Backlash is one of several discontinuities found in different kinds of systems. It can be found in actuators of different types, such as mechanical and hydraulic, giving way to unwanted effects in the system behavior. Proportional integral (PI) loop shaping control design implementing a describing function to find the limit cycle oscillations and the appropriate control gain selection by particle swarm optimization is developed. Therefore, a frequency domain approach is implemented for the control of nonlinear system of any kind such as robotics, mechatronics, and other kind of mechanisms, electrical motors, etc. Finally, in order to corroborate the theoretical background explained in this chapter, the stabilization of a cart-pendulum system with the proposed control strategy is shown.
Top1. Introduction
Recent decades have witnessed many important developments related to the design of nonlinear systems for many practical applications. Several inspiring approaches have been proposed, such as optimal control, nonlinear feedback control, adaptive control, sliding mode control, nonlinear dynamics, chaos control, chaos synchronization control, fuzzy logic control, fuzzy adaptive control, fractional order control, and robust control and their integrations (Azar & Vaidyanathan, 2015a,b,c, 2016; Azar & Zhu, 2015; Meghni et al, 2017a,b,c; Boulkroune et al, 2016a,b; Ghoudelbourk et al., 2016; Azar & Serrano, 2015a,b,c,d, 2016a,b, 2017; Azar et al., 2017a,b,c,d; Azar 2010a,b, 2012; Mekki et al., 2015; Vaidyanathan & Azar, 2015a,b,c,d, 2016a,b,c,d,e,f,g, 2017a,b,c; Zhu & Azar, 2015; Grassi et al., 2017; Ouannas et al., 2016a,b, 2017a,b,c,d,e,f,g,h,I,j; Singh et al., 2017; Vaidyanathan et al, 2015a,b,c; Wang et al., 2017; Soliman et al., 2017; Tolba et al., 2017).
Backlash is a phenomenon found in different kinds of actuators such as mechanical and hydraulic, generally it occurs when the contact of two mating gears does not match and this give way to many unwanted effects on the systems provoking problems to the whole mechanical system. There are different kinds of mechanical systems to be stabilized (Azar & Serrano, 2014, 2015a,b,c,d, 2016a,b, 2017; Silva & Erraz, 2006), where the stabilization and control of mechanical system with backlash is done by several techniques such as conventional PID control and PI loop shaping control obtaining the optimal outcomes by the proposed strategy. Then another example can be found in Azar & Serrano (2015c) where the stabilization of the Furuta pendulum is done by a second order sliding mode control and adaptive sliding mode control considering that is important to mention these kinds of mechanical systems because of the proposed strategy shown in this chapter can be implemented for any kind of mechanism. Many researchers have proposed several solutions respect to the control and stability issues of these systems with input nonlinearities; taking into account that due to the complexity of the nonlinear model, traditional control strategies fail in most of the cases, and therefore it is necessary to design nonlinear control strategies or implement modified traditional ones. The objective of this chapter is to explain diverse control strategies for systems with this kind of nonlinearity. The intention is to show some nonlinear control techniques that have been developed and propose different approaches to solve this problem.