Proportional control equation

The maintenance of constant liquid level in the reflux drum can be expressed by the following proportional control equation... 

Example 10.4. Consider again the three-CSTR system. We have already developed its doscdloop characteristic equation with a proportional controller [Equation (10.22)]. 

The time constant of the controller R C is called reset time and is the time interval in which the controller output changes by an amoimt equal to the input change or deviation. Note that, when the offset returns to zero. Equation 24.7 reduces to that describing pure proportional control. Equation 24.4. 

If we have a second order system, we can derive an analytical relation for the controller. If we have a proportional controller with a second order process as in Example 5.2, the solution is unique. However, if we have, for example, a PI controller (2 parameters) and a first order process, there are no unique answers since we only have one design equation. We must specify one more design constraint in order to have a well-posed problem. 

Example 7.2 If we have only a proportional controller (i.e., one design parameter) and real negative open-loop poles, the Routh-Hurwitz criterion can be applied to a fairly high order system with ease. For example, for the following closed-loop system characteristic equation ... 

Example 7.4 Consider a system with a proportional controller and a first order process but with dead time. The closed-loop characteristic equation is given as... 

We can now state the problem in more general terms. Let us consider a closed-loop characteristic equation 1 + KCG0 = 0, where KCG0 is referred to as the "open-loop" transfer function, G0l- The proportional gain is Kc, and G0 is "everything" else. If we only have a proportional controller, then G0 = GmGaGp. If we have other controllers, then G0 would contain... 

Example 7.4A. This time, let s revisit Example 7.4 (p. 7-8), which is a system with dead time. We would like to know how to start designing a PI controller. The closed-loop characteristic equation with a proportional controller is (again assuming the time unit is in min)... 

In the simplest scenario, we can think of the equation as a unity feedback system with only a proportional controller (i.e., k = Kc) and G(s) as the process function. We are interested in finding the roots for different values of the parameter k. We can either tabulate the results or we can plot the solutions s in the complex plane—the result is the root-locus plot. 

Consider the case of a proportional controller, which is required to maintain a desired reactor temperature, by regulating the flow of coolant. Neglecting dynamic jacket effects, the reactor heat balance can then be modified to include the effect of the varying coolant flow rate, hj, in the model equation as ... 

The proportional-integral control equation, as given in Section 2.3.2.2, is ... 

The equation describing a proportional controller in the time domain is ... 

C. THIRD-ORDER OPENLOOP UNSTABLE PROCESS. If an additional lag is added to the system and a proportional controller is used, the closedloop characteristic equation becomes... 

Also shown in Fig. 16.1 is the W plot when only proportional controllers are used. Note that the curves with P controllers start on the positive real axis. However, with PI controllers the curves start on the negative real axis. This is due to the two integrators, one in each controller, which give 180 degrees of phase angle lag at low frequencies. As shown in Eq. (16.3), the product of the and B2 controllers appears in the closedloop characteristic equation. 

It can be seen from equation 7.3 and Fig. 7.7 that the controller output will continue to increase as long as e > 0. With proportional control an error (offset) had to be maintained so that the controlled variable (i.e. the temperature at Y—Fig. 7.1) could be kept at a new control point after a step change in load, i.e. in the inlet temperature of the cold stream. This error was required in order to produce an additional output from the proportional controller to the control valve. However, with PI control, the contribution from the integral action does not return to zero with the error, but remains at the value it has reached at that time. This contribution provides the additional output necessary to open the valve wide enough to keep the level at the desired value. No continuous error (i.e. no offset) is now necessary to maintain the new steady state. A quantitative treatment of this is given later (Section 7.9.3). 

Control action due to the derivative mode occurs only when the error is changing (equation 7.4). The presence of the derivative mode contributes an additional output, KD(de/dt), to the final control element as soon as there is any change in error. When the error ceases to change, derivative action no longer occurs (Fig. 7.8). The effect of this is similar to having a proportional controller with a high gain... 

Referring to Fig. 7.36, for a proportional controller Gc(j) = Kc (equation 7.62). Assume for simplicity that the time constants of G,(j) (the final control element) and H(j) (the measuring element) are negligible compared with that of G2(j) (the... 

Thus, if Vit represents the set point of the controller, Vij the measured value and R /R2 the proportional gain, then equation 7.261 is the equivalent of the relationship for proportional control action given by equation 7.2. The proportional potentiometer enables the proportional gain to be varied. 

Figure 3.8 shows the evolution of the product purity X23 (the main performance indicator of the process) for different step changes in Mrsp. The operation of the process is stabilized by the three proportional controllers in Equation (3.35). Clearly, these responses indicate that the slow dynamics of the process is nonlinear the implementation of a nonlinear supervisory controller for this process is thus highly desirable. 

Note that, owing to the underlying algebraic constraints in the DAE system that describes the slow dynamics, the holdups Mb, Me, and Mr are not independent (there are only two linearly independent constraints among the three holdups, i.e., 0 = wr — and 0 = k2w2 — r, where u, U2, and kr are determined by the proportional control laws in Equation (3.35)). Thus, controlling one of the holdups (e.g., Mb) amounts to regulating the total material holdup in the process. 

Following a similar procedure to the one employed above, it is easy to verify that we obtain a model that approximates the fast dynamics of the system in Figure 4.2, in the form of Equation (4.20). Also, it can be verified that only 2N + 8 of the 2N + 9 steady-state constraints that correspond to the fast dynamics are independent. After controlling the reactor holdup Mr, the distillate holdup Md, and the reboiler holdup MB with proportional controllers using respectively F, D, and B as manipulated inputs, the matrix Lb (x) is nonsingular, and hence the coordinate change... 

We will first concentrate on studying the process dynamics, so let us consider a numerical experiment that consists of starting a dynamic simulation of the process from initial conditions that are slightly perturbed from the nominal, steady-state values of the state variables. Although material holdups are stabilized using the proportional controllers in Equation (4.40), in view of the process-level operating objective stated above, this can be considered an open-loop simulation. 

Inspecting Equation (5.29), we notice that three of the state variables (namely, Mr, My, and Ml) are material holdups, which act as integrators and render the system open-loop unstable. Our initial focus will therefore be a pseudo-open loop analysis consisting of simulating the model in Equation (5.29) after the holdup of the reactor, and the vapor and liquid holdup in the condenser, have been stabilized. This task is accomplished by defining the reactor effluent, recycle, and liquid-product flow rates as functions of Mr, My, and Ml via appropriate control laws (specifically, via the proportional controllers (5.42) and (5.48), as discussed later in this section). With this primary control structure in place, we carried out a simulation using initial conditions that were slightly perturbed from the steady-state values in Table 5.1. 

The first (distributed) layer of the control structure proposed in Section 5.4 was implemented as described in Equation (5.42), i.e., by stabilizing the holdups of the units within the recycle loop (the reactor and the vapor phase of the condenser) with proportional control laws. The liquid holdup in the condenser... 

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