Closed-loop damping ratio

The Process Reaction Method assumes that the optimum response for the closed-loop system occurs when the ratio of successive peaks, as defined by equation (3.71), is 4 1. From equation (3.71) it can be seen that this occurs when the closed-loop damping ratio has a value of 0.21. The controller parameters, as a function of R and D, to produce this response, are given in Table 4.2. 

Transient response criteria Analytical derivation Derive closed-loop damping ratio from a second order system characteristic polynomial. Relate the damping ratio to the proportional gain of the system. 

Example 5.2 Derive the closed-loop transfer function of a system with proportional control and a second order overdamped process. If the second order process has time constants 2 and 4 min and process gain 1.0 [units], what proportional gain would provide us with a system with damping ratio of 0.7 ... 

While we have the analytical results, it is not obvious how choices of integral time constant and proportional gain may affect the closed-loop poles or the system damping ratio. (We may get a partial picture if we consider circumstances under which KcKp 1.) Again, we ll defer the analysis... 

By and large, a quarter decay ratio response is acceptable for disturbances but not desirable for set point changes. Theoretically, we can pick any decay ratio of our liking. Recall Section 2.7 (p. 2-17) that the position of the closed-loop pole lies on a line governed by 0 = cos C In the next chapter, we will locate the pole position on a root locus plot based on a given damping ratio. 

In terms of controller design, the closed-loop poles (or now the root loci) also tell us about the system dynamics. We can extract much more information from a root locus plot than from a Routh criterion analysis or a s = jco substitution. In fact, it is common to impose, say, a time constant or a damping ratio specification on the system when we use root locus plots as a design tool. 

Where the root locus intersects the 0.7 damping ratio line, we should find, from the result returned by rlocfind (), the proportional gain to be 1.29 (1.2944 to be exact), and the closed-loop poles at -0.375 0.382j. The real and imaginary parts are not identical since cos 10.7 is not exactly 45°. 

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. 

If saturation is not a problem, the proportional gain Kc = 7.17 (point B) is preferred. The corresponding closed-loop pole has a faster time constant. (The calculation of the time period or frequency and confirmation of the damping ratio is left as homework.)... 

To achieve a damping ratio of 0.8, we can find that the closed-loop poles must be at -4.5 3.38j (using a combination of what we learned in Example 7.5 and Fig. 2.5), but we can cheat with MATLAB and use root locus plots ... 

Hence, our first step is to use root locus to find the closed-loop poles of a PI control system with a damping ratio of 0.8. The MATLAB statements to continue with Example 4.7B are ... 

A generally used set of criteria for good control is that the controlled variable in response to a unit step change in set point (a) overshoot by not more than 20 per cent of the step and (b) damp out with a subsidence ratio of about one-third. This behavior is approximated by many systems if the closed-loop frequency response and the corresponding open-loop frequency response have certain simple characteristics. Since the closed-loop frequency response characteristics can be determined readily from the open-loop frequency response, the latter characteristics of simple control systems can be used as a convenient basis for design. 

The damping ratio preferably should be at least unity. A low damping ratio, as shown by equations (16.12) and (16.13), causes severe peaking in the frequency response in the vicinity of the closed-loop natural frequency, I/t. Flow and level regulation in that frequency range be very poor. 

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