Analysis of Partial-ISS Property and Unification between ISS and Partial-ISS for Time-Varying Systems
[1]
T. Binazadeh, School of Electrical and Electronic Engineering, Shiraz University of Technology, Shiraz, Iran.
The notion of input-to-state stability (ISS) provides a framework to investigate the boundedness of all state variables of a nonlinear system in the presence of bounded inputs and initial states. In this paper, the problem of partial stability that is stability with respect to only some of the state variables is considered. Since, partial stability finds applications in many engineering problems, some new concepts like partially bounded, partially ultimately bounded and partial-input-to-state stability (partial-ISS) are introduced in this paper. These concepts provide an extension of partial stability, in the presence of bounded inputs and initial states. Moreover, some sufficient Lyapunov-like conditions are derived to check the partial-ISS property for the nonlinear systems and in this regard a theorem is proposed and proofed. These conditions motivate checking the partial-ISS property by investigating a Lyapunov function (which is called partial-ISS Lyapunov function) for the given nonlinear system. Additionally, the unification between partial-ISS and ISS property for nonlinear time-varying systems is presented. Finally, two examples are given to show the way of investigate the partial-ISS property.
Partial Stability, Partially bounded, Partial-Input-to-State Stability, Partial-ISS Lyapunov Function
[1]
V.V. Rumyantsev, On asymptotic stability and instability of motion with respect to a part of the variables, J. Appl. Math. Mech., 35(1) (1971) 10-30.
[2]
V. Chellaboina, WM. Haddad, A unification between partial stability and stability theory for time-varing systems, IEEE Cont. Syst. Mag., 22(6) (2002) 66-75.
[3]
VI. Vorotnikov, Partial stability and control, Boston: Birkhauser, 1998.
[4]
T. Binazadeh, MJ. Yazdanpanah, Partial stabilization of uncertain nonlinear systems, ISA T., 51 (2012) 298-303.
[5]
M. H. Shafiei, T. Binazadeh, Partial Stabilization-based Guidance, ISA T., 51 (2012) 141-145.
[6]
W. Hu, J. Wang, X. Li, An approach of partial control design for system control and synchronization, Chaos Soliton. Fract., 39(3) (2009) 1410-1417.
[7]
T. Binazadeh, M. J., Yazdanpanah, Robust partial control design for nonlinear control systems: A guidance application, ImechE, Int. J. Syst. Cont. Eng., 226 (2011) 233-242.
[8]
T. Binazadeh, M. J., Yazdanpanah, Partial stabilization approach to 3-dimensional guidance law design, J Dyn. Syst-T ASME, 133 (2011) 064504 (4 pages).
[9]
M. H. Shafiei, T. Binazadeh, Application of partial sliding mode in guidance problem, ISA T., 52(2), 192-197 (2013)
[10]
D., Angeli, E. D. Sontag, Y.,Wang, A characterization of integral input-to-state stability, IEEE T. Automat. Cont., 45(6) (2000) 1082–1097.
[11]
Z.P. Jiang, M.Arcak, Robust global stabilization with ignored input dynamics: an input-to-state stability (ISS) small-gain approach, IEEE T. Automat. Cont., 46 (9) (2001) 1411 – 1415.
[12]
D., Angeli, E. D. Sontag, Y.,Wang, Input-to-state stability with respect to inputs and their derivatives, Int. J. Robust Nonlin., 13(11) (2003) 1035–1056.
[13]
Z. P. Jiang, M., Arcak, Input-to-state stability of time-delay systems: a link with exponential stability, IEEE T. Automat. Cont., 53(6) (2008) 1526-1531.
[14]
E. D., Sontag, Input to State Stability: Basic Concepts and Results, Nonlin. Optim. Cont. Theory, Lecture notes in mathematic, (2008) 163-220.
[15]
C., Ning, Y., He, M., Wu, Q., Liu, J., She, Input-to-state stability of nonlinear systems based on an indefinite Lyapunov function, Sys. & Cont. Let., 61(12) (2012)1254-1259
[16]
H., Zhang, P. M., Dower, Computation of tight integral input-to-state stability bounds for nonlinear systems, Sys. & Cont. Let., 62(4) (2013) 355-365
[17]
E.D. Sontag, Y. Wang, Notions of input to output stability, Sys. & Cont. Let., 38(45) (1999) 235-248.
[18]
H.K. Khalil, Nonlinear Systems. 3rd Edition, Prentice-Hal, 2002.
[19]
H. J. Marques, Nonlinear Control Systems. John Wiley & Sons, 2003.
[20]
W.M. Haddad, V. Chellaboina, Nonlinear Dynamical Systems and Control, A Lyapunov-Based Approach, Princeton University Press, 2008
