Initially it is essential to examine the theoretical basis for the concept of stability in order to identify the various conditions which characterize a stable system. After determining the stability conditions, the Graphical User Interface Development Environment (GUIDE) module of MATLAB will be studied to examine the process of developing a Graphical User Interface for a MATLAB application which determines the stability of a given system.
The BIBO (Bounded Input Bounded Output) concept of stability states that if a bounded input to the system produces a bounded output, then the system is stable (Ogata. K. 1997). It is important to determine the physical significance of stability both in the time domain and in the frequency domain. In the time domain, for continuous functions to be BIBO stable, an integral of their impulse response should exist. Similarly for discrete functions in the time domain, we should be able to sum up the impulse responses of the discrete function (Ogata. K. 1995). To determine stability in the frequency domain, we consider the Laplace transform (used for converting continous functions from time domain to frequency domain and vice versa) for the continous signals and Z transform (used for converting discrete functions from time domain to frequency domain and vice versa) for the discrete signals. ...
If the region of convergence Z transform includes the unit circle then the system is stable. A system is stable if and only if all its poles are lying within the unit circle.
Significance of the transfer function, poles and zeros
The transfer function is an illustration of the relationship between the input to a system and the output of the system (Ogata. K. 1997). It accurately represents a system which is time invariant and where the output varies from the input in a linear fashion. For determining the stability of such a system, the transfer function of a continous system is often depicted as ratio of the Laplace transforms of the output to the input. Similarly the transfer function of a discrete system is often depicted as ratio of the Z transforms of the output to the input (Ogata. K. 1997). The poles of the system are those complex frequencies where the transfer function tends to infinity. The zeros are those points where the complex frequencies of the transfer function collapses to zero. Poles and zeros plotted on the real and imaginary axes give an insight. that the scale of the transfer function will be bigger when it approaches the poles and smaller when it is closer to the zeros. The poles and zeros can either be real or complex. When complex, they occur in complex conjugate pairs. For a system to be realizable physically, the total number of poles should be atleast equal to the number of zeros (Ogata. K. 1995).
Root Locus Diagram
The root locus is the locus or the path traced by the poles of a transfer function as the total system gain is varied. A single input single output system can be analysed well by looking at its root locus diagram (Ogata. K. 1997). Some relevant aspects of a root locus diagram are, it is symmetric about the real axis as