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10. Dynamic Behavior of Closed-Loop Control Systems

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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.

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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
• Feedback control loop( or closed-loop system) ; the combination of the process, the feedback controller, and the instruments.

10.1 Block Diagram Representation

Process

Where the subscripts , , and refer respectively to the wall of the heating coil, and to its steam and process sides.

10.1.1 Process

Figure 10.2. Inputs and output of the process.

• Dynamic model of a steam-heated, stirred tank:

Where is defined by

• Assume that .
• Assume that the dynamics of the heating coil are negligible since the dynamics are fast compared to the dynamics of the tank contents.  Left side of (10.2) equal to zero!

Substituting (10.4) into (10.1) gives

where

Figure 10.3. Block diagram of the process.

If  negligible dynamics.

• Steady-state gain depends on the input and output ranges of the thermocouple-transmitter combination.

10.1.2 Thermocouple and Transmitter

• Assume that the dynamic behavior of the thermocouple and transmitter can be approximated by a first-order transfer function:

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.

10.1.3 Controller

• Assume that a proportional plus integral controller is used

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.

Complete block diagram

Figure 10.8. Block diagram for the entire control system.

= Controlled variable

= 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

• The standard notations.
• In Figure 10.9. each variable is the Laplace Transform of a deviation variable.
• Feedforward path: the signal path from to through blocks ,

and .

• Feedback path: the signal path from to the comparator through .

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

By successive substitution,

Figure 10.12. Equivalent block diagram.

10.2.1 Block Diagram Reduction

• It is often convenient to combine several blocks into a single block.
• Example

Figure 10.11. Three blocks in series.

10.2.2 Set Point Changes( = Servo Problem)

Figure 10.9. Standard block diagram of a feedback control system

• Desired closed-loop transfer function,

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)

• Desired closed-loop transfer function,

10.3 Closed-loop Responses of Simple Control Systems

In this section, we consider the dynamic behavior of several elementary control problems for load variable and set-point changes.

• For the liquid-level control system

Figure 10.10. Liquid-level Control System

q1: the load variable.

q2: the manipulated variable.

Assumption:

1. The liquid density r and the cross-sectional area of the tank A are

constant.

2. The flow-head relation is linear, q3 = h/R.

3. The transmitter and control valves have negligible dynamics.

4. Pneumatic instruments are used.

• Mass balance for the tank contents.
• Transfer Function

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 .

• Block diagram for level control system

Figure 10.11. Block diagram for level control system

Proportional Control and Set-Point Change

If a proportional controller is used, then Gc(s) = Kc .

The closed-loop transfer function for set-point changes is given by

where,

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)

The offset ( steady-state error) is defined as

Since KOL =Kc Kv Kp Km

Kc : , KOL : , offset : 

if Kc , offset  0

Proportional Control and Load Changes

The closed-loop transfer function for load changes is given by

where,

The closed-loop response to step change of magnitude M in load

The same situation can be observed as set point change case

PI Control and Load Changes

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

For a unit step change in load,

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

PI Control of an Integrating Process

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 process and load transfer functions are given by

The closed-loop transfer function for load changes

where, K4 = -tI / KcKvKm

KOL = KcKvKpKm , Kp = - 1/A

The effect of tI

tI: 

: 

 closed-loop responses: less oscillatory

• The effect of Kc

Kc:   z4:   closed-loop responses: less oscillatory

• The effect of Kc for the stable process except the integrating process

Kc:   closed-loop responses: more oscillatory