# A. James Clark School of Engineering

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### Browsing A. James Clark School of Engineering by Author "Abed, Eyad H."

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- ItemActive Stabilization of Rotating Stall: A Bifurcation-Theoretic Approach(1992) Liaw, Der-Cherng; Abed, Eyad H.; ISRActive control of the onset of stall instabilities in axial flow compressors is pursued using bifurcation analysis of a dynamical model proposed by Moore and Greitzer (1986). This model consists of three ordinary differential equations with state variables being the mass flow rate, pressure rise, and the amplitude of the first harmonic mode of the asymmetric component of the flow. The model is found to exhibit a stationary (pitchfork) bifurcation at the inception of stall, resulting in hysteresis. Using throttle opening as a control, analysis of the linearized system at stall shows that the critical mode (zero eigenvalue) is unaffected by linear feedback. Hence, nonlinear tools must be used to achieve stabilization. A quadratic feedback control law using the measurement of asymmetric dynamics is proposed which stabilizes the bifurcation and eliminates the undesirable hysteretic behavior.
- ItemAnalysis and Control of Period Doubling Bifurcation in Buck Converters Using Harmonic Balance(1998) Fang, Chung-Chieh; Abed, Eyad H.; ISRPeriod doubling bifurcation in buck convertersis studied by using the harmonic balance method.A simple dynamic model of a buck converter in continuous conduction modeunder voltage mode or current mode controlis derived.This model consists of the feedback connection of a linear system and a nonlinear one.An exact harmonic balance analysis is usedto obtain a necessary and sufficient condition fora period doubling bifurcation to occur.If such a bifurcation occurs,the analysis also provides information on its exact location.Using the condition for bifurcation,a feedforward control is designed thateliminates a period doubling bifurcation.This results in a wider range of allowed source voltage,and also in improved output voltage regulation.
- ItemApplication of Center Manifold Reduction to System Stabilization(1991) Liaw, Der-Cherng; Abed, Eyad H.; ISRThe Center Manifold Theorem is applied to the local feedback stabilization of nonlinear systems in critical cases. The paper addresses two particular critical cases, for which the system linearization at the equilibrium point of interest is assumed to possess either a simple zero eigenvalue or a complex conjugate pair of simple, pure imaginary eigenvalues. In either case, the noncritical eigenvalue are taken to be stable. The results on stabilizability and stabilization are given explicitly in terms of the nonlinear model of interest in its original form, i.e., before reduction to the center manifold. Moreover, the formulation given in this paper uncovers connections between results obtained using the center manifold reduction and those of an alternative approach.
- ItemBifurcation Analysis of Nonuniform Flow Patterns in Axial-Flow Gas Compressors(1992) Adomaitis, Raymond A.; Abed, Eyad H.; ISRWe study the transition from steady, spatially uniform-flow to nonuniform and time-dependent gas axial velocity profiles in an axial flow compression system. Local bifurcation analysis of the uniform-flow solution reveals a series of bifurcations to traveling waves of different mode number as a function of throttle opening. The number of bifurcating modes is found to depend on the gas viscosity parameter, an effect introduced in this work. Using the local approximations of the bifurcating solutions as starting points of our numerical analysis, we uncover a complicated scenario of secondary bifurcations ultimately resulting in parameter ranges where locally asymptotically stable stalled-flow solutions of different mode number coexist.
- ItemBifurcation Analysis of Surge and Rotating Stall in Axial Flow Compressors(1990) Abed, Eyad H.; Houpt, Paul K.; Hosny, Wishaa M.; ISRIn this paper, the surge and rotating stall post instability behaviors of anial flow compressors are investigated from a bifurcation-theoretic perspective. A sequence of local and global bifurcations of the nonlinear system dynamics is uncovered. This includes a previously unknown global bifurcation of a pair of large amplitude periodic solutions. Resulting from this bifurcation are a stable oscillation ("surge") and an unstable oscillation ("antisurge"). The latter oscillation is found to have a deciding significance regarding post-instability behavior experienced by the compressor. These results are used to reconstruct Greitzer's (1976) findings regarding the manner in which post-instability behavior depends on system parameters. Moreover, the results provide significant new insight deemed valuable in the prediction, analysis and control of stall instabilities in gas turbine jet engines.
- ItemBifurcation Control of chaotic Dynamical Systems(1992) Wang, H.O.; Abed, Eyad H.; ISRA nonlinear system which exhibits bifurcations, transient chaos, and fully developed chaos is considered, with the goal of illustrating the role of two ideas in the control of chaotic dynamical systems. The first of these ideas is the need for robust control, in the sense that, even with an uncertain dynamic model of the system, the design ensures stabilization without at the same time changing the underlying equilibrium structure of the system. Secondly, the paper shows how focusing on the control of primary bifurcations in the model can result in the taming of chaos. The latter is an example of the 'bifurcation control' approach. When employed along with a dynamic feedback approach to the equilibrium structure preservation issue noted above, this results in a family of robust feedback controllers by which one can achieve various types of 'stability' for the system.
- ItemBifurcation Control of Nonlinear Systems(1990) Abed, Eyad H.; Fu, Jyun-Horng; Lee, Hsien-Chiarn; Liaw, Der-Cherng; ISRBifurcation control is discussed in the context of the stabilization of high angle-of-attach flight dynamics. Two classes of stabilization problems for which bifurcation control is useful are discussed. In the first class, which is emphasized in this presentation, a nonlinear control system operates at an equilibrium point which persists only under very small perturbations of a parameter. Such a system will tend to exhibit a jump, or divergence, instability in the absence of appropriate control action. In the second class of systems, an instance of which arises in a tethered satellite system model [14], eigenvalues of the system linearization appear on (or near) the imaginary axis in the complex plane, regardless of the values of system parameters or admissible linear feedback gains.
- ItemBifurcations, Chaos and Crises in Power System Voltage Collapse(1992) Wang, Hua O.; Abed, Eyad H.; Hamdan, Anan M. A.; ISRBifurcations occurring in power system models exhibiting voltage collapse have been the subject of several recent studies. Although such models have been shown to admit a variety of bifurcation phenomena, the view that voltage collapse is triggered by possibly the simplest of these, namely by the (static) saddle node bifurcation of the nominal equilibrium, has been the dominant one. The authors have recently shown that voltage collapse can occur "prior" to the saddle node bifurcation. In the present paper, a new dynamical mechanism for voltage collapse is determined: the boundary crisis of a strange attractor or synonymously a blue sky bifurcation. This determination is reached for an example power system model akin to one studied in several recent papers. The identified mechanism for voltage collapse amounts to the disappearance of a strange attractor through collision with a coexisting saddle equilibrium point. This mechanism results in solution trajectories containing both an oscillatory component (as predicted by recent analytical work), and a sharp, steady drop in voltage (as observed in the field). More generally, blue sky bifurcations (not necessarily chaotic) are identified as important mechanisms deserving further consideration in the study of voltage collapse.
- ItemBorder Collision Bifurcation Control of Cardiac Alternans(2003) Hassouneh, Munther A.; Abed, Eyad H.; Abed, Eyad H.; ISRThe quenching of alternans is considered using a nonlinear cardiac conduction model. The model consists of a nonlinear discrete-time piecewise smooth system. Several authors have hypothesized that alternans arise in the model through a period doubling bifurcation. In this work, it is first shown that the alternans exhibited by the model actually arise through a period doubling border collision bifurcation. No smooth period doubling bifurcation occurs in the parameter region of interest. Next, recent results of the authors on feedback control of border collision bifurcation are applied to the model, resulting in control laws that quench the bifurcation and hence result in alternan suppression.
- ItemClosed-Loop Monitoring Systems for Detecting Incipient Instability(1998) Kim, Taihyun; Abed, Eyad H.; ISRMonitoring systems are proposed for the detection of incipientinstability in uncertain nonlinear systems. The work employsgeneric features associated with the response to noise inputsof systems bordering on instability. These features, called "noisy precursors" in the work of Wiesenfeld, also yield information onthe type of bifurcation that would be associated with thepredicted instability. The closed-loop monitoring systems proposedin the paper have several advantages over simple open-loop monitoring.The advantages include the ability to influence the frequencies atwhich the noisy precursors are observed, and the ability tosimultaneously monitor and control the system.
- ItemControl of Nonlinear Phenomena at the Inception of Voltage Collapse(1993) Wang, H.O.; Abed, Eyad H.; Adomaitis, Raymond A.; Hamdan, Anan M. A.; ISRNonlinear Phenomena, including bifurcations and chaos, occurring in power system models exhibiting voltage collapse have been the subject of several recent studies. These nonlinear phenomena have been determined to be crucial factors in the inception of voltage collapse in these models. In this paper, the problem of controlling voltage collapse in the presence of these nonlinear phenomena is addressed. The work focuses on an example power system model that has been studied in several recent papers. The bifurcation control approach is employed to modify the bifurcations and to suppress chaos. The control law is shown to result in improved performance of the system for a greater range of parameter values.
- ItemControllability of Multiparameter Singularly Perturbed Systems.(1988) Abed, Eyad H.; ISRThe controllability of a general linear time-invariant multiparameter singularly perturbed system is studied with no a priori assumptions on the relative magnitudes of the small parameters. It is shown how Kokotovic and Haddad's result on persistence of controllability under singular perturbations in the single parameter case extends to this more general setting. The separation of the system eigenvalues into 'slow' end 'fast' groups is proved here for the first time and employed in the analysis. It is found that one does not expect controllability for all sufficiently small values of the parameters, but conditions are given for this property to hold for almost all sufficiently small values of the small parameters. Moreover, one can describe the set in parameter space for which the system is not controllable.
- ItemDecomposition and Stability of Multiparameter Singular Perturbation Problems.(1985) Abed, Eyad H.; ISRTime-scale separation and stability of linear time-varying and time-invariant multiparameter singular perturbation problems are analyzed. The first problem considered in the paper is that of deriving upper bounds on the small parasitic parameters ensuring the existence of an invertible, bounded transtormation exactly separating test and slow dynamics. This problem is most interesting for the time-varying case. The analysis of this problem in the time-varying case requires the two time-scale setting introduced by H.IC. Khalil and P.V. Kokotovic (SLIAM J. Control Optim., 17, 56-65, l979). This entails that the mutual ratios of the small parameters are bounded by known positive constants. The second problem considered is to derive parameter bounds ensuring that the system in question is uniformly asymptotically stable. The results on decomposition are used to facilitate the derivation of these latter bounds. Fortunately, the analysis of decomposition and stability questions for time- invariant multiparameter singular perturbation problems requires no restriction on the relative magnitudes of the small parameters. A concept of 'strong D-stability' is introduced and shown to greatly simplify the stability analysis of time- invariant multiparameter problems.
- ItemDelay, elasticity, and stability trade-offs in rate control(2003) Ranjan, Priya; Abed, Eyad H.; La, Richard J.; ISR; CSHCNWe adopt the optimization framework for rate allocation problem proposed by Kelly and characterize the stability condition with arbitrary communication delay. We demonstrate that there is a fundamental trade-off between the price elasticity of demand of users and responsiveness of the resources through a choice of price function as well as between the system stability and resource utilization.
- ItemDiscrete-Time Integral Control of PWM DC-DC Converters(1998) Fang, Chung-Chieh; Abed, Eyad H.; ISRDiscrete-time integral control of thePWM DC-DC converters is considered using recentlydeveloped general sampled-data power stage models. Either the voltage or current of the power stage can be controlled.Both state feedback and output feedback are addressed.Line and load regulation are achieved through a discrete-time integrator.By adding an analog filter, the average value of output can be regulated.State observers are constructed for converters belonging to a certain class.
- ItemDynamic Bifurcations in a Power System Model Exhibiting Voltage Collapse(1992) Abed, Eyad H.; Alexander, James C.; Wang, H.; Hamdan, Anan M. A.; Lee, Hsien-Chiarn; ISRDynamic bifurcations, including Hopf and period-doubling bifurcations, are found to occur in a power system dynamic model recently employed in voltage collapse studies. The occurrence of dynamic bifurcations is ascertained in a region of state and parameter space linked with the onset of voltage collapse. The work focuses on a power system model studied by Dobson and Chiang (1989). The presence of the dynamic bifurcations, and the resulting implications for dynamic behavior, necessitate a re- examination of the role of saddle node bifurcations in the voltage collapse phenomenon. The bifurcation analysis is performed using the reactive power demand at a load bus as the bifurcation parameter. Due to numerical ill-conditioning, a reduced-order model is employed in some of the computations. It is determined that the power system model under consideration exhibits two Hopf bifurcations in the vicinity of the saddle node bifurcation. Between the Hopf bifurcations, i.e., in the "Hopf window," period-doubling bifurcations are found to occur. Simulations are given to illustrate the various types of dynamic behaviors associated with voltage collapse for the model. In particular, it is seen that an oscillatory transient may play a role in the collapse.
- ItemFamilies of Liapunov Functions for Nonlinear Systems in Critical Cases(1990) Fu, Jyun-Horng; Abed, Eyad H.; ISRLiapunov functions are constructed for nonlinear systems of ordinary differential equations whose linearized system at an equilibrium point possesses either a simple zero eigenvalue or a complex conjugate pair of simple, pure emaginary eigenvalues. The construction is explicit, and yields parametrized families of Liapunov functions for such systems. In the case of a zero eigenvalue, the Liapunov functions contain quadratic and cubic terms in the state. Quadratic terms appear as well for the case of a pair of pure imaginary eigenvalues. Predictions of local asymptotic stability using these Liapunov functions are shown to coincide with those of pertinent bifurcation-theoretic calculations. The development of the paper is carried out using elementary properties of multilinear functions. The Liapunov function families thus obtained are amendable to symbolic computer coding.
- ItemFamilies of Liapunov Functions for Nonlinear Systems in Critical Cases(1991) Fu, Jyun-Horng; Abed, Eyad H.; ISRLiapunov functions are constructed for nonlinear systems of ordinary differential equations whose linearized system at an equilibrium point possesses either a simple zero eigenvalue or a complex conjugate pair of simple, pure imaginary eigenvalues. The construction is explicit, and yields parametrized families of Liapunov functions for such systems. In the case of a zero eigenvalue, the Liapunov functions contain quadratic and cubic terms in the state. Quartic terms appear as well for the case of a pair of pure imaginary eigenvalues. Predictions of local asymptotic stability using these Liapunov functions are shown to coincide with those of pertinent bifurcation-theorectic calculations. The development of the paper is carried out using elementary properties of multilinear functions. The Liapunov function families thus obtained are amenable to symbolic computer coding.
- ItemFeedback Control of Bifurcation and Chaos in Dynamical Systems(1993) Abed, Eyad H.; Wang, H.O.; ISRFeedback control of bifurcation and chaos in nonlinear dynamical systems is discussed. The article summarizes some of the recent work in this area, including both theory and applications. Stabilization of period doubling bifurcations and of the associated route to chaos is considered. Open problems in bifurcation control are noted.
- ItemFeedback Control of Border Collision Bifurcations in Piecewise Smooth Systems(2002) Hassouneh, Munther A.; Abed, Eyad H.; Abed, Eyad H.; ISRFeedback controls that stabilize border collision bifurcations are designed for piecewise smooth systems undergoing border collision bifurcations. The paper begins with a summary of the main results on border collision bifurcations, and proceeds to a study of stabilization of these bifurcations for one-dimensional systems using both static and dynamic feedback. The feedback can be applied on one side of the border, or on both sides. To achieve robustness to uncertainty in the border itself, a simultaneous stabilization problem is stated and solved. In this problem, the same control is applied on both sides of the border. Dynamic feedback employing washout filters to maintain fixed points is shown to lead to stabilizability for a greater range of systems than static feedback. The results are obtained with a focus on systems in normal form.