Introduction
Specific nonlinear dynamic system behavior is complex anomalies that can induce qualitative changes in the system's steady-state trajectory such as chaos, bifurcations, co-existing attractors, and fractal basin boundaries [
1‐
7]. The research field of chaotic system analysis and control has seen rapid development in recent years. The chaos appears in many mechanical and electrical systems such as lasers, nonlinear optical systems, optimization techniques, biological systems, chemical reactions, cancer treatments, fluids flow, moreover, many other applications [
8‐
16]. Practically, the chaos and bifurcation are harmful to these systems as they can lead them to be unstable or make undesirable behavior, so control techniques and methodologies are required to reduce or eliminate the harmful chaotic effect. Consequently, studying chaos behavior and suppressing it has been the major target of researchers for practical nonlinear systems. Consequently, there are numerous methods to control chaotic systems, such as the (Ott, Grebogi and Yorke) OGY method [
31], feed-back linearization method [
32‐
34], a real-time cycle to cycle variable slope compensation method [
35], time-delay feedback control [
36], developing Floquet theory [
26,
30,
37‐
39], fuzzy control [
41,
43,
45,
46].
The stability boundary of the chaotic system relies on invariant sets like equilibrium points, periodic, quasi-periodic, and chaotic orbits are representative. In this work, the periodic and chaotic orbits are investigated to study the stability behavior of the dynamic systems. Periodic orbits are powerful tools in studying and analyzing the dynamic systems as they provide a periodic solution which shows a closed orbit in phase-space. While the chaotic orbits have an unlimited number of points in the phase portrait and its trajectory is a random-like but bounded behavior and does not intersect with each other.
The popular methods used for analyzing chaos are the Floquet theory and the Lyapunov exponent method which depends on the Poincaré map. Poincaré map creates a complex structure with a discrete state-space which is a smaller dimension than the original continuous dynamic scheme. The stability of the Poincaré map depends on the map’s Jacobian matrix, but the calculation process of the Jacobian matrix may be complex. This interpretation of the Poincaré map is seldom used as it requires developing information for a limited cycle location [
24,
25]. While, the Floquet theory depends on the State Transition Matrix (STM) for the full cycle [
34,
35,
37‐
39]. Both methods give the same results; however, the Floquet theory fits well with mechanical switching systems and simpler to be implemented as compared to the Lyapunov exponent method [
25‐
30].
Chaotic systems are deterministic, nonlinear, and very sensitive to the initial condition. Accordingly, long-term prediction of chaotic systems becomes impossible, on the other hand, they are controllable systems [
17‐
19]. While, the change in specific behavior of the system is known as a bifurcation [
19,
20]. The bifurcation phenomenon describes the fundamental alteration in the dynamics of nonlinear systems under parameter variation. As the theory of bifurcation is a tool to help to understand equilibrium loss and its consequences for complex behavior. There are many types of periodic orbit local bifurcation such as period-doubling bifurcation, and Hopf bifurcation [
20‐
23].
DC (direct current) motor is used in many applications and nearly most mechanical movement that can be observed around us. As it is found in several domestic appliances such as HD cameras, smart toys, mixing machines, and in advanced orthopedic bone drilling [
47‐
51]. Recently, the analysis of the non-linear characteristic of an electric DC motor has become an emerging research topic. The drive system consists of a converter and a controller which produce pulse width modulation (PWM) to adjust the voltage value of the DC motor. PWM’s switching action makes the entire drive system to be time-varying and nonlinear, because the operation of the system during the ON state differs from its operation in the OFF state. The nominal steady state of the PMDC drive is the period-1 orbit, but altering some drive parameters will lead to a new attracting orbit that is periodic, quasi-periodic, or chaotic [
39].
Zadeh [
40] introduced the fundamental of the fuzzy logic controller (FLC) in 1965, many researchers dedicated themselves to this topic to find innovative strategies for improving the performance of the FLC system and ensuring its reliability. So, FLC can eliminate the non-linearity effects of a DC motor simply and comfortably. Designing FLC also does not need a systematic procedure or a mathematical model of a system. Besides, it improves performance and provides superior outcomes than other techniques. Hence, FLC gives better results with position control and DC/DC converters [
42] and [
44], stabilizes the induction motor [
45], removes chaos in converters [
41], and controls complicated, undefined chaotic systems Parameters [
43,
46].
Most researchers deal with the impact of changing the input voltage of PMDC motors or the controller gains on the chaos behavior [
20,
30,
35,
41]. Another research gap is that most control techniques deal with eliminating chaos and bifurcation without taking into consideration the performance of the system [
32,
33,
37‐
39]. Therefore, the contribution of this work is to investigate another important factor in the PMDC motor which is the impact of changing the load torque on chaos behavior with taking into consideration the performance of the system. Through this work, the nominal state for the PMDC motor was explored while changing load torque, which results in losing stability and changing the trajectory from period-1 to period-2 orbit. Also, changing the load torque causes chaos. So, two techniques were used to overcome these problems. The first technique is developing Floquet theory, where the result shows that the system is still in period-1 (nominal state) orbit with changing the load torque and the system suffers from overshoot and oscillations. The second technique is the Proportional Integral (PI) fuzzy controller, where the simulation result shows that the system also still in a nominal state without overshoot or oscillations and eliminates the steady-state error. Simulation results were performed using MATLAB / SIMULINK software.
The outline of the paper is as follows:—the mathematical model, dynamic behavior of PMDC drive, and analysis of the stability of period-1 orbit are devoted in "
Preliminaries", while "
The controller" introduces the control of the system using a fuzzy PI controller and the next section summarizes the conclusion of the results.
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