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Über dieses Buch

This monograph presents theoretical methods involving the Hamilton–Jacobi–Bellman formalism in conjunction with set-valued techniques of nonlinear analysis to solve significant problems in dynamics and control. The emphasis is on issues of reachability, feedback control synthesis under complex state constraints, hard or double bounds on controls, and performance in finite time. Guaranteed state estimation, output feedback control, and hybrid dynamics are also discussed. Although the focus is on systems with linear structure, the authors indicate how to apply each approach to nonlinear and nonconvex systems. The main theoretical results lead to computational schemes based on extensions of ellipsoidal calculus that provide complete solutions to the problems. These computational schemes in turn yield software tools that can be applied effectively to high-dimensional systems. Ellipsoidal Techniques for Problems of Dynamics and Control: Theory and Computation will interest graduate and senior undergraduate students, as well as researchers and practitioners interested in control theory, its applications, and its computational realizations.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Linear Control Systems

Abstract
This chapter gives an exposition of control theory for linear systems with emphasis on items and techniques given in a form appropriate for topics in forthcoming chapters. It introduces problems of reachability and optimal target control under constraints, as well as time-optimal control. Indicated are solution approaches to open-loop control that involve the moment problem, the maximum principle, and the duality methods of convex analysis.
Alexander B. Kurzhanski, Pravin Varaiya

Chapter 2. The Dynamic Programming Approach

Abstract
This chapter describes general schemes of the Dynamic Programming approach. It introduces the notion of value function and its role in these schemes. They are dealt with under either classical conditions or directional differentiability of related functions, leaving more complicated cases to later chapters. Here the emphasis is on indicating solutions to forward and backward reachability problems for “linear-convex” systems and the design of closed-loop control strategies for optimal target and time-optimal feedback problems.
Alexander B. Kurzhanski, Pravin Varaiya

Chapter 3. Ellipsoidal Techniques: Reachability and Control Synthesis

Abstract
This chapter describes the ellipsoidal techniques for control problems introduced in earlier chapters. We derive formulas for reachability sets using the properties of ellipsoids and relations from convex analysis. The formulas are derived through inductive procedures. They allow calculation of both external and internal ellipsoidal approximations of forward and backward reachability sets with any desired level of accuracy. The approximations are illustrated on examples explained in detail, then followed by ellipsoid-based formulas for problems of reachability and control synthesis.
Alexander B. Kurzhanski, Pravin Varaiya

Chapter 4. Solution Examples on Ellipsoidal Methods: Computation in High Dimensions

Abstract
In this chapter we describe solution examples for controlled systems that illustrate the contents of Chaps. 1– 3. These include the multiple integrator, planar Newtonian motions and calming down a chain of springs. Special sections are devoted to relevant computational formulas and high-dimensional systems. Also discussed are possible degeneracy effects in computation and the means of their avoidance.
Alexander B. Kurzhanski, Pravin Varaiya

Chapter 5. The Comparison Principle: Nonlinearity and Nonconvexity

Abstract
This chapter introduces generalizations and applications of the presented approach prescribed earlier to nonlinear systems, nonconvex reachability sets and systems subjected to non-ellipsoidal constraints. The key element for these issues lies in the Comparison Principle for HJB equations which indicates schemes of approximating their complicated solutions by arrays of simpler procedures. Given along these lines is a deductive derivation of ellipsoidal calculus in contrast with previous inductive derivation.
Alexander B. Kurzhanski, Pravin Varaiya

Chapter 6. Impulse Controls and Double Constraints

Abstract
In the first section of this chapter we deal with the problem of feedback impulse control in the class of generalized inputs that may involve delta functions and discontinuous trajectories in the state space. Such feedback controls are not physically realizable. The second section thus treats the problem of feedback control under double constraints: both hard bounds and integral bounds. Such solutions are then used for approximating impulse controls by bounded “ordinary” functions.
Alexander B. Kurzhanski, Pravin Varaiya

Chapter 7. Dynamics and Control Under State Constraints

Abstract
The topics of this chapter are problems of reachability and system dynamics under state constraints in the form of reach tubes. Indicated are general approaches based on the Hamiltonian formalism and a related Comparison Principle. Further emphasis is on the dynamics of linear systems under hard bounds on the controls and system trajectories. A detailed solution is presented based on ellipsoidal approximations of bounded trajectory tubes.
Alexander B. Kurzhanski, Pravin Varaiya

Chapter 8. Trajectory Tubes State-Constrained Feedback Control

Abstract
This chapter begins with the theory of trajectory tubes which are necessary elements of realistic mathematical models for controlled processes and their evolutionary dynamics. We then deal with the evolution in time of state-constrained forward and backward reachability tubes also known as “viability tubes.” The backward tubes are then used to design feedback controls under state constraints that may also appear in the form of obstacles to be avoided by system trajectories.
Alexander B. Kurzhanski, Pravin Varaiya

Chapter 9. Guaranteed State Estimation

Abstract
This chapter deals with the problem of set-membership or “guaranteed” state estimation. The problem is to estimate the state of a dynamic process from partial observations corrupted by unknown but bounded noise in the system and measurement inputs (in contrast with stochastic noise). The problem is treated in both continuous and discrete time. Comparison with stochastic filtering is also discussed.
Alexander B. Kurzhanski, Pravin Varaiya

Chapter 10. Uncertain Systems: Output Feedback Control

Abstract
This chapter finalizes some results of earlier chapters for systems that operate under set-membership uncertainty. Its aim is to emphasize a successful application of previously described techniques to such systems. The chapter thus gives a concise presentation of solution techniques for the problem of output feedback control based on available measurements under set-membership uncertainty.
Alexander B. Kurzhanski, Pravin Varaiya

Chapter 11. Verification: Hybrid Systems

Abstract
This chapter deals with a specific class of hybrid systems which combine controlled continuous dynamics through switching from one available motion to another due to discrete-time logical commands. Solutions to the reachability problem and their verification are indicated, followed by computational schemes, The application of impulse controls to the switching process is described. Examples of various difficulty are worked out. The chapter is to demonstrate applicability of methods of this book to hybrid systems.
Alexander B. Kurzhanski, Pravin Varaiya

Backmatter

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