Abstract
This paper proposes a new tool for quantized nonlinear control design of dynamic systems transformable into the dynamically perturbed strict-feedback form. To address the technical challenges arising from measurement and actuator quantization, a new approach based on set-valued maps is developed to transform the closed-loop quantized system into a large-scale system composed of input-to-state stable (ISS) subsystems. For each ISS subsystem, the inputs consist of quantization errors and interacting states, and moreover, the ISS gains can be assigned arbitrarily. Then, the recently developed cyclic-small-gain theorem is employed to guarantee input-to-state stability with respect to quantization errors and to construct an ISS-Lyapunov function for the closed-loop quantized system. Interestingly, it is shown that, under some realistic assumptions, any n-dimensional dynamically perturbed strict-feedback nonlinear system can be globally practically stabilized by a quantized control law using 2n three-level dynamic quantizers.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Brockett RW, Liberzon D (2000) Quantized feedback stabilization of linear systems. IEEE Trans Autom Control 45: 1279–1289
Miller RK, Mousa MS, Michel AN (1988) Quantization and overflow effects in digital implementation of linear dynamic controllers. IEEE Trans Autom Control 33: 698–704
Delchamps DF (1990) Stabilizing a linear system with quantized state feedback. IEEE Trans Autom Control 35: 916–924
Fu M, Xie L (2005) The sector bound approach to quantized feedback control. IEEE Trans Autom Control 50: 1698–1711
Elia N, Mitter SK (2001) Stabilization of linear systems with limited information. IEEE Trans Autom Control 46: 1384–1400
Ceragioli F, De Persis C (2007) Discontinuous stabilization of nonlinear systems: quantized and switching controls. Syst Control Lett 56: 461–473
De Persis C (2009) Robust stabilization of nonlinear systems by quantized and ternary control. Syst Control Lett 58: 602–608
Liberzon D, Hespanha JP (2005) Stabilization of nonlinear systems with limited information feedback. IEEE Trans Autom Control 50: 910–915
Liberzon D, Nešić D (2007) Input-to-state stabilization of linear systems with quantized state measurements. IEEE Trans Autom Control 52: 767–781
Grüne L (2002) Asymptotic behavior of dynamical and control systems under perturbation and discretization. Lecture Notes in Mathematics, vol 1783. Springer, Berlin
Liberzon D (2003) Hybrid feedback stabilization of systems with quantized signals. Automatica 39: 1543–1554
Liu T, Jiang ZP, Hill DJ (2012) A sector bound approach to feedback control of nonlinear systems with state quantization. Automatica 48: 145–152
Teel AR, Praly L (1995) Tools for semiglobal stabilization by partial state and output feedback. SIAM J Control Optim 33: 1443–1488
De Persis C (2005) n-bit stabilization of n-dimensional nonlinear systems in feedforward form. IEEE Trans Autom Control 50: 299–311
Liberzon D (2003) Switching in systems and control. Birkhäuser, Boston
Ledyaev YS, Sontag ED (1999) A Lyapunov characterization of robust stabilization. Nonlinear Anal 37: 813–840
Freeman RA, Kokotović PV (1996) Robust nonlinear control design: state-space and Lyapunov techniques. Birkhäuser, Boston
Krstić M, Kanellakopoulos I, Kokotović PV (1995) Nonlinear and adaptive control design. Wiley, New York
Sontag ED (2007) Input to state stability: basic concepts and results. In: Nistri P, Stefani G (eds) Nonlinear and optimal control theory. Springer, Berlin, pp 163–220
Liberzon D (2006) Quantization, time delays, and nonlinear stabilization. IEEE Trans Autom Control 51: 1190–1195
Liberzon D (2008) Observer-based quantized output feedback control of nonlinear systems. In: Proceedings of the 17th IFAC World Congress, pp 8039–8043
Jiang ZP, Teel AR, Praly L (1994) Small-gain theorem for ISS systems and applications. Math Control Signals Syst 7: 95–120
Nešić D, Liberzon D (2009) A unified framework for design and analysis of networked and quantized control systems. IEEE Trans Autom Control 54: 732–747
Jiang ZP, Wang Y (2008) A generalization of the nonlinear small-gain theorem for large-scale complex systems. In: Proceedings of the 7th World congress on intelligent control and automation, pp 1188–1193
Liu T, Hill DJ, Jiang ZP (2011) Lyapunov formulation of ISS cyclic-small-gain in continuous-time dynamical networks. Automatica 47: 2088–2093
Liu T, Jiang ZP, Hill DJ (2012) Small-gain based output-feedback controller design for a class of nonlinear systems with actuator dynamic quantization. IEEE Trans Autom Control (in press)
Jiang ZP, Mareels IMY, Wang Y (1996) A Lyapunov formulation of the nonlinear small-gain theorem for interconnected systems. Automatica 32: 1211–1214
Teel AR Input-to-state stability and the nonlinear small-gain theorem. Private communication
Potrykus HG, Allgöwer F, Qin SJ (2003) The character of an idempotent-analytic nonlinear small gain theorem. In: Positive systems. Lecture notes in control and information science. Springer, Berlin, pp 361–368
Dashkovskiy S, Rüffer BS, Wirth FR (2007) An ISS small-gain theorem for general networks. Math Control Signals Syst 19: 93–122
Dashkovskiy S, Rüffer BS, Wirth FR (2010) Small gain theorems for large scale systems and construction of ISS Lyapunov functions. SIAM J Control Optim 48: 4089–4118
Rüffer BS (2007) Monotone systems, graphs, and stability of large-scale interconnected systems. Ph.D. thesis, University of Bremen, Germany. http://nbn-resolving.de/urn:nbn:de:gbv:46-diss000109058
Heemels WPMH, Weiland S (2007) Input-to-state stability and interconnections of discontinuous dynamical systems. Tech. Rep. DCT report 2007-104, Eindhoven University of Technology. http://repository.tue.nl/656556
Heemels WPMH, Weiland S (2008) Input-to-state stability and interconnections of discontinuous dynamical systems. Automatica 44: 3079–3086
Filippov AF (1988) Differential equations with discontinuous righthand sides. Kluwer Academic Publishers, Dordrecht
Federer H (1969) Geometric measure theory. Springer, New York
Hespanha JP, Liberzon D, Angeli D, Sontag ED (2005) Nonlinear norm-observability notions and stability of switched systems. IEEE Trans Autom Control 50: 154–168
Praly L, Jiang ZP (1993) Stabilization by output feedback for systems with ISS inverse dynamics. Syst Control Lett 21: 19–33
Jiang ZP (1999) A combined backstepping and small-gain approach to adaptive output-feedback control. Automatica 35: 1131–1139
Liu T, Jiang ZP, Hill DJ (2011) Quantized output-feedback control of nonlinear systems: a cyclic-small-gain approach. In: Proceedings of the 30th Chinese Control Conference, pp 487–492
Jiang ZP, Mareels IMY (1997) A small-gain control method for nonlinear cascade systems with dynamic uncertainties. IEEE Trans Autom Control 42: 292–308
Sharon Y, Liberzon D (2012) Input to state stabilizing controller for systems with coarse quantization. https://netfiles.uiuc.edu/liberzon/www/research/yoav-coarse-quant-iss.pdf (in press)
Praly L, Wang Y (1996) Stabilization in spite of matched unmodeled dynamics and an equivalent definition of input-to-state stability. Math Control Signals Syst 9: 1–33
Liu T, Jiang ZP, Hill DJ (2011) Quantized output-feedback control of nonlinear systems: a cyclic-small-gain approach. Tech. rep., The Australian National University. http://users.cecs.anu.edu.au/tengfei/TR.pdf
Karafyllis I, Jiang ZP (2011) Stability and stabilization of nonlinear systems. Springer, Berlin
Yakubovich VA, Leonov GA, Gelig AK (2004) Stability of stationary sets in control systems with discontinuous nonlinearities. World Scientific Press, Singapore
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported in part under the Australian Research Councils Discovery funding scheme (project number: FF0455875), in part by a seed grant from NYU and POLY, and by NSF grant DMS-0906659.
Rights and permissions
About this article
Cite this article
Liu, T., Jiang, ZP. & Hill, D.J. Quantized stabilization of strict-feedback nonlinear systems based on ISS cyclic-small-gain theorem. Math. Control Signals Syst. 24, 75–110 (2012). https://doi.org/10.1007/s00498-012-0079-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00498-012-0079-x