  First-order optimization solvers, such as the Fast Gradient Method, are
increasingly being used to solve Model Predictive Control problems in
resource-constrained environments. Unfortunately, the convergence rate of these
solvers is significantly affected by the conditioning of the problem data, with
ill-conditioned problems requiring a large number of iterations. To reduce the
number of iterations required, we present a simple method for computing a
horizon-independent preconditioning matrix for the Hessian of the condensed
problem. The preconditioner is based on the block Toeplitz structure of the
Hessian. Horizon-independence allows one to use only the predicted system and
cost matrices to compute the preconditioner, instead of the full Hessian. The
proposed preconditioner has equivalent performance to an optimal preconditioner
in numerical examples, producing speedups between 2x and 9x for the Fast
Gradient Method. Additionally, we derive horizon-independent spectral bounds
for the Hessian in terms of the transfer function of the predicted system, and
show how these can be used to compute a novel horizon-independent bound on the
condition number for the preconditioned Hessian.
