Improved homoclinic predictor for Bogdanov-Takens bifurcation

An improved homoclinic predictor at a generic codim 2 Bogdanov-Takens (BT) bifucation is derived. We use the classical "blow-up" technique to reduce the canonical smooth normal form near a generic BT bifurcation to a perturbed Hamiltonian system. With a simple perturbation method, we derive explicit rst- and second-order corrections of the unperturbed homoclinic orbit and parameter value. To obtain the normal form on the center manifold, we apply the standard parameter-dependent center manifold reduction combined with the normalization, that is based on the Fredholm solvability of the homological equation. By systematically solving all linear systems appearing from the homological equation, we remove an ambiguity in the parameter transformation existing in the literature. The actual implementation of the improved predictor in MatCont and numerical examples illustrating its eciency are discussed

Stability Analysis and Robust Controller Design of Indirect Vector Controlled Induction Motor

The thesis considers stability analysis and controller design through different performance measures for indirect vector controlled induction motor (IVCIM).These problems are known to be complex due to nonlinearity, large order and multi-loop scenario. Some new approaches and results on IVCIM are proposed in this work. IVCIM dynamics is well known for having different bifurcation behavior, viz., saddle-node, Hopf, Bogdanov–Takens and Zero–Hopf bifurcations due to rotor resistance variation. These bifurcations affect the control performance and may lead to stalling or permanent damage of motor. A numerical analysis of these bifurcations for proportional integral (PI) controlled IVCIM is made in this thesis using full-order induction motor model (stator dynamics is included). This analysis aids to determine the allowable bifurcation parameter variation range as well as suitable choice of speed-loop gains to avoid these. Some new observations on the bifurcation behavior are made. Simulation and experimental results are presented validating the bifurcation behaviors. For improving dynamic performance in the presence of load torque and rotor resistance variation, a new method for designing PI gains is proposed for IVCIM. The inner-loop current PI controllers are tuned simultaneously along with the speed controller. This method is implemented using a static output feedback scheme in which iterative linear matrix inequality (ILMI) based∞control technique is employed. Such a design makes stator currents and speed response to be robust against rotor resistance and load variations. A comparison between proposed design and a conventional one is shown using simulation and experimental results that validate the superiority of the proposed approach. Owing to multi-loop and nonlinear system behavior, IVCIM dynamics is known to have coupling in between the two inner-loop stator current components (flux and torque). Such coupling affects the dynamic torque output of the motor. Decoupling of the stator currents are important for smoother torque response of IVCIM. Conventionally, additional feedforward decoupler is used to take care of the coupling that requires exact knowledge of the motor parameters and additional circuitry or signal processing. A method is proposed to design the regulating PI gains while minimizing coupling without any requirement of additional decoupler. The variation of the coupling terms for change in load torque is considered as the performance measure. The same ILMI based∞control design approach is used to obtain the controller gains. A comparison between the conventional feedforward decoupling and proposed decoupling scheme is presented through simulation and experimental results that establish the effectiveness of the proposed method riding over its simplicity. Finally, since the PI controller can yield limited performance, a dynamic controller is designed for the IVCIM drive system. In the design process, iron-loss dynamics are incorporated into induction motor model to fetch benefit through better performance. A sequential design method is used for the controller design in which, first, the inner-loop controllers are designed. The designed inner-loop controllers is then used for designing the outer speed-loop controller. The proposed design employs ILMI based∞control design for dynamic output feedback controller that makes stator currents and speed response to be robust against disturbances. A comparison among proposed dynamic controller design, PI controller and compensator design is shown using simulation and experimental results demonstrate enhanced performance of the proposed controller and suitability for industrial purpose

PI-controlled bioreactor as a generalized Lienard system

It is shown that periodic orbits can occur in Cholette's bioreactor model working under the influence of a PI-controller. We find a diffeomorphic coordinate transformation that turns this controlled enzymatic reaction system into a generalized Lienard form. Furthermore, we give sufficient conditions for the existence and uniqueness of limit cycles in the new coordinates. We also perform numerical simulations illustrating the possibility of the existence of a local center (period annulus). A result with possible practical applications is that the oscillation frequency is a function of the integral control gain parameterComment: 15 pages, 5 figures, accepted version at Computers & Chem. En

Dynamics and Stability of Permanent-Magnet Synchronous Motor

The aim of this article is to explore the dynamic characteristics and stability of the permanent-magnet synchronous motor (PMSM). PMSM equilibrium local stability condition and Hopf  bifurcation condition, pitchfork bifurcation condition, and fold bifurcation condition have been derived by using the Routh-Hurwitz criterion and the bifurcation theory, respectively. Bifurcation curves of the equilibrium with single and double parameters are obtained by continuation method. Numerical simulations not only confirm the theoretical analysis results but also show one kind of codimension-two-bifurcation points of the equilibrium. PMSM, with or without external load, can exhibit rich dynamic behaviors in different parameters regions. It is shown that if unstable equilibrium appears in the parameters regions, the PMSM may not be able to work stably. To ensure the PMSMs work stably, the inherent parameters should be designed in the region which has only one stable equilibrium

PI-controlled bioreactor as a generalized Liénard system

"It is shown that periodic orbits can emerge in Cholette’s bioreactor model working under the influence of a PI-controller. We find a diffeomorphic coordinate trans-formation that turns this controlled enzymatic reaction system into a general-ized Lie´nard form. Furthermore, we give sufficient conditions for the existence and uniqueness of limit cycles in the new coordinates. We also perform numerical simu-lations illustrating the possibility of the existence of a local center (period annulus). A result with possible practical applications is that the oscillation frequency is a function of the integral control gain parameter.

Computational dynamical systems analysis : Bogdanov-Takens points and an economic model

The subject of this thesis is the bifurcation analysis of dynamical systems (ordinary differential equations and iterated maps). A primary aim is to study the branch of homoclinic solutions that emerges from a Bogdanov-Takens point. The problem of approximating such branch has been studied intensively but neither an exact solution was ever found nor a higher-order approximation has been obtained. We use the classical ``blow-up'' technique to reduce an appropriate normal form near a Bogdanov-Takens bifurcation in a generic smooth autonomous ordinary differential equations to a perturbed Hamiltonian system. With a regular perturbation method and a generalization of the Lindstedt-Poincare' perturbation method, we derive two explicit third-order corrections of the unperturbed homoclinic orbit and parameter value. We prove that both methods lead to the same homoclinic parameter value as the classical Melnikov technique and the branching method. We show that the regular perturbation method leads to a ``parasitic turn'' near the saddle point while the Lindstedt-Poincare' solution does not have this turn, making it more suitable for numerical implementation. To obtain the normal form on the center manifold, we apply the standard parameter dependent center manifold reduction combined with the normalization, using the Fredholm solvability of the homological equation. By systematically solving all linear systems appearing from the homological equation, we correct the parameter transformation existing in the literature. The generic homoclinic predictors are applied to explicitly compute the homoclinic solutions in the Gray-Scott kinetic model. The actual implementation of both predictors in the MATLAB continuation package MatCont and five numerical examples illustrating its efficiency are discussed. Besides, the thesis discusses the possibility to use the derived homoclinic predictor of generic ordinary differential equations to continue the branches of homoclinic tangencies in the Bogdanov-Takens map. The second part of this thesis is devoted to the application of bifurcation theory to analyze the dynamic and chaotic behaviors of a nonlinear economic model. The thesis studies the monopoly model with cubic price and quadratic marginal cost functions. We present fundamental corrections to the earlier studies of the model and a complete discussion of the existence of cycles of period 4. A numerical continuation method is used to compute branches of solutions of period 5, 10, 13 and 17 and to determine the stability regions of these solutions. General formulas for solutions of period 4 are derived analytically. We show that the solutions of period 4 are never linearly asymptotically stable. A nonlinear stability criterion is combined with basin of attraction analysis and simulation to determine the stability region of the 4-cycles. This corrects the erroneous linear stability analysis in previous studies of the model. The chaotic and periodic behavior of the monopoly model are further analyzed by computing the largest Lyapunov exponents, and this confirms the above mentioned results