The PID (Proportional, Integral and Derivative) controller is feedback controller which aims to make system output follow desired response (Hägglund & Åström, 1995). The controller is defined by following equation:
where u is input control signal to the plant model and e is the control error which is difference between desired and actual system output used to generate the control action. As shown by equation the controller consists of proportional, integral, and derivative terms. Gain K, integral time Ti and derivative time Td are all parameters of PID controller.
We can obtain proportional to the error controller by setting Ti= ∞ and Td=0. And the equation 1 is reduced to
where ub is a bias or reset introduced for the case when error is zero.
Proportional controller is the control action on present system error. For this controller, steady state error is always present but can be regulated by gain K. As K increases the error decreases but oscillations increase at the same time. For large errors proportional controller behaves similar to on-off controller.
Integral controller is used to reduce or completely eliminate steady state error. The action of this controller can be considered as a mechanism which resets proportional controller’s bias term. Oscillations can also be reduced. The action of integral has more effect and is faster with increasing Ti value, but system output becomes oscillatory as well. Integral controller is the control action on past performance error. This controller ensures that output of the system will be identical to the reference signal in steady state.
As derivative control action is oriented on future system error it produces predictions using linear extrapolation for time Td. This controller aims to improve the stability of the closed-loop and due to process dynamics takes some time before its effect can be noticed. The controller increases damping and decreases it again with large values of derivative time. As time increases oscillations elevate as well. There is one significant drawback that derivative control has, for signals with high frequency an ideal derivative has large gain. The problem is that input control signal to the plant will have large variations due to high frequency noise. This can be resolved by using first-order filter for an ideal derivative with particular time constant.
Conventional methods for designing controller can be applied to PID controller, however, there are numerous methods are developed specifically for PID tuning. The most popular methods were provided by Ziegler and Nichols based on step response and frequency response of the system. Furthermore, there exist on-line algorithms to tune PID parameters including fuzzy logic, neural network and genetic algorithm.
PID algorithm serves as a basis for more than 90% of controllers in industry (Shoujun & Weiguo, 2011). PID is most commonly used controller for motor drive systems. It is also used in robotics, for example, as a position control in laparoscopic surgery robots (Song et al., 2015) or performance control for the pneumatic artificial muscle manipulator (Ahn & Thanh, 2005).
Ahn, K., & Thanh, T. (2005). Nonlinear PID control to improve the control performance of the
pneumatic artificial muscle manipulator using neural network. Journal Of Mechanical Science And Technology, 19(1), 106-115. http://dx.doi.org/10.1007/bf02916109
Hägglund, T., & Åström, K. (1995). PID Controllers: Theory, Design, and Tuning (2nd ed.).
Shoujun, S., & Weiguo, L. (2011). Application of Improved PID Controller in Motor Drive
System. In T. Mansour, PID Control, Implementation and Tuning (1st ed., pp. 91-92). InTech.
Song, S., Moon, Y., Lee, D., Ahn, C., Jo, Y., & Choi, J. (2015). Comparative Study of Fuzzy
PID Control Algorithms for Enhanced Position Control in Laparoscopic Surgery Robot. Journal Of Medical And Biological Engineering, 35(1), 34-44. http://dx.doi.org/10.1007/s40846-015-0003-1