Abstract
The emission of sounds is an important means of communication between animals, which helps them to find a partner, establish their territory, and even alert other members of a population about a possible threat. In human beings, the emission of sounds allows an even more specialized communication, because it possibly render a wide exchange of ideas and knowledge. The most important process related to voice emission is the movement of vocal folds. Therefore, their proper functioning is essential for the emission of sounds. Vocal folds possess a muscular tissue, and are located inside the larynx. When air passes through them, they vibrate, thus emitting the sound by which we communicate. The vocal folds are elastic fibers that distend or relax under the action of the larynx muscles, thus modulating and modifying the sound as we speak or sing, for example. This complex process is modeled by a system of differential equations. In this paper, we present a study about asymmetric vocal folds, where a difference in their stiffness is considered. The objective of this study is to synchronize the movement of vocal folds caused by that asymmetry through the optimal linear control method, showing their functioning for different values of stiffness for each vocal fold. In this way, the control design has shown to be efficient, producing a good functioning of the vocal folds and rendering the emission of sound possible even in diseased vocal folds.
Highlights
Non-linear dynamic systems are mathematical models for several problems in physics, chemistry, biology, economy, engineering, and correlated areas of knowledge (Chavarette, Balthazar, Rafikov, & Hermini, 2009; Zhang, Liu, & Wang, 2009; Gupta, Dalal, & Mishra, 2014; 2015; Ferreira, Chavarette, & Peruzzi, 2017; Mishra, Sen, & Mohapatra, 2017). These non-linear dynamic systems are usually associated with non-linear ordinary differential equations
When studying the behavior of dynamic systems that are modeled by ordinary differential equations, some typical behaviors are found, such as periodic, chaotic, and synchronized (Rafikov, Balthazar, & Tusset, 2008; Zhang et al, 2009)
This paper aims at synchronizing the movement of vocal folds through optimal linear control (LQR), which presented satisfactory results in many cases analyzed (Berry, Herzel, Titze, Krischer, & Story, 1994; Berry, Herzel, & Titze, 1996; Giovanni et al, 1999; Giovanni, Ouakinine, & Triglia, 1999; Tusset, Baltthazar, Chavarette, & Felix, 2012; Chavarette, 2011)
Summary
Non-linear dynamic systems are mathematical models for several problems in physics, chemistry, biology, economy, engineering, and correlated areas of knowledge (Chavarette, Balthazar, Rafikov, & Hermini, 2009; Zhang, Liu, & Wang, 2009; Gupta, Dalal, & Mishra, 2014; 2015; Ferreira, Chavarette, & Peruzzi, 2017; Mishra, Sen, & Mohapatra, 2017). These non-linear dynamic systems are usually associated with non-linear ordinary differential equations. Natural systems such as fireflies blinking, cardiac pacemakers, and neurons firing are prone to synchronization (Pikovsky, Rosenblum, & Kurths, 2003)
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