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].

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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
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=
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and F, :P-r *Hl :
[]
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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:
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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