《Dynamic Systems Models: New Methods of Parameter and State Estimation》

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《Dynamic Systems Models: New Methods of Parameter and State Estimation》
动态系统模型：参数和状态估计的新方法
作者：
Josif A. Boguslavskiy (deceased)
State Scientific Research Institute
of Automated Systems
Moscow
Russia
出版社：Springer
出版时间：2016年

《Dynamic Systems Models: New Methods of Parameter and State Estimation》

《Dynamic Systems Models: New Methods of Parameter and State Estimation》

《Dynamic Systems Models: New Methods of Parameter and State Estimation》

《Dynamic Systems Models: New Methods of Parameter and State Estimation》

目录
1 Linear Estimators of a Random Parameter Vector . . . . . . . . . . . . 1
1.1 Linear Estimator, Optimal in the Root-Mean-Square Sense . . . . 1
1.2 Vector Measure of Nonlinearity of Vector Y1 in Relation
to Vector h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Decomposition of Path of Observations to the Recurrence
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Recurrent Form of Algorithm for Estimator Vector . . . . . . . . . 9
1.5 Problem of Optimal Linear Filtration . . . . . . . . . . . . . . . . . . . 13
1.6 Problem of Linear Optimal Recurrent Interpolation
(Problem of Optimal Smoothing) . . . . . . . . . . . . . . . . . . . . . . 16
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Basis of the Method of Polynomial Approximation . . . . . . . . . . . . 19
2.1 Extension Sets of Observations: The Heuristic Path
for Nonlinear Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 The Statistical Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Polynomial Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Calculating Statistical Moments and Choice
of Stochastic Measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Fragment of Program of Modified Method of Trapezoids . . . . . 27
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Polynomial Approximation and Optimization of Control . . . . . . . . 29
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Problem of Polynomial Approximation of a Given Function . . . 30
3.3 Applied Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.1 Detection of a Polynomial Function . . . . . . . . . . . . . . . 32
3.3.2 Approximation Errors for a State Vector
of Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Polynomial Approximation in Control
Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Optimization of Control by a Linear System: Linear
and Quadratic Optimality Criteria . . . . . . . . . . . . . . . . . . . . . 38
v
3.6 Approximate Control Optimization
for a Nonlinear Dynamic System . . . . . . . . . . . . . . . . . . . . . . 41
3.7 Polynomial Approximation with Random Errors . . . . . . . . . . . 42
3.8 Identification of a ‘‘Black Box’’. . . . . . . . . . . . . . . . . . . . . . . 43
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 Polynomial Approximation Technique Applied
to Inverse Vector-Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 The Problem of Polynomial Approximation
of an Inverse Vector-Function . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 A Case Where Multiple Root Vectors Exist Along
with Partitioning of the a Priori Domain . . . . . . . . . . . . . . . . . 53
4.4 Correctness of the Estimator Algorithm and a Way of Taking
Random Observation Items into Account . . . . . . . . . . . . . . . . 54
4.5 Implementations of the Polynomial Approximation
Technique Applied to the Inverse Vector-Function. . . . . . . . . . 56
4.6 Numerical Solutions of Underdetermined and Overdetermined
Systems of Linear Algebraic Solutions . . . . . . . . . . . . . . . . . . 59
4.7 Solving Simultaneous Equations with Nonlinearities
Expressed by Integer Power Series. . . . . . . . . . . . . . . . . . . . . 63
4.8 Solving Simultaneous Equations with Nonlinearities Expressed
by Trigonometric Functions, Exponentials, and Functions
Including Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.9 Solving a Two-Point Boundary Value Problem
for a System of Nonlinear Differential Equations. . . . . . . . . . . 66
4.10 The System of Algebraic Equations
with Complex-Valued Roots . . . . . . . . . . . . . . . . . . . . . . . . . 68
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5 Identification of Parameters of Nonlinear Dynamic Systems;
Smoothing, Filtration, Forecasting of State Vectors . . . . . . . . . . . . 71
5.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 Heuristic Schemes of a Simple Search
and an Organized Search . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.3 Mathematical Model to Test Algorithms. . . . . . . . . . . . . . . . . 75
5.4 Organized Search with the MATLAB Function fmins . . . . . . . . 77
5.5 System of Implicit Algebraic Equations . . . . . . . . . . . . . . . . . 79
5.6 Contraction Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.7 Computational Scheme of Organized Search
in Bayes Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.8 Smoothing, Filtration, and Forecasting (SFF) by Observations
in Noise for a Nonlinear Dynamic System . . . . . . . . . . . . . . . 88
vi Contents
5.8.1 Mathematical Model of Dynamic System
and Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.8.2 Conceptual Algorithm for Smoothing, Filtration,
and Forecasting (SFF Algorithm). . . . . . . . . . . . . . . . . 90
5.8.3 Qualitative Comparison of SFF Algorithm
and PUK Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.8.4 Recurrent form of the SFF (RSFF) Algorithm . . . . . . . . 94
5.8.5 About Computation of a Priori First and Second
Statistical Moments . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.8.6 Evaluation of the Initial Conditions and Parameter
of the Van der Pol Equation . . . . . . . . . . . . . . . . . . . . 97
5.8.7 Smoothing and Filtration for a Model of a Two-Level
Integrator with Nonlinear Feedback . . . . . . . . . . . . . . . 98
5.8.8 The Solution of a Problem of a Filtration
by the EKF Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 100
5.8.9 Identification of Velocity Characteristic of the
Integrator and of the Nonlinearity
of the Type ‘‘Backlash’’ . . . . . . . . . . . . . . . . . . . . . . . 100
5.9 A Servo-System with a Relay Drive and Hysteresis Loop. . . . . 102
5.10 Evaluation of Principal Moments of Inertia of a Solid . . . . . . . 103
5.11 Nonlinear Filtration at Bounded Memory of Algorithm . . . . . . 105
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6 Estimating Status Vectors from Sight Angles . . . . . . . . . . . . . . . . 109
6.1 Space Object Status Vector Evaluation . . . . . . . . . . . . . . . . . . 109
6.1.1 Equations of Motion and Observation Data Model. . . . . 110
6.1.2 Scheme of Estimator . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.1.3 Model Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Estimation of the Air- and Space-Craft Status Vector, Local
Vertical Orientation Angles, and AC-Borne
Sighting System Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2.1 Primary Navigation Errors and Formulation
of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.2.2 Navigation Parameters: The Nonlinear
Estimation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2.3 Calculation Model and Estimation Results . . . . . . . . . . 121
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7 Estimating the Parameters of Stochastic Models . . . . . . . . . . . . . . 125
7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.2 The Basic Structure of the Algorithm-Estimator . . . . . . . . . . . 128
7.3 Statistics and Empirical Frequencies. . . . . . . . . . . . . . . . . . . . 129
7.4 The Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.5 A Bayesian Statistical Construction . . . . . . . . . . . . . . . . . . . . 135
Contents vii
7.6 Estimating Hidden Markov Model Parameters
by the Algorithm-Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.7 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.7.1 Maximum Likelihood Method of Observing
Instants of Direct Transitions . . . . . . . . . . . . . . . . . . . 141
7.7.2 Algorithm of Estimating Observation States
in Instants of Indirect Transitions . . . . . . . . . . . . . . . . 143
7.7.3 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 145
7.8 Introduction and Statement of the Problem . . . . . . . . . . . . . . . 148
7.8.1 Basic Scheme of the Proposed Nonlinear
Filtration Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.8.2 Effective Work of Nonlinear Filtration Algorithm
at Estimating States of a Nominal Model Markov
Random Process if Random Observation Errors
are Large and Uniformly Distributed in ½100; 100 . . . 154
7.9 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.10 Fundamentals of the Method . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.11 Parameter Estimation for a Nonlinear STGARCH Model . . . . . 162
7.12 Parameter Estimation for a Multivariate MGARCH Model . . . . 164
7.13 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8 Designing Motion Control to a Target Point of Phase Space . . . . . 169
8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
8.2 Setting Boundary Value Problems and Problem-Solving
Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
8.3 Necessary and Sufficient Conditions
for Time-Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.4 The Stages of the Calculation Process . . . . . . . . . . . . . . . . . . 176
8.5 Near-Circular Orbit Correction in Minimum Practicable
Time Using Micro-Thrust Operation of Two Engines . . . . . . . . 177
8.6 Correcting the Near-Circular Orbit and Position
of the Earth Satellite Vehicle in Minimum Practicable
Time Using Micro-Thrust Operation of Two Engines . . . . . . . . 182
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
9 Inverse Problem of Dynamics: The Algorithm for Identifying
the Parameters of an Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
9.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
9.2 Statement of the Problem and Basic Scheme
of the Identification Algorithm . . . . . . . . . . . . . . . . . . . . . . . 189
9.3 Identification of Aerodynamic Coefficients of the Pitching
Motion for a Pseudo F-16 Aircraft . . . . . . . . . . . . . . . . . . . . . 193
9.3.1 Pitching Motion Equations . . . . . . . . . . . . . . . . . . . . . 194
viii Contents
9.3.2 Parametric Model of Aerodynamic
Forces and Moments . . . . . . . . . . . . . . . . . . . . . . . . . 198
9.3.3 Transient Processes of Characteristics
of Nominal Motions. . . . . . . . . . . . . . . . . . . . . . . . . . 199
9.3.4 Estimating Identification Accuracy of 48 Errors
of Aerodynamic Parameters of the Aircraft . . . . . . . . . . 200
9.4 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

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