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

There is an increasing demand for dynamic systems to become safer and more reliable This requirement extends beyond the normally accepted safety-critical systems such as nuclear reactors and aircraft, where safety is of paramount importance, to systems such as autonomous vehicles and process control systems where the system availability is vital It is clear that fault diagnosis is becoming an important subject in modern control theory and practice Robust Model-Based Fault Diagnosis for Dynamic Systems presents the subject of model-based fault diagnosis in a unified framework It contains many important topics and methods; however, total coverage and completeness is not the primary concern The book focuses on fundamental issues such as basic definitions, residual generation methods and the importance of robustness in model-based fault diagnosis approaches In this book, fault diagnosis concepts and methods are illustrated by either simple academic examples or practical applications The first two chapters are of tutorial value and provide a starting point for newcomers to this field The rest of the book presents the state of the art in model-based fault diagnosis by discussing many important robust approaches and their applications This will certainly appeal to experts in this field Robust Model-Based Fault Diagnosis for Dynamic Systems targets both newcomers who want to get into this subject, and experts who are concerned with fundamental issues and are also looking for inspiration for future research The book is useful for both researchers in academia and professional engineers in industry because both theory and applications are discussed Although this is a research monograph, it will be an important text for postgraduate research students world-wide The largest market, however, will be academics, libraries and practicing engineers and scientists throughout the world

Robust Model-Based Fault Diagnosis for Dynamic Systems

1. Introduction. 2. Linear Algebra. 3. Linear Systems. 4. H2 and Ha Spaces. 5. Internal Stability. 6. Performance Specifications and Limitations. 7. Balanced Model Reduction. 8. Uncertainty and Robustness. 9. Linear Fractional Transformation. 10. m and m- Synthesis. 11. Controller Parameterization. 12. Algebraic Riccati Equations. 13. H2 Optimal Control. 14. Ha Control. 15. Controller Reduction. 16. Ha Loop Shaping. 17. Gap Metric and ...u- Gap Metric. 18. Miscellaneous Topics. Bibliography. Index.

/pdf/essentials-of-robust-control-3kaks6yk3h.pdf

Essentials of Robust Control

This augmented edition of a respected text teaches the reader how to use linear quadratic Gaussian methods effectively for the design of control systems. It explores linear optimal control theory from an engineering viewpoint, with step-by-step explanations that show clearly how to make practical use of the material. The three-part treatment begins with the basic theory of the linear regulator/tracker for time-invariant and time-varying systems. The Hamilton-Jacobi equation is introduced using the Principle of Optimality, and the infinite-time problem is considered. The second part outlines the engineering properties of the regulator. Topics include degree of stability, phase and gain margin, tolerance of time delay, effect of nonlinearities, asymptotic properties, and various sensitivity problems. The third section explores state estimation and robust controller design using state-estimate feedback. Numerous examples emphasize the issues related to consistent and accurate system design. Key topics include loop-recovery techniques, frequency shaping, and controller reduction, for both scalar and multivariable systems. Self-contained appendixes cover matrix theory, linear systems, the Pontryagin minimum principle, Lyapunov stability, and the Riccati equation. Newly added to this Dover edition is a complete solutions manual for the problems appearing at the conclusion of each section.

Optimal Control: Linear Quadratic Methods

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Feedback control is developed for the class of flexible systents described by the generalized wave equation with damping. The control force distribution is provided by a number of point force actuators and the system displacements and/or their velocities are measured at various points. A feedback controller is developed for a finite number of modes of the flexible system and the controllability and observability conditions necessary for successful operation are displayed. The control and observation spillover due to the residual (uncontrolled) modes is examined and the combined effect of control and observation spillover is shown to lead to potential instabilities in the closed-loop system. Some remedies for spillover, including a straightforward phase-locked loop prefilter, are suggested to remove the instability mechanism. The concepts of this paper are illustrated by some numerical studies on the feedback control of a simply-supported Euler-Bernoulli beam with a single actuator and sensor.

Feedback control of flexible systems

Since mechanically flexible systems are distributed-parameter systems, they are infinite-dimensional in theory and, in practice, must be modelled by large-dimensional systems. The fundamental problem of actively controlling very flexible systems is to control a large-dimensional system with a much smaller dimensional controller. For example, a large number of elastic modes may be needed to describe the behavior of a flexible satellite; however, active control of all these modes would be out of the question due to onboard computer limitations and modelling error. Consequently, active control must be restricted to a few critical modes. The effect of the residual (uncontrolled) modes on the closed-loop system is often ignored. In this paper, we consider the class of flexible systems that can be described by a generalized wave equation,u
tt+Au=F, which relates the displacementu(x,t) of a body Θ inn-dimensional space to the applied force distributionF(x,t). The operatorA is a time-invariant symmetric differential operator with a discrete, semibounded spectrum. This class of distributed parameter systems includes vibrating strings, membranes, thin beams, and thin plates. The control force distribution
 
$$F(x,t) = \sum\limits_{i = 1}^M { \delta (x - x_i )f_i (t)} $$

 is provided byM point force actuators located at pointsx
i on the body. The displacements (or their velocities) are measured byP point sensorsy
i(t)=u(z
j,t), oru
t(z
j,t),j=1, 2, ...,P, located at various pointsz
j along the body. We obtain feedback control ofN modes of the flexible system and display the controllability and observability conditions required for successful operation. We examine the control and observation spillover due to the residual modes and show that the combined effect of spillover can lead to instabilities in the closed-loop system. We suggest some remedies for spillover, including a straightforward phase-locked loop prefilter, to remove the instability mechanism. To illustrate the concepts of this paper, we present the results of some numerical studies on the active control of a simply supported beam. The beam dynamics are modelled by the Euler-Bernoulli partial differential equation, and the feedback controller is obtained by the above procedures. One actuator and one sensor (at different locations) are used to control three modes of the beam quite effectively. A fourth residual mode is simulated, and the destabilizing effect of control and observation spillover together on this mode is clearly illustrated. Once observation spillover is eliminated (e.g., by prefiltering the sensor outputs), the effect of control spillover alone on this system is negligible.

https://link.springer.com/content/pdf/10.1007/BF00932903.pdf

Active control of flexible systems

This paper presents a mathematical framework for discussion of large space structure (LSS) control theory. Within this framework, current trends in LSS control theory and related topics in general control science are surveyed.

Trends in large space structure control theory: Fondest hopes, wildest dreams

which relates the displacements u(x,t) of the equilibrium position of a flexible structure ft (a bounded open connected set with smooth boundary dfi in /7-dimensiona l space R") to the applied force distribution F(x,t). The mass density m(x) is a positive function of the location x on the structure. The change of variables u(x,t) ^u(x,t)/m(x) l/2 eliminates m(x) without changing the properties of Eq. (1) and, henceforth, assume m(x) = \. The non-negative real number £ is the damping coefficient of the structure; it is quite small for LSS. The operator A is a symmetric, non-negative, time-invariant differential operator with compact resolvent and a square root A'/2. The domain D(A) of A contains all sufficiently differentiable functions which satisfy the appropriate boundary conditions for the LSS. D(A) is dense in the Hilbert space //=L 2 (Q) with the usual inner product (.,.)

Direct Velocity Feedback Control of Large Space Structures

This article considers an adaptive control scheme previously developed for region 2 control of a variable speed wind turbine. In this paper, the question of theoretical stability of the torque controller is addressed, showing that the rotor speed is asymptotically stable under the torque control law in the constant wind speed input case and L/sub 2/ stable with respect to time-varying wind input. Further, a method is derived for selecting /spl gamma//sub /spl Delta/M/ in the gain adaptation law to guarantee convergence of the adaptive gain M to its optimal value M*.

Mark J. Balas

Papers

Feedback control of flexible systems

Active control of flexible systems

Trends in large space structure control theory: Fondest hopes, wildest dreams

Direct Velocity Feedback Control of Large Space Structures

Control of variable-speed wind turbines: standard and adaptive techniques for maximizing energy capture