按一下以編輯母片標題樣式,按一下以編輯母片文字樣式,第二層,第三層,第四層,第五層,*,非 線 性 控 制,Nonlinear Control,林心宇,長庚大學電機工程學系,2010春,非 線 性 控 制Nonlinear Contro,1,教 師 資 料,教師：林心宇,Office Room:,工學大樓六樓,Telephone:Ext.3221,E-mail:shinylinmail.cgu.edu.tw,Office Hour:2:00 4:00 pm,Thursday,教 師 資 料教師：林心宇,2,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.,教 科 書Textbook:,3,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 Theory,Elementary Differential Equations,課程目標及背景需求1.介紹如何以Phase Portrait,4,評 量 標 準,作業(20%),正式考試 2 次(各40%),評 量 標 準作業(20%),5,Chapter 1,Introduction,Chapter 1Introduction,6,1.1 Why Nonlinear Control？,1.1 Why Nonlinear Control？,7,-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,-Linear control methods rely,8,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.,Analysis of hard nonlineariti,9,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.,Dealing with Model Uncertaint,10,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,Design SimplicityGood nonline,11,1.2 Nonlinear System Behavior,1.2 Nonlinear System Behavior,12,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.,Nonlinearities Inherent(natu,13,Linear Systems,Linear time-invariant(LTI)control systems,of the form,with,x,being a vector of states and,A,being the system matrix.,Linear SystemsLinear time-inva,14,Properties of LTI systems,Unique equilibrium point,if,A,is nonsingular,Stable if all eigenvalues of,A,have negative real parts,regardless of initial conditions,General solution can be solved analytically,Properties of LTI syste,15,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,Common Nonlinear System Behavi,16,Example 1.2:A first-order system,with,x,(0)=,x,0,.Its linearization is,with solution,x,(,t,)=,x,0,e,t,:general solution,can be solved analytically.,Unique equilibrium point at,x,=0.,Stable regardless of initial condition.,Example 1.2:A first-order sys,17,-Integrating equation,dx/(x+x,2,)=dt,Tow equilibrium points,x=,0 and,x,=1.,Qualitative behavior strongly depends on its,initial condition.,-Integrating equation dx/(x,18,Figure 3.1:,Responses of the linearized system(a)and the nonlinear system(b),Figure 3.1:Responses of the l,19,Stability of Nonlinear Systems May Depend on Initial Conditions:,Motions starting with,1 converges.,Motions starting with 1 diverges.,Stability of Nonlinear Systems,20,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.,Properties of LTI Systems:,21,Stability of Nonlinear Systems May Depend on Input Values:,A bilinear system,converges.,diverges.,Stability of Nonlinear Systems,22,Oscillations of fixed amplitude and fixed,period without external excitation.,Example 1.3:Van der Pol Equation,where,m,c,and,k,are positive constants.,II.Limit Cycles,Oscillations of fixed ampli,23,-Limit cycle,The trajectories starting from both outside and inside converge to this curve.,Figure 2.8:,Phase portrait of the Van der Pol equation,-Limit cycleFigure 2.8:Phase,24,A mass-spring-damper system with a,position-dependent damping coefficient,2,c,(,x,2,-1),For large,x,2,c,(,x,2,-1)0:the damper removes,energy from the system-convergent tendency.,For small,x,2,c,(,x,2,-1)0:the damper adds,energy to the system-divergent tendency.,A mass-spring-damper system w,25,Neither grow unboundedly nor decay to zero.,-Oscillate independent of initial conditions.,Neither grow unboundedly nor d,26,-As parameters changed,the stability of the,equilibrium point can change.,-critical,or,bifurcation,values:,Values