Closed loops--Stability and control modes (part 2)

Integral control

Integral action is used to control towards no offset in the output signal. So, it controls towards no error (ERR = 0). Integral control is normally used to assist proportional control. We call the combination of both PI-control.

Integral and proportional with integral formula:

Formula for I-control: Single Flow Loop - Proportional (P) Control ~ Flow Control This exercise will introduce the main control action of controllers - proportional control.

Formula for PI-control:

INT T is the integral time constant.

Since integral control (I-control) integrates the error over time, the control action grows larger the longer the error persists. This integration of the error takes place until no error exists. Every integral action has a phase lag of 90°. This phase shift has a destabilizing effect. For this reason, we rarely use I-control without P-control.

Integral action:

Let us review a few principles of calculus and trigonometry in relation to integral calculation, especially the integration of a sine wave. +=+=+=+ the phase lag of the integral calculation on a sine wave. The same effect exists if integral action is used in a closed loop control system. The integral action adds to the existing phase lag. The maximum of the integrated sine wave is when the sine wave swings back.

y =sine a y =sine adt Integral action has a phase lag

+=+=+=+ The phase shift of the integration action

If we consider a 'steady-state' value exists for the ERR term, then the integral output will, at the completion of each of its time constants, INT , T increase its output value by ERR × KC in the form of a ramp.

===

SP change Control-mode output for various settings of controller adjustments

0 Time a 5a 4a 2a 20% Proportional band 100% Proportional band 200% Proportional band Integral action only, with 100% PB Proportional action a:

Time, unit periods Time, unit periods Time, unit periods 2 × TINT 1 × TINT 0.5 × TINT

+=+=+=+ 5(a) Integral relationships and output

===

Integral action in practice: In practice, as the integral output increases and passes through the process the PV will move towards the SP value and the ERR term will reduce in magnitude. This will reduce the rate-of-change during the integral time interval, resulting in the classic first-order 'curve' response shown. If the rate-of-change or the value on TINT is too small, along with the 90° phase lag in the integral action, oscillations may occur, i.e., in effect, applying over-correction-in-time to the value of the offset term.

If this happens with a closed loop control system in the industry, we have a stability problem.

The conclusion that we get from this is that we have to be careful in the use of integral control if we have a closed loop control system which has a tendency towards instability.

Integral control eliminates offset at the expense of stability

Exercise 3 Single Flow Loop - Integral (I) Control ~ Flow Control

This exercise will introduce the integral control action of controllers.

Exercise 4 Single Flow Loop - Proportional and Integral (PI) Control ~ Flow Control

This exercise will introduce the combination of the proportional and integral control action of controllers.

===

Control mode Relation between controlled variable and output Change in output caused by ERR step of 0 (Time ) (Time ) Sample of ERR curve during line-out (based on assumed prop. curve) On-Off Control Output 0 SP 100% Control output; 100% Variable in % of span. For a step across setpoint 100 Output for negative ERR Output for positive ERR 0

0 Continuous Proportional Relation adjustable to select span of variable for full span of controller output Narrow span; Span of 100% Span over 100% Variable; Variable; Variable For a step in negative direction of deviation Narrow PB 100% PB; PB over 100% Narrow PB Will be offset Wide PB offset

Integral Relation adjustable to select speed at which proportional effect on output is repeated by the integral action, with a constant ERR Integral effect Prop. Constant Integral effect Prop. positioning

Output after 1 min, one repeat per min.

Output after 1 min, two repeat per min.

Integral's contribution to output for a step from zero ERR Level of prop. correction Level of prop. correction Integral correction for 1R/M Integral correction for 2R/M min Integral combined with 'Wide PB' above: Slow R/M Faster rise Fast R/M Offset eliminated Period Longer than with narrow PB 100; Constant ERR

+=+=+=+ (b) Integral action in practice

===

Derivative control

The only purpose of derivative control is to add stability to a closed loop control system.

The magnitude of derivative control (D-control) is proportional to the rate of change (or speed) of the PV. Since the rate of change of noise can be large, using D-control as a means of enhancing the stability of a control loop is done at the expense of amplifying noise. As D-control on its own has no purpose, it’s always used in combination with P-control or PI-control. This results in a PD-control or PID-control. PID-control is mostly used if D-control is required.

Derivative formula:

Formula for D-control: +

TDER is the derivative time constant.

Again, using the principles of calculus and trigonometry in relation to the derivative calculation, especially the case of differentiation of a sine wave we can derive the following principles. +=+=+=+ the phase lead of derivative calculation on a sine wave. The same effect exists if derivative action is used in a closed loop control system.

y =cosine a =sine a dt Derivative action has a phase lead dy dt

+=+=+=+ Phase shift of differentiation

Derivative action can remove part or all of an existing phase lag. This is theoretically achieved by the output of the derivative function going immediately to an infinite value when the ERR value is seen to change.

Derivative action in practice:

In practice the output will be changed to +8 times the value of the change of the ERR value. Then the output will decrease at a rate of 63.2% in every derivative time unit.

===

SP change Control-mode output for various settings of controller adjustments a 5a 20% PB 100% PB Derivative action 200% PB Proportional action a: Time 8a; 8a; 63.2% of drop 63.2% of drop 11 Time in minutes Time in minutes 1 × TDER 2 × TDER

+=+=+=+(a): Derivative relationships and output

===

Summary of integral and derivative functional relationships:

Integration can be considered as charging a capacitor, from a constant voltage source, via a resistor. The voltage across the capacitor rises from a zero value in an exponential form.

This being caused by the difference between the supply and capacitor voltage reducing in time.

Derivative control has no functionality on its own:

Derivative action is in essence the inverse of the example for integral action. Taking a fully charged capacitor and discharging it through a resistor results in an exponential decay, as the difference in capacitor voltage reduces from its maximum value to zero.

At first glance, it would appear that the integral and derivative functions, one being the inverse of the other, would effectively cancel out each other. However it has to be remembered that the ERR term is dynamic and constantly changing.

There is a fairly strict ratio between TINT and TDER and the process or loop time TPROC. these relationships being explained earlier ('system tuning procedures').

General Single Loop With Interactive PID (Real Form) - Introduction to Derivative (D) Control This exercise will introduce the Derivative Control action of controllers

Proportional, integral and derivative modes

Most controllers are designed to operate as PID-controllers.

Enabling/disabling integral and derivative functions:

• If no derivative action is wanted, TDER (derivative time constant) has to be set to zero.

• If no integral action is wanted, TINT (integral time constant) has to be set to a large value (999 min, E.g.).

Most controllers work as an I-controller only if K is set to zero. In such cases, a unit gain of 1 is active for integral action only: The concept of a PID-controller.

In 'Tuning of controllers in closed loop control', we will review the most common methods for tuning of P-controllers, PI-controllers and PID-controllers. At this stage you should be aware of the balancing act necessary to optimize the control action.

+=+=+=+ Block diagram of an ideal PID-controller

ISA vs Allen Bradley

The PID functions, considered within a digital (PLC) system, equate to a process where the output of a controller is designed to drive the process variable (PV) toward the setpoint (SP) value. The difference between the PV and SP values is the system error value, upon which the PID functions operate. The greater the error value the greater the output signal.

ISA (Instrument Society of America) has a set of rules that make the P, I and D functions dependent on each other, and E.g., the Allen Bradley PLC system operates either on ISA (dependent) or independent gains.

PID relationships and related interactions

P-control is the principle method of control and should do most of the work. I-control is added carefully just to remove the offset left behind by P-control. D-control is there for stability only. It should be set up so that its stabilizing effect is larger than the destabilizing effect of I-control. In cases where there is no tendency towards instability, D-control is not used. This includes most flow applications.

Exercise: Practical Introduction into Stability Aspects Gives practical experience on the topics of closed loop stability.

Applications of process control modes

Proportional mode (P): The most basic form of control. This can be used if the resultant offset in the output is constant and acceptable. Varied by the controller gain C. K

Proportional and integral mode (PI): Integral control can be added to the proportional control to remove the offset from the output.

This can be used if there are no stability problems such as in a tight flow control loop.

Proportional, integral and derivative mode (PID):

This is a full 3-term controller, used where there is instability caused by the integral mode being used. The derivative function amplifies noise and this must be considered when using the full three terms.

Proportional and derivative mode (PD): This mode is used when there are excessive lag or inertia problems in the process.

Integral mode (I): This mode is used almost exclusively in the primary controller in a cascaded configuration. This is to prevent the primary controllers output from performing a 'Step change' in the event of the controllers setpoint being moved.

Typical PID controller outputs:

+=+=+=+ Typical controller outputs

NEXT: Principles of Digital Process Control
PREV: part 1

All related articles   Top of Page   Home