Classical proper PID controllers are designed for linear time invariant plants whose transfer functions are rational functions of sα, where 0<α<1, and s is the Laplace transform variable.
The reason for this is that the classical rate laws are rational functions of the variables and they are built upon different types of simplifying assumptions on the detailed mechanism of the reactions.
If w is non-odd and non-even, then w ( z ) = Q 1 ( ℘ ( z ) ) + ℘ ( z ) ′ Q 2 ( ℘ ( z ) ), where Q 1 and Q 2 are rational functions with the form of (1.8).
Therefore both conclusions (1) and (2) give w ( z ) = Q 1 ( ℘ ( z ) ) + ℘ ( z ) ′ Q 2 ( ℘ ( z ) ), where Q 1 and Q 2 are rational functions with the form of (3.11).
Suppose that (f z)) is a solution to the equation bigl(f' z bigr)^{n}=frac{A qz,f qz))}{B z,f z))}, (1.9) where (A z,y)) and (B z,y)) are rational functions with meromorphic coefficients of growth (S r,f)) such that (A z,y)) and (B z,y)) are irreducible in y.
Suppose that (f z)) is a solution to the equation sum_{s=1}^{n}alpha_{s} z f^{(lambda_{s})} z)= frac {A qz,f qz))}{B z,f z))}, (1.8) where (A z,y)) and (B z,y)) are rational functions with meromorphic coefficients of growth (S r,f)) such that (A z,y)) and (B z,y)) are irreducible in y.
Let c be a complex constant satisfying (|c|>1), and suppose that f is a nonconstant meromorphic solution of a functional equation of the form Abigl(cz,f(cz bigr)=Bbigl z,f z bigr), where (A z,y)) and (B z,y)) are rational functions with meromorphic coefficients of growth (S r,f)) such that (A z,y)) and (B z,y)) are irreducible in y.
Suppose that f is a transcendental meromorphic solution of the equation f qz f' z)=Rbigl z,f z bigr)=frac{P z,f z))}{Q z,f z))}, (6) where (qinmathbb{C}), (|q|>1), and P, Q are relatively prime polynomials in f over the field of rational functions satisfying (p=deg_{f} P), (t=deg_{f} Q), (d=p-tgeq4), where the coefficients of P, Q are rational functions in z.
where and, are rational functions.
