Browsing by Author "Liaw, Der-Cherng"
Now showing 1 - 8 of 8
Results Per Page
Sort Options
Item Active 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.Item Application 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.Item Bifurcation 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.Item Feedback Stabilization via Center Manifold Reduction with Application to Tethered Satellites(1990) Liaw, Der-Cherng; Abed, E.H.; ISRCenter manifold reduction has recently been introduced as a tool for design of stabilizing control laws for nonlinear systems in critical cases. In this dissertation, the center manifold approach is elaborated for general such nonlinear systems in several critical cases of interest, and the results are applied to the control of tethered satellite systems (TSS). In addition, to address stability questions for satellite deployment via TSS, we obtain new results in finite-time stability theory. The critical cases considered in the general feedback stabilization studies include the cases in which the system linearization possesses a simple zero eigenvalue (of multiplicity one or two), a pair of simple pure imaginary eigenvalues, one zero eigenvalues along with a pair of simple pure imaginary eigenvalues, and two pairs of simple pure imaginary eigenvalues. The calculations involve center manifold reduction, normal form transformations, and Liapunov function construction for critical systems. These calculations are explicit. The tethered satellite systems considered here consist of a satellite and subsatellite connected by a tether, in orbit around the Earth. The Lagrangian formulation of dynamics is used to obtain a nonlinear system of ordinary differential equations for TSS dynamics. For simplicity, a rigid, massless tether is assumed. Linear analysis reveals the presence of critical eigenvalues in the station- keeping mode of operation. This renders useful results on stabilization in critical cases to this application. The control variable assumed is tether tension feedback. Besides the design of stabilizing station-keeping controllers, stability of deployment and instability of retrieval are also shown for a constant angle deployment/retrieval scheme.Item Nonlinear Dynamics of Axial Flow Compressors: A Parametric Study(1991) Liaw, Der-Cherng; Adomaitis, Raymond A.; Abed, Eyad H.; ISRThis paper presents the analysis of the dynamics of a representative axial flow compressor model, emphasizing the influence of two important control parameters. These are a nondimensional parameter B (previously introduced by Greitzer (1976) as a primary determinant of post-stall behavior) and the setting of the throttle line. Greitzer's lumped-parameter model is employed in this study, with a specific choice of compressor and throttle characteristics. Our analysis shows the presence of a rich variety of global as well as local bifurcations as the two control parameters are varied. The analysis leads to a characterization of compressor operation into three major zones: the stalled zone, the pre-stall zone, and the normal (unstalled) zone. Simulation results demonstrate the qualitatively different dynamical behaviors within each regime of parameter space.Item Nonlinear Dynamics of Axial-Flow Compressors: A Parametric Study(1992) Adomaitis, Raymond A.; Liaw, Der-Cherng; Abed, Eyad H.; ISRAnalysis of the post-instability dynamical behavior of an axial- flow compression system model is carried out in a bifurcation- theoretic setting. Using global analysis techniques, we uncover the sequence of bifurcations in parameter space which allows us to rigorously determine whether the compressor stalls or surges when the throttle is slowly closed beyond the instability margin. Using these computational techniques, we also determine the conditions under which stalled and/or surging flow solutions coexist with the desired uniform-flow operating point and quantify the perturbations which destabilize this operating point.Item Stabilization of Tethered Satellites During Station-Keeping.(1988) Liaw, Der-Cherng; Abed, Eyad H.; ISRAfter deriving a set of dynamic equations governing the dynamics of a Tethered Satellite System (TSS), stabilizing tension control laws in feedback form are derived. The tether is assumed rigid and massless, and the equations of motion are derived using the system Lagrangian. It is observed that, to stabilize the system, tools from stability analysis of critical nonlinear systems must be applied. This paper employs tools related to the Hopf Bifurcation Theorem in the construction of the stabilizing control laws, which may be taken purely linear. Simulations illustrate the nature of the conclusions, and show that nonlinear terms in the feedback can be used to significantly improve the transient response.Item Tethered Satellite System Stability.(1989) Liaw, Der-Cherng; Abed, Eyad H.; ISRIssues of stability of the Tethered Satellite System (TSS) during station-keeping, deployment and retrieval are considered. The basic nonlinear equations of motion of the TSS are derived using the system Lagrangian. Using the Hopf bifurcation theorem, tension control laws are established which guarantee the stability of the system during the station-keeping mode. A constant angle control method is hypothesized for subsatellite deployment and retrieval. It is proved that this control law results in stable deployment but unstable retrieval. An enhanced control law for deployment is also proposed, which entails use of the constant angle method followed by a station-keeping control law once the tether length is sufficiently near the desired value. Simulations are given to illustrate the conclusions.