An efficient implementation of partial condensing for Nonlinear Model Predictive Control

Frison, Gianluca; Kouzoupis, Dimitris; Jørgensen, John Bagterp; Diehl, Moritz

doi:10.1109/cdc.2016.7798946

Cited by 30 publications

(18 citation statements)

References 11 publications

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“…In Axehill (2014Axehill ( , 2015 partial condensing is introduced as a way of combining the sparse and the condensed formulations to choose the best condensation strategy, which often is in between the traditional sparse and the condensed formulations. This has later been studied further in Kouzoupis et al (2015), Frison (2015), Frison et al (2016), Quirynen (2017). How to exploit the structure associated with the cftoc problem even for the dense formulation has been investigated in Axehill and Morari (2012) and Frison and Jørgensen (2013).…”

Section: Constrained Finite-time Optimal Control Problemmentioning

confidence: 94%

Structure-Exploiting Numerical Algorithms for Optimal Control

Nielsen¹

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Cover illustration: An illustration of the structure of the Karush-Kuhn-Tucker matrix for the unconstrained finite-time optimal control problems considered in this thesis. Thank you Jonas Linder for helping me with the design.Linköping studies in science and technology. Dissertations.No. 1848 Tillägnas hela min underbara familj Structure-Exploiting Numerical Algorithms for Optimal Control AbstractNumerical algorithms for efficiently solving optimal control problems are important for commonly used advanced control strategies, such as model predictive control (mpc), but can also be useful for advanced estimation techniques, such as moving horizon estimation (mhe). In mpc, the control input is computed by solving a constrained finite-time optimal control (cftoc) problem on-line, and in mhe the estimated states are obtained by solving an optimization problem that often can be formulated as a cftoc problem. Common types of optimization methods for solving cftoc problems are interior-point (ip) methods, sequential quadratic programming (sqp) methods and active-set (as) methods. In these types of methods, the main computational effort is often the computation of the second-order search directions. This boils down to solving a sequence of systems of equations that correspond to unconstrained finite-time optimal control (uftoc) problems. Hence, high-performing second-order methods for cftoc problems rely on efficient numerical algorithms for solving uftoc problems. Developing such algorithms is one of the main focuses in this thesis. When the solution to a cftoc problem is computed using an as type method, the aforementioned system of equations is only changed by a low-rank modification between two as iterations. In this thesis, it is shown how to exploit these structured modifications while still exploiting structure in the uftoc problem using the Riccati recursion. Furthermore, direct (non-iterative) parallel algorithms for computing the search directions in ip, sqp and as methods are proposed in the thesis. These algorithms exploit, and retain, the sparse structure of the uftoc problem such that no dense system of equations needs to be solved serially as in many other algorithms. The proposed algorithms can be applied recursively to obtain logarithmic computational complexity growth in the prediction horizon length. For the case with linear mpc problems, an alternative approach to solving the cftoc problem on-line is to use multiparametric quadratic programming (mp-qp), where the corresponding cftoc problem can be solved explicitly off-line. This is referred to as explicit mpc. One of the main limitations with mp-qp is the amount of memory that is required to store the parametric solution. In this thesis, an algorithm for decreasing the required amount of memory is proposed. The aim is to make mp-qp and explicit mpc more useful in practical applications, such as embedded systems with limited memory resources. The proposed algorithm exploits the structure from the qp problem in the parametric solution in order to reduc...

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Section: Constrained Finite-time Optimal Control Problemmentioning

confidence: 94%

Structure-Exploiting Numerical Algorithms for Optimal Control

Nielsen¹

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“…It is also possible to use partial condensing [21] to obtain a smaller but still sparse QP problem. Computation efficiency improvement using partial condensing has been reported in [22], [23].…”

Section: Algorithm Basicsmentioning

confidence: 99%

MATMPC - A MATLAB Based Toolbox for Real-time Nonlinear Model Predictive Control

Bruschetta

Picotti

Beghi

2019

2019 18th European Control Conference (ECC)

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In this paper we introduce MATMPC, an open source software built in MATLAB for nonlinear model predictive control (NMPC). It is designed to facilitate modelling, controller design and simulation for a wide class of NMPC applications. MATMPC has a number of algorithmic modules, including automatic differentiation, direct multiple shooting, condensing, linear quadratic program (QP) solver and globalization. It also supports a unique Curvature-like Measure of Nonlinearity (CMoN) MPC algorithm. MATMPC has been designed to provide state-of-the-art performance while making the prototyping easy, also with limited programming knowledge. This is achieved by writing each module directly in MATLAB API for C. As a result, MATMPC modules can be compiled into MEX functions with performance comparable to plain C/C++ solvers. MATMPC has been successfully used in operating systems including WINDOWS, LINUX AND OS X. Selected examples are shown to highlight the effectiveness of MATMPC.

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“…Each problem dimension, eg, n x for the number of states, is defined node‐wise and represented by an array of N e + 1 integers. The varying dimensions of the nodes can be useful for, eg, dynamic systems with multiple phases or partially condensed QPs . The sparsity pattern defined by the tree graph is not required to have the usual symmetry arising in multistage MPC (see Figure A) but it can have any arbitrary tree structure.…”

Section: Software Implementationmentioning

confidence: 99%

“…The varying dimensions of the nodes can be useful for, eg, dynamic systems with multiple phases or partially condensed QPs. 35,36 The sparsity pattern defined by the tree graph is not required to have the usual symmetry arising in multistage MPC (see Figure 1A) but it can have any arbitrary tree structure. § 3acts with the main algorithm via a set of function pointers, common to all different implementations of the same module.…”

Section: Software Implementationmentioning

confidence: 99%

“…It also requires M symmetric matrix-matrix multiplications that induce another first two phases can be executed in M parallel threads. If n cpu is the number of available computational units, we can replace M with max(1, M n cpu ) in (34)-(36). ‡ The expression in(35) assumes that forward Cholesky factorizations are used in the previous step of the algorithm.…”

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confidence: 99%

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A dual Newton strategy for tree‐sparse quadratic programs and its implementation in the open‐source software treeQP

Kouzoupis

Klintberg

Frison

et al. 2019

Intl J Robust & Nonlinear

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Summary This paper presents a dual Newton scheme for tree‐sparse quadratic programs as they may arise in the field of stochastic programming. Previous work suggests to introduce auxiliary variables to decompose the tree into scenarios and use Newton's method to solve a dual problem formulation. Following a different approach, we apply the same principle directly on the tree‐sparse problem, avoiding the increase in dimensionality. In combination with a tailored algorithm for the calculation of the step direction, which is typically the most expensive operation per iteration, the proposed algorithm achieves a linear complexity in the number of nodes and supports parallel processing of the tree branches in a stage‐wise fashion. An open‐source implementation of the presented dual Newton strategy is publicly available in treeQP, a toolbox of open‐source solvers for tree‐sparse quadratic programs.

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An efficient implementation of partial condensing for Nonlinear Model Predictive Control

Cited by 30 publications

References 11 publications

Structure-Exploiting Numerical Algorithms for Optimal Control

Structure-Exploiting Numerical Algorithms for Optimal Control

MATMPC - A MATLAB Based Toolbox for Real-time Nonlinear Model Predictive Control

A dual Newton strategy for tree‐sparse quadratic programs and its implementation in the open‐source software treeQP

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