Describing function

About: Describing function is a research topic. Over the lifetime, 1742 publications have been published within this topic receiving 26702 citations.

Real‐time forecasting with a conceptual hydrologic model: 1. Analysis of uncertainty

Abstract: The optimal control of watershed systems requires accurate real-time short-term forecasts of river flows. For the first time, this paper formulates a large, nonlinear conceptual model (the National Weather Service catchment model) in a mode amenable to analysis of uncertainty and the utilization of real-time information (measurements, forecasts, guesses) to update system states and improve streamflow predictions. The proposed methodology is based on the state space formulation of the equations describing the hydrologic model and the assumption of sources of uncertainty in the data and in the model structure. The first two moments of random variables are estimated in a computationally efficient way using on-line linear estimation techniques. Linearization of functional relationships is performed with the uncommon but powerful multiple-input describing function technique for the most strongly nonlinear responses and Taylor expansion for the rest. The linear feedback rule developed is based on the Kalman filter.

Control Systems with Friction

Abstract: Friction-related problems are frequently encountered in control systems. This thesis treats three aspects of such problems: modeling, analysis, and friction compensation. A new dynamic friction model is presented and investigated. The model is described by a first order nonlinear differential equation with a reasonable number of parameters, yet it captures most of the experimentally observed friction phenomena. The model is suitable both for simulation purposes and control design. Analysis of friction-generated limit cycles in control systems is the second topic of the thesis. A distinction is made between limit cycles with and without periods of sticking. Oscillations without sticking where the velocity is zero only for single time instants can be treated as oscillations in relay-feedback systems for which tools are available. These tools are in the thesis extended to oscillations with sticking where the velocity is kept at zero for a period of time by the friction. The new tools give a procedure for exact computation of shape and stability of limit cycles caused by friction. The procedure requires the solution of a nonlinear equation system and that the feasibility of the solution is checked. The method is applied to several examples and comparisons are made with describing function analysis. The thesis also treats friction compensation based on the new model. A friction force observer is developed which enables model based friction compensation. The observer can be combined with traditional linear compensators. Stability theorems are given which allows a wide range of controller designs. The compensation scheme is applied to an example where the performance is studied with respect to model errors and disturbances. The resulting control error is thoroughly investigated. It is described how a simple statistical analysis of the error can give information on the success of the friction compensation. Furthermore the error during zero velocity crossings provides information on how model parameters should be changed.

The human operator as a servo system element

Abstract: This paper considers the role of human elements in certain closed loop control systems. A quantitative description of human dynamics useful to control system designers is essential to understanding and analysis of such systems. Accordingly, the human behavior description must be expressed in terms which are compatible with conventional descriptions of control system components. This compatibility is achieved by the use of a quasi-linear mathematical model for the human operator. The model is composed of two components—a describing function and remnant. The describing function, which for a linear system is identical with the conventional transfer function, is established to characterize that portion of the operator's output which is linearly correlated with his input. The input, upon which the describing function is based, is selected on the basis of a priori estimates of the nature of certain human non-linear behavior. Human output power which cannot be characterized by the operation of the describing function on the input is designated as the remnant. After presenting the analytic basis for measurements of human dynamics, steady-state describing functions measured by the various experiments in the field are discussed and the adaptive, optimalizing behavior of the human operator is demonstrated. The remnants are also discussed and plausible sources for their origin are postulated. Knowledge of the range of parameter adjustment of which the human operator is capable in his adaptation as well as knowledge of his criteria for adjusting these parameters enables the designer to specify input functions and operator controlled dynamics compatible with both desirable human operator behavior and good system performance. By judiciously trading off system complexity against operator preferences, while still making proper engineering use of the human operator's adaptability, a control system may be optimized for both performance and reliability.