AME Unit 11 Q2

Proportional band (PB) and gain are two different ways to express the same concept in a proportional controller. They are inversely related to each other.  

1. Proportional Gain and Proportional Band Explained – Technical Articles – Control.com

Proportional Band (PB)

• Defines the range of process variable error over which the controller output changes from minimum to maximum.   1. Proportional Gain and Proportional Band Explained – Technical Articles – Control.com control.com
• Expressed as a percentage of the input span.   1. Proportional Gain and Proportional Band Explained – Technical Articles – Control.com control.com
• A higher PB means the controller is less sensitive to changes in the process variable.

Gain

• Represents the amplification factor of the controller.
• It’s the ratio of the change in controller output to the change in process variable error.   1. Proportional-only Control | Flow Measurements and Reynolds Numbers | Textbook control.com
• A higher gain means the controller is more sensitive to changes in the process variable.

Mathematical Relationship

• Gain = 100 / PB
• PB = 100 / Gain

In summary:

• A wide proportional band corresponds to a low gain.
• A narrow proportional band corresponds to a high gain.

Example:

• If the proportional band is 50%, the gain is 2.   1. Fundamentals of PID Control – International Society of Automation (ISA) www.isa.org
• If the gain is 4, the proportional band is 25%.

Understanding the relationship between proportional band and gain is essential for tuning a proportional controller to achieve desired performance.

Understanding the System

Before we dive into how to adjust the gain, let’s clarify the system. A float and lever proportional system typically consists of:

• Float: Rises and falls with the water level.   1. Ballcock – Wikipedia en.wikipedia.org
• Lever: Connected to the float, amplifies the float’s movement.
• Valve: Controlled by the lever, regulates water inflow.

The gain of the system is essentially the ratio of the valve opening change to the water level change.

Increasing Gain

To increase the gain of the system, you need to amplify the effect of the float’s movement on the valve. Here are some methods:

1. Increase Lever Arm Length:
   • Lengthen the arm of the lever connected to the float. This will magnify the float’s movement, resulting in a larger valve opening change for a given water level variation.
2. Decrease Valve Stem Diameter:
   • A thinner valve stem will open or close more rapidly for the same lever movement, increasing the system’s responsiveness.
3. Reduce Friction:
   • Minimize friction in the lever and valve mechanism. This ensures that the system responds quickly to changes in water level.

Decreasing Gain

To decrease the gain, you need to dampen the effect of the float’s movement on the valve:

1. Decrease Lever Arm Length:
   • Shorten the arm of the lever connected to the float. This will reduce the amplification of the float’s movement.
2. Increase Valve Stem Diameter:
   • A thicker valve stem will open or close more slowly for the same lever movement, reducing the system’s responsiveness.
3. Increase Friction:
   • Introduce friction into the lever or valve mechanism. This will slow down the system’s response to water level changes.

Important Considerations:

• Stability: Increasing the gain too much can lead to instability and oscillations in the water level.
• Over-shoot: A high gain might cause the water level to overshoot the desired level before settling.
• Under-shoot: A low gain might result in slow response times and the water level staying below the desired level.
• System Dynamics: The physical characteristics of the tank, water flow, and valve characteristics also influence the system’s behavior.

Additional Tips:

• Experimentation: Carefully adjust the system parameters while observing the water level response.
• Damping: Consider adding damping elements to the system to reduce oscillations.
• Feedback: Incorporate feedback mechanisms to improve system performance.

By carefully adjusting the lever arm length, valve stem diameter, and friction, you can effectively increase or decrease the gain of your float and lever control system to achieve the desired water level control.

When the outflow from a water tank is increased, the water level in the tank will decrease.

This is a fundamental principle of fluid dynamics: the rate of change of volume in a container is equal to the inflow rate minus the outflow rate.

Factors Affecting the Water Level Decrease Rate:

• Magnitude of the outflow increase: A larger increase in outflow will lead to a faster decrease in water level.
• Inflow rate: If the inflow rate remains constant, the decrease in water level will be more pronounced.
• Tank size: A larger tank will experience a slower decrease in water level compared to a smaller tank for the same outflow increase.

System Response:

• Float and lever system: The float will descend as the water level drops, causing the lever to move and reduce the valve opening, thereby decreasing the inflow rate.
• Control system response: Depending on the control system’s settings, it might increase the inflow rate to compensate for the increased outflow and maintain a desired water level.

In essence, an increased outflow will create an imbalance between inflow and outflow, leading to a drop in water level until a new equilibrium is reached.

Increasing the controller gain in response to an increased outflow will generally lead to a faster return to the desired steady-state tank level.

Explanation:

• Gain is a measure of the controller’s sensitivity to the error between the desired and actual tank level.
• Increasing the gain means the controller will react more strongly to deviations from the setpoint.
• When the outflow increases, the tank level begins to drop. A higher gain controller will detect this error more rapidly and increase the inflow accordingly.

Potential Issues:

• Overshoot: While a higher gain can improve response time, it can also lead to overshoot, where the tank level rises above the setpoint before settling.
• Oscillations: Excessive gain can cause the system to oscillate around the setpoint, making it difficult to maintain a stable level.
• Instability: In extreme cases, too high a gain can destabilize the system, leading to uncontrolled fluctuations in the tank level.

Conclusion:

While increasing the controller gain can improve the system’s response to disturbances, it’s essential to find the optimal value to balance responsiveness with stability. Often, a combination of proportional, integral, and derivative (PID) control is used to achieve better performance.

Integral action is a control mode that accumulates the error over time. This means that the controller output is not only proportional to the current error but also to the sum of the past errors.  

1. Proportional–integral–derivative controller – Wikipedia

2. Integral Control Action

In the context of a water tank system, introducing integral action would have the following effects:

Benefits:

• Eliminates steady-state error: One of the primary advantages of integral action is its ability to eliminate steady-state error. This means that the tank level will eventually reach the desired setpoint without any offset.   1. The PID Controller & Theory Explained – NI – National Instruments www.ni.com2. [Solved] The effect of integral controller on the steady state error – Testbook testbook.com
• Improved disturbance rejection: Integral action helps the system to recover more quickly from disturbances, such as an increase in outflow.

Potential Drawbacks:

• Overshoot and oscillation: Excessive integral action can lead to overshoot and oscillation of the tank level.
• Windup: If the controller output saturates (e.g., the valve is fully open or closed), the integral term can accumulate a large error, leading to a large overshoot when the output becomes unsaturated.

Overall Improvement:

By carefully tuning the integral term, it’s possible to significantly improve the system’s performance. A well-tuned integral action can help to eliminate steady-state error, reduce settling time, and improve disturbance rejection without causing excessive overshoot or oscillation.

Combining proportional and integral control (PI control) is often a good starting point for many control applications, including water tank systems. The proportional term provides quick response to changes in the error, while the integral term eliminates steady-state error.