Transformation standard-controllable--2020
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The document finds the state feedback gain matrix K for a controllable system to place the closed loop poles at specific locations. It provides the system model in controllable canonical form. It then calculates the characteristic equation for the closed loop system with feedback. By setting the characteristic equation equal to the desired closed loop pole locations, it solves for the feedback gain matrix K.
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This document discusses coordinate systems and vector calculus concepts needed for electromagnetic field theory. It introduces Cartesian, cylindrical, and spherical coordinate systems. It explains that vector integration requires defining appropriate differential elements (length, area, volume) that vary based on the coordinate system. It also introduces concepts of gradient, divergence, and curl - vector operators used to take derivatives of vector fields. The gradient represents the maximum rate of change, divergence measures flux, and curl represents rotational nature. Expressions for these operators are given in the three coordinate systems.
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Fields Lec 5&6
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Fields Lec 4
Electrical stimulation can be used for many applications including vision restoration, epilepsy control, tremor control, cardiac pacing, and more. Magnetic fields are generated by moving electric charges. The Biot-Savart law describes the magnetic field generated by a current element, while Ampere's law relates the magnetic field to the current passing through a closed loop. Materials respond differently to magnetic fields based on properties like diamagnetism, paramagnetism, and ferromagnetism. Ferromagnetic materials have domains that can align with an external magnetic field, allowing the material to retain magnetization.
Fields Lec 3
Here are some examples of FDA-approved therapeutic devices that use direct current (DC) electric fields:
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- Transcutaneous electrical nerve stimulators (TENS) - Apply electric currents to stimulate nerves for pain relief and muscle rehabilitation.
- Iontophoresis devices - Use low-level electrical currents to drive ionized drug molecules through the skin for local drug delivery.
- Cardioversion/defibrillation devices - Apply controlled electric shocks to the heart to treat irregular heart rhythms like atrial fibrillation or ventricular fibrillation.
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Transformation standard-controllable--2020
- 1. l
- 2. -U- q,^r-{^ ,.h R^{h , ,$,tffi'*"J-";"s{e.. a^-- T*ro^^ o'r^', .-d^_ ' (:4'A^) t"*., e'$'*^* t""'.rNL1u-,""E 15=r-^^ -. *^ Y'=-1 a^^ Y'= ,-^. r r,s.Q...*...*- -1* C5 rtq- ,* tu' c=h C! t- G, $:hf = r=tcnl r = L o- - <-,f = aC*il NS = ts s- ( sl -{ -s l{=) a 4lt N.L qe €'e = v)ir..Q;. s ftt -] er=lq I I, lt.G,, r L" J A |I:l- itl =*s.,!F y" ID
- 3. 3/, TD N^x ler =$-s =,=,ir: :r] [rli.': Li";r :N^, k-"t=: Q." = [1 [] <" ^ a- - q q I -( q - *-----€ ^G o.oo**f, U-*;.5 t= 's= 1, I Lr.q-l = wl : R-=C) =L41 - err-, Tq,.^ , rc-v4^S^"_
- 4. rD Given the controllable system described by x(k+,,= [| i +] -r.H] u(k), y(k)=rl-1 zl Find the state feedback gain matrix K so as to locate the closed loop poles at zt,z: 0.4 !0.6 j and 4 = -0.2 Solution: - coN:rH G*H G*c*Hl:[j, & ,1] coN.r =l], j, il:Wl P-'=V;:,1::[-f -i'iland P- ts !:,#] The standard controllable form of the system before feedback is given by i (k + L) : P-t5 P fr(k) + P-thu(k) Thus, we have i(k +1) - p-r c p r(k) + p-, h ,G): [*3 i i] ,ur. [?] ,n, The closed loop characteristic equation is given by '3+ur* + azz +a3 z,'-;2).**'6?, l'o.rtl-0'6i ) (z +0'2) Therefore, R = l(ocs- as ) (xz- az) (xr- ar) I : [ -1.896 1.36 1.4] u(k):: r(k) - fr ft(k) : r(k) - fr P-1x(k) :.> or K: fr p-t K: [ 1.696 -0.864 4.528]
- 5. r? all q ;-.t qt^^h NdN_,*_Ur*-R^" rF..t".u h^J S"* &{^n o.re,se+R$Y)N}}:-:'[ v^p.rr r'-r o^^ til -".h . - o a^* ^^* N-.- e- .r',g-, x^^x .1 . **- iq*^.q. lq^ -A"*'k" { rt K^A {e- 6c^N- [. **ai,S.- c-, l :s={sS:.nIYHffiNffi*u$s^ ^"^^S