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1、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 Isidor
2、i,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 1Introduction1.1 Why Nonlinear
3、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 t
4、he 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 sys
5、tem 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 res
6、ult comes from the fact that nonlinear controller designs are often deeply rooted in the physics of the plants.-Example:pendulum1.2 Nonlinear System BehaviorNonlinearities-Inherent(natural):Coulomb friction between contacting surfaces.-Intentional(artificial):adaptive control laws.-Continuous-Discon
7、tinuous:Hard nonlinearities (backlash.,Hysteresis.)cannot be locally approximated by linear function.Linear SystemsLinear 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 nonsingular
8、Stable if all eigenvalues of A have negative real parts,regardless of initial conditionsGeneral solution can be solved analyticallyCommon Nonlinear System Behaviors Nonlinear systems frequently have more than one equilibrium point(an equilibrium point is a point where the system can stay forever wit
9、hout moving).I.Multiple Equilibrium PointsExample 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)=dt-Tow equilibrium
10、points,x=0 and x=1.-Qualitative behavior strongly depends on its initial condition.Figure 3.1:Responses of the linearized system(a)and the nonlinear system(b)Stability of Nonlinear Systems May Depend on Initial Conditions:-Motions starting with 1 converges.-Motions starting with 1 diverges.Propertie
11、s 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 s
12、ystem ,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 po
13、rtrait 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 unbou
14、ndedly 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 BehaviorsIII.B
15、ifurcations-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 sp
16、lits into 3 points(),as shown in Figure 1.5(a).is a critical bifurcation value.Figure 1.5:(a)a pitchfork bifurcation (b)a Hopf bifurcation-The system output is extremely sensitive to initial conditions.-Essential feature:the unpredictability of the system output.Common Nonlinear System BehaviorsIV.C
17、haosSimple 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 CourseI.Phase plane analysisII.II.Lyapunov theoryIII.III.Feedback linearizationIV.IV.Sliding controlV.VI.Adaptive control