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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.

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.

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.

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 Airplane, which is a transformation of the s-plane. Time responses are discussed with respect to pole positions in the complex Airplane 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

This report presents the conceptual development of a life extending control system where the objective is to achieve high performance and structural durability of the plant. A life extending controller is designed for a reusable rocket engine via damage mitigation in both the fuel and oxidizer turbines while achieving high performance for transient responses of the combustion chamber pressure and the O2/H2 mixture ratio. This design approach makes use of a combination of linear and nonlinear controller synthesis techniques and also allows adaptation of the life extending controller module to augment a conventional performance controller of a rocket engine. The nonlinear aspect of the design is achieved using nonlinear parameter optimization of a prescribed control structure.

This paper presents the conceptual development of a life-extending control system where the objective is to achieve high performance and structural durability of the plant. A life-extending controller is designed for a reusable rocket engine via damage mitigation in both the fuel (H2) and oxidizer (O2) turbines while achieving high performance for transient responses of the combustion chamber pressure and the O2/H2 mixture ratio. The design procedure makes use of a combination of linear and nonlinear controller synthesis techniques and also allows adaptation of the life-extending controller module to augment a conventional performance controller of the rocket engine. The nonlinear aspect of the design is achieved using non-linear parameter optimization of a prescribed control structure. Fatigue damage in fuel and oxidizer turbine blades is primarily caused by stress cycling during start-up, shutdown, and transient operations of a rocket engine. Fatigue damage in the turbine blades is one of the most serious causes for engine failure.

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.

This paper develops a simplified continuum (continuous with respect to time, stress, etc.) fatigue damage model for use in critical design, Life Extending Control and fault prognosis. The work is based on the local strain cyclic damage modeling method. New nonlinear explicit equation forms of cyclic damage in terms of stress amplitude are derived to facilitate the continuum modelling. Stress based continuum models are derived. Extension to plastic strain-strain rate models is also presented. Progress toward a non-zero mean stress based is presented. Also new nonlinear explicit equation forms in terms of stress amplitude are derived for this case. Application of the various models to design, control, and fault prognosis is considered.

The potential for a revolutionary step in the durability of reusable rocket engines is made possible by the combination of several emerging technologies. The recent creation and analytical demonstration of life extending (or damage mitigating) control technology enables rapid rocket engine transients with minimum fatigue and creep damage. This technology has been further enhanced by the formulation of very simple but conservative continuum damage models. These new ideas when combined with recent advances in multidisciplinary optimization provide the potential for a large (revolutionary) step in reusable rocket engine durability. This concept has been named the robust rocket engine concept (RREC) and is the basic contribution of this paper. The concept also includes consideration of design innovations to minimize critical point damage.

Three primary issues will drive the design and control used in next generation reuseable rocket engines. In addition to steady-state and dynamic performance, the requirements for increased durability, reliability and operability (with faults) will dictate which new controls and design technologies and features will be brought to bear. An array of concepts which have been brought forward will be tested against the measures of cost and benefit as reflected in the above 'ilities'. This paper examines some of the new concepts and looks for metrics to judge their value.

This paper develops a simplified continuum (continuous wrp to time, stress, etc.) fatigue damage model for use in Life Extending Controls (LEC) studies. The work is based on zero mean stress local strain cyclic damage modeling. New nonlinear explicit equation forms of cyclic damage in terms of stress amplitude are derived to facilitate the continuum modeling. Stress based continuum models are derived. Extension to plastic strain-strain rate models are also presented. Application of these models to LEC applications is considered. Progress toward a nonzero mean stress based continuum model is presented. Also, new nonlinear explicit equation forms in terms of stress amplitude are also derived for this case.

Calculations are presented showing that a substantial improvement in service life of a reusable rocket engine can be achieved by an insignificant reduction in the system dynamic performance. The paper introduces the concept of damage mitigation and formulates a continuous-time model of fatigue damage dynamics. For control of complex mechanical systems, damage prediction and damage mitigation are carried out based on the available sensory and operational information such that the plant can be inexpensively maintained and safely and efficiently steered under diverse operating conditions. The results of simulation experiments are presented for transient operations of a reusable rocket engine.

New control concepts for the next generation of advanced air-breathing and rocket engines and hypersonic combined-cycle propulsion systems are analyzed. The analysis provides a database on the instrumentation technologies for advanced control systems and cross matches the available technologies for each type of engine to the control needs and applications of the other two types of engines. Measurement technologies that are considered to be ready for implementation include optical surface temperature sensors, an isotope wear detector, a brushless torquemeter, a fiberoptic deflectometer, an optical absorption leak detector, the nonintrusive speed sensor, and an ultrasonic triducer. It is concluded that all 30 advanced instrumentation technologies considered can be recommended for further development to meet need of the next generation of jet-, rocket-, and hypersonic-engine control systems.

The application of advanced control concepts to airbreathing engines may yield significant improvements in aircraft/engine performance and operability. Accordingly, the NASA Lewis Research Center has conducted screening studies of advanced control concepts for airbreathing engines to determine their potential impact on turbine engine performance and operability. The purpose of the studies was to identify concepts which offered high potential yet may incur high research and development risk. A target suite of proposed concepts was formulated by NASA and industry. These concepts were evaluated in a two phase study to quantify each concept's impact on desired engine characteristics. To aid in the evaluation, three target aircraft/engine combinations were considered: a military high performance fighter mission, a high speed civil transport mission, and a civil tiltrotor mission. Each of the advanced control concepts considered in the study were defined and described. The concept's potential impact on engine performance was determined. Relevant figures of merit on which to evaluate the concepts were also determined. Finally, the concepts were ranked with respect to the target aircraft/engine missions.

The application of advanced control concepts to airbreathing engines may yield significant improvements in aircraft/engine performance and operability. Accordingly, the NASA Lewis Research Center has conducted screening studies of advanced control concepts for airbreathing engines to determine their potential impact on turbine engine performance and operability. The purpose of the studies was to identify concepts which offered high potential yet may incur high research and development risk. A target suite of proposed concepts was formulated by NASA and industry. These concepts were evaluated in a two phase study to quantify each concept's impact on desired engine characteristics. To aid in the evaluation, three target aircraft/engine combinations were considered: a military high performance fighter mission, a high speed civil transport mission, and a civil tiltrotor mission. Each of the advanced control concepts considered in the study were defined and described. The concept's potential impact on engine performance was determined. Relevant figures of merit on which to evaluate the concepts were also determined. Finally, the concepts were ranked with respect to the target aircraft/engine missions.

The concept of damage-mitigating control is to minimize fatigue (as well as creep and corrosion) damage of critical components of mechanical structures while simultaneously maximizing the system dynamic performance. Given a dynamic model of the plant and the specifications for performance and stability robustness, the task is to synthesize a control law that would meet the system requirements and, at the same time, satisfy the constraints that are imposed by the material and structural properties of the critical components. The authors present the concept of damage-mitigating control systems design with the following objectives: (1) to achieve high performance with a prolonged life span; and (2) to systematically update the controller as the new technology of advanced materials evolves. The major challenge is to extract the information from the material properties and then utilize this information in a mathematical form so that it can be directly applied to robust control synthesis for mechanical systems. The basic concept of damage-mitigating control is illustrated using a relatively simplified model of a space shuttle main engine.

The concept of Life Extending Control (LEC) is introduced. Possible extensions to the cyclic damage prediction approach are presented based on the identification of a model from elementary forms. Several candidate elementary forms are presented. These extensions will result in a continuous or differential form of the damage prediction model. Two possible approaches to the LEC based on the existing cyclic damage prediction method, the measured variables LEC and the estimated variables LEC, are defined. Here, damage estimates or measurements would be used directly in the LEC. A simple hydraulic actuator driven position control system example is used to illustrate the main ideas behind LEC. Results from a simple hydraulic actuator example demonstrate that overall system performance (dynamic plus life) can be maximized by accounting for component damage in the control design.

The notion of controller partitioning is described. Conditions are developed under which the input/output behavior of a multi-input multi-output centralized controller can be exactly matched by two separate subsystem controllers interconnected through output crossfeed. A systematic method is developed for determining a controller partitioning which best approximates the input/output behavior of the centralized controller for the general case when the exact matching conditions are not satisfied. The controller partitioning procedure is demonstrated for a centralized integrated flight/propulsion controller designed in a previous study.

Several components of intelligence are defined. Within the context of these definitions an intelligent control system for rocket engines is described. The description includes a framework for development of an intelligent control system, including diagnostics, coordination, and direct control. Some current results and issues are presented.

The issues of Chemical Rocket Engine Control are broadly covered. The basic feedback information and control variables used in expendable and reusable rocket engines, such as Space Shuttle Main Engine, are discussed. The deficiencies of current approaches are considered and a brief introduction to Intelligent Control Systems for rocket engines (and vehicles) is presented.

The National Aeronautics and Space Administration is actively involved in the development of enabling technologies that will lead towards aircraft with new/enhanced maneuver capabilities such as Short Take-Off Vertical Landing (STOVL) and high angle of attack performance. Because of the high degree of dynamic coupling between the airframe and propulsion systems of these types of aircraft, one key technology is the integration of the flight and propulsion control. The NASA Lewis Research Center approach to developing Integrated Flight Propulsion Control (IFPC) technologies is an in-house research program referred to as IMPAC (Integrated Methodology for Propulsion and Airframe Control). The goals of IMPAC are to develop a viable alternative to the existing integrated control design methodologies that will allow for improved system performance and simplicity of control law synthesis and implementation, and to demonstrate the applicability of the methodology to a supersonic STOVL fighter aircraft. Based on some preliminary control design studies that included evaluation of the existing methodologies, the IFPC design methodology that is emerging at the Lewis Research Center consists of considering the airframe and propulsion system as one integrated system for an initial centralized controller design and then partitioning the centralized controller into separate airframe and propulsion system subcontrollers to ease implementation and to set meaningful design requirements for detailed subsystem control design and evaluation. An overview of IMPAC is provided and detailed discussion of the various important design and evaluation steps in the methodology are included.

The concept of Life Extending Control is defined. Life is defined in terms of mechanical fatigue life. A brief description is given of the current approach to life prediction using a local, cyclic, stress-strain approach for a critical system component. An alternative approach to life prediction based on a continuous functional relationship to component performance is proposed.Base on cyclic life prediction an approach to Life Extending Control, called the Life Management Approach is proposed. A second approach, also based on cyclic life prediction, called the Implicit Approach, is presented. Assuming the existence of the alternative functional life prediction approach, two additional concepts for Life Extending Control are presented.

The concept of Life Extending Control is defined. Life is defined in terms of mechanical fatigue life. A brief description is given of the current approach to life prediction using a local, cyclic, stress-strain approach for a critical system component. An alternative approach to life prediction based on a continuous functional relationship to component performance is proposed. Based on cyclic life prediction, an approach to life extending control, called the Life Management Approach, is proposed. A second approach, also based on cyclic life prediction, called the implicit approach, is presented. Assuming the existence of the alternative functional life prediction approach, two additional concepts for Life Extending Control are presented.

Discussed here is research at NASA Lewis in the area of propulsion controls as driven by trends in advanced aircraft. The objective of the Lewis program is to develop the technology for advanced reliable propulsion control systems and to integrate the propulsion control with the flight control for optimal full-system control.