Engineering topics

Hartley, Tom T.

Publications and source records attributed to Hartley, Tom T..

At least 19 records

NASA NTRS2004

Control of Initialized Fractional-Order Systems

Fractional-Order systems, or systems containing fractional derivatives and integrals, have been studied by many in the engineering area. Additionally, very readable discussions, devoted specifically to the subject, are presented by Oldham and Spanier, Miller and Ross, and Pudlubny (1999a). It should be noted that there are a growing number of physical systems whose behavior can be compactly described using fractional system theory. Of specific interest to electrical engineers are long lines, electrochemical processes, dielectric polarization, colored noise, viscoelastic materials, and chaos. With the growing number of applications, it is important to establish a theory of control for these fractional-order systems, and for the potential use of fractional-order systems as feedback compensators. This topic is addressed in this paper. The first section discusses the control of fractional-order systems using a vector space representation, where initialization is included in the discussion. It should be noted that Bagley and Calico and Padovan and Sawicki both present a fractional state-space representation, which do not include the important historic effects. Incorporation of these effects based on the initialized fractional calculus is presented . The control methods presented in this paper are based on the initialized fractional order system theory. The second section presents an input-output approach. Some of the problems encountered in these sections are: a) the need to introduce a new complex plane to study the dynamics of fractional-order systems, b) the need to properly define the Laplace transform of the fractional derivative, and c) the proper inclusion of the initialization response in the system and control formulation. Following this, the next section generalizes the proportional-plus-integral-control (PI-control) and PID-control (PI-plus- derivative) concepts using fractional integrals. This is then further generalized using general fractional- order compensators. Finally the compensator concept is generalized by the use of a continuum of fractions in the compensator via the concept of order-distributions. The last section introduces fractional feedback in discrete-time.

Hartley, Tom T.↗

NASA NTRS2003

Optimal Battery Charging for Damage Mitigation

Our control philosophy is to charge the NiH2 cell in such a way that the damage incurred during the charging period is minimized, thus extending its cycle life. This requires nonlinear dynamic model of NiH2 cell and a damage rate model. We must do this first. This control philosophy is generally considered damage mitigating control or life-extending control. This presentation covers how NiH2 cells function, electrode behavior, an essentialized model, damage mechanisms for NiH2 batteries, battery continuum damage modeling, and battery life models. The presentation includes graphs and a chart illustrating how charging a NiH2 battery with different voltages and currents affects damages the battery and affects its life. The presentation concludes with diagrams of control system architectures for tracking battery recharging.

Hartley, Tom T.↗

NASA NTRS2002

Control of Initialized Fractional-Order Systems

Due to the importance of historical effects in fractional-order systems, this paper presents a general fractional-order control theory that includes the time-varying initialization response. Previous studies have not properly accounted for these historical effects. The initialization response, along with the forced response, for fractional-order systems is determined. Stability properties of fractional-order systems are presented in the complex w-plane, which is a transformation of the s-plane. Time responses are discussed with respect to pole positions in the complex w-plane and frequency response behavior is included. A fractional-order vector space representation, which is a generalization of the state space concept, is presented including the initialization response. Control methods for vector representations of initialized fractional-order systems are shown. Nyquist, root-locus, and other input-output control methods are adapted to the control of fractional-order systems. Finally, the fractional-order differintegral is generalized to continuous order-distributions that have the possibility of including a continuum of fractional orders in a system element.

Hartley, Tom T.↗

NASA NTRS2002

Variable Order and Distributed Order Fractional Operators

Many physical processes appear to exhibit fractional order behavior that may vary with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. This paper develops the concept of variable and distributed order fractional operators. Definitions based on the Riemann-Liouville definitions are introduced and behavior of the operators is studied. Several time domain definitions that assign different arguments to the order q in the Riemann-Liouville definition are introduced. For each of these definitions various characteristics are determined. These include: time invariance of the operator, operator initialization, physical realization, linearity, operational transforms. and memory characteristics of the defining kernels. A measure (m2) for memory retentiveness of the order history is introduced. A generalized linear argument for the order q allows the concept of "tailored" variable order fractional operators whose a, memory may be chosen for a particular application. Memory retentiveness (m2) and order dynamic behavior are investigated and applications are shown. The concept of distributed order operators where the order of the time based operator depends on an additional independent (spatial) variable is also forwarded. Several definitions and their Laplace transforms are developed, analysis methods with these operators are demonstrated, and examples shown. Finally operators of multivariable and distributed order are defined in their various applications are outlined.

Lorenzo, Carl F.↗

NASA NTRS2002

Optimal Charging of Nickel-Hydrogen Batteries for Life Extension

We are exploring the possibility of extending the cycle life of battery systems by using a charging profile that minimizes cell damage. Only nickel-hydrogen cells are discussed at this time, but applications to lithium-ion cells are being considered. The process first requires the development of a fractional calculus based nonlinear dynamic model of the specific cells being used. The parameters of this model are determined from the cell transient responses. To extend cell cycle life, an instantaneous damage rate model is developed. The model is based on cycle life data and is highly dependent on cell voltage. Once both the cell dynamic model and the instantaneous damage rate model have been determined, the charging profile for a specific cell is determined by numerical optimization. Results concerning the percentage life extension for different charging strategies are presented. The overall procedure is readily adaptable to real-time implementations where the charging profile can maintain its minimum damage nature as the specific cell ages.

Hartley, Tom T.↗

NASA NTRS2000

R-Function Relationships for Application in the Fractional Calculus

The F-function, and its generalization the R-function, are of fundamental importance in the fractional calculus. It has been shown that the solution of the fundamental linear fractional differential equation may be expressed in terms of these functions. These functions serve as generalizations of the exponential function in the solution of fractional differential equations. Because of this central role in the fractional calculus, this paper explores various intrarelationships of the R-function, which will be useful in further analysis. Relationships of the R-function to the common exponential function, e(t), and its fractional derivatives are shown. From the relationships developed, some important approximations are observed. Further, the inverse relationships of the exponential function, el, in terms of the R-function are developed. Also, some approximations for the R-function are developed.

Lorenzo, Carl F.↗

NASA NTRS2000

Initialized Fractional Calculus

This paper demonstrates the need for a nonconstant initialization for the fractional calculus and establishes a basic definition set for the initialized fractional differintegral. This definition set allows the formalization of an initialized fractional calculus. Two basis calculi are considered; the Riemann-Liouville and the Grunwald fractional calculi. Two forms of initialization, terminal and side are developed.

Lorenzo, Carl F.↗

NASA NTRS1999

Fractional System Identification: An Approach Using Continuous Order-Distributions

This paper discusses the identification of fractional- and integer-order systems using the concept of continuous order-distribution. Based on the ability to define systems using continuous order-distributions, it is shown that frequency domain system identification can be performed using least squares techniques after discretizing the order-distribution.

Hartley, Tom T.↗

NASA NTRS1999

Generalized Functions for the Fractional Calculus

Previous papers have used two important functions for the solution of fractional order differential equations, the Mittag-Leffler functionE(sub q)[at(exp q)](1903a, 1903b, 1905), and the F-function F(sub q)[a,t] of Hartley & Lorenzo (1998). These functions provided direct solution and important understanding for the fundamental linear fractional order differential equation and for the related initial value problem (Hartley and Lorenzo, 1999). This paper examines related functions and their Laplace transforms. Presented for consideration are two generalized functions, the R-function and the G-function, useful in analysis and as a basis for computation in the fractional calculus. The R-function is unique in that it contains all of the derivatives and integrals of the F-function. The R-function also returns itself on qth order differ-integration. An example application of the R-function is provided. A further generalization of the R-function, called the G-function brings in the effects of repeated and partially repeated fractional poles.

Lorenzo, Carl F.↗

NASA NTRS1999

The Vertical Linear Fractional Initialization Problem

This paper presents a solution to the initialization problem for a system of linear fractional-order differential equations. The scalar problem is considered first, and solutions are obtained both generally and for a specific initialization. Next the vector fractional order differential equation is considered. In this case, the solution is obtained in the form of matrix F-functions. Some control implications of the vector case are discussed. The suggested method of problem solution is shown via an example.

Lorenzo, Carl F.↗

NASA NTRS1998

Intitialization, Conceptualization, and Application in the Generalized Fractional Calculus

This paper provides a formalized basis for initialization in the fractional calculus. The intent is to make the fractional calculus readily accessible to engineering and the sciences. A modified set of definitions for the fractional calculus is provided which formally include the effects of initialization. Conceptualizations of fractional derivatives and integrals are shown. Physical examples of the basic elements from electronics are presented along with examples from dynamics, material science, viscoelasticity, filtering, instrumentation, and electrochemistry to indicate the broad application of the theory and to demonstrate the use of the mathematics. The fundamental criteria for a generalized calculus established by Ross (1974) are shown to hold for the generalized fractional calculus under appropriate conditions. A new generalized form for the Laplace transform of the generalized differintegral is derived. The concept of a variable structure (order) differintegral is presented along with initial efforts toward meaningful definitions.

Lorenzo, Carl F.↗

NASA NTRS1998

A Solution to the Fundamental Linear Fractional Order Differential Equation

This paper provides a solution to the fundamental linear fractional order differential equation, namely, (sub c)d(sup q, sub t) + ax(t) = bu(t). The impulse response solution is shown to be a series, named the F-function, which generalizes the normal exponential function. The F-function provides the basis for a qth order "fractional pole". Complex plane behavior is elucidated and a simple example, the inductor terminated semi- infinite lossy line, is used to demonstrate the theory.

Hartley, Tom T.↗

NASA NTRS1998

Insights Into the Fractional Order Initial Value Problem via Semi-Infinite Systems

This paper considers various aspects of the initial value problem for fractional order differential equations. The main contribution of this paper is to use the solutions to known spatially distributed systems to demonstrate that fractional differintegral operators require an initial condition term that is time-varying due to past distributed storage of information.

Hartley, Tom T.↗

NASA NTRS1998

To Err is Normable: The Computation of Frequency-Domain Error Bounds from Time-Domain Data

This paper exploits the relationships among the time-domain and frequency-domain system norms to derive information useful for modeling and control design, given only the system step response data. A discussion of system and signal norms is included. The proposed procedures involve only simple numerical operations, such as the discrete approximation of derivatives and integrals, and the calculation of matrix singular values. The resulting frequency-domain and Hankel-operator norm approximations may be used to evaluate the accuracy of a given model, and to determine model corrections to decrease the modeling errors.

Hartley, Tom T.↗

NASA NTRS1998

A Method for Generating Reduced-Order Linear Models of Multidimensional Supersonic Inlets

Simulation of high speed propulsion systems may be divided into two categories, nonlinear and linear. The nonlinear simulations are usually based on multidimensional computational fluid dynamics (CFD) methodologies and tend to provide high resolution results that show the fine detail of the flow. Consequently, these simulations are large, numerically intensive, and run much slower than real-time. ne linear simulations are usually based on large lumping techniques that are linearized about a steady-state operating condition. These simplistic models often run at or near real-time but do not always capture the detailed dynamics of the plant. Under a grant sponsored by the NASA Lewis Research Center, Cleveland, Ohio, a new method has been developed that can be used to generate improved linear models for control design from multidimensional steady-state CFD results. This CFD-based linear modeling technique provides a small perturbation model that can be used for control applications and real-time simulations. It is important to note the utility of the modeling procedure; all that is needed to obtain a linear model of the propulsion system is the geometry and steady-state operating conditions from a multidimensional CFD simulation or experiment. This research represents a beginning step in establishing a bridge between the controls discipline and the CFD discipline so that the control engineer is able to effectively use multidimensional CFD results in control system design and analysis.

Chicatelli, Amy↗

NASA NTRS1997

A Method for Generating Reduced Order Linear Models of Supersonic Inlets

For the modeling of high speed propulsion systems, there are at least two major categories of models. One is based on computational fluid dynamics (CFD), and the other is based on design and analysis of control systems. CFD is accurate and gives a complete view of the internal flow field, but it typically has many states and runs much slower dm real-time. Models based on control design typically run near real-time but do not always capture the fundamental dynamics. To provide improved control models, methods are needed that are based on CFD techniques but yield models that are small enough for control analysis and design.

Chicatelli, Amy↗

NASA NTRS1996

Dimension Determination of Precursive Stall Events in a Single Stage High Speed Compressor

This paper presents a study of the dynamics for a single-stage, axial-flow, high speed compressor core, specifically, the NASA Lewis rotor stage 37. Due to the overall blading design for this advanced core compressor, each stage has considerable tip loading and higher speed than most compressor designs, thus, the compressor operates closer to the stall margin. The onset of rotating stall is explained as bifurcations in the dynamics of axial compressors. Data taken from the compressor during a rotating stall event is analyzed. Through the use of a box-assisted correlation dimension methodology, the attractor dimension is determined during the bifurcations leading to rotating stall. The intent of this study is to examine the behavior of precursive stall events so as to predict the entrance into rotating stall. This information may provide a better means to identify, avoid or control the undesirable event of rotating stall formation in high speed compressor cores.

Bright, Michelle M.↗

NASA NTRS1996

Chaos in a Fractional Order Chua System

This report studies the effects of fractional dynamics in chaotic systems. In particular, Chua's system is modified to include fractional order elements. Varying the total system order incrementally from 2.6 to 3.7 demonstrates that systems of 'order' less than three can exhibit chaos as well as other nonlinear behavior. This effectively forces a clarification of the definition of order which can no longer be considered only by the total number of differentiations or by the highest power of the Laplace variable.

Lorenzo, Carl F.↗