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Self-balancing Car Kit For Arduino Robot Project 9: PID Principle

Posted by Fiona Su on

The Proportion of deviation, Integral and the Differential are linearly combined to form a control quantity, then control object by this control quantity. Such a controller is called a PID controller.
In analog control systems, the most common control law for controllers is PID control.
The block diagram of the common analog PID control system is shown below.


The system consists of an analog PID controller and a controlled object.
In the figure, r(t) is a given value; y(t) is the actual output value of the system;
control deviation e(t) equals the given value subtracts the actual output value.
e(t) = r(t) − y(t)
e(t) is the input of the PID control; u(t) is used as the output of the PID controller and the input of the controlled object.
So the control law of the analog PID controller is follows:
Kp: the proportional coefficient of the controller;
Ti: the integral time of the controller, also called the integral coefficient;
Td: the differential time of the controller, also called the differential coefficient.

(1) Proportion part
The mathematical formula of the proportional part is: Kp×e(t)
In the analog PID controller, the proportional link is to react to the moment of deviation.

Once the deviation occurs, the controller immediately produces a control action that causes the control amount to change in the direction of decreasing the deviation.

The strength of the control depends on the proportional coefficient Kp.
The larger the proportional coefficient Kp, the stronger the control effect, the faster the transition process, the smaller the static deviation of the control process.

However, the larger the Kp, the more likely it is to oscillate and to destroy the system stability.
Therefore, the selection of the proportional coefficient Kp must be appropriate to achieve a small transition time and a small and stable static difference.

(2) Integral part
The mathematical formula of the integral part is:


From the formula of the integral part, we can know that as long as there is a deviation, its control effect will continue to increase.

Only when the deviation e(t)=0, its integral can be a constant, and the control effect is not a Increased constant.
It can be seen that the integral part can eliminate the deviation of the system.

Although the adjustment function of the integral link will eliminate the static error, it will reduce the response speed of the system and increase the overshoot of the system.

The larger the integral constant Ti is, the weaker the accumulation of the integral is. At this time, the system does not oscillate during the transition.
Increasing the integral constant Ti can slow down the elimination process of the static error, and the time required to eliminate the deviation is also longer; however, can reduce the amount of overshoot and improve the stability of the system.

When Ti is small, the integral action is strong; oscillation may occur during the system transition time, but the time required to eliminate the deviation is short. Therefore, Ti must be determined according to the specific requirements of actual control.

(3) Differential part
The mathematical formula of the differential part is:


In addition to the desire to eliminate static errors, the actual control system requires an accelerated adjustment process.
At the moment when the deviation occurs, or at the moment of the deviation change, not only the immediate response (the role of the proportional link ) is required, but also the appropriate correction is given in advance according to the trend of the deviation.

In order to achieve this, a differential link can be added to the PI controller to form a PID controller.
The role of the differential link is to prevent changes in the deviation.
It is controlled according to the trend of change (speed of change). The faster the deviation changes, the larger the output of the differential controller, and the correction can be made before the deviation value becomes larger.

The introduction of differential action will help to reduce the overshoot, overcome the oscillation, and stabilize the system, especially for the high-order system, which speeds up the tracking speed of the system.

However, the effect of differentiation is sensitive to the noise of the input signal.
For those systems with higher noise, the differential signal is generally not used, or the input signal is filtered before the differential action.

The effect of the differential portion is determined by the differential time constant Td
The larger the Td, the stronger the effect of suppressing the variation of the deviation e(t);
The smaller the Td, the weaker the effect of its resistance to the deviation e(t).
The differential part obviously has a great effect on the stability of the system.
By appropriately selecting the differential constant Td, the differential action can be optimized.

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