Measurement Products PID control theory made easy Optimising plant performance with modern process controllers 1. What are the main types of control applications?
Introduction 1. What are the main types of control and rubber processing applications Accurate control is critical to every applications? such as tyre manufacture. In such process. As a means of ensuring that The wide variety of industrial processes processes, the main goal is to ensure tasks such as production, distribution in use today means there are many that specific quantities of products and treatment processes are carried different types of control systems, each are produced in a specific way, under out under the right conditions for the of which are geared towards their own specific conditions and for a specific right amount of time and in the right particular application. period of time. With a single error quantities, control devices form a crucial potentially rendering an entire batch part of virtually every industrial process. It is, however, possible to roughly of products unusable, tight control of In the past, many traditional control segment these processes into three process conditions is critical. processes relied on either mechanical main categories, these being: or solid state electronic technology, – Continuous processes. As their with pneumatics being used to produce – Discrete processes. These types name suggests, these processes are changes in the process, such as in valve of processes are most commonly those which produce a continuous control installations for example. operated in manufacturing and stream of products over a continuous With advances in computerised control packaging applications, where period of time. Examples include technology, there are now a host of new discrete items are being produced large scale production processes in possibilities for the way that processes and handled. This category would the chemicals, refining and plastics can be monitored and controlled. These also include robotic assembly industries. In such applications, the include the use of complex algorithms processes, for example in the car- processes tend to be largely constant, capable not only of reacting to making industry. being geared towards the production changes in process conditions but also of a specific product. increasingly predicting them as well, – Batch processes. Batch processes enabling corrective action to be taken are widely used throughout many Of course, many applications may automatically. industries, obvious examples being actually combine all of these types of food and beverage, pharmaceutical processes, which will have an impact on the control system or systems used.
2 Control theory l White paper 2. Types of control systems:
2. Types of control systems: To obtain a more accurate or more – Improved ability to help track and There are two types of traditional control adaptive control, however, it is trace faults / problems systems: necessary to use a closed-loop system, – Open-loop control systems where the output of the system is fed Some systems use closed-loop and – Closed-loop control systems back to the inputs of the controller. open-loop control simultaneously. In these systems, the open-loop control 2.1 Open loop control systems: 2.2 Closed loop control systems: is termed as ‘feedforward’ and serves Open loop controllers are also known Modern control theory is based on to help improve reference tracking as ‘non-feedback controllers’. These feedback – i.e. signals from a process performance. controllers compute their input into a that can be used to control it more system based on a specific model / effectively. A closed-loop controller uses 2.4 How do closed-loop control known set of conditions. They do not feedback to control states or outputs of systems work? use data from the process to change a system or process. The term ‘closed- As stated above, closed loop controllers their output or vary the process and loop’ comes from the information work by using the sensed value from are unable to compensate for any path in the system – process inputs a process to assess and, if needs be, disturbances in process conditions. to a system have an effect on the alter the process conditions. Instead, any variations in process process outputs, which is measured conditions would need to be achieved with sensors and processed by the A closed-loop or feedback controller by an operator manually adjusting the controller. The result (the control signal) works in the following way: A sensor final control element. is used as an input to the process, of some type measures the output of closing the loop. the system and feeds it back to be As such, open loop control systems compared against a reference value. are best suited to processes where 2.3 Benefits of closed-loop control The controller takes the difference there is a relatively stable set of systems: between the output and the reference operating conditions or where these Closed-loop controllers offer several key value and uses it to change the inputs conditions can be governed by a known advantages over open-loop controllers, to the system to help compensate for relationship. Their inherent simplicity including: the difference. and low cost makes open loop control – Ability to adapt to changing conditions systems ideally suited to simple – e.g. changing loads, process This is called a single-input, single processes where conditions are variations away from the established output (SISO) control system. Those unlikely to change and where feedback norm with multiple inputs and multiple inputs/ is not critical. – Ability to stabilise unstable processes outputs (MIMO) are also common.
Control theory l White paper 3 3. PID controllers
A. B.
PV 100 100%
75 75%
ariable Note. Control Set point Proportional 50% 50 Output Band = 50% action set
ocess V to reverse Pr Engineering Range 25 25%
0 0% Time Time Fig 1. Proportional action
3. PID controllers The derivative term is used to provide Definitions: The PID controller is probably the most damping or shaping of the response. Proportional – Proportional control widely-used type of feedback controller. The action of derivative is to cater for gives a change in output that is PID stands for Proportional-Integral- disturbances and sudden changes. In proportional to the deviation of the Derivative, referring to the three terms effect, it is used to predict what is going process variable from the set point. operating on the error signal to produce to happen within the process and takes The range over which the output is a control signal. quicker action than the integral term to adjusted from 0 to 100% is called the correct it. proportional band. The proportional The desired parameters/conditions band is expressed as a percentage of for a closed loop system are normally PID controllers are the most well- the engineering range. achieved by tuning the system to the established class of control system. inherent conditions without specific When proportional-only control is knowledge of a plant model. Stability used, manual reset is required. In the can often be ensured using only the diagram above the manual reset is proportional term. Pure proportional set to 50%. This means that when the action will result in control offset. process variable is ‘at set point’ the control output will be 50%. Adjusting The integral term eliminates the offset. the manual reset value has the effect of It does this by repeating the action of shifting the proportional band up and the proportional band every integral time down. This has the effect of changing constant. This enables the system to the control output when the process recover more quickly from a disturbance variable is ‘at set point’. The manual in conditions. reset would normally be adjusted by an operator every time the set point is changed, or when an offset is present for any reason.
4 Control theory l White paper 4. Common control terms
A. B.
Set point Integral Response = 10% Output ariable Proportional Step = 10% ocess V
Pr Note. Control action set to reverse
Time Time
Fig 2. Integral action (TI) Integral time constant
Integral – Due to limitations with the control output in the same direction amount that it has changed due to proportional-only control, which appear as the proportional action, until the proportional action, when a constant as an offset between the process offset or error is removed. deviation is present. Fig 2B shows the variable and the set point, it is often effect that a 20 second integral action necessary to introduce an integral The integral action is expressed as the would have on the control output. action. If an offset is present, the time taken for the output to change integral action will continue to change (due to the integral action) by the same
A. B. Time period being evaluated by controller
Proportional + Derivative
Proportional Only X% ariable Output Note. Control action ocess V X% Y%
Pr set to reverse
Time Time Derivative Action Time Fig 3. Derivative action
Derivative – Derivative action can be is dependant on its setting. Derivative be due to the proportional action after used with proportional control, with action is set as a time constant and the derivative time period. This change or without integral. It has the effect of has the following effect on the control in output shown on the diagram as Y% making changes to the control output, output. Taking only the proportional part is then added to the current control dependant on the rate of change of the of the output as shown by the lower output value as derivative action X%. process variable. The amount of effect line on Fig 3B, the controller calculates that derivative action has on the output forward to find out what the output will
Control theory l White paper 5 4. Common control terms
ZN - Open Loop Method ZN - Closed Loop Method
y t Set poin Tc
r t a 1. Proportional only control 2. Make small setpoint change and observe response 1. Apply a step change to the controller output and monitor the PV 3. Increase/decrease gain and repeat step until constant sustained 2. Draw tangent at point of highest rate of change oscillation is obtained 3. Calculate intercepts with x and y axes 4. Gain = critical (ultimate) gain, period of oscillation = critical period
4. Common control terms On the other hand, for more sensitive Despite its simplicity, the Ziegler Getting the right control for a process processes where a system change Nichols control loop method can take isn’t just a case of finding the right could have a major impact on process a long time to perform and also has controller. Even when the choice of conditions, a more gentle PID algorithm the potential to create uncontrolled controller has been made, operators would be required, acting more slowly oscillations, which affects control need to have a full understanding of over a longer period. stability. Traditionally, a manual tuning all of the issues that can affect both process involved an engineer first short-term and long-term performance To date, the most useful approach is that running a Ziegler Nichols test and noting to enable them to achieve the optimum set down by JG Ziegler and NB Nichols, the results on a strip chart. Using this control for their process. The following who worked for the Taylor Instrument data, they would then try to interpret is an explanation of some of the key company in Rochester, New York, which the characteristics of the process which terms that are likely to be encountered was later to become part of ABB. would then be used to help tune the during both the selection and operation control loop. Not surprisingly, this was a of a control device. In 1942 they published a paper entitled time-consuming process that was open ‘Optimum Settings for Automatic to error. 4.1 Loop tuning Controllers’, which set out an approach Loop tuning is a complicated issue and for the tuning of PID parameters which As technology has evolved, this manual much attention has been paid to trying was later to be adopted industry-wide. process can now be carried out to define and address it over the years. This approach involved both a test to automatically using the autotune feature define the speed and manner in which that can be found on most modern In tuning a PID loop, the key challenge a process responds to a change in controllers. is to strike a balance between the control, and formulas to enable the response of the controller and the results to be used to correctly tune a characteristics of the process. In controller. a process that is relatively slow to respond to a system change, for The paper proposed two techniques example, the PID algorithm would for loop tuning, namely a closed-loop need to be set up to aggressively and and open-loop method. The principles immediately respond in the event of of these methods still provide the any potential changes in the process foundation of many of the autotune or where the setpoint may have been algorithms used in many modern adjusted by the operator. industrial applications.
6 Control theory l White paper 4.2 Autotuning
1. Monitors noise and calculates 3. When the process exceeds the 5. When consistent oscillation is a hysteresis value hysteresis value the output is established the experiment is stopped stepped down
Period
3. 5. Amplitude SP PV 6. 1.
2.
4.
2. User defined initial step in the 4. Automatically adjusts output amplitude 6. The critical period and amplitude are output process starts to move so PV does not exceed level needed to recorded and used to calculate the away from the setpoint isolate from process noise optimum settings
4.2 Autotuning This is achieved in the following way: distinguish the process value oscillation Almost every modern controller will from the process noise. include an autotuning function. This 1. The controller begins by monitoring function effectively enables an operator the inherent background noise in the This process of stepping the output to automatically tune a controller to a process and creating an hysteresis up and down is continued with the process even where they have little or value, whereby low and high value are process variable passing the hysteresis no knowledge of that process. established either side of the setpoint thresholds until a consistent oscillation based on the minimum and maximum is established. At this point, the ABB’s ControlMaster autotuner uses values that are required to achieve a autotuner records the duration of the a technique developed by Astron and reliable response. oscillation and its amplitude, from which Hagglund from Lund University, which the critical period and critical gain are further refined the Ziegler Nichols closed 2. A user-defined step in the controller determined. loop tuning model. This technique output is then made, causing the basically enables a stable oscillation process to start to move away from the The information created by the autotune to be ascertained by stepping the setpoint. The response of the system is about the process is then used to process variable up and down within an measured. calculate the PID settings for the established hysteresis band. application. 3. When the process exceeds the When the autotuner is first started, the hysteresis value, the output is then Key advantages of autotuning: controller switches out the normal PID stepped down to find the lower value. – No detailed knowledge of the process algorithm and replaces it with a non- is required linear function whereby the controller The autotuner is designed to minimise – Tuning process is executed under will step up and down a prescribed the amount of process disturbance. tight feedback control parameter range until it establishes a After the initial output step, all – The tuning time is short compared consistent control range for the process. subsequent output steps both up and with the closed loop response time down are automatically adjusted until – Automatically detects processes the minimum amount of disturbance where the use of Derivative is not is achieved while still being able to appropriate
Control theory l White paper 7 4.3 Gain scheduling
Selected Controller Parameters (PID) PID Parameter Table
Gain scheduling reference (GS Ref)
SP O/P Controller Process PV PV
4.3 Gain scheduling The above diagram shows how gain conditions means it has been proven to For linear processes, where the scheduling acts as a system with produce significant improvements in the process characteristics do not change two feedback loops. The inner loop control of a process. significantly with time or load conditions, is comprised of the process and the then using the autotuner to set fixed controller while the ‘outer loop’ adjusts Key advantages of gain scheduling: PID or PI parameters will probably be the controller parameters based on the sufficient to ensure effective control. operating conditions. – Improves control of non-linear processes In the case of non-linear processes, The ControlMaster provides up to three – Eliminates the need to have PID however, being limited to a single independent sets of PID parameters. settings detuned to the worst case to set of fixed parameters can become The choice of a specific set is controlled avoid instability problematic, as they have to be set for by the gain scheduling reference signal – Is based on a measurable variable the worst case in order to find the best (GSRef) or gain scheduling source, that correlates well with the process overall response. This means that the which can be assigned to any analog dynamics controller has to be given a very low signal, both external or internal, – Three independent sets of PID gain so that it will not cause instability available in the controller. Two limits are parameters can be defined problems when the process is at its provided which set the points at which – The reference signal can be any highest gain. The consequence of this is the controller will alternate between the analog signal that very sluggish control results when three different PID sets. To help simplify – Two dedicated limits can be the controller is operating where the set-up and operation, these limits are programmed in engineering units for process gain is at its lowest. expressed in engineering units. ease of use – Capable of responding to rapid Gain scheduling overcomes these Compared to other adaption changes in operating conditions problems by enabling users to opt techniques, gain scheduling may seem for more than one set of control relatively simple. However, its ability to parameters. respond to rapid changes in operating
8 Control theory l White paper 4.4 Adaptive control
Least squares Process model estimation
Calculate PID
Band pass filter Band pass filter
SP OP PV PID Controller Process
4.4 Adaptive control With the ControlMaster, the need to is one of the areas identified and set by As described above, gain scheduling is have extensive prior knowledge of the auto tuner. well-suited to the control of non-linear the process to help with set-up of processes which behave in a predictable the adaptive control function can be After the signals have been filtered, a way and where the characteristics can eliminated by using the autotuner. mathematical process known as ‘least be correlated to a variable that can be Running the autotuner will help provide squares estimation’ is applied to update easily measured. both good initial PID values and also a the coefficients of the mathematical number of other parameters required model. Using this information, the However, this is not the case for for the proper operation of the adaptive algorithm then calculates the processes which are subject either to adaptive algorithm. appropriate PID settings that will provide rapid and/or unpredictable changes. the optimum control of the model, In such processes, the solution is to The above diagram illustrates the with the controller settings then being have an adaptive control function, model monitoring process. At the modified accordingly. such as that incorporated within the first stage, the control output and the To ensure the model is only adjusted ControlMaster. process variable signals are put through when necessary, the ControlMaster a narrow band pass filter. This allows includes a built-in supervisory logic The adaptive controller works by only signal data at a specific frequency function. This ensures the estimates are creating a mathematical model of the to pass through, with any parts of only updated when there is information process which is then used to estimate the signal that have a higher or lower available and prevents the model from the optimum controller settings for frequency being blocked. This prevents being modified when the process is the process. This model is continually unwanted high frequency process noise under good control. updated by using data gained from and low frequency load disturbances comparing the performance of the that could otherwise distort the process Similarly, when the supervisory logic model against real-life operating model. The centre frequency of the filter detects that a load a disturbance has conditions. is linked to the process dynamics and occurred, it restricts the adaption
Control theory l White paper 9 4.4 Adaptive control
Ref
PI D Ref
74.1 3 2.7 3rd set
823.5
35.2 18 0.0 2nd set
237.5
58.5 4 2.5 1st set
process until the initial effect of the Adaptive control can also be combined Key advantages of adaptive control: disturbance has passed, enabling with gain scheduling to deliver benefits the data from the disturbance to be for processes which are non-linear – Helps control of processes with incorporated into adjusting the model. but which vary with time or other changing dynamics conditions. Here, the adaptive algorithm – Fully integrated with auto tuner to The supervisor also checks the automatically builds individual internal automatically provide initial process parameters generated by the least models for each segment of the gain knowledge square algorithm to ensure that they scheduler. Using these, the controller – Creates and updates an internal are reasonable. automatically updates the appropriate model of the process it is trying to set of PID parameters depending control The supervisory logic function can also on where in the process curve it is – Filters the control output and the be useful for detecting potential fault operating, as shown in the above process variable signals so that conditions such as sticking valves. In diagram: unwanted information from noise and the event of a sudden significant change local disturbances are removed in the control parameters, it will also In this example, with the GS reference – Signals are analysed in a least- stop the adaption process and warn at around 500, a detected increase in squares estimator, with the resulting the user. the gain of the process will result in it information used to update the automatically calculating new PID values process model As the adaptive controller uses the for the second set of parameters only, – Works out the PID settings that would same process dynamics as the auto leaving the others unmodified. work best with the internal model tuner, the user has the option of using PI control only or to calculate controller settings suitable for processes where the deadtime is relatively long.
10 Control theory l White paper 4.5 Deadtime compensation
OP
τ = L L + Tc
PV Deadtime (L)
Time Constant (Tc)
4.5 Deadtime compensation larger, making the process increasingly ‘Deadtime’ occurs where the variable difficult to control with just a simple PID being measured does not respond to a algorithm. step change in the controller output for a certain period of time. It commonly To overcome this, some form of occurs in applications where material is predictive control is needed. The classic being transported via a pipeline or on solution is known as a ‘Smith Predictor’. a conveyor. An example is a pH dosing This is a deadtime compensating process. The dosing pump is often controller which makes a prediction located some distance upstream from based on the controller output and an the sensor, resulting in a delay between internal simulation of the process. the controller increasing the dose and the resulting effect being seen at the However, this approach has a number sensor. The time it takes for the dose of disadvantages, the biggest of which to travel down the pipeline creates an is its complexity. To obtain the process effective deadtime. model needed for the algorithm, a systematic identification process To address this, a formula has been is needed which needs at least the devised called the controllability ratio. process gain, time constant and the The controllability ratio is defined as the deadtime all to be specified. Combined ratio of the process deadtime divided with the PI algorithm, operators may by the deadtime plus the dominant time find themselves having to tune a constant. Its purpose is to ascertain total of five parameters. Furthermore, how easily a particular process can be unlike standard PID control, the Smith controlled. Predictor cannot be easily manually tuned by trial and error. Small values of the ratio are easy to control. As the deadtime becomes more significant compared to the time constant though - that is, as the periods of deadtime get longer - the ratio gets
Control theory l White paper 11 4.6 Predictive PI (pPI) control - overcoming the Smith Predictor complexity
SP OP PV pPI Controller Process
+
Dead Model time - Σ
I +
Model without P dead time
4.6 Predictive PI (pPI) control – the process variable will also start to overcoming the Smith Predictor slowly respond. Although the control is complexity reasonable, the response will tend to be The new ControlMaster range employs a very sluggish. deadtime compensation controller which is much simpler to use. Because the pPI controller can predict what will happen, it is able to raise Called Predictive PI (pPI) control, it is the control output much more quickly based on the work of Tore Hagglund of without the problems of overshoot Lund University in Sweden. Although it or instability. Once the deadtime has shares the same structure as the Smith elapsed, the process variable then Predictor, the controller differs by having approaches the setpoint much more two process model parameters that are quickly. determined automatically based simply on the Proportional and Integral values. Key advantages of predictive PI (pPI) control: The consequence of this is that, as – Much simpler to use than a Smith for a standard PID controller, there are Predictor just three parameters that need to be – Uses same structure as a Smith tuned, namely the proportional term, Predictor, but needs no process the integral term and an estimate of the model to be specified process deadtime. – It creates its own process model from gain, integral time and deadtime 4.6.1 PI response vs pPI – Ideal where the deadtime is more than A key benefit of predictive PI control is double the dominant time constant its ability to greatly reduce the impact – Can be manually tuned using a simple of deadtime compared to standard step response test PI control. With PI control, where a step change is made to the control setpoint, the control output will slowly increase. As the deadtime is exceeded,
12 Control theory l White paper 4.7 Feedforward Control
PID PID SP PV SP PV
PID OP PID OP FF x Gain +
OP OP
FT Flow meter Dosing AT pH Transmitter FT Flow meter Dosing AT pH Transmitter Pump Pump
4.7 Feedforward Control from the dosing pump and using this Key advantages of feedforward Feedforward control offers a highly information as a feedforward signal to control: effective way of compensating for help achieve an output proportional – Helps prevent large disturbances in measurable disturbances before they to the fluid flow rate. By enabling the output. Disturbances are detected can affect the process variable. As dosing rate to immediately track any before they can affect the process such, it can often help deliver significant changes in flow rate, the possibility of – Helps speed up response times and improvements in control quality. potentially expensive or even dangerous eliminate delays associated with over or under dosing is eliminated. feedback control The benefits of feedback control were outlined earlier in this paper. Although it The below diagram highlights the simple helps to inform the process, one of its principle of feedforward control. Here, inherent drawbacks is that corrective the feedforward signal, in this case the action cannot be taken until the output flow rate, is scaled by a gain factor and deviates from the setpoint. then added to the normal PID output. The gain itself can either be set at a The following diagram shows a practical fixed value by the user or adjusted example of a dosing control installation. dynamically using the ControlMaster’s If the flow rate of the solution being adaptive feedforward function. treated changes, then the dose amount also needs to be changed accordingly. Using adaptive gain confers a number of benefits. Firstly, it eliminates the need If only feedback control is used, the to manually calculate the required gain, controller will only realise that something reducing both the time and complexity has changed once the solution reaches of set up and removing the the chance the pH sensor. Any action by the of human error. Secondly, it ensures that controller to adjust the dosing output good control is maintained irrespective will therefore be too late to have of any disturbances to the system, any immediate effect on the solution either due to time; any changes in the currently being measured. properties of the solution being treated; or any other potential reasons. This situation can be resolved by measuring the flow rate upstream
Control theory l White paper 13 5.0 Introducing the ABB ControlMaster
5.0 Introducing the ABB also be performed via DTM-based PC control, which allows the ControlMaster ControlMaster configuration software. Configuration to automatically adjust its control The new ABB ControlMaster builds on files can be transferred to the response to suit variations in process the highly successful Commander family ControlMaster via its front-mounted response. Dual loop control makes it to offer an even wider range of control infrared port or stored locally on a PC. possible to use one ControlMaster to and indicator functions in just four easy control two processes. The inclusion of to specify versions. A key feature of the The ControlMaster range offers a additional auto tuner, math, logic, gain CM15 indicator and the CM10, CM30 choice of communications options scheduling, totalizer, custom lineariser and CM50 controllers is their full color – Ethernet, Modbus, and Profibus. and delay timer functions provide TFT display, providing operators with Ethernet communications provide the powerful problem-solving functionality. a clear and comprehensive display of ability for users to be automatically process status and key information. notified of critical process events via With just four models in the range, Adjustment of the display via the email or remotely monitor the controller product selection for basic through to customisation feature enables the and process via the ControlMaster’s advanced applications has never been operator to tailor their view to their integrated webserver by simply using a easier. Initial model selection is made requirements, while a chart display standard web browser. For integration based on panel cut-out and then I/O provides short term trending information. with larger control systems, the Modbus and control functionality selected to suit (RTU and TCP) and Profibus options application requirements. When required The use of a TFT display also makes enable access to real time data on field upgrades can be easily performed the ControlMaster range extremely easy process values and device status. using plug and play function keys and I/O to configure. Configuration menus are cards. The common design of all of the displayed in clear text, with no complex Other new features include additional four models provides standard operation codes and abbreviations found on control options alongside the standard and configuration across the range. controllers with lesser displays. Further cascade, ratio and feedforward time savings during configuration are control functions to help bring even All products in the ControlMaster range made possible by the use of built- the trickiest processes under control. feature full IP66 and NEMA4X protection in application templates, enabling Predictive control algorithms including as standard. These high levels of protec- operators to simply select the template dead time compensation, makes the tion makes the range suitable for a wide best matched to their requirements, ControlMaster ideal for applications with range of industrial control applications, with the ControlMaster automatically long dead-times such as pH dosing from arduous oil and gas facilities configuring its I/O, display and control applications. Added control efficiency through to the washdown environments strategy to suit. Configuration can is enabled by the inclusion of adaptive in the food and beverage industry.
14 Control theory l White paper 6.0 Examples of applications for the ABB ControlMaster
ABB’s ControlMaster process controllers are suitable for use across a wide range of applications. Typical configutrations can include:
Further information For more information about ABB’s controllers, visit www.abb.com/intrumentation or email: [email protected] ref. ‘controllers’.
Control theory l White paper 15 Notes
16 Control theory l White paper Notes
Control theory l White paper 17 Notes
18 Control theory l White paper Notes
Control theory l White paper 19 Contact us www.abb.com visit: information product more For www.abb.com/contacts visit: contact ABB local your Tofind parties or utilization of its contents – contents its of utilization or parties third to disclosure reproduction, Any therein. contained illustrations and matter subject the in and document this in rights all reserve We document. this in information of lack possible or errors potential for whatsoever responsibility any accept not does ABB prevail. shall particulars agreed the orders, purchase to regard With notice. prior without document this of contents the modify or changes technical make to right the reserve We Notes: Printed in UK (07.2011) (07.2011) UK in Printed reserved rights All ABB 2011 Copyright© owners. rightful their of property the are trademarks and copyright All ABB. of consent written prior without forbidden is – parts in or whole in
PG/RandC/001-EN/Control theory white paper/July 2011