Lecture 11 state observer-2020
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This document describes designing an observer for a system. The observer is designed to have poles at z1=0, z2=0, and z3=-0.1. The initial state of the system is [1;0] and the initial state of the observer is [0;1]. The error between the actual and observed state is calculated for the first 5 samples, and is [5.0259; 1.2579; -0.2619; 0.0262; 0.0022].
![a
9rtD
k-tS.Lv-fr : GLcA -1.$
c- qA = c Lrc,a, -tcA
*f k 1r+se-**..
CLw = q ecfi
uA ,tea- "X a""- e,cto.r e-t t&o**e,o q.^,.) V- u^re^an^+-
e_cr. = G qc
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".
L.x
.
<:
(
K=n e( =iet
, Q^"- &o.,io.r 1 e^* e,,.c5rr "r,U-* c'
q *^^." ^=^N L ,.,.xo,uoR^'-,(&-.c-L
x^-osix q .o,R. *=A. i*^.U--R"- *-S -"*"),* **L$
c=
:*.i :1 .+ q= F),Js:1 .rQi.*-;,
L-o-ut] - -r L:
t:='Lt]
k^J t. P--5q= -.-^J.ir- a.o , wr<J-egL*< . $o(e-oec,..
(- Q^*
6,,5
sn^c- s-o-.N-e..L*.^,...*Tt., to'' I
(=:1":.;1 -i q=ti.y[ilL. .-:..uJ
q"" $e-
t""- e/sfar <rL,, ..rte^^" *V;
o-U=.Q*k -:*](https://image.slidesharecdn.com/lecture11state-observer-2020-200826112510/85/Lecture-11-state-observer-2020-2-320.jpg)
![tv- =
c$
tu- =
q ttt .^c + K* L q -.q1 k
cT,c
F"* rua.J*uo- *,t o..k-cc*. x.-1rr "q, ^Q**-*,
K. 't tr *.d*r --"a^*"S^*-eJstoc- U&te-e^ q.
*t *^hiE . ,u-
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' R*** e,o *- l' - i-co,a. "^ s ],eso
<r
xt o.A ,.de, .orsQ. .
.h--F" (. * c r
uye-
X"t
xcwt -trv- = L*ciL.N a. K &.q -*
e- lr** =
b
-(](https://image.slidesharecdn.com/lecture11state-observer-2020-200826112510/85/Lecture-11-state-observer-2020-3-320.jpg)
![ril,
Consider the system described by
l- 1 -L2 21 t21
x(k+ t, =
[j., I _T]
x@)+
[_or]",u,,
y(k) - [-o.s 0 -t]x(k),
a) Design an observer whose poles are located ot 21= Q, zz = 0 and za = -0.1) for this system
b)- Irthe initial states orthe system x(0):
[i]
* orthe observer i(0) :
[jl],determine
the error between the actual state and observed state for the initial 5 samples.
Solution:
Let Gr = o' : [-i, ? -S'tt r-o'sr
l. z -o.s -, l'
and Hr : c'=
t -l l
Then, the controllability matrix is given by
coNr = [Hr Gr*Hr Gr2 H,] =
[l:
::
+]
Itsinverse coNl-' :[, -3.u
jrl t R=t-2 0 l]
l-z o 1l
I ft 1 t-z o 1r
Hence
'-':lr{'rr"1: [: 1:3 -%]
r,:p-,*G*p :[:"
:B:i ]rll:r: _:, l,']
:[_,, ':j lj]
=
[| L i],
and F, :P-r *Hl :
[]
t-2 0.5 -0.5 r
And t=[-0,
i -', I](https://image.slidesharecdn.com/lecture11state-observer-2020-200826112510/85/Lecture-11-state-observer-2020-6-320.jpg)
![ri?
Observer characteristic equation : * * ( z + 0.1) : f + O.l *
f, :[1 -4 1.1]
l-2 0 1l
Kr=fl -4 l.ll l0 -0.5 0l:[4.6 1.45 -3.4]
I o -o.s -4]
r 4.61
Ke: K I:
[:;1il
(G_Ke*c): *:l:!:; -i'
*il
a) e(k+l) = Ge e(k);
",t, { l]
t -38 I t-26.L91 I 2.6L9 r t-0.26191 t 0.0262 1
e(r)=l s.ozsl, e(2)=lzszsl, e(g)=l-o.zraal, e(+)=l o.ozra l, e(s)=l o.oozz I
l, zs.+ I l_ tt.+o I I -t.t+o I l- 0.1146l L-o.orzsl](https://image.slidesharecdn.com/lecture11state-observer-2020-200826112510/85/Lecture-11-state-observer-2020-7-320.jpg)
Lecture 11 state observer-2020
- 1. s&. Css,6-ec L' tl .&^,t*.1. .ryr =^n rrt . ursr L yn.&J -& ^*"qu *y "o. _-",4 se^,. . e^.^dc a._,.Fr- ^ aqrr , v- *^",o* L.ri . q"* VO **&^.'^ * o*. yns"- R^- . q^^ er**& .*_ -t q_r ttt-* s.^^ . -f*^^,1,.-'e^, e- i v & *k-^ . ;$-L-'u.- -'^^T rr"gt ays ( Lr= 9...A_r."eA ttq : (, r-c -t -( fe*- r 'rh ,-E"K e,rge5ver .r',-L*:.-* t Lcv-t, : G., .A.1 ,''' tr-tq a, .= c- Lu4 " S""- o5** t'** q^^P {l*tl"-tq. *(q o^,'. Xt .. ^-GuJr .-^ ^^wN . *.*-[Q-- N"h- oL=***"- ,,.,-,5N- .)=,U^r.e- s.qc-J,ry*" T,cA IeI.t^ ...c *^"tA u*^.o- :.! zo e,; *^48-u.,. N..*I-..^aN--"- 5t-'u^ u 5t"^^ . S*i.*u.- R.* + o 5g*^^q q.Aq- .^tri* R*"" t^x" ern{..'* ' ta - r*trQ*^^U"o, *va {il".a-}. **srllr<- . BN r -r^,,.ce- ^}Q.- ...NtV--^,f x-t g,,"^^, ce- **1L. e-Jrtl;... R^"-N t, o"^ d- a;o t^l*- EF <,C = r-c -C,,e, st-{5(+,* trq*.btr t
- 2. a 9rtD k-tS.Lv-fr : GLcA -1.$ c- qA = c Lrc,a, -tcA *f k 1r+se-**.. CLw = q ecfi uA ,tea- "X a""- e,cto.r e-t t&o**e,o q.^,.) V- u^re^an^+- e_cr. = G qc e( :SE-C=*1 ". L.x . <: ( K=n e( =iet , Q^"- &o.,io.r 1 e^* e,,.c5rr "r,U-* c' q *^^." ^=^N L ,.,.xo,uoR^'-,(&-.c-L x^-osix q .o,R. *=A. i*^.U--R"- *-S -"*"),* **L$ c= :*.i :1 .+ q= F),Js:1 .rQi.*-;, L-o-ut] - -r L: t:='Lt] k^J t. P--5q= -.-^J.ir- a.o , wr<J-egL*< . $o(e-oec,.. (- Q^* 6,,5 sn^c- s-o-.N-e..L*.^,...*Tt., to'' I (=:1":.;1 -i q=ti.y[ilL. .-:..uJ q"" $e- t""- e/sfar <rL,, ..rte^^" *V; o-U=.Q*k -:*
- 3. tv- = c$ tu- = q ttt .^c + K* L q -.q1 k cT,c F"* rua.J*uo- *,t o..k-cc*. x.-1rr "q, ^Q**-*, K. 't tr *.d*r --"a^*"S^*-eJstoc- U&te-e^ q. *t *^hiE . ,u- ^r.^x*-R.n-.[*so( {o.. eansr-tr-> ' R*** e,o *- l' - i-co,a. "^ s ],eso <r xt o.A ,.de, .orsQ. . .h--F" (. * c r uye- X"t xcwt -trv- = L*ciL.N a. K &.q -* e- lr** = b -(
- 4. 4rr YKWil T,H,^* qA^A' *Je.-(;-ffi w* d, -"or-q-^^a.. a*-ee- &-* * e$* *)'"-^1aiN *u+ u*e-*)*^^ $ilhnt 1 ale- v'**u-,.^oJ r*' -tt-cr-. h, Wsiffi$***^"^*'x-cx-= SuAr.ut h *tf { ,i,q, c-v-. 8.,^^, g.,,, ao:e.*.r x,oi-*^XiL vl ( {. .** *Y^t *- *' d <--'t""'v"^'^ *^ *^'' c='
- 5. tI Sx aq**,V (e- ..)e- LV. dk^*- Se^^ ur< = c, -c*.1 rq t"e..r*(-fr ^:*^,Til*J:: : _K,1* .fu+.- N* *"k*.'q"ffi)[ffie&^:'UNt,.x_ K*= "I . - ' $r^/v1 3s-- r:rs.-e'Lsexves T"v-, q,* brr*o**r "tr^^^ 1i*^5'.,*s*;t ,.a-"N* L os.N.ffiffik-*;".** o*n u*r r* V-u. ^o13qJ **n*.d. -^-_L lL", U..- *w --* .e_ .r *x* tr^ ^t GN^.L-.-^ o,., ,iiE (.+.lc *tr^ r .-, .ra* ^ ""sslt - -."*di :il "^ d.x=re. --,.^'.te-..r, *r*" *ns_ VN S.* xR, _ru *q,- H;ffivec'*.vv*Ail*c
- 6. ril, Consider the system described by l- 1 -L2 21 t21 x(k+ t, = [j., I _T] x@)+ [_or]",u,, y(k) - [-o.s 0 -t]x(k), a) Design an observer whose poles are located ot 21= Q, zz = 0 and za = -0.1) for this system b)- Irthe initial states orthe system x(0): [i] * orthe observer i(0) : [jl],determine the error between the actual state and observed state for the initial 5 samples. Solution: Let Gr = o' : [-i, ? -S'tt r-o'sr l. z -o.s -, l' and Hr : c'= t -l l Then, the controllability matrix is given by coNr = [Hr Gr*Hr Gr2 H,] = [l: :: +] Itsinverse coNl-' :[, -3.u jrl t R=t-2 0 l] l-z o 1l I ft 1 t-z o 1r Hence '-':lr{'rr"1: [: 1:3 -%] r,:p-,*G*p :[:" :B:i ]rll:r: _:, l,'] :[_,, ':j lj] = [| L i], and F, :P-r *Hl : [] t-2 0.5 -0.5 r And t=[-0, i -', I
- 7. ri? Observer characteristic equation : * * ( z + 0.1) : f + O.l * f, :[1 -4 1.1] l-2 0 1l Kr=fl -4 l.ll l0 -0.5 0l:[4.6 1.45 -3.4] I o -o.s -4] r 4.61 Ke: K I: [:;1il (G_Ke*c): *:l:!:; -i' *il a) e(k+l) = Ge e(k); ",t, { l] t -38 I t-26.L91 I 2.6L9 r t-0.26191 t 0.0262 1 e(r)=l s.ozsl, e(2)=lzszsl, e(g)=l-o.zraal, e(+)=l o.ozra l, e(s)=l o.oozz I l, zs.+ I l_ tt.+o I I -t.t+o I l- 0.1146l L-o.orzsl