Mathematical Theory of Control Systems Design

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1 Mathematical Theory of Control Systems Design by V. N. Afarias'ev, V. B. Kolmanovskii and V. R. Nosov Moscow University of Electronics and Mathematics, Moscow, Russia KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

2 Preface Introduction xix xxi PART ONE. STABILITY OF CONTROL SYSTEMS Chapter I. Continuous and Discrete Deterministic Systems 3 1. Basic Definitions of Stability Theory for Continuous systems 3 1. Stability 3 2. Asymptotic stability 8 3. Other stability definitions Lyapunov's Direct Method Examples of the Use of Lyapunov's Method Stability of the motion of a shell Motion of a rigid body fastened at one point Development of Lyapunov's Method The Barbashin-Krasovskii theorem The Matrosov criterion The comparison principle Stability with respect to part of the variables (partial stability) Stability of Linear Time-Invariant Systems The Routh-Hurwitz criterion Frequency criteria of stability Automatic frequency control system of a heterodyne receiver Linear single-circuit automatic control systems Robust stability Stability of Linear Time-Varying Equations On the method of "frozen" coefficients Systems with an almost constant matrix Linear systems with periodic coefficients Equation of the second order with periodic coefficients Parametric resonance in engineering applications 42

3 7. Lyapunov Functions for Linear Time-Invariant Systems and First Approximation Stability Lyapunov's matrix equation Stability in the first approximation Rotation of a shell Time-Varying equations of the first approximation Synthesis of Control Systems for the Manipulator Robot Single-link manipulator robot The Cyclone-type robot 50 9 Use of the Logarithmic Norm in Stability Theory The definition and properties of the logarithmic norm Stability of nonlinear systems Use of Degenerate Lyapunov Functions Stability of Discrete Systems Lyapunov's direct method Linear time-invariant equations Stability in the first approximation Stability with respect to specified variables Unconditional minimization of functions 67 Main Results and Formulas of Chapter I 69 Chapter II. Stability of Stochastic Systems Introduction ' Some Preliminaries from Probability Theory and the Theory of Stochastic processes Basic probability space Random variables Stochastic processes Stochastic Integrals and Stochastic Differential Equations The stochastic integrals of Ito and Stratonovich The Ito formula Markov diffusion processes Linear stochastic equations Definition of Stochastic Stability Application of Lyapunov's Direct Method Sufficient stability conditions Stability in mean square of linear systems Scalar equations of the nth order Stability in Probability of Satellite Motion The pitch stability of a symmetric satellite in a circular orbit The yaw angle stability of a satellite in a circular equatorial orbit 100 Main Results and Formulas of Chapter II 101

4 Exercises of Part One 105 PART TWO. CONTROL OF DETERMINISTIC SYSTEMS Chapter III. Description of Control Problems Introduction Statement of the Optimal Control Problem The equations of evolution of a system The functional to be minimized (cost functional) Trajectory constraints Control constraints Joint constraints Examples of Optimal Control in Engineering Optimal control of an electric motor Optimization of the characteristics of nuclear reactors Optimal control of spacecraft 145 Main Results and Formulas of Chapter III 150 Chapter IV. The Classical Calculus of Variations and Optimal Control Problems with a Variable End Point and Fixed Time Main assumptions Cauchy's formula Necessary conditions for optimal control The Boltz problem Optimal Control of Linear Systems with a Quadratic Functional Necessary conditions for optimal control Construction of an optimal control Matrix Riccati equation The scalar case Optimal control of wire reeling Necessary Conditions for Optimal Control. The Method of Lagrange Multipliers The method of Lagrange multipliers Fixed initial and terminal moments, and fixed initial state Fixed initial and terminal moments, and variable initial and terminal states Problems with fixed values of some state variables at the initial and terminal moments Problems with an unspecified terminal moment The Chaplygin problem 175

5 7. Maximization of the rocket velocity immediately before putting the rocket into a rectilinear trajectory 177 Main Results and Formulas of Chapter IV 180 Chapter V. The Maximum Principle Problems with a Variable Terminal Point and Prescribed Transfer Time The Mayer problem The Boltz problem About solutions of maximum principle equations Rotation of a motor shaft through a maximum angle Problems with an Unspecified Terminal Moment Reduction to a Mayer problem Necessary conditions for the optimality of systems that are linear in a scalar control variable Multivariable control Time-invariant systems Transfer of a system from one manifold to another manifold Control problems with isoperimetric constraints Sufficiency of the maximum principle Connection between the maximum principle and the classical calculus of variations The maximum principle for discrete systems Practical Applications of the Maximum Principle Optimal configuration of a nuclear reactor Control of a motion with regulated friction Problem of the soft landing on the moon Problem of the planar transfer of a space vehicle from one circular orbit to another Control of Ecological Systems Equations describing the evolution of a single population Communities of at least two species Statements of typical control problems for ecological systems Time-optimal control of the "predator-prey" system 227 Main Results and Formulas of Chapter V 234 Chapter VI. Linear Control Systems A Time-Optimal Problem Evaluation of the number of switch points The damping of the material point The damping of a pendulum 244

6 4. "Control of the rotation of an axially symmetric space vehicle The controllability set Controllability of Linear Systems Controllability of linear time-invariant systems Controllability of linear time-varying systems Canonical form of linear time-invariant control systems Canonical form of linear time-varying control systems The Hautus criterion for controllability Controllability of a two-link manipulator Observation in Linear Systems. Observers Statement of an observation problem. Duality of control and observation problems ' On a method of determining the state vector Observer of maximum order Observer of reduced order (the Luenberger observer) Observer of reduced order in the stabilization system of an aircraft Linear Feedback Control Systems Various ways of describing feedback control systems. The realization problem Criteria of performance for SISO-systems Criteria of performance for MIMO-systems. The Hardy spaces H 2 and #«, Fundamentals of tfoo-theory Statement of the problem of constructing an Hoooptimal controller Estimates of the Hoo and Hi norms of the transfer matrix of an auxiliary system The ifoo-problem for a static controller The general case of a dynamic controller Robust stability Zeros of a Linear Time-Invariant System and Their Use 292 Main Results and Formulas of Chapter VI 297 Chapter VII. Dynamic Programming Approach. Sufficient Conditions for Optimal Control The Bellman Equation and its Properties The principle of dynamic programming. Heuristic derivation of The Bellman equation Determining an F-control with the help of the dynamic programming approach 303

7 3. Connection between the dynamic programming approach and the maximum principle Determining F-control in the problem of damping the motion of a rigid body Optimal procedure for reducing the power of a nuclear reactor The linear-quadratic problem Control on an Unbounded Time Interval. Stabilization of Dynamical Systems Problem statements Lyapunov's direct method for the optimal stabilization Problem Exponential stabilization Stabilization of the motion of a manipulator robot Stabilization of Linear Systems Time-varying linear-quadratic problems The use of the method of successive approximations for determining optimal control Time-invariant linear-quadratic equation The algebraic Riccati equation Stabilization of Quasilinear Systems Quasioptimal stabilization and estimation of its error Adaptive stabilization Sufficient Conditions for Optimal Control Using Auxiliary Functions Conditions for optimal control Sufficient conditions for the existence of a minimizing sequence Problems with unspecified time 337 Main Results and Formulas of Chapter VII 339 Chapter VIII. Some Additional Topics of Optimal Control Theory Existence of Optimal Control Problem statement and the main assumptions Main theorem Analysis of the conditions of the main theorem Singular Optimal Control The definition and determination of singular controls Optimality of singular control Generalization of the Kelley and Kopp-Moyer conditions Chattering Mode Sliding Optimal Mode 365 Main Results and Formulas of Chapter VIII 372

8 Exercises of Part Two 375 PART THREE. OPTIMAL CONTROL OF DYNAMICAL SYSTEMS UNDER RANDOM DISTURBANCES Chapter IX. Control of Stochastic Systems. Problem Statements and Investigation Techniques Statements of Control Problems for Stochastic Systems Equations of motion of a system Control constraints Cost functional Dynamic Programming Approach The Bellman function The Bellman equation Connection between the Bellman function and the Bellman equation Stochastic Linear-Quadratic Problem on a Finite Time Interval Linear-quadratic problems in the case of an accurate measurement of phase coordinates Linear-quadratic problems under incomplete information Optimal program control in linear-quadratic problems Linear-quadratic problem for Gaussian and Poisson disturbances Control of wire reeling with regard to random disturbances Control on an Unbounded Time Interval. Stabilization of Stochastic Control Systems Problem statement Application of Lyapunov's direct method to optimal stabilization problems Stabilization of linear stochastic systems Approximate Methods for Determining Optimal Control Description of the algorithm of successive approximations Zero approximation estimate First approximation estimate Higher-order approximation estimates 429 Main Results and Formulas of Chapter IX 431 Chapter X. Optimal Control on a Time Interval of Random Duration Time-Optimal Control 435

9 1. Statement of time-optimal problems in dynamical systems under random disturbances Existence of an admissible control An algorithm for constructing optimal control Time-optimal control of the motion of a rigid body Numerical construction of the time-optimal control of the motion of a material point Time-Optimality for a Gyrostat Problem statement The existence of an admissible control Construction of an optimal control Control Problems with Stochastic Functional Problem statement and method of solution Optimal control of the motion of a material point involving a stochastic cost functional Maximization of the probability that a point stays in a given region Stochastic control of the motion of a simple pendulum Control of a rigid body, given the stochastic cost functional Maximization of the mean time for the system staying within a given region. Determination of optimal control for the motion of a rigid body 463 Main Results and Formulas of Chapter X 466 Chapter XI. Optimal Estimation of the State of the System Estimation Problems Involving Random Disturbances Statement of an optimal estimation problem Linear estimation Optimal estimation of Gaussian variables Linear estimation of stationary processes. The Wiener-Hopf equation The Kalman filter Problem statement The dual problem of optimal control Equation for the estimation error Equation for the optimal estimate Stability of the filter Filtering problem with constant parameters Filtering problem with degenerate noise in the observation channel Optimal extrapolation Optimal interpolation The discrete Kalman filter Some Relations of Nonlinear Filtering Theory 488

10 1. Statement of the general filtering problem The Stratonovich differential equations for the conditional probability distribution Conditionally Gaussian processes Quasioptimal and quasilinear filtering (the scalar case) 493r 5. Quasilinear filtering (the multidimensional case) 499 Main Results and Formulas of Chapter XI,503 Chapter XII. Optimal Control of the Observation Process Optimization of the Observation Process Problem statement. Basic relations Construction of optimal observation laws minimizing terminal variance 1>13 3. An example of observation control with integral cost functional Optimal pulse observation laws Pulsed observation of a material point Optimal noise control in the observation channel Optimal Combination of Control and Observation Linear-quadratic problem A deterministic scalar system A stochastic scalar system 534 Main Results and Formulas of Chapter XII 542 Exercises of Part Three 543 PART FOUR. NUMERICAL METHODS IN CONTROL SYSTEMS Chapter XIII. Linear Time-Invariant Control Systems Stability of Linear Time-Invariant Systems Precise methods for solving the complete eigenvalue problem Iterative methods The Routh-Hurwitz criterion The Zubov method of a functional transformation of a matrix Methods of Solving the Lyapunov Equation Preliminary remarks The series method Method of the matrix sign function Application of the QR-algorithm Construction of stabilizing control Computation of the covariance matrix Controllability and Observability 570

11 xvi 4. Linear-Quadratic Time-Invariant Problem of Optimal stabilization Reduction to a sequence of Lyapunov equations Use of the QR-algorithm Landing control of a Boeing-747 airplane 576 Main Results and Formulas of Chapter XIII 579 Chapter XIV. Numerical Methods for the Investigation of Nonlinear Control Systems Analysis of Transients. The Runge-Kutta Methods On numerical methods of investigating systems One-step methods Error estimates for Runge-Kutta methods Estimation of solution errors for stable equations Standard programs Analysis of Transients. Multistep Methods General definitions Particular multistep formulas Computational scheme for multistep formulas Error estimation The Butcher formulas Stiff Systems of Equations Numerical Methods for the Design of Optimal Control in the Linear-Quadratic Problem Determination of an optimal F-control for timevarying systems Time-invariant linear-quadratic problem on a finite interval Optimal stabilization problems and methods for solving the algebraic Riccati equation The numerical design of the Kalman filter 613 Main Results and Formulas of Chapter XIV 617 Chapter XV. Numerical Design of Optimal Control Systems On Numerical Methods of Solving Optimal Control Problems Reduction to Nonlinear Programming Reduction to a Boundary-Value Problem Boundary-value problem of the maximum principle The Newton method of solving a boundary-value problem An example of reducing an optimal control problem to a boundary-value problem Solution of Linear Boundary-Value Problems 625

12 xvii 1. Reduction of an optimal control problem to a linear boundary-value problem Reduction of a linear boundary-value problem to Cauchy problems Method of the transfer of boundary conditions The Abramov method The Shatrovskii Method of Successive Improvements of Control The Fedorenko Method of Reducing to a Linear Programming Problem 635 Main Results and Formulas of Chapter XV 639 Exercises of Part Four 641 General References 657 Subject Index 663

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