Time constant: closed-loop

Like direct synthesis, xc is the closed-loop time constant and our only tuning parameter. The first order function raised to an integer power of r is used to ensure that the controller is physically realizable. 2 Again, we would violate this intention in our simple example just so that we can obtain results that resemble an ideal PID controller. 

We want tight control of these important quantities for economic and operational reasons. Hence we should select manipulated variables such that the dynamic relationships between the controlled and manipulated variables feature small time constants and deadtimes and large steady-state gains. The former gives small closed-loop time constants and the latter prevents problems with the rangeability of the manipulated variable (control valve saturation t... 

The closed-loop time constant is smaller than the open-loop time constant. That is, proportional action makes the closed-loop process respond faster than the open-loop process. 

Figure 8.8 gives the root locus plot for a proportional controller. The three curves start at - 10 on the real axis (only one complex root is shown). Two of the loci go off at 60 angles and cross the imaginary axis at 17.3 (the ultimate frequency) vdien the gain is 6 (the ultimate gain). For a closedloop damping coefficient of 0.3 (the radial line), the closed-loop time constant is about 0.085 hours. 

If a proportional feedback temperature controller is used, calculate the controller gain Kc that yields a closedloop damping coefficient of 0.707, and calculate the closed-loop time constant of the system when ... 

Figure 2 shows the frequency response of the linearized model from the scaled disturbance fco to the scaled output yo. As seen from the figure the crossover frequency (Od 0.2 rad/min, implying that we need to attenuate disturbances up to this frequency. With feedback control this would correspond to a closed-loop time-constant around 5 min. This can be related to the process time-constant which is about 170 min. To check whether there exist any fundamental properties of the process that will limit the achievable bandwidth we compute the zeros of... 

The results shown in Figure 14.7 reveal fairly large closed-loop time constants. This occurs because of the large holdup of retentate in the five modules relative to flie small flowrate of retentate. The total retentate holdup in the live modules is 4.5 m. The flowrate of distillate to the pervaporation unit is 57.4kmol/h, which corresponds to 54 L/min. Thus the residence time of retentate is about 80 min. 

A structured model of the process (typically a Laplace tramsfer function) is used directly in a design method such as pole-placement or internal model control (IMG) to yield expressions for the controfier parameters that are functions of the process model parameters and some user-specified parameter related to the desired performance, e.g. a desired closed-loop time constant. These approaches to PID design carry restrictions on the allowable model structure, although it has been shown that a wide range of types of processes can be accommodated if the PID controller is augmented with a first order filter in series. An example of this design approach may be found in Rivera et al. (1986). 

The parameter C is set to be 1 and fd is used to adjust the performance, where the desired closed-loop time constant is P y. Recall that a smaller 0 corresponds to a faster desired closed-loop system response speed. The PID controller parameters, calculated using an estimate of the desired closed-loop settling time given by Ta = 7 l7i, are shown in Table 6.3 for three diflferent choices for 0. Closed-loop simulations are shown in Figmre 6.15... 

We refer to as the normaJized desired closed-loop time constant and denote it ais fcj. Therefore, the desired closed-loop time constant and its normaJized form are related by = dfd. For a step setpoint change, we would expect the process output to take approximately (5fci -I- l)d = (5tcj -I- d) time units to reaich the new setpoint vaJue. 

Note that the normalized PID controller parameters Kc, f/ and fu are only dependent on the ratio of the time constant to delay, L, and the normalized desired closed-loop time constant, fd-... 

Although Equations (7.11)-(7.25) give analytical solutions for the PID controller parameters, it would be even more convenient for the user to have tuning rules requiring a minimum number of calculations. In order to present a range of desired closed-loop response speeds, we have chosen six different values for the normalized closed-loop time constant id = 4, 2, 1.33, 1, 0.8 and 0.67. The corresponding value of the parameter a can be simply calculated as a = - and the aw tual desired closed-loop time constant is given by... 

To apply the new timing rules, the user must first identify the process dynamics in terms of JC, T and d. Then, a value for the normalized desired closed-loop time constant Td must be selected. This choice of fd would be dictated by the desired response speed of the closed-loop system (i.e. Ta = (5fci -f- l)d) with due consideration for the acceptable stability margins. This information is available for both PI and PID settings in Table 7.1 and Figures 7.1-7.4. With the selected value for fd and the value identified for i = 7, the normalized controller parameters Kc, f/, fo) are calcidated from Table 7.1. The final step is to evaluate the actual controller parameters for implementation according to Equations (7.11), (7.15) and (7.21), i.e. Kc = -,Ti = dfj TD — dfu-... 

Both PI and PID controllers have been designed for this process for three values of fd = 2, 1, and 0.67, which correspond in this case to desired closed-loop time constants t. of 10, 5 and 3.3, respectively. Table 7.2 presents the actual controller parameters for these choices of fd- Figure 7.5 shows the closed-loop responses with PI control and Figure 7.6 shows the closed-loop responses under PID control, for a unit step setpoint change followed by a negative unit step load disturbance. The closed-loop system has been simulated with the derivative action, including a first order filter with time constant equal to O.Itd, applied only to the measurement. 

In the derivation of the rules, the damping factor has been chosen equal to either 0.707 or 1 to produce two sets of tuning rules, with the parameter /3 being used to adjust the closed-loop response speed. The desired closed-loop time constant is t = /3d. /3 has been varied for both cases from 1 to 17 and the corresponding normalized PID controller parameters Kc, ti and td have been calculated. Prom this information, polynomial functions have been fit to produce explicit solutions for the normalized controller parameters as a function of j3. 

The root locations also provide an indication of how rapid the transient response will be. A real root at 5- = -a corresponds to a closed-loop time constant of T = 1/a, as is evident from Eqs. 11-85 and 11-86. Thus, real roots close to the imaginary (vertical) axis result in slow responses. Similarly, complex roots near the imaginary axis correspond to slow response modes. The farther the complex roots are away from the real axis, the more oscillatory the transient response will be (see Example 11.14). However, the process zeros also influence the response, as discussed in Chapter 6. 

Consider three values of the desired closed-loop time constant ic = 10- Evaluate the controllers for unit... 

In analogy with the DS method, is the desired closed-loop time constant. Parameter r is a positive integer. The usual choice is r = 1. 

From Eq. 16-11 the closed-loop time constant for the inner loop is 0.2 min. In contrast, the conventional feedback control system has a time constant of 1 min because in this case, Y2 s)IYsp2 s) = Gy = 5/(5 + 1). Thus, cascade control significantly speeds up the response of Y2. Using a proportional controller in the primary loop (G -i = K ), the characteristic equation becomes... 

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