Abstract
This chapter deals with a new approach to robust control design for a class of nonlinearly affine control systems. The dynamic models under consideration are described by implicit differential equations in the presence of additive bounded uncertainties. The proposed robust feedback design procedure is based on an extended version of the classical invariant ellipsoid technique. In this book, this extension is called the attractive ellipsoid method. The stability/robustness analysis of the resulting closed-loop system involves a modified descriptor approach associated with the usual Lyapunov-type methodology. The theoretical schemes elaborated in our contribution are finally illustrated by a simple computational example.
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Basin, M., & Calderon-Alvarez, D. (2010). Optimal filtering over linear observations with unknown parameters. Journal of the Franklin Institute, 347, 988–1000.
Basin, M., Rodriguez-Gonzalez, J., & Fridman, L. (2007). Optimal and robust control for linear state-delay systems. Journal of the Franklin Institute, 344, 830–845.
Boyd, S., Ghaoui, E., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. Philadelphia: SIAM.
Chistyakov, V. (2007). Nonlocal theorems on existence of solutions of differential-algebraic equations of index 1. Russian Mathematics, 51, 71–76.
Dahleh, M., Pearson, J., & Boyd, J. (1988). Optimal rejection of persistent disturbances, robust stability, and mixed sensitivity minimization. IEEE Transactions on Automatic Control, 33, 722–731.
Dai, L. (1989). Singular control systems. Berlin: Springer.
Fridman, E. (2006). Descriptor discretized Lyapunov functional method: analysis and design. IEEE Transactions on Automatic Control, 51, 890–897.
Fridman, E. (2010). A refined input delay approach to sampled-data control. Automatica, 46, 421–427.
Haddad, W., & Chellaboina, V. (2008). Nonlinear Dynamical Systems and Control. Princeton: Princeton University Press.
Karafyllis, I., & Jiang, Z.-P. (2011). Stability and stabilization of nonlinear systems. London: Springer.
Khalil, H. (2002). Nonlinear systems. Upper Saddle River: Prentice Hall.
Kunkel, P., & Mehrmann, V. L. (2006). Differential-algebraic equations: analysis and numerical solutions. Switzerland: EMS Publishing House.
Kurzhanski, A., & Veliov, V. (1994). Modeling techniques and uncertain systems. New York: Birkhäuser.
Masubuchi, I., Kamitane, Y., Ohara, A., & Suda, N. (1997). h ∞ control for descriptor systems: A matrix inequalities approach. Automatica, 33(4), 669–673.
Mera, M., Poznyak, A., & Azhmyakov, V. (2011). On the robust control design for a class of continuous-time dynamical systems with a sample-data output. In Proceedings of the IFAC World Congress.
Michel, A., Hou, L., & Liu, D. (2007). Stability of dynamical systems. New York: Birkhäuser.
Polyak, B. T., & Topunov, M. V. (2008) Suppression of bounded exogenous disturbances: Output feedback. Automation and Remote Control, 69, 801–818. ellipsoids/polyak2008.pdf.
Poznyak, A., Azhmyakov, V., & Mera, M. (2011). Practical output feedback stabilisation for a class of continuous-time dynamic systems under sample-data outputs. International Journal of Control, 84, 1408–1416.
Reis, T., & Stykel, T. (2011). Lyapunov balancing for passivity-preserving model reduction of RC circuits. SIAM Journal of Applied Dynamic Systems, 10, 1–34.
Rheinbolt, W. (1981). On the existence and uniqueness of solutions of nonlinear semi-implicit differential-algebraic equations. Nonlinear analysis. Theory, Methods and Applications, 16, 642–661.
Rheinbolt, W. (1988). Differential-algebraic systems as differential equations on manifolds. Mathematics of Computation, 43, 473–482.
Takaba, K. (2002). Stability analysis of interconnected implicit systems based on passivity. In Proceedings of the IFAC 15th World Congress, Barcelona, Spain.
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Poznyak, A., Polyakov, A., Azhmyakov, V. (2014). Robust Control of Implicit Systems. In: Attractive Ellipsoids in Robust Control. Systems & Control: Foundations & Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-09210-2_7
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DOI: https://doi.org/10.1007/978-3-319-09210-2_7
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