## 1 Introduction  
The primary structural components that experience external forces are usually modeled as continuous cantilevered beams in engineering structures such as vehicles, aircrafts, buildings, and bridges. Such structures commonly have dynamic behavior, which cannot be modeled perfectly and occur from unexpected external sources. These influences could induce undesirable vibrations, resulting in fatigue and likely severe failure. Smart structures with active control methods are one possible solution for isolating these detrimental vibrations, but the control system itself and mistuned control parameters can cause instabilities. Therefore, it is important to design control methods that are inherently insensitive to parameter changes to attenuate structural vibrations. Lead zirconate titanate (PZT) patches have a number of real-world applications for active control for reducing noise, vibration, and harshness (NVH). Some automotive parts have thin walls and can transmit unexpected noise and vibration from many sources. Thus, there have been many efforts to apply PZT patches to automotive parts such as the body panels [1], muffler system [2], and powertrain oil pan [3]. PZT patches are also used for mechanical and aerodynamic NVH reduction applications, such as aircraft fin-tips [4], rotorcraft blades [5], and aeroelastic flutter [6]. PZT patches are also used in active fault detections and energy harvesting devices. Piezoelectric shunt circuits can control vibrations from the actuator arms of hard disk drives [7], and they can be used for detecting or fixing cracked beams [8; 9]. A hybrid-type switching control method has been used in order to isolate unwanted vibration as well as to improve the energy efficiency [10]. Active vibration isolation techniques have been implemented with piezoelectric actuators and sensors for the functionally graded material beams [11]. A smart structure with shunted piezoelectric patches was constructed, and topology optimization was performed for maximizing the electromechanical coupling factor in terms of the damping efficiency, piezo patch geometries, and patch locations [12]. A visualization technique was proposed to identify and reduce the vibrational energy of a 2D plate using a constrained damping layer [13]. To achieve satisfactory vibration attenuation, the structures and vibrational responses should be analyzed. An analytical solution was introduced for a piezoelectric bimorph cantilever beam with suitable boundary conditions [14]. The Airy stress function can also be used to study other different piezoelectric cantilevers, including multilayer piezoelectric cantilevers under different loads. The active vibration control of parametrically excited systems with resonance was examined to avoid the parametric instability, and an active control system was introduced, which includes a beam subjected to an axial harmonic load [15]. An enhanced method was proposed to stabilize the system via velocity feedback and pole placement by relocating the transition curves. Finally, generally layered smart beams were observed with a one-dimensional finite element method [16]. This state space model is based on first-order shear deformation beam theory with Hermite shape functions and the effect of external electric loads. Appropriate control algorithms must be designed for individual systems to achieve desirable performance. Piezoelectric shunt circuits have been applied extensively for the semi-active control of vibrating structures, but a single piezoelectric shunt circuit can deal with only one vibration mode at a time [17]. Many methods have been explored to manage more than one mode, such as multiple piezoelectric shunt circuits [18], synchronized switch voltage damping [19], and periodic piezoelectric shunts [20]. However, these techniques are still passive methods, for which adjustment is needed in advance. Thus, a number of active control schemes have been examined.