Eigenstructure Control Algorithms: Applications to aircraft/rotorcraft handling qualities design
Eigenstructure control involves modification of both the eigenvalues and eigenvectors of a system using feedback. Based on this key concept, algorithms are derived for the design of control systems using controller structures such as state feedback, output feedback, observerbased dynamic feedback, implicit and explicit modelfollowing, etc. The simpletouse algorithms are well suited to evolve practical engineering solutions. The design of control laws for modern flybywire high performance aircraft/rotorcraft offers some unique design challenges. The control laws have to provide a satisfactory interface between the pilot and the vehicle that results in good handling qualities (HQ) in precision control tasks. This book, through detailed aircraft and rotorcraft design examples, illustrates how to develop practical, robust flight control laws to meet these HQ requirements. This book demonstrates that eigenstructure control theory can be easily adapted and infused into the aircraft industry's stringent design practices; therefore practicing flight control engineers will find it useful to explore the use of the new design concepts discussed. The book, being interdisciplinary in nature, encompassing control theory and flight dynamics, should be of interest to both control and aeronautical engineers. In particular, control researchers will find it interesting to explore an extension of the theory to new multivariable control problem formulations. Finally, the book should be of interest to graduate/doctoral students keen on learning a multivariable control technique that is useful in the design of practical control systems.
Inspec keywords: helicopters; eigenstructure assignment; observers; aerospace control; control system synthesis
Other keywords: aircraft lateral directional handling quality design; aircraft state variable; aircraft flutter control system design; eigenstructure control algorithm; aircraft longitudinal handling quality design; flight control system design; modal canonical observer; flight mechanics; rotorcraft handling quality design; eigenstructure synthesis algorithm; eigenstructure assignment characterisation; model following control system
Subjects: Simulation, modelling and identification; Control system analysis and synthesis methods; Aerospace control
 Book DOI: 10.1049/PBCE074E
 Chapter DOI: 10.1049/PBCE074E
 ISBN: 9781849192590
 eISBN: 9781849192606
 Page count: 304
 Format: PDF

Front Matter
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1 Introduction
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This book addresses handling quality (HQ) design issues for smallamplitude precision flying tasks. Eigenstructure assignment algorithms is used to derive these control laws. Chapter 8 briefly reviews the relevant HQ metrics and flight control design guidelines, usually used by the industry, as applicable to the flight control applications of Chapters 911. Chapter 9 discusses in detail the development of lateraldirectional HQ design process of a military aircraft. Chapter 10 addresses the control of the pitch axis of an aircraft. Chapter 11 deals with the complex rotorcraft handling problem. The decoupling of the pitchrollyaw axes is achieved by formulating a dynamic feedback structure, based on functional observers developed in Chapter 6. The forward path attitude command system design is based on the tunable explicit model following control algorithm developed in Chapter 7. Chapter 12, not related to HQ design, discusses the application of eigenstructure assignment techniques to control of aeroelastic modes of a flight vehicle. A wind tunnel model flutter control problem is addressed.

2 Eigenstructure assignment characterisation
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In this chapter, the eigenstructure synthesis using state variable feedback has been analysed. It is shown that the solution scheme is simple and just involves solving a set of linear equations to synthesise each eigenvector for a specified eigenvalue. The modal characterisation elegantly maps the n*m arbitrary feed back gain parameters to the arbitrary selection of neigenvalues and n*(m 1) eigenvector elements (total of n*m parameters). Since the system dynamic response is fully characterised by its eigenstructure, the direct control of the eigenstructure as discussed in this chapter provides a useful basis for feedback control design. The special mode decoupling eigenvector structures assignable, as illustrated in Example 2.1, demonstrates the unique advantage of the direct eigenvector element selection formulation alluded to in Chapter 1.

3 Eigenstructure synthesis algorithm
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This chapter discusses eigenstructure assignment. The algorithm sequentially constructs the specified eigenstructure while allowing maximal flexibility in the specification of eigenvalues/eigenvectors and assures the generation of an ndimensional eigenspace required for the solution. The numerical condition of the eigenstructure being synthesised, which is closely related to the robustness of the resulting solution, is transparent to the designer, through the σparameters, as the eigenvectors are sequentially synthesised. The simplicity of the algorithm makes it well suited for computer implementation as an iterative/interactive design tool for multivariable control system synthesis.

4 Eigenstructure assignment by output feedback
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The eigenstructure assignment using state variable feedback is analysed. It is interesting to examine the extent of eigenstructure assign ment that is possible if only output feedback is employed (number of outputs less than the number of states). At the outset, the pole placement problem with output feedback has received much attention by researchers over the years. In the state feedback case, all system poles can be arbitrarily assigned if the system is controllable. Unfortunately such a definitive statement in case of output feed back continues to be elusive.

5 Robust eigenstructure assignment
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The design of feedback systems based on a 'mathematical model' of the plant should be insensitive to uncertainty associated with the parameters of the model. This is required since the system model parameters, in most cases, are only a best estimate of the real plant. The concepts of gain and phase margins evolved to guarantee closedloop stability in the presence of specified uncertainty of the plant gain/phase characteristics. Thus, the design of 'robust' feedback systems has been extensively studied in the control literature. In the multivariable control setting, a wide range of robustness metrics has been postulated to character ise system behaviour under plant ignorance. The nature of the feedback design technique also influences the selection of appropriate robustness criteria, and this is also true for eigenstructure assignment. During the characterisation of eigenstructure assignment using state and output feedback, it was emphasised that for the solution to exist, the eigenvector matrix (X) must be nonsingular. In this chapter this nonsingularity of X will be numerically quantified and its relationship to the robustness of the resulting solution will be explained.

6 Modal canonical observers
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In this chapter a unified approach to the design of observers has been detailed. The algorithm developed is applicable to all categories of observers studied in the literature. The key concept of eigenstructure assignment developed in Chapter 2 is used in the construction of the observer. The observer eigenvalues define the observer stability matrix F, and the eigenvectors define the trans formation matrix T. Defining the stability matrix in the modal canonical form facilitates formulation of a computationally simple observer design algorithm. Further this modal canonical observer structure also enjoys advantages in real time implementation since (i) the stability matrix F has a minimum parameter representation and (ii) only a bank of firstorder (real eigenvalues) or secondorder (complex conjugate eigenvalues) filters needs to be implemented. Existence conditions of the observer with arbitrary eigenvalues are derived using the Kronecker canonical structure properties of the resulting singular rectangular matrix pencils. The computationally stable SCF is used to extract these structural properties. However, construction of the SCF is only required to check the existence of the solution for a specified observer problem and is not needed for design optimisation of the observer.

7 Model following control systems
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In this chapter the model following control problem has been formulated as an eigenstructure assignment problem. For the special case where the plant and model have the same number of states, both EMF and IMF control schemes can be adopted. The equivalence between the two schemes has been established. For this special case, the constraints on the plant and model stability matrices for PMF are also defined. In cases where model selection flexibility exists, these constraints can be used to advantage to improve tracking performance. The command tracker problem formulation at the outset tacitly assures only steadystate tracking of the output variables. This leads to the possibility of unacceptable transient response errors. This is especially true if the model and plant orders are the same. This appears to be a major limitation of the scheme. However, the key concept of considering the elements of the model output matrix as free design parameters, as proposed in this chapter, alleviates this problem. This allows finetuning of transient/steadystate mismatch errors in selected response variables by formulating a suitable optimisation problem. This tunable CGT design scheme substantially enhances the capability of the controller to optimise modelmatching errors.

8 Flight control system design guidelines
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In this chapter, the basic specifications for design of flight control systems have been outlined. It should be pointed out that historically, flight vehicle HQ guidelines evolved as a basis for the design of a new aircraft with desir able dynamic characteristics using only low authority stability augmentation systems. With the modern full authority highgain flybywire flight control systems, most of these specifications can easily be achieved. However, other design problems associated with such highgain/highbandwidth feedback systems, namely (i) stability margins, (ii) interaction of structural modes with rigid body dynamics and (iii) actuator rate saturation during largeamplitude manoeuvres in the presence of severe atmospheric turbulence, etc. take centre stage in the control law design process. Brief specifications/guidelines to meet these practical control system design requirements have also been outlined. The guidelines detailed in this chapter will be used for the design studies in Chapters 911. Additional useful flight control design practices have also been well documented in Reference 17.

9 Aircraft lateraldirectional handling qualities design
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This chapter discusses the eigenstructure assignment technique. It is to improve the lateraldirectional dynamics of an aircraft has been illustrated. It is shown that using the aircraft industry's preferred feedback sensor set of roll rate, yaw rate and lateral acceleration.

10 Aircraft longitudinal handling qualities design
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In this chapter, the pitch axis controller design of an aircraft is studied. If the aircraft has only one control surface in the pitch axis, eigenstructure synthesis is not possible and the design reduces to a simple eigenvalue assignment problem. Since the aircraft dynamic performance is dictated by the shortperiod mode,the natural frequency and damping of the mode can be modified by feedback to improve the response characteristics. The Gibson's time and frequency domain criteria are usually satisfied by proper design of the command filter. If the aircraft is equipped with more than one control surface in the pitch axis, some control can be exercised in shaping the eigenvectors. Again since the main interest is in shaping the shortperiod mode response, the complex conjugate pair of eigenvectors associated with the shortperiod mode needs to be synthesised. In this chapter, it is shown that the multivariable pitch axis design can be conveniently formulated as a model following control problem since the design objectives can be easily translated into an appropriate dynamic model. Examples of both IMF and EMF control design formulations are included to highlight the utility of eigenstructure assignment algorithms. Thus, for the pitch axis design, using multiple inputs, optimisation techniques play a major role both in constructing ideal models and in the subsequent design of the controllers to match the performance of these ideal models.

11 Rotorcraft handling qualities design
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In this chapter the design of a control system to improve the flying characteris tics of a BO105 helicopter model is addressed. The study is aimed at meeting the level 1 handling qualities requirements specified in ADS33EPRF for the pitch, roll and yaw axes. The control system consists of two parts, namely: (i) a feedback controller to reduce the severe interaxis coupling and (ii) a command path controller that shapes the response to pilot control inputs to reduce pilot workload when the helicopter is operated in a degraded UCE.

12 Aircraft flutter control system design
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This chapter is primarily based on eigenstructure assignment and classical control techniques used in the AFW program, with additional extension to improve upon the results presented therein. In this chapter, problem of stabilising aeroelastic modes has been examined. The study of the flutter mechanism, from a feedback control point of view, reveals that associated with each pair of interacting elastic modes (poles) participating in flutter, there exists a zero called the critical zero. This zero determines the achievable improvement in the damping of the unstable flutter mode. The role of EAC to optimally deploy multiple control effectors for this control problem is also highlighted.

Appendix A: Relevant flight mechanics models
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This appendix covers the following topics: aircraft lateraldirectional state variable model, aileron to rudder interconnect characteristics, and aircraft response to turbulence disturbance.

Appendix B: F8C aircraft state variable models
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The F8C aircraft state variable models are used for numerical examples used in the book. The linear perturbation rigid body models of the aircraft in Reference 1 are given in Cartesian (rectangular) coordinates. These are converted to spherical (polar) coordinates (convenient for stability and control analysis). The flight control system (FCS) hardware elements add phase lags associated with (i) sensors, (ii) actuators, (iii) filters to suppress structural mode response and (iv) digital computer delays. These effects have to be accounted in the assessment of the feedback system performance.

Appendix C: BO105 helicopter state variable models
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This appendix includes the following topics: BO105 helicopter dynamics, rotor model, state variables scaling, control variables scaling and flight control system hardware.

Appendix D: Properties of singular matrix pencils
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The system matrix pencil plays an important role in characterising many control theoretical properties of multivariable state space system. In particular, the characterisation of multivariable zeros is of interest in the design of observers (Chapter 6). A brief definition of multivariable zeros is included here for reference.

Appendix E: Conversion of FPS units to SI units
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The aircraft parameters and flight condition data used in the application chapters 912 have been given in FPS units. This has been done in order to preserve the original authenticated data of the referenced sources. In order to facilitate easy conversion of these data to SI units, the appropriate conversionfactors from FPS to SI units are given in this appendix.

Back Matter
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