A journal of IEEE and CAA , publishes high-quality papers in English on original theoretical/experimental research and development in all areas of automation
Volume 7 Issue 4
Jun.  2020

IEEE/CAA Journal of Automatica Sinica

  • JCR Impact Factor: 11.8, Top 4% (SCI Q1)
    CiteScore: 17.6, Top 3% (Q1)
    Google Scholar h5-index: 77, TOP 5
Turn off MathJax
Article Contents
Pierluigi Di Franco, Giordano Scarciotti and Alessandro Astolfi, "Stability of Nonlinear Differential-Algebraic Systems Via Additive Identity," IEEE/CAA J. Autom. Sinica, vol. 7, no. 4, pp. 929-941, July 2020. doi: 10.1109/JAS.2020.1003219
Citation: Pierluigi Di Franco, Giordano Scarciotti and Alessandro Astolfi, "Stability of Nonlinear Differential-Algebraic Systems Via Additive Identity," IEEE/CAA J. Autom. Sinica, vol. 7, no. 4, pp. 929-941, July 2020. doi: 10.1109/JAS.2020.1003219

Stability of Nonlinear Differential-Algebraic Systems Via Additive Identity

doi: 10.1109/JAS.2020.1003219
More Information
  • The stability analysis for nonlinear differential-algebraic systems is addressed using tools from classical control theory. Sufficient stability conditions relying on matrix inequalities are established via Lyapunov Direct Method. In addition, a novel interpretation of differential-algebraic systems as feedback interconnection of a purely differential system and an algebraic system allows reducing the stability analysis to a small-gain-like condition. The study of stability properties for constrained mechanical systems, for a class of Lipschitz differential-algebraic systems and for an academic example is used to illustrate the theory.


  • loading
  • 1Loosely speaking, the index indicates the number of time-differentiations required to reduce a DAE system to a system of ordinary differential equations, see [11] for a precise definition.
    2See [24] for detail on the transformation of fully-implicit DAE systems to the semi-explicit form and vice versa.
    3The dependence of $ h $ on the algebraic variable $ w $ is explicit only when the index is one. However, with some abuse of notation, we use $ h(x, w) $ for any index.
    4Throughout the paper all mappings are assumed to be smooth.
    5We consider the notion of “classical” solution as formulated in [27].
    6See Hadamard’s Lemma [33].
    7The matrices $ A_{i j} $ , for $ i = 1, 2 $ and $ j = 1, 2 $ , are not uniquely defined.
    8Again, see Hadamard’s Lemma [33].
    9See [31] for the concept of generalized $ {\cal{L}}_2 $ -gain.
    10In [37] $ a $ is constant.
  • [1]
    W. Blajer, “Index of differential-algebraic equations governing the dynamics of constrained mechanical systems,” Appl. Math. Model., vol. 16, no. 2, pp. 70–77, Feb. 1992. doi: 10.1016/0307-904X(92)90083-F
    O. Khatib, “A unified approach for motion and force control of robot manipulators: The operational space formulation,” IEEE J. Rob. Autom., vol. 3, no. 1, pp. 43–53, Feb. 1987. doi: 10.1109/JRA.1987.1087068
    A. Kumar and P. Daoutidis, “Control of nonlinear differential algebraic equation systems: An overview,” in Nonlinear Model Based Process Control, R. Berber and C. Kravaris, Eds. Dordrecht, Netherlands: Springer, 1998, pp. 311–344.
    R. Riaza, Differential-Algebraic Systems: Analytical Aspects and Circuit Applications. London, UK: World Scientific, 2008.
    T. Fliegner, H. Nijmeijer, and Ü. Kotta, “Some aspects of nonlinear discrete-time descriptor systems in economics,” in Predictability and Nonlinear Modelling in Natural Sciences and Economics, J. Grasman and G. van Straten, Eds. Dordrecht, Netherlands: Springer, 1994, pp. 581–590.
    P. Fritzson, Principles of Object-Oriented Modeling and Simulation With Modelica 2.1. Piscataway, USA: IEEE Press, 2004.
    M. Arnold, “DAE aspects of multibody system dynamics,” in Surveys in Differential-Algebraic Equations IV, A. Ilchmann and T. Reis, Eds, Cham, Germany: Springer, 2017.
    P. Di Franco, G. Scarciotti, and A. Astolfi, “A globally stable algorithm for the integration of high-index differential-algebraic systems,” IEEE Trans. Autom. Control, vol. 65, no. 5, pp. 2107–2122, May 2020. doi: 10.1109/TAC.2019.2939638
    T. Berger, “On observers for nonlinear differential-algebraic systems,” IEEE Trans. Autom. Control, vol. 64, no. 5, pp. 2150–2157, May 2019. doi: 10.1109/TAC.2018.2866438
    J. C. Arceo, M. Sánchez, V. Estrada-Manzo, and M. Bernal, “Convex stability analysis of nonlinear singular systems via linear matrix inequalities,” IEEE Trans. Autom. Control, vol. 64, no. 4, pp. 1740–1745, Apr. 2019. doi: 10.1109/TAC.2018.2854651
    V. Mehrmann, Index Concepts for Differential-Algebraic Equations. Berlin Heidelberg, Germany: Springer, 2015, pp. 676–681.
    L. Y. Dai, Singular Control Systems. Berlin, Heidelberg, Germany: Springer-Verlag, 1989.
    N. H. McClamroch, “Feedback stabilization of control systems described by a class of nonlinear differential-algebraic equations,” Syst. Control Lett., vol. 15, no. 1, pp. 53–60, Jul. 1990. doi: 10.1016/0167-6911(90)90044-U
    A. Kumar and P. Daoutidis, “Feedback control of nonlinear differential-algebraic-equation systems,” AIChE J., vol. 41, no. 3, pp. 619–636, Mar. 1995. doi: 10.1002/aic.690410319
    T. N. Chang and E. J. Davison, “Decentralized control of descriptor systems,” IEEE Trans. Autom. Control, vol. 46, no. 10, pp. 1589–1595, Oct. 2001. doi: 10.1109/9.956054
    J. Åslund and E. Frisk, “An observer for non-linear differential-algebraic systems,” Automatica, vol. 42, no. 6, pp. 959–965, Jun. 2006. doi: 10.1016/j.automatica.2006.01.026
    H. S. Wu and K. Mizukami, “Lyapunov stability theory and robust control of uncertain descriptor systems,” Int. J. Syst. Sci., vol. 26, no. 10, pp. 1981–1991, Oct. 1995. doi: 10.1080/00207729508929149
    P. G. Wang and J. Zhang, “Stability of solutions for nonlinear singular systems with delay,” Appl. Math. Lett., vol. 25, no. 10, pp. 1291–1295, Oct. 2012. doi: 10.1016/j.aml.2011.11.029
    D. F. Coutinho, A. S. Bazanella, A. Trofino, and A. S. Silva, “Stability analysis and control of a class of differential-algebraic nonlinear systems,” Int. J. Robust Nonlinear Control, vol. 14, no. 16, pp. 1301–1326, Nov. 2004. doi: 10.1002/rnc.950
    K. Takaba, N. Morihira, and T. Katayama, “H control for descriptor systems: A J-spectral factorization approach,” in Proc. 33rd IEEE Conf. Decision and Control, Lake Buena Vista, USA, 1994, pp. 2251–2256.
    I. Masubuchi, Y. Kamitane, A. Ohara, and N. Suda, “H control for descriptor systems: A matrix inequalities approach,” Automatica, vol. 33, no. 4, pp. 669–673, Apr. 1997. doi: 10.1016/S0005-1098(96)00193-8
    H. S. Wang, C. F. Yung, and F. R. Chang, “H control for nonlinear descriptor systems,” IEEE Trans. Automatic Control, vol. 47, no. 11, pp. 1919–1925, Nov. 2002.
    L. Y. Sun and Y. Z. Wang, “An undecomposed approach to control design for a class of nonlinear descriptor systems,” Int. J. Robust Nonlinear Control, vol. 23, no. 6, pp. 695–708, Apr. 2013. doi: 10.1002/rnc.2790
    C. W. Gear, “Differential-algebraic equation index transformations,” SIAM J. Sci. Stat. Comput., vol. 9, no. 1, pp. 39–47, Jan. 1988. doi: 10.1137/0909004
    P. Di Franco, G. Scarciotti, and A. Astolfi, “A note on the stability of nonlinear differential-algebraic systems,” IFAC-PapersOnLine, vol. 50, no. 1, pp. 7421–7426, Jul. 2017. doi: 10.1016/j.ifacol.2017.08.1501
    P. Di Franco, G. Scarciotti, and A. Astolfi, “A disturbance attenuation approach for the control of differential-algebraic systems,” in Proc. IEEE Conf. Decision and Control, Miami Beach, FL, USA, 2018, pp. 4695–4700.
    P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solution. Zurich, Switzerland: European Mathematical Society, 2006.
    D. C. Tarraf and H. H. Asada, “On the nature and stability of differential-algebraic systems,” in Proc. American Control Conf., Anchorage, USA, 2002, pp. 3546–3551.
    P. Di Franco, G. Scarciotti, and A. Astolfi, “On the stability of constrained mechanical systems,” in Proc. IEEE 56th Annu. Conf. Decision and Control, Melbourne, Australia, 2017, pp. 3170–3174.
    P. Di Franco, G. Scarciotti, and A. Astolfi, “Stabilization of differential-algebraic systems with Lipschitz nonlinearities via feedback decomposition,” in Proc. 18th European Control Conf., Naples, Italy, 2019.
    A. Rapaport and A. Astolfi, “Practical L2 disturbance attenuation for nonlinear systems,” Automatica, vol. 38, no. 1, pp. 139–145, Jan. 2002. doi: 10.1016/S0005-1098(01)00176-5
    J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “State-space solutions to standard H2 and Hcontrol problems,” IEEE Trans. Autom. Control, vol. 34, no. 8, pp. 831–847, Aug. 1989. doi: 10.1109/9.29425
    J. Nestruev, Smooth Manifolds and Observables. New York, USA: Springer, 2003.
    M. D. S. Aliyu and E. K. Boukas, “H filtering for nonlinear singular systems,” IEEE Trans. Circuits Syst. I:Regular Pap., vol. 59, no. 10, pp. 2395–2404, Oct. 2012. doi: 10.1109/TCSI.2012.2189038
    J. Sjöoberg, K. Fujimoto, and T. Glad, “Model reduction of nonlinear differential-algebraic equations,” IFAC Proc., vol. 40, no. 12, pp. 176–181, 2007. doi: 10.3182/20070822-3-ZA-2920.00030
    A. Rapaport and A. Astolfi, “A remark on the stability of interconnected nonlinear systems,” IEEE Trans. Autom. Control, vol. 49, no. 1, pp. 120–124, Jan. 2004. doi: 10.1109/TAC.2003.821407
    A. Isidori, Nonlinear Control Systems II, E. D. Sontag and M. Thoma, Eds. London, UK: Springer, 1995.
    J. LaSalle, “Some extensions of Liapunov’s second method,” IRE Trans. Circuit Theory, vol. 7, no. 4, pp. 520–527, Dec. 1960. doi: 10.1109/TCT.1960.1086720
    C. W. Gear, B. Leimkuhler, and G. K. Gupta, “Automatic integration of Euler-Lagrange equations with constraints,” J. Comput. Appl. Math., vol. 12-13, pp. 77–90, May 1985. doi: 10.1016/0377-0427(85)90008-1
    C. Führer and B. J. Leimkuhler, “Numerical solution of differential-algebraic equations for constrained mechanical motion,” Numerische Mathematik, vol. 59, no. 1, pp. 55–69, Dec. 1991. doi: 10.1007/BF01385770
    D. Koenig, “Observer design for unknown input nonlinear descriptor systems via convex optimization,” IEEE Trans. Autom. Control, vol. 51, no. 6, pp. 1047–1052, Jun. 2006. doi: 10.1109/TAC.2006.876807
    L. N. Zhou, C. Y. Yang, and Q. L. Zhang, “Observers for descriptor systems with slope-restricted nonlinearities,” Int. J. Autom. Comput., vol. 7, no. 4, pp. 472–478, Nov. 2010. doi: 10.1007/s11633-010-0529-1
    M. K. Gupta, N. K. Tomar, and S. Bhaumik, “Observer design for descriptor systems with Lipschitz nonlinearities: An LMI approach,” Nonlinear Dyn. Syst. Theory, vol. 14, no. 3, pp. 291–301, Jan. 2014.
    M. Abbaszadeh and H. J. Marquez, “A generalized framework for robust nonlinear H filtering of Lipschitz descriptor systems with parametric and nonlinear uncertainties,” Automatica, vol. 48, no. 5, pp. 894–900, May 2012. doi: 10.1016/j.automatica.2012.02.033
    W. W. Hager, “Updating the inverse of a matrix,” SIAM Rev., vol. 31, no. 2, pp. 221–239, 1989. doi: 10.1137/1031049
    H. K. Khalil, Nonlinear Systems. Upper Saddle River, USA: Prentice Hall, 1996.


    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索


    Article Metrics

    Article views (1503) PDF downloads(152) Cited by()


    • Representation of DAE systems as feedback interconnection.
    • Stability analysis forDAE systems via Lyapunov Method and Small Gain-like arguments.
    • Stability analysis for nonlinear mechanical systems with holonomic constraints.
    • Stability analysis of Lipschitz DAE systems.


    DownLoad:  Full-Size Img  PowerPoint