system of differential equations f(t) : input u(t) and v(t) : outputs to be found system of constant...
TRANSCRIPT
SYSTEM OF DIFFERENTIAL EQUATIONS
vdt
du
fvudt
dv 3120
f(t) : Input
u(t) and v(t) : Outputs to be found
System of constant coefficient differential equations with two unknowns
* First order derivative terms are on the left hand side
* Non-derivative terms are on the right hand side
fv
u
v
u
1
0
3120
10
BuAxx
x : State variable matrix (nx1)
A : System matrix (nxn)
u : Input matrix (mx1)
B : Matrix with dimension (nxm)
Example:
y33.5+x70.7-f=3850y-8175x+x5.8
y65.5-x33.5=3850x-7350y+y3.2
f(t):Input x(t) and y(t) : Outputs to be found
f.
v
u
y
x
....
....
v
u
y
x
0
170
0
0
5205109229611203
85212866351409
1000
0100
BuAxx
ux and vy ux and
vy
Let’s use the variables
BuAxx
s X(s) - x0 = A X(s) + B U(s)
s I X(s) – x0 = A X(s) + B U (s)
[s I – A] X(s) = x0 + B U (s)
X(s) = [s I – A]-1 x0 + [s I – A]-1 B U (s)
Homogenous part Particular part
Laplace Transformation:
3120
10
s0
0s]AsI[
D(s)=det[sI-A]=s2+3s+120
)s(D
1
s120
13s]AsI[ 1
3s120
1s
0AsI : Eigenvalue equation
s
4
1
0
s120
13s
3
2
s120
13s
)s(V
)s(U)s(D
f1
0
v
u
3120
10
v
u
At t=0 u=-2 , v=3 ; F(s) = -4/s
)s(sD
4
)s(D
3s2)s(U
)s(D
4
)s(D
240s3)s(V
X(s) = [s I – A]-1 x0 + [s I – A]-1 B U (s)
Example:
3120
10
s0
0s]AsI[ D(s)=det[sI-A]=s2+3s+120
)s(D
1
s120
13s]AsI[ 1
3s120
1s
0AsI : Eigenvalue equation
s
4
1
0
s120
13s
3
2
s120
13s
)s(V
)s(U)s(D
f1
0
v
u
3120
10
v
u
At t=0 u=-2 , v=3 ; F(s) = -4/s
)s(sD
4
)s(D
3s2)s(U
)s(D
4
)s(D
240s3)s(V
X(s) = [s I – A]-1 x0 + [s I – A]-1 B U (s)
Example:
With Matlab :
Calculation of eigenvalues:
a=[0,1 ; -120,-3] ; eig(a)
syms s;a=[0,1;-120,-3] ; i1=eye(2);a1=inv(s*i1-a);pretty(a1)
f1
0
v
u
3120
10
v
u
Determination of matrix [sI-A]1: