Rien Quirynen

Auto Generation of Implicit Integrators for Embedded NMPC with Microsecond Sampling Times

Abstract Algorithms for fast real-time Nonlinear Model Predictive Control (NMPC) for mechatronic systems face several challenges. They need to respect tight real-time constraints and need to run on embedded control hardware with limited computing power and memory. A combination of efficient online algorithms and code generation of explicit integrators was shown to be able to overcome these hurdles. This paper generalizes the idea of code generation to Implicit Runge-Kutta (IRK) methods with efficient sensitivity generation. It is shown that they often outperform existing auto-generated Explicit Runge-Kutta (ERK) methods. Moreover, the new methods allow to treat Differential Algebraic Equation (DAE) systems by NMPC with microsecond sampling times.

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.

Experimental Validation of Combined Nonlinear Optimal Control and Estimation of an Overhead Crane

Abstract This paper validates the combination of nonlinear model predictive control and moving horizon estimation to optimally control an overhead crane. Real-time implementation of this combined optimal control and estimation approach with execution times far below the sampling time was realized through the use of automatic code generation. Besides experiments that reflect good point-to-point performance, the approach showed to be good in disturbance rejection as well as in servo-tracking.

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.

Aircraft control based on fast non-linear MPC & multiple-shooting

For extreme maneuvers, feasible flight trajectories can be difficult to compute. If the aircraft is controlled based on linear approximations of the system dynamics that are computed along infeasible trajectories, poor control actions and violations of the flight envelope constraints can result. This paper proposes a Non-linear Model Predictive Control (NMPC) approach, to handle extreme maneuvers, respect the flight envelope, handle actuator failure, and perform emergency obstacle avoidance with a single, non-hierarchical controller that can be implemented in real time.

From linear to nonlinear MPC: bridging the gap via the real-time iteration

ABSTRACT Linear model predictive control (MPC) can be currently deployed at outstanding speeds, thanks to recent progress in algorithms for solving online the underlying structured quadratic programs. In contrast, nonlinear MPC (NMPC) requires the deployment of more elaborate algorithms, which require longer computation times than linear MPC. Nonetheless, computational speeds for NMPC comparable to those of MPC are now regularly reported, provided that the adequate algorithms are used. In this paper, we aim at clarifying the similarities and differences between linear MPC and NMPC. In particular, we focus our analysis on NMPC based on the real-time iteration (RTI) scheme, as this technique has been successfully tested and, in some applications, requires computational times that are only marginally larger than linear MPC. The goal of the paper is to promote the understanding of RTI-based NMPC within the linear MPC community.

Nonlinear MPC for a two-stage turbocharged gasoline engine airpath

Innovative charging concepts such as two-stage turbocharging for gasoline engines, cause high demands on the process control due to the complex, nonlinear system behavior. For complex, nonlinear systems Nonlinear Model-based Predictive Controllers (NMPC) offer a high potential. They are capable of handling coupled multiple-input systems while achieving high control quality and respecting constraints of the system. In the case of turbocharging, considerations to protect components can introduce the necessity to constrain certain system values. This paper presents a two-stage turbocharged gasoline airpath modeling approach which is suited to be used in a NMPC implementation. The control implementation is based on direct optimal control using an online Sequential Quadratic Programming (SQP) type algorithm. For validating the control performance, simulations are conducted. The computation time of the algorithm is determined by implementation on a control prototyping platform for validation of the real-time capability.

Lifted implicit integrators for direct optimal control

Nonlinear Model Predictive Control (NMPC) relies on solving an Optimal Control Problem (OCP) online at every sampling time. The discretization of the continuous time dynamics requires the deployment of some numerical integration method. To that end, implicit integrators are often preferred when stiff or implicitly defined dynamics are present in the system. Implicit integration schemes, however, are typically more expensive to implement than explicit methods. This paper presents a novel lifting method for implicit integrators which improves their computational efficiency and accuracy in the context of Newton-type optimization algorithms. Similar to the standard lifted Newton, the proposed lifting method requires a marginal implementation effort. This novel approach has been implemented in the ACADO code generation software, and its efficiency illustrated using a nontrivial control example. An improved convergence and a computational speedup of about factor 2 are reported.

Inexact Newton-Type Optimization with Iterated Sensitivities

This paper presents and analyzes an Inexact Newton-type optimization method based on Iterated Sensitivities (INIS). A particular class of Nonlinear Programming (NLP) problems is considered, where a subset of the variables is defined by nonlinear equality constraints. The proposed algorithm considers any problem-specific approximation for the Jacobian of these constraints. Unlike other inexact Newton methods, the INIS-type optimization algorithm is shown to preserve the local convergence properties and the asymptotic contraction rate of the Newton-type scheme for the feasibility problem, yielded by the same Jacobian approximation. The INIS approach results in a computational cost which can be made close to that of the standard inexact Newton implementation. In addition, an adjoint-free (AF-INIS) variant of the approach is presented which, under certain conditions, becomes considerably easier to implement than the adjoint based scheme. The applicability of these results is motivated, specifically for dynamic optimization problems. In addition, the numerical performance of a specific open-source implementation is illustrated.

Robust MPC via minmax differential inequalities

This paper is concerned with tube-based model predictive control (MPC) for both linear and nonlinear, input-affine continuous-time dynamic systems that are affected by time-varying disturbances. We derive a min-max differential inequality describing the support function of positive robust forward invariant tubes, which can be used to construct a variety of tube-based model predictive controllers. These constructions are conservative, but computationally tractable and their complexity scales linearly with the length of the prediction horizon. In contrast to many existing tube-based MPC implementations, the proposed framework does not involve discretizing the control policy and, therefore, the conservatism of the predicted tube depends solely on the accuracy of the set parameterization. The proposed approach is then used to construct a robust MPC scheme based on tubes with ellipsoidal cross-sections. This ellipsoidal MPC scheme is based on solving an optimal control problem under linear matrix inequality constraints. We illustrate these results with the numerical case study of a spring-mass-damper system.