Continuous Control Systems: a review

PI and PID Controllers

The PI (proportional + integral) and PID (proportional + integral + derivative) controllers are widely used for the control of continuous-time systems.

An extremely rich literature has been dedicated to design methods and parameters adjustment of these controllers. Also note that there are several structures for PI and PID controllers (with different transfer function and tuning parameters).
Synthesis methods for PI and PID controllers have been developed and implemented (see the references at the end of the chapter). These methods can be divided into two categories: a) methods using frequency and time characteristics of the plant (non–parametric model) and b) methods using the plant transfer function (parametric model).
In this section, we shall only present basic schemes for PI and PID controllers as an introduction to the digital PI and PID controllers.

1. 
   PI Controller
   

In general PI controllers have as input the difference between the reference and the measured output and as output the control signal delivered to the actuator (see Figure 1.15). A typical transfer function of a PI controller is

(s)  ^
1 ^ K (T_is 1)

_{H R} K _1  _ 

s
 ^{T i} 
^T i ^s
(1.3.1)

in which K is called the proportional gain and T_i the integral action of the PI controller. There also exist, however also PI controllers with independent actions, i.e.

_{H R} (s)  K _p
 1
^T i ^s

In certain situations the proportional action may operate only on the measured output.

2. 
   PID Controller
   

The transfer function of a typical PID controller is



1

_{H PID} (s)  K ^1  



^Td ^{s }
(1.3.2)

 _Ti s
1  ^Td s ^

N
 
 

in which K specifies the proportional gain, T_i characterizes the integral action, T_d characterizes the derivative action and 1 + (T_d / N) s introduces a filtering effect on the derivative action (low-pass filter).

By summing up the three terms, the transfer function given by Equation 1.3.2 can also be rewritten as




^{ } T ^{ }
Tⁱ T^d _ 

^{K 1  s}T_i ^{
d   s2} _^{Ti Td}  _ ^

^H PID
 



(s) 
N _{ } N _{ }


 _Td 

(1.3.3)

T_i ^s^{1 } _N ^s 
 

Several structures for PID controllers exist. In addition there are situations when the proportional and derivative actions act only on the measured output.

4. Concluding Remarks

The behavior of controlled plants around an operating point can in general be described by linear dynamic models. Linear dynamic models are characterized in the time domain by linear differential equations and in the frequency domain by transfer functions.

Control systems are closed-loop systems that contain the plant, the controller and the feedback connection. For these systems, the control applied to the plant is a function of the difference between the desired value and the measured value of the controlled variable. Control systems are characterized by a dynamic model that depends upon the structure and the coefficients of the plant and controller transfer functions.
The desired control performances can be expressed in terms of the desired characteristics of the dynamic model of the closed-loop system (ex.: transfer function with specified coefficients). This allows the synthesis of the controller if the plant model is known.
The plot in the complex plane of the transfer function of the open-loop system (controller + plant), also called Nyquist plot, plays an important role in the assessment of controller qualities. In particular, it allows studying of the stability and robustness properties of the closed-loop system.

5. Notes and References

Many books deal with the fundamentals of continuous-time control. Among the different titles we can mention:

Takahashi Y., Rabins M., Auslander D. (1970) Control. Addison Wesley, Readings, Mass
Franklin G., Powell J.D. (1986) Feedback Control of Dynamic Systems. Addison Wesley, Reading, Mass
Ogata K. (1990) Modern Control Engineering (second edition). Prentice Hall, N.J Kuo B.C. (1991) Automatic Control Systems (sixth edition). Prentice Hall, N.J

PID controller adjustment techniques are discussed in:

Ziegler J.G., Nichols N.B. (1942) Optimum Settings for Automatic Controllers.
Trans. ASME, vol. 64, pp. 759-768
Shinskey F.G. (1979) Process Control Systems. McGraw-Hill, N.Y.
Aström K.J., Hägglund I. (1995) PID Controllers Theory, Design and Tuning, 2nd edition. ISA, Research Triangle Park, N.C., U.S.A
Voda A., Landau I.D. (1995a) A method for the auto-calibration of PID controllers. Automatica, vol. 31, no. 1, pp. 45-53.
Voda A., Landau I.D. (1995b) The auto-calibration of PI controllers based on two frequency measurements. Int. J. of Adaptive Control and Signal Processing, vol. 9, no. 5, pp. 395-422
as well as in (Takahashi et al. 1970) and (Ogata 1990).