Nonlinear controller synthesis based on inverse describing function technique in the MATLAB environment

doi:10.1016/j.advengsoft.2005.09.009

Advances in Engineering Software

Volume 37, Issue 6, June 2006, Pages 370-374

https://doi.org/10.1016/j.advengsoft.2005.09.009 Get rights and content

Abstract

In this research, a new software based on a systematic nonlinear controller design technique for nonlinear systems is developed. This software is in the MATLAB environment, and it is based on inverse describing functions in order to arrive at nonlinear gains of the controller. The software may be used to synthesize nonlinear controllers for nonlinear systems without restriction on system order, the number, type, and arrangement of nonlinear terms. The goal is to design an output feedback system that is a solution to a robust design such that sensitivity of the feedback control system with respect to amplitude of the excitation signal and the operating regimes would be at a minimum.

Introduction

The describing function theory for synthesis of a controller for use with hard nonlinear systems has received considerable attention [1], [2], [3], [4], [5], [6]. Those nonlinear systems that one is unable to linearize are called hard nonlinear systems; otherwise, the systems are said to be soft. The design logic based on describing function models considers three different cases. In the first case, the nonlinear system may be characterized by one operating regime, and the designed controllers are said to be single-range [4], [5]. In the second case, the nonlinear system may be characterized by two different operating regimes [6], and in this case the designed controllers are said to be dual-range. Finally, in the third case, the nonlinear system is characterized by many operating regimes (more than two) [2] and the designed controllers are said to be multi-range. In this research, concentration is on the third case, and the necessary and required software modules, in the MATLAB environment, are developed and presented; the software tools are based on the method developed in Ref. [2], and the interested reader may send an email to the first author to receive a copy of the software.

Section snippets

Background

The goal of this paper is to present the software for synthesis of multi-range nonlinear controllers for use with highly nonlinear systems. The end result of the design synthesis activity is a nonlinear feedback system whose behavior is insensitive to various operating regimes of interest in a near-optimum fashion. The synthesis method is based on a set of sinusoidal-input describing function (SIDF) models. With reference to items (a)–(f) noted below [1], [2], it may be concluded that SIDF

Software and the method of nonlinear controller synthesis

The synthesis method is systematic, and it consists of 11 steps [2]. These steps are described below.

Step 1
In this step, a set of values for amplitude levels and frequencies of the excitation signal, which fall into operating regimes of interest, is selected. The set of amplitudes is denoted {a_i}, and the set of frequencies is denoted {ω_k}.
Step 2
With the use of the MATLAB command, which is developed in this research, obtain the input–output frequency models of the nonlinear plant. These models are denoted G

Demonstration example problem

The nonlinear plant that a nonlinear controller is synthesized for is shown in Fig. 1.

Motor saturation is modeled by two linear regions with slopes m₁ and m₂ separated by a break point at δ. For this problem, m₁=5.0 Nm/ν, δ=0.5 ν, and m₂=1.0 Nm/ν. Friction is modeled via the following relations. $T_{m} = {\begin{array}{l} T_{e} - f_{v} \dot{θ} - f_{c} sig (\dot{θ}), & | T_{e} | > f_{c} \\ T_{e} - f_{v} \dot{θ} - f_{c} sig (\dot{θ}), & \dot{θ} \neq 0 \\ 0.0, & | T_{e} | < f_{c} and \dot{θ} = 0.0 \end{array}$ where in the above f_v=0.1 Nm-s/rad, f_c=1 Nm, and the moment of inertia is J=0.01 kg-m². The computer model of the complete system, in

Summary and conclusions

In this research, a set of software for synthesis of a nonlinear controller for use with highly nonlinear systems in the MATLAB environment is developed. The application of this software to a nonlinear position control system was demonstrated. The conclusion of this research is that method of controller synthesis, based on inverse describing function method, results in nonlinear feedback systems that are fairly insensitive to the amplitude of the excitation command; therefore, a robust design

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There are more references available in the full text version of this article.

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Nonlinear controller synthesis based on inverse describing function technique in the MATLAB environment

Abstract

Introduction

Section snippets

Background

Software and the method of nonlinear controller synthesis

Demonstration example problem

Summary and conclusions

Adv Eng Softw

Adv Eng Softw

Adv Eng Softw

A systematic nonlinear controller design method based on quasilinear system models

Proceedings of American Control Conference, San Francisco, CA, USA

Nonlinear compensator synthesis via sinusoidal-input describing functions

Proceedings of American Control Conference, Boston, MA, USA