Graduationwoot

Dragon Notes

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  UNDER CONSTRUCTION

Control Systems:
Control Design




Exogenous Input Control [FSFB]

Dynamic system design accounting for exogenous inputs (in Full-state Feedback (FSFB) systems):

\(\ds\boxed{\bn{G}_0 = (\bn{C}\bn{A}_c^{-1}\bn{B})^{-1}\bn{C}\bn{A}_c^{-1}\bn{E} = \bn{B}^{\t{#}}\bn{E}}\)
Goal is to zero-out observed output in response to exogenous input \(x_0\) - i.e., \(\bn{y}=\bn{Cx}=0\), where \(\t{dim}(\bn{y}) = \t{dim}(\bn{u})\)
  • - Must satisfy \(\bn{C}\bn{A}_c^{-1}\bn{B}\bn{G}_0 = \bn{C}\bn{A}_c^{-1}\bn{E}\)
  • - \(\bn{B}^{\t{#}} =\) left pseudo-inverse of \(\bn{B}\); \(\bn{B}^{\t{#}}\bn{B} = \bn{I}\)
  • - \(\bn{C}\bn{A}_c^{-1}\bn{B} = \) square matrix
  • - If \(\bn{E}=k\bn{B}\), then \(\bn{G}_0 = k\bn{I}\)
Full-state feedback control system
\(\ds\dot{\bn{x}}=\bn{Ax}+\bn{Bu}+\bn{Ex}_0,\ \bn{u}=-\bn{Gx}-\bn{G}_0\bn{x}_0\Rightarrow\)
\(\dsup\dot{\bn{x}}=(\bn{A}-\bn{BG})\bn{x}+(\bn{E}-\bn{BG}_0)\bn{x}_0\)
\(\dsup\bn{A}_c=\bn{A}-\bn{BG}\)

Root Locus

The root locus is a representation of the paths of closed-loop poles as the gain is varied. More generally, it describes the change in roots of a system as a certain parameter is varied.

[#locus branches \(=\) #cl-p's] Branch \(=\) the path a pole traverses as gain is varied; there's one branch per closed-loop pole
[Real-axis symmetry] Complex closed-loop poles not in conjugate pairs imply polynomials with complex coefficients, not realizable by physical systems
[Real-axis segments] Real-axis angle is \(\deg{180}\) for regions left of an odd # of ol-pz's, and \(\deg{0}\) otherwise (alternating). (Angular contributions to points on the real axis cancel for conjugate-pair & left-lying ol-pz's)
[Path: poles \(\rightarrow\) zeros] Branches emerge from finite & infinite ol-poles and terminate at finite & infinite ol-zeros.
[Infinity branches] \(\boxed{\sigma_a=\frac{\sum\t{finite poles }-\sum\t{ finite zeros}}{\t{#finite poles }-\t{ #finite zeros}}}\) \(\boxed{\theta_a=\frac{(2k+1)\pi} {\t{#finite poles }-\t{ #finite zeros}}}\) Branches asymptotically approach straight lines as the locus approaches infinity, with the lines' real-axis intercepts given by \(\sigma_a\) and angle \(\theta_a\) wrt. pos. real axis.
[\((-K)^{1/r}\)] Rate of loci approach to zeros at infinity, where \(r = \t{deg}(\t{Den})-\t{deg}(\t{Num})=\) 'relative degree' of denominator w.r.t. numerator \((=\t{#finite poles }-\t{ #finite zeros})\)
Root locus, second order system

Nyquist Diagram

Graphically relates cl system stability to ol frequency response and ol-p location; obtained by mapping an infinite semicircle enclosing the rhp through the ol transfer function.

  • Nyquist Criterion: \(Z=0\ra \t{Stability}\), where \(Z = P - N\)
  • \(Z=\) # of enclosed open-loop zeros (or closed-loop poles)
  • \(P=\) # of enclosed open-loop poles (or closed-loop zeros)
  • \(N=\) # of counterclockwise revolutions about \(s = -1\)
  • - The Nyquist diagram is a plot of \(F(s)-1=G(s)H(s)\) with \(s=j\omega\) traversing an infinite rhp-semicirle (frequency response)
  • - Distance between origin and a point on the plot is the magnitude of \(G(s)H(s)\)
Nyquist diagram of \(\s{}G(s) = 1/(s+2)(s+4)\)
is shown to the right (\(\s{}H(s)=1\))





Dragon Notes,   Est. 2018     About

By OverLordGoldDragon