Second-order numerator-dynamics systems: Effects of initial discontinuities



Linear second-order systems with numerator-dynamics are considered and their transient responses are investigated for discontinuous inputs. Characteristic parameters for second-order systems with numerator dynamics are identified. The solution profiles of these systems depend upon the value of one such parameter relative to that of the others on the number line. Effects of initial jump discontinuities of the inputs on the correctness of the solutions are also treated. A methodology for analysis of discontinuities is presented. Initial discontinuities of the calculated response either get accounted for in the input function or the initial condition of the input function for all but the following case. For the impulse response alone, that too among inherent numerator-dynamics systems only, an initial di scontinuity cannot be accounted for in the input. The value of that discontinuity is thus required for correct solutions of these cases. The results are general and extendable to different inputs and higher order systems.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Coughanowr, D.R., Process Systems Analysis and Control, New York: McGraw-Hill, 1991.Google Scholar
  2. 2.
    Harriott, P., Process Control, India: Tata McGraw-Hill Ed., 1972.Google Scholar
  3. 3.
    Luyben, W.L., Process Modeling, Simulation and Control for Chemical Engineers, New York: McGraw-Hill, 1990.Google Scholar
  4. 4.
    Bequette, B.W., Process Dynamics: Modeling, Design and Simulation, Upper Saddle River: Prentice Hall PTR, 1998.Google Scholar
  5. 5.
    Bequette, B.W., Process Dynamics: Modeling, Design and Simulation, New Jersey: Prentice Hall PTR, Upper Saddle River, 2002.Google Scholar
  6. 6.
    Hwang, C. and Hwang, J.H., A New Step Iterative Method for Optimal Reduction of Linear SISO Systems, J. Franklin Inst., 1996, vol. 133, no. 5, pp. 631–645.CrossRefGoogle Scholar
  7. 7.
    Padma, S.R. and Chidambaram, M., Simple Method of Calculating Set Point Weighting Parameter for Unstable Systems with a Zero, Comp. Chem. Eng., 2004, vol. 28, no. 11, pp. 2433–2437.CrossRefGoogle Scholar
  8. 8.
    Strogatz, S.H., Love-Affairs and Differential Equations, Math. Magazine, 1988, vol. 61, p. 35.Google Scholar
  9. 9.
    You, K.H. and Lee, E.B., Time Maximum Disturbance Switch Curve Isochrones of Linear Second-Order with Numenator-Dynamics, J. Franklin Inst., 2000, vol. 337, no. 6, p. 725–742.CrossRefGoogle Scholar
  10. 10.
    You, K.H. and Lee, E.B., BIBO Stability Integral for Second-Order Systems with Numenator-Dynamics, Automatica, 2000, vol. 36, no. 11, pp. 1693–1699.CrossRefGoogle Scholar
  11. 11.
    Calabree, G., Finite Differencing Second-Order Systems Describing Black-Hole Space Times, Phys. Rev. D: Particles, Fields, Gravitation, and Cosmology, 2005, vol. 71, no. 2, pp. 1–4.Google Scholar
  12. 12.
    Garay, M.A.B., Melendzand, J.U., and Berriel, M.C.H., Didactic Prototype of Feed-Back Automatic Control, Advances en Ingenieria Qumica, 1998, vol. 7, no. 2, pp. 139–142.Google Scholar
  13. 13.
    Hall, I.A.M., Human Pilot as a Servo Problem, J. R. Aeronautical Soc., 1963, vol. 67, p. 351.Google Scholar
  14. 14.
    Hoffer, M.S. and Resnick, W., A Study of Agitated Liquid/Liquid Dispersions: Dynamic Response of Dispersion Geometry to Changes in Composition and Temperature, Trans. Inst. Chem. Eng., 1979, vol. 57, no. 1, pp. 1–7.Google Scholar
  15. 15.
    Panda, R.C., Estimation of Parameters of Under-Damped Second-Order Plus Dead-Time Systems Using Relay Feedback, Comp. Chem. Eng., 2006, vol. 30, no. 5, pp. 425–434.CrossRefGoogle Scholar
  16. 16.
    Stephanopoulos, G., Chemical Process Control: An Introduction to Theory and Practice, Englewood Cliffs: Prentice-Hall, 1984.Google Scholar
  17. 17.
    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
  18. 18.
    Bulgakova, N.M. and Burakov, I.M., Non-Linear Hydrodynamic Waves: Effect of Equation of State, Phys. Rev. E: Statistical, Non-Linear, and Soft Matter Phys., 2004, vol. 70, no. 3–2, pp. 036303-1–036303-5.Google Scholar
  19. 19.
    Hirshberg, J., Alksne, A., Colburn, D.S., Bame, S.J., and Hundhausen, A.J., Observation of Solar Flare Induced Interplanetary Shock and Helium-Enriched Drive Gas, J. Geophys. Res., 1970, vol. 75, no. 1, pp. 1–15.CrossRefGoogle Scholar
  20. 20.
    Karelsky, K.V, Papkov, V.V, and Petrosyan, AS., The Initial Discontinuity Decay Problem for Shallow Water Equations on Slopes, Phys. Lett. A, 2000, vol. 271, nos. 5, 6, pp. 349–357.CrossRefGoogle Scholar
  21. 21.
    Pederson, H. and Tanoff, M., Solving Parabolic PDE’s with Initial Discontinuities: Application to Mixing with Chemical Reactions, Comp. Chem. Eng., 1982, vol. 6, no. 3, pp. 197–207.CrossRefGoogle Scholar
  22. 22.
    Pons, P. and Blasquez, G., Transient Response of Capacitative Pressure Sensors and Actuators, A: Physical, 1992, vol. A32, nos. 1–3, pp. 616–621.Google Scholar
  23. 23.
    Poon, T.W., Yip, S., Ho, P.S., and Abraham, F.F., Ledge Interactions and Stress Relaxations on Silicon (001) Stepped Surfaces, Phys. Rev. B: Condens. Matter Mat. Phys., 1992, vol. 45, no. 7, pp. 3521–3531.Google Scholar
  24. 24.
    Shao, Z., Kong, D., and Li, Y., Global Solutions with Shock Waves to the Generalized Riemann Problem for a Class of Quasilinear Hyperbolic Systems of Balance Laws, Nonlinear Analysis, 2006, vol. 64, no. 11, pp. 2575–2603.CrossRefGoogle Scholar
  25. 25.
    Levenspiel, O., Chemical Reaction Engineering, New York: John Wiley and Sons, 1999.Google Scholar
  26. 26.
    Al-Hayani, W. and Casaiiss, L., On the Applicability of the Adomian Method to Initial Value Problems with Discontinuities, Appl. Math. Lett., 2006, vol. 19, no. 1, pp. 22–31.CrossRefGoogle Scholar
  27. 27.
    Benchhra, M., Ntonyas, S.K., and Vahab, A.O., Extremal Solutions of Second-Order Impulsive Dynamic Equations on Time Scales, J. Math. Anal. Appl., 2006, vol. 324, no. 1, pp. 425–434.CrossRefGoogle Scholar
  28. 28.
    Casasus, L. and Al-Hayani, W., The Decomposition Method for Ordinary Differential Equations with Discontinuities, Appl. Math. Comput., 2002, vol. 131, nos. 2–3, pp. 245–251.CrossRefGoogle Scholar
  29. 29.
    Karelsky, K.V. and Petrosyan, A.S., Particular Solutions and Riemann Problem for Modified Shallow Water Equations, Fluid Dynamic Res., 2006, vol. 38, no. 5, pp. 339–358.CrossRefGoogle Scholar
  30. 30.
    Kelevedjiev, P. and Seman, J., Extince of Solutions to Initial Value Problems for First-Order Differential Equations, Nonlinear Anal., 2004, vol. 57, nos. 7–8, pp. 879–889.CrossRefGoogle Scholar
  31. 31.
    Wylie, C.R. and Barrett, L.C., Advanced Engineering Mathematics, New Delhi: Tata McGraw-Hill, 1995.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Department of Chemical EngineeringThapar UniversityPatialaIndia

Personalised recommendations