Closed loops--Stability and control modes (part 1)

Goals

• Indicate what stability is, and mathematically what causes instability

• Describe the function and use of proportional, integral and derivative control and various combinations of these terms

• Indicate what problems in closed loop control are caused by and how to correct them.

The industrial process in practice

We have seen the basic principles of closed loop control in the previous section. A control action is calculated, based on the deviation of the PV from the desired value of control as defined by the SP (ERR = PV- SP).

We have to consider the industrial process as it works in the real world. As an example of this, which we will now review, is a feed heater which is used to heat up material before it’s fed into a distillation column.

+=+=+=+ Temperature control of a feed heater: MV Fuel Air Pressure T1 T2 Disturbances flow, temperature Inlet Outlet PV

The objective of the system is temperature control of the outlet temperature (T2) that should be kept constant. The manipulated variable is the fuel valve position. It should be noted, that for economic and environmental reasons, cross limiting control of the combustion is normally required to minimize the output of carbon monoxide. In this example for simplicity, we will neglect cross limiting control totally and manipulate the valve position directly.

This example of feed heater control will serve as an example for us to look into the practical implications of stability, different control modes, control strategies and practical exercises. For this reason we will first have a closer look into the basic dynamic behavior and the most common disturbances of the process which affect this control system.

Dynamic behavior of the feed heater

There are two major types of systems lag, control and disturbance, that effect the dynamic behavior of this heater system.

Control lag:

A lag between positioning of the fuel valve and the outlet temperature exists. The main reason for this lag can be seen by virtue of the fact that not all feed material in the heater will be heated up at the same time after a change of the fuel valve position. Some part of the feed material in the heater at the time of fuel valve change will leave the heater shortly after and some other part later. A minor deadtime is also a part of the control reaction.

Disturbance lags:

The impact of disturbances on the outlet temperature also has a lag action. Every disturbance has its own lag time constant. Most disturbances have a minor deadtime as well.

Note: There is no measurable difference between two high order lags one with a minor deadtime and the other without.

Major disturbances of the feed heater

There are four Major disturbances that can, and will be considered as being critical to the stable operation of the system, these being:

Fuel flow pressure changes: Increasing pressure increases the fuel flow and results in a higher outlet temperature (T2) and vice versa.

Feed flow changes: Since the feed heater serves another (unpredictable) process downstream of it, there is no way of keeping the feed flow constant. The feed flow depends totally on the need for material by the following process. An increase in the feed flow (demanded by the downstream process) decreases the outlet temperature and vice versa.

Feed inlet pressure changes: If the feed material is in the form of gas, this becomes an important issue. It’s important to know the mass-flow rather than the volumetric flow of the feed material. With increasing pressure we increase the mass flow which results in a decrease of the outlet temperature and vice versa.

Feed inlet temperature changes: The higher the inlet temperature, the less we have to heat. An increase in inlet temperature results in an increase of the outlet temperature and vice versa.

Stability

We have stability in a closed loop control system if we have no continuous oscillation. We must not confuse the problems and the different effects that disturbances, noise signals and instability have on a system. A noisy and disturbed signal may show up as a varying trend, but it should never be confused with loop instability.

The criteria for stability are these two conditions:

1. The loop gain (KLOOP) for the critical frequency <1

2. Loop phase shift for the critical frequency <180°. 5.5.1 Loop gain for critical frequency

Consider the situation where the total gain of the loop for a signal with that frequency has a total loop phase shift of 180°. A signal with this frequency is decaying in magnitude, if the gain for this signal is below 1. The other two alternatives are:

1. Continuous oscillations which remain steady (loop gain = 1)

2. Continuous oscillations which are increasing, or getting worse (loop gain >1).

Loop phase shift for critical frequency:

Consider the situation where the total phase shift for a signal with frequency that has a total loop gain of 1. A signal with this phase shift of 180° will generate oscillations if the loop gain is greater than 1.

Note:

• Increasing the gain or phase shift destabilizes a closed loop, but makes it more responsive or sensitive.

• Decreasing the gain or phase shift stabilizes a closed loop at the expense of making it more sluggish.

• The gain of the loop (KLOOP) determines the offset value of the controller and offset varies with setpoint changes.

===

Process (Kc =0.5 xKP =2 Total gain = 1) Disturbance +1EU

Disturbance; Disturbance on PV = (+2EU, s) ERR=SP-PV=N-PV= N-2EU =-2EU OP = ERR× Kc =-2 × 0.5 = -1

Dist(1) = +1 => fedback as -1 PV = 0 => +1 => -1 => +1 = oscillation; Process

+=+=+=+ Increasing Instability with a 180º phase shift (and gain = >1)

===

Proportional control

This is the principal means of control. The automatic controller needs to correct the controllers OP, with an action proportional to ERR. The correction starts from an OP value at the beginning of automatic control action.

Proportional error and manual value: We will call this starting value MANUAL. In the past, this has been referred to as 'manual reset'. In order to have an automatic correction made, that means correcting from the MANUAL starting term, we always need a value of ERR. Without an ERR value there is no correction and we go back to the value of MANUAL. We therefore always need a small 'left over' error to keep the corrective control up. This left over error is called the offset. ERR0 is the error value we would have without any control at all.

KC is the gain applied to scale the size of the control action based on ERR. LOOP is the total loop gain which is the product of controller gain (KC) and process gain (KP). The only tuning constant for proportional control is KC (controller gain). The larger we make the value of KC, the more difficult or sensitive (reduced stability) is the control of the system.

With larger values of KC, the offset value becomes smaller. If the gain is made too large, we may face a stability problem. The following relationships follow from the above:

Proportional relationships:

1. OP = KC × ERR + MANUAL

2. LOOP K = KC × KP

3. Offset = ERR0 / LOOP (1) K + ? C LOOP; LOOP ERR = SP PV OP = ERR PV = ERR = SP PV = SP ERR; ERR + ERR = SP ERR (1 + ) = SP

LOOP At a steady state ERR = SP/(1 + ) K The error term (ERR) is defined as 'error = Indicated - Ideal' and is produced as: ERR () = SP () PV () ttt - Although this indicates that the setpoint (SP) can be time-variable, in most process control problems it’s kept constant for long periods of time. For a proportional controller the output is proportional to this error signal, being derived as: CC OP ( )= + ( ) tPKEt Where C; C; OP = The controller output = The controller output bias, or MANUAL starting value = The controller gain (usually dimensionless) = The ERROR value.

This leads the way to evaluating a set of concepts for proportional control.

Evaluation of proportional control concepts:

• The controller gain (KC) can be adjusted to make the controller output (OPC), changes as sensitive as desired to differences that occur between the SP and PV values.

• The sign of KC can be chosen (+ or -) to make OPC either increase or decrease as the deviation or ERR value increases.

In proportional controllers, the MANUAL or starting value of the OUTPUT is adjustable. Since the controller output equals the value of MANUAL when the error value is zero (SP = PV), the value of MANUAL is adjusted so that the controller output and consequently the manipulated variable, MV, are at their nominal steady-state values.

E.g., if the controller output drives a valve, MANUAL is adjusted so that the flow through the valve is equal to the nominal steady-state value when ERR = 0.

The gain KC is then adjusted and for general controllers it’s dimensionless that is the terms MANUAL and ERR have the same unit terms of measurement.

The disadvantage of proportional controllers is that they are unable to eliminate the steady-state errors that occur after a setpoint or a sustained load change. 5.6.4 Proportional band

A controllers proportional band is usually defined, in percentage terms, as the ratio of the input value, or PV to a full or 100% change in the controller output value or MV. Its relationship to proportional, or controller gain (KC) is given by:

Proportional: Proportional band %: when MV 100%. ?=

As shown, if the PB, or proportional band, of a controller is set at 100% (KC = 1) then a full change of the PV, or input, from 0 to 100% will result in a change of the MV, or output, from 0 to 100%, resulting in 100% of valve motion or operation.

Ranges of proportional bands:

If the PB is set at 20% (KC = 5) then a change in the PV, or input, from 40 to 60% will result in the same change of the MV, or output, from 0 to 100%. With the same resultant motion of the valve from fully closed to fully open. Likewise, a PB value of 500% (KC = 0.2) will result in the MV, or output, changing from 40 to 60% when the PV, or input, changes from 0 to 100%.

High percentage values of the PB therefore constitute a less sensitive response from the controller while low percentage values result in a more sensitive response.

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