Simple On-Off Control for First Orders and Overdamped Systems

The bang-bang principle of operation is the on/off control with hysteresis, as the diagram shows the system will turn on when the input [e2] is greater than H but, it will hold on until [e2] is less than L. 

Normally this is all the bang-bang functionality, but we have added a feedback inner loop at the on/off output through a first order filter. This filter is implemented mathematically in the algorithm. 

This first order filter (FOF) will receive the status of the bang-bang (BB) (hysteresis) output block that is a 1 (one) or 0 (cero) depending if the output is on or off. The result will be a capacitance charge or discharge depending on the state of the Bang-Bang block output. The tendence of the FOF output will be K when the BB output is on and will tend to cero when the output is off.

The output of the FOF is subtracted of the error [e] producing a reduced [e2] that will tend to K, this means that when the error (e=SP-PV) is greater for enough time, e2 will tend to e-Kr, therefore e2 will be greater than H turning the output of the BB to on.

The final control element will stay open until the plant reaches the difference stated by e2 ≤ L the output of the BB will turn off even when the plant has not yet reached the Setpoint, but then the feedback filter will begin a discharge provoking an increase on e2. The discharge will continue until e2 reaches H that will turn again the output of BB.

The process will continue to cycle between on/off until the PV reaches SP.

Tunning the Bang-Bang

Tunning method is quite intuitive. But is best to begin with small gain in the feedback filter and few units of error to turn on the control and few units to turn off the control. It can begin centered over cero. For example, start with [H = 0.5% of the control variable] and [L=- 0.5% of the control variable]. Make K=0 to see how the system reacts to a pure bang-bang.

The sample time should be equal to the periodic function call for the Bang-Bang function. This parameter is important because it will determine the system timing for the following steps.

Check the overshoot after the system stabilizes, make [K] equal to overshoot and [Tao] equal to the dead time (t0) to estimate this time (does not have to be precise) measure the time between the output of the actuator activates and when you can see a perceivable change of 0.5% in the control variable. 

This will set up the Bang-Bang for the first pass tunning. 

By observing the system capacity, you may reduce the gap between [H] and [L] to reduce magnitude of the fluctuations around the set point [SP].

You can also reduce or increase the on/off period by modifying [Tao] on the control. This will enhance the precision of the system output but will create and increase of on/off cycles in the actuator.

Be aware that if the actuator has a mechanical operation such as a relay, contactor or solenoid valve, the life of the actuator will be inverse of the number of on/off cycles, so you will need to make a trade off among life of the actuator and precision of the control.

Cooling Bang-Bang (Actuator to decrease the control variable)

The algorithm to decrease the control variable such as cooling in a heated tank or pressure release valve in a pressure tank is very similar to the heating diagram but with some changes as shown in figure 2:

The difference in the algorithm is that the decrease output is multiplied by -1 before feeding the input of the inner loop filter. The control different parameters, now the hysteresis turns on the decrease actuator when the e2 is less than de SP at DecON point and will turn off the decrease actuator when the e2 is greater than DecOFF point.

Heating and cooling Bang-Bang

The last case for Bang-Bang is when you have both actuators for the same system, this is the case of application such as chrome plating or galvanizing where the systems need to be at a certain temperature to begin the process but once the process is in operation, the process generates more heat due to the electro plating current and hence needs to be cooled.

In this case we need to use the Bang-Bang in dual mode as shown in figure 3:

As you can see in figure 8, we have 2 hysteresis, one to drive the increase actuator and one to drive the decrease actuator. The key to tune this option is to make sure that the [DecON] parameter is always less than the [IncOn] and is preferable that the off point of both parameters does not cross each other. Is preferable that both hysteresis are set up as mutually exclusive.

Tools & prerequisites

Software:

• Studio 5000 Logix Designer v35.
• FactoryTalk View ME v12.00.00.

Hardware:

• ControlLogix Logix (1756-L83) FW v35.
• CompactLogix I/O Adapter (5069-AEN2TR).
• CompactLogix Universal Analog Input Card (5069-IY4).
• CompactLogix 16 24VDC output card (5069-OB16F).
• PanelView Plus 7 Performance 700 (2711P-T7C22A9P).

Test System Components:

• 1 Pc 8 liters aquarium.
• 1 pc RTD PT100 Temperature Sensor.
• 1 pc Solid State Relay 24 VDC input coil / 125 VAC output.
• 1 pc 400 watts heating resistor.

Files:

• BB.acd Logix Designer Addon Instruction

Inputs & Outputs for the testing system.

The input outputs used are:

• 1 PT100 RTD temperature sensor connected to Remote Compact I/O input in slot 3 input 0 (5069-IY4). Controller tag: Compact_Ethernet_Adapter:3:I.Ch00.Data.
• 2 DC output connected to 24 DC output card in Remote Compact I/O in slot 2 output 0 & 1. Compact_Ethernet_Adapter:2:O.Pt00.Data & Compact_Ethernet_Adapter:2:O.Pt01.Data.

Bang-Bang Add On Instruction (AOI).

The Bang-Bang Add On Instruction contains the instruction parameters, input variables and output variables to make the control work as well as the logic. The parameters that are input and output are shown in the faceplate made for Panelview Plus 7.

AOI Structure

The AOI instruction has a defined data type to name each control system by its unique name. In the case of this example the name of the system is “Acuarium” that has the {aoi_Bang_Bang} data type.

Enable/Disable Parameters:

Parameter List and description.

[BB_is_ON] Boolean input variable to turn the control to ON or OFF.

[Has_inc_element] Boolean input to indicate if control has variable increase actuator.

[Has_dec_element] Boolean input to indicate if control has variable decrease actuator.

Operation parameters:

[PV] Process variable: This is a real floating-point parameter. You should assign the scaled variable of the input card to this variable.

[SP] Set Point: This is a real floating-point parameter. This variable will be set in the HMI or any input device that sets the reference for the control.

Calibration parameters:

[Inc_On] Real floating-point parameter. This is part of the hysteresis of the Bang-Bang that if the input error to the hysteresis greater than this value, the output to the increase the control variable is turned on.

[Inc_Off] Real floating-point parameter. This is part of the hysteresis of the Bang-Bang that if the input error to the hysteresis is less than this value, the output to the increase the control variable is turned off.

[Dec_On] Real floating-point parameter. This is part of the hysteresis of the Bang-Bang that if the input error to the hysteresis is less than this value, the output to the decrease the control variable is turned on.

[Dec_Off] Real floating-point parameter. This is part of the hysteresis of the Bang-Bang that if the input error to the hysteresis is greater than this value, the output to the decrease the control variable is turned off.

[tsample] Real floating-point parameter. This parameter should be equal to the time defined for the scan time of the periodic task for the Bang-Bang.

[Tao] Real floating-point parameter. This is the time constant in seconds for the first order filter at the inner loop feedback.

[K] Real floating-point parameter. This is the gain that the output of the first order system will tend after long period of time (normally 7 times Tao). If K = 3, then after 7 times Tao, the filter output will be near 3.

Monitoring Parameters

[e] Real floating-point variable. This indicates the difference between the setpoint and the process variable ([e] = [SP] – [PV]).

[e2] Real floating-point variable. Is the difference between the error [e] and the output of the first order filter [FB_Filter_Out].

[FB_Filter_In] Real floating-point variable. Is the input to the feedback filter which is the result of adding [INC_OUT] – [DEC_OUT] output parameters. These parameters are the actual outputs to actuator elements. Because [INC_OUT] and [DEC_OUT] parameters are Boolean, to add these parameters is required to use integer variables before assigning to the Filter input.

[FB_Filter_Out] Real floating-point variable. Is the output of the feedback filter which is the result of the digitalization of the transfer function of a first order system. For simplification purposes, the first order differential equation is transformed to digital sampling with Euler method.

The filter equation is as follows:

Output Parameters

[INC_OUT] Boolean Output Parameter. This is the output that must be connected to the output card to drive the Increase control variable actuator.

[DEC_OUT] Boolean Output Parameter. This is the output that must be connected to the output card to drive the decrease control variable actuator.

AOI internal variables

The Add On Instruction has other internal variables required to execute the instruction such as conversion from Boolean to Integer. We will not describe each of these variables in this document as the AOI instruction is open for reviewing and editing.

Panelview plus 7 screens

Panelview 7 machine edition screen is provided as face plate for the Bang-Bang Control.

The face plate is provided as a sample for a system and is included in the documentation package.

Previous Knowledge

Definition of a First Order and Second Order Critically Damped or Overdamped System. You can find the complete document here.