Estimators for Uncertain Dynamic Systems

When solving the control and design problems in aerospace and naval engi neering, energetics, economics, biology, etc., we need to know the state of investigated dynamic processes. The presence of inherent uncertainties in the description of these processes and of noises in measurement devices leads to the necessity to construct the estimators for corresponding dynamic systems. The estimators recover the required information about system state from mea surement data. An attempt to solve the estimation problems in an optimal way results in the formulation of different variational problems. The type and complexity of these variational problems depend on the process model, the model of uncertainties, and the estimation performance criterion. A solution of variational problem determines an optimal estimator. Howerever, there exist at least two reasons why we use nonoptimal esti mators. The first reason is that the numerical algorithms for solving the corresponding variational problems can be very difficult for numerical imple mentation. For example, the dimension of these algorithms can be very high.

Klappentext

The optimal estimation problems for linear dynamic systems, and in particular for systems with aftereffect, reduce to different variational problems. The type and complexity of these variational problems depend on the process model, the model of uncertainties, and the estimation performance criterion. A solution of a variational problem determines an optimal estimator. In addition, frequently the optimal algorithm for one noise model must operate under another, more complex assumption about noise. Hence, simplified algorithms must be used. It is important to evaluate the level of nonoptimality for the simplified algorithms. Since the original variational problems can be very difficult, the estimate of nonoptimality must be obtained without solving the original variational problem. br/ In this book, guaranteed levels of nonoptimality for simplified estimation and control algorithms are constructed. To obtain these levels the duality theory for convex extremal problems is used. br/ emAudience:/em This book will be of interest to applied mathematicians, researchers and engineers who deal with estimation and control systems. The material, which requires a good knowledge of calculus, is also suitable for a two-semester graduate or postgraduate course.

Inhalt

1. Guaranteed Parameter Estimation.- 1. Simplest Guaranteed Estimation Problem.- 2. Continuous Measurement Case.- 3. Linear Programming.- 4. Necessary and Sufficient Conditions for Optimality.- 5. Dual Problem and Chebyshev Approximation.- 6. Combined Model for Measurement Noise.- 7. Least-Squares Method in Guaranteed Parameter Estimation.- 8. Guaranteed Estimation with Anomalous Measurement Errors.- 9. Comments to Chapter 1.- 10. Excercises to Chapter 1.- 2. Guaranteed Estimation in Dynamic Systems.- 1. Lagrange Principle and Duality.- 2. Uncertain Deterministic Disturbances.- 3. Conditions for Optimality of Estimator.- 4. Computation of Estimators.- 5. Optimality of Linear Estimators.- 6. Phase Constraints in Guaranteed Estimation Problem.- 7. Comments to Chapter 2.- 8. Excercises to Chapter 2.- 3. Kalman Filter in Guaranteed Estimation Problem.- 1. Level of Nonoptimality for Kaiman Filter.- 2. Bound for the Level of Nonoptimality.- 3. Derivation of Main Result.- 4. Kaiman Filter with Discrete Measurements.- 5. Proofs for the Case of Discrete Measurements.- 6. Examples for the Bounds of Nonoptimality Levels.- 7. Comments to Chapter 3.- 8. Excercises to Chapter 3.- 4. Stochastic Guaranteed Estimation Problem.- 1. Optimal Stochastic Guaranteed Estimation Problem.- 2. Approximating Problem. Bound for the Level of Nonoptimality.- 3. Derivation of Main Result for Stochastic Problem.- 4. Discrete Measurements in Stochastic Estimation Problem.- 5. Examples for Stochastic Problems.- 6. Kaiman Filter under Uncertainty in Intensities of Noises.- 7. Comments to Chapter 4.- 8. Excercises to Chapter 4.- 5. Estimation Problems in Systems with Aftereffect.- 1. Pseudo-Fundamental Matrix and Cauchy Formula.- 2. Guaranteed Estimation in Dynamic Systems with Delay.- 3. Level of Nonoptimality in Stochastic Problem.- 4. Simplified Algorithms for Mean-Square Filtering Problem.- 5. Control Algorithms for Systems with Aftereffect.- 6. Reduced Algorithms for Systems with Weakly Connected Blocks.- 7. Comments to Chapter 5.- 8. Excercises to Chapter 5.