The Greenhouse Model on Paper · Python problem 413405

A greenhouse controller models how the air temperature y responds to the heater power x, one sample a minute, with a difference equation in the Level 2 form:

y[k] = c[0] * y[k-1] + c[1] * y[k-2] + ...  +  d[0] * u[k] + d[1] * u[k-1] + ...

where u[k] = x[k - delay]: the heat reaches the sensor only delay minutes after the heater switches (dead time). The feedback weights are c (c[i] multiplies y[k-1-i]) and the input weights are d.

To get the transfer function, write every delay as a power of z^-1 (y[k-1] becomes z^-1 * Y, x[k-delay-j] becomes z^-(delay+j) * X) and move every y term to the left:

Y \left(1 - c_0 z^{-1} - c_1 z^{-2} - \dots\right) = X \, z^{-\text{delay}} \left(d_0 + d_1 z^{-1} + \dots\right)

H(z) = \frac{Y}{X} = \frac{B(z)}{A(z)} = \frac{z^{-\text{delay}} \left(d_0 + d_1 z^{-1} + \dots\right)}{1 - c_0 z^{-1} - c_1 z^{-2} - \dots}

The usual convention, used in the rest of this level, writes

a_0 y_k + a_1 y_{k-1} + a_2 y_{k-2} + \dots = b_0 x_k + b_1 x_{k-1} + \dots

with a[0] = 1: the feedback weights appear in a with the opposite sign.

Write to_transfer(c, d, delay) that returns the pair (b, a): a has len(c) + 1 entries starting with 1, and b has delay + len(d) entries (delay zeros, then the weights d).

Examples

Input:  c = [0.9], d = [0.05], delay = 2
Output: ([0, 0, 0.05], [1, -0.9])
Explanation: y[k] = 0.9 y[k-1] + 0.05 x[k-2], so y[k] - 0.9 y[k-1] = 0.05 x[k-2]:
H(z) = 0.05 z^-2 / (1 - 0.9 z^-1).

Input:  c = [1.2, -0.5], d = [0.1, 0.2], delay = 0
Output: ([0.1, 0.2], [1, -1.2, 0.5])
Explanation: the ride-height filter of Level 2.

Constraints

answers are compared with a tolerance of 1e-6; lists may be given as tuples

Goals

Turn a difference equation into the coefficient lists b and a of its transfer function
Move the feedback terms to the left-hand side, which flips their signs
Write a pure delay of d samples as d leading zeros in b