Inexact Newton based Lifted Implicit Integrators for fast Nonlinear MPC

Nonlinear Model Predictive Control (NMPC) requires the online solution of an Optimal Control Problem (OCP) at every sampling instant. In the context of multiple shooting, a numerical integration is needed to discretize the continuous time dynamics. For stiff, implicitly defined or differential-algebraic systems, implicit schemes are preferred to carry out the integration. The Newton-type optimization method and the implicit integrator then form a nested Newton scheme, solving the optimization and integration problem on two different levels. In recent research, an exact lifting technique was proposed to improve the computational efficiency of the latter framework. Inspired by that work, this paper presents a novel class of lifted implicit integrators, using an inexact Newton method. An additional iterative scheme for computing the sensitivities is proposed, which provides similar properties as the exact lifted integrator at considerably reduced computational costs. Using the example of an industrial robot, computational speedups of up to factor 8 are reported. The proposed methods have been implemented in the open-source ACADO code generation software

Inexact Newton based Lifted Implicit Integrators for fast Nonlinear MPC

Nonlinear Model Predictive Control (NMPC) requires the online solution of an Optimal Control Problem (OCP) at every sampling instant. In the context of multiple shooting, a numerical integration is needed to discretize the continuous time dynamics. For stiff, implicitly defined or differential-algebraic systems, implicit schemes are preferred to carry out the integration. The Newton-type optimization method and the implicit integrator then form a nested Newton scheme, solving the optimization and integration problem on two different levels. In recent research, an exact lifting technique was proposed to improve the computational efficiency of the latter framework. Inspired by that work, this paper presents a novel class of lifted implicit integrators, using an inexact Newton method. An additional iterative scheme for computing the sensitivities is proposed, which provides similar properties as the exact lifted integrator at considerably reduced computational costs. Using the example of an industrial robot, computational speedups of up to factor 8 are reported. The proposed methods have been implemented in the open-source ACADO code generation software

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

An Interior Point Algorithm for Optimal Coordination of Automated Vehicles at Intersections

In this paper, we consider the optimal coordination of automated vehicles at intersections under fixed crossingorders. We state the problem as a Direct Optimal Control problem, and propose a line-search Primal-Dual Interior Point algorithm with which it can be solved. We show that the problem structure is such that most computations required to construct the search- direction and step-size can be performed in parallel on-board the vehicles. This is realized through the Schur-complement of blocks in the KKT-matrix in two steps and a merit-function with separa- ble components. We analyze the communication requirements of the algorithm, and propose a conservative approximation scheme which can reduce the data exchange. We demonstrate that in hard but realistic scenarios, reductions of almost 99% are achieved, at the expense of less than 1% sub-optimality

A Newton algorithm for distributed Semi-Definite Programs using the primal-dual interior-point method

This paper considers the problem of solving convex decomposable Semi-Definite Programs (SDPs) in a distributed fashion. The SDP subproblems are solved locally, while the constraints coupling the different local problems are introduced in the local cost functions using a Lagrange relaxation. The local problems are solved via the primal-dual interior-point method, taking steps along the Nesterov-Todd directions, while the feasibility of the coupling constraints is improved along the central path by taking Newton iterations on the multipliers associated to the Lagrange relaxation. The local factorisations involved in computing the Nesterov-Todd directions are re-used to construct gradients and Hessians for the Lagrange multipliers. The local factorisations are also re-used to construct linear predictors for both the local primal-dual variables and the multipliers, which improve significantly the tracking of the central path

A Newton algorithm for distributed Semi-Definite Programs using the primal-dual interior-point method

This paper considers the problem of solving convex decomposable Semi-Definite Programs (SDPs) in a distributed fashion. The SDP subproblems are solved locally, while the constraints coupling the different local problems are introduced in the local cost functions using a Lagrange relaxation. The local problems are solved via the primal-dual interior-point method, taking steps along the Nesterov-Todd directions, while the feasibility of the coupling constraints is improved along the central path by taking Newton iterations on the multipliers associated to the Lagrange relaxation. The local factorisations involved in computing the Nesterov-Todd directions are re-used to construct gradients and Hessians for the Lagrange multipliers. The local factorisations are also re-used to construct linear predictors for both the local primal-dual variables and the multipliers, which improve significantly the tracking of the central path

An analysis of the Directional-Modifier Adaptation algorithm based on Optimal Experimental Design

The modifier approach has been extensively explored and offers a theoretically-sound and practically-useful method to deploy real-time optimization. The recent directional-modifier adaptation algorithm offers a heuristic to tackle the modifier approach. The directional-modifier adaptation algorithm, supported by strong theoretical properties and the ease of deployment in practice, proposes a meaningful compromise between process optimality and quickly improving the quality of the estimation of the gradient of the process cost function. This paper proposes a novel view of the directional-modifier adaptation algorithm, as an approximation of the optimal trade-off between the underlying experimental design problem and the process optimization problem. It moreover suggests a minor modification in the tuning of the algorithm, so as to make it a more genuine approximatio

A distributed algorithm for NMPC-based wind farm control

Nonlinear Model Predictive Control (NMPC) has been identified as a highly promising control technique for wind turbine generators, and has been shown to be realtime feasible for the control of individual wind turbines. The potential benefit of performing control at the wind farm level is well understood, and has been recently investigated in the literature. Likewise, the extension of NMPC from wind turbine to wind farm control is highly desirable, but very challenging since it requires solving large non-convex optimal control problems in real time. It has not been considered so far in the literature. This paper proposes a first contribution in that direction, using a distributed optimisation approach based on the Lagrange relaxation. The proposed algorithm requires a very limited amount of additional computations when compared to controlling the wind turbines individually via NMPC. The problem of smoothing the wind farm power output is considered here