Evolutionary Search for Limit Cycle and Controller Design in Multivariable Nonlinear Systems

Abstract

A feature of many practical control systems is a Multi‐Input Multi‐Output (MIMO) interactive structure with one or more gross nonlinearities. A primary controller design task in such circumstances is to predict and ensure the avoidance of limit cycling conditions followed by achieving other design objectives. This paper outlines how such a system may be investigated using the Sinusoidal Input Describing Function (SIDF) philosophy quantifying magnitude, frequency and phase of any possible limit cycle operation. While Sinusoidal Input Describing function is a suitable linearization technique in the frequency domain for assessment of stability and limit cycle operation, it can not be employed in time domain. In order to be able to incorporate the time domain requirements in an overall controller design technique, the appropriate linearization technique suggested here is the Exponential Input Describing Function (EIDF). First, an evolutionary search based on a multi‐objective formulation is employed for the direct solution of the harmonic balance system matrix equation. The search is based on Multi‐Objective Genetic Algorithms (MOGA) and is capable of predicting specified modes of theoretically possible limit cycle operation. Second, the design requirements in time as well as frequency domain are formulated by a set of constraint inequalities. A numerical synthesis procedure also based on Multi‐Objective Genetic Algorithm is employed to adjust the initial compensator parameters to meet the imposed constraints. Robust stability and robust performance are investigated with respect to linearization uncertainty within the context of multiobjective formulation. In order to make the Genetic Algorithm (GA) search more amenable to design trade‐off between different and often contradictory specifications, a weighted sum of the functions is introduced. This criterion is subsequently optimized subject to the nonlinear system dynamics and a set of design requirements. Examples of use are given to illustrate the effectiveness of the proposed approach.

Archives of Control Sciences

“…Growing interest in prediction of limit cycles in 2 × 2 nonlinear systems has been closely noticed among researchers for several decades [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. The problem is remarkable and more complex in the presence of memory type nonlinearity [8,19,24,[34][35][36].…”

“…4 6 (4 + 2K 1 + 2K 2 ) (4K 1 K 2 )s 3 6(9 + K 2 ) − (4 + 2K 1 + 2K 2 ) 6 = a 6(8K 1 + K 2 ) − 4K 1 K 2 6 = b s 2 a(4 + 2K 1 + 2K 2 ) − 6b a = c 4K 1 K 2 s 1 bc − a(4K 1 K 2 ) c s 0 4K 1 K 2 For s 0 row: 4K 1 K 2 0This implies K 1 and K 2 are positive. For s 3 row:6(9 + K 2 ) − (4 + 2K 1 + 2K 2 ) 49K 1 − 20K 2 − 7K 1 K 2 − 50 0.…”

Pareto‐Set Based Multi‐Objective Dynamic Reactive Power and Voltage Control

“…For this reason, finding its Pareto-optimal solution is an ideal method [12,13]. some objectives contradict each other, more solutions are expected in order to adapt to different situations.…”

“…In the literature, numerous approaches [1][2][3][4][5][6][7][8][9][10][11][12] have been reported to solve the dynamic Volt/VAR control problem. In [1], a dynamic reactive power optimization model for medium-high voltage distribution system was established, and the model could be simplified and solved by pretreatment to discrete control variables and dynamic loads.…”

“…some objectives contradict each other, more solutions are expected in order to adapt to different situations. For this reason, finding its Pareto‐optimal solution is an ideal method . In this paper, a multi‐objective dynamic Volt/VAR control model was created and the NSGA‐II is improved to solve the model.…”