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

Abstract

Abstract: For the numerical solution of an ill-posed positive linear system wecombine the methods from invariant manifold theory and sliding mode control the-ory, developing an affine nonlinear dynamical system with a positive control forceand with the residual vector as being a gain vector. This system is proven asymp-totically stable to the zero residual vector by using an argument from the Lyapunovstability theory. We find that the system fast tends to the sliding surface and thenmoves 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 ofan ill-posed linear system. It is interesting that even under a random noise with anintensity 10 5 we can obtain a quite accurate solution of the linear Hilbert prob-lem with dimension n = 500. For this highly ill-conditioned problem the numberof iterations is still smaller than 100. Numerical tests, including the inverse prob-lems of backward heat conduction problem and Cauchy problems, confirm thatthe present SMCA has superior computational efficiency and accuracy even for ahighly ill-conditioned linear equations system under a large noise.Keywords: Ill-posed linear equations, Invariant manifold, Sliding mode controlmethod, Asymptotically stable1 IntroductionIn this paper we propose a robust and easily-implemented algorithm to solve thefollowing linear equations system:Bx=b; (1)where B2R