Control Loops

Open loop vs closed loop The terms open-loop control and closed-loop control are often not clearly distinguished. Therefore, the difference between open-loop control and closed-loop control is demonstrated in the following example of a room heating system. In the case of openloop control of the room temperature

according to Figure 1.1 the outdoor

Figure 1.1: Open-loop control of a room heating system temperature

will be measured by a temperature sensor and fed into a control device. In

the case of changes in the outdoor temperature

(

disturbance

) the control device

adjusts the heating flow according to the characteristic of Figure 1.2 using the motor M and the valve V. The slope of this characteristic can be tuned at the control device. If the room temperature is changed by opening a window ( disturbance ) this will not influence the position of the valve, because only the outdoor temperature will influence the heating flow. This control principle will not compensate the effects of all disturbances.

Figure 1.2: Characteristic of a heating control device for three different tuning sets (1, 2, 3) In the case of closed-loop control of the room temperature as shown in Figure 1.3 the room temperature

is measured and compared with the set-point value

, (e.g.

). If the room temperature deviates from the given set-point value, a controller (C) alters the heat flow . All changes of the room temperature , e.g. caused by opening the window or by solar radiation, are detected by the controller and removed.

Figure 1.3: Closed-loop control of a room heating system The block diagrams of the open-loop and the closed-loop temperature control systems are shown in Figures 1.4 and 1.5, and from these the difference between open- and closedloop control is readily apparent.

Figure 1.4: Block diagram of the open-loop control of the heating system

Figure 1.5: Block diagram of the closed-loop control of the heating system The order of events to organise a closed-loop control is characterised by the following steps:

•

Measurement of the controlled variable ,

•

Calculation of the control error (comparison of the controlled variable with the set-point value ), Processing of the control error such that by changing the manipulated variable the control error is reduced or removed.

•

Comparing open-loop control with closed-loop control the following differences are seen: Closed-loop control • • •

shows a closed-loop action (closed control loop); can counteract against disturbances (negative feedback); can become unstable, i.e. the controlled variable does not fade away, but grows (theoretically) to an infinite value.

Open-loop control • • •

shows an open-loop action (controlled chain); can only counteract against disturbances, for which it has been designed; other disturbances cannot be removed; cannot become unstable - as long as the controlled object is stable.

Summarising these properties we can define: Systems in which the output quantity has no effect upon the process input quantity are called open-loop control systems. Systems in which the output has an effect upon the process input quantity in such a manner as to maintain the desired output value are called closed-loop control systems.

The classical three-term PID controller

We have seen in section 7.1 that proportional feedback control can reduce error responses but that it still allows a non-zero steady-state error for a proportional system. In addition, proportional feedback increases the speed of response but has a much larger transient overshoot. When the controller includes a term proportional to the integral of the error, then the steady-state error can be eliminated, as shown in section 7.2. But this comes at the expense of further deterioration in the dynamic response. Addition of a term proportional to the derivative of the error can damp the dynamic response. Combined, these three kinds of actions form the classical PID controller, which is widely used in industry. This principle mode of action of the PID controller can be explained by the parallel connection of the P, I and D elements shown in Figure 8.1. From this diagram the transfer function of the PID controller is (8.1)

Figure 8.1: Block diagram of the PID controller The controller variables are gain integral action time derivative action time

Eq. (8.1) can be rearranged to give (8.2)

These three variables , and are usually tuned within given ranges. Therefore, they are often called the tuning parameters of the controller. By proper choice of these tuning parameters a controller can be adapted for a specific plant to obtain a good behaviour of the controlled system. If follows from Eq. (8.2) that the time response of the controller output is

(8.3)

Using this relationship for a step input of , i.e. , the step response of the PID controller can be easily determined. The result is shown in Figure 8.2a. One has to observe that the length of the arrow weight of the impulse.

of the D action is only a measure of the

Figure 8.2: Step responses (a) of the ideal and (b) of the real PID controller In the previous considerations it has been assumed that a D behaviour can be realised by the PID controller. This is an ideal assumption and in reality the ideal D element cannot be realised (see section 3.3). In real PID controllers a lag is included in the D behaviour. Instead of a D element in the block diagram of Figure 8.1 a function

element with the transfer

(8.4)

is introduced. From this the transfer function of the real PID controller or more precisely of the

controller follows as

(8.5)

Introducing the controller tuning parameters and it follows (8.6)

The step response of the controller is shown in Figure 8.2b. This response from gives a large rise, which declines fast to a value close to the P action, and then migrates into the slower I action. The P, I and D behaviour can be tuned independently. In commercial controllers the 'D step' at can often be tuned 5 to 25 times larger than the 'P step'. A strongly weighted D action may cause the actuator to reach its maximum value, i.e. it reaches its 'limits'. As special cases of PID controllers one obtains for: a) the PI controller with transfer function (8.7)

b) the ideal PD controller with the transfer function (8.8)

and the

controller with the transfer function (8.9)

c)

and

the P controller with the transfer function (8.10)

The step responses of these types of controllers are compiled in Figure 8.3. A pure I controller may also be applied and this has the transfer function (8.11)

Figure 8.3: Step responses of the PID controller family

The measure of the quality of the transient response of a PID controlled system can be performed by calculating an integral performance index as shown in section 7.3.2. The best controller is one that has the minimum performance index. When this performance

index is a minimum for a specified input, the system performance is said to be optimal. When the input signal is specified the quadratic performance index

can be calculated

for a given plant transfer function as a function of the tuning parameters, e.g. and

,

,

.

The mathematical calculation of this performance index for given values of the tuning parameters is simple as shown in section 7.3.3. But getting the optimal parameters is a non-trivial task. Though computerised optimisation algorithms are available to calculate the optimal parameter setting, for the case of quadratic performance indices a mathematical analysis is possible. The approach shown in section A.7 gives more insight into the controller settings and can be applied to all types of plants and PID controllers. In the following the command and disturbance behaviour of a control system with a real PID controller and a plant with the transfer function (8.12)

will be investigated. The response of the control error to step changes command input and

in the

in the plant input is

For the plant (Eq. (8.12)) and the real PID controller (Eq. (8.6)) one obtains (8.13)

which is in the form of Eq. (7.40) or Eq. (A.52) for Applying the analysis shown in section A.7 to the diagrams in Figure 8.4,

. performance index one gets the

Figure 8.4: Stability and performance diagram for step changes (a) in the command input ( , ) and (b) in the plant input ( , ) separately for the command and disturbance inputs. The integral action time constant is normalized by

. These diagrams are shown for the optimal value

of the derivative action time constant. The filter time constant is . The diagrams show a rather rectangular stability area that makes tuning of and for a fixed easy from the stability point of view. But the performance characteristics are quite different. The optimal parameters for the two cases differ by about a factor of two. Therefore, an optimal tuned controller is in general never optimally tuned for command

Advantages and disadvantages of the different types of controllers In the following the disturbance behaviour is investigated using the controllers introduced in section 8.1. Their parameters are tuned optimally according to the performance index from section 7.3.2. The plant is given by Eq. (8.12). Figure 8.5 shows for the different types of controller the responses to a step disturbance variable , which is normalised by relation

of the controlled

. These curves indicate that because

the

is valid.

For discussing these curves the term settling time according to section 7.3.1 is used, which is related to the steady state of the uncontrolled case (8.14)

In addition, the different cases should be compared with respect to the normalised maximum overshoot

.

The different cases are discussed below: a) The P controller shows a relatively high maximum overshoot settling time

as well as a steady-state error

, a long

.

b) The I controller has a higher maximum overshoot than the P controller due to the slowly starting I behaviour, but no steady-state error. c)

The PI controller fuses the properties of the P and I controllers. It shows a maximum overshoot and settling time similar to the P controller but no steadystate error. d) The real PD controller according to Eq. (8.9) with has a smaller maximum overshoot due to the 'faster' D action compared with the controller types mentioned under a) to c). Also in this case a steady-state error is visible, which is smaller than in the case of the P controller. This is because the PD controller generally is tuned to have a larger gain due to the positive phase shift of the D action. For the results shown in Figure 8.5 the gain for the P controller is of

and for the PD controller

. The plant has a gain

.

e) The PID controller according to Eq. (8.6) with fuses the properties of a PI and PD controller. It shows a smaller maximum overshoot than the PD controller and has no steady state error due to the I action. The qualitative concepts of this example are also relevant to other type of plants with delayed proportional behaviour. This discussion has given some first insights into the static and dynamic behaviour of control loops.

Disturbance feed-forward on the controller According to Figure 11.1 the disturbance will feed via the transfer function the controller, which will compensate the influence of the disturbance.

to

Figure 11.1: Block diagram of the feed-forward on the controller From this diagram the controlled variable directly follows as (11.1)

With some manipulations one obtains from this (11.2)

which gives (11.3)

where for brevity the argument is omitted. From the transfer functions of Eq. (11.3) one can see that the characteristic equation is (11.4)

with regard to disturbance behaviour and (11.5)

with regard to the reference behaviour. The disturbance will be fully compensated if (11.6)

from which the required transfer function for the feed-forward element is (11.7)

This approach can only be realised by a controller if the pole excess of than that of polynomial

is not larger

. Otherwise a total compensation is not possible. Moreover, the must be Hurwitzian.

For the frequent case that the disturbance and control behaviour are equal, i.e. the case of , the transfer function of the feed-forward elements is

(11.8)

As the total compensation of a disturbance in a plant with P behaviour is only possible by a controller with I behaviour, the transfer function of the feed-forward element, according to Eq. (11.8), should thus show ideal D behaviour. If there is a PI controller in the loop, the feed-forward element must be designed as a

element.

Often the feed-forward element cannot be realised as ideally designed according to Eqs. (11.7) or (11.8), because elements. Also in these cases a

, besides pure I behaviour, normally contains delay element is recommended.

Disturbance feed-forward on the manipulated variable The configuration with feed-forward on the manipulated variable or on the actuator, respectively, is shown in Figure 11.2. From this for the controlled variable it follows that

and after rearranging (11.9)

or (11.10)

Figure 11.2: Block diagram of the feed-forward on the manipulated variable The characteristic equations are the same as in the previous case with feed-forward to the controller. For the ideal compensation of the disturbances it follows from Eq. (11.9) and (11.10) that (11.11)

from which the transfer function of the feed-forward element follows is (11.12)

For the special case of

, where the disturbance acts directly at the plant input,

the compensation is performed by

directly at the plant input..

Similarly as in the case of Eq. (11.7) the realisation of the feed-forward element according to Eq. (11.12) is not possible if (11.13)

with

and

is valid, as

must be realised by PD elements.

Also in the case of a non-minimum phase behaviour of or of instability of the Eq. (11.12) cannot be realised, as the required feed-forward element is unstable. In those cases in which a dynamical compensation according to Eq. (11.12) is not possible one must be content with a static compensation using a P element (11.14)

where

and

are the gains of the transfer functions

and

.

Figure 11.3 shows disturbance feed-forward

Figure 11.3: Examples of disturbance feed-forward configurations (a) on the controller and (b) on the actuator of a steam superheater temperature control system configurations on (a) the controller and (b) the manipulated variable, for the case of a temperature control system of a steam superheater (SH) in a power station. The steam temperature at the superheater outlet is the controlled variable. The manipulated variable is the cooling water flow in the spray-water cooler (C). Fluctuations of the steam flow have an influence on the steam temperature and are treated as disturbances. The steam flow (disturbance ) is measured and fed via the manipulated variable, respectively.

or

to the controller or to

Figure 8.5: Behaviour of the normalised controlled variable at the input to the plant

; controllers

for step disturbance for different types of