Underdamped second-order system

General case for an underdamped second-order system... 

The response of an underdamped second-order system to a unit step change may be shown to be ... 

Example 6il2. A general underdamped second-order system is forced by a unit step function ... 

Which one the three controllers, P, PI, PID, would give more robust closed-loop response to an underdamped second-order system ... 

For process control problems the underdamped form is frequently encountered in investigating the properties of processes under feedback control. Control systems are sometimes designed so that the controlled process responds in a manner similar to that of an underdamped second-order system (see Chapter 12). Next we develop the relation for the step response of all three classes of second-order processes. 

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. 

Starting with Eq. 5-51, derive expressions for the following response characteristics of the underdamped second-order system. 

The root locus diagram can be used to provide a quick estimate of the transient response of the closed-loop system. The roots closest to the imaginary axis correspond to the slowest response modes. If the two closest roots are a complex conjugate pair (as in Fig. 11.28), then the closed-loop system can be approximated by an underdamped second-order system as follows. 

The curves in Fig. J.14 are similar to the corresponding plots for an underdamped second-order system (see Sections 5.4 and 14.3.3). For Fig. 14.3 a peak amplitude ratio of 1.25 corresponds to a damping coefficient of... 

With the expectation that the second order system may exhibit underdamped behavior, we rewrite the closed-loop function as... 

The point of the last two calculations is that a simple second order system may become extremely underdamped, but it never goes unstable. 

Note that the system remains stable in all cases, as it should for a first or second order system. One final question Based on the design guidelines by which the system should respond faster than the process and the system should be slightly underdamped, what are the ranges of derivative and integral time constants that you would select for the PD, PI, and PID controllers And in what region are the desired closed-loop poles ... 

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. 

The examples above have illustrated a very important point the higher the order of the system, the worse the dynamic response of the closedloop system. The hrst-order system is never underdamped and cannot be made closedloop unstable for any value of gain. The second-order system becomes underdamped as gain is increased but never goes unstable. Third-order (and higher) systems can be made closedloop unstable. 

Discuss the overdamped, critically damped, and underdamped responses of a second-order system. Identify their distinguishing characteristics. 

Remark. In subsequent chapters (Part IV), our objective during the design of a controller will be proper selection of the corresponding C and T values, so that the overshoot is small, the rise time short, the decay ratio small, and the response time short. We will realize that it will not be possible to achieve all these objectives for the same values of C and t, and that an acceptable compromise should be defined. Good understanding of the underdamped behavior of a second-order system will help tremendously in the design of efficient controllers. 

If the time constants xPI and xP2 are equal, we have two equal poles. Therefore, noninteracting capacities always result in an overdamped or critically damped second-order system and never in an underdamped system. The response of two noninteracting capacities to a unit step change in the input will be given by eq. (11.7) for the overdamped case, or eq. (11.8) for the critically damped. Instead of eq. (11.7), we can use the following equivalent form for the response ... 

Such a process can exhibit underdamped behavior, and consequently it cannot be decomposed into two first-order systems in series (interacting or noninteracting) with physical significance, like the systems we examined in previous sections. They occur rather rarely in a chemical process, and they are associated with the motion of liquid masses or the mechanical translation of solid parts, possessing (1) inertia to motion, (2) resistance to motion, and (3) capacitance to store mechanical energy. Since resistance and capacitance are characteristic of the first-order systems, we conclude that the inherently second-order systems are characterized by their inertia to motion. The three examples in Appendix 11A clearly demonstrate this feature. 

For the selection of the best values for Kc and t/ we will use simple criteria stemming from the underdamped response of a second-order system. Select the one-quarter decay ratio criterion. From eq. (11.12) we know that... 

Figure 14.3 shows the Bode plots for overdamped (C>1), critically damped ( = 1), and underdamped (0< = 1) processes as a function of cot. The low-frequency limits of the second-order system are identical to those of the first-order system. However, the limits are different at high frequencies, cot 1. 

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.)... 

In contrast to Eq. (6-22), we can dictate a second order underdamped system response ... 

Example 6.4 Derive the controller function for a system with a second order overdamped process but an underdamped system response as dictated by Eq. (6-27). 

The technique of using the damp ratio hne 0 = cos in Eq. (2-34) is apphed to higher order systems. When we do so, we are implicitly making the assumption that we have chosen the dominant closed-loop pole of a system and that this system can be approximated as a second order underdamped function at sufficiently large times. For this reason, root locus is also referred to as dominant pole design. 

Derive an analytical relationship between openloop oiajumum log modulus and damping coeflident for a second-order underdamped openloop system with a gain of unity. Show that a damping coeflident of 0.4 corresponds to a maximum log modulus of +2J dedbels. 

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. 

Example 11.4 demonstrates very clearly how the simple first-order dynamic behavior of a tank can change to that of a second-order when a proportional-integral controller is added to the process. Also, it indicates that the control parameters Kc and r can have a very profound effect on the dynamic behavior of the system, which can range from an underdamped to an overdamped response. 

The model for a variable capacitance pressure transducer was developed in Appendix 11A and is given by eq. (13.9). It shows that the system is inherently second-order and can exhibit underdamped response. What does this mean for the applicability of such device ... 

Systems with inherent second-order dynamics can exhibit oscillatory (underdamped) behavior but are rather rare in chemical processes. In this appendix we present three simple units which can be encountered in chemical plants and which possess second-order dynamics. 

In Appendix 11A we found that simple manometers, externally mounted level indicators, variable capacitance transducers, and pneumatic valves exhibit inherent second-order dynamic behavior. Design all the systems above so that they exhibit underdamped behavior with a decay ratio equal to. In other words, find the conditions that the values of the physical parameters of these systems should satisfy in order for the device to exhibit underdamped behavior with decay ratio... 

Step responses of a second-order underdamped system. 

The second-order underdamped system is probably the most important transfer function that we need to translate into the frequency domain. Since we often design... 

---