Thank you for your response.
Coincedentally I was just progressing along a similar route myself. What I came
up with was
eqs = eq.coeffs()
solution = {}
solution[K_C] = sympy.solve(eqs[1], K_C)[0]
solution[tau_D] = sympy.solve(eqs[0], tau_D)[0].subs(solution)
solution[tau_I] = sympy.solve(eqs[2], tau_I)[0].subs(solution).simplify()
This matches your method. So this has saved me some tedious algebra, but not
the effort of finding the order in which to evaluate the equations. I wish that
sympy could do this automatically. I seem to remember Sage being able to solve
this set of equations, but I can’t find the worksheet now.
> On 23 Feb 2016, at 18:35, Carsten Knoll <[email protected]> wrote:
>
> Hi,
>
> I already had similar problems. The system of equation is of order 3
> which might be too hard. But there is a linear part and if this is
> solved first and then plugged into the remaining 2 equations, sympy can
> manage it.
>
> I documented my attempt here:
>
> https://gist.github.com/cknoll/c03dcf8443c0409d37da
>
>
> (Generally, I think sympy.solve could sometimes be a little bit smarter,
> to use such structural properties.)
>
>
>
> On 02/23/2016 06:43 AM, Carl Sandrock wrote:
>> 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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