Institute for Systems Research
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Item New Results on Modal Participation Factors: Revealing a Previously Unknown Dichotomy(2007) Hashlamoun, Wael A.; Hassouneh, Munther A.; Abed, Eyad H.This paper presents a new fundamental approach to modal participation analysis of linear time-invariant systems, leading to new insights and new formulas for modal participation factors. Modal participation factors were introduced over a quarter century ago as a way of measuring the relative participation of modes in states, and of states in modes, for linear time-invariant systems. Participation factors have proved their usefulness in the field of electric power systems and in other applications. However, in the current understanding, it is routinely taken for granted that the measure of participation of modes in states is identical to that for participation of states in modes. Here, a new analysis using averaging over an uncertain set of system initial conditions yields the conclusion that these quantities(participation of modes in states and participation of states in modes) should not be viewed as interchangeable. In fact, it is proposed that a new definition and calculation} replace the existing ones for state in mode participation factors, while the previously existing participation factors definition and formula should be retained but viewed only as mode in state participation factors. Examples are used throughout the paper to illustrate the issues addressed and results obtained.Item On Participation Factors for Linear Systems(1999) Abed, Eyad H.; Lindsay, David; Hashlamoun, Wael A.; ISRParticipation factors are nondimensional scalars that measure the interaction betweenthe modes and the state variables of a linear system. Since their introduction byVerghese, P'erez-Arriaga and Schweppe, participationfactors have been used for analysis, order reduction and controller design in a variety of fields.In this paper, participation factors are revisited, resulting in new definitions. The aim ofthese definitions is to achieve a conceptual framework that doesn't hinge on anyparticular choice of initial condition. The initial condition is modeled as an uncertainquantity, which can be viewed either in a set-valued or a probabilistic setting.If the initial condition uncertainty obeys a symmetry condition, the new definitionsare found to reduce to the original definition of participation factors.
Item Stationary Bifurcation Control for Systems with Uncontrollable Linearization(1999) Taihyun Kim; Abed, Eyad H.; ISRStationary bifurcation control is studied under the assumption thatthe critical zero eigenvalue is uncontrollable for thelinearized system. The development facilitates explicit constructionof feedback control laws that render the bifurcation supercritical.Thus, the bifurcated equilibria in the controlled system are guaranteedstable.Bothpitchfork bifurcation and transcritical bifurcation are addressed.The results obtained forpitchfork bifurcations apply to general nonlinear models smoothin the state and the control. For transcritical bifurcations,the results require the system to be affine in the control.
Item Sampled-Data Modeling and Analysis of the Power Stage of PWM DC-DC Converters(1999) Fang, Chung-Chieh; Abed, Eyad H.; Abed, Eyad H.; ISRThe power stage of the PWM DC-DC converter is modeledand analyzed using the sampled-data approach.The work addressescontinuous and discontinuous conduction mode under voltage mode control,and continuous conduction mode under current mode control.For each configuration, nonlinear and linearized sampled-data models andcontrol-to-output transfer function are derived.Using this approach, both current mode control and discontinuous conduction modecan be handled systematically in a unified framework,making the modeling for these cases simpler than with the use of averaging.The results of this paper are similar to the results of Tymerski,but they are presented in a simpler manner tailored to facilitate immediate application to specific circuits.
It is shown howsampling the output at certain instants improves the obtained phase response.Frequency responses obtained from the sampled-data model aremore accurate than those obtained from various averaged models. In addition, a new ("lifted")continuous-time switching frequency-dependent model of the power stage isderived from the sampled-data model. Detailed examples illustrate themodeling tools presented here, and also provide a means of comparingresults obtained from the sampled-data approach with those obtainedfrom averaging.
Item Sampled-Data Modeling and Analysis of Closed-Loop PWM DC-DC Converters(1999) Fang, Chung-Chieh; Abed, Eyad H.; Abed, Eyad H.; ISRSampled-data analysis of converters has been a topic of investigationfor the past two decades. However, this powerful tool is not widelyused in control loop design or in closed-loop performance validation.Instead, averaged models are typically used for control loopdesign, while detailed simulations are used for validatingclosed-loop performance. This paper makes several contributions tothe sampled-data modeling and analysis of closed-loop PWM DC-DC converters,with the aim of increasing appreciation and use of the method.General models are presented in a unified and simple manner, while removingsimplifying approximations present in previous work. These models applyboth for current mode control and voltage mode control.The general models are nonlinear. They are used toobtain {it analytical} linearized models, which are in turn employedto obtain local stability results.
Detailed examplesillustrate the modeling and analysis in the paper,and point to situations in which the sampled-data approachgives results superior to alternate methods.For instance, it is shownthat the sampled-data approach will reliablypredict the (local) stability of aconverter for which averaging or simulation predicts instability.
Item Local Bifurcations in PWM DC-DC Converters(1999) Fang, Chung-Chieh; Abed, Eyad H.; Abed, E. H.; ISRA general sampled-data model of PWM DC-DC converters is employed tostudy types of loss of stability of the nominal (periodic)operating condition andtheir connection with local bifurcations.In this work, the nominal solution's periodic natureis accounted for via the sampled-data model.This results in moreaccurate predictions of instability and bifurcation than can be obtained using the averaging approach.The local bifurcations of the nominal operating conditionstudied here are period-doublingbifurcation, saddle-node bifurcation, and Neimark bifurcation.Examples of bifurcations associated with instabilities in PWM DC-DC convertersare given.In particular, input filter instability is shown to be closely related tothe Neimark bifurcation.Item Closed-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.Item Guardian Maps and the Generalized Stability of Parametrized Families of Matrices and Polynomials(1988) Saydy, Lahcen; Tits, Andre; Abed, Eyad H.; ISRThe generalized stability of families of real matrices and polynomials is considered. (Generalized stability is meant in the usual sense of confinementof matrix eigenvalues or polynomial zeros to a prespecified domain in the complex plane, and includes Hurwitz and Schur stability as special cases.)"Guardian maps" and "semiguardian maps" are introduced as a unifying tool for the studyof this problem. Basically these are scalar maps that vanish when theirmatrix or polynomial argument loses stability. Such maps are exhibited for a wide variety of cases of interest corresponding to a generalized stability with respect to domains of the complex plane.
In the case of one- and two-parameter families of matrices or polynomials, concise necessary and sufficient conditions for generalized stability arederived. For the general multiparameter case, the problem is transformedinto one of checking that a given map is nonzero for the allowedparameter values.
Note: This is TR 88-69-r1. A previous version of this report, TR 88-69, had a different title.