Now we will describe the tuning of feedback controllers, that is the adjustment of the controller parameters to match the characteristics (or personality) of the rest of the components of the loop. We will look at three different specification of control loop performance:

1. Quarter decay ratio response

2. Minimum error integral

3. Controller synthesis

This latter method, in addition to providing some simple controller-tuning relationships , will give us some insight into the selection of the proportional, integral and derivate modes for various process transfer functions.

QUARTER DECAY RATIO RESPONSE BY ULTIMATE GAIN

This pioneer method, also known as the close-loop or on-line tuning method, was proposed by Ziegler and Nichols in 1942. like all the other tuning methods, it consists of two steps:

1. Determination of the dynamic characteristics, or personality, of the control loop
2. Estimation of the controller tuning parameters that produce a desired response for the dynamic characteristic determined in the first step, in other words, matching the personality of the controller to that of the other elements in the loop.

In this method the dynamic characteristic of the process are represented by the ultimate gain of a proportional controller and the ultimate period of oscillation of the loop. We usually determinate the ultimate gain and period from the actual process by the following procedure:

• Switch off the integral and derivative modes of the feedback controller so as to have a proportional controller.
• With the controller in automatic (i.e., the loop closed ), increase the proportional gain (or reduce the proportional band) until the loop oscillates with constant amplitude. Record the value of the gain that produces sustained oscillation. To prevent the loop from going unstable, smaller increments in gain are made as the ultimate gain is approached.
• From a time recording of the controlled variable such as the figure below, the period of oscillation is measured and recorded as T the ultimate period.

Response of the loop with the controller gain set equal to the ultimate gain K. T is the ultimate period

For the desired response of the close loop, Ziegler and Nichols specified a decay ratio of one-fourth. The decay ratio is the ratio of the amplitudes of two successive oscillations. It should be independent of the input to the system and should depend only on the roots of the characteristic equation for the loop.

MINIMUM ERROR INTEGRAL CRITERIA

Because the quarter decay ratio tuning parameters are not unique, a substantial research project was conducted under Professor Paul W. Murrill and Cecil L. Smith to develop tuning relationship that were unique. Their specification of the closed-loop response is basically a minimum error or deviation of the controlled variable from its set point. The error is a function of time for the duration of the response, so the sum of the error at each instant of time must be minimized.

Because the tuning relationship are intended to minimize the integral of the error, their use is referred to as minimum error integral tuning. However the integral of the error cannot be minimized directly, because a very large negative error would be the minimum.

Definition of error integrals for disturbance and for set point changes.

CONTROLLERS SYNTHESIES

In the preceding sections, we have taken the approach of tuning a feedback controller by adjusting the parameters of the proportional-integral-derivative (PID) control structure. In this section, we will take a different approach to controller design, that of controller synthesis, which is performed as follows:

Given the transfer function of the components of a feedback loop, synthesize the

controller required to produce a specific closed-loop response.

Although we get no assurance that the controller resulting from our procedure can be built in practice, we stand to gain some insight into the selection of the various controller modes and their tunings.

Let us consider the simplified block diagram in which the transfer functions of all the loop components other than the controller have been lumped into a single block

From the block diagram algebra, the transfer function of the closed loop is:

C (s) / R (s) = G_s (s) G(s )/ 1 + G_s(s) G(s)

Next we use this expression to solve for the controller transfer function

G_c(s) = 1 / G(s) . C(s) / R(s) / 1 – C(s) / R(s)

This is the controller formula synthesis. It gives us the controller transfer function G_c(s) from the process transfer function G(s) and the specified closed-loop response C(s) / R(s). The resulting controller is:

G_c(s) = 1 / G(s) . 1 / 1-1 = 1 / G(s) . 1 / 0

This says that in order to force the output to equal the set point at all times, the controller gain must be infinite. In other words, perfect control cannot be achieved with feedback control. This is because any feedback corrective action must be based on an error. The controller synthesis formula results in different combinations of closed-loop response specifications and process transfer functions.

The advantage of the controller synthesis is that it allows the engineer to achieve a specified response by adjusting a single parameter, the gain regardless of controller modes involved. The tunable gain is a disadvantage, because the formulas do not provide the “ball-park” value for it.