Chapter II Summary of literature review in the fields of solar energy and related
2.2 Basic Control theory concepts
2.2.3 Some Traditional and Advanced Control Techniques
2.2.3.3 PID Feedback Control System
A good example of a feedback control system is that using a PID controller, which is the most widely used closed loop control system in industrial process control, with a share of about 95%, according to Astrom K. J. and Hagglund T. H as cited by Jinghua Zhong (Zhong, 2006).
The PID controller is composed of three terms, which can be distinguished in the following expression:
%? |z ) |} ziq ) |22
2 (2.2.3.3.1)
That may also be represented as:
%? |~z )
} ziq ) 22 (2.2.3.3.2) or:
%? |'z )@
} ziq ) 22
2 4 (2.2.3.3.3)
where Kp, Ki and Kd, are the proportional, the integral and the derivative gains respectively. Ti and Td, are the integral and derivative times, respectively. The variable e is the error, on which the PID controller action is calculated, which will be eventually applied to the system so as to drive it into the desired behaviour. Its block diagram is as follows:
Following is the Laplace transform of the above PID controller:
%"8# ? O O (2.2.3.3.4) or otherwise:
%"8# ? |'1 )@
) 284 (2.2. .3.3.5)
A PID feedback control system is the one that uses a PID controller, whose block I = |} ziq
P = |z
D=|22 2
e Vm
Figure 26 - Block diagram of a PID controller
diagram is as follows:
where Vs is the set point variable (the desired value), Vm is the manipulated variable; Vc is the actual process output value (as a result of the manipulated variables action over the process plus any possible disturbances), e is the error (difference between Vs and Vc). The manipulated variable Vm is the PID controller output and represents the controller’s action to null the error e.
It is worth noting that it is not always necessary that a control system will use all these terms to achieve the desired performance characteristics. Some will need only one of the three controllers most likely the proportional or the integral controllers only, while some others may need a combination of the two, most likely PI or PD.
The individual effects of each one of the PID components on the control actions are the following:
• P – The proportional term outputs a proportion of the error, thus giving a reaction to the current error to the control the system. However, the proportional component introduces an offset error into the system.
• I – The integral component provides a reset action against the offset error created by the proportional component. It performs so by integrating (summing) all the ‘historical’ errors . But as a result, it may lead to what is called integral windup, where the integral term, keeps increasing indefinitely.
• D – the derivative term reduces the rate of change of the error e. This may lead to noise amplification producing system instability.
2.2.3.3.1 PID Tuning
The manner in which the individual PID terms affect the overall manipulated variable, depends on the respective gain. The gains are tuning parameters, whose optimal values are the ones that guaranties short rise time, null or small overshoot, fast settling time and small (ideally null) steady state error. Tuning gains may be determined as follows:
a) Analytically, if a mathematical model exists. Additionally, a computer aided calculation may be used; well addressed in the literature, as in (Ogata, 2002).
b) Adjusting the system behaviour manually in real time (done by an expert such as an experienced technician);
Vm Process or VC Plant
Vs -
e I = |} ziq P = |z
D=|22 2
Figure 27 - Block diagram of a PID feedback control system
c) With the help of many well established methods, well considered in various textbooks, like the reaction curve Ziegler-Nichols method, well addressed by various authors like Leigh (Leigh, 1988), (Ogata, 2002), etc.
d) Through self tuning methods, by using computer based methods embedded inside the controller implementation, like the evolutionary concepts (genetic algorithms) used by Jin-Sung Kim et al. (Kim, et al., 2008).
2.2.3.3.1.1 Ziegler-Nichols reaction curve method
The reaction curve method (known as the First Method) for determining the optimal P, PI and PID controller gains, is only applicable to processes with neither integrators nor dominant complex-conjugate poles (Ogata, 2002). This method consists of obtaining the delay time L and the time constant T from the plant’s open loop unit step transient response, whose graphical representation resembles the S-shaped curve of Figure 28. The PID gains are then found by computing the relations from Table 2.2 below.
Figure 28 - Reaction curve (plant’s open loop transient response to unit step input)
In Figure 28, Vc(t) is the measured process output (the S-shaped curve); tg is a tangent at the S-curve inflection point; L is the delay time and T the time constant.
Resulting gains may need further
Kp Ti Td
P T/L ∞ 0
PI 0.9*T/L L/0.3 0
PID 1.2 * T/L 2L L/2
Table 2.1 – Ziegler – Nichols reaction curve coefficients (Ogata, 2002).
performance specifications. Such specifications are normally application-specific.
They will, in turn, tell how far to take and what to focus in the fine tuning.
Fine tuning is performed by further adjusting each one of Kp, Ki, and Kd until a desired overall response is obtained. When performing the fine tuning, one should take into account that the increase of a given gain, aiming at improving a certain characteristic may lead to worsening of other characteristics, namely:
T t
VC(t)
VS
VC(t)
tg
L
Kp – when increased: decreases the rise time, however, this may worsen offset (steady-state) error.
Ki – when increased: decreases the steady-state error but may lead to integral windup, where the integral term (summing of the errors), keeps increasing indefinitely.
Kd – when increased: it reduces the overshoot and settling time, though it may perform amplification of noise induced into the feedback loop, resulting in system instability.
The best fine tuning process, however, is that performed by a computer program, either performed outside the control system or built in as a self tuning mechanism.
Figure 29 shows a closed loop unit step response curve (Matlab generated) of a hypothetical process, whose open loop transfer function is
"8# ? UO1@ O3
and whose closed loop (with PID controller) transfer function is
"8# ? 6.322382)188)12.811 84)683)11.322382)188)12.811
and PID gains are Kp=18, Ki=18/1.405, Kd=18x0.35124. The PID gains (and thereby the closed loop transfer function) were found through the second Ziegler-Nichols tuning method (Ogata, 2002). It can be observed that, some of the performance characteristics, namely the overshoot (ratio is more than 50%) will need improvement, by further fine tuning steps, as suggested above.
Figure 29 - Closed loop step response of a system