CiNii Books - Discrete linear control systems

One service mathematics has rendered the 'Bt mm, ...* si j'avait su comment en revenir, human race. It has put common sense back je n'y serais point alIe.' Jules Verne where it belongs. on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense'. Eric T. Bell able to do something with it. O. Heavisidc Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Table of Contents

1 Basic concepts and statement of problems in control theory.- 1.1 Initial Premises.- 1.2 Basic concepts of control theory.- 1.2.1 The control object.- 1.2.2 Control algorithm.- 1.2.3 Control objective.- 1.3 Modelling of control objects and their general characteristics.- 1.3.1 State equations of discrete processes.- 1.3.2 Observability and controllability.- 1.3.3 Linear proces.- 1.4 Precising the statement of the control problem.- 1.4.1 Classification of control objectives.- 1.4.2 Optimisation of control.- 1.4.3 Observations on selection of control strategies.- 2 Finite time period control.- 2.1 Dynamic programming.- 2.1.1 Statement of the optimization problem.- 2.1.2 Description of the Dynamic programming methods.- 2.1.3 Bellman's equation.- 2.1.4 Example: Linear-quadratic deterministic system.- 2.1.5 Generalisation of Bellman's equation for infinite time control problems.- 2.2 Stochastic control systems.- 2.2.1 Statement of the problem.- 2.2.2 Dependence of the optimal solution on the choice of the admissible control strategies.- 2.3 Stochastic dynamic programming.- 2.3.1 Description of the method.- 2.3.2 Bellman's equation for stochastic control systems.- 2.3.3 Example: Linear quadratic problem with randomly varying coefficients and observable states of the control object.- 2.3.4 Example: Linear stationary object with control delay.- 2.4 Bayesian control strategy.- 2.4.1 Bayesian approach to the optimization problem.- 2.4.2 A posteriori distribution and Bayesian formula.- 2.4.3 Regularity in Bayesian control strategy.- 2.4.4 Recursive formulae for computations of a posteriori distributions.- 2.5 Linear quadratic Gaussian Problem.- 2.5.1 Statement of the problem.- 2.5.2 Conditional Gaussism of the states and sufficient statistics.- 2.5.3 Bayesian control strategy.- 2.A Appendix.- 2.A.1 General forms of probability theory.- 2.A.2 Convergence of random variables.- 2.P Proofs of lemmas and theorems.- 2.P.1 Proof of the theorem 2.1.1.- 2.P.2 Proof of the theorem 2.1.2.- 2.P.3 Proof of the theorem 2.3.1.- 2.P.4 Proof of the lemma 2.3.1.- 2.P.5 Proof of the theorem 2.3.2..- 2.P.6 Proof of the lemma 2.4.1.- 2.P.7 Proof of the theorem 2.4.1.- 2.P.8 Proof of the theorem 2.4.2..- 3 Infinite time period control.- 3.1 Stabilitzation of dynamic systems using Liapunov's method.- 3.1.1 Description of Liapunov's method.- 3.1.2 Stabilization of linear systems with observable states.- 3.1.3 Stabilization of linear systems with unobservable states.- 3.2 Discrete form for analytical design of regulators.- 3.2.1 Statement of the problem.- 3.2.2 Reduction of the optimization problem to the solvability of the matrix Riccati equation.- 3.2.3 Lur'e equation and a few of its properties.- 3.2.4 Analytical design of regulators in the presence of additive noise.- 3.3 Transfer function method in linear optimization problem.- 3.3.1 Statement of the linear optimization problem.- 3.3.2 Transfer functions of control systems and their properties.- 3.3.3 Geometrical interpretation of the linear optimization problem.- 3.3.4 Weiner - Kolmogorov method for conditional minimization of a quadratic functional.- 3.3.5 Parametrization of the set of transfer functions.- 3.3.6 Design of the optimal regulator for the object expressed in the standard form.- 3.3.7 Correspondence between transfer function method and method of Lur'e solving equation.- 3.3.8 Design of the optimal regulator for control object equations expressed through 'input-output' variables.- 3.4 Limiting optimal control of stochastic processes.- 3.4.1 Sufficient conditions for optimality of admissible control strategies.- 3.4.2 Statement of the limiting linear quadratic optimal control problem.- 3.4.3 Solvability of the optimization problem.- 3.4.4 Design of optimal linear regulators through transfer function method.- 3.4.5 Formulation of the limited optimal control problems using Riccati equation.- 3.5 Minimax control.- 3.5.1 Statement of the minimax control problem.- 3.5.2 Control system transfer function and its properties.- 3.5.3 Geometrical interpretation of the minimax control problem.- 3.5.4 Properties of the sets in geometrical interpretation of the optimization problem.- 3.5.5 Statement of the basic result.- 3.5.6 The Properties of the optimal regulator.- 3.5.7 A few generalisations.- 3.A Appendix.- 3.A.1 Frequency theorem.- 3.P Proofs of the lemmas and theorems.- 3.P. 1 Proof of the theorem 3.1.1.- 3.P.2 Proof of the theorem 3.1.2.- 3.P.3 Proof of the theorem 3.1.3.- 3.P.4 Proof of the theorem 3.1.4.- 3.P.5 Proof of the theorem 3.2.1.- 3.P.6 Proof of the theorem 3.2.2.- 3.P.7 Proof of the theorem 3.3.1.- 3.P.8 Proof of the lemma 3.3.1.- 3.P.9 Proof of the theorem 3.3.2.- 3.P.10 Proof of the lemma 3.3.2.- 3.P.11 Proof of the theorem 3.3.3.- 3.P.12 Proof of the lemma 3.3.3.- 3.P.13 Proof of the lemma 3.3.4.- 3.P.14 Proof of the theorem 3.4.1.- 3.P.15 Proof of the theorem 3.4.2.- 3.P.16 Proof of the theorem 3.4.3.- 3.P.17 Proof of the theorem 3.4.4.- 3.P.18 Proof of the theorem 3.4.5.- 3.P.19 Proof of the theorem 3.4.1.- 3.P.20 Proof of the lemma 3.4.2.- 3.P.21 Proof of the theorem 3.4.6.- 3.P.22 Proof of the theorem 3.4.7.- 3.P.23 Proof of the lemma 3.4.3.- 3.P.24 Proof of the theorem 3.5.1.- 3.P.25 Proof of the theorem 3.5.2.- 4 Adaptive linear control systems with bounded noise.- 4.1 Fundamentals of adaptive control.- 4.1.1 Adaptive control strategy.- 4.1.2 Identification method in adaptive control.- 4.2 Existence of adaptive control strategy in a minimax control problem.- 4.2.1 Statement of the problem.- 4.2.2 Synthesis of an adaptive control strategy.- 4.2.3 Examples.- 4.3 Self-tuning systems.- 4.3.1 Self-tuning with no disturbance.- 4.3.2 Self-tuning in the presence of disturbance.- 4.3.3 Adaptive control with bounded disturbance in the control object.- 4.3.4 Method of recursive objective inequalities in an adaptive tracking problem.- 4.P Proofs of the lemmas and theorems.- 4.P.1 Proof of the lemma 4.2.1.- 4.P.2 Proof of the theorem 4.2.1..- 4.P.3 Proof of the theorem 4.3.1.- 4.P.4 Proof of the theorem 4.3.2.- 4.P.5 Proof of the theorem 4.3.3.- 4.P.6 Proof of the theorem 4.3.4.- 5 The problem of dynamic system identification.- 5.1 Optimal recursive estimation.- 5.1.1 Formulation of the estimation problems.- 5.1.2 Duality of the estimation and optimal control problems.- 5.1.3 Solution of the matrix linear quadratic cost optimization problem.- 5.1.4 The Kalman-Bucy filter.- 5.1.5 Optimal properties of the Kalman-Bucy filter.- 5.2 The Kalman-Bucy filter for tracking the parameter drift in dynamic systems.- 5.2.1 Optimal tracking of the parameter drift in presence of Gaussian disturbances.- 5.2.2 Asymptotic properties of the Kalman-Bucy filter.- 5.3 Recursive estimation.- 5.3.1 Forecasting methods of identification.- 5.3.2 Selection of forecasting models.- 5.3.3 Recursive schemes for estimation.- 5.4 Identification of a linear control object in the presence of correlated noise.- 5.4.1 Uniqueness of the minimum of the forecasting performance criterion.- 5.4.2 Modification of the estimation algorithm.- 5.4.3 Consistency of the estimates of the identification algorithm.- 5.4.4 Identification of linear systems with known spectral density of noise.- 5.5 Identification of control objects using test signals.- 5.5.1 Statement of the identification problem.- 5.5.2 Introduction of the estimation parameter.- 5.5.3 Estimation algorithm.- 5.5.4 Consistency of the estimates.- 5.P Proofs of lemmas and theorems.- 5.P.1 Proof of the theorem 5.1.1.- 5.P.2 Proof of the lemma 5.1.1.- 5.P.3 Proof of the lemma 5.1.2.- 5.P.4 Proof of the lemma 5.1.3.- 5.P.5 Proof of the theorem 5.2.1.- 5.P.6 Proof of the theorem 5.2.2.- 5.P.7 Proof of the theorem 5.2.3.- 5.P.8 Proof of the theorem 5.2.4.- 5.P.9 Proof of the lemma 5.4.1.- 5.P.10 Proof of the lemma 5.4.2.- 5.P.11 Proof of the theorem 5.4.1.- 5.P.12 Proof of the theorem 5.4.2.- 5.P.13 Proof of the lemma 5.5.1.- 5.P.14 Proof of the theorem 5.5.1.- 6 Adaptive control of stochastic systems.- 6.1 Dual control.- 6.1.1 Bayesian approach to adaptive control problems.- 6.1.2 Adaptive version of the Gaussian linear quadratic control problems, with observable vector states.- 6.1.3 Bayesian control strategy.- 6.1.4 Recursive relations for a posteriori distributions.- 6.2 Initial synthesis of adaptive control strategy in presence of the correlated noise.- 6.2.1 Adaptive optimal control for a performance criterion dependent on events.- 6.2.2 Direct method of adaptive control formulation.- 6.2.3 Adaptive optimization of the unconditional performance criterion.- 6.3 Design of the adaptive minimax control.- 6.3.1 Statement of the problem.- 6.3.2 Formulation of the adaptive control strategy.- 6.P Proofs of the lemmas and the theorems.- 6.P.1 Proof of the theorem 6.1.1.- 6.P.2 Proof of the theorem 6.1.2.- 6.P.3 Proof of the lemma 6.2.1.- 6.P.4 Proof of the lemma 6.2.2.- 6.P.5 Proof of the lemma 6.2.3.- 6.P.6 Proof of the lemma 6.2.4.- 6.P.7 Proof of the lemma 6.2.5.- 6.P.8 Proof of the theorem 6.2.1.- 6.P.9 Proof of the theorem 6.2.2.- 6.P.10 Proof of the theorem 6.2.3.- 6.P.11 Proof of the lemma 6.2.6.- 6.P.12 Proof of the theorem 6.2.4.- 6.P.13 Proof of the lemma 6.2.7.- 6.P.14 Proof of the theorem 6.2.5.- 6.P.15 Proof of the theorem 6.3.1.- Comments.- References.- Operators and Notational Conventions.

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Discrete linear control systems

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