Nonlinear Control: Introduction to Essentially Nonlinear Phenomena in Nonlinear Systems, Slides of Nonlinear Control Systems

Download Nonlinear Control: Introduction to Essentially Nonlinear Phenomena in Nonlinear Systems and more Slides Nonlinear Control Systems in PDF only on Docsity! Outline Motivation Reference Books Topics Introduction Nonlinear Control Lecture 1: Introduction Department of Electrical Engineering Fall 2011 Farzaneh Abdollahi Nonlinear Control Lecture 1 1/15 Docsity.com Outline Motivation Reference Books Topics Introduction Motivation Reference Books Topics Introduction Examples Farzaneh Abdollahi Nonlinear Control Lecture 1 2/15 Docsity.com Outline Motivation Reference Books Topics Introduction Essentially nonlinear phenomena I Subharmonic,harmonic or almost periodic oscillations: A stable linear system under a periodic input output with the same frequency; A nonlinear system under a periodic input can oscillate with submultiple or multiple frequency of input or almost-periodic oscillation. I Chaos: A nonlinear system may have a different steady-state behavior which is not equilibrium point, periodic oscillation or almost-periodic oscillation. This chaotic motions exhibit random, despite of deterministic nature of the system. I Multiple modes of behavior: A nonlinear system may exhibit multiple modes of behavior based on type of excitation: I an unforced system may have one limit cycle. I Periodic excitation may exhibit harmonic, subharmonic,or chaotic behavior based on amplitude and frequency of input. I if amplitude or frequency is smoothly changed, it may exhibit discontinuous jump of the modes as well. Farzaneh Abdollahi Nonlinear Control Lecture 1 5/15 Docsity.com Outline Motivation Reference Books Topics Introduction I Linear systems: can be described by a set of ordinary differential equations and usually the closed-form expressions for their solutions are derivable. Nonlinear systems: In general this is not possible It is desired to make a prediction of system behavior even in absence of closed-form solution. This type of analysis is called qualitative analysis. I Despite of linear systems, no tool or methodology in nonlinear system analysis is universally applicable their analysis requires a wide verity of tools and higher level of mathematic knowledge I ∴ stability analysis and stabilizablity of such systems and getting familiar with associated control techniques is the basic requirement of graduate studies in control engineering. I The aim of this course are I developing a basic understanding of nonlinear control system theory and its applications. I introducing tools such as Lyapunov’s method analyze the system stability I Presenting techniques such as feedback linearization to control nonlinear systems. Farzaneh Abdollahi Nonlinear Control Lecture 1 6/15 Docsity.com Outline Motivation Reference Books Topics Introduction Reference Books I Text Book: Nonlinear Systems, H. K. Khalil, 3rd edition, Prentice-Hall, 2002 I Other reference Books: I Applied Nonlinear Control, J. J. E. Slotine, and W. Li, Prentice-Hall, 1991 I Nonlinear System Analysis, M. Vidyasagar, 2nd edition, Prentice-Hall, 1993 I Nonlinear Control Systems, A. Isidori, 3rd edition Springer-Verlag, 1995 Farzaneh Abdollahi Nonlinear Control Lecture 1 7/15 Docsity.com Outline Motivation Reference Books Topics Introduction I Most of our analysis are dealing with unforced state equations where u does not present explicitly in Equ (1): ẋ = f (t, x) I In unforced state equations, input to the system is NOT necessarily zero. I Input can be a function of time: u = γ(t), a feedback function of state: u = γ(x), or both u = γ(t, x) where is substituted in Equ (1). I Autonomous or Time-invariant Systems: ẋ = f (x) (3) I function of f does not explicitly depend on t. I Autonomous systems are invariant to shift in time origin, i.e. changing t to τ = t − a does not change f . I The system which is not autonomous is called nonautonomous or time-varying. Farzaneh Abdollahi Nonlinear Control Lecture 1 10/15 Docsity.com Outline Motivation Reference Books Topics Introduction I Equilibrium Point x = x∗ I x∗ in state space is equilibrium point if whenever the state starts at x∗, it will remain at x∗ for all future time. I for autonomous systems (3), the equilibrium points are the real roots of equation: f (x) = 0. I Equilibrium point can be I Isolated: There are no other equilibrium points in its vicinity. I a continuum of equilibrium points Farzaneh Abdollahi Nonlinear Control Lecture 1 11/15 Docsity.com Outline Motivation Reference Books Topics Introduction Pendulum I Employing Newton’s second law of motion, equation of pendulum motion is: ml θ̈ = −mg sin θ − kl θ̇ l : length of pendulum rod; m: mass of pendulum bob; k : coefficient of friction; θ: angle subtended by rod and vertical axis I To obtain state space model, let x1 = θ, x2 = θ̇: ẋ1 = x2 ẋ2 = − g l sinx1 − k m x2 Farzaneh Abdollahi Nonlinear Control Lecture 1 12/15 Docsity.com