# Bounding Computational Complexity under Cost Function Scaling in Predictive Control

@article{McInerney2019BoundingCC, title={Bounding Computational Complexity under Cost Function Scaling in Predictive Control}, author={Ian McInerney and Eric C. Kerrigan and George A. Constantinides}, journal={ArXiv}, year={2019}, volume={abs/1902.02221} }

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…

## One Citation

Modeling Round-off Error in the Fast Gradient Method for Predictive Control

- Computer Science, Mathematics2019 IEEE 58th Conference on Decision and Control (CDC)
- 2019

A method for determining the smallest precision required to have algorithmic stability of an implementation of the Fast Gradient Method when solving a linear Model Predictive Control problem in fixed-point arithmetic and proposes a metric for measuring the amount of round-off error the FGM iteration can tolerate before becoming unstable.

## References

SHOWING 1-10 OF 36 REFERENCES

The double description method for the approximation of explicit MPC control laws

- Mathematics2008 47th IEEE Conference on Decision and Control
- 2008

A new interpolation method is introduced based on recent work on barycentric interpolation that operates on implicitly-defined convex sets, meaning that the optimal solution of the parametric problem is not required.

A linear MPC algorithm for embedded systems with computational complexity guarantees

- Computer Science, Mathematics2014 18th International Conference on System Theory, Control and Computing (ICSTCC)
- 2014

A linear MPC scheme for embedded systems based on the dual fast gradient algorithm for solving the corresponding control problem is proposed by appropriately deriving tight convergence estimates of order O(1/k2) for an average primal sequence generated by the proposed numerical optimization algorithm.

Computational Complexity Certification for Real-Time MPC With Input Constraints Based on the Fast Gradient Method

- Computer Science, MathematicsIEEE Transactions on Automatic Control
- 2012

The main focus is on Nesterov's fast gradient method's a priori computational complexity certification which consists of deriving lower iteration bounds such that a solution of pre-specified suboptimality is obtained for any possible state of the system.

Optimal preconditioning and iteration complexity bounds for gradient-based optimization in model predictive control

- Computer Science2013 American Control Conference
- 2013

This paper shows how to optimally precondition the optimization data by solving a semidefinite program, where optimally refers to the preconditionsing that minimizes an explicit iteration complexity bound.

Fixed-Point Implementation of a Proximal Newton Method for Embedded Model Predictive Control

- Computer Science, Engineering
- 2014

An implementation with fixed-point arithmetic of a proximal Newton method to solve optimization problems arising in input-constrained MPC, showing the robustness of the algorithm with respect to finite-precision computations.

A sub-optimal receding horizon control strategy for constrained linear systems

- EngineeringProceedings of the 2003 American Control Conference, 2003.
- 2003

A novel receding horizon control strategy for constrained linear systems, based on a suboptimal solution to the fixed horizon optimization problem considered in Model Predictive Control, which delivers a feasible solution without excessively compromising performance.

Model Predictive Control Tuning by Controller Matching

- MathematicsIEEE Transactions on Automatic Control
- 2010

This technical note provides two methods for selecting the MPC weight matrices so that the resulting MPC controller behaves as the given linear controller, therefore solving the posed inverse problem of controller matching, and is globally asymptotically stable.

On the asymptotic properties of the Hessian in discrete-time linear quadratic control

- MathematicsProceedings of the 2004 American Control Conference
- 2004

This paper studies the asymptotic properties of the Hessian in discrete-time linear quadratic optimal control. We show that the singular values of the Hessian converge, in a well defined sense, to…

The Optimal Sampling Pattern for Linear Control Systems

- Computer Science, MathematicsIEEE Transactions on Automatic Control
- 2014

The optimal sampling problem is formulated and a necessary condition for the optimality of a set of sampling instants in the linear quadratic regulator problem is derived and a new quantization-based sampling strategy is proposed that is computationally tractable and capable of achieving near-optimal cost.

Optimal control: linear quadratic methods

- Mathematics
- 1990

This augmented edition of a respected text teaches the reader how to use linear quadratic Gaussian methods effectively for the design of control systems. It explores linear optimal control theory…