**2. PMDC motor model**

Using supply current, motor speed and supply voltage relationship, we have the PMDC motor mathematical model as:

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


### **Table 1.**

*The parameters of motor and their acceptable limits of variation.*

$$\frac{di}{dt} = -\frac{R}{L}i - \frac{k\_b}{L}\omega + \frac{1}{L}u \tag{1}$$

$$\frac{d\alpha}{dt} = \frac{1}{L}k\_m i - \frac{1}{J}k\_f \alpha \tag{2}$$

Here, *R* and *L* are resistance and inductance of motor coil, *kb* is denoted as the constant of back Electromotive force, *J* is rotor inertia and *kf* is friction constant.

The representation of SS (state space) matrix of similar set of the equations is given by:

$$
\frac{d\boldsymbol{\alpha}}{dt} \begin{bmatrix} \boldsymbol{i} \\ & \boldsymbol{\alpha} \end{bmatrix} = \begin{bmatrix} -\frac{R}{L} & -\frac{K\_b}{L} \\ \frac{K\_m}{L} & -\frac{K\_f}{J} \end{bmatrix} \begin{bmatrix} \boldsymbol{i} \\ \boldsymbol{\alpha} \end{bmatrix} + \begin{bmatrix} \mathbf{1} \\ & \mathbf{L} \\ \mathbf{0} \end{bmatrix} \tag{3}
$$

$$\mathbf{y}(t) = [\mathbf{0}\ \mathbf{1}] \begin{bmatrix} & i \\ & \alpha \end{bmatrix} + [\mathbf{0}]u(t) \tag{4}$$

The overall transfer function for the speed and input voltage, is expressed as:

$$\frac{\omega(\mathfrak{s})}{\mathfrak{u}(\mathfrak{s})} = \frac{K\_m}{fL\mathbb{S}^2 + (\mathcal{J}\mathbb{R} + L\mathbb{K}\_f)\mathfrak{s} + K\_m\mathbb{K}\_b + RK\_f} \tag{5}$$

The transfer function of PMDC motor model is considered for a specified output speed and input voltage provided. The unknown motor parameters are prone to the variations due to aging of motor and its wear and tear. The controller must be built to offer resilience for each motor parameter within acceptable variation limits. The parameters of nominal motor are chosen from the standard motor of Maxon-RE35. The variation limits and motors parameters are given in **Table 1**. The proposed controller design for disturbance rejection in this work is based on the presumption that the PMDC motor internal parameter variations are within tolerable limits as specified for Maxon-RE35. The proposed Robust Controller design is discussed in subsequent section.
