Discrete Linear Control Systems

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.