## 1 Introduction  
There are two broad groups of dc (direct current)) motors: brushed dc motors and brushless dc motors. The first group has brushes and a commutator, while the second one does not. The advantage of brushed dc motors is that they can be easily controlled. Because of this, they have been the most popular motor type in the past, and they continue to be used in many applications today. Brushed dc motors can be used for either low or high power applications. Usually, low-power brushed dc motors are used in domestic applications, although they may also be used in some industry sectors, such as the automotive industry. On the other hand, high-power brushed dc motors are mainly used in industrial applications. There are two types of observers for detecting the speed of a brushed dc motor: conventional observers and sensorless observers. Conventional observers are devices such as encoders, tachometers, and Hall sensors, among others. They are independent devices attached to the motor shaft. Consequently, they can be a source of failures and they increase cost and maintenance [1]. Sensorless observers are algorithms that estimate speed and/or position of motors from a voltage or current signal of the motor. The main advantages of sensorless observers include decreased maintenance, reduced number of connections, lower system cost, and easier miniaturization of the system. The use of sensorless observers for measuring motor speed is divided into two groups [2]: those based on the dynamic model, and those based on the ripple component:* The observers based on the dynamic model estimate the speed using a model of the brushed dc motor. A linear model is usually used [3; 4], but this type of model has the problem that the parameters used, such as resistance, inductance, and back electromotive force (back-EMF), change under different operating conditions [5; 6]. In this situation, the parameters are adjusted for one specific work condition and they are not modified. Consequently, the observers work correctly when operating conditions are similar to the specified conditions, but they do not work as well when the operating conditions differ from the specified conditions. A first solution for this is to dynamically estimate the parameters of the model [7; 8; 9], but this generates a complex model that is usually nonlinear. A second solution is to use a nonlinear model of the brushed dc motor [10; 11; 12], or a technique that indirectly models the motor, such as Neural Networks [13; 14] and the Kalman filter [15; 16]. The problem with these solutions is that they have a high computational cost, and the estimation of the parameters used is not an easy task. A third solution is to use a method with a simple model that is only valid in a specific condition. For example, the method can model the motor only when the supply turns off [17]. The problem with this method is that the speed can only be measured if the condition is met. * The observers based on the ripple component use a part of the ac current signal, which is called the ripple component [2; 18]. This component is the result of two effects. The first effect is produced because the electromotive force induced in each coil has a sinusoidal shape, and it is not perfectly rectified by the brush-commutator system [19]. The second effect is produced because the brushes in the commutator sometimes short two adjacent commutator segments, joining the terminals of the coils connected to these commutator segments, resulting in peaks in the current [20]. Both effects produce undulations in the current. The number of undulations per second, or ripple frequency, is related to motor speed [21]. Many methods in the literature use the ripple component [2; 22; 23; 24; 25; 18]. The detection of undulations in sensorless observers based on the ripple component can be difficult because of noise [26]. Noise produces ghost undulations, which are unobservable undulations of the ripple component.