Control of Complex and Uncertain Systems

Control of Complex and Uncertain Systems examines feedback controller design for dynamical systems. It gives a compact presentation of the evolution of feedback theory and feedback design methods, before presenting a new theoretical approach to feedback control.
Traditionally, the lack of standard procedures for the design of feedback controls that will be subject to uncertain conditions, coupled with a limited understanding of how feedback mechanisms will react to those uncertain conditions, has presented a problem.
This book presents a standard principle, as well as concrete methods, for feedback design. It equally aims to increase our understanding of how feedback controllers react to uncertain forces. In doing so it lays the theoretical basis for a new generation of perfect automatic systems.
Topics covered include:
- Synthesis of Nonlinear Controllers
- Higher Degree Sliding Modes
- Theory of Coordinate-Operator Feedback / Theory of Operator-Coordinate Feedback
- Physical Foundations of the Compensation for Disturbances, and Stabilization of Forced Motion in Binary Systems
- Signal Differentiation
- Suboptimal Stabilization of an Uncertain Object

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.