**2. Heliostat Orientation Control**

The orientation control system is presented in Figure 1. The control system modifies the angular position of two DC motors connected to the axes of the heliostat through two worm drive mechanisms to guide the heliostat to the desired position. A microcontroller unit (MCU) calculates the position of the sun and the desired angles of the heliostat in order to reflect the solar radiation on a specific target by using the geographic position of the heliostat and the current time and date values. Afterwards, a position control algorithm calculates the error between the desired and current angular position of the heliostat axes by using two rotary encoders in order to obtain a control signal which orients the heliostat by using two motor drivers that allow the bidirectional control of the DC motors.

**Figure 1.** Orientation control diagram.

### *2.1. DC Motor Mathematical Model*

A DC motor can be described by using the equivalent model shown in Figure 2. The reduced transfer function of the armature-controlled DC motor is given by (1) [32].

$$G(s) = \frac{\Theta(s)}{V\_a(s)} = \frac{k\_m}{s[R\_a(Js+b) + k\_bk\_m]}\tag{1}$$

where *km* represents the motor torque constant and *kb* represents the back electromotive-force constant.

**Figure 2.** Equivalent model of a permanent magne<sup>t</sup> brushed DC motor.

The mathematical model of the DC motor can be estimated by using the step signal response method with the motor speed response under a fixed voltage. The transfer function of the position and speed model can be described by (2) and (3).

$$\frac{\Theta(\mathbf{s})}{V\_d(\mathbf{s})} = \frac{\frac{k\_w}{f\mathcal{R}\_d}}{\mathbf{s}\left(\mathbf{s} + \frac{b\mathcal{R}\_d + k\_p k\_w}{f\mathcal{R}\_d}\right)} = \frac{\mathcal{C}\_k}{\mathbf{s}\left(\mathbf{s} + \mathcal{C}\_p\right)}\tag{2}$$

$$\frac{\Omega(s)}{V\_a(s)} = \frac{\mathbb{C}\_k}{s + \mathbb{C}\_p} = \frac{\frac{\mathbb{C}\_k}{\mathbb{C}\_p}}{\frac{1}{\mathbb{C}\_p}s + 1} = \frac{K}{\tau s + 1} \tag{3}$$

where *Ck* and *Cp* are fixed parameters, τ represents the time constant, and *K* represents the steady-state gain of the system.

The steady-state gain is the ratio of the output and the input in steady-state [33] and is given by (4).

$$K = \frac{\alpha\_{\rm s}}{\mu\_{\rm stcp}} = \frac{\mathbb{C}\_k}{\mathbb{C}\_p} \tag{4}$$

where ω*s* represents the steady speed of the DC motor and *ustep* represents the step input signal.

Finally, the transfer function of the DC motor is given by (5).

$$G(\mathbf{s}) = \frac{\Theta(\mathbf{s})}{V\_a(\mathbf{s})} = \frac{\frac{\omega\_s}{u\_{\text{stup}}}}{s(\mathbf{s} + \frac{1}{\pi})} \tag{5}$$
