تجزیه و تحلیل سیستم از طریق نقشه های عملکرد

While there are a number of visual methods common to the design and analysis of dynamic systems, they tend to be specific to their application and limited in the amount of information that they yield. This paper explores a visualization technique, titled the performance map, which is derived from the Julia set commonly used in the visualization of iterative chaos. Performance maps are generated via digital computation, and require a minimum of a priori knowledge of the system under evaluation. By the use of color-coding, these images convey a wealth of information to the informed user about dynamic behaviors of a system that may be hidden from all but the expert analyst.

There are a number of graphical techniques used in the classical design and analysis of dynamic systems, including time graphs, phase portraits, the root locus method, Bode diagrams and Nyquist plots. These are powerful analytic tools, yet each is constrained by its own set of limitations and tends to be rigidly focused, with limited utility. Because their graphical output is intentionally simple, each has a definite limitation in the amount of information that can be conveyed in a single image. And finally, each can require further mathematical analysis, which may be difficult for ill-defined or highly nonlinear systems. This paper presents a method, termed the performance map, designed to visualize the behavior of a system's dynamic performance across variations in system parameters. This technique is derived from the familiar Julia set, which is used to visualize the occurrence of chaos in iterative mathematical formulae. The resulting visualization is color-coded, providing an immediate wealth of information to the knowledgeable viewer. Furthermore, because the performance map is developed algorithmically, via digital computation, its generation requires only moderate a priori knowledge and mathematical analysis. This paper begins with a brief overview of the Julia set technique, in which the rules employed to color the image pixels is emphasized. The scheme is then adapted to fit the analysis of control and other dynamical systems by parameter mapping and appropriate rule selection. To demonstrate the effectiveness and veracity of the performance map, a simulation of a readily verified linear feedback system is considered. The generated map is demonstrated to accurately display multiple effects of parameter values on the system solution. The paper concludes with a performance map analysis of a complex system.

This paper has presented a method, derived from a visualization technique common in the study of chaos, which can be used to visualize the effects of variations in parameters on a system's response. Derived from the Julia set, the performance map is generated via digital computation, and with minimal need for rigorous mathematical analysis. By using simple color-coding, a wealth of useful information is portrayed to the knowledgeable viewer about system performance. A performance map can be readily generated for systems that are highly nonlinear, multi-variable, or chaotic and any number of rules may be employed in the test for pixel coloration (Russell & Alpigini, 1997b) and knowledge acquisition. The technique was first demonstrated using an easily verified second-order linear system and shown to yield the expected results. Next, maps of a “real world” model, namely the Pegasus turbofan engine model, were examined using a similar methodology. The true strength of the performance map paradigm rests in its flexibility. Any type of system motion or artifact can be visualized providing that a suitable performance rule can be developed. For example, unstable oscillations and intermittent spikes can be readily detected by rules based on system derivatives. Such motions are often a type of “intermittent” or “transient” chaos (Moon, 1992) which could easily be dismissed as simple noise or perhaps “ringing” in a control system. In addition, system motions can be quantified in state space using the fractal information dimension (Russell, 1994) with pixel coloration assigned according to ranges of the value of that dimension. Such a measurement yields a map that reflects changes in the complexity of a system's response as parameters are varied across an interval (Alpigini & Russell, 1998). Alternately, a rule set can be derived using rough set theory (Alpigini, 2002). The resulting rough performance map, while an approximation, is still suitable for tuning a system for stable operation. The current focus of this research is on the generation of N-dimensional performance maps. While visualizing three parameters has been successfully performed, it is desirable to be able to visualize more. To affect this, parallel processing is under investigation. It is believed that this adaptation will enable the performance map to move from mathematical models to real-time systems.