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It means that the criteria are not valid if any of the coefficients of the characteristic equation is exponential or complex.Affiliations: Department of Electrical and Computer Engineering, University of Houston, Houston, Texas 77204, USA | Department of Physiology and Biophysics, Biomedical Engineering Center, Office of Academic Computing, University of Texas Medical Branch, Galveston, Texas 77555, USAĪbstract: In this paper, we analyze in detail the bounded-input/bounded-output (BIBO) stability of the nonlinear fuzzy proportional-integral (PI) control systems developed in Ying, Siler, and Buckley (1990).
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The transfer function provides the poles and zeroes, which can be plotted graphically. We can also use the value of s to construct a plot. It means that the variable s is a complex number. The poles and zeroes are shown in the below plot. It means that the above transfer function includes two zeroes and one pole.
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Here, there are two terms in the numerator and one term in denominator. 1/infinite = 0.īoth zeroes and poles can be real or complex quantities. It is because it is a part of denominator. Poles are the value of s for which the transfer function is infinite. The roots of the denominator can be represented as ?p1, -p2, -p3. The roots of the numerator can be represented as ?z1, -z2, -z3. Similarly, the value of s for which the denominator of the given transfer function is equated to zero is known as the system's pole. The value of s for which the numerator of the given transfer function is equated to zero is known as zeroes of the system. Here, z represents zeroes, and p represents poles. In terms of poles and zeroes, we can represent the transfer function as: Stability with respect to the location of poles and zeroes When the output tends to infinity it becomes unstable. The output in this case, increases with time. The positive exponential graph (Y(t) = e^t) is shown below: Let's consider a case of positive exponential graph. The above graph shows that the output decreases and reaches value 0.Hence, the system is stable. Now, for the condition to be stable, the output also needs to be bounded. The input in both cases is e^-t (inverse exponential function). Let us consider two systems described the following equations: It means that if the input is zero, the output should also be zero irrespective of any initial conditions.