Introduction
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. The imaginary part is plotted on the Y axis. The frequency is swept as a parameter, resulting in a plot per frequency. Alternatively, in polar coordinates, the gain of the transfer function is plotted as the radial coordinate, while the phase of the transfer function is plotted as the angular coordinate.
Theory
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 poles and zeros of the transfer function by the angle at which the curve approaches the origin.
When drawn by hand, a cartoon version of the Nyquist plot is sometimes used, which shows the shape of the curve, but where coordinates are distorted to show more details in regions of interest. When plotted computationally, one needs to be careful to cover all frequencies of interest. This typically means that the parameter is swept logarithmically, in order to cover a wide range of values.
Importance of Nyquist stability criteria
The Nyquist criterion is widely used in electronics and control system engineering, as well as other fields, for designing and analysing systems with feedback. While Nyquist is one of the most general stability tests, it is still restricted to linear, time-invariant (LTI) systems. Non-linear systems must use more complex stability criteria, such as Lyapunov or the circle criterion. While 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.
The importance of Nyquist stability lies in the fact that it can also be used to determine the relative degree of system stability by producing the so-called phase and gain stability margins. These stability margins are needed for frequency domain controller design techniques.
The Nyquist method is used for studying the stability of linear systems with time delay.
For a SISO feedback system the closed-loop transfer function is given by where
Where G(s) represents the system and H(s) is the feedback element. Since the system poles are determined as those values at which its transfer function becomes infinity, it follows that the closed-loop system poles are obtained by solving the following equation
which, in fact, represents the
system characteristic equation.
In the following equation, we consider the complex function:
The Nyquist stability test is obtained by applying the Cauchy principle of argument to the complex function D(s).
We state the Nyquist theorem in two parts:
Open loop stable systems: When G(s)H(s) has no poles in the R.H.S , the encirclement of G(s)H(s) contour corresponding to Nyquist path around the critical point (-1,0) must be zero i.e the G(s)H(s) contour should not encircle the critical point. Only then the closed loop system is stable.
Open loop unstable systems:If G(s)H(s) has P poles in the R.H.S of s plane, then the G(s)H(s) contour corresponding to the Nyquist path should go round the critical point(-1,0) P times and in the direction opposite to the nyquist path for a stable system . Actually Z=N+P represents the closed loop poles in the R.H.S of a plane and if Z does not come out to be zero then system is unstable.
Manual
Scilab is a programming language associated with a rich collection of numerical algorithms covering many aspects of scientific computing problems.
Whatever your platform is (i.e. Windows, Linux or Mac), Scilab binaries can be downloaded directly from the Scilab homepage
or from the
Download area.
Scilab elementary mathematical operators:
+ addition
- subtraction
* multiplication
/ right division, i.e. x/y =xy-1
\ left division, i.e. x\y = x-1y
^ power, i.e. xy
** power (same as ^)
' transpose conjugate
In this scilab program
syslin defines a linear system as a list and checks consistency of data. dom='c' for a continuous time system, dom='d' for a discrete time system, n for a sampled system with sampling period n (in seconds), dom=[] if the time domain is undefined.
clf is used to reset a figure to its default values.
Nyquist command displays the symetric graph (positive and negative frequencies).
Procedure
- Install scilab in your PC.
- Write the program code given below in the command window.
Program
G=syslin('c',(s-2)/(s+1))
H=1
nyquist(G*H)
Output :
