Quantitative design of a class of nonlinear systems with parameter uncertainty

Abstract

This paper considers the case in which a linear time‐invariant (LTI) but uncertain plant suffers from nonlinearities y=n(x) which can be expressed as y=Kn+η(x), |η(x)|≤M, with K, a possibly uncertain scalar. This covers a large and very important class of nonlinearities encountered in practice such as friction, backlash, dead zone and quantization. Quantitative design techniques are presented for this class for the satisfaction of specifications. Special attention is paid to the avoidance of limit cycles using describing function theory, although the design method is also amenable of application using other stability criteria such as the circle criteria. Numerical examples are developed illustrating the design procedure.

A QFT framework for nonlinear robust stability

“…The easiest way to use Equation (7) in avoiding the existence of multiple equilibrium points and limit cycles is to derive conditions over Gð joÞ to satisfy the second inequality, which is a sufficient condition. This second condition can be embedded in the QFT framework using (6), where the function Nða; oÞ is substituted by N 1 ða 0 ; a 1 Þ ¼ ½0; 1, a similar condition to the obtained in Section 2. As a result, boundaries given in Figure 3 are not only valid for a symmetric dead zone, but also for any asymmetric dead zone.…”

“…Once the multiplier Wð joÞ is fixed and N is identified with the complex locus defined by the circles cðoÞ þ rðoÞU given by (14), it is straightforward to obtain boundaries for Gð joÞ. It suffices to apply the inequality (6) with N given by the family of circles.…”

“…Although no details about computation will be given here, a realistic example is developed in the following. The nonlinear system in this example is borrowed from Example 1 in Reference [6]. It represents an electric motor driving a load through a gear in the presence of backlash.…”

“…This is the sense of stability that will be adopted here. To the knowledge of authors, the nonlinear stability problem has been only mentioned in the more recent QFT literature in the work [6], where the Circle Criterion is suggested as a way to incorporate I/O stability in nonlinear QFT designs, but it is not developed so far. Related work about the use of the Circle Criterion in QFT is given in Reference [7], where a particular case is used for analysing the effect of a time-varying gain on closed-loop performance.…”

Nonlinear quantitative stability

“…In the framework of QFT, these systems have been first studied in Reference [14]. In the following, the quantitative stability method is applied to these types of systems, although a deeper treatment needs a separate analysis.…”