Control of Complex and Uncertain Systems

I. Principles Underlying the Design of Automatic Control Systems.- 1. Principles Underlying the Design of Linear Automatic Control Systems.- 1.1. Statement of a Control Problem and Preliminaries.- 1.2. Load Control Principle.- 1.3. Principle of Disturbance Control.- 1.4. Compensation Principle in an Indirect Measurement of Disturbance.- 1.5. Double-Channel Principle.- 1.6. The K-Image Method or the Method of an Internal Model.- 1.7. High Gain Factor.- 1.7.1. Statement of the problem, its peculiarities and the idea of its solution.- 1.7.2. Problems and limitations of the high gain feedback method.- 1.7.3. On the robustness of systems with a high gain factor.- 1.7.4. The state space method in the analysis of systems with a high gain factor.- 1.7.5. Geometrical interpretation of systems with a high gain factor.- 1.7.6. The effect produced by an amplitude constraint on systems with a high gain factor.- 1.8. Bibliographical Comments.- 2. Synthesis of Nonlinear Controllers.- 2.1. Relay Feedback.- 2.1.1. Basic concepts.- 2.1.2. Sliding mode at a point.- 2.1.3. Switching mode.- 2.1.4. On the robustness of the switching mode.- 2.1.5. Relay stabilization of an object with self-regulation.- 2.1.6. Stabilization of an object with a high relative order.- 2.1.7. Robust stabilization: discontinuity, continuity, and information about the state.- 2.1.8. Robust stabilization of an object of the first relative order.- 2.1.9. Sliding mode on an interval.- 2.1.10. Real sliding mode on an interval.- 2.1.11. Relay stabilization of a generalized object.- 2.2. Stabilization of an Object with an Uncertain Operator.- 2.2.1. Generalities.- 2.2.2. Principle of cascade control.- 2.2.3. The structure of objects with cascade control.- 2.2.4. Stabilization of interval objects.- 2.2.5. Interval stability.- 2.2.6. General features of the adaptive stabilization theory.- 2.3. Stabilization by a Variable Structure Controller.- 2.3.1. An astatic tracking system.- 2.3.2. Second-order astatism.- 2.3.3. Astatism of order m.- 2.3.4. A variable structure astatic tracking system.- 2.3.5. Sliding mode throughout a straight line.- 2.3.6. Analysis of robustness of VSS relative to parametric perturbations.- 2.3.7. VSS in the presence of an external force.- 2.3.8. Quasirelay representation of a ?-cell.- 2.3.9. Limitations and drawbacks of the VSS theory and the related problems.- 2.4. Bibliographical Comments.- II. New Types of Feedback.- 3. General Aspects of the Theory of New Types of Feedback.- 3.1. Introductory Remarks.- 3.2. System of Basic Concepts.- 3.2.1. Operator signal.- 3.2.2. Types of dynamical objects.- 3.2.3. Binary operation.- 3.2.4. Types of control elements.- 3.2.5. New types of feedback.- 3.3. Structural Synthesis of Binary Systems.- 3.3.1. Stabilization problem.- 3.3.2. Nonlinear feedback as a means of suppressing uncertainty.- 3.3.3. Filtration problem.- 4. Theory of Coordinate-Operator Feedback.- 4.1. Stabilization of a Second-Order Object with Unknown Parameters and an External Action.- 4.1.1. The scalarization principle and the equation of an object in the error space.- 4.1.2. Some remarks concerning the statement of the problem and its generalizations.- 4.1.3. The coordinate-operator phase space.- 4.2. CO-Algorithms of Stabilization.- 4.2.1. Direct compensation.- 4.2.2. Asymptotic estimation or an indirect measurement of the O-perturbation.- 4.2.3. Compensation for a wave O-perturbation.- 4.2.4. Relay CO-stabilization.- 4.2.5. Remark concerning the robustness of systems with relay CO-feedback.- 4.2.6. Linear CO-algorithms of stabilization.- 4.2.7. Integro-relay CO-algorithm of stabilization.- 5. Higher-Degree Sliding Modes.- 5.1. Preliminaries from the Theory of Sliding Modes.- 5.1.1. Equations of sliding.- 5.1.2. On the invariance of an equation of sliding relative to disturbances which satisfy the matching condition.- 5.1.3. Equations of real sliding.- 5.1.4. Remarks concerning the degree of sliding.- 5.2. Algorithms of Second-Degree Sliding.- 5.2.1. Asymptotic algorithms of the second-degree sliding.- 5.2.2. Discontinuous asymptotic algorithms for the second-degree sliding.- 5.2.3. Finite algorithms of second-degree sliding: linear feedback.- 5.2.4. Finite algorithms of second-degree sliding: relay feedback.- 5.2.5. Twisting algorithm.- 5.3. Output Finite Stabilization.- 6. Theory of Operator Feedback.- 6.1. The Purpose of Operator Feedback.- 6.2. Motion Equations in the Coordinate-Operator Space.- 6.3. Statical Operator Feedback.- 6.3.1. Statical operator and coordinate-operator feedbacks.- 6.3.2. Statical operator and dynamical coordinate-operator feedbacks.- 6.3.3. Inertial coordinate-operator feedback.- 6.3.4. Inertial-relay coordinate-operator feedback.- 6.3.5. Inertial-relay coordinate-operator feedback with an unknown parameter in the control.- 6.3.6. Integral-relay coordinate-operator feedback.- 7. Theory of Operator-Coordinate Feedback.- 7.1. Dynamical Statism and Operator-Coordinate Feedback.- 7.2. Motion Equations for an Operator-Coordinate Object.- 7.3. Statical OC-Controller.- 7.4. Integral OC-Controller.- 7.5. The Main Properties and Specific Features of Binary Stabilization Systems with Different Types of Feedback.- 7.6. Discontinuous OC-Feedback.- 7.6.1. Integral-relay OC-controller.- 7.6.2. Second-degree sliding modes in an OC-loop.- 8. Constraints, Physical Foundations of the Compensation for Disturbances, and Stabilization of Forced Motion in Binary Systems.- 8.1. Constraints Imposed on the Operator Variable.- 8.2. On the Global Behavior of a Binary System.- 8.3. Physical Foundations of the Compensation for Uncertainty.- 8.4. On the Compensation for the Coordinate Disturbance.- 9. Signal Differentiation.- 9.1. Statement of the Differentiation Problem.- 9.1.1. Filtration.- 9.1.2. RC-circuit.- 9.1.3. Discrete-difference approximations.- 9.2. Tracking Differentiating Systems.- 9.2.1. A linear differentiator.- 9.2.2. Relay differentiator.- 9.2.3. Variable-structure differentiator.- 9.3. Tracking Asymptotic Binary Differentiator.- 9.4. Finite Binary Differentiator.- 9.5. Nonstandard Differentiating Systems.- 9.5.1. Differentiator with a "small" amplitude of discontinuities.- 9.5.2. Nonstandard binary differentiator.- 9.5.3. The results of discrete simulation of a nonstandard binary differentiator.- 10.Suboptimal Stabilization of an Uncertain Object.- 10.1. Statement of the Optimal Stabilization Problem.- 10.2. Example of an Optimal Stabilization Problem under Uncertainty.- 10.3. Optimal Stabilization "in the Mean".- 10.4. Minimax Optimal Stabilization.- 10.5. Stabilization with the Use of the Standard Model and an Error Feedback with High Gain.- 10.6. Stabilization by the Methods of the Binary Control Theory.- 10.6.1. A variable structure system.- 10.6.2. Binary stabilization with an integral CO-feedback.- 10.6.3. Stabilization with the use of a second-degree sliding mode.- 10.7. Reduction of the Suboptimal Stabilization Problem to the Problemof Asymptotic Invariance.- 10.7.1. Main concepts of the theory of asymptotic invariance.- 10.7.2. Suboptimal linearly quadratic stabilization.- Conclusion.