1 PID Control Schemes

So far, we design controller/compensator based on models of plants(transfer function, Bode plot).
However, in many practical situations, we deal with very complicated plants whose mathematical models are difficult to obtain. Hence, the analytical approach of designing a controller is not possible.
In this chapter, we shall introduce tuning rules for basic PID controllers.
More than half of industrial plants today still employ PID control schemes.

The standard form of PID controller is \[C(s) = K_p\left(1+\frac{1}{T_is}+T_ds\right) = K_p+\frac{K_I}{s}+K_Ds.\]
P(Proportional) action: P-action provides an action which depends on the instantaneous value of the control error. It can be used to improve speed of response and steady-state error but with limited performance.
I (Integral) action: I-action gives a controller output that is proportional to the accumulated error, which implies that it is a slow reaction control mode. It is used to achieve zero-steady error in the presence of a step reference and/or disturbance. The shortcomings are: its pole at the origin is detrimental to loop stability and undesirable effect in the presence of actuator saturation, known as wind-up.
D (Derivative) action: It acts on the changing rate of the control error, sometimes referred to as a predictive mode. Its limitation is the tendency to yield large control signals in response to high frequency control errors such as error induced by set-point changes or measurement noises.

The process of selecting the controller parameters \(K_p\), \(T_i\) and \(T_d\) is known as PID Tuning.
Two classical methods for determining the parameters of PID controllers were presented by Ziegler-Nichols in 1942.
- Step response method
- Self-oscillation method
\(K_p\), \(T_i\) and \(T_d\) from Ziegler-Nichols rule often serve as the starting point for tuning procedures used by manufacturers and process industry.
It typically yields an aggressive gain and overshoot. Further tuning is needed if minimizing overshoot is desirable.

Many plants, particularly those in the process industries has step response of S-shape (reaction curve):
- open-loop stable
- no integrator
- no dominant complex poles
An experiment for deriving step response can be designed as
- With the plant in open-loop, let the plant run at normal operating condition with constant input \(u_0\) and steady state output \(y_0\) .
- At the initial time \(t_0\) , apply a step change to the plant, from \(u_0\) to \(u_\infty\) (in the range of 10 to 20% of full scale).
- Record the output response to get the reaction curve. In the figure, the m.s.t. stands for the maximum slope tangent.
- Compute \[K = \frac{y_\infty-y_0}{u_\infty-u_0},\,L=t_1-t_0,T = t_2-t_1.\] \(K\) is the system gain. \(L\) and \(T\) are often called the apparent dead time and the apparent time constant.
The PID parameters are then chosen according to the table

Controller	\(K_p\)	\(T_i\)	\(T_d\)
P	\(T/KL\)
PI	\(0.9T/KL\)	\(3L\)
PID	\(1.2T/KL\)	\(2L\)	\(0.5L\)

Consider a plant with transfer function \[G(s) = \frac{5}{(s+1)^3}.\]

From the unit step response, it is known that \[K = 5/1 = 5,\,L = 0.8,\,T = 4.5-0.8 = 3.7.\]
The PID parameters given by Ziegler-Nichols tuning rule is \[K_p = 1.11,\,T_i = 1.60,\,T_d = 0.40.\]

Set the integrator and derivative gain to \(0\).
Increase the proportional gain \(K_p\) from \(0\), until the closed-loop system starts oscillation.
Record the critical propotional gain \(K_{cr}\) and the oscillation period \(P_{cr}\).
Notice that if the open-loop transfer function of the plant is given, then
- The critical gain is the gain margin, i.e., \(K_{cr} = K_g\).
- The critical oscillation period is related to the phase crossover frequency: \[P_{cr} = 2\pi/\omega_\phi.\]
Set the PID controller according to

Controller	\(K_p\)	\(T_i\)	\(T_d\)
P	\(0.5K_{cr}\)
PI	\(0.45K_{cr}\)	\(P_{cr}/1.2\)
PID	\(0.6K_{cr}\)	\(0.5P_{cr}\)	\(0.125P_{cr}\)

Consider the same plant with transfer function \[G(s) = \frac{5}{(s+1)^3}.\]
The phase crossover happens when \(\angle G(j\omega_\phi) = -180^\circ\), which implies that \[\angle 1+j\omega_\phi = 60^\circ\Rightarrow\omega_\phi = \sqrt{3}.\]
The corresponding magnitude is \(|G(j\sqrt{3})| = 5/8\) and hence the gain margin/critical gain is \(1.6\).
The critical period is \(2\pi/\sqrt{3} = 3.63\).
The PID parameters according to Ziegler-Nichols rule is \[K_p = 0.96,\, T_i=1.81,\,T_d = 0.453.\]

Ziegler-Nichols tuning typically yields an aggressive gain and overshoot, which may be unacceptable in some applications.
However, it can serve as a starting point for finer tuning.
For example, by increasing \(T_i\) and \(T_d\), we can expect the overshoot will be reduced.
However, for certain applications where the measurement noise is significant, we need to be extra careful when increasing \(T_d\).
Ziegler-Nichols tuning rules have been widely used in process control where the plant dynamics are unknown. When the plant model is available, other controller design methods exists.
Generally, to apply the step response method, one needs to obtain the S-shape response. Plants with complicated dynamics but no integrators are usually the cases.
The self-oscillation method requires the plant to be forced into oscillation. This can be expensive and dangerous.

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