Steadystate error

The basic purpose of integral action is to drive the process back to its setpoint when it has been disturbed. A proportional controller will not usually return the controlled variable to the setpoint when a load or setpoint disturbance occurs. This permanent error (SP — PM) is called steadystate error or offset. Integral action reduces the offset to zero. 

The steadystate error is another time-domain specification. It is not a dynamic specification, but it is an important performance criterion. In many loops (but not all) a steadystate error of zero is desired, i.e, the value of the controlled variable should eventually level out at the setpoint. 

B. PROPORTIONAL CONTROLLER. The output of a proportional controller changes only if the error signal changes. Since a load change requires a new control-valve position, the controller must end up with a new error signal. This means that a proportional controller usually gives a steadystate error or offset. This is an inherent limitation of P controllers and why integral action is usually added. 

Steadystate error is not always undesirable. In many level control loops the absolute level is unimportant as long as the tank does not run dry or overflow. Thus a proportional controller is often the best type for level control We will discuss this in more detail in Sec. 7.3. 

C. PROPORTIONAL-INTEGRAL (PI) CONTROLLER. Most control loops use PI controllers. The integral action eliminates steadystate error in T (see Fig. 7.11c). The smaller the integral time r, the faster the error is reduced. But the system becomes more underdamped as t( is reduced. If it is made too small, the loop becomes unstable. 

The usual steadystate performance specification is zero steadystate error. We will show below that this steadystate performance depends on both the system (process and controller) and the type of disturbance. This is different from the question of stability of the system which, as we have previously shown, is only a function of the system (roots of the characteristic equation) and does not depend on the input. 

Thus the steadystate error is reduced by increasing, the controller gain. 

If the steadystate error is to go to zero, the term l/s(l + Gm, <,)) must go to zero as s goes to zero. This requires that B(,)Gj f(,) must contain a 1/s term. Double integration is needed to drive the steadystate error to zero for a ramp input (to make the output track the changing setpoint). 

Repeat Prob. 10.6 using a proportional feedback controller [parts (h) and (d)]. Will there be a steadystate error in the closedloop system for (n) a step change in setpoint or (h) a step change in feed rate Fq ... 

This pulse transfer function has a zero at z = a and a pole at z = +1. It cannot produce any phase-angle advance since the pole lies to the right of the zero (a is less than 1). The pole at -I-1 is equivalent to integration (pole at s = 0 in continuous systems) which drives the system to zero steadystate error for step disturbances. 

To find the steadystate value of the error, we will use the final-value theorem from Chap. 9. 

---