Technology Aviation

Technology Aviation

Example 4.1

The given transfer function is as follows:

G(s) = G(s) = G(s) =

G(s) =

G(s) =

Now, we will use a combination of simple Matlab commands to obtain rise time, settling time and overshoot. The Matlab code will be as follows.

Code:

(File name: response.m)

sys = tf([1],[0.0001 0.0126 0.27255 1.26 1]);

step(sys)

S = stepinfo(sys,'RiseTimeLimits',[0.01,0.99])

Output:

S = Rise Time: 4.7778

Settling Time: 4.1965

Settling Min: 0.9902

Settling Max: 0.9997

Overshoot: 0

Undershoot: 0

Peak: 0.9997

Peak Time: 8.4716

Observations:

Rise Time: 4.7778

Settling Time: 4.1965

Overshoot: 0

Step response output of the given transfer function

Calculating a and L from step response of the given transfer function

From step response over Simulink we obtained the values of a= 0.11 and L = 0.16 by drawing tangents over the response curve. The Ziegler-Nichols' closed loop method is based on experiments executed on an established control loop (a real system or a simulated system), see Figure below.

Ziegler-Nichols' closed loop method

The tuning procedure is as follows:

1. Bring the process to (or as close to as possible) the speci?ed operating point of the control system to ensure that the controller during the tuning is “feeling” representative process dynamic and to minimize the chance that variables during the tuning reach limits. You can bring the process to the operating point by manually adjusting the control variable, with the controller in manual mode, until the process variable is approximately equal to the set point.

2. Turn the PID controller into a P controller by setting set Ti = 8 and Td = 0. Initially set gain Kp = 0. Close the control loop by setting the controller in automatic mode.

3. Increase Kp until there are sustained oscillations in the signals in the control system, e.g. in the process measurement, after an excitation of the system. (The sustained oscillations correspond to the system being on the stability limit.) This Kp value is denoted the ultimate (or critical) gain, Kpu. The excitation can be a step in the set point. This step must be small, for example 5% of the maximum set point range, so that the process is not driven too far away from the operating point where the dynamic properties of the process may be di?erent. On the other hand, the step must not be too small, or it may be di?cult to observe the oscillations due to the inevitable measurement noise.

It is important that Kpu is found without the control signal being driven to any saturation limit (maximum or minimum value) during the oscillations. If such limits are reached, you will ?nd that there will be sustained oscillations for any (large) value of Kp, e.g. 1000000, and the resulting Kp-value (as calculated from the Ziegler-Nichols' formulas, cf. Table 1) is useless (the control system will probably be unstable). One way to say this is that Kpu must be the smallest Kp value that drives the control loop into sustained oscillations.

4. Measure the ultimate (or critical) period Pu of the sustained oscillations.

5. Calculate the controller parameter values according to Table 1, and use these ...