Underdamped system

Testing of metal-enclosed switchgear assemblies 14/447 For underdamped systems. 0 < rj < I... 

It is possible to identify the mathematieal model of an underdamped seeond-order system from its step response funetion. 

Consider a unity-gain K = ) seeond-order underdamped system responding to an input of the form... 

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

From the solution of the underdamped step response (0 < f < 1), we can derive the following characteristics (Fig. 3.2). They are useful in two respects (1) fitting experimental data in the measurements of natural period and damping factor, and (2) making control system design specifications with respect to the dynamic response. 

The elimination of the offset is usually at the expense of a more underdamped system response. The oscillatory response may have a short rise time, but is penalized by excessive overshoot or exceedingly long settling time. 1... 

In this section, we will derive the closed-loop transfer functions for a few simple cases. The scope is limited by how much sense we can make out of the algebra. Nevertheless, the steps that we go through are necessary to learn how to set up problems properly. The analysis also helps us to better understand why a system may have a faster response, why a system may become underdamped, and when there is an offset. When the algebra is clean enough, we can also make observations as to how controller settings may affect the closed-loop system response. The results generally reaffirm the qualitative statements that we ve made concerning the characteristics of different controllers. 

The key is to recognize that the system may exhibit underdamped behavior even though the open-loop process is overdamped. The closed-loop characteristic polynomial can have either real or complex roots, depending on our choice of Kc. (This is much easier to see when we work with... 

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

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.)... 

One reason why this approximation works is that process unit operations are generally open-loop stable, and many are multi-capacity in nature. Reminder Underdamped response of the system is due to the controller, which is taken out in the open-loop step test. 

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). 

With the IMC tuning setting in Example 5.7B, the resulting time response plot is (very nicely) slightly underdamped even though the derivation in Example 6.4 predicates on a system response without oscillations. Part of the reason lies in the approximation of the dead time function, and part of the reason is due to how the system time constant was chosen. Generally, it is important to double check our IMC settings with simulations. 

If we increase further the value of Kg, the closed-loop poles will branch off (or breakaway) from the real axis and become two complex conjugates (Fig. E7.5). No matter how large Kc becomes, these two complex conjugates always have the same real part as given by the repeated root. Thus what we find are two vertical loci extending toward positive and negative infinity. In this analysis, we also see how as we increase Kc, the system changes from overdamped to become underdamped, but it is always stable. 

There are two important steps that we must follow. First, make sure you go through the MATLAB tutorial (Session 6) carefully to acquire a feel on the probable shapes of root locus plots. Secondly, test guidelines 3 and 4 listed above for every plot that you make in the tutorial. These guidelines can become your most handy tool to deduce, without doing any algebra, whether a system will exhibit underdamped behavior. Or in other words, whether a system will have complex closed-loop poles. 

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. 

We also want to have a slightly underdamped system with a reasonably fast response and a damping ratio of 0.7. How should we design the controller To restrict the scenario for illustration, we consider (a) Xi = 3 min, and (b) x = 2/3 min. (a)... 

For a process or system that is sufficiently underdamped, Z, < 1/2, the magnitude curve will rise above the low frequency asymptote, or the polar plot will extend beyond the K-radius circle. 

We can design a controller by specifying an upper limit on the value of Mpco. The smaller the system damping ratio C the larger the value of Mpco is, and the more overshoot, or underdamping we expect in the time domain response. Needless to say, excessive resonance is undesirable. 

From the characteristic polynomial, it is probable that we ll get either overdamped or underdamped system response, depending on how we design the controller. The choice is not clear from the algebra, and this is where the root locus plot comes in handy. From the perspective of a root-locus plot, we can immediately make the decision that no matter what, both Zq and p0 should be larger than the value of l/xp in Gp. That s how we may "steer" the closed-loop poles away from the imaginary axis for better system response. (If we know our root locus, we should know that this system is always stable.)... 

Example 4.7B Let us revisit the two CSTR-in-series problem in Example 4.7 (p. 4-5). Use the inlet concentration as the input variable and check that the system is controllable and observable. Find the state feedback gain such that the reactor system is very slightly underdamped with a damping ratio of 0.8, which is equivalent to about a 1.5% overshoot. 

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

These authors also reported theoretical calculations of this frequency-dependent rotational relaxation. The theory of Auer et al. [98] using the quadratic electric field map, originally developed for HOD/D2O, was extended to the H0D/H20 system [52]. As before [38], the orientation TCF was calculated for those molecules within specified narrow-frequency windows (those selected in the experiment) at t = 0. TCFs for selected frequency windows, up to 500 fs, are shown in Fig. 8. One sees that in all cases there is a very rapid decay, in well under 50 fs, followed by a pronounced oscillation. The period of this oscillation appears to be between about 50 and 80 fs, which corresponds most likely to underdamped librational motion [154]. Indeed, the period is clearly longer on the blue side, consistent with the idea of a weaker H bond and hence weaker restraining potential. At 100 fs the values of the TCFs show the same trend as in experiment, although the theoretical TCF loses... 

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 above equation is called the characteristic equation of the system. It is the system s most important dynamic feature. The values of s that satisfy Eq. (6.67) are called the roots of the characteristic equation (they are also called the eigenvalues of the system). Their values, as we will shortly show, will dictate if the system is fast or slow, stable or unstable, overdamped or underdamped. Dynamic analysis and controller design consists of finding out the values of the roots of the characteristic equation of the system and changing their values to give the desired response. The rest of this book is devoted to looking at roots of characteristic equations. They are an extremely important concept that you should fully understand. ... 

For C < 1 (underdamped system). Things begin to get interesting when the damping coefficient is less than unity. Now the term inside the square root in Eq. 

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