Moritz Diehl

A Lyapunov Function for Economic Optimizing Model Predictive Control

Standard model predictive control (MPC) yields an asymptotically stable steady-state solution using the following procedure. Given a dynamic model, a steady state of interest is selected, a stage cost is defined that measures deviation from this selected steady state, the controller cost function is a summation of this stage cost over a time horizon, and the optimal cost is shown to be a Lyapunov function for the closed-loop system. In this technical note, the stage cost is an arbitrary economic objective, which may not depend on a steady state, and the optimal cost is not a Lyapunov function for the closed-loop system. For a class of nonlinear systems and economic stage costs, this technical note constructs a suitable Lyapunov function, and the optimal steady-state solution of the economic stage cost is an asymptotically stable solution of the closed-loop system under economic MPC. Both finite and infinite horizons are treated. The class of nonlinear systems is defined by satisfaction of a strong duality property of the steady-state problem. This class includes linear systems with convex stage costs, generalizing previous stability results and providing a Lyapunov function for economic MPC or MPC with an unreachable setpoint and a linear model. A nonlinear chemical reactor example is provided illustrating these points.

Efficient Numerical Methods for Nonlinear MPC and Moving Horizon Estimation

This overview paper reviews numerical methods for solution of optimal control problems in real-time, as they arise in nonlinear model predictive control (NMPC) as well as in moving horizon estimation (MHE). In the first part, we review numerical optimal control solution methods, focussing exclusively on a discrete time setting. We discuss several algorithmic ”building blocks” that can be combined to a multitude of algorithms. We start by discussing the sequential and simultaneous approaches, the first leading to smaller, the second to more structured optimization problems. The two big families of Newton type optimization methods, Sequential Quadratic Programming (SQP) and Interior Point (IP) methods, are presented, and we discuss how to exploit the optimal control structure in the solution of the linear-quadratic subproblems, where the two major alternatives are “condensing” and band structure exploiting approaches. The second part of the paper discusses how the algorithms can be adapted to the real-time challenge of NMPC and MHE. We recall an important sensitivity result from parametric optimization, and show that a tangential solution predictor for online data can easily be generated in Newton type algorithms. We point out one important difference between SQP and IP methods: while both methods are able to generate the tangential predictor for fixed active sets, the SQP predictor even works across active set changes. We then classify many proposed real-time optimization approaches from the literature into the developed categories.

qpOASES: a parametric active-set algorithm for quadratic programming

Many practical applications lead to optimization problems that can either be stated as quadratic programming (QP) problems or require the solution of QP problems on a lower algorithmic level. One relatively recent approach to solve QP problems are parametric active-set methods that are based on tracing the solution along a linear homotopy between a QP problem with known solution and the QP problem to be solved. This approach seems to make them particularly suited for applications where a-priori information can be used to speed-up the QP solution or where high solution accuracy is required. In this paper we describe the open-source C++ software package qpOASES, which implements a parametric active-set method in a reliable and efficient way. Numerical tests show that qpOASES can outperform other popular academic and commercial QP solvers on small- to medium-scale convex test examples of the Maros-Mészáros QP collection. Moreover, various interfaces to third-party software packages make it easy to use, even on embedded computer hardware. Finally, we describe how qpOASES can be used to compute critical points of nonconvex QP problems.

Time-Optimal Path Tracking for Robots: A Convex Optimization Approach

This paper focuses on time-optimal path tracking, a subproblem in time-optimal motion planning of robot systems. Through a nonlinear change of variables, the time-optimal path tracking problem is transformed here into a convex optimal control problem with a single state. Various convexity-preserving extension are introduced, resulting in a versatile approach for optimal path tracking. A direct transcription method is presented that reduces finding the globally optimal trajectory to solving a second-order cone program using robust numerical algorithms that are freely available. Validation against known examples and application to a more complex example illustrate the versatility and practicality of the new method.

An auto-generated real-time iteration algorithm for nonlinear MPC in the microsecond range

In this paper, we present an automatic C-code generation strategy for real-time nonlinear model predictive control (NMPC), which is designed for applications with kilohertz sample rates. The corresponding code export module has been implemented within the software package ACADO Toolkit. It is capable of exporting fixed step-size integrators together with their sensitivities as well as a real-time Gauss-Newton method. Here, we employ the symbolic representation of optimal control problems in ACADO in order to auto-generate plain C-code which is optimized for final production. The exported code has been tested for model predictive control scenarios comprising constrained nonlinear dynamic systems with four states and a control horizon of ten samples. The numerical simulations show a promising performance of the exported code being able to provide feedback in much less than a millisecond.

CasADi -- A symbolic package for automatic differentiation and optimal control

We present CasADi, a free, open-source software tool for fast, yet efficient solution of nonlinear optimization problems in general and dynamic optimization problems in particular. To the developer of algorithms for numerical optimization and to the advanced user of such algorithms, it offers a level of abstraction which is notably lower, and hence more flexible, than that of algebraic modeling languages such as AMPL or GAMS, but higher than working with a conventional automatic differentiation (AD) tool.CasADi is best described as a minimalistic computer algebra system (CAS) implementing automatic differentiation in eight different flavors. Similar to algebraic modeling languages, it includes high-level interfaces to state-of-the-art numerical codes for nonlinear programming, quadratic programming and integration of differential-algebraic equations. CasADi is implemented in self-contained C++ code and contains full-featured front-ends to Python and Octave for rapid prototyping. In this paper, we present the AD framework of CasADi and benchmark the tool against AMPL for a set of nonlinear programming problems from the CUTEr test suite.

Online Walking Motion Generation with Automatic Foot Step Placement

The goal of this paper is to demonstrate the capacity of model predictive control (MPC) to generate stable walking motions without the use of predefined footsteps. Building up on well-known MPC schemes for walking motion generation, we show that a minimal modification of these schemes allows designing an online walking motion generator that can track a given reference speed of the robot and decide automatically the footstep placement. Simulation results are proposed on the HRP-2 humanoid robot, showing a significant improvement over previous approaches.

Predictive control of a real-world Diesel engine using an extended online active set strategy

Abstract In order to meet tight emission limits Diesel engines are nowadays equipped with additional hardware components like an exhaust gas recirculation valve and a variable geometry turbocharger. Conventional engine control units use two SISO control loops to regulate the exhaust gas recirculation valve and the variable geometry turbocharger, although their effects are highly coupled. Moreover, these actuators are subject to physical constraints which seems to make an advanced control approach like model predictive control (MPC) the method of choice. In order to deal with MPC sampling times in the order of milliseconds, we employed an extension of the recently developed online active set strategy for controlling a real-world Diesel engine in a closed-loop manner. The results show that predictive engine control based on online optimisation can be accomplished in real-time – even on cheap controller hardware – and leads to increased controller performance.

Online walking gait generation with adaptive foot positioning through Linear Model Predictive control

Building on previous propositions to generate walking gaits online through the use of linear model predictive control, the goal of this paper is to show that it is possible to allow on top of that a continuous adaptation of the positions of the foot steps, allowing the generation of stable walking gaits even in the presence of strong perturbations, and that this additional adaptation requires only a minimal modification of the previous schemes, especially maintaining the same linear model predictive form. Simulation results are proposed then on the HRP-2 humanoid robot, showing a significant improvement over the previous schemes.

Distributed Spectrum Management Algorithms for Multiuser DSL Networks

Modern digital subscriber line (DSL) networks suffer from crosstalk among different lines in the same cable bundle. This crosstalk can lead to a major performance degradation. By balancing the transmit power spectra, the impact of crosstalk can be minimized leading to spectacular performance gains. This is referred to as spectrum management. In this paper, a unifying perspective is presented on distributed spectrum management algorithms based on the Karush-Kuhn-Tucker (KKT) conditions. Furthermore, novel distributed algorithms are presented within the same KKT framework. The proposed distributed algorithms consist of local water-filling-like algorithms running in the individual modems, controlled by the spectrum management center. Extensive simulation results show that the proposed algorithms perform very well for several multi-user ADSL and VDSL scenarios.