# Multiple-Input Describing Functions and Nonlinear System Design

Written by: Arthur Gelb (Analytic Sciences Corporation) and Wallace E. Vander Velde (MIT)

This free online textbook was published in 1968, however, it is one of three required textbooks for the Estimation and Control of Aerospace Systems course at MIT. It is an undergraduate level course on control systems in their Aeronautics and Astronautics program. Students should have taken a “first course on systems.” Students should also have had two semesters of calculus and physics.

The course syllabus includes learning objectives which include:

• understanding of classical control system design
• understanding of robustness issues arising in control system analysis and design
• understanding the role of nonlinearities in control systems and to be able to analyze basic nonlinearities using phase-plane and describing function techniques
• designing simple nonlinear (bang/bang) systems in the phase plane
• understanding space applications
• understanding basic nonlinear phenomena, such as jump resonance phenomena
• understanding modeling and identification issues through laboratory activities

Multiple-Input Describing Functions and Nonlinear System Design was written by Arthur Gelb of The Analytic Sciences Corporation and Wallace E. Vander Velde, professor of Aeronautics and Astronautics at MIT. It was written with both students and practicing control engineers in mind. The author believes this control systems textbook could serve as a complete reference work in the subject area for professionals. (Can’t comment if that’s still the case but clearly MIT believes this work continues to provide value to their students.)

Chapter 1: Nonlinear Systems and Describing Functions
1.0 Introduction
1.1 Nonlinear-system Representation
1.2 Behavior of Nonlinear Systems
1.3 Methods of Nonlinear-system Study
1.4 The Describing Function Viewpoint
1.5 A Unified Theory of Describing Functions
Chapter 2: Sinusoidal-input Describing Function (DF)
2.0 Introduction
2.1 Asymptotic Methods for the Study of Nonlinear Oscillations
2.2 Equivalent Linearization and the DF
2.3 DF Calculation for Frequency-independent Nonlinearities
2.4 DF Calculation for Frequency-dependent Nonlinearities
2.5 Synthesis of DFs
2.6 Techniques for Approximate Calculation of the DF
2.7 DF Inversion
Chapter 3: Steady-state Oscillations in Nonlinear Systems
3.0 Introduction
3.1 Determination of Limit Cycles
3.2 Limit Cycle Stability
3.3 Frequency Response of Non-limit-cycling Nonlinear Systems
3.4 Application to a Time-optimal Computer Control System
3.5 Linear and Nonlinear Loop Compensation Techniques
3.6 Treatment of Multiple Nonlinearities
3.7 Accuracy of the DF Approximation
3.8 Exact Methods for Relay Control Systems
Chapter 4: Transient Oscillations in Nonlinear Systems
4.0 Introduction
4.1 Analytic Description of Transient Oscillations
4.2 Relation to Other Work
4.3 Solution of the Equations Defining the Oscillation
4.4 Limit Cycle Dynamics
4.5 Limit Cycle Stability
Chapter 5: Two-sinusoid-input Describing Function (TSIDF)
5.0 Introduction
5.1 TSIDF Calculation
5.2 Subharmonic Oscillations
5.3 Frequency Response Counterexamples
5.4 Multiple Limit Cycles
5.5 Incremental-input Describing Function
Chapter 6: Dual-input Describing Function (DIDF)
6.0 Introduction
6.1 Mathematical Formulation of the DIDF
6.2 DIDF Calculation
6.3 Forced Response of Limit Cycling Nonlinear Systems
6.4 A Scheme for Parameter-adaptive Control
6.5 Application to an Adaptive Missile Roll Control System
6.6 Limit Cycles in Systems with an Asymmetric Nonlinearity
6.7 Artificial Dither and Signal Stabilization
6.8 TSIDF Calculation via the D F of a DIDF
6.9 Basis for Higher-order Approximations
Chapter 7: Random and Other Signals in Nonlinear Systems
7.0 Introduction
7.1 Statistical Linearization
7.2 Calculation of Random-input Describing Functions (RIDFs)
7.3 Feedback Systems with Random Signals and Noise
7.4 Feedback Systems with Random and Other Inputs
7.5 Alternative Approach for Nonlinearities with Memory
Chapter 8: Nonoscillatory Transients in Nonlinear Systems
8.0 Introduction
8.1 Transient-input Describing Function
8.2 Linear-system Approximations
8.3 Transient Response of Non-limit-cycling Nonlinear Systems
8.4 Design Procedure for a Specified Transient Response
8.5 Exponential-input Describing Function
Chapter 9: Oscillations in Nonlinear Sampled-data Systems
9.0 Introduction
9.1 Limit Cycles in Sampled Two-level Relay Systems
9.2 Limit Cycles in Other Sampled Nonlinear Systems
9.3 Stability of Limit Cycle Modes
9.4 Exact Verification of Limit Cycle Modes
9.5 Limit Cycles in Pulse-width-modulated Systems
Appendix A: Amplitude-ratio-Decibel Conversion Table
Appendix B: Table of Sinusoidal-input Describing Functions (DFs)
Appendix C: Table of Dual-input Describing Functions (DIDFs)
Appendix D: Table of Two-sinusoid-input Describing Functions (TSIDFs
Appendix E: Table of Random-input Describing Functions (RIDFs)
E.1 Gaussian-input RIDFs
E.2 Gaussian-plus-bias-input RIDFs
E.3 Gaussian-plus-bias-plus-sinusoid-input RIDFs
Appendix F: Table of Sampled Describing Functions
Appendix G: Analytical Justification for the Filter Hypothesis
Appendix H: Introduction to Probability and Random Processes

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Multiple-Input Describing Functions and Nonlinear System Design 