Abstract
This paper is concerned with an evolutionary search for limit cycle operation in a class of nonlinear systems. In the first part, single input single output (SISO) systems are investigated and sinusoidal input describing function (SIDF) is extended to those cases where the key assumption in its derivation is violated. Describing function matrix (DMF) is employed to take into account the effects of higher harmonic signals and enhance the accuracy of predicting limit cycle operation. In the second part, SIDF is extended to the class of nonlinear multiinput multioutput (MIMO) systems containing separable nonlinear elements of any general form. In both cases linearized harmonic balance equations are derived and the search for a limit cycle is formulated as a multiobjective problem. Multiobjective genetic algorithm (MOGA) is utilized to search the space of parameters of theoretically possible limit cycle operations. Case studies are presented to demonstrate the effectiveness of the proposed approach.
Highlights
The theory of linear dynamic systems is well understood and is widely applied to many fields of engineering such as robotics, processes control, ship stirring to name a few
The second part of the paper further extends the SIDF techniques to a class of multiloop nonlinear systems in which the nonlinear elements are separable from the linear part
Let C be an analogous vector of complex Fourier coefficients of the nonlinearity output y, an infinite matrix N the describing function matrix may be defined as follows: The 0th column is given by Nk0 C a0 /a0, The 1st column is given by Nk1 Ck a1 − Ck a0 /a1, where, for example, the 3.1, 2.1 element represents the amplitude of the 3rd output harmonic divided by that of the 1st input harmonic
Summary
The theory of linear dynamic systems is well understood and is widely applied to many fields of engineering such as robotics, processes control, ship stirring to name a few. With the advent of fast and powerful digital computers, research for a more precise and accurate analysis of nonlinear systems has grown considerably 1–3 One such method which traditionally has been applied is the replacement of nonlinear behavior with a quasi-linear gain called describing function DF 4. The second part of the paper further extends the SIDF techniques to a class of multiloop nonlinear systems in which the nonlinear elements are separable from the linear part In both cases, emphasis is placed on the multiobjective formulation of predicting the limit cycle operation and the subsequent solution of the harmonically linearised system equations by the multiobjective genetic algorithms MOGA
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