Global stabilization of switched nonlinear systems in non-triangular form and its application. (English) Zbl 1293.93636
Summary: This paper investigates the problem of global stabilization of switched nonlinear systems in non-triangular form whose subsystems are not assumed to be asymptotically stabilizable. The use of multiple Lyapunov functions (MLFs) method permits removal of a common restriction in which the nonlinear structures in the non-switched nonlinear systems are restricted to a triangular structure when applying backstepping. Using the MLFs method and the adding a power integrator technique, we design state-feedback controllers for individual subsystems and construct a switching law to guarantee asymptotic stability of the closed-loop switched system. As an application of the proposed design method, the global stabilization problem of a continuously stirred tank reactor (CSTR) system and two inverted pendulums which cannot be handled by the existing methods is investigated.
MSC:
| 93D20 | Asymptotic stability in control theory |
| 93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |
| 93D30 | Lyapunov and storage functions |
| 93C15 | Control/observation systems governed by ordinary differential equations |
Keywords:
global stabilization; switched nonlinear systems in non-triangular form; multiple Lyapunov functions (MLFs); state-feedback controller; guarantee asymptotic stability; closed-loop switched systemReferences:
| [1] | Adesina, A. A.; Adewale, K. E.P., Steady-state cubic autocatalysis in an isothermal stirred tank, Ind. Eng. Chem. Res., 30, 430-434 (1991) |
| [2] | Alvarez-Ramirez, J., Observers for a class of continuous tank reactors via temperature measurement, Chem. Eng. Sci., 50, 1393-1399 (1995) |
| [3] | Alvarez-Ramirez, J.; Femat, R., Robust PI stabilization of a class of chemical reactors, Syst. Control Lett., 38, 219-225 (1999) · Zbl 0985.93040 |
| [4] | Branicky, M. S., Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Autom. Control, 43, April (4), 475-482 (1998) · Zbl 0904.93036 |
| [5] | Bacciotti, A., Stabilization by means of state space depending switching rules, Syst. Control Lett., 53, 195-201 (2004) · Zbl 1157.93480 |
| [6] | Cheng, D.; Guo, L.; Lin, Y.; Wang, Y., Stabilization of switched linear systems, IEEE Trans. Autom. Control, 50, May (5), 661-666 (2005) · Zbl 1365.93389 |
| [7] | Cao, M.; Morse, A. S., Dwell-time switching, Syst. Control Lett., 59, 57-65 (2010) · Zbl 1186.93005 |
| [8] | Colaneri, P.; Geromel, J. C.; Astolfi, A., Stabilization of continuous time switched nonlinear systems, Syst. Control Lett., 57, 95-103 (2008) · Zbl 1129.93042 |
| [9] | Chen, Z.; Huang, J., Global robust stabilization of cascaded polynomial systems, Syst. Control Lett., 47, 445-453 (2002) · Zbl 1106.93336 |
| [10] | Dashkovskiy, S.; Pavlichkov, S., Global uniform input-to-state stabilization of large-scale interconnections of MIMO generalized triangular form switched systems, Math. Control Signals Syst., 24, 135-168 (2012) · Zbl 1238.93095 |
| [11] | El-Farra, N. H.; Mhaskar, P.; Christofides, P. D., Output feedback control of switched nonlinear systems using multiple Lyapunov functions, Syst. Control Lett., 54, 1163-1182 (2005) · Zbl 1129.93497 |
| [12] | Feng, W.; Zhang, J., Input-to-state stability of switched nonlinear systems, Sci. China Ser. F Inf. Sci., 51, 12, 1992-2004 (2008) · Zbl 1291.93284 |
| [13] | Freeman, R. A.; Kokotović, P. V., Robust Control of Nonlinear Systems (1996), Birkhauser: Birkhauser Boston · Zbl 0863.93075 |
| [14] | Fissore, D., Robust control in presence of parametric uncertaintiesobserver-based feedback controller design, Chem. Eng. Sci., 63, 1890-1900 (2008) |
| [15] | Goebel, R.; Sanfelice, R. G.; Teel, A. R., Hybrid dynamical systems, IEEE Control Syst. Mag., 29, 28-93 (2009) · Zbl 1395.93001 |
| [16] | Ghommam, J.; Mnif, F.; Benali, A.; Derbel, N., Asymptotic backstepping stabilization of an underactuated surface vessel, IEEE Trans. Control Syst. Technol., 14, November (6), 1150-1157 (2006) |
| [17] | Ge, S. S.; Hang, C. C.; Zhang, T., Nonlinear adaptive control using neural networks and its application to CSTR systems, J. Process Control, 9, 313-323 (1998) |
| [18] | Hespanha, J. P., Root-mean-square gains of switched linear systems, IEEE Trans. Autom. Control, 48, November (11), 2040-2045 (2003) · Zbl 1364.93213 |
| [19] | Hong, Y.; Wang, J.; Cheng, D., Adaptive finite-time control of nonlinear systems with parameteric uncertainty, IEEE Trans. Autom. Control, 51, May (5), 858-862 (2006) · Zbl 1366.93290 |
| [20] | Han, T.; Ge, S. S.; Lee, T., Adaptive neural control for a class of switched nonlinear systems, Syst. Control Lett., 58, 109-118 (2009) · Zbl 1155.93414 |
| [21] | Krstić, M.; Kanellakopoulos, I.; Kokotović, P., Nonlinear and Adaptive Control Design (1995), Wiley, Interscience · Zbl 0763.93043 |
| [22] | Kravaris, C.; Palanki, S., Robust nonlinear state feedback under structured uncertainty, AIChE J., 34, 1119-1127 (1988) |
| [23] | Korobov, V. I.; Pavlichkov, S. S., Global properties of the triangular systems in the singular case, J. Math. Anal. Appl., 342, 1426-1439 (2008) · Zbl 1141.93023 |
| [24] | Liberzon, D., Switching in Systems and Control (2003), Birkhauser: Birkhauser Boston, MA · Zbl 1036.93001 |
| [25] | Long, L.; Zhao, J., Global stabilization for a class of switched nonlinear feedforward systems, Syst. Control Lett., 60, 734-738 (2011) · Zbl 1226.93114 |
| [26] | Long, L.; Zhao, J., \(H_\infty\) control of switched nonlinear systems in p-normal form using multiple Lyapunov functions, IEEE Trans. Autom. Control, 57, May (5), 1285-1291 (2012) · Zbl 1369.93253 |
| [29] | Lu, W.; Doyle, J., \(H_\infty\) control of nonlinear systemsa convex characterization, IEEE Trans. Autom. Control, 40, September (9), 1668-1675 (1995) · Zbl 0836.93014 |
| [30] | McClamroch, N. H.; Kolmanovsky, I., Performance benefits of hybrid control design for linear and nonlinear systems, Proc. IEEE, 88, July (7), 1083-1096 (2000) |
| [31] | Mojica-Nava, E.; Quijano, N.; Rakoto-Ravalontsalama, N.; Gauthier, A., A polynomial approach for stability analysis of switched systems, Syst. Control Lett., 59, 98-104 (2010) · Zbl 1186.93035 |
| [32] | Ma, R.; Zhao, J., Backstepping design for global stabilization of switched nonlinear systems in lower triangular form under arbitrary switchings, Automatica, 46, 1819-1823 (2010) · Zbl 1218.93075 |
| [33] | Parulekar, S. J., Modal analysis and optimization of isothermal autocatalytic reactions, Chem. Eng. Sci., 53, 2379-2394 (1998) |
| [34] | Pavlichkov, S. S.; Ge, S. S., Global stabilization of the generalized MIMO triangular systems with singular input-output links, IEEE Trans. Autom. Control, 54, August (8), 1794-1806 (2009) · Zbl 1367.93532 |
| [35] | Qian, C.; Lin, W., Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization, Syst. Control Lett., 42, 185-200 (2001) · Zbl 0974.93050 |
| [36] | Qian, C.; Lin, W., A continuous feedback approach to global strong stabilization of nonlinear systems, IEEE Trans. Autom. Control, 46, July (7), 1061-1079 (2001) · Zbl 1012.93053 |
| [37] | Raptis, I. A.; Valavanis, K. P.; Moreno, W. A., A novel nonlinear backstepping controller design for helicopters using the rotation matrix, IEEE Trans. Control Syst. Technol., 19, March (2), 465-473 (2011) |
| [38] | Sepulchre, R.; Janković, M.; Kokotović, P., Constructive Nonlinear Control (1996), Springer-Verlag: Springer-Verlag London · Zbl 1067.93500 |
| [39] | Schwartz, C. A.; Maben, E., A minimum energy approach to switching control for mechanical systems, (Morse, A. S., Control Using Logic-Based Switching (1997), Springer-Verlag: Springer-Verlag New York), 142-150 · Zbl 0875.93019 |
| [41] | Sakamoto, N.; van der Schaft, A. J., Analytical approximation methods for the stabilizing solution of the Hamilton-Jacobi equation, IEEE Trans. Autom. Control, 53, November (10), 2335-2350 (2008) · Zbl 1367.93535 |
| [42] | Spooner, J. T.; Passino, K. M., Decentralized adaptive control of nonlinear systems using radial basis neural networks, IEEE Trans. Autom. Control, 44, November (11), 2050-2057 (1999) · Zbl 1136.93363 |
| [43] | Salehi, S.; Shahrokhi, M., Adaptive fuzzy backstepping approach for temperature control of continuous stirred tank reactors, Fuzzy Sets Syst., 160, 1804-1818 (2009) · Zbl 1175.93133 |
| [44] | Viel, F.; Jadot, F.; Bastin, G., Robust feedback stabilization of chemical reactors, IEEE Trans. Autom. Control, 42, April (4), 473-481 (1997) · Zbl 0878.93049 |
| [45] | Wu, J., Stabilizing controllers design for switched nonlinear systems in strict-feedback form, Automatica, 45, 1092-1096 (2009) · Zbl 1162.93030 |
| [46] | Yazdi, M. B.; Jahed-Motlagh, M. R., Stabilization of a CSTR with two arbitrarily switching modes using model state feedback linearization, Chem. Eng. J., 155, 838-843 (2009) |
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