Oscillatory second-order response

In Chapter 14 we examined the dynamic characteristics of the response of closed-loop systems, and developed the closed-loop transfer functions that determine the dynamics of such systems. It is important to emphasize again that the presence of measuring devices, controllers, and final control elements changes the dynamic characteristics of an uncontrolled process. Thus nonoscillatory first-order processes may acquire oscillatory behavior with PI control. Oscillatory second-order processes may become unstable with a PI controller and an unfortunate selection of Kc and t,. 

If we assume that an oscillatory system response can be fitted to a second order underdamped function. With Eq. (3-29), we can calculate that with a decay ratio of 0.25, the damping ratio f is 0.215, and the maximum percent overshoot is 50%, which is not insignificant. (These values came from Revew Problem 4 back in Chapter 5.)... 

The hrsl-order system considered in the previous section yields well-behaved exponential responses. Second-order systems can be much more exciting since they can give an oscillatory or underdamped response. 

General Second-Order Element Figure 8-3 illustrates the fact that closed-loop systems can exhibit oscillatory behavior. A general second-order transfer function that can exhibit oscillatory behavior is important for the study of automatic control systems. Such a transfer function is given in Fig. 8-15. For a unit step input, the transient responses shown in Fig. 8-16 result. As can be seen, when t, < 1, the response oscillates and when t, < 1, the response is S-shaped. Few open-loop chemical processes exhibit an oscillating response most exhibit an S-shaped step response. 

Experiments showed that arterioles tend to perform damped oscillatory contractions in response to external stimuli [17]. This behavior may be captured by a second-order differential equation of the form ... 

Another aspect of nonlinearity is a change in the mean burning rate, which is of importance in producing secondary peaks of the mean pressure. It is easy to write formulas for this change if the response is assumed to be entirely quasisteady. For example, if equation (7-41) is taken to apply under conditions of oscillatory pressure, then with m = mo(l -f m ) and p — p(l -f p ), where mo is the value of m at p = 0 (so that m 0 at finite amplitude), we find through expansion to second order (by use of the average p = 0) that... 

Manometers and pressure springs may be described dynamically to a first approximation by second-order differential equations for which the roots of the characteristic equation are conjugate complex. As shown in Section III, 8, lc, since the roots are complex, these systems have an oscillatory mode, and the response of the system to step forcing, for example, is a damped sinusoid. 

Figure 15.5 shows the three general types of dynamic behavior of a second-order process, which can also be nsed to describe the dynamic behavior of feedback systems overdamped, critically damped, and nnderdamped. Overdamped behavior is characterized by a monotonic approach to steady state. Underdamped behavior is characterized by an oscillatory approach to steady state. A critically damped response marks the boundary between overdamped and underdamped behavior. 

Depending on the value of the damping factor for the uncontrolled second-order system, eq. (14.23b) shows that 1. If > 1, the overdamped response of the closed-loop system is very sluggish. Therefore, we prefer to increase the value of Ke and make < 1. Then the closed-loop response reacts faster but it becomes oscillatory. Also, by increasing Kc, the offset decreases. 

The time-domain behavior of the zero-input response of a circuit is related to the frequency-domain property of resonance. In the case of a second-order circuit, its zero-input response will be overdamped, critically damped, or underdamped, depending on the value of the circuit s components. If the components are such that the response is highly underdamped, the circuit is said to be in resonance, and its zero-input response will be oscillatory in nature and will not decay rapidly. The relative proximity of the poles of the circuit s transfer function H(s) to the j axis accounts for this oscillatory behavior. To see this, note that each distinct pair of complex poles in H s) contributes to yz R(t) a term having the following form ... 

Control system designers sometimes attempt to make the response of the controlled variable to a set-point change approximate the ideal step response of an underdamped second-order system, that is, make it exhibit a prescribed amount of overshoot and oscillation as it settles at the new operating point. When damped oscillation is desirable, values of in the range 0.4 to 0.8 may be chosen. In this range, the controlled variable y reaches the new operating point faster than with = 1.0 or 1.5, but the response is much less oscillatory (settles faster) than with = 0.2. 

From the analysis of second-order transfer functions in Chapter 5, we know that the closed-loop response is oscillatory for 0 < 4 < 1. Thus, Eq. 11-72 indicates that the degree of oscillation can be reduced by increasing either Kc(Kc > 0) or t/. The effect of t/ is familiar, because we have noted previously that increasing T/ tends to make closed-loop responses less oscillatory. However, the effect of Kc is just the opposite of what normally is observed. In most control problems, increasing Kc tends to produce a more oscillatory response. However, (11-72) indicates that increasing Kc results in a less oscillatory response. This anomalous behavior is due to the integrating nature of the process (cf. Eq. 11-66). 

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