Although boundary control of linear partial differential equations has become an important research area, there is still no readily available simulation tool to help researchers analyze and design. In this article, a simulation method for some typical boundary control problems, combining symbolic math and a numerical method, is presented with application examples. In the intermediate steps of the simulation, an important by-product, transfer function of the controlled system, can be obtained, which makes the design of more advanced boundary controllers possible and much easier.
Morgül, Ö . 2002. An exponential stability result for the wave equation . Automatica38: 731-735 .
2.
Morgül, Ö . 2001. Stabilization and disturbance rejection for the beam equation . IEEE Transactions on Automatic Control46 (12): 1913-1918 .
3.
Conrad, F. , and Ö. Morgül . 1998. On the stability of a flexible beam with a tip mass . SIAM Journal on Control and Optimization36 (6): 1962-1986 .
4.
Morgül, Ö . 1998. Stabilization and disturbance rejection for the wave equation . IEEE Transactions on Automatic Control43 (1): 89-95 .
5.
Guo, B.-Z.2001. Riesz basis approach to the stabilization of a flexible beam with a tip mass . SIAM Journal on Control and Optimization39 (6): 1736-1747 .
6.
Guo, B.-Z.2002. Riesz basis property and exponential stability of controlled Euler-Bernoulli beam equations with variable coefficients . SIAM Journal on Control and Optimization40 (6): 1905-1923 .
7.
Partial differential equation (PDE) tool box user’s guide. 2002. Natick, MA: The Mathworks, Inc.
8.
Mitchell, A. R. , and D. Griffiths . 1980. The finite difference method in partial differential equations. New York: John Wiley .
9.
Jeffrey, A.2002. Advanced engineering mathematics. New York: Harcourt/Academic Press .
10.
Symbolic math toolbox user’s guide. 2002. Natick, MA: The Mathworks, Inc.
11.
Duffy, D. G.1993. On the numerical inversion of Laplace transforms: Comparison of three new methods on characeristic problems from applications . ACM Transactions on Mathematical Software19: 333-359 .
12.
Brančík, L. 1999. Programs for fast numerical inversion of Laplace transforms in Matlab language enviroment . In Konference MATLAB ’99 ZCU Plzen, pp. 27-39 .
13.
Liang, J. , Y. Chen , and B.-Z. Guo . 2003. A new boundary control method for beam equation with delayed boundary measurement using modified Smith predictors . In Proceedings of the 41st IEEE Conference on Decision and Control, December, Maui, HI, pp. 809-814 .
14.
Brančík, L. 1998. The fast computing method of numerical inversion of Laplace transforms using FFT algorithm . In Proceedings of the 5th EDS 98 International Conference, June, Brno, Czech Republic, pp. 97-100 .
15.
Brančík, L. 1999. An improvement of FFT-based numerical ILT procedure by application of ε-algorithm. In Sborník přednášek moderní směry výuky elektrotechniky a elektroniky STO-7, Brno, Září, pp. 196-9.
16.
Nigmatullin, R. R.1986. Realization of the generalized transfer equation in a medium with fractal geometry . Physical Status Solidi B133: 425-430 .
17.
Wyss, W.1986. The fractional diffusion equation . Journal of Mathematical Physics27 (11): 2782-2785 .
18.
Schneider, W. R. , and W. Wyss . 1989. Fractional diffusion and wave equation . Journal of Mathematical Physics30 (1): 134-144 .
19.
Mainardi, F. , and P. Paradisi . 1997. A model of diffusive waves in viscoelasticity based on fractional calculus . In Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, CA.
20.
Agrawal, O. P.2002. Solution for a fractional diffusion-wave equation defined in a bounded domain . Nonlinear Dynamics29: 145-155 .
21.
Liang, J. , Y. Chen , B. M. Vinagre , and I. Podlubny . 2004. Boundary stabilization of a fractional wave equation via a fractional order boundary controller . In The First IFAC Symposium on Fractional Derivatives and Applications (FDA’04), July, Bordeaux, France.
22.
Caputo, M.1967. Linear models of dissipation whose Q is almost frequency independent—II . Geophysics Journal of the Royal Astronomical Society13: 529-539 .
23.
Podlubny, I.1999. Fractional differential equations. New York: Academic Press .
24.
Kassimali, A.1993. Structural analysis (Pws-Kent series in Engineering). Boston, MA: PWS Publishing .
25.
Chen, G. , M. C. Delfour , A. M. Krall , and G. Payre . 1987. Modelling, stabilization and control of serially connected beams . SIAM Journal on Control and Optimization25: 526-546 .
26.
Datko, R. , J. Lagnese , and M. P. Polis . 1986. An example on the effect of time delays in boundary feedback stabilization of wave equations . SIAM Journal on Control and Optimization24: 152-156 .
27.
Datko, R.1993. Two examples of ill-posedness with respect to small time delays in stabilized elastic systems . IEEE Transactions on Automatic Control38 (1): 163-166 .
28.
Levine, W. , ed. 1996. The control handbook. Boca Raton, FL: CRC Press .
