Sébastien Gros

A Lyapunov function for periodic economic optimizing Model Predictive Control

Model Predictive Control (MPC) schemes are commonly using reference-tracking cost functions, which have attractive properties in terms of stability and numerical implementation. However, many control applications have clear economic objectives that can be used directly as the NMPC cost function. Such NMPC schemes are labelled Economic NMPC. Unfortunately, Economic NMPC schemes suffer from some drawbacks. In particular, stability results for economic NMPC are still very sparse. A Lyapunov function for Economic NMPC was first proposed in [1] for problems having a steady-state optimum. The present paper develops a further generalization and clarification of these results for periodic systems.

Optimizing Control based on Output Feedback

In the framework of process optimization, the use of measurements to compensate the effect of uncertainty has re-emerged as an active area of research. One of the ideas therein is to adapt the inputs in order to track the active constraints and push certain sensitivities to zero. In perturbation- based optimization, the sensitivities are evaluated by perturbation of the inputs and measurement of the cost function, which can be experimentally time consuming. However, since more measurements (typically the outputs) than just the cost function are available, the idea developed in this paper is to incorporate the outputs in a measurement-based optimization framework. This is done using an extension to the neighboring-extremal scheme for the case of output measurements. If measurement noise can be neglected, the approach is shown to converge to the optimum in at most two input updates. The effect of measurement noise is also investigated. The strength of neighboring-extremal output feedback for optimization is illustrated on a continuous chemical reactor example.

Nonlinear MPC and MHE for Mechanical Multi-Body Systems with Application to Fast Tethered Airplanes

Mechanical applications often require a high control frequency to cope with fast dynamics. The control frequency of a nonlinear model predictive controller depends strongly on the symbolic complexity of the equations modeling the system. The symbolic complexity of the model equations for multi-body mechanical systems can often be dramatically reduced by using representations based on non-minimal coordinates, which result in index-3 differential-algebraic equations (DAEs). This paper proposes a general procedure to efficiently treat multi-body mechanical systems in the context of MHE & NMPC using non-minimal coordinate representations, and provides the resulting computational times that can be achieved on a tethered airplane system using code generation.

Modeling of Airborne Wind Energy Systems in Natural Coordinates

This paper presents a modeling approach for AWE systems that allows for developing models of low symbolic complexity and low nonlinearity. The approach is based on multi-body modeling, using natural coordinates and algebraic constraints as a representation of the system evolution. This paper shows how to build such models for AWE systems in the Lagrangian framework and how to efficiently incorporate a non-singular representation of the pose (i.e. 3D orientation) of the wing. The proposed modeling technique is illustrated on a single-wing AWE system for power generation and rotating start-up, and for a dual-wing AWE system.

An economic NMPC formulation for wind turbine control

Model Predictive Control (MPC) is a strong candidate for the control of large Multi-MegaWatt Wind Turbine Generators. Several MPC and some Nonlinear MPC scheme have been proposed in the literature, based on reference-tracking objective functions. While the resulting schemes offer very promising results, the difficulty of tuning a reference-tracking NMPC scheme for performance is likely to be a hindrance to the industrial success of NMPC-based WTG control. Because they directly maximize the system performance, economic NMPC schemes are more straightforward to tune. Economic NMPC schemes present, however, some known difficulties that are a serious obstacle to their real-time deployment. This paper presents an economic NMPC formulation for maximizing the generated power of wind turbine generators, which does not suffer from such difficulties. The relationship between the proposed and more classical reference-tracking approaches is formally established.

Neighboring Extremal Control for Singular Dynamics Optimization Problems II: Multiple-input Case

A powerful approach for dynamic optimisation in the presence of uncertainty is to incorporate measurements into the optimisation framework so as to track the optimum. For non-singular control problems, this can be done by tracking active constraints along boundary arcs and using neighbouring-extremal (NE) control along interior arcs. Essentially, NE control forces the first-order variation of the necessary conditions of optimality (NCO) to zero. In this article, an extension of NE control to singular control problems is proposed. This article focuses on single-input systems, while the extension to multiple-input systems is investigated in the companion paper. The idea is to design NE controllers from successive time differentiations of the first-order variation of the NCO. Approximate NE feedback laws are also proposed, which are both easily implementable and tractable from a real-time optimisation perspective. These developments are illustrated by the case study of a semi-batch chemical reactor.

Numerical Trajectory Optimization for Airborne Wind Energy Systems Described by High Fidelity Aircraft Models

In order to study design tradeoffs in the development of an AWE system, it is useful to develop a code to optimize a trajectory for arbitrary objective function and constraints. We present a procedure for using direct collocation to optimize such a trajectory where a model is specified as a set of differential–algebraic equations. The six degree of freedom single-kite, pumping-mode AWE model developed in Chap. 10 is summarized, and two typical periodic optimal control problems are formulated and solved: maximum power and number of cycles per retraction. Finally, a procedure for optimally transitioning between two fixed trajectories is presented.

Efficient NMPC for nonlinear models with linear subsystems

Real-time optimal control algorithms for fast, mechatronic systems need to be run on embedded hardware and they need to respect tight timing constraints. When using nonlinear models, the simulation and generation of sensitivities forms a computationally demanding part of any algorithm. Automatic code generation of Implicit Runge-Kutta (IRK) methods has been shown to reduce its CPU time significantly. However, a typical model also shows a lot of structure that can be exploited in a rather elegant and efficient way. The focus of this paper is on nonlinear models with linear subsystems. With the proposed model formulation, the new auto generated integrators can be considered a powerful generalization of other solvers, e.g. those that support quadrature variables. A speedup of up to 5 - 10 is shown in the integration time for two examples from the literature.

Orbit control for a power generating airfoil based on nonlinear MPC

The Airborne Wind Energy paradigm proposes to generate energy by flying a tethered airfoil across the wind flow. An essential problem posed by Airborne Wind Energy is the control of the tethered airfoil trajectory during power generation. Tethered flight is a fast, strongly nonlinear, unstable and constrained process, motivating control approaches based on fast Nonlinear Model Predictive Control. In this paper, a computationally efficient 6-DOF control model for a high performance, large-scale, rigid airfoil is proposed. A control scheme based on receding-horizon Nonlinear Model Predictive Control to track reference trajectories is applied to the proposed model. In order to make a real-time application of Nonlinear Model Predictive Control possible, a Real-Time Iteration scheme is proposed and its performance investigated.

Neighbouring-extremal control for singular dynamic optimisation problems. Part I: single-input systems

Dynamic optimisation provides a unified framework for improving process operations while taking operational constraints into account. In the presence of uncertainty, measurements can be incorporated into the optimisation framework for tracking the optimum. For non-singular control problems, neighbouring-extremal (NE) control can be used to force the first-order variation of the necessary conditions of optimality (NCO) to zero along interior arcs. An extension of NE control to singular control problems has been proposed in the companion paper for single-input problems. In this article, a generalisation to multiple-input systems is presented. In order for these controllers to be tractable from a real-time optimisation perspective, an approximate NE feedback law is proposed, whose application guarantees, under mild assumptions, that the first-order variation of the NCO converges to zero exponentially. The performance of multi-input NE control is illustrated by the case study of a steered car.