A fast well-conditioned interior point method for predictive control

Abstract

Abstract-Interior point methods (IPMs) have proven to be an efficient way of solving quadratic programming problems in predictive control. A linear system of equations needs to be solved in each iteration of an IPM. The ill-conditioning of this linear system in the later iterations of the IPM prevents the use of an iterative method in solving the linear system due to a very slow rate of convergence; in some cases the solution never reaches the desired accuracy. In this paper we propose the use of a well-conditioned, approximate linear system, which increases the rate of convergence of the iterative method. The computational advantage is obtained by the use of an inexact Newton method along with the use of novel preconditioners. Numerical results indicate that the computational complexity of our proposed method scales quadratically with the number of states and linearly with the horizon length.

A warm-start interior-point method for predictive control

“…In this subsection, we will briefly describe the δ-active set based infeasible IPM proposed by Shahzad et al (2010), and extend that work to propose a warm-start strategy. Note that if s k i → 0 as k → ∞, then (5b) indicates that the i th inequality constraint is active.…”

“…To avoid this ill-conditioning in the later stage of an IPM, the following scheme is proposed by Shahzad et al (2010). Permute the matrix A k according to the δ-active and δ−inactive constraints as …”

“…The linear system is either solved by direct or iterative methods. A well-conditioned approximation of this linear system is proposed by Shahzad, Kerrigan, and Constantinides (2010), in which an approximate search direction is calculated to reduce the computations by an iterative method. In this paper we have extended that work to design a warm-start strategy in an IPM.…”

Model predictive control for deeply pipelined field-programmable gate array implementation: algorithms and circuitry

“…In our current implementation, the latency is O(T 2 ) because the number of iterations of the linear solver depends on T , and we have not completely exploited a full parallelization of the solver. Future work will investigate the implementation of a pre-conditioner [20] and further use of parallelization to further reduce the latency.…”

“…In addition, these methods are easy to parallelize since they mostly consist of large matrix-vector multiplications. Furthermore, it is possible to derive pre-conditioners that can help reduce the number of iterations needed to solve the system to the required accuracy [20], and control the ill-conditioning of the matrices towards the later iterations of the interior-point method. As a consequence, simple primal-dual methods, where a single system of equations is solved, could be more suited to the FPGA fabric.…”