There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

‡We describe the use of singular value decomposition in transforming genome-wide expression data from genes 3 arrays space to reduced diagonalized ‘‘eigengenes’’ 3 ‘‘eigenarrays’’ space, where the eigengenes (or eigenarrays) are unique orthonormal superpositions of the genes (or arrays). Normalizing the data by filtering out the eigengenes (and eigenarrays) that are inferred to represent noise or experimental artifacts enables meaningful comparison of the expression of different genes across different arrays in different experiments. Sorting the data according to the eigengenes and eigenarrays gives a global picture of the dynamics of gene expression, in which individual genes and arrays appear to be classified into groups of similar regulation and function, or similar cellular state and biological phenotype, respectively. After normalization and sorting, the significant eigengenes and eigenarrays can be associated with observed genome-wide effects of regulators, or with measured samples, in which these regulators are overactive or underactive, respectively.

Singular Value Decomposition for Genome-Wide Expression Data Processing and Modeling

Computational Analysis of Microarray Data

Stability in Aviation. By Prof. G. H. Bryan, F.E.S. Pp. X + 192. 5S. net. 1911. (Macmillan.)

From Introduction: "The subject of this paper is so broad in scope that a large volume might be devoted to it. In a short paper of this kind it is possible simply to sketch in the high lights of aircraft engine design showing the development to date, the possibilities of the future, and the underlying fundamental principles. Summarizing this development and referring to the graph (Fig.1), we that there is now a water-cooled engine in every power from 150 to 800 HP. and an air-cooled engine in the 200 to 400 HP. classes."

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Aircraft Engine Design

3Hollkamp, J. J., “Multimodal Passive Vibration Suppressionwith Piezoelectrics,” AIAA Paper 93-1683, 1993. 4Hollkamp, J. J., and Starchville, T. F., “A Self-Tuning Piezoelectric Vibration Absorber,” Journal of Intelligent Materials Systems and Structures, Vol. 5, No. 4, 1994, pp. 559–566. 5Rew, K.-H., Han, J. H., and Lee, I., “Adaptive Multimodal Vibration Control of Winglike Composite Structure Using Adaptive Positive Position Feedback,” AIAA Paper 2000-1422, 2000.

Level Flight Trim and Stability Analysis Using Extended Bifurcation and Continuation Procedure

The bifurcation and continuation methodology has evolved over the last two decades into a powerful tool for the analysis of trim and stability problems in aircraft flight dynamics. Over the years, bifurcation methods have been employed to deal with a variety of problems in aircraft dynamics, such as predicting high angle of attack behavior, especially spin, and studying instabilities in rolling maneuvers. The bifurcation methodology has served as a tool for the design of flight control systems, and is promising to be a useful tool in the aircraft design, simulation, testing, and evaluation process. In the present paper, we describe the state-of-the-art in the use of bifurcation and continuation methods for the analysis of aircraft trim and stability with a few illustrative examples. Both the standard and extended bifurcation analysis procedures are discussed and typical results for instabilities in high-α flight and in inertia-coupled roll maneuvers are shown. This is followed by several problems in nonlinear flight dynamics where bifurcation and continuation methods have been fruitfully applied to yield effective solutions. Finally, the use of bifurcation theory to arrive at analytical instability criteria is demonstrated for the aircraft roll coupling and wing rock problems. 76 references have been cited in the text.

Use of Bifurcation and Continuation Methods for Aircraft Trim and Stability Analysis - A State-of-the-Art

In this paper, bifurcation analysis-based methodology is used in conjunction with a sliding-mode-based control
algorithm to construct and simulate maneuvers for a nonlinear, 6 degree-of-freedom, F-18 high-alpha research
vehicle aircraft model. Three different types of maneuvers, namely, a minimum-radius level turn, velocity vector roll,
and spin recovery to a level flight condition, are attempted to demonstrate the usefulness of the proposed approach. The procedure involves constructing the desired maneuvers using constrained bifurcation analysis-based
methodology. The results obtained from bifurcation analysis provide the reference inputs for the sliding mode
controller to switch the aircraft between desired flight conditions. Robustness of the sliding mode controller is also examined by introducing uncertainties in aerodynamic parameters. Closed-loop simulation results are later
presented to show the effectiveness of the proposed technique.

Aircraft Maneuver Design Using Bifurcation Analysis and Sliding Mode Control Techniques

In this brief, a new guidance law for the defense missile of nonmaneuverable aircraft is formulated based on dynamic game considerations. First, a simple differential game of protecting a static target in 2-D, involving simple motions for the attacker and defender, is introduced. The analysis is then extended to a moving noncooperative target in 2-D, in view of the fact that a nonmaneuverable aircraft would not be able to cooperate with the defender. A heuristic solution for the game is proposed and tested, and the results of the 2-D analysis are then extended to 3-D to formulate a new guidance law for the defense missile called the command to optimal interception point (COIP) guidance law. The validity of the new guidance law is checked using trajectory and envelope simulations, built with high-fidelity 6-DOF models using the computer-aided design of aerospace concepts in C++ framework. Performance comparisons are shown between the COIP guidance law and the recently proposed airborne command to line-of-sight (A-CLOS) guidance law. The results show that the performances of COIP and A-CLOS guidance laws are almost identical in a coplanar engagement scenario, but the COIP law has the additional advantage of working with only position information, without the knowledge of motion of the players. In addition, in a noncoplanar engagement case studied, the defense missile is shown to achieve intercept using the COIP guidance law, but fails when using the A-CLOS guidance law.

A New Guidance Law for the Defense Missile of Nonmaneuverable Aircraft

S PIN is an autorotational state of aircraft at high angles of attack characterized by large rotation rate in yaw when compared with rates in roll and pitch. In a typical spin, an aircraft rotates about its center of gravity and an axis perpendicular to Earth descending vertically at high speed following a downward corkscrew path [1]. It is one of themost dangerous phenomena encountered bymany of the modern fighter aircraft required to fly in high-angle-of-attack flight regimes. Improper functioning of aerodynamic control surfaces during spin makes it difficult for the pilot to control the aircraft, leading to many a problem, such as spatial disorientation and uncontrolled motion, resulting in fatal accidents and subsequent loss of aircraft. Spin states for many high-angle-of-attack aircraft models have been computed by using bifurcation analysis and continuation technique methodology. Bifurcation analysis results provide the onset points of bifurcations or the critical values of control surface deflections that may inadvertently or voluntarily land an aircraft into spin. Standard control inputs using proper deflection of rudder to control yaw rate with simultaneous application of elevator to reduce the angle of attack have been recommended for spin recovery [1]. Design of new-generation fighter aircraft with new configurations, however, cannot rely on standard piloting strategies. Instead, a uniform approach to design aircraft-model-based recovery strategies is called for. An introduction to spin problem and needs to design nonlinear controllers to recover aircraft from spin has been reported recently in [2]. Identifying spin and level-trim states from a bifurcation analysis of a nonlinear aircraft model, Raghavendra et al. [2] developed a nonlinear dynamic inversion (NDI)-techniquebased spin-recovery controller. NDI-technique-based methods have emerged as popular control design techniques in aircraft flight dynamics. Controllers based on the NDI techniques, however, come with certain associated disadvantages, such as separation of dynamics based on timescales, resulting in complicated control architecture, lack of robustness due to external uncertainties and unmodeled plant dynamics, and inability to achieve control saturation limits [3]. In this Note, we present a sliding-mode (SM) controller based on variable-structure control technique for spin recovery of aircraft. Variable-structure-technique-based controllers have been found to be robust in the presence of system uncertainties and external disturbances, and, usually result in simpler control algorithms [4]. Controller presented in this Note uses results from a bifurcation analysis of the high-angle-of-attack research vehicle (HARV) model of F-18 available in literature [2]. This Note is organized as follows. In Sec. II, a brief description of the aircraft model with reference states for SM controller design is presented. In Sec. III, SM control design technique is explained. Results and discussions are presented in Sec. IV and conclusions follow thereafter in Sec. V.

Nandan K. Sinha

Papers

Level Flight Trim and Stability Analysis Using Extended Bifurcation and Continuation Procedure

Use of Bifurcation and Continuation Methods for Aircraft Trim and Stability Analysis - A State-of-the-Art

Aircraft Maneuver Design Using Bifurcation Analysis and Sliding Mode Control Techniques

A New Guidance Law for the Defense Missile of Nonmaneuverable Aircraft

Aircraft Spin Recovery Using a Sliding-Mode Controller