Question 14.4: Although the previous example portrays the manner in which c...

In this example, the problem of Example 14.3 is solved by use of aPIDcontroller, but a more directed approach is used to select the parameters. Recall that a PID controller gives rise to a second-order polynomial in the numerator of the open-loop transfer function:

G_{(s)}=\frac{\frac{12k_{\rho } }{T_{i} }(T_{i}T_{d}s^{2}+T_{i}s+1 ) }{s(6s^{3}+11s^{2}+6s+1 ) }  .

The second-order numerator means that there are two zeros of the open-loop transfer function. If those zeros are placed (by the appropriate selection of T_{i} and T_{d}) in such a way that they coincide with poles of the transfer function, those poles will, in effect, be canceled. As long as the poles that are canceled are not the fastest poles in the system, then overall dynamic performance will be improved. Further, as long as the pole at the origin is not the one canceled, then the appropriate steady-state performance will not be compromised as well. The poles of this transfer function are

0,

−0.3333,

−0.5,

−1.0.

Hence, if the controller parameters are chosen such that the zeros correspond with the middle two poles, dynamic performance will be improved without sacrificing steadystate behavior.

The second-order polynomial whose roots are −1/3 and −1/2 is

6s^{2}+5s+1 .

When the coefficients are equated with the coefficients of the second-order polynomial in the numerator of G(s), it is easy to solve for T_{i} and T_{d}:

Now the only remaining task is to choose the proportional gain k_{p}. Again, a trialand- error approach could be used, this time much more easily because there is only one parameter to vary and there is a high probability that one could achieve nearly optimal performance for these choices of T_{i} and T_{d}. However, the root-locus method provides a more systematic approach.

Begin with the new open-loop transfer function based on the new values of T_{i} and T_{d}:

G_{(s)}=\frac{0.4k_{p}(6s^{2}+5s+1) }{s(6s^{3}+11s^{2}+6s+1 ) }  .

Now generate the root-locus plot for this transfer function and find the highest value of k_{p} for which the system shows no overshoot. In MATLAB, the root-locus plot of this transfer function is easily generated, as shown in Fig. 14.20.

The plot is a little difficult to interpret because so much of the loci sit on the axes. The ×’s represent the four open-loop poles (at 0, −0.333, −0.5, and −1). The circles represent the two zeros (chosen to cancel the poles at −0.333 and −0.5). The two loci begin at the other poles (at k_{p} = 0) and move toward each other until they meet at 0.5. At that point, the loci depart from the real axis and move out vertically as k_{p} approaches infinity.

The point of departure corresponds to a maximum value of k_{p} for which there is no overshoot. The value of gain that corresponds to zero overshoot is 0.625. Summarizing, the PID parameters that should satisfy the design criteria are

Figure 14.21 shows the step response for the system obtained by use of these parameters. When compared with the step response of Fig. 14.19, the step response of Fig. 14.21 can be seen to be much smoother, and reaches steady state in approximately the same time.

One final note is in order before leaving this example. The astute reader will note that the much-touted Ziegler–Nichols tuning rules suggest a set of gains that appear quite far from those that satisfy our design requirements. The reason lies in the assumptions under which those tuning rules were developed. In particular, the system is assumed to be dominated by a time delay and a first-order response. The system in this example is a third-order system with no delay. At the time Ziegler and Nichols did their work, the chemical process industry dominated the field of industrial control systems and the model they assumed was a very good choice for a wide variety of systems under study. Finally, however, it is important to point out that the tuning rules provided gains that led to a stable response and provided a much better starting point for the design work than did a random combination of gains.