1 / 12

Measures of System Performance

This mini-lecture explores essential concepts of system performance metrics, focusing on controllability, observability, and participation factors. Key discussions include the influence of states on system modes, the role of right and left eigenvectors in analyzing system dynamics, and the importance of participation factors in determining machine contributions to various modes. By examining controllability and observability matrices, we gain insights into decision-making for control variables and the interpretation of initial conditions in system perturbations, facilitating a better understanding of system behavior and performance.

nita
Télécharger la présentation

Measures of System Performance

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Measures of System Performance Starrett Mini-Lecture #6

  2. Questions • Which states are influenced by certain modes? • Which states are capable of influencing certain modes?

  3. Measures of System Structure • Right & Left Eigenvectors • Controllability & Observability • Participation Factors

  4. Controllability-Type Information • How can we control a mode or which states influence certain modes? • Needed for deciding which variables to control or modulate • Left Eigenvectors => zo = L xo • Controllability Matrix => B' = L B

  5. Controllability Index • z’ = Lz + LBDu = Lz + B’ DuDy = CRDx + DDu • kth row of B' = 0 => mode k is uncontrollable • Elements of rows tell which machines have the most influence • Left eigenvectors play a key role

  6. Observability-Type Information • Which machines are most influenced by certain modes? • Needed for deciding which variables to use as input for controller feedback • Right Eigenvectors => x(t) = R z(t) • Observability Matrix => C' = C R

  7. Observability Index • z’ = Lz + LBDu = Lz + B’ Du Dy = CRDx + DDu = C’Dx + DDu • ith column of C' = 0 => mode i is unobservable • Elements of columns tell which machines "show" the modes the most • Right eigenvectors play a key role

  8. Participation Factors • Which machines participate in which modes? • Two-way measures -- contain both controllability and observability information • Unit-less & independent if EV scaling • Assumes a "super-position-like" analysis

  9. Participation Factor Theory • Assume we are able to perturb only one state of the system (apply an initial condition along the axis of the kth state variable) • Initial Cond. xo = [0 0 ... 0 1 0 ... 0]T • Jordan IC zo = Lxo = [l1k l2k l3k ... lnk]T • Jordan solution => zi(t) = likel1t

  10. Transform Back to x-Space Using Dx = R z • xk(t) = rk1 l1k el1t + rk2 l2k el2t + ... + rkn lnk elnt • recall at t = 0 => xko = 1 = rk1 l1k + rk2 l2k + ... + rkn lnk • Participation Factor pki = rki lik • pki tells the extent of machine k's participation in mode i

  11. Linearizing a System • Taylor series for a function, g(x) • gTS(x) = 1[dg/dx](x-xo) + (1/2)[d2g/dx2] (x-xo)2 + (1/6)[d3g/dx3] (x-xo)3 + …

  12. Use Matrix Form • Dx’ = F(x) • Dx’ = F1(Dx) + F2(Dx) + F3(Dx) + ...Dx’ = A Dx + F2(Dx) + F3(Dx) + ... • Truncating the series and taking only the linear terms yields the familiar formDx’ = A Dx

More Related