5.10 Summary

The objective of this chapter was to provide an introduction to feedback control. We showed how to convert an instrumentation and control diagram to a control block diagram. Block diagram manipulations were used to develop a closed-loop transfer function. Transfer function representations for standard PID controllers were developed. The Routh array was used to test for the necessary and sufficient conditions for closed-loop stability based on the characteristic equation of the closed-loop transfer function. We noted that proportional controllers exhibit offset for setpoint and disturbance changes, motivating the development of controllers with integral action. Open-loop stable processes generally require that the controller gain be the same sign as the process gain, while open-loop unstable processes often require that the controller gain be the opposite sign of the process gain.

It is suggested that you work some of the problems in the student exercises section. It is particularly important to be able to find the closed-loop relationship (transfer function) between any two signals on a control block diagram.

What has not been discussed in this chapter is the issue of robustness. We have presented methods of analysis that assume that the process transfer function is exactly known. In reality, a process model is only an approximate representation of the dynamic behavior of the process. A set of tuning parameters that is stable under one operating condition may be unstable under another. Chapter 7 presents tuning methods that account for uncertainty in process parameters. Also, the model-based methods developed in Chapters 8 and 9 allow for detuning parameters based on model parameter uncertainty.

Common terms used in this chapter are as follows:

Characteristic eqn:	denominator polynomial for closed-loop analysis
Fail-closed:	valve closes on loss of instrument air (air-to-open)
Fail-open:	valve opens on loss of instrument air (air-to-close)
Manipulated input:	also called the controller output (CO)
Necessary condition:	must be satisfied for a chance of stability
Offset:	difference between setpoint and long-term process output
Process output:	also called the process variable (PV)
Routh stability:	stability test based on the closed-loop characteristic equation
Sufficient condition:	guarantees stability (if necessary condition is satisfied)

The tuning parameters are as follows:

kc:	proportional gain
tI:	integral time
tD:	derivative time
PB:	proportional band	(PB + 100/kc)

The block diagram (Laplace domain) variables are as follows:

c(s):	controller output
e(s):	error
r(s):	setpoint (reference signal)
u(s):	manipulated variable
y(s):	process output
ym(s):	measured output

The transfer functions are as follows:

gc(s):	controller
gv(s):	valve
gp(s):	process
gm(s):	measurement
gCL(s):	closed-loop

The controller types are as follows:

P:	proportional
PI:	proportional-integral
PID:	proportional-integral-derivative

The relationships are as follows:

gc(s) = kc	P control
	PI control
	"Ideal" PID control
	"Real" PID control (series)
	"Real" PID control (parallel)
	Ideal derivative of output rather than error
	Real derivative of output rather than error
	Closed-loop transfer function (CLTF)
	CLTF simplified
CLCE = 1 + gp(s)gv(s)gc(s)gm(s)	Closed-loop characteristic equation
CLCE = 1+ gp(s)gc(s)	Simplified CLCE