Динамикалық жүйенің тұрақтылығын графиктік әдіспен анықтау

Graphical method of determining the stability of a dynamical system

In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer de at Siemens in 1930 Additionally, other stability criteria like Lyapunov methods can also be applied for non linear systems. Although Nyquist is a graphical technique, it only provides a limited amount of intuition for why a system is stable or unstable, or how to modify an unstable system to be stable. Techniques like Bode plots, while less general, are sometimes a more useful design tool.

A Nyquist plot is a parametric plot of a frequency response used in automatic control and signal processing. The most common use of Nyquist plots is for assessing the stability of a system with feedback. In Cartesian coordinates, the real part of the transfer function is plotted on the X axis while the imaginary part is plotted on the Y axis. The frequency is swept as a parameter, resulting in one point per frequency. The same plot can be described using polar coordinates, where gain of the transfer function is the radial coordinate, and the phase of the transfer function is the corresponding angular coordinate. The Nyquist plot is named after Harry Nyquist, a former engineer at Bell Laboratories. Assessment of the stability of a closed loop negative feedback system is done by applying the Nyquist stability criterion to the Nyquist plot of the open loop system (i. e. the same system without its feedback loop). This method is easily applicable even for systems with delays and other non rational transfer functions, which may appear difficult to analyze with other methods. Stability is determined by looking at the number of encirclements of the point (−1, 0). The range of gains over which the system will be stable can be determined by looking at crossings of the real axis. The Nyquist plot can provide some information about the shape of the transfer function. For instance, the plot provides information on the difference between the number of zeros and poles of the transfer function

If the open loop transfer function has a zero pole of multiplicity , then the Nyquist plot has a discontinuity at During further analysis it should be assumed that the phasor travels times clockwise along a semicircle of infinite radius. After applying this rule, the zero poles should be neglected, i. e. if there are no other unstable poles, then the open loop transfer function should be considered stable. If the open loop transfer function is stable, then the closed loop system is unstable, if and only if, the Nyquist plot encircle the point −1 at least once. If the open loop transfer function is unstable, then for the closed loop system to be stable, there must be one counter clockwise encirclement of −1 for each pole of in the right half of the complex plane. The number of surplus encirclements (N + P greater than 0) is exactly the number of unstable poles of the closed loop system. However, if the graph happens to pass through the point , then deciding upon even the marginal stability of the system becomes difficult and the only conclusion that can be drawn from the graph is that there exist zeros on the axis.