modelado en el espacio matlab.pdf
TRANSCRIPT
-
>> syms s t C
>> LHS = laplace(3*diff(sym('c(t)'),2) +5*diff (sym('c(t)'))+ 1*sym('c(t)'))
LHS =
3*s^2*laplace(c(t), t, s) - 3*D(c)(0) - 3*s*c(0) - 5*c(0) + 5*s*laplace(c(t), t, s) + laplace(c(t), t, s)
>> newLHS = 3*s^2*C + 5*s*C + C
newLHS =
3*C*s^2 + 5*C*s + C
>> ut=exp(0*t)
ut =
1
>> RHS=laplace(8*ut)
RHS =
8/s
>> C = solve (newLHS-RHS, C)
C =
8/(3*s^3 + 5*s^2 + s)
>> pretty(C)
8
---------------
3 2
3 s + 5 s + s
-
() =1
( 1)+
2( 2)
+3
( 3)+
() =1.547
( + 1.4343)
9.547
( + 0.2324)+
8
( 0)+ 0
>> b = [0 0 0 8] b = 0 0 0 8 >> a = [3 5 1 0] a = 3 5 1 0 >> [r, p, k] = residue (b, a) r = 1.5470 -9.5470 8.0000 p = -1.4343 -0.2324 0 k = [] >> t=0:0.5:40 >> y= 1.547*exp(-1.4343*t) - 9.547*exp(-0.2324*t) + 8 >> plot(t,y)
>> num = [0 0 8] num = 0 0 8 >> den = [3 5 1] den = 3 5 1 >> sys=tf(num,den) sys = 8 --------------- 3 s^2 + 5 s + 1 >> step(sys) >> pole(sys) ans = -1.4343 -0.2324
-
>> syms s t C
>> LHS = laplace(3*diff(sym('c(t)'),2) +5*diff (sym('c(t)'))+ 1*sym('c(t)'))
LHS =
3*s^2*laplace(c(t), t, s) - 3*D(c)(0) - 3*s*c(0) - 5*c(0) + 5*s*laplace(c(t), t, s) + laplace(c(t), t, s)
>> newLHS = 3*s^2*C + 5*s*C + C
newLHS =
3*C*s^2 + 5*C*s + C
>> ut=exp(0*t)
ut =
1
>> RHS=laplace(8*ut)
RHS =
8/s
>> C = solve (newLHS-RHS, C)
C =
8/(3*s^3 + 5*s^2 + s)
>> G=tf([8],[3 5 1])
G =
8
---------------
3 s^2 + 5 s + 1
Continuous-time transfer function.
>> sys2=canon(ss(G),'compan')
sys2 =
a =
x1 x2
x1 0 -0.3333
x2 1 -1.667
-
b =
u1
x1 1
x2 0
c =
x1 x2
y1 0 2.667
d =
u1
y1 0
Continuous-time state-space model.
>> step(sys2)
-
+ 3 + 2 + 5 + 8 = 7
>> syms s t C
>> LHS = laplace(diff(sym('c(t)'),4) + 3*diff (sym('c(t)'),3)+ 2*diff (sym('c(t)'),2) + 5*diff (sym('c(t)'))
+ 8*sym('c(t)'))
LHS =
2*s^2*laplace(c(t), t, s) - 5*c(0) - 2*D(c)(0) - 3*D(D(c))(0) - 2*s*c(0) - D(D(D(c)))(0) +
3*s^3*laplace(c(t), t, s) + s^4*laplace(c(t), t, s) - 3*s*D(c)(0) - 3*s^2*c(0) - s^3*c(0) - s*D(D(c))(0) -
s^2*D(c)(0) + 5*s*laplace(c(t), t, s) + 8*laplace(c(t), t, s)
>> newLHS = 2*s^2*C + 3*s^3*C + s^4*C + 5*s*C + 8*C
newLHS =
C*s^4 + 3*C*s^3 + 2*C*s^2 + 5*C*s + 8*C
>> ut=exp(0*t)
ut =
1
>> RHS=laplace(7*ut)
RHS =
7/s
>> C = solve (newLHS-RHS, C)
C =
7/(s*(s^4 + 3*s^3 + 2*s^2 + 5*s + 8))
>> pretty(C)
7
------------------------------
4 3 2
s (s + 3 s + 2 s + 5 s + 8)
>> G=tf([7],[1 3 2 5 8])
-
= [
0008
1005
0102
0013
]
= [
0007
]
= [1 0 0 0]
= 0
G =
7
-----------------------------
s^4 + 3 s^3 + 2 s^2 + 5 s + 8
Continuous-time transfer function.
>> sys2=canon(ss(G),'compan')
sys2 =
a =
x1 x2 x3 x4
x1 0 0 0 -8
x2 1 0 0 -5
x3 0 1 0 -2
x4 0 0 1 -3
b =
u1
x1 1
x2 0
x3 0
x4 0
c =
x1 x2 x3 x4
y1 0 0 0 7
d =
u1
y1 0
Continuous-time state-space model.
-
= [
0008
1005
0102
0013
] = [
0007
] = [1 0 0 0] D=0
>> step(sys2)
= [
0008
1005
0102
0013
] + [
0007
]
= [1 0 0 0] +
-
>> A=[0 1 0 0; 0 0 1 0; 0 0 0 1; -8 -5 -2 -3]
A =
0 1 0 0
0 0 1 0
0 0 0 1
-8 -5 -2 -3
>> B=[0;0;0;7]
B =
0
0
0
7
>> C=[1 0 0 0]
C =
1 0 0 0
>> D=[0]
D =
0
>> sys=ss(A,B,C,D)
sys =
a =
x1 x2 x3 x4
x1 0 1 0 0
x2 0 0 1 0
x3 0 0 0 1
x4 -8 -5 -2 -3
-
b =
u1
x1 0
x2 0
x3 0
x4 7
c =
x1 x2 x3 x4
y1 1 0 0 0
d =
u1
y1 0
Continuous-time state-space model.
>> tf(sys)
ans =
7
-----------------------------
s^4 + 3 s^3 + 2 s^2 + 5 s + 8
Continuous-time transfer function.
>> pole(sys)
ans =
-2.4829 + 0.0000i
-1.4892 + 0.0000i
0.4860 + 1.3883i
0.4860 - 1.3883i
-
= [
0008
1005
0102
0013
] = [
0007
] = [1 0 0 0] D = 0
>> A=[0 1 0 0; 0 0 1 0; 0 0 0 1; -8 -5 -2 -3]
A =
0 1 0 0
0 0 1 0
0 0 0 1
-8 -5 -2 -3
-
>> B=[0; 0; 0; 7]
B =
0
0
0
7
>> C=[1 0 0 0]
C =
1 0 0 0
>> D=[0]
D =
0
>> I=eye(4)
I =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
>> syms s
>> G=C*inv(s*I-A)*B+D
G =
7/(s^4 + 3*s^3 + 2*s^2 + 5*s + 8)
>> pretty(G)
7
--------------------------
4 3 2
s + 3 s + 2 s + 5 s + 8