Boris Houska

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.

Optimal Control of Towing Kites

In this paper we present a challenging application of periodic optimal control. A kite that is towing a ship into a given target direction should fly optimal loops. We show how to find the maximum average tractive force by controlling the roll angle of the towing kite taking into account that the wind is increasing with the altitude over the sea. The optimal control problem for this highly nonlinear and unstable system has periodicity constraints, free initial values, and a free cycle duration. For its solution, we use MUSCOD-II, an optimal control package based on the direct multiple shooting method. Finally, we discuss the influence of an important design parameter, the effective glide ratio of the kite

Experimental validation of nonlinear MPC on an overhead crane using automatic code generation

Recent advances in improving the efficiency of nonlinear model predictive control (MPC) algorithms have made them suited for challenging mechatronic applications that require high sampling rates. We demonstrate this fact by applying a highly efficient nonlinear MPC algorithm to a laboratory-scale overhead crane setup, featuring a fast moving cart and a winch mechanism. The aim is to perform optimized point-to-point motions with varying line length while respecting actuator limits. In order to solve the resulting optimization problems in less than one millisecond, an automatically generated Gauss-Newton real-time iteration algorithm is employed. We show experimental results illustrating the control performance of the closed-loop system as well as the efficiency of the nonlinear MPC algorithm.

Multi-objective Optimal Control of Chemical Processes using ACADO Toolkit

Abstract Many practical chemical engineering problems involve the determination of optimal trajectories given multiple and conflicting objectives. These conflicting objectives typically give rise to a set of Pareto optimal solutions. To enhance real-time decision making efficient approaches are required for determining the Pareto set in a fast and accurate way. Hereto, the current paper illustrates the use of the freely available toolkit ACADO Multi-Objective ( www.acadotoolkit.org ) on several chemical examples. The rationale behind ACADO Multi-Objective is the integration of direct optimal control methods with scalarisation-based multi-objective methods enabling the exploitation of fast deterministic gradient-based optimisation routines.

Robustness and stability optimization of power generating kite systems in a periodic pumping mode

In this paper we formulate and solve optimal control problems for power generating kite systems. Here, the kite generates energy by periodically pulling a generator on the ground while flying fast in a crosswind direction. We are searching for an intrinsically open-loop stable trajectory such that the kite generates as much power as possible without needing feedback, while neither the kite nor the cable should touch the ground in the presence of wind turbulence. As the wind turbulences are unknown, robustness aspects need to be taken into account. The formulation of the associated optimal control problem makes use of periodic Lyapunov differential equations in order to guarantee local open-loop stability while robustness aspects are regarded in a linear approximation. The main result of this paper is that open-loop stable kite orbits exist and that open-loop stability only costs approximately 23% compared to the power-optimal unstable orbit.

Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs

This paper presents a framework for constructing and analyzing enclosures of the reachable set of nonlinear ordinary differential equations using continuous-time set-propagation methods. The focus is on convex enclosures that can be characterized in terms of their support functions. A generalized differential inequality is introduced, whose solutions describe such support functions for a convex enclosure of the reachable set under mild conditions. It is shown that existing continuous-time bounding methods that are based on standard differential inequalities or ellipsoidal set propagation techniques can be recovered as special cases of this generalized differential inequality. A way of extending this approach for the construction of nonconvex enclosures is also described, which relies on Taylor models with convex remainder bounds. This unifying framework provides a means for analyzing the convergence properties of continuous-time enclosure methods. The enclosure techniques and convergence results are illustrated with numerical case studies throughout the paper, including a six-state dynamic model of anaerobic digestion.

Robust nonlinear optimal control of dynamic systems with affine uncertainties

In this paper we present novel strategies to formulate and solve nonlinear robust optimal control problems for dynamic systems which are affine in the uncertainty. We suggest the definition of a constrained Lyapunov differential equation providing robustness interpretations with respect to L2-bounded disturbances in the context of inequality state constraints. This interpretation allows us to compute the robust counterpart formulation for optimal control problems which are affine in the uncertainty. Furthermore, we demonstrate the applicability of the presented formulation for a numerical test example: a crane should carry a mass from one to another point while an unknown force excites the open-loop controlled system. The robustly optimized input allows us to control the mass to a target region while satisfying inequality constraints on the worst-case excitation.

Branch-and-Lift Algorithm for Deterministic Global Optimization in Nonlinear Optimal Control

This paper presents a branch-and-lift algorithm for solving optimal control problems with smooth nonlinear dynamics and potentially nonconvex objective and constraint functionals to guaranteed global optimality. This algorithm features a direct sequential method and builds upon a generic, spatial branch-and-bound algorithm. A new operation, called lifting, is introduced, which refines the control parameterization via a Gram–Schmidt orthogonalization process, while simultaneously eliminating control subregions that are either infeasible or that provably cannot contain any global optima. Conditions are given under which the image of the control parameterization error in the state space contracts exponentially as the parameterization order is increased, thereby making the lifting operation efficient. A computational technique based on ellipsoidal calculus is also developed that satisfies these conditions. The practical applicability of branch-and-lift is illustrated in a numerical example.

Multi-objective optimal control of dynamic bioprocesses using ACADO Toolkit

The optimal design and operation of dynamic bioprocesses gives in practice often rise to optimisation problems with multiple and conflicting objectives. As a result typically not a single optimal solution but a set of Pareto optimal solutions exist. From this set of Pareto optimal solutions, one has to be chosen by the decision maker. Hence, efficient approaches are required for a fast and accurate generation of the Pareto set such that the decision maker can easily and systematically evaluate optimal alternatives. In the current paper the multi-objective optimisation of several dynamic bioprocess examples is performed using the freely available ACADO Multi-Objective Toolkit (http://www.acadotoolkit.org). This toolkit integrates efficient multiple objective scalarisation strategies (e.g., Normal Boundary Intersection and (Enhanced) Normalised Normal Constraint) with fast deterministic approaches for dynamic optimisation (e.g., single and multiple shooting). It has been found that the toolkit is able to efficiently and accurately produce the Pareto sets for all bioprocess examples. The resulting Pareto sets are added as supplementary material to this paper.

Symmetric algorithmic differentiation based exact Hessian SQP method and software for Economic MPC

Economic Model Predictive Control (EMPC) is an advanced receding horizon based control technique which optimizes an economic objective subject to potentially nonlinear dynamic equations as well as control and state constraints. The main contribution of this paper is an algorithmic differentiation (AD) based real-time EMPC algorithm including a software implementation in ACADO Code Generation. The scheme is based on a novel memory efficient, symmetric AD approach for real-time propagation of second order derivatives. This is used inside a tailored multiple-shooting based SQP method, which employs a mirrored version of the exact Hessian. The performance of the proposed auto-generated EMPC algorithm is demonstrated for the optimal control of a nonlinear biochemical reactor benchmark case-study. A speedup of a factor more than 2 can be shown in the CPU time for integration and Hessian computation of this example.