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A Sliding Mode Control Algorithm for Solving an Ill-posed Positive Linear System

Chein-Shan Liu1

Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. E-mail:

Computers, Materials & Continua 2014, 39(2), 153-178.


For the numerical solution of an ill-posed positive linear system we combine the methods from invariant manifold theory and sliding mode control theory, developing an affine nonlinear dynamical system with a positive control force and with the residual vector as being a gain vector. This system is proven asymptotically stable to the zero residual vector by using an argument from the Lyapunov stability theory. We find that the system fast tends to the sliding surface and then moves with a sliding mode, such that the resultant sliding mode control algorithm (SMCA) is robust against large noise and stable to find the numerical solution of an ill-posed linear system. It is interesting that even under a random noise with an intensity 10-5 we can obtain a quite accurate solution of the linear Hilbert problem with dimension n = 500. For this highly ill-conditioned problem the number of iterations is still smaller than 100. Numerical tests, including the inverse problems of backward heat conduction problem and Cauchy problems, confirm that the present SMCA has superior computational efficiency and accuracy even for a highly ill-conditioned linear equations system under a large noise.


Cite This Article

C. Liu, "A sliding mode control algorithm for solving an ill-posed positive linear system," Computers, Materials & Continua, vol. 39, no.2, pp. 153–178, 2014.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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