Part I: Feedback vs Feedforward
A. The boiler system will have water entering the tank to be heated for a steam supply that
varies by demand. For the boiler system the controlled variable will be the fluid height,
the manipulated variable will be the feedwater flow, and the disturbance variable is the
steam outlet.
B. The boiler system will be controlled with a combined feedforward-feedback control [1].
This provides advantages of feedforward by reducing the effects of measurable
disturbances, while implementing corrections from measurement or disturbance errors
with the feedback control aspect [2]. A feedback-only control system would be the
cheapest and simplest option, but the combined method has other big advantages that are
a trade-off for the higher cost. Additionally, a combined system has a higher safety value
as it can take “proactive action” to a disturbance before a safety error occurs.
C. There are some other control options such as a manual control and an on/off controller.
The manual control would rely on an operator always being present to monitor the fluid
height and adjust as necessary. Some downsides to this method is the cost and time
needed to have the labor and could have unforeseen human errors. This method may be
utilized in a start-up, shutdown, or emergency procedure, but should not be relied on for
long-term usage. The on/off controller would be simple and cheap but would provide
very little attention to detail. The controller would control the pump to either on or off to
fill the tank to a set level of water. The biggest problem with this method is the damage it
may cause to the equipment, as the controller will turn the pump 100% on anytime the
water level deviates from the set point. As steam will be pulled out of the tank at
variables times, the switch will rapidly be turning on and off as it wobbles around the set
water level. This method is the simplest feedback control that needs to be advanced
before it could be realistically implemented.
Part II: Designing, building, implementing
D. Figure 1 below depicts the combined feedback and feedforward control scheme for the
boiler system. The goal of the controls is to maintain the water level in the tank as water
can be pumped into the tank and the steam outlet demand varies [1].
Figure 1. The P&ID diagram for the combined feedback and feedforward control scheme for the boiler system.
The system units are the water pump and boiler drum with a heating element in the
bottom of the drum to produce steam. The water into the drum is controlled by a level
control valve that is determined by feedforward control from the disturbance variable and
correct with feedback control by monitoring the tank level. There is also a manual bypass
valve for startup, shutdown, and emergencies.
E. The approximate capital is found by the given list of instrumentation needed for the
Figure 1 design, which is compiled in Table 1 as a comparison between the combined
controls and a feedback only control scheme [1].
Table 1. Instrumentation cost comparison between the combined and feedback only control schemes.
Combined Feedback Only
Fluid Height Sensor
$1,000
$1,000
Pneumatically Actuated Fail-open Valve for Liquid Water
$800
$800
Feedback Controller
$75
$75
Feedforward Controller
$500
Current-to-Pressure Transducer
$300
$300
Vapor Flow Rate Sensor
$500
Manual Control Valve for Liquid Water
$500
$500
Total
$3,675
$2,675
As mentioned in part B, the combined feedback and feedforward control is more
expensive due to its additional instrumentation for a more precise control. The feedback
only scheme is simpler and approximately $1,000 cheaper in this scenario from one less
controller and sensor. However, if the Figure 1 design included more safety valves, then
both control methods may cost an additional couple thousand dollars. One improvement
would be to add a boiler vent with a fail-open valve to let steam out of the drum in an
over-pressurized emergency. The other safety valve that could be added is a fail-closed
valve to the heating element if it is heated through a hot liquid or gas to prevent a
runaway temperature in the case of an emergency. Both situations are out of the scope of
this project as the steam line may have safety valves in place downstream and there is
little information on the heating element in the boiler system.
F. The transfer function for the tank process is derived from the mass balance.
𝑑𝑚
𝑑𝑉
𝑑ℎ(𝑡)
= 𝜌𝑙
= 𝜌𝑙 𝐴
= 𝑚𝑖 − 𝑚𝑒 = 𝜌𝑙 𝑞𝑖 − 𝜌𝑣 𝑞𝑒
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑ℎ(𝑡) 𝑞𝑖 𝜌𝑣 𝑞𝑒
𝐹𝑖𝑛𝑎𝑙 𝑀𝐵:
= −
𝑑𝑡
𝐴
𝜌𝑙 𝐴
2
2
𝐴 = 𝜋𝑟 = 7.065 𝑚
𝑆𝑎𝑡𝑢𝑟𝑎𝑡𝑒𝑑 𝑙𝑖𝑞𝑢𝑖𝑑 𝑎𝑛𝑑 𝑤𝑎𝑡𝑒𝑟 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑖𝑒𝑠 𝑎𝑡 12.5 𝑝𝑠𝑖 [3]:
𝑘𝑔
𝜌𝑣 0.5137 𝑚3
=
= 0.0005343
𝑘𝑔
𝜌𝑙
961.5 3
𝑚
𝑈𝑠𝑒 𝑡ℎ𝑒 𝑑𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒: 𝑢(𝑡) = 𝑢′ (𝑡) + 𝑢̅
𝑑(ℎ′ (𝑡) + ℎ̅) (𝑞 ′ 𝑖 (𝑡) + 𝑞̅𝑖 )
𝜌𝑣 (𝑞 ′ (𝑡) + 𝑞̅𝑒 )
=
−( ) 𝑒
𝑑𝑡
𝐴
𝜌𝑙
𝐴
′ (𝑡)
′ (𝑡)
′ (𝑡)
𝑞
𝑑ℎ
𝜌𝑣 𝑞
= 𝑖
−( ) 𝑒
𝑑𝑡
𝐴
𝜌𝑙
𝐴
𝑇𝑎𝑘𝑒 𝑡ℎ𝑒 𝐿𝑎𝑝𝑙𝑎𝑐𝑒 𝑇𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚:
1 ′
−𝜌𝑣
𝐻 ′ (𝑠) =
𝑄 𝑖 (𝑠) + (
) 𝑄 ′ 𝑒 (𝑠)
𝐴𝑠
𝜌𝑙 𝐴𝑠
𝐻 ′ (𝑠)
1
1
𝑃𝑟𝑜𝑐𝑒𝑠𝑠 𝑇𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛: 𝐺𝑃 = ′
=
=
𝑄 𝑖 (𝑠) 𝐴𝑠 7.065𝑠
G. The best option for the FB controller configuration is PI control. The reason why
derivative control is not included is because of the system properties of noisy data from
water entering the tank and constant boiling. The constant slight change in the tank will
trigger the derivative control to send erratic signals to the control valve which will
eventually damage the pump. On the other hand, integral control is needed to balance out
the proportional control, which alone could cause a large offset in the level height due to
the integrating nature of the process. The control requirements of this system demand that
the liquid level is maintained and not let the tank overfill or dry up. The integral control
corrects the offset of proportional control. This allows the controller to adjust the pump to
send water to the tank around the set point of 1.2 meters.
H. The transfer function for the disturbance variable is derived from the mass balance.
𝑇ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑖𝑠 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑚𝑎𝑠𝑠 𝑏𝑎𝑙𝑎𝑛𝑐𝑒 𝑖𝑛 𝑝𝑎𝑟𝑡 𝐹
𝑎𝑓𝑡𝑒𝑟 𝑡ℎ𝑒 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠 𝑎𝑛𝑑 𝑙𝑎𝑝𝑙𝑎𝑐𝑒 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚 ℎ𝑎𝑣𝑒 𝑏𝑒𝑒𝑛 𝑎𝑝𝑝𝑙𝑖𝑒𝑑:
1 ′
−𝜌𝑣
𝐻 ′ (𝑠) =
𝑄 𝑖 (𝑠) + (
) 𝑄 ′ 𝑒 (𝑠)
𝐴𝑠
𝜌𝑙 𝐴𝑠
𝐻 ′ (𝑠)
−𝜌𝑣 −7.562 ∙ 10−5
𝐷𝑖𝑠𝑡𝑢𝑟𝑏𝑎𝑛𝑐𝑒 𝑇𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛: 𝐺𝐷 = ′
=
=
𝑄 𝑒 (𝑠) 𝜌𝑙 𝐴𝑠
𝑠
I. For tuning of the FB control, the feedback loop transfer functions need to be derived. The
transfer functions for the fluid height sensor and transducer are from the output over the
input of the instrument design spans [1].The process transfer function is calculated in Part
F and the control valve transfer function is modeled as a 1st order valve using the
instrument information [1].
20 − 4
𝐹𝑙𝑢𝑖𝑑 𝐻𝑒𝑖𝑔ℎ𝑡 𝑆𝑒𝑛𝑠𝑜𝑟 𝑇𝐹: 𝐺𝑚 =
= 8.889
1.9 − 0.1
25 − 0
𝑇𝑟𝑎𝑛𝑠𝑑𝑢𝑐𝑒𝑟 𝑇𝐹: 𝐺𝐼𝑃 =
= 1.563
20 − 4
𝑔𝑎𝑙
𝑚3
100 𝑚𝑖𝑛 ∙ 0.0378
−0
𝑔𝑎𝑙
𝐾𝑣
0.03154
15 − 3 𝑝𝑠𝑖𝑔
𝐶𝑜𝑛𝑡𝑟𝑜𝑙 𝑉𝑎𝑙𝑣𝑒 𝑇𝐹: 𝐺𝑣 =
=
=
1𝑚𝑖𝑛
𝜏𝑣 𝑠 + 1
0.0833𝑠 + 1
5𝑠𝑒𝑐 ∙ 60𝑠𝑒𝑐 ∙ 𝑠 + 1
Next, multiply all the forward transfer functions together to get the combined transfer
function.
0.06196
𝐺̃ = 𝐺𝑃 𝐺𝑚 𝐺𝐼𝑃 𝐺𝑣 =
𝑠(0.0833𝑠 + 1)
Since this is an integrating process with a variable disturbance variable, it is best to use
the SIMC method for tuning. 𝐺̃ can be turned into a integrator time delay form with
Skogestad’s half rule. This allows IMC tuning relation of M or N or G in Table 12.1 of
Process Dynamics and Control [2]. Case G is a possibility because with an infinite time
constant the equation simplifies to the integrating process.
𝐾 −𝜃𝑠
𝐾 ∗ 𝑒 −𝜃𝑠 𝜏 𝑒
𝐺=
=
𝑎𝑛𝑑 𝑡𝑜 𝑠ℎ𝑜𝑤 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑛𝑔 𝑝𝑟𝑜𝑐𝑒𝑠𝑠 𝑙𝑒𝑡 𝜏 → ∞.
1
𝜏𝑠 + 1
𝑠+𝜏
𝐾 −𝜃𝑠
𝑒
𝐾 ∗ 𝑒 −𝜃𝑠
𝑇ℎ𝑢𝑠 𝑡ℎ𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑖𝑛 𝑡ℎ𝑒 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 𝑖𝑠 𝑔𝑜𝑛𝑒 𝑎𝑛𝑑 𝐺 = 𝜏
=
,
𝑠
𝑠
0.06196 −0.0833𝑠
𝑤ℎ𝑖𝑐ℎ 𝑟𝑒𝑠𝑢𝑙𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑇𝐹: 𝐺̃ =
𝑒
𝑤ℎ𝑒𝑟𝑒 𝜃 = 0.0833 & 𝐾 ∗ = 0.06196
𝑠
This shows that with some algebraic manipulation case G can be used to find correct
tuning values for the feedback transfer function.
1
𝜏
1
1
𝐾𝑐 = ∙
= ∗∙
= 96.87
𝐾 𝜏𝑐 + 𝜃 𝐾 𝜏𝑐 + 𝜃
The SIMC tuning method specifically modifies the IMC method by changing the integral
time limitations.
𝜏𝐼 = 4(𝜏𝐶 + 𝜃) = 4(0.0833 + 0.0833) = 0.6664 𝑚𝑖𝑛
1
The final tuned feedback controller is 𝐺𝐶 = 96.87 (1 + 0.6664𝑠).
J. The feedforward controller transfer functions combines all the transfer functions in its loop,
including the disturbance variable. The only variable undefined so far is the vapor flowrate
sensor transfer function.
20 − 4
𝐺𝑡 =
= 0.8
20 − 0
−𝐺𝐷
𝐺𝑓𝑓 =
= 0.01299(0.0833𝑠 + 1) = 0.0018082𝑠 + 0.01299
𝐺𝑃 𝐺𝐼𝑃 𝐺𝑣 𝐺𝑡
K. The block diagram was created in Simulink to simulate the control response on the
manipulated and control variables. All the transfer functions inputted in the software
match the calculations shown in the previous parts.
Figure 2. Block diagram for the combined feedback and feedforward control schematic of the boiler system.
L. The block diagram in Part K was ran to show the given results in Figure 3 and 4. The two
conditions to simulate were a step change in the demand of steam and a slight raise in set
point. In order to see the outcomes, the changes were timed at different intervals to
compare to the baseline.
Figure 3. Increase in steam demand by 3 cubic meters per minute. The blue line shows the valve response and
yellow shows the water level of the tank.
In Figure 3, the manipulated variable (blue line) has a spike indicating that the valve is
being opened and a small amount of additional water is being supplied. The controlled
variable (yellow line) is the water level and this simulation showed that there was no
change. This simulation showed that the feedforward mathematical model is accurate as
the disturbance was “anticipated” and corrected before it could affect the tank level.
Figure 4. Simulation results for change in setpoint. At time 30 the setpoint was increased by 0.2m. The input for this
part of the simulation had to be converted to mA due to the block diagram setup so 1.778 was the final step value in
order to give a 0.2m setpoint increase.
At time 30 in the simulation, we can identify the valve reaction as a PI control. There is
an immediate spike to open the valve in order to reach the setpoint, but after a second the
valve closes and quickly stabilizes. With the control variable represented in yellow, the
simulation showed a small spike and quick stabilization to meet the new setpoint.
M. The settling times for the changes are within constraints of the project as they are
adjusting in minutes. Also the tank will never overfill or empty with this setup as the
water level variable has precise control and does not overshoot after the increase.
However, there is a failed constraint for the maximum pump capacity. This controller
spikes the pump outlet above the constrained 100GPM. Additionally, the steam demand
is going to be greater than the 5kg/min constraint as steam has density near 0.51 kg/m3 at
12.5 psi.
N. To improve this design, I would need adjust the tuning on the controllers. The
mathematical model developed rapidly adjusts the systems for changes, but the trade-off
for speed violates the pump constraints. To improve the constraints’ performance, the PI
control would need to extend its settling time in order to gradually increase the pump
flowrate to not exceed 100GPM. This will simply extend the settling time above the
current 2 minutes. Also, the tank should be increased in size or have an adjusted
operating pressure in order to handle 8 cubic meters per minute of pressurized steam.
References
[1] “Fluid Level Control in a Boiler” [PDF]
[2] D. E. Seborg, T. F. Edgar, D. A. Mellichamp, and F. J. Doyle, Process Dynamics and
Control, 4th ed. Hoboken, Nj: Wiley, 2017.