Loading Document1 Four input-output transfer functions are listed below. Describe the systems they represent in state variable form, then draw the simulation diagram.
a) S+α1 b) S+αS+β c) S2+2ζωS+ω2S+β d) S2+2ζ2ω2S+ω22S2+2ζ1ω1S+ω12
a) x1˙=−αx1+u
y=x1
b) x1˙=−αx1+(β−α)u
y=x1+u
c) ⎣⎡x˙1x˙2⎦⎤=⎣⎡−2ζω−ω210⎦⎤⎣⎡x1x2⎦⎤+⎣⎡1β⎦⎤
y=x1
d) ⎣⎡x˙1x˙2⎦⎤=⎣⎡−2ζ2ω2−ω2210⎦⎤⎣⎡x1x2⎦⎤+⎣⎡2ζ1ω1−2ζ2ω2ω12−ω22⎦⎤u
y=x1+u
2-a Select a suitable set of state variables for the system whose transfer function is
T(S)=S3+9S2+24S+20S+3
observable canonical form
⎣⎢⎢⎢⎡x1˙x2˙x3˙⎦⎥⎥⎥⎤=⎣⎢⎢⎢⎡−9−24−20100010⎦⎥⎥⎥⎤⎣⎢⎢⎢⎡x1x2x3⎦⎥⎥⎥⎤+⎣⎢⎢⎢⎡013⎦⎥⎥⎥⎤u
y=[100]⎣⎢⎢⎢⎡x1x2x3⎦⎥⎥⎥⎤
controllable canonical form
S3+9S2+24S+20S+3=S3+9S2+24S+201×S+3
⎣⎢⎢⎢⎡x1˙x2˙x3˙⎦⎥⎥⎥⎤=⎣⎢⎢⎢⎡00−2010−2401−9⎦⎥⎥⎥⎤⎣⎢⎢⎢⎡x1x2x3⎦⎥⎥⎥⎤+⎣⎢⎢⎢⎡001⎦⎥⎥⎥⎤u
y=[310]⎣⎢⎢⎢⎡x1x2x3⎦⎥⎥⎥⎤
factored form
S3+9S2+24S+20S+3=(S+2)2(S+5)S+3
=S+2S+3×S+21×S+51
⎣⎢⎢⎢⎡x1˙x2˙x3˙⎦⎥⎥⎥⎤=⎣⎢⎢⎢⎡−5001−2001−2⎦⎥⎥⎥⎤⎣⎢⎢⎢⎡x1x2x3⎦⎥⎥⎥⎤+⎣⎢⎢⎢⎡011⎦⎥⎥⎥⎤u
y=[100]⎣⎢⎢⎢⎡x1x2x3⎦⎥⎥⎥⎤
2-b A system is described by
⎣⎡x1˙x2˙⎦⎤=⎣⎡−2−110⎦⎤⎣⎡x1x2⎦⎤+⎣⎡31⎦⎤u(t)
If x(0)=⎣⎡110⎦⎤ and u(t)=0 find x(t)
x(t)=L−1[SI−A]−1⋅x(0)
SI=⎣⎡S00S⎦⎤ A=⎣⎡−2−110⎦⎤ SI−A=⎣⎡S+21−1S⎦⎤
[SI−A]−1=(S+1)21⎣⎡S−11S+2⎦⎤
x(S)=(S+1)21⎣⎡S−11S+2⎦⎤⎣⎡110⎦⎤
=(S+1)21⎣⎡S+1010S+19⎦⎤=⎣⎢⎢⎡(S+1)2S+10(S+1)210S+19⎦⎥⎥⎤=⎣⎢⎢⎡S+11+(S+1)29S+110+(S+1)29⎦⎥⎥⎤
x1(t)=e−t+9te−t and x2(t)=10e−t+9te−t
4-a Define the controllability and observability.
A system is said to be controllable at time t0 if it is possible by means of a control signal to transfer the system from any initial state x(t0) to any other state in a finite interval of time.
A system is said to be observable at time t0 if it is possible to determine the initial state x(t0) from the observation of the output over a finite time interval.
4-b Investigate the observability condition for linear time invariant system
Consider the system x˙=Ax and y=Cx
where x=state vector(n−vector)
y= output vector(m−vector)
A=n×n matrix
C:m×n matrix
y(t)=CeAtx(0) where eAt is the state transition matrix.
if we expand eAt into a power series
eAt=K=0∑n−1AK⋅αK(t)K!tK
y(t)=C×K=0∑n−1{AK×αK(t)}×x(0)
for the system to be observable, the initial state x(0) can be determined given the output over a range of time.
we know that αK(t)can't be zero because t>0
that means the nm by n matrix ⎣⎢⎢⎢⎢⎢⎢⎡CCA⋮CAn−1⎦⎥⎥⎥⎥⎥⎥⎤must be of rank n.
in other form [C∗A∗C∗⋯A∗n−1C∗]must be of rank n.
4-d for the system T(S)=u(S)y(S)=(S+7)(S+15)(S+67)S+a
Determine the value of a, so that the system is either uncontrollable or unobservable.
T(S)=S3+89S2+1579S+7035S+a
the controllable canonical form is
A=⎣⎢⎢⎢⎡00−703510−157901−89⎦⎥⎥⎥⎤B=⎣⎢⎢⎢⎡001⎦⎥⎥⎥⎤C=[a10]
controllability matrix [BABA2B]=⎣⎢⎢⎢⎡00101−891−896342⎦⎥⎥⎥⎤ is full rank so the system is always controllable.
AT=⎣⎢⎢⎢⎡010001−7035−1579−89⎦⎥⎥⎥⎤CT=⎣⎢⎢⎢⎡a10⎦⎥⎥⎥⎤
observability matrix [CTATCT(AT)2CT]=⎣⎢⎢⎢⎡a100a1−7035−1579a−89⎦⎥⎥⎥⎤
det=a(a(a−89)+1579)−7035=a3−89a2+1579a−7035
has zeros a=7,15,67 thus will make the system unobservable.