The simplest controller is the proportional control described in Subsection 14.5.3:
T_{C}(s) =k_{p} .
The open-loop transfer function, combining the controller and the system under control, is
G(s) = T_{C}(s) T_{p}(s)=\frac{2k_{p} }{6s^{3} + 11s^{2} + 6s + 1} .
For a unit step input, the steady state error is determined according to Eq. (14.10):
e_{ss} =\underset{s\rightarrow 0 }{\lim } \frac{1(1/s)}{1 + G(s)H(s)} .
e_{ss} =\underset{s\rightarrow \infty }{\lim } \frac{1}{1 + G(s)} =\frac{1}{1 + 2k_{p} } .
This result indicates that, regardless of the value of kp, the specification for steadystate error will never be met. Therefore a different control structure is desired. In Subsection 14.5.3, the PI control law is presented. The main characteristic of this control law is that it increases the type of the system and hence improves steady state performance. The original system is a type 0 system. If the controller is a PI controller, then the combined system will be type 1, and, according to Table 14.1, the steady-state error to a step input will be zero. To achieve the desired transient response, further analysis is required.
The controller transfer function is
T_{C}(s)=k_{p} \left(1+\frac{1}{T_{i}s } \right) ,
and the combined open-loop transfer function is
G(s)=\frac{\frac{2k_{p} }{T_{i} }{\left(T_{i}s+1\right) } }{s\left(6s^{3} + 11s^{2} + 6s + 1\right) } .
As a starting point for the values of kp and Ti, the Ziegler–Nichols tuning rules are used. Figure 14.15 shows a step response of the original system, computed with MATLAB’s step command. Indicated on the plot are the values of R and Lfor this response, as defined in Fig. 14.12. From the plot in Fig. 14.15, the following characteristics can be measured:
L = 1.5 s,
R = 0.27 m/s.
The Ziegler–Nichols tuning rules for PI controllers suggest the following values for gains:
k_{p} = 2.2,
T_{i} = 4.95 s.
To test whether or not the second design criterion is satisfied with these values, the following steps in MATLAB will generate a closed-loop step response:
