**3. H∞ based robust controller design for disturbance rejection**

The proposed controller design model should meet the desired specifications within bounds of internal parameter variations. Figure below shows the variation in the gain of PMDC plant against variations of internal parameters. Similar variations are represented in the domain of frequency via bode plot, that are shown in **Figures 1** and **2**. Henceforth, PID controller is designed to address variation in the model and provide desired specifications for closed loop performance.

This chapter discusses the target specifications for better disturbance rejection and simultaneously set the point tracking. Here, we assume that the control bandwidth, exhibits increasing slope at the crossover frequency compared to disturbance rejection, allowing better gain within the bandwidth. The higher slope is described as lesser phase margin resulting in acceptable overshoot in response to set point. In order to match competing requirements of the tracking rejection and disturbance, the PID controller with 2-DOF (Degree of Freedom) is used as transfer function.

$$u = K\_P(br - \jmath) + \frac{K\_i}{\mathfrak{s}}(r - \jmath) + \frac{K\_D \mathfrak{s}}{1 + T\_f \mathfrak{s}}(cr - \jmath) \tag{6}$$

**Figure 1.** *The variations in PMDC gain of motor plant for uncertain and nominal internal parameter variations.*

**Figure 2.** *Bode plot of plant model subjected to the internal parameter variations.*

*PID Gain Tuning for Robust Control of PMDC Motor for External Disturbance Rejection with… DOI: http://dx.doi.org/10.5772/intechopen.102546*

**Figure 3.** *Block diagram with external disturbances (load variations).*

2-DOF (Degree of Freedom) PID controllers contain weighing coefficients associated with derivative and proportional terms. These weighing coefficients facilitate effective disturbance rejection restraining maximization of the over-shoot under optimal conditions. The PID controller of 2-DOF works better in moderating the changes arising in the reference signal or control-signal.

The close loop control system is considered as shown in **Figure 3**. Here, the external disturbances such as load variations on motor shafts and its torque variations are added that influences on the motor parametric variations.
