# Zero Average Dynamic Controller for Speed Control of DC Motor
## 1 Introduction  
Power converters can efficiently transform electrical energy from the generation source to users and help meet electricity needs in the power grid [1]. In a direct current (DC) microgrid, numerous power converters are configured as interfaces between renewable energies and loads [2]. Microgrid is emerging as an efficient solution for the high integration of distributed renewable energies, energy storage systems, and flexible loads in modern power systems [3]. Digital signal processing (DSP) provides greater flexibility than analog designs [4]. Some advantages are that it provides a straightforward implementation of nonlinear controllers, quasi-sliding mode control (QSMC), and advanced control techniques [5]. Other advantages are that it provides low power consumption [6], reduces the use of external passive elements, gives low sensitivity to parameter changes, and allows various configuration options in pulse width modulation (PWM) [7, 8]. However, quantization and discretization can cause periodic bands [9], chaotic behavior, oscillations, subharmonics, and limit cycles [10]. Additionally, the time delays in the output of the controllers can produce instability [11]. Thus, the performance of a well-designed discretized quasi-sliding mode control (DQSMC) may be affected when executed by a digital controller [12] due to the finite sampling frequency. Nevertheless, research on DQSMC is overly limited [13]. Authors in [10] expose the limit cycles in power electronic converters controlled with DPWM (Digital Pulse Width Modulation). They propose some conditions on the control law and the quantization limits to avoid such oscillations. The Fixed Point InducControl (FPIC) technique helps stabilize orbits and chaos [14], and the parameters of the power converter can be estimated as presented in [15; 16]. The minimum sampling time and quantization resolution criteria for digital controller parameters have been established in [17]. Research has been conducted since the 1990s related to chaotic behavior in electric motors. For instance, some research papers have been presented that show stabilization of such chaotic behaviors [18]. Recently, techniques such as Zero Average Dynamics (ZAD) and FPIC have been used to control DC motors, as presented in [19; 20]. The FPIC technique allows researchers to reduce chaos and maintain a low error; however, the system is not robust and the steady-state error increases when internal parameters change [21]. The ZAD control technique has been validated in [22], both for the pulse to the center and pulse to the side. The author in [21] also proposed using the FPIC control technique to control chaotic systems applied to a DC-DC converter and DC-AC [23]. ZAD has been used to regulate different types of systems. For example, in [24], a dynamic sliding surface is used for the ZAD control technique as an alternative in a quasi-sliding mode. The control with ZAD has also been used to implement a buck converter in a Field Programmable Gate Array (FPGA) [25], applying ZAD with a pulse on the side fulfills the requirement of fixed switching frequency [26; 27]. Additionally, studies on the transition from periodicity to chaos in a buck DC-DC converter operated with the ZAD control technique have been carried out in [28; 29; 30], revealing bifurcations due to period-doubling and corner collision, as well as chaotic phenomena, chaotic bands, and their doubling. Power converters help regulate the speed of DC motors to perform high-precision tasks [31]. However, some chaotic behavior in motor speed requires special control to regulate variations and maintain the motors operating efficiently. Advanced control methods, e.g., sliding mode control (SMC) and predictive control, allow effective motor control, with high output tracking accuracy, disturbance rejection capability, and fast response.