Publication Date


Document Type


First Advisor

Bobis, James P.

Degree Name

M.S. (Master of Science)

Legacy Department

Department of Electrical Engineering


Chaotic behavior in systems; Nonlinear control theory


Chaotic behavior, commonly encountered in many otherwise orderly systems, is nonlinear and does not converge. The output is highly dependent on the initial conditions, and any error in the selection of the initial conditions results in totally unpredictable output. This paper studies the Lorenx equations, a chaotic system in three-dimensional space. The location of system attractors is solved mathematically and then illustrated using phase plane plots and a reiterative computer program. In order to eliminate the effect of chaotic behavior in the Lorenz equations, input state linearization is successfully employed by designing a control input which cancels the nonlinearities to create a linear system of equations which converges. Phase plane plots are used to illustrate convergence in the resulting system and Bode plots are used to find the gain and phase margins for the system. In a second controller design, the Lorenz equations are linearized and produce a stable harmonic oscillation. A Proportional-Integral-Deravitive (PID) controller is designed, using pole placement, to cause the oscillations to converge. The output of the new system is analyzed using phase plane plots and velocity versus time traces and stability analysis in the frequency domain measures the gain and phase margins. Finally, a unit step input is added to closed loop system and a graph of the output versus time illustrated stability, controllability, and convergence.


Includes bibliographical references (leaf [131]).


173 pages




Northern Illinois University

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