Controller gain

Controller gain Load transfer function gain Measurement gain Process gain... 

The controller gain is inversely proportional to the process gain for constant dead time and time constant. 

Time-Delay Compensation Time delays are a common occurrence in the process industries because of the presence of recycle loops, fluid-flow distance lags, and dead time in composition measurements resulting from use of chromatographic analysis. The presence of a time delay in a process severely hmits the performance of a conventional PID control system, reducing the stability margin of the closed-loop control system. Consequently, the controller gain must be reduced below that which could be used for a process without delay. Thus, the response of the closed-loop system will be sluggish compared to that of the system with no time delay. 

Detuning a controller (e.g., using a smaller controller gain or a larger reset time) tends to reduce control loop interactions by sacrificing the performance for the detuned loops. This approach may be acceptable if some of the controlled variables are faster or less important than others. 

Eodt (13-174) where V and are initial values, Kc and T are respectively feed-back-controller gain and feedback-reset time for integr action, and E is the error or deviation from the set point as given by ... 

Basic process control system (BPCS) loops are needed to control operating parameters like reactor temperature and pressure. This involves monitoring and manipulation of process variables. The batch process, however, is discontinuous. This adds a new dimension to batch control because of frequent start-ups and shutdowns. During these transient states, control-tuning parameters such as controller gain may have to be adjusted for optimum dynamic response. 

Proportional control, gain A lfV/m) Control signal U s) = KfX is) - ZmCv))... 

Control problem To select the controller gain K to achieve the settling time and tachogenerator constant to provide critical damping. 

Hence, to give a GM of 2 and a PM of 50°, the controller gain must be set at 1.0. If it is doubled, i.e. multiplied by the GM, then the system just becomes unstable. Check using the Routh stability criterion ... 

Figure 6.27 (see also Appendix 1, fig627.m) shows the Nichols chart for K = 4 (controller gain K = 1). These are the settings shown in the Bode diagram in Figure 6.23(a), curve (i), and (b), where... 

Equation set (9.104) approximates to an inverse square law, and increases the controller gains at low speeds, where the control surfaces are at their most insensitive. 

Using a GA with a population of 10 members, find the values of the controller gain K and the tachogenerator constant that maximizes the fitness function... 

Luyben (1973) (see simulation example RELUY) also demonstrates a reactor simulation including the separate effects of the measuring element, measurement transmitter, pneumatic controller and valve characteristics which may in some circumstances be preferable to the use of an overall controller gain term. 

Proportional control can be based on the temperature of the third stage. Here FO is the base flow rate, KC is the proportional controller gain, and TSET is the temperature set point. Note that in order to guard against the unrealistic condition of negative flow, a limiter condition on F should be inserted into the DYNAMIC region. This can be accomplished with ISIM by the following statement... 

Study the influence of control, both by varying the controller gain, Kq, and the set point temperature, Tjet- Compare the results with the conclusions stated by Luus (1972). Note whether the amplitude of the reactor oscillations is greater or smaller than those of the disturbances. 

In this chapter, however, our objective is more restricted. We will purposely choose simple cases and make simplifying assumptions such that the results are PID controllers. We will see how the method helps us select controller gains based on process parameters (/. e., the process model). The method provides us with a more rational controller design than the empirical tuning relations. Since the result depends on the process model, this method is what we considered a model-based design. 

From Eq. (6-20), it is immediately clear that we cannot have an ideal servo response where C/R = 1, which would require an infinite controller gain. Now Eq. (6-21) indicates that C/R cannot be some constant either. To satisfy (6-21), the closed-loop response C/R must be some function of s, meaning that the system cannot respond instantaneously and must have some finite response time. 

This is an equation that we will use to retrieve the corresponding PID controller gains. For now, we substitute Eq. (6-28a) in an equation around the process,... 

Table 6.3. Summary of methods to select controller gains... 

The complete Routh array analysis allows us to find, for example, the number of poles on the imaginary axis. Since BIBO stability requires that all poles lie in the left-hand plane, we will not bother with these details (which are still in many control texts). Consider the fact that we can calculate easily the exact roots of a polynomial with MATLAB, we use the Routh criterion to the extent that it serves its purpose.1 That would be to derive inequality criteria for proper selection of controller gains of relatively simple systems. The technique loses its attractiveness when the algebra becomes too messy. Now the simplified Routh-Hurwitz recipe without proof follows. 

The Hurwitz test requires that Kc > -6 or simply Kc > 0 for positive controller gains. 

Root locus method gives us a good indication of the transient response of a system and the effect of varying the controller gain. However, we need a relatively accurate model for the analysis, not to mention that root locus does not handle dead time as well. 

Since few things are exact in this world, we most likely have errors in the estimation of the process and the dead time. So we only have partial dead time compensation and we must be conservative in picking controller gains based on the characteristic polynomial 1 + GCG = 0. 

There are two very useful MATLAB features. First, we can overlay onto the root locus plot lines of constant damping factor and natural frequency. These lines help us pick the controller gain if the design specification is in terms of the frequency or the damping factor. 

Establish limits on the controller gain. Usually applies to relatively simple systems with the focus on the proportional gain. Need be careful on interpretation when the lower limit on proportional gain is negative. 

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