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PID Transfer Function
PID (Proportional-Integral-Derivative) controllers are a widely used control strategy in engineering, specifically in process control. They are used to control the output of a process based on feedback from sensors. The transfer function of a PID controller is a mathematical model that describes the relationship between the input and output signals of the controller.
Three Definitions for Transfer Function of PID Controller
Three widely used definitions for transfer function of PID controller in the literature of control theory are:
- standard (or non-interacting) form with transfer function \(C(s)=K(1+\frac{1}{T_is}+T_ds)\)
- series (or classical, interacting, interactive) form with transfer function \(C(s)=K(1+\frac{1}{T_is})(1+T_ds)\)
- parallel form with transfer function \(C(s)=k_p+\frac{k_i}{s}+k_ds\)
where \(s\) is the Laplace transform variable. Note that almost all of the PID parameter tuning tables assume that the transfer function of PID controller is in the standard form (see here for more details). But, Matlab/Simulink and many control textbooks use the parallel form for defining the transfer function of PID controller.
The only reason for the importance of series form is that in the early pneumatic PID controllers this form of the transfer function of PID controller could be implemented. When Ziegler and Nichols were developing their tuning methods they also assumed this form of controller. In today real world applications the series form is obsolete and not used anymore.
The reason why the series form is also called “interacting” form can be understood from its transfer function which can be expanded as follows
\[C(s)=K+\frac{KT_d}{T_i}+\frac{K}{T_is}+KT_ds\]
The first and second terms in the right hand side of the above equation determine the proportional term of controller. As it can be seen, the proportional gain is a function of integral and derivative time constants, which is undesired since it means that it cannot be adjusted independently. Figure 1 shows the block diagram of parallel and series forms for PID controller.
Figure 1: (a) parallel form, (b) series form for PID controller
The series (interacting) form can always be transformed to the standard (non-interacting) form but the opposite of this statement is not necessarily true. The standard form can always be transformed to the parallel form and vice versa.
The Gains in PID Transfer Function
The term \(K\) in standard and series forms is called “controller gain”. In the parallel form \(k_p\) is the proportional gain, \(k_i\) is the integral gain, and \(k_d\) is the derivative gain.
The proportional gain term \(k_p\) contributes to the output signal in proportion to the error between the desired setpoint and the measured process variable. The integral gain term \(k_i\) contributes to the output signal in proportion to the accumulated error over time. The derivative gain term \(k_d\) contributes to the output signal in proportion to the rate of change of the error over time.
More About Derivative Term
Note that the transfer function of ideal derivative term equals \(s\) which has two limitations for real world applications. First, it cannot be used in online applications since it is a non-causal system, and second, it amplifies the sensor noise.
To overcome these limitations, in practice the derivative term is implemented in series with a lowpass filter. It means that the transfer function of derivative term is considered as \(\frac{s}{1+s/N}\) where \(N\) determines the bandwidth of the lowpass filter. In this case the filter bandwidth, \(N\) can also be considered as a tuning parameter. Hence, in practice the PID controller actually has 4 tuning parameters.
The available PID tuning tables do not have any column for tuning the bandwidth of the derivative (D) term of PID controller. The best way to do this is to use the pidtune function of Matlab.
The parallel form of the discrete-time PID controller transfer function of is as follows
\[C(z)=k_p+\frac{k_iz^{-1}}{1-z^{-1}}+k_d(1-z^{-1})\]
where \(z\) is the \(z\)-transform variable.
Applications of PID Transfer Function
The transfer function of a PID controller can be used to analyze and design the controller. Specifically, the transfer function can be used to determine stability, frequency response, and performance metrics such as overshoot and settling time.
PID controllers are widely used in industry due to their simplicity, robustness, and effectiveness. They can be tuned to achieve desired performance specifications by adjusting the gains \(k_p\), \(k_i\), and \(k_d\). Proper tuning of a PID controller can result in stable and accurate control of a process.
In conclusion, the transfer function of a PID controller is a mathematical model that describes the relationship between the input and output signals of the controller. It is a key tool for analyzing and designing PID controllers for process control applications.
See here for an internationally recognized famous book on tuning PID controllers which is written by Prof. Astrom and Hagglund.
Written by Farshad Merrikh-Bayat, updated on March 3, 2023