Nonlinear Control Lecture 1: Introduction

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OutlineMotivationReference BooksTopicsIntroductionNonlinear ControlLecture 1: IntroductionFarzaneh AbdollahiDepartment of Electrical EngineeringAmirkabir University of TechnologyFall 2011Farzaneh AbdollahiNonlinear ControlLecture 11/15

OutlineMotivationReference BooksTopicsIntroductionMotivationReference BooksTopicsIntroductionExamplesFarzaneh AbdollahiNonlinear ControlLecture 12/15

OutlineMotivationReference BooksTopicsIntroductionMotivationsIA system is called linear if its behavior set satisfies linear superpositionlaws: .i.e. z1 , z2 B and constant c R z1 z2 B, cz1 BIA nonlinear system is simply a system which is not linear.IPowerful tools founded based on superposition principle make analyzingthe linear systems simple.IAll practical systems posses nonlinear dynamics.ISometimes it is possible to describe the operation of physical systems bylinear model around its operating pointsILinearized system can provide us an approximate behavior of thenonlinear systemIBut in analyzing the overall system behavior, often linearized modelinadequate or inaccurate.Farzaneh AbdollahiNonlinear ControlLecture 13/15

OutlineMotivationReference BooksTopicsIntroductionILinearization is an approximation in the neighborhood of an operatingsystemit can only predict local behavior of nonlinear system. (No inforegarding nonlocal or global behavior of system)IDue to richer dynamics of nonlinear systems comparing to the linear ones,there are some essentially nonlinear phenomena that can take place onlyin presence of nonlinearityEssentially nonlinear phenomenaIIIIFinite escape time: The state of linear system goes to infinity as t ;nonlinear system’s state can go to infinity in finite time.Multiple isolated equilibria: linear system can have only one isolatedequilibrium point which attracts the states irrespective on the initial state;nonlinear system can have more than one isolated equilibrium point, thestate may converge to each depending on the initial states.Limit cycle: There is no robust oscillation in linear systems. To oscillatethere should be a pair of eigenvalues on the imaginary axis which due topresence of perturbations it is almost impossible in practice; For nonlinearsystems, there are some oscillations named limit cycle with fixed amplitudeand frequency.Farzaneh AbdollahiNonlinear ControlLecture 14/15

OutlineMotivationReference BooksTopicsIntroductionEssentially nonlinear phenomenaISubharmonic,harmonic or almost periodic oscillations: A stable linearsystem under a periodic inputoutput with the same frequency;A nonlinear system under a periodic inputcan oscillate withsubmultiple or multiple frequency of input or almost-periodic oscillation.IChaos: A nonlinear system may have a different steady-state behaviorwhich is not equilibrium point, periodic oscillation or almost-periodicoscillation. This chaotic motions exhibit random, despite of deterministicnature of the system.Multiple modes of behavior: A nonlinear system may exhibit multiplemodes of behavior based on type of excitation:IIIIan unforced system may have one limit cycle.Periodic excitation may exhibit harmonic, subharmonic,or chaotic behaviorbased on amplitude and frequency of input.if amplitude or frequency is smoothly changed, it may exhibit discontinuousjump of the modes as well.Farzaneh AbdollahiNonlinear ControlLecture 15/15

OutlineMotivationReference BooksTopicsIntroductionILinear systems: can be described by a set of ordinary differentialequations and usually the closed-form expressions for their solutions arederivable. Nonlinear systems: In general this is not possibleIt isdesired to make a prediction of system behavior even in absence ofclosed-form solution. This type of analysis is called qualitative analysis.IDespite of linear systems, no tool or methodology in nonlinear systemanalysis is universally applicabletheir analysis requires a wide verity oftools and higher level of mathematic knowledgeI stability analysis and stabilizablity of such systems and getting familiarwith associated control techniques is the basic requirement of graduatestudies in control engineering.The aim of this course areIIIIdeveloping a basic understanding of nonlinear control system theory and itsapplications.introducing tools such as Lyapunov’s method analyze the system stabilityPresenting techniques such as feedback linearization to control nonlinearsystems.Farzaneh AbdollahiNonlinear ControlLecture 16/15

OutlineMotivationReference BooksTopicsIntroductionReference BooksIText Book: Nonlinear Systems, H. K. Khalil, 3rd edition,Prentice-Hall, 2002IOther reference Books:IApplied Nonlinear Control, J. J. E. Slotine, and W. Li,Prentice-Hall, 1991INonlinear System Analysis, M. Vidyasagar, 2nd edition,Prentice-Hall, 1993INonlinear Control Systems, A. Isidori, 3rd editionSpringer-Verlag, 1995Farzaneh AbdollahiNonlinear ControlLecture 17/15

OutlineMotivationReference BooksTopicsIntroductionTopicsTopicIntroduction, Phase plane, AnalysisStability TheoryInput-to-state, I/O stabilityPassivityFeedback ControlFeedback linearizationSliding ModeBack SteppingFarzaneh AbdollahiNonlinear ControlDateWeeks 1,2Weeks 3-5Weeks 6,7Week 8Weeks 9,10Weeks 11,12Weeks 13,14Weeks 15,16Lecture 1RefsChapters 1-3Chapters 4,8,9Chapters 4,5Chapter 10Chapter 12Chapters 12,138/15

OutlineIMotivationReference BooksTopicsIntroductionAt this course we consider dynamical systems modeled by a finitenumber of coupled first-order ordinary differential equations:ẋ f (t, x, u)(1)where x [x1 , . . . , xn ]T : state vector, u [u1 , . . . , up ]T : inputvector, and f (.) [f1 (.), . . . , fn (.)]T : a vector of nonlinear functions.IEuq. (1) is called state equation.IAnother equation named output equation:y h(t, x, u)Iwhere y [y1 , . . . , yq ]T : output vector.Equ (2) is employed for particular interest in analysis such asIII(2)variables which can be measured physicallyvariables which are required to behave in a desirable mannerEqus (1) and (2) together are called state-space model.Farzaneh AbdollahiNonlinear ControlLecture 19/15

OutlineIMotivationReference BooksTopicsIntroductionMost of our analysis are dealing with unforced state equations where udoes not present explicitly in Equ (1):ẋ f (t, x)IIIIn unforced state equations, input to the system is NOT necessarily zero.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).Autonomous or Time-invariant Systems:ẋ f (x)III(3)function of f does not explicitly depend on t.Autonomous systems are invariant to shift in time origin, i.e. changing t toτ t a does not change f .The system which is not autonomous is called nonautonomous ortime-varying.Farzaneh AbdollahiNonlinear ControlLecture 110/15

OutlineIMotivationReference BooksTopicsIntroductionEquilibrium Point x x IIIx in state space is equilibrium point if whenever the state starts at x , itwill remain at x for all future time.for autonomous systems (3), the equilibrium points are the real roots ofequation: f (x) 0.Equilibrium point can beFarzaneh AbdollahiIIIsolated: There are no other equilibrium points in its vicinity.a continuum of equilibrium pointsNonlinear ControlLecture 111/15

OutlineMotivationReference BooksTopicsIntroductionPendulumIEmploying Newton’s second law of motion, equation ofpendulum motion is:ml θ̈ mg sin θ kl θ̇l: length of pendulum rod;m: mass of pendulum bob;k: coefficient of friction;θ: angle subtended by rod and vertical axisITo obtain state space model,let x1 θ, x2 θ̇:Farzaneh Abdollahiẋ1 x2gkẋ2 sinx1 x2lmNonlinear ControlLecture 112/15

OutlineMotivationReference BooksTopicsIntroductionPendulumITo find equilibrium point: ẋ1 ẋ2 00 x2gk0 sinx1 x2lmIThe Equilibrium points are at (nπ, 0) for n 0, 1, 2, . . .IIIPendulum has two equilibrium points: (0, 0) and (π, 0),Other equilibrium points are repetitions of these two which correspond tonumber of pendulum full swings before it restsPhysically we can see that the pendulum rests at (0, 0), but hardlymaintain rest at (π, 0)Farzaneh AbdollahiNonlinear ControlLecture 113/15

OutlineMotivationReference BooksTopicsIntroductionTunnel Diode CircuitIThe tunnel diode is characterized by iR h(vR )IThe energy-storing elements areC and L which assumed are linear anddiLCtime-invariant iC C dvdt , vL L dt .IEmploying Kirchhoff’s current law:iC iR iL 0IEmploying Kirchhoff’s voltage law:vC E RiL vL 0Ifor state space model, let x1 vC , x2 iL andu E as a constant input:ẋ1 ẋ2 Farzaneh Abdollahi1[ h(x1 ) x2 ]C1[ x1 Rx2 u]LNonlinear ControlLecture 114/15

OutlineMotivationReference BooksTopicsIntroductionTunnel Diode CircuitITo find equilibrium point: ẋ1 ẋ2 00 0 I1[ h(x1 ) x2 ]C1[ x1 Rx2 u]LEquilibrium points depends on Eand Rx2 h(x1 ) II1E x1RRFor certain E and R, it may have 3 equilibriumpoints (Q1 , Q2 , Q3 ).if E , and same Ronly Q3 exists.if E , and same Ronly Q1 exists.IFarzaneh AbdollahiNonlinear ControlLecture 115/15

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 Abd

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