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We present a framework for upper bounding the number of iterations requiredby first-order optimization algorithms implementing constrained LQRcontrollers. We derive new bounds for the condition number and extremaleigenvalues of the primal and dual Hessian matrices when the cost function isscaled. These bounds are horizon-independent, allowing for their use withreceding, variable and decreasing horizon controllers. We considerably relaxprior assumptions on the structure of the weight matrices and assume only thatthe system is Schur-stable and the primal Hessian of the quadratic program (QP)is positive-definite. Our analysis uses the Toeplitz structure of the QPmatrices 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 complexityof trading off the input energy used against the state deviation. An examplesystem shows a three-times increase in algorithm iterations between the twoextremes, with the state 2-norm decreased by only 5% despite a greatlyincreased state deviation penalty.

Bounding Computational Complexity under Cost Function Scaling in Predictive Control