Analysis and design of nonlinear feedback systems

Abstract

The paper discusses methods, primarily those using frequency-domain techniques, for the analysis and design of nonlinear feedback control systems. The behavioural properties peculiar to nonlinear feedback systems are first discussed. This is followed by a review and discussion of the applicability of absolute stability criteria and describing-function methods for single-variable and multivariable systems. The calculation of limit cycles in relay systems, time-domain methods of analysis and simulation techniques are then considered. Finally, a few applications are considered which highlight both the applicability and limitations of the available analytical techniques for nonlinear systems. IntroductionThe aim of this paper is to give an overview of methods available for the analysis and design of nonlinear systems. As a consequence of the distinct behaviour of nonlinear systems, most methods of analysis are directed at solutions to specific problems, such as system stability or the existence of limit cycles. Unlike the case for linear systems, the approach may be of limited usefulness for answering other questions of concern, such as, for example, the time-response behaviour. Further, in several methods, the assumptions made are based on the expected form of the solution, so that some knowledge of the possible forms of nonlinear behaviour is a requirement for the analyst. Because of the limited applicability and the approximations often involved in methods of nonlinear analysis, it is important for the engineer to be familiar with a variety of techniques, to appreciate their limitations, and to know how to combine their useful attributes. The nonlinear features of a system may originate inherently or intentionally. Inherent nonlinearity is an inseparable characteristic of the laws governing the operation of the system to be controlled. This situation has frequently been handled by considering the linearised performance about one or more system operating points, often resulting in loss of accuracy, unreliable predictions and the need for costly testing. Intentional nonlinearity, on the other hand, is deliberately introduced into the design of the system as the best way to achieve the desired performance. Typical examples of this are the use of on/off time-optimal position controls [1 ] and the application of jet thrusters in the attitude control loops of current communication [2], and some scientific [3], satellites. The increasing use of microprocessor controllers will, without doubt, result in the more frequent use of nonlinear control strategies as, in contrast to analogue controllers, they allow complex nonlinear algorithms to be easily implemented. The requirements placed on process-controller algorithms to achieve both satisfactory control of set-point changes and regulation are rarely compatible, and thus an opportunity for the use of nonlinear techniques clearly exists. Some nonlinear controllers which have recently become available [4,5] have been designed primarily on the basis of mimicking the behaviour ...

Analyses and designs of a nonlinear mechanical scanning control system

“…The light damped system can be easily controlled with nonlinear elements; e.g., saturation elements in forward path or high gain with dead zone in feedback path, to produce limit cycle [3][4][5][6]. The saturation element provides phase lag to destabilize the system while the saturation level cramps the diverging inclination.…”

On the Periodic Solutions in One Dimensional Cellular Nonlinear Networks Based on Josephson Junctions (JJ's)

“…It is possible to use the describing function method not only to get an indication about the existence of the periodic solution, but also about their stability. Using the graphical criterion and following [12], [13] we can give the analytical condition for the stability of periodic solution, which is: for a real valued describing function, periodic solution is predicted to be stable if: We shall continue our stability analysis with definition of the equilibrium points of our the model considered. For the sake of simplicity let us rewrite system (15) Then we shall define the equilibrium points of our model: Definition 2.…”

Limit Cycle Predictions of Nonlinear Multivariable Feedback Control Systems with Large Transportation Lags

“…Assume the nonlinearity is symmetric, then the DC component F o is equal to zero. In general, fundamental components P 1 and R 1 are used to describe the nonlinearity [7][8][9][10]. Therefore, there is a modeling error between describing function and the real nonlinear element.…”

“…However, if nonlinearities are described by the sinusoidal input describing function with fundamental components [7][8][9][10], then modeling errors of the found A i by inverting describing functions of N i (a i ) make magnitudes of (10) and (12) are not equal to unities exactly. Therefore, M θ2 {10} = M θ2…”

“…These methods are based upon the graphical or numerical solutions of the linearized harmonic-balance equations [1][2][3][4][5][6][7][8][9][10]. It has been shown that for multivariable systems, over arbitrary ranges of amplitudes (A i ), frequency (ω) and phases (θ i ), an infinite number of possible solutions may exist.…”