PID Control System: Proportional Integral Derivative Control

 Both for analog and digital control systems, the PID controller is a hero for its robustness and simplicity. As our goal in modern precision-oriented industrial processes, fast response and stability of a process or system are highly dependent on the PID controller. The feedback control loop mechanism is utilized in the PID controller.

What is a PID Controller?

The controller that calculates the error between the target point and the measured process variable using an effective feedback control loop mechanism is known as a PID controller. More precisely, to have better understanding, PID controller is such a kind of device that does the operation of controlling the process variables (temperature, pressure, flow rate, level). To obtain the actual output, closed-loop feedback is used. The proportional, integral, and derivative actions are combined by parallel connection in a PID controller. In the mechanism, an "error value" e(t) is on continuous calculation, and the correction is employed with proportional, integral, and derivative terms.

How PID Controller Came:

Engineers used manual control and simple feedback mechanisms before PID. But there were overshoots and instability. Afterwards, James Clerk Maxwell (1868) published first mathematical analysis of feedback systems. Later on, Elmer Sperry worked on ship steering systems and early forms of proportional and derivative control were introduced. Reacting to current error (proportional) and also focusing on past behavior (integral) and anticipated future error (derivative) were studied by Nicolas Minorsky who worked for automatic ship steering to provide U.S. Navy. After that event, the use of automatic PID controller has been frequent and vast.

Explanation of  P, I and D Terms:

P(Proportional): An amplifier-attenuator enables the action. The term is proportional to the error e(t).

Mathematical expression: P_out​ = Kp​⋅e(t)

Kp​ = Proportional gain

However, instability and oscillations can occur, though the response is fast for higher proportional gain.

Figure 1: Functional block of proportional action.

       

I(Integral): A pure integrator produces the action. To eliminate the offset or steady state error is the main purpose here. Past errors are summed over time.

Figure 2: Functional block of integral action.

Mathematical expression: I_out​ = K_i ​∫e(t)dt

K_i​ = Integral gain

Overshoot and oscillatory behavior can be produced by a higher Integral gain.

 D(Derivative): A pure derivation conducts the action. Stability is increased and overshoot is minimized by creating the damping error, where the higher the constant value, the higher the chance of amplifying noise.

Figure 3: Functional block of derivative action.

Mathematical expression: D_out​= K_d​⋅ 

 K_d​ = Derivative gain

Types of PID Controller:

Basic types of controllers are: On-Off, Proportional, and PID controller.

On-Off Controller: This controller only deals with two states: On and Off. There remains no middle state. When the process value crosses the set point, it goes on and when is lower, it goes off. There is no stability with its output. Manually used latching relay is used here which is inconvenient.

Proportional Controller: The controller reduces the power while the process value approaches to the set point and tries to keep the output stable. The on-off cycle is hindered, rather, proportional action is taken, keeping the output on and off for a small time of interval.

PID Controller: It works through proportional, integral, and derivative control. Changes or differences between process value and set point are compensated. Therefore, stable output is maintained, and desired accuracy is satisfied.

Block Diagram of PID Controller:

A closed-loop feedback control loop is engaged to keep this controller operable. In the error calculation block, set point and process variable are given as input and the error is defined by their difference. After that, the correction is done in P, I, and D block. The the outputs are summed and also fed back to closed the loop.

Figure 4: Block diagram of PID controller

Working Principle of PID Controller:

How Does PID Manage Challenges?

PID controller ensures zero error between the process value and the set point. The control is not limited like ON-Off controller. PID gives detailed control and stable output. The usage of a low-cost basic ON-OFF controller allows for only two control states: fully ON or entirely OFF. It is used for limited control applications in which these two control states are sufficient to achieve the control aim. However, the oscillating nature of this control limits its use, and hence it is being superseded by PID controllers.PID controllers use closed-loop procedures to maintain an output with zero error between the process variable and the setpoint/desired output. PID employs three basic control characteristics, which are discussed below. The output of the PID control with respect to time:

u(t)= Kp​⋅e(t) + K_i ​∫e(t)dt + K_d​⋅ 

It’s the sum of the P, I, and D terms.

Proportional Control: The proportional parts do the correction, and this correction is proportional to the error. For example, a student who appears in the exam hall will write faster according to the amount of time left. The output of a proportional or P-controller is proportionate to the current error, e (t). It contrasts the actual value or feedback process value with the intended or set point. The output is obtained by multiplying the resulting error by a proportional constant. This controller output is 0 if the error value is zero.

Benefits of P control: It responds fast. Operation is simple.

Limitation: Steady state error can interrupt.

Integral Control: Past errors are summed up in this case. The student accumulates small, persistent errors and increases the speed of writing, having lessons from past mistakes- this is a good example of it. In the event of a negative error, integral control reduces its output. It impacts the system's stability and restricts response time. By lowering the integral gain, Ki, the response speed is accelerated. The I-controller output is restricted to a certain range while utilizing the PI controller in order to overcome the integral wind up conditions, which occur when the integral output continues to increase even at zero error state because of nonlinearities in the plant.

Benefits of I control: Steady-state error is eliminated.

Limitation: Steady state error can interrupt. Slow response and overshoot may mislead.

Derivative Control: The prediction of error’s behavior in the future is the work of this control, actually. It looks like adjusting one’s speed and looking ahead. The I-controller lacks the capacity to forecast how errors will behave in the future. Therefore, after the setpoint is altered, it responds normally. By predicting how the mistake will behave in the future, the D-controller solves this issue. Its output is determined by multiplying the derivative constant by the rate at which the error changes with respect to time. It increases system response by providing the output with a kick start. The controller is larger than the PI controller in the response of D, and the output settling time is likewise shorter. By making up for the phase lag brought on by the I-controller, it increases the system's stability. The speed of response increases as the derivative gain rises.

Benefits of P control: It responds fast. Transient response and stability are improved.

Limitation: Noise can increase if not tuned properly. It can’t perform well in steady-state conditions.

We could obtain the system's intended response by integrating these three controllers. PID algorithms are designed differently by various vendors.

Tuning a PID Controller:

Several methods are employed in PID control. They have their own specification, control operations, advantages, and limitations. The methods are as follows:

1. Manual Tuning: Kp is increased, keeping K_i = 0 and K_d​ = 0. After having a response, K_i  is added for removing steady-state error and then is  K_d employed for reducing overshoot. Using trial and error method, manual tuning proceeds for the optimum response.

2. Ziegler–Nichols Method: The methods suggests presetting the values of the parameters of the controller. The suggested chart:

Figure 4: Presetting the parameters according to Ziegler–Nichols Method.

 Ultimate gain and ultimate period are identified by this method. For future tuning, it is a famous method and a good starting.

3. Tyreus-Luyben Tuning: It’s similar to Ziegler-Nichols but designed for systems with significant time delays. It offers more stable and robust control.

 4. Cohen-Coon Tuning: With long dead times and large time constants, the method is effective.

 5. Software-Based Auto-Tuning: Auto-tuning functionalities are rendered by many modern industrial controllers and software packages. These algorithms automatically calculate optimal PID parameters by employing disturbance.

6. Model-Based Tuning: For analytically deriving optimal PID parameters, advanced control techniques can be used. It can be accomplished only if a mathematical model of the process is available.

Practical Considerations:

Noise Filtering: Low-pass filters are used to mitigate high-frequency noise amplified by derivative action.

Deadband Compensation: Mechanical slack or digital quantization is initiated.

Anti-windup: When actuators saturate (output limit reached), it prevents the integral term from accumulating excessively.

Digital Implementation:

PID algorithm is implemented discretely in PLC or microcontrollers, which are digital control systems. The equation of this type of control:

Where $k$ = Discrete time step

$T$ = Sampling time.

Applications:

• Motor Control: For position, speed, and torque regulation of servo and stepper motor control systems.
• HVAC Systems: For energy-efficient climate control in buildings.
• Medical Devices: In ventilators, and temperature regulation in incubators.
• Industrial Automation: In chemical plants, temperature control in furnaces for process control.
• Robotics: For precise control in robotic arms.
• Aerospace: In aircraft and spacecraft, altitude control.

Advantages of PID Controller:

•  Simplicity: Simple design and implementation
•  Less Requirement: No requirement for a detailed model of the system.
• Robustness: PID controllers are remarkably robust and can handle a wide range of process variations.
• Versatility: Works well for a wide range of systems
• Friendly in Use: Easy to tune and understand.
• Effectiveness: Effective in both analog and digital domains.

Limitations and Considerations of PID

Controller:

• Delay: Manual tuning may be time-consuming and can’t offer optimum control.
• Non-Linearity: Performance is not satisfactory with non-linear or time-varying systems
•  Sensitivity: Sensitive to measurement noise.
• Tuning Complexity: Iterative tuning may be needed, which causes complexity.
• Dead Time Issue: Processes with significant dead time (the time it takes for a change in input to affect the output) can pose challenges.

Future Enhancements:

Adaptive PID, Fuzzy logic-driven PID, Neural Network PID (Combination of classical control and machine learning), and Model Predictive Control (MPC) are the upcoming leading future trends of PID controllers. Advanced adaptation will certainly make these controllers with fewer limitations and turn them into a workhorse of future control system engineering.

Keep learning something new always, and enjoy!