Dimitris Kouzoupis

First-order methods in embedded nonlinear model predictive control

Several algorithms based on Nesterovs fast gradient method have been recently proposed in the literature for use in linear model predictive control (MPC). Their simple algorithmic schemes have attracted much attention for MPC applications on embedded hardware. The purpose of this paper is to investigate the suitability of these algorithms in a nonlinear MPC setup. We assess the necessary numerical modifications and analyze the additional online computational complexity. We illustrate our findings by combining different first-order methods with the real-time iteration (RTI) scheme for nonlinear MPC and using them to control a model of an inverted pendulum.

BLASFEO: Basic Linear Algebra Subroutines for Embedded Optimization

Basic Linear Algebra Subroutines for Embedded Optimization (BLASFEO) is a dense linear algebra library providing high-performance implementations of BLAS- and LAPACK-like routines for use in embedded optimization and small-scale high-performance computing, in general. A key difference with respect to existing high-performance implementations of BLAS is that the computational performance is optimized for small- to medium-scale matrices, i.e., for sizes up to a few hundred. BLASFEO comes with three different implementations: a high-performance implementation aimed at providing the highest performance for matrices fitting in cache, a reference implementation providing portability and embeddability and optimized for very small matrices, and a wrapper to standard BLAS and LAPACK providing high performance on large matrices. The three implementations of BLASFEO together provide high-performance dense linear algebra routines for matrices ranging from very small to large. Compared to both open-source and proprietary highly tuned BLAS libraries, for matrices of size up to about 100, the high-performance implementation of BLASFEO is about 20--30% faster than the corresponding level 3 BLAS routines and two to three times faster than the corresponding LAPACK routines.

Towards proper assessment of QP algorithms for embedded model predictive control

With model predictive control (MPC) becoming a viable approach for advanced feedback control at very fast sampling times, a plethora of methods for solving quadratic programming (QP) problems on embedded computing hardware has been proposed. While most of these methods seem to be useful and superior to competing approaches on particular problem instances, very little effort has been put into a proper benchmarking on a non-trivial number of MPC problems. This paper is intended to help filling this gap by (i) briefly discussing the most important aspects for assessing the suitability of a certain QP method for an MPC problem at hand, (ii) providing a concise overview of about a dozen different QP algorithms that have been proposed for use in MPC, (iii) describing a general benchmarking framework for comparing the numerical performance of the different QP algorithms, and (iv) providing preliminary benchmarking results based on different performance metrics. Numerical performance of the various algorithms is assessed by means of a suitable collection of benchmark problems, taken from both academic studies and industrial applications of MPC.

An efficient implementation of partial condensing for Nonlinear Model Predictive Control

Partial (or block) condensing is a recently proposed technique to reformulate a Model Predictive Control (MPC) problem into a form more suitable for structure-exploiting Quadratic Programming (QP) solvers. It trades off horizon length for input vector size, and this degree of freedom can be employed to find the best problem size for the QP solver at hand. This paper proposes a Hessian condensing algorithm particularly well suited for partial condensing, where a state component is retained as an optimization variable at each stage of the partially condensed MPC problem. The optimal input-horizon trade-off is investigated from a theoretical point of view (based on algorithms flop count) as well as by benchmarking (in practice, the performance of linear algebra routines for different matrix sizes plays a key role). Partial condensing can also be seen as a technique to replace many operations on small matrices with fewer operations on larger matrices, where linear algebra routines perform better. Therefore, in case of small-scale MPC problems, partial condensing can greatly improve performance beyond the flop count reduction.

An efficient SQP algorithm for Moving Horizon Estimation with Huber penalties and multi-rate measurements

Moving Horizon Estimation (MHE) is a powerful, yet computationally expensive approach for state and parameter estimation that is based on online optimization. In applications with multi-rate measurements that may include outliers, the Huber penalty is often a better candidate for the MHE objective than the commonly used Euclidean norm. Treating this non-smooth objective in Newton-type optimization typically requires the use of slack variables that would in turn increase the problem size significantly. As an alternative, we propose a novel algorithm that combines Sequential Convex Programming (SCP) and Sequential Quadratic Programming (SQP) techniques in an effort to reduce the computational complexity. The proposed implementation is tailored to embedded applications, as it combines state-of-the-art numerical tools and efficient C code. We demonstrate the performance of the algorithm on a real-world state estimation problem where the position and orientation of a single propeller aircraft are estimated using GPS and IMU measurement data.

A quadratically convergent primal decomposition algorithm with soft coupling for nonlinear parameter estimation

Distributed algorithms for sparse, large-scale optimization problems are preferable over centralized solvers when the computational units are physically far apart from each other or the problem size is too large for the available memory. However, most distributed methods sacrifice convergence speed for simpler computations. In this paper, we propose a novel algorithm for a certain class of nonconvex, separable optimization problems that combines both distributed computations and locally quadratic convergence. It is based on the principle of primal decomposition with exact Hessian information but uses soft coupling between the agents to avoid global calculations and adapt faster to online data changes. An important application field of the presented method is nonlinear parameter estimation, where increasing the number of experiments may lead to problem dimensions that are prohibitive for conventional solvers. We assess the performance of our method on the identification of an Airborne Wind Energy (AWE) system using real-world experimental data.

A block based ALADIN scheme for highly parallelizable direct Optimal Control

Nonlinear Model Predictive Control (NMPC) requires the online solution of a nonlinear Optimal Control Problem (OCP) at each sampling instant. This paper presents a novel, block based and highly parallelizable algorithm which solves nonlinear OCPs using a recently proposed Augmented Lagrangian based method (ALADIN). The latter employs techniques from standard Sequential Quadratic Programming (SQP) methods within a more parallelizable framework. An implementation tailored to optimal control is proposed where Nonlinear Programs (NLPs) are solved approximately and concurrently on each stage while a centralized consensus step is used to update the dual variables of the coupling constraints. The implementation also comprises algorithmic concepts to extend the parallelizability of the consensus step and a blocking technique to accelerate convergence. The performance of the resulting scheme is illustrated using as benchmark example the control of an overhead crane.