Bounding Computational Complexity under Cost Function Scaling in Predictive Control

TL;DR

This paper develops a system-theoretic framework to bound the iteration count of first-order optimization algorithms in constrained LQR control, accounting for cost scaling and system properties, enabling better complexity analysis.

Contribution

It introduces horizon-independent bounds for the Hessian spectrum in scaled cost functions, relaxing prior assumptions and linking spectral properties to system transfer functions.

Findings

Bounds are horizon-independent and applicable to variable horizons.

Scaling the cost affects iteration count and state deviation.

Example shows a three-fold increase in iterations with minimal state deviation change.

Abstract

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.