Figure 10.1. Temperature control system or stirred-tank heater with steam heating.( ----- electrical signals; pneumatic signal). 10. Dynamic Behavior of Closed-Loop Control Systems.
Figure 10.1. Temperature control system or stirred-tank heater with steam heating.( ----- electrical signals; pneumatic signal)10. Dynamic Behavior of Closed-Loop Control Systems
10.1 Block Diagram Representation
Where the subscripts , , and refer respectively to the wall of the heating coil, and to its steam and process sides.
Figure 10.2. Inputs and output of the process.
Substituting (10.4) into (10.1) gives
Figure 10.3. Block diagram of the process.
10.1.2 Thermocouple and Transmitter
Figure 10.4. Block diagram for the thermocouple
and temperature transmitter.
where and are the Laplace transforms of the controller output and error signal . Note that and are electrical signals which have the units of [mA] while is dimensionless.
Figure 10.5. Block diagram for the controller.
Since transducers are usually designed to have linear characteristics and negligible(fast) dynamics Assume that the transducer transfer function merely consists of a steady-state gain :
10.1.5 Control Valve
The flow through the valve is a non-linear function of the signal to the valve actuator. However, a first-order transfer function usually provides an adequate model for operation of an installed valve in the vicinity of a nominal steady-state.
10.1.4 Current-to-Pressure(I/P) Transducer
Figure 10.6. Block diagram for
the I/P transducer.
Figure 10.7. Block diagram for
the control valve.
Figure 10.8. Block diagram for the entire control system.
= Manipulated variable
= Load variable
= Controller output
= Error signal
= Change in due to
= Internal set point(used by controller)
= Measured value of
= Set point
= Change in due to
= Set point
= Process transfer function
= Transfer function for final control element
= Steady-state gain for
= Load transfer function
= Transfer function for measuring element and transmitter
10.2 Closed-Loop Transfer Functions
and must be preserved. Thus, and must be related by the expression.
Figure 10.10. Alternative form of the standard block diagram
of a feedback control system
Figure 10.12. Equivalent block diagram.
10.2.1 Block Diagram Reduction
Figure 10.11. Three blocks in series.
Figure 10.9. Standard block diagram of a feedback control system
A comparison of (10.23) and (10.25) indicate that both closed-loop transfer functions have the same denominator.
The denominator is written as where is the open-loop transfer function, .
10.2.2 Load Changes( = Regulatory Problem)
In this section, we consider the dynamic behavior of several elementary control problems for load variable and set-point changes.
Figure 10.10. Liquid-level Control System
q2: the manipulated variable.
1. The liquid density r and the cross-sectional area of the tank A are
2. The flow-head relation is linear, q3 = h/R.
3. The transmitter and control valves have negligible dynamics.
4. Pneumatic instruments are used.
Where, KP= R, t = RA
Assuming that the dynamics of the level transmitter and control valve, the corresponding transfer functions can be written as Gm(s) = Km and Gv(s) = Kv .
Figure 10.11. Block diagram for level control system
If a proportional controller is used, then Gc(s) = Kc .
The closed-loop transfer function for set-point changes is given by
KOLis the open-loop gain, KOL =Kc Kv Kp Km (KOL > 0 for stability → chapter 11)
Thus since t1 < t , one consequence of feedback control is that it enables the controlled process to respond more quickly than the uncontrolled process. The reason for the introduction of feedback control
The closed-loop response to a unit step change of magnitude M in set point is given by
Figure 10.12. Step response for proportional control (set-point change)
Since KOL =Kc Kv Kp Km
Kc : , KOL : , offset :
if Kc , offset 0
The closed-loop transfer function for load changes is given by
The closed-loop response to step change of magnitude M in load
The same situation can be observed as set point change case
The closed-loop transfer function for load change is given by
This transfer function can be rearranged as a second-order one.
where, K3 = tI / KcKvKm
because of the exponential term in (10.39).
For set-point change, offset will be zero too!
Addition of Integral action
Elimination of offset for step changes of load and set-point
Figure 10.13. Liquid-level control system with pump in exit line
This system differs from the previous example in two ways
1. The exit line contains a pump
2. The manipulated variable is the exit flow rate rather than an inlet flow rate.
The closed-loop transfer function for load changes
where, K4 = -tI / KcKvKm
KOL = KcKvKpKm , Kp = - 1/A
closed-loop responses: less oscillatory
Kc: z4: closed-loop responses: less oscillatory
Kc: closed-loop responses: more oscillatory