Ahmad Harb

On nonlinear control design for autonomous chaotic systems of integer and fractional orders

Abstract In this paper, we address the problem of chaos control for autonomous nonlinear chaotic systems. We use the recursive “backstepping” method of nonlinear control design to derive the nonlinear controllers. The controller effect is to stabilize the output chaotic trajectory by driving it to the nearest equilibrium point in the basin of attraction. We study two nonlinear chaotic systems: an electronic chaotic oscillator model, and a mechanical chaotic “jerk” model. We demonstrate the robustness of the derived controllers against system order reduction arising from the use of fractional integrators in the system models. Our results are validated via numerical simulations.

Nonlinear chaos control in a permanent magnet reluctance machine

Abstract The dynamics of a permanent magnet synchronous machine (PMSM) is analyzed. The study shows that under certain conditions the PMSM is experiencing chaotic behavior. To control these unwanted chaotic oscillations, a nonlinear controller based on the backstepping nonlinear control theory is designed. The objective of the designed control is to stabilize the output chaotic trajectory by forcing it to the nearest constant solution in the basin of attraction. The result is compared with a nonlinear sliding mode controller. The designed controller that based on backstepping nonlinear control was able to eliminate the chaotic oscillations. Also the study shows that the designed controller is mush better than the sliding mode control.

BIFURCATIONS IN A POWER SYSTEM MODEL

Bifurcations are performed for a power system model consisting of two generators feeding a load, which is represented by an induction motor in parallel with a capacitor and a combination of constant power and impedance PQ load. The constant reactive power and the coefficient of the reactive impedance load are used as the control parameters. The response of the system undergoes saddle-node, subcritical and supercritical Hopf, cyclic-fold, and period-doubling bifurcations. The latter culminate in chaos. The chaotic solutions undergo boundary crises. The basin boundaries of the chaotic solutions may consist of the stable manifold of a saddle or an unstable limit-cycle. A nonlinear controller is used to control the subcritical Hopf and the period-doubling bifurcations and hence mitigate voltage collapse.

Controlling Hopf bifurcation and chaos in a small power system

Abstract For the power systems, the stabilization and tracking of voltage collapse trajectory, which involves severe nonlinear and nonstationary (unstable) features, is somewhat difficult to achieve. In this paper, we choose a widely used three-bus power system to be our case study. The study shows that the system experiences a Hopf bifurcation point (subcritical point) leads to chaos throughout period-doubling route. A model-based control strategy based on global state feedback linearization (GLC) is applied to the power system to control the chaotic behavior. The performance of GLC is compared with that for a nonlinear state feedback control.

Chaos control of third-order phase-locked loops using backstepping nonlinear controller

Abstract Previous study showed that a third-order phase-locked loop (PLL) with sinusoidal phase detector characteristics experienced a Hopf bifurcation point as well as chaotic behavior. As a result, this behavior drives the PLL to the out-of-lock (unstable) state. The analysis was based on a modern nonlinear theory such as bifurcation and chaos. The main goal of this paper is to control this chaotic behavior. A nonlinear controller based on the theory of backstepping is designed. The study showed the effectiveness of the designed nonlinear controller in controlling the undesirable unstable behavior and pulling the PLL back to the in-lock state.

Chaos and bifurcation in a third-order phase locked loop

Abstract The chaos induced in a new type of phase locked loop (PLL) having a second-order loop filter is investigated. The system under consideration is modeled as a third-order autonomous system with sinusoidal phase detector characteristics. The modern of nonlinear theory such as bifurcation and chaos is applied to a third-order of PLL. A method is developed to quantitatively study the type of bifurcations that occur in this type of PLL’s. The study showed that PLL experiencing a Hopf bifurcation point as well as chaotic behaviour. The method of multiple scales is used to find the normal form near the Hopf bifurcation point. The point is found to be supercritical one.

Nonlinear control of permanent magnet stepper motors

Abstract The nonlinear dynamics of a permanent magnet stepper motor is studied by means of modern nonlinear theories such as bifurcation and chaos. A three-phase stepper motor is considered as a case study in this paper. The study shows that the system experiences a dynamic bifurcation (Hopf bifurcation) at high frequencies. Since this kind of motors is widely used in some important applications such as printers, disk drives, process control systems, X–Y records, and robotics, controlling such instabilities is the main concern of this paper. A nonlinear robust model-reference controller is introduced. The study shows how to stabilize the system, while having a satisfactory performance, even in the case when some of the motor parameters were uncertain.

Application of Bifurcation Theory to Subsynchronous Resonance in Power Systems

A bifurcation analysis is used to investigate the complex dynamics of a heavily loaded single-machine-infinite-busbar power system modeling the characteristics of the BOARDMAN generator with respect to the rest of the North-Western American Power System. The system has five mechanical and two electrical modes. The results show that, as the compensation level increases, the operating condition loses stability with a complex conjugate pair of eigenvalues of the Jacobian matrix crossing transversely from the left- to the right-half of the complex plane, signifying a Hopf bifurcation. As a result, the power system oscillates subsynchronously with a small limit-cycle attractor. As the compensation level increases, the limit cycle grows and then loses stability in a secondary Hopf bifurcation, resulting in the creation of a two-period quasiperiodic subsynchronous oscillation, a two-torus attractor. On further increases of the compensation level, the quasiperiodic attractor collides with its basin boundary, resulting in the destruction of the attractor and its basin boundary in a bluesky catastrophe. Consequently, there are no bounded motions. The results show that adding damper windings may induce subsynchronous resonance.

TRANSMISSION PLANNING USING ADMITTANCE APPROACH AND QUADRATIC PROGRAMMING

ABSTRACT A method for transmission networks planning is proposed using quadratic programming and the admittance approach. The cost of investment and cost of losses, load flow and security constraints and the interest and inflation rates are included. The developed method can be used for static and dynamic modes of transmission planning.

A bifurcation analysis of subsynchronous oscillations in power systems

A bifurcation analysis is used to investigate the complex dynamics of a heavily loaded turbine-generator system connected to an infinite busbar through a series capacitor-compensated transmission line. It reveals the existence of self-excited subsynchronous torsional oscillations of a 5-mass rotor, which may eventually lead to the destruction of the shaft or the loss of synchronism of the generator. Specifically, we show that, as the capacitor-compensation value increases and reaches a critical value, called supercritical Hopf bifurcation, the system around the operating point undergoes small single-period oscillations with constant amplitude. This in turn results in a small limit-cycle attractor. As the compensation level increases further, the amplitude of oscillation grows until a secondary Hopf bifurcation is reached. There, the oscillations characterize themselves by two incommensurate periods and bounded amplitudes, signifying the transformation of the limit cycle into two-period quasiperiodic motion called a two-torus attractor. When the capacitor-compensation level passes a third critical value, the amplitude of oscillations becomes unbounded following the destruction of the two-torus attractor and its basin of attraction in a so-called bluesky catastrophe. Interestingly, this scenario repeats itself in the vicinity of three supercritical Hopf bifurcations. The bifurcation analysis is validated with numerical solutions of the differential equations that govern the power system.