Statistical approach to estimation Examine the multivariate Gaussian distribution: (x-)\v-(x-) Multivariant f(x)= Minimize (x-u)'v(x-u) gives largest probability density By minimizing the argument of the exponential in the probability density function, we maximize the likelihood of the estimates(MLE)
Fall 2001 16.313-1 Introduction Root locus methods have Advantages k Good indicator if transient response k Explicity shows location of all closed-loop poles Trade-offs in the design are fairly clear Disadvantages k Requires a transfer function model(poles and zeros) k Difficult to infer all performance metrics k Hard to determine response to steady-state(sinusoids
Interpretations With noise in the system, the model is of the form =AC+ Bu+ Buw, y= Ca +U And the estimator is of the form =Ai+ Bu+L(y-9,y=Ci e Analysis: in this case: C-I=[AT+ Bu+Buw-[Ac+ Bu+L(y-gI A(-)-L(CI-Ca)+B
Full-state Feedback Control How do we change the poles of the state-space system? Or, even if we can change the pole locations Where do we put the poles? Linear Quadratic Regulator Symmetric Root Locus How well does this approach work? Copyright [2001 by JOnathan dHow
Class expectations This is a graduate level class. There is no final exam Grading in the class is based on homework (75%) and on a final written report(25%) The report will be revised during semester and should be 2000-3000 words(8-10 double spaced pages)
MG!∈R- ORDER5)sT6As RREL丹T(0sHP5 BETWEEN1MER6soN5毛 TAST∈P兵 ND THE POLE LOCAT(0NS心EK ALCULATE0 FoR A SECOND-ORDER SYSTEM GuES60工 NSIGHTS AcTuALLy Gooo APPRoXIMATIONS foR MAwy HIGHER- ORDER SYSTEMS BECAUSE THEIR
In going from the state space model i(t)=A.(t)+ Bu(t y(t)= Ca(t)+ Du(t) to the transfer function G(s)=C(sI -A)-B+D need to form the inverse of the matrix(sI- A)-a symbolic inverse- not easy at all For simple cases, we can use the following
