I am attempting to work a problem from a textbook in sympy, but sympy fails
to find a solution which appears valid. For interest, it is the design of a
PID controller using direct synthesis with a second order plus dead time
model.
The whole problem can be reduced to finding K_C, tau_I and tau_D which will
make
K_C*(s**2*tau_D*tau_I + s*tau_I + 1)/(s*tau_I) = (s**2*tau_1*tau_2 + s*tau_1 +
s*tau_2 + 1)/(K*s*(-phi + tau_c))
for given tau_1, tau_2, K and phi.
I have tried to solve this by matching coefficients:
import sympy
s, tau_c, tau_1, tau_2, phi, K = sympy.symbols('s, tau_c, tau_1, tau_2,
phi, K')
target = (s**2*tau_1*tau_2 + s*tau_1 + s*tau_2 + 1)/(K*s*(-phi + tau_c))
K_C, tau_I, tau_D = sympy.symbols('K_C, tau_I, tau_D', real=True)
PID = K_C*(1 + 1/(tau_I*s) + tau_D*s)
eq = (target - PID).together()
eq *= sympy.denom(eq).simplify()
eq = sympy.poly(eq, s)
sympy.solve(eq.coeffs(), [K_C, tau_I, tau_D])
This returns an empty matrix. However, the textbook provides the following
solution:
booksolution = {K_C: 1/K*(tau_1 + tau_2)/(tau_c - phi),
tau_I: tau_1 + tau_2,a
tau_D: tau_1*tau_2/(tau_1 + tau_2)}
Which appears to satisfy the equations I'm trying to solve:
[c.subs(booksolution).simplify() for c in eq.coeffs()]
returns
[0, 0, 0]
Can I massage this into a form which sympy can solve? What am I doing wong?
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