  We present a framework for upper bounding the number of iterations required
by first-order optimization algorithms implementing constrained LQR
controllers. We derive new bounds for the condition number and extremal
eigenvalues of the primal and dual Hessian matrices when the cost function is
scaled. These bounds are horizon-independent, allowing for their use with
receding, variable and decreasing horizon controllers. We considerably relax
prior assumptions on the structure of the weight matrices and assume only that
the system is Schur-stable and the primal Hessian of the quadratic program (QP)
is positive-definite. Our analysis uses the Toeplitz structure of the QP
matrices to relate their spectrum to the transfer function of the system,
allowing for the use of system-theoretic techniques to compute the bounds.
Using these bounds, we can compute the effect on the computational complexity
of trading off the input energy used against the state deviation. An example
system shows a three-times increase in algorithm iterations between the two
extremes, with the state 2-norm decreased by only 5% despite a greatly
increased state deviation penalty.
