In this paper the software environment and algorithm collection ACADO Toolkit is presented, which implements tools for automatic control and dynamic optimization. It provides a general framework for using a great variety of algorithms for direct optimal control, including model predictive control as well as state and parameter estimation. The ACADO Toolkit is implemented as a self-contained C++ code, while the object-oriented design allows for convenient coupling of existing optimization packages and for extending it with user-written optimization routines. We discuss details of the software design of the ACADO Toolkit 1.0 and describe its main software modules. Along with that we highlight a couple of algorithmic features, in particular its functionality to handle symbolic expressions. The user-friendly syntax of the ACADO Toolkit to set up optimization problems is illustrated with two tutorial examples: an optimal control and a parameter estimation problem.ACADO TOOLKIT 299 such advanced controllers. Thus, efficient and reliable optimization algorithms for performing this step-possibly on embedded hardware-are of great interest.Searching the literature, we can find a number of optimization algorithms which have been implemented for solving OCPs. We can only discuss some of the most common packages: Let us start the list with the open-source package IPOPT [1, 2], originally developed by Andreas Wachter and Larry Biegler, which implements an interior point algorithm for the optimization of large-scale differential algebraic systems. It can be combined with collocation methods for the discretization of the continous dynamic system while a filter strategy is implemented as a globalization technique. IPOPT is written in C/C++ and Fortran, but uses modeling languages such as AMPL or MATLAB in order to provide a user interface and to allow automatic differentiation.Furthermore, a MATLAB package named PROPT [3] receives more and more attention. PROPT is a commercial tool, developed by the Tomlab Optimization Inc.. PROPT solves optimal control problems based on collocation techniques, while using existing NLP solvers such as KNITRO, CONOPT, SNOPT or CPLEX. Owing to the MATLAB syntax, the package PROPT is more user-friendly than IPOPT-at the price that it is not open-source.Recently, another open-source code has been published by Brian C. Fabien [4] under the name dsoa. This package is written in C/C++ and discretizes differential algebraic systems based on implicit Runge-Kutta methods. Unfortunately, the package does only implement single-shooting methods, which is often not advisable for nonlinear OCPs. On the optimization level, sequential quadratic programming techniques are employed.Similar to dsoa, the proprietary package MUSCOD-II, originally developed by Daniel Leineweber [5], is suitable for solving OCPs. MUSCOD-II discretizes the differential algebraic systems based on backward differentiation formula (BDF) or Runge Kutta integration methods and uses Bock's direct multiple shooting [6]. Sequential quadratic pro...

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SUMMARYNearly all algorithms for linear model predictive control (MPC) either rely on the solution of convex quadratic programs (QPs) in real time, or on an explicit precalculation of this solution for all possible problem instances. In this paper, we present an online active set strategy for the fast solution of parametric QPs arising in MPC. This strategy exploits solution information of the previous QP under the assumption that the active set does not change much from one QP to the next. Furthermore, we present a modification where the CPU time is limited in order to make it suitable for strict real-time applications. Its performance is demonstrated with a challenging test example comprising 240 variables and 1191 inequalities, which depends on 57 parameters and is prohibitive for explicit MPC approaches. In this example, our strategy allows CPU times of well below 100 ms per QP and was about one order of magnitude faster than a standard active set QP solver.

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.

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