, and their Lyapunov characterizations, Systems Control Lett., 38 (1999), pp. 209-217. · Zbl 0986.93036 |
| [4] | D. Angeli, E. D. Sontag, and Y. Wang, A characterization of integral input-to-state stability, IEEE Trans. Automat. Control, 45 (2000), pp. 1082-1097. · Zbl 0979.93106 |
| [5] | C. Cai and A. R. Teel, Input-output-to-state stability for discrete-time systems, Automatica J. IFAC, 44 (2008), pp. 326-336. · Zbl 1283.93244 |
| [6] | A. Girard and G. J. Pappas, Approximation metrics for discrete and continuous systems, IEEE Trans. Automat. Control, 52 (2007), pp. 782-798. · Zbl 1366.93032 |
| [7] | K. A. Grasse, Simulation and bisimulation of nonlinear control systems with admissible classes of inputs and disturbances, SIAM J. Control Optim., 46 (2007), pp. 562-584. · Zbl 1141.93009 |
| [8] | K. A. Grasse, A connection between simulation relations and feedback transformations in nonlinear control systems, Systems Control Lett., 61 (2012), pp. 631-637. · Zbl 1250.93043 |
| [9] | K. A. Grasse and N. Ho, Simulation relations and controllability properties of linear and nonlinear control systems, SIAM J. Control Optim., 53 (2015), pp. 1346-1374. · Zbl 1316.93021 |
| [10] | J. P. Hespanha, D. Liberzon, D. Angeli, and E. D. Sontag, Nonlinear norm-observability notions and stability of switched systems, IEEE Trans. Automat. Control, 50 (2005), pp. 154-168. · Zbl 1365.93349 |
| [11] | J. P. Hespanha, D. Liberzon, and A. S. Morse, Supervision of integral-input-to-state stabilizing controllers, Automatica J. IFAC, 38 (2002), pp. 1327-1335. · Zbl 1172.93394 |
| [12] | C. J. Himmelberg, Measurable relations, Fund. Math., 87 (1975), pp. 53-72. · Zbl 0296.28003 |
| [13] | A. Isidori, Nonlinear Control Systems, 3rd ed., Springer, New York, 1995. · Zbl 0878.93001 |
| [14] | A. J. Krener and A. Isidori, Linearization by output injection and nonlinear observers, Systems Control Lett., 3 (1983), pp. 47-52. · Zbl 0524.93030 |
| [15] | M. Krichman, E. D. Sontag, and Y. Wang, Input-output-to-state stability, SIAM J. Control Optim., 39 (2001), pp. 1874-1928. · Zbl 1005.93044 |
| [16] | R. Li, Y. Hong, and X. Wang, Nonlinear norm-observability and simulation of control systems, Systems Control Lett., 105 (2017), pp. 14-19. · Zbl 1372.93046 |
| [17] | H. Lin, Mission accomplished: An introduction to formal methods in mobile robot motion planning and control, Unmanned Systems, 2 (2014), pp. 201-216. |
| [18] | R. Marino, Adaptive observers for single output nonlinear systems, IEEE Trans. Automat. Control, 35 (1990), pp. 1054-1058. · Zbl 0729.93016 |
| [19] | R. Marino and P. Tomei, Global adaptive output-feedback control of nonlinear systems, Part I: Linear parameterization, IEEE Trans. Automat. Control, 38 (1993), pp. 17-32. · Zbl 0783.93032 |
| [20] | M. A. Müller and D. Liberzon, Input/output-to-state stability and state-norm estimators for switched nonlinear systems, Automatica J. IFAC, 48 (2012), pp. 2029-2039. · Zbl 1257.93088 |
| [21] | G. J. Pappas, Bisimilar linear systems, Automatica J. IFAC, 39 (2003), pp. 2035-2047. · Zbl 1045.93033 |
| [22] | G. J. Pappas, G. Lafferriere, and S. Sastry, Hierarchically consistent control systems, IEEE Trans. Automat. Control, 45 (2000), pp. 1144-1160. · Zbl 0971.93004 |
| [23] | G. J. Pappas and S. Simić, Consistent abstractions of affine control systems, IEEE Trans. Automat. Control, 47 (2002), pp. 745-756. · Zbl 1364.93324 |
| [24] | G. Pola, C. Manes, A. J. van der Schaft, and M. D. D. Benedetto, Bisimulation equivalence of discrete-time stochastic linear control systems, IEEE Trans. Automat. Control, 63 (2018), pp. 1897-1912. · Zbl 1423.93367 |
| [25] | G. Pola and P. Tabuada, Symbolic models for nonlinear control systems: Alternating approximate bisimulations, SIAM J. Control Optim., 48 (2009), pp. 719-733. · Zbl 1194.93020 |
| [26] | P. Prabhakar, G. Dullerud, and M. Viswanathan, Stability preserving simulations and bisimulations for hybrid systems, IEEE Trans. Automat. Control, 60 (2015), pp. 3210-3225. |
| [27] | P. Prabhakar and J. Liu, Bisimulations for input-output stability of hybrid systems, in Proceedings of the IEEE 55th Conf. Decision and Control, 2016, Piscataway, NJ, pp. 5515-5520. |
| [28] | P. Prabhakar, J. Liu, and R. M. Murray, Pre-orders for reasoning about stability properties with respect to input of hybrid systems, in Proc. of the International Conf. on Embedded Software, Piscataway, NJ, 2013, pp. 1-10. |
| [29] | M. Rungger and P. Tabuada, A notion of robustness for cyber-physical systems, IEEE Trans. Automat. Control, 61 (2016), pp. 2108-2123. · Zbl 1359.93434 |
| [30] | R. G. Sanfelice, Input-output-to-state stability tools for hybrid systems and their interconnections, IEEE Trans. Automat. Control, 59 (2014), pp. 1360-1366. · Zbl 1360.93636 |
| [31] | E. D. Sontag, Comments on integral variants of ISS, Systems Control Lett., 34 (1998), pp. 93-100. · Zbl 0902.93062 |
| [32] | E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd ed., Springer, New York, 1998. · Zbl 0945.93001 |
| [33] | E. D. Sontag and Y. Wang, Output-to-state stability and detectability of nonlinear systems, Systems Control Lett., 29 (1997), pp. 279-290. · Zbl 0901.93062 |
| [34] | E. D. Sontag and Y. Wang, Notions of input to output stability, Systems Control Lett., 38 (1999), pp. 235-248. · Zbl 0985.93051 |
| [35] | P. Tabuada and G. J. Pappas, Bisimilar control affine systems, Systems Control Lett., 52 (2004), pp. 49-58. · Zbl 1157.93300 |
| [36] | Y. Tang and Y. Hong, Hierarchical distributed control design for multi-agent systems using approximate simulation, Acta Automat. Sinica, 39 (2013), pp. 868-874. |
| [37] | A. J. van der Schaft, Equivalence of dynamical systems by bisimulation, IEEE Trans. Automat. Control, 49 (2004), pp. 2160-2172. · Zbl 1365.93212 |
| [38] | K. Yang and H. Ji, Hierarchical analysis of large-scale control systems via vector simulation function, Systems Control Lett., 102 (2017), pp. 74-80. · Zbl 1377.93010 |
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. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
