Efficient Interior Point Methods for Multistage Problems Arising in Receding Horizon Control

Receding horizon control requires the solution of an optimization problem at every sampling instant. We present efficient interior point methods tailored to convex multistage problems, a problem class which most relevant MPC problems with linear dynamics can be cast in, and specify important algorithmic details required for a high speed implementation with superior numerical stability. In particular, the presented approach allows for quadratic constraints, which is not sup- ported by existing fast MPC solvers. A categorization of widely used MPC problem formulations into classes of different complexity is given, and we show how the computational burden of certain quadratic or linear constraints can be decreased by a low rank matrix forward substitution scheme. Implementation details are provided that are crucial to obtain high speed solvers. We present extensive numerical studies for the proposed methods and compare our solver to three well-known solver packages, outperforming the fastest of these by a factor 2-5 in speed and 3-70 in code size. Moreover, our solver is shown to be very efficient for large problem sizes and for quadratically constrained QPs, extending the set of systems amenable to advanced MPC formulations on low-cost embedded hardware

Distributed Model Predictive Consensus via the Alternating Direction Method of Multipliers

We propose a distributed optimization method for solving a distributed model predictive consensus problem. The goal is to design a distributed controller for a network of dynamical systems to optimize a coupled objective function while respecting state and input constraints. The distributed optimization method is an augmented Lagrangian method called the Alternating Direction Method of Multipliers (ADMM), which was introduced in the 1970s but has seen a recent resurgence in the context of dramatic increases in computing power and the development of widely available distributed computing platforms. The method is applied to position and velocity consensus in a network of double integrators. We find that a few tens of ADMM iterations yield closed-loop performance near what is achieved by solving the optimization problem centrally. Furthermore, the use of recent code generation techniques for solving local subproblems yields fast overall computation times.Comment: 7 pages, 5 figures, 50th Allerton Conference on Communication, Control, and Computing, Monticello, IL, USA, 201

Exploiting Chordality in Optimization Algorithms for Model Predictive Control

In this chapter we show that chordal structure can be used to devise efficient optimization methods for many common model predictive control problems. The chordal structure is used both for computing search directions efficiently as well as for distributing all the other computations in an interior-point method for solving the problem. The chordal structure can stem both from the sequential nature of the problem as well as from distributed formulations of the problem related to scenario trees or other formulations. The framework enables efficient parallel computations.Comment: arXiv admin note: text overlap with arXiv:1502.0638

Bayesian model predictive control: Efficient model exploration and regret bounds using posterior sampling

Tight performance specifications in combination with operational constraints make model predictive control (MPC) the method of choice in various industries. As the performance of an MPC controller depends on a sufficiently accurate objective and prediction model of the process, a significant effort in the MPC design procedure is dedicated to modeling and identification. Driven by the increasing amount of available system data and advances in the field of machine learning, data-driven MPC techniques have been developed to facilitate the MPC controller design. While these methods are able to leverage available data, they typically do not provide principled mechanisms to automatically trade off exploitation of available data and exploration to improve and update the objective and prediction model. To this end, we present a learning-based MPC formulation using posterior sampling techniques, which provides finite-time regret bounds on the learning performance while being simple to implement using off-the-shelf MPC software and algorithms. The performance analysis of the method is based on posterior sampling theory and its practical efficiency is illustrated using a numerical example of a highly nonlinear dynamical car-trailer system

Approximate Dynamic Programming via Sum of Squares Programming

We describe an approximate dynamic programming method for stochastic control problems on infinite state and input spaces. The optimal value function is approximated by a linear combination of basis functions with coefficients as decision variables. By relaxing the Bellman equation to an inequality, one obtains a linear program in the basis coefficients with an infinite set of constraints. We show that a recently introduced method, which obtains convex quadratic value function approximations, can be extended to higher order polynomial approximations via sum of squares programming techniques. An approximate value function can then be computed offline by solving a semidefinite program, without having to sample the infinite constraint. The policy is evaluated online by solving a polynomial optimization problem, which also turns out to be convex in some cases. We experimentally validate the method on an autonomous helicopter testbed using a 10-dimensional helicopter model.Comment: 7 pages, 5 figures. Submitted to the 2013 European Control Conference, Zurich, Switzerlan

Rate analysis of inexact dual first order methods: Application to distributed MPC for network systems

In this paper we propose and analyze two dual methods based on inexact gradient information and averaging that generate approximate primal solutions for smooth convex optimization problems. The complicating constraints are moved into the cost using the Lagrange multipliers. The dual problem is solved by inexact first order methods based on approximate gradients and we prove sublinear rate of convergence for these methods. In particular, we provide, for the first time, estimates on the primal feasibility violation and primal and dual suboptimality of the generated approximate primal and dual solutions. Moreover, we solve approximately the inner problems with a parallel coordinate descent algorithm and we show that it has linear convergence rate. In our analysis we rely on the Lipschitz property of the dual function and inexact dual gradients. Further, we apply these methods to distributed model predictive control for network systems. By tightening the complicating constraints we are also able to ensure the primal feasibility of the approximate solutions generated by the proposed algorithms. We obtain a distributed control strategy that has the following features: state and input constraints are satisfied, stability of the plant is guaranteed, whilst the number of iterations for the suboptimal solution can be precisely determined.Comment: 26 pages, 2 figure