非线性控制Nonlinear Control概要课件.ppt
非 線 性 控 制Nonlinear Control,林心宇長庚大學電機工程學系2010春,教 師 資 料,教師：林心宇Office Room:工學大樓六樓Telephone:Ext.3221E-mail:shinylinmail.cgu.edu.twOffice Hour:2:00 4:00 pm,Thursday,教 科 書,Textbook:Jean-Jacques E.Slotine and Weiping Li,Applied Nonlinear Control,Pearson Education Taiwan Ltd.,1991.Reference:Alberto Isidori,Nonlinear Control Systems,Springer-Verlag,1999.,課程目標及背景需求,1.介紹如何以Phase Portrait及Lyapunov Method分析非線性系統穩定性及控制器的設計。2.介紹Feedback Linearization,Sliding Control及Adaptive Control等方法。背景需求Linear System TheoryElementary Differential Equations,評 量 標 準,作業(20%)正式考試 2 次(各40%),Chapter 1,Introduction,1.1 Why Nonlinear Control？,-Linear control methods rely on the key assumption of small range operation for the linear model to be valid.-Nonlinear controllers may handle the nonlinearities in large range operation directly.,Improvement of Existing Control Systems,Analysis of hard nonlinearities,Linear control assumes the system model is linearizable.Hard nonlinearities:nonlinearities whose discontinuous nature does not allow linear approximation.Coulomb friction,saturation,dead-zones,backlash,and hysteresis.,Dealing with Model Uncertainties,In designing linear controllers,we assume that the parameters of the system model are reasonably well known.In real world,control problems involve uncertainties in the model parameters.The model uncertainties can be tolerated in nonlinear control.,Design Simplicity,Good nonlinear control designs may be simpler and more intuitive than their linear counterparts.This result comes from the fact that nonlinear controller designs are often deeply rooted in the physics of the plants.Example:pendulum,1.2 Nonlinear System Behavior,Nonlinearities,Inherent(natural):Coulomb friction between contacting surfaces.Intentional(artificial):adaptive control laws.Continuous Discontinuous:Hard nonlinearities(backlash.,Hysteresis.)cannot be locally approximated by linear function.,Linear Systems,Linear time-invariant(LTI)control systems,of the formwith x being a vector of states and A being the system matrix.,Properties of LTI systems Unique equilibrium point if A is nonsingularStable if all eigenvalues of A have negative real parts,regardless of initial conditionsGeneral solution can be solved analytically,Common Nonlinear System Behaviors,Nonlinear systems frequently have more than one equilibrium point(an equilibrium point is a point where the system can stay forever without moving).,I.Multiple Equilibrium Points,Example 1.2:A first-order systemwith x(0)=x0.Its linearization is with solution x(t)=x0et:general solution can be solved analytically.Unique equilibrium point at x=0.Stable regardless of initial condition.,-Integrating equation dx/(x+x2)=dtTow equilibrium points,x=0 and x=1.Qualitative behavior strongly depends on its initial condition.,Stability of Nonlinear Systems May Depend on Initial Conditions:Motions starting with 1 converges.Motions starting with 1 diverges.,Properties of LTI Systems:In the presence of an external input u(t),i.e.,with-Principle of superposition.-Asymptotic stability implied BIBO stability in the presence of u.-Sinusoidal input lead to a sinusoidal output of the same frequency.,Stability of Nonlinear Systems May Depend on Input Values:A bilinear system,converges.,diverges.,Oscillations of fixed amplitude and fixed period without external excitation.Example 1.3:Van der Pol Equationwhere m,c and k are positive constants.,II.Limit Cycles,-Limit cycleThe trajectories starting from both outside and inside converge to this curve.,Figure 2.8:Phase portrait of the Van der Pol equation,A mass-spring-damper system with a position-dependent damping coefficient 2c(x2-1)For large x,2c(x2-1)0:the damper removes energy from the system-convergent tendency.For small x,2c(x2-1)0:the damper adds energy to the system-divergent tendency.,Neither grow unboundedly nor decay to zero.-Oscillate independent of initial conditions.,-As parameters changed,the stability of the equilibrium point can change.-critical or bifurcation values:Values of the parameters at which the qualitative nature of the systems motion changes.,Common Nonlinear System Behaviors,III.Bifurcations,Topic of bifurcation theory:Quantitative change of parameters leading to qualitative change of system properties.-Undamped Duffing equation(the damped Duffing Equation is,which may represent a mass-damper-spring system with a hardening spring).,-As varies from+to-,one equilibrium point splits into 3 points(),as shown in Figure 1.5(a).is a critical bifurcation value.,The system output is extremely sensitive to initial conditions.Essential feature:the unpredictability of the system output.,Common Nonlinear System Behaviors,IV.Chaos,Simple Nonlinear system Two almost identical initial conditions,Namely,and-The two responses are radically different after some time.,Figure 1.6:Chaotic behavior of a nonlinear system,Outlines of this Course,Phase plane analysisII.Lyapunov theoryIII.Feedback linearizationIV.Sliding controlVI.Adaptive control,
