Control, Identification, and Input Optimization

TL;DR: In this paper, the authors present an approach for optimal control and filtering of In-homogeneous suspended cable systems with two unknown parameters and Vector Measurement methods for linear two-point boundary value problems.

Abstract: I. Introduction.- 1. Introduction.- 1.1. Optimal Control.- 1.2. System Identification.- 1.3. Optimal Inputs.- 1.4. Computational Preliminaries.- Exercises.- II. Optimal Control and Methods for Numerical Solutions.- 2. Optimal Control.- 2.1. Simplest Problem in the Calculus of Variations.- 2.1.1. Euler-Lagrange Equations.- 2.1.2. Dynamic Programming.- 2.1.3. Hamilton-Jacobi Equations.- 2.2. Several Unknown Functions.- 2.3. Isoperimetric Problems.- 2.4. Differential Equation Auxiliary Conditions.- 2.5. Pontryagin's Maximum Principle.- 2.6. Equilibrium of a Perfectly Flexible Inhomogeneous Suspended Cable.- 2.7. New Approaches to Optimal Control and Filtering.- 2.8. Summary of Commonly Used Equations.- Exercises.- 3. Numerical Solutions for Linear Two-Point Boundary-Value Problems..- 3.1. Numerical Solution Methods.- 3.1.1. Matrix Riccati Equation.- 3.1.2. Method of Complementary Functions.- 3.1.3. Invariant Imbedding.- 3.1.4. Analytical Solution.- 3.2. An Optimal Control Problem for a First-Order System.- 3.2.1. The Euler-Lagrange Equations.- 3.2.2. Pontryagin's Maximum Principle.- 3.2.3. Dynamic Programming.- 3.2.4. Kalaba's Initial-Value Method.- 3.2.5. Analytical Solution.- 3.2.6. Numerical Results.- 3.3. An Optimal Control Problem for a Second-Order System.- 3.3.1. Numerical Methods.- 3.3.2. Analytical Solution.- 3.3.3. Numerical Results and Discussion.- Exercises.- 4. Numerical Solutions for Nonlinear Two-Point Boundary-Value Problems.- 4.1. Numerical Solution Methods.- 4.1.1. Quasilinearization.- 4.1.2. Newton-Raphson Method.- 4.2. Examples of Problems Yielding Nonlinear Two-Point Boundary-Value Problems.- 4.2.1. A First-Order Nonlinear Optimal Control Problem.- 4.2.2. Optimization of Functionals Subject to Integral Constraints.- 4.2.3. Design of Linear Regulators with Energy Constraints.- 4.3. Examples Using Integral Equation and Imbedding Methods.- 4.3.1. Integral Equation Method for Buckling Loads.- 4.3.2. An Imbedding Method for Buckling Loads.- 4.3.3. An Imbedding Method for a Nonlinear Two-Point Boundary-Value Problem.- 4.3.4. Post-Buckling Beam Configurations via an Imbedding Method.- 4.3.5. A Sequential Method for Nonlinear Filtering.- Exercises.- III. System Identification.- 5. Gauss-Newton Method for System Identification.- 5.1. Least-Squares Estimation.- 5.1.1. Scalar Least-Squares Estimation.- 5.1.2. Linear Least-Squares Estimation.- 5.2. Maximum Likelihood Estimation.- 5.3. Cramer-Rao Lower Bound.- 5.4. Gauss-Newton Method.- 5.5. Examples of the Gauss-Newton Method.- 5.5.1. First-Order System with Single Unknown Parameter.- 5.5.2. First-Order System with Unknown Initial Condition and Single Unknown Parameter.- 5.5.3. Second-Order System with Two Unknown Parameters and Vector Measurement.- 5.5.4. Second-Order System with Two Unknown Parameters and Scalar Measurement.- Exercises.- 6. Quasilinearization Method for System Identification.- 6.1. System Identification via Quasilinearization.- 6.2. Examples of the Quasilinearization Method.- 6.2.1. First-Order System with Single Unknown Parameter.- 6.2.2. First-Order System with Unknown Initial Condition and Single Unknown Parameter.- 6.2.3. Second-Order System with Two Unknown Parameters and Vector Measurement.- 6.2.4. Second-Order System with Two Unknown Parameters and Scalar Measurement.- Exercises.- 7. Applications of System Identification.- 7.1. Blood Glucose Regulation Parameter Estimation.- 7.1.1. Introduction.- 7.1.2. Physiological Experiments.- 7.1.3. Computational Methods.- 7.1.4. Numerical Results.- 7.1.5. Discussion and Conclusions.- 7.2. Fitting of Nonlinear Models of Drug Metabolism to Experimental Data.- 7.2.1. Introduction.- 7.2.2. A Model Employing Michaelis and Menten Kinetics for Metabolism.- 7.2.3. An Estimation Problem.- 7.2.4. Quasilinearization.- 7.2.5. Numerical Results.- 7.2.6. Discussion.- Exercises.- IV. Optimal Inputs for System Identification.- 8. Optimal Inputs.- 8.1. Historical Background.- 8.2. Linear Optimal Inputs.- 8.2.1. Optimal Inputs and Sensitivities for Parameter Estimation.- 8.2.2. Sensitivity of Parameter Estimates to Observations.- 8.2.3. Optimal Inputs for a Second-Order Linear System.- 8.2.4. Optimal Inputs Using Mehra's Method.- 8.2.5. Comparison of Optimal Inputs for Homogeneous and Nonhomogeneous Boundary Conditions.- 8.3. Nonlinear Optimal Inputs.- 8.3.1. Optimal Input System Identification for Nonlinear Dynamic Systems.- 8.3.2. General Equations for Optimal Inputs for Nonlinear Process Parameter Estimation.- Exercises.- 9. Additional Topics for Optimal Inputs.- 9.1. An Improved Method for the Numerical Determination of Optimal Inputs.- 9.1.1. Introduction.- 9.1.2. A Nonlinear Example.- 9.1.3. Solution via Newton-Raphson Method.- 9.1.4. Numerical Results and Discussion.- 9.2. Multiparameter Optimal Inputs.- 9.2.1. Optimal Inputs for Vector Parameter Estimation.- 9.2.2. Example of Optimal Inputs for Two-Parameter Estimation.- 9.2.3. Example of Optimal Inputs for a Single-Input, Two-Output System.- 9.2.4. Example of Weighted Optimal Inputs.- 9.3. Observability, Controllability, and Identifiability.- 9.4. Optimal Inputs for Systems with Process Noise.- 9.5. Eigenvalue Problems.- 9.5.1. Convergence of the Gauss-Seidel Method.- 9.5.2. Determining the Eigenvalues of Saaty's Matrices for Fuzzy Sets.- 9.5.3. Comparison of Methods for Determining the Weights of Belonging to Fuzzy Sets.- 9.5.4. Variational Equations for the Eigenvalues and Eigenvectors of Nonsymmetric Matrices.- 9.5.5. Individual Tracking of an Eigenvalue and Eigenvector of a Parametrized Matrix.- 9.5.6. A New Differential Equation Method for Finding the Perron Root of a Positive Matrix.- Exercises.- 10. Applications of Optimal Inputs.- 10.1. Optimal Inputs for Blood Glucose Regulation Parameter Estimation.- 10.1.1. Formulation Using Bolie Parameters for Solution by Linear or Dynamic Programming.- 10.1.2. Formulation Using Bolie Parameters for Solution by Method of Complementary Functions or Riccati Equation Method.- 10.1.3. Improved Method Using Bolie and Bergman Parameters for Numerical Determination of the Optimal Inputs.- 10.2. Optimal Inputs for Aircraft Parameter Estimation.- Exercises.- V. Computer Programs.- 11. Computer Programs for the Solution of Boundary-Value and Identification Problems.- 11.1. Two-Point Boundary-Value Problems.- 11.2. System Identification Problems.- References.- Author Index.