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To obtain the transient response C(t), it is necessary to find the inverse of Eq. (14.3). This requires obtaining the roots of the denominator of Eq. (14.2), which is third-order. We can no longer find these roots as easily as we did for the secondorder systems by use of the quadratic formula. However, in principle they can always be obtained by algebraic methods. It is appmnt that the roots of the denominator depend on the particular values of the time constants and K,. These roots determine the nature of the transient response, according to the rules presented in Fig. 3.1 and Table 3.1. It is of interest to examine the nature of the response for the control system of Fig. 14.1 as K, is varied, assuming the time constants ~1~72, and 73 to be fixed. To be specific, consider the step response for ~1 = 1,~ = 1, and 73 = 5 for several values of K, . without going into the detailed calculations at this time, the results of inversion of Eq. (14.3) are shown as response curves in Fig. 14.2. From these response curves, it is seen that, as K, increases, the system response becomes more oscillatory. In fact, beyond a certain value of K,, the successive amplitudes of the response grow rather than decay; this type of response is called unstable. Evidently, for some values of K,, there is a pair of roots corresponding to s4 and s; of Fig. 3.1. As control system designers, we are clearly interested in being able to determine quickly the values of K, that give unstable responses, such as that corresponding to K, = 12 in Fig. 14.2.
FIGURE 14-2
Response of control system of Fig. 14-1 for a unit-step change in set point.
LINEAR CLOSED-UX)P
SYSTEMS
If the order of Eq. (14.2) had been higher than three, the calculations necessary to obtain Fig. 14.2 would have been even more difficult. In the next chapter, on root-locus methods, a powerful graphical tool for finding the necessary roots will be developed. In this chapter, the focus is on developing a clearer understanding of the concept of stability. In addition, we shall develop a quick test for detecting roots having positive real parts, such as ~4 and sq* in Fig. 3.1.
Definition of Stability (Linear Systems)
For our purposes, a stable system will be defined as one for which the output response is bounded for all bounded inputs. A system exhibiting an unbounded response to a bounded input is unstable. This definition, although somewhat loose, is adequate for most of the linear systems and simple inputs that we shall study. A bounded input function is a function of time that always falls within certain bounds during the course of time. For example, the step function and sinusoidal function are bounded inputs. The function f(t) = t is obviously unbounded. Although the definition of an unstable system states that the output becomes unbounded, this is true only in the mathematical sense. An actual physical system always exhibits bounds or restraints. A linear mathematical model (set of linear differential equations describing the system) from which stability information is obtained is meaningful only over a certain range of variables. For example, a linear control valve gives a linear relation between flow and valve-top pressure only over the range of pressure (or flow) corresponding to values between which the valve is shut tight or wide open. When the valve is wide open, for example, further change in pressure to the diaphragm will not increase the flow. We often describe such a limitation by the term saturation. A physical system, when unstable, may not follow the response of its linear mathematical model beyond certain physical bounds but rather may saturate. However, the prediction of stability by the linear model is of utmost importance in a real control system since operation with the valve shut tight or wide open is clearly unsatisfactory control.
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