Infinite-Horizon Value Check

Problem

Compute the closed-form infinite-horizon LQR value for a scalar system.

With s = b ** 2 and beta = R * (1 - a ** 2) - s * Q, return (-beta + sqrt(beta ** 2 + 4 * s * R * Q)) / (2 * s), the positive root of the algebraic Riccati equation. If s is 0, return Q.

Hint

This is the fixed point of the Riccati recursion, obtained from a quadratic equation.

Test cases

case 1

in: [1,1,1,1]

expected: 1.618033988749895

case 2

in: [0.9,0.5,1,0.5]

expected: 1.7577914214416384

case 3

in: [1,0,2,1]

expected: 2

Starter codePython

def infinite_horizon_value(a, b, Q, R):
    # Your code here
    pass

3 test cases · run your code against them in the editor