 # Dragon Notes UNDER CONSTRUCTION
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# 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 $$\bn{G}_0 = k\bn{I}$$ $$\ds\dot{\bn{x}}=\bn{Ax}+\bn{Bu}+\bn{Ex}_0,\ \bn{u}=-\bn{Gx}-\bn{G}_0\bn{x}_0\Rightarrow$$
$$\dsup\dot{\bn{x}}=(\bn{A}-\bn{BG})\bn{x}+(\bn{E}-\bn{BG}_0)\bn{x}_0$$
$$\dsup\bn{A}_c=\bn{A}-\bn{BG}$$

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 $$(=\t{#finite poles }-\t{ #finite zeros})$$ 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 ($$\s{}H(s)=1$$) 