Integral control action

The parameter X/ is called the integral time constant. For a rgactor it is the order of magnitude at the space-time. Integral control action can also be expressed by the coupled set of equations... 

Integral control action alone is expected to make the response of the closed-loop system more sluggish. 

The reader can verify easily that for the regulator problem the integral control action produces a second-order closed-loop response and leads again to zero offset. 

Equation (14.26b) indicates that the form of the closed-loop response (i.e., overdamped, critically damped, underdamped) depends on the values of the controller gain Kc and reset time t/. Therefore, tuning the integral control action for the appropriate values of Kc and T is an important question and will be discussed in Chapters 16 and 18. 

Integral control action makes a process (a) faster or slower (b) more oscillatory or less (c) with larger deviations from the set point or smaller Explain your answers. 

Input/Output interface, 557-61 Input-output models, 81 discrete-time, 609-26 examples, 81-85, 162, 163, 166 using Laplace transforms, 159-66 Input variables, 12-14 Integral of absolute error, 302 Integral control action, 273, 277-78 advantages and drawbacks, 274-75, 307... 

Equation (14.25) indicates an important effect of the integral control action,... 

What is the condition that X and Xj are negative It is that Kc > 0 and 4AKj > 0 (and the condition of 4AKj < Kc is also satisfied) which is, of course, always satisfied. Thus, it is clear that the integral control action always prevents the offset, since X and X2 are always negative. 

For integral control action, the controller output depends on the integral of the error signal over time,... 

An inherent disadvantage of integral control action is a phenomenon known as reset windup. Recall that the integral mode causes the controller output to change as long as e(t ) 0 in Eq. 8-8. When a sustained error... 

The function of derivative control action is to anticipate the future behavior of the error signal by considering its rate of change. In the past, derivative action was also referred to as rate action, pre-act, or anticipatory control For example, suppose that a reactor temperature increases by 10 °C in a short period of time, say, 3 min. This clearly is a more rapid increase in temperature than a 10 °C rise in 30 min, and it could indicate a potential runaway situation for an exothermic reaction. If the reactor were under manual control, an experienced plant operator would anticipate the consequences and quickly take appropriate corrective action to reduce the temperature. Such a response would not be obtainable from the proportional and integral control modes discussed so far. Note that a proportional controller reacts to a deviation in temperature only, making no distinction as to the time period over which the deviation develops. Integral control action is also ineffective for a sudden deviation in temperature, because the corrective action depends on the duration of the deviation. 

The addition of integral control action eliminates offset but tends to make the response more oscillatory. Adding derivative action reduces both the degree of oscillation and the response time. The use of P, PI, and PID controllers does not always result in oscillatory process responses the nature of the response depends on the choice of the controller settings T/, and T/)) and the process dynamics. However, the responses in Fig. 8.12 are representative of what occurs in practice. 

For a liquid-level control system similar to that in Fig. 11.22, Appelpolscher has argued that integral control action... 

The IhcS term provides integral control action and thus eliminates offset. Design parameter provides a convenient controller tuning parameter that can be used to make the controller more aggressive (small t ) or less aggressive (large t ). 

Note that this controller also contains integral control action. 

When integral control action is added to a proportional-only controller, Kc should be reduced. The further addition of derivative action allows Kc to be increased to a value greater than that for proportional-only control. 

Because offset is not important in averaging level control, it is reasonable to use a proportional-only controller. But if integral control action is desired, St. Clair (1993) recommends the following PI controller settings for averaging level control ... 

Conversely, the addition of integral control action can produce closed-loop instability. Curve b in Fig. 14.10 indicates that an unstable closed-loop system occurs for Gc(s) = 0.4 (1 + l/0.2s), because ARql > 1 when ql = -180°. To find Kcm for t/ = 0.2 min, we note that (o<. depends on t/ but not on K, because Kc has no effect on curve... 

The MPC control law in Eq. 20-56 can be interpreted as a multivariable, proportional control law based on the predicted error rather than the conventional control error (set point-measurement). The control law utilizes the latest measurement y(k) because it appears in the expressions for the corrected prediction y(A ), and thus also in the predicted unforced error, E k + 1). Furthermore, the MPC control law in Eq. (20-56) implicitly contains integral control action because u tends to change until the unforced error E becomes zero. Thus, offset is eliminated for set-point changes or sustained disturbances. 

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