Theoretical Foundations of Chemical Engineering

, Volume 49, Issue 5, pp 612–621 | Cite as

Discontinuity analysis for the treatment of nonlinear lumped-parameter systems for singular inputs



Different approaches suggested for the treatment of nonlinear systems subjected to singular inputs are either approximate or difficult to apply. In this paper, discontinuity analysis is used for this treatment, and is found to represent a good compromise between accuracy and ease of applicability. An example of non-isothermal CSTR representing nonlinear lumped-parameter chemical engineering systems is considered for the treatment. In the treatment, the integration of a model is carried out across the time of jump discontinuity by including singularity in the cause. This allows a priori estimation of initial conditions. The estimated initial conditions are then used for the initialization of the numerical solution of the model, whose singular input variables are set at their pre-initial values. The accuracy of the procedure in providing a reliable estimation of the system behavior is quantified by comparison with the numerical solutions initialized through the application of physical balances to the initial effects of singularity on the system, and with the numerical solutions obtained through pulse approximation for impulse. The procedure has simple and uniform applicability in comparison to the application of physical balances, and is substantially accurate than the commonly used pulse approximation method.


Dynamic simulation Impulse Nonlinear Process modeling Systems analysis 


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  1. 1.
    Ahuja, S., Second-order numerator-dynamics systems: Effects of initial discontinuities, Theor. Found. Chem. Eng., 2010, vol. 44, pp. 300–308.CrossRefGoogle Scholar
  2. 2.
    Ahuja, S., Effects of initial discontinuities on nonlinear systems represented by differential equations with terms containing differentials of the input function, Chem. Eng. Comm., 2011, vol. 198, pp. 760–782.CrossRefGoogle Scholar
  3. 3.
    Ahuja, S., Discontinuity analysis for the treatment of lumped-parameter chemical engineering systems for singular inputs, Ph D Thesis, Thapar University, Patiala, India, 2013.Google Scholar
  4. 4.
    Alopaeus, V., Lavi, H., and Aitamaa, J., A dynamic model for plug-flow reactor state profiles, Comput. Chem. Eng., 2008, vol. 32, pp. 1494–1506.CrossRefGoogle Scholar
  5. 5.
    Pour, N.D., Huang, B., and Shah, S.L., Subspace approach to identification of step-response model from closed-loop data, Ind. Eng. Chem. Res., 2010, vol. 49, pp. 8558–8567.CrossRefGoogle Scholar
  6. 6.
    Chaves, M., Sontag, E.D., and Dinerstein, R.J., Optimal length and signal amplification in weakly activated signal transduction cascades, J. Phys. Chem. B, 2004, vol. 108, pp. 15311–15320.CrossRefGoogle Scholar
  7. 7.
    Nauman, E.B., Residence time theory, Ind. Eng. Chem. Res., 2008, vol. 47, pp. 3752–3766.CrossRefGoogle Scholar
  8. 8.
    Čermáková, J., Siyakatshana, N., Silar, F., Kudrna, V., Jahoda, M., and Machon, V., Comparison of residence time distributions of liquid for different types of input signal using a stimulus-response technique, Chem. Pap., 2003, vol. 57, pp. 427–431.Google Scholar
  9. 9.
    Tondeur, D., Kabir, H., Luo, L.A., and Granger, J., Multicomponent adsorbtion equllibria from impulse response chromatography, Chem. Eng. Sci., 1996, vol. 51, pp. 3781–3799.CrossRefGoogle Scholar
  10. 10.
    Lee, P.J., Vítkovský, J.P., Lambert, M.F., Simpson, A.R., and Liggett, J., Leak location in pipe lines using the impulse response function, J. Hydraul. Res., 2007, vol. 45, pp. 643–652.CrossRefGoogle Scholar
  11. 11.
    Ramasamy, M. and Sundaramoorthy, S., PID controller tuning for desired closed-loop responses for SISO systems using impulse response, Comput. Chem. Eng., 2008, vol. 32, pp. 1773–1788.CrossRefGoogle Scholar
  12. 12.
    Silva, R., Sbarbaro, D., and Barra, B.A.L., Closedloop process identification under PI control: A time domain approach, Ind. Eng. Chem. Res., 2006, vol. 45, pp. 4671–4678.CrossRefGoogle Scholar
  13. 13.
    Zhang, W., Ou, L., and Gu, D., Algebraic solution to H2 control problems, Ind. Eng. Chem. Res., 2006, vol. 45, pp. 7151–7162.CrossRefGoogle Scholar
  14. 14.
    Sarrico, C.O.R., The multiplication of distributions and the Tsodyks model of synapses dynamics, Int. J. Math. Anal., 2012, vol. 6, pp. 999–1014.Google Scholar
  15. 15.
    Brigola, R. and Singer, P., On initial conditions, generalized functions and Laplace transform, Electr. Eng., 2009, vol. 91, pp. 9–13.Google Scholar
  16. 16.
    Lundberg, K.H., Miller, H.R., and Trumper, R.L., Initial conditions, generalized functions, and the Laplace transform, IEEE Control Syst. Mag., 2007, vol. 27, pp. 22–35.CrossRefGoogle Scholar
  17. 17.
    Muralidhar, G.S., Bovik, A.C., and Markey, M.K., Noise analysis of a new singularity index, IEEE Trans. Signal Process., 2013, vol. 61, pp. 6150–6163.CrossRefGoogle Scholar
  18. 18.
    Gezici, S., Kobayashi, H., Poor, H.V., and Molisch, A.F., Performance evaluation of impulse radio UWB system with pulse-based polarity randomization, IEEE Trans. Signal Process., 2005, vol. 53, pp. 2537–2549.CrossRefGoogle Scholar
  19. 19.
    Makila, P.M., A note on the Laplace transform method for initial value problems, Int. J. Control, 2006, vol. 79, pp. 36–41.CrossRefGoogle Scholar
  20. 20.
    Pilipchuk, V.N., Application of special non-smooth temporal transformations to linear and nonlinear systems under discontinuous and impulsive excitation, Nonlinear Dyn., 1999, vol. 18, pp. 203–234.CrossRefGoogle Scholar
  21. 21.
    Orlov, Y., Schwartz’ distribution in nonlinear setting: Applications to differential equations, filtering and optimal control, Math. Prob. Eng., 2002, vol. 8, pp. 367–387.CrossRefGoogle Scholar
  22. 22.
    Luyben, W.L., Process Modeling, Simulation and Control for Chemical Engineers, New York: McGraw-Hill, 1996, 2nd ed.Google Scholar
  23. 23.
    Bequette, B.W., in Process Dynamics: Modeling, Analysis, and Simulation, New Jersey: Prentice Hall, 1998, pp. 507–514, pp. 559–562.Google Scholar

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© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Department of Chemical EngineeringThapar UniversityPatialaIndia

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