A module for symbolic manipulation of linear, time-invariant control systems.
import sympy
import ltisym
# define a transfer function and convert to state space:
sympy.var('s a0 a1 a2 b0 b1 b2')
tf = ltisym.TransferFunction(
(a2*s**2 + a1*s + a0) /
(s**3 + b2*s**2 + b1*s + b0)
)
ss = ltisym.StateSpace(tf)
print(tf)
print(ss)
# define a state space and convert to transfer function:
var('a11 a12 a21 a22 b1 b2 c1 c2 d1')
a = Matrix([[a11, a12],
[a21, a22]])
b = Matrix([[b1],
[b2]])
c = Matrix([[c1, c2]])
d = Matrix([[d1]])
ss = ltisym.StateSpace(a, b, c, d)
tf = ltisym.TransferFunction(ss)
print(ss)
print(tf)
# Compound system from block diagram components (cascade example):
"""
IEEE AC8B Exciter (simplified)
.--------.
vref .->| Kpr |----.
| | '--------' |
+ v | + v .--------. .--------.
,-. e | .--------. + ,-. pid | Ka | | Ke |
( S )----+->| Kir/s |->( S )---->| ------ |----->| ------ |---> vfd
`-' | '--------' `-' | 1+s*Ta | vr | 1+s*Te |
- ^ | .--------. + ^ '--------' '--------'
| | | s*Kdr | | (x2) (x3)
vt '->| ------ |----'
| 1+s*Tr |
'--------'
(x0, x1)
"""
sympy.var('s Kpr Kir Kdr Tdr Ka Ta Ke Te')
# define component transfer functions:
tf1 = ltisym.TransferFunction(Kpr + Kir/s + Kdr*s / (1 + s*Tdr))
tf2 = ltisym.TransferFunction(Ka / (1 + s*Ta))
tf3 = ltisym.TransferFunction(Ke / (1 + s*Te))
# convert to state space models:
ss1 = ltisym.StateSpace(tf1)
ss2 = ltisym.StateSpace(tf2)
ss3 = ltisym.StateSpace(tf3)
# connect systems using the StateSpace.cascade function:
ss_ac8b = ss1.cascade(ss2).cascade(ss3)
# create transfer function version of the whole system:
tf_ac8b = ltisym.TransferFunction(ss_ac8b)
# display models:
print(tf_ac8b)
print(ss_ac8b)tf to ss test
-------------
G(s) =
2
a0 + a1*s + a2*s
----------------------
2 3
b0 + b1*s + b2*s + s
A =
[-b2 -b1 -b0]
[ ]
[ 1 0 0 ]
[ ]
[ 0 1 0 ]
B =
[1]
[ ]
[0]
[ ]
[0]
C= [a2 a1 a0]
D= [0]
ss to tf test
-------------
A =
[a11 a12]
[ ]
[a21 a22]
B =
[b1]
[ ]
[b2]
C = [c1 c2]
D = [0]
G(s) =
-(b1*(a21*c2 - c1*(a22 - s)) + b2*(a12*c1 - c2*(a11 - s)))
-----------------------------------------------------------
a12*a21 - (a11 - s)*(a22 - s)
cascade test
------------
A =
⎡ -1 ⎤
⎢ ─── 0 0 0 ⎥
⎢ Tdr ⎥
⎢ ⎥
⎢ 1 0 0 0 ⎥
⎢ ⎥
⎢ Kdr Kir -1 ⎥
⎢- ──── + Kir ─── ─── 0 ⎥
⎢ 2 Tdr Ta ⎥
⎢ Tdr ⎥
⎢ ⎥
⎢ Ka -1 ⎥
⎢ 0 0 ── ───⎥
⎣ Ta Te⎦
B =
⎡ 1 ⎤
⎢ ⎥
⎢ 0 ⎥
⎢ ⎥
⎢Kdr ⎥
⎢─── + Kpr⎥
⎢Tdr ⎥
⎢ ⎥
⎣ 0 ⎦
C =
⎡ Ke⎤
⎢0 0 0 ──⎥
⎣ Te⎦
D = [0]