The Multi-Parametric Toolbox is a collection of algorithms for modeling, control, analysis, and deployment of constrained optimal controllers developed under Matlab. It features a powerful geometric library that extends the application of the toolbox beyond optimal control to various problems arising in computational geometry. The new version 3.0 is a complete rewrite of the original toolbox with a more flexible structure that offers faster integration of new algorithms. The numerical side of the toolbox has been improved by adding interfaces to state of the art solvers and by incorporation of a new parametric solver that relies on solving linear-complementarity problems. The toolbox provides algorithms for design and implementation of real-time model predictive controllers that have been extensively tested. • modeling of dynamical systems, • MPC-based control synthesis,

a b s t r a c tThis paper presents an investigation of how Model Predictive Control (MPC) and weather predictions can increase the energy efficiency in Integrated Room Automation (IRA) while respecting occupant comfort. IRA deals with the simultaneous control of heating, ventilation and air conditioning (HVAC) as well as blind positioning and electric lighting of a building zone such that the room temperature as well as CO 2 and luminance levels stay within given comfort ranges. MPC is an advanced control technique which, when applied to buildings, employs a model of the building dynamics and solves an optimization problem to determine the optimal control inputs. In this paper it is reported on the development and analysis of a Stochastic Model Predictive Control (SMPC) strategy for building climate control that takes into account the uncertainty due to the use of weather predictions.As first step the potential of MPC was assessed by means of a large-scale factorial simulation study that considered different types of buildings and HVAC systems at four representative European sites. Then for selected representative cases the control performance of SMPC, the impact of the accuracy of weather predictions, as well as the tunability of SMPC were investigated. The findings suggest that SMPC outperforms current control practice.

Abstract-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 supported 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.

Abstract-One of the most critical challenges facing society today is climate change and thus the need to realize massive energy savings. Since buildings account for about 40% of global final energy use, energy efficient building climate control can have an important contribution. In this paper we develop and analyze a Stochastic Model Predictive Control (SMPC) strategy for building climate control that takes into account weather predictions to increase energy efficiency while respecting constraints resulting from desired occupant comfort. We investigate a bilinear model under stochastic uncertainty with probabilistic, time varying constraints. We report on the assessment of this control strategy in a large-scale simulation study where the control performance with different building variants and under different weather conditions is studied. For selected cases the SMPC approach is analyzed in detail and shown to significantly outperform current control practice.

This paper proposes to use Nesterov's fast gradient method for the solution of linear quadratic model predictive control (MPC) problems with input constraints. The main focus is on the method's a priori computational complexity certification which consists of deriving lower iteration bounds such that a solution of pre-specified suboptimality is obtained for any possible state of the system. We investigate cold-and warm-starting strategies and provide an easily computable lower iteration bound for cold-starting and an asymptotic characterization of the bounds for warm-starting. Moreover, we characterize the set of MPC problems for which small iteration bounds and thus short solution times are expected. The theoretical findings and the practical relevance of the obtained lower iteration bounds are underpinned by various numerical examples and compared to certification results for a primal-dual interior point method.

We characterize the maximum controlled invariant (MCI) set for discrete-as well as continuous-time nonlinear dynamical systems as the solution of an infinite-dimensional linear programming problem. For systems with polynomial dynamics and compact semialgebraic state and control constraints, we describe a hierarchy of finite-dimensional linear matrix inequality (LMI) relaxations whose optimal values converge to the volume of the MCI set; dual to these LMI relaxations are sum-of-squares (SOS) problems providing a converging sequence of outer approximations to the MCI set. The approach is simple and readily applicable in the sense that the approximations are the outcome of a single semidefinite program with no additional input apart from the problem description. A number of numerical examples illustrate the approach. Introduction.Given a controlled dynamical system described by a differential (continuous-time) or difference (discrete-time) equation, its maximum controlled invariant (MCI) set is the set of all initial states that can be kept within a given constraint set ad infinitum using admissible control inputs. This set goes by many other names in the literature, e.g., viability kernel in viability theory [5], or (A, B)-invariant set in the linear case [13].Set invariance is a ubiquitous and essential concept in dynamical systems theory as far as both analysis and control synthesis is concerned. In particular, by its very definition, the MCI set determines fundamental limitations of a given control system with respect to constraint satisfaction. In addition, there is a very tight link between invariant sets and (control) Lyapunov functions. Indeed, sublevel sets of a Lyapunov function give rise to invariant sets. Conversely, at least in the linear case, any controlled invariant set gives rise to a control Lyapunov function, and therefore these sets can be readily used to design stabilizing control laws; see, e.g., [9] for a general treatment and, e.g., [17,26] for applications in model predictive control design.The problem of (maximum) controlled invariant set computation for discretetime systems has been a topic of active research for more than four decades. The central tool in this effort has been the contractive algorithm of [7] and its expansive counterpart [18]. For an exhaustive survey and historical remarks see the survey [9] and the book [12].

This paper investigates the relations between three different properties, which are of importance in optimal control problems: dissipativity of the underlying dynamics with respect to a specific supply rate, optimal operation at steady state, and the turnpike property. We show in a continuous-time setting that if along optimal trajectories a strict dissipation inequality is satisfied, then this implies optimal operation at this steady state and the existence of a turnpike at the same steady state. Finally, we establish novel converse turnpike results, i.e., we show that the existence of a turnpike at a steady state implies optimal operation at this steady state and dissipativity with respect to this steady state. We draw upon a numerical example to illustrate our findings. (Milan Korda), colin.jones@epfl.ch (Colin N. Jones), dominique.bonvin@epfl.ch (Dominique Bonvin). 1 We remark that occasionally turnpike phenomena are denoted by varying names: [1,29] refer to turnpikes as a dichotomy of optimal control problems, while [24] uses the Preprint

This paper presents a new formulation and synthesis approach for stabilizing cooperative distributed model predictive control (MPC) for networks of linear systems, which are coupled in their dynamics. The controller is defined by a network-wide constrained optimal control problem, which is solved online by distributed optimization. The main challenge is the definition of a global MPC problem, which both defines a stabilizing control law and is amenable to distributed optimization, i.e., can be split into a number of appropriately coupled subproblems. For such a combination of stability and structure, we propose the use of a separable terminal cost function, combined with novel time-varying local terminal sets. For synthesis, we introduce a method that allows for constructing these components in a completely distributed way, without central coordination. The paper covers the nominal case in detail and discusses the extension of the methodology to reference tracking. Closed-loop functionality of the controller is illustrated by a numerical example, which highlights the effectiveness of the proposed controller and its time-varying local terminal sets.