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In robust control under state constraints the set of admissible inputs is usually considered as given, under the assumption that the actuators have been already designed. However, if the input set is too small any controller will fail in stabilizing the closed-loop system while satisfying all prescribed constraints for some initial states of interest, or vice versa the chosen actuators may be over-sized. To handle this issue, in this paper we address the problem of computing the smallest input constraint set such that the closed-loop system is stabilizable from a prescribed set of initial states while respecting all constraints. We focus our attention on linear systems with additive disturbances, and develop the algorithm based on recursive feasibility of robust model predictive control. We demonstrate the results using numerical examples, in which we consider different metrics for the input constraint set selection.

This paper proposes a method for solving optimization problems in which the decision-maker cannot evaluate the objective function, but rather can only express apreferencesuch as “this is better than that” between two candidate decision vectors. The algorithm described in this paper aims at reaching the global optimizer by iteratively proposing the decision maker a new comparison to make, based on actively learning a surrogate of the latent (unknown and perhaps unquantifiable) objective function from past sampled decision vectors and pairwise preferences. A radial-basis function surrogate is fit via linear or quadratic programming, satisfying if possible the preferences expressed by the decision maker on existing samples. The surrogate is used to propose a new sample of the decision vector for comparison with the current best candidate based on two possible criteria: minimize a combination of the surrogate and an inverse weighting distance function to balance between exploitation of the surrogate and exploration of the decision space, or maximize a function related to the probability that the new candidate will be preferred. Compared to active preference learning based on Bayesian optimization, we show that our approach is competitive in that, within the same number of comparisons, it usually approaches the global optimum more closely and is computationally lighter. Applications of the proposed algorithm to solve a set of benchmark global optimization problems, for multi-objective optimization, and for optimal tuning of a cost-sensitive neural network classifier for object recognition from images are described in the paper. MATLAB and a Python implementations of the algorithms described in the paper are available athttp://cse.lab.imtlucca.it/~bemporad/glis.

We present a general-purpose solver for convex quadratic programs based on the alternating direction method of multipliers, employing a novel operator splitting technique that requires the solution of a quasi-definite linear system with the same coefficient matrix at almost every iteration. Our algorithm is very robust, placing no requirements on the problem data such as positive definiteness of the objective function or linear independence of the constraint functions. It can be configured to be division-free once an initial matrix factorization is carried out, making it suitable for real-time applications in embedded systems. In addition, our technique is the first operator splitting method for quadratic programs able to reliably detect primal and dual infeasible problems from the algorithm iterates. The method also supports factorization caching and warm starting, making it particularly efficient when solving parametrized problems arising in finance, control, and machine learning. Our open-source C implementation OSQP has a small footprint, is library-free, and has been extensively tested on many problem instances from a wide variety of application areas. It is typically ten times faster than competing interior-point methods, and sometimes much more when factorization caching or warm start is used. OSQP has already shown a large impact with tens of thousands of users both in academia and in large corporations. © 2020 Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society. ISSN:1867-2957 ISSN:1867-2949

We present a semi-explicit formulation of model predictive controllers for hybrid systems with feasibility guarantees. The key idea is to use a machine-learning approach to learn a compact predictor of the integer/binary components of optimal solutions of the multiparametric mixed-integer linear optimization problem associated with the controller, so that, on-line, only a linear programming problem must be solved. In this scheme, feasibility is ensured by a simple rule-based engine that corrects the binary configuration only when necessary. The performance of the approach is assessed on a well known benchmark for which explicit controllers based on domain-specific knowledge are already available. Simulation results show how our proposed method considerably lowers computation time without deteriorating closed-loop performance.

Active-set (AS) methods for quadratic programming (QP) are particularly suitable for real-time optimization, as they provide a high-quality solution in a finite number of iterations. However, they add or remove one constraint at each iteration, which makes them inefficient when the number of constraints and variables grows. Block principal pivoting (BPP) methods perform instead multiple changes in the working set in a single iteration, resulting in a much faster execution time. The infeasible primal–dual method proposed by Kunisch and Rendl (KR) (Kunisch and Rendl, 2003) is a BPP method for box-constrained QP that is particularly attractive when reducing the time for finding an accurate solution is crucial, such as in linear model predictive control (MPC) applications. However, the method is guaranteed to converge only under very restrictive sufficient conditions, and tight bounds on the worst-case complexity are not available. For a given set of box-constrained QP’s that depend on a vector of parameters, such as those that arise in linear MPC, this paper proposes an algorithm that computes offline the exact number of iterations and flops needed by the KR method in the worst-case, and the region of the parameter space for which the method converges or is proved to cycle.

This paper proposes a new algorithm for solving convex quadratic programs (QP) subject to linear inequality and equality constraints. The method extends an approach recently proposed by the author for solving strictly convex QP's using nonnegative least squares, by making it numerically more robust and able to handle also the nonstrictly convex case, equality constraints, and warm starting from an initial guess. Robustness is achieved by introducing an outer proximal-point iteration scheme that regularizes the problem without altering the solution, and by adaptively weighting the least squares problems encountered while solving the problem. The performance of the resulting QP solver in terms of speed and robustness in the single precision arithmetic is assessed against other optimization algorithms tailored to embedded model predictive control applications.

Energy disaggregation aims at reconstructing the power consumed by each electric appliance available in a household from the aggregate power readings collected by a single-point smart meter. With the ultimate goal of fully automatizing this procedure, we first estimate a set of jump models, each of them describing the consumption behaviour of each electric appliance. By representing the total power consumed at the household level as the sum of the outputs of the estimated jump models, a filtering algorithm, based on dynamic programming, is then employed to reconstruct, in an iterative way, the power consumption at an individual appliance level.

The deployment of hybrid model predictive control (MPC) in practical applications requires primarily an efficient and robust on-line Mixed-Integer Quadratic Programming (MIQP) solver that runs in real time. In this paper we propose a new algorithm for solving MIQP problems which is particularly tailored to solve small-scale MIQPs, such as those that arise in embedded hybrid MPC applications. The algorithm couples a branch and bound (B&B) scheme with a recently proposed numerically robust Quadratic Programming (QP) solver based on nonnegative least squares (NNLS) and proximal-point iterations. The resulting MIQP solver supports positive semidefinite Hessian matrices, often appearing in hybrid MPC formulations, and warm starts with respect to both binary and real variables. We show that the speed of execution of our solver is comparable with state-of-the-art commercial solvers, while it is relatively simple to code in an embedded control system.

Active-set methods are recognized to often outperform other methods in terms of speed and solution accuracy when solving small-size quadratic programming (QP) problems, making them very attractive in embedded linear model predictive control (MPC) applications. A drawback of active-set methods is the lack of tight bounds on the worst-case number of iterations, a fundamental requirement for their implementation in a real-time system. Extensive simulation campaigns provide an indication of the expected worst-case computation load, but not a complete guarantee. This paper solves such a certification problem by proposing an algorithm to compute the exact bound on the maximum number of iterations and floating point operations required by a state-of-the-art dual active-set QP solver. The algorithm is applicable to a given QP problem whose linear term of the cost function and right-hand side of the constraints depend linearly on a vector of parameters, as in the case of linear MPC. In addition, a new solver is presented that combines explicit and implicit MPC ideas, guaranteeing improvements of the worst-case computation time. The ability of the approach to exactly quantify memory and worst-case computation requirements is tested on a few MPC examples, also highlighting when online optimization should be preferred to explicit MPC.

Summary This paper proposes stochastic model predictive control as a tool for hedging derivative contracts (such as plain vanilla and exotic options) in the presence of transaction costs. The methodology combines stochastic scenario generation for the prediction of asset prices at the next rebalancing interval with the minimization of a stochastic measure of the predicted hedging error. We consider 3 different measures to minimize in order to optimally rebalance the replicating portfolio: a trade-off between variance and expected value of hedging error, conditional value at risk, and the largest predicted hedging error. The resulting optimization problems require solving at each trading instant a quadratic program, a linear program, and a (smaller-scale) linear program, respectively. These can be combined with 3 different scenario generation schemes: the lognormal stock model with parameters recursively identified from data, an identification method based on support vector regression, and a simpler scheme based on perturbation noise. The hedging performance obtained by the proposed stochastic model predictive control strategies is illustrated on real-world data drawn from the NASDAQ-100 composite, evaluated for a European call and a barrier option, and compared with delta hedging.

The execution of a hybrid model predictive controller (MPC) on an embedded platform requires solving a Mixed-Integer Quadratic Programming (MIQP) in real time. The MIQP problem is NP-hard, which poses a major challenge in an environment where computational and memory resources are limited. To address this issue, we propose the use of accelerated dual gradient projection (GPAD) to find both the exact and an approximate solution of the MIQP problem. In particular, an existing GPAD algorithm is specialized to solve the relaxed Quadratic Programming (QP) subproblems that arise in a Branch and Bound (B&B) method for solving the MIQP to optimality. Furthermore, we present an approach to find a suboptimal integer feasible solution of a MIQP problem without using B&B. The GPAD algorithm is very simple to code and requires only basic arithmetic operations which makes it well suited for an embedded implementation. The performance of the proposed approaches is comparable with the state of the art MIQP solvers for small-scale problems.

For all its successes, Reinforcement Learning (RL) still struggles to deliver formal guarantees on the closed-loop behavior of the learned policy. Among other things, guaranteeing the safety of RL with respect to safety-critical systems is a very active research topic. Some recent contributions propose to rely on projections of the inputs delivered by the learned policy into a safe set, ensuring that the system safety is never jeopardized. Unfortunately, it is unclear whether this operation can be performed without disrupting the learning process. This paper addresses this issue. The problem is analysed in the context of $Q$-learning and policy gradient techniques. We show that the projection approach is generally disruptive in the context of $Q$-learning though a simple alternative solves the issue, while simple corrections can be used in the context of policy gradient methods in order to ensure that the policy gradients are unbiased. The proposed results extend to safe projections based on robust MPC techniques., Comment: Accepted at IFAC 2020

In this paper we propose a lightweight neural network architecture that is able to learn the binary components of the optimal solution of a class of multiparametric mixed-integer quadratic programming (MIQP) problems, such as those that arise from hybrid model predictive control formulations. The predictor provides a binary warm-start to a specifically designed branch and bound (B&B) algorithm to quickly discover an integer-feasible solution of the given MIQP, with the aim of reducing the overall solution time required to find the global optimal solution on line.

Reinforcement Learning (RL) has demonstrated a huge potential in learning optimal policies without any prior knowledge of the process to be controlled. Model Predictive Control (MPC) is a popular control technique which is able to deal with nonlinear dynamics and state and input constraints. The main drawback of MPC is the need of identifying an accurate model, which in many cases cannot be easily obtained. Because of model inaccuracy, MPC can fail at delivering satisfactory closed-loop performance. Using RL to tune the MPC formulation or, conversely, using MPC as a function approximator in RL allows one to combine the advantages of the two techniques. This approach has important advantages, but it requires an adaptation of the existing algorithms. We therefore propose an improved RL algorithm for MPC and test it in simulations on a rather challenging example.

In nonlinear regression choosing an adequate model structure is often a challenging problem. While simple models (such as linear functions) may not be able to capture the underlying relationship among the variables, over-parametrized models described by a large set of nonlinear basis functions tend to overfit the training data, leading to poor generalization on unseen data. Piecewise-affine (PWA) models can describe nonlinear and possible discontinuous relationships while maintaining simple local affine regressor-to-output mappings, with extreme flexibility when the polyhedral partitioning of the regressor space is learned from data rather than fixed a priori. In this paper, we propose a novel and numerically very efficient two-stage approach for PWA regression based on a combined use of (i) recursive multi-model least-squares techniques for clustering and fitting linear functions to data, and (ii) linear multi-category discrimination, either offline (batch) via a Newton-like algorithm for computing a solution of unconstrained optimization problems with objective functions having a piecewise smooth gradient, or online (recursive) via averaged stochastic gradient descent.

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.

This technical note proposes an active set method based on nonnegative least squares (NNLS) to solve strictly convex quadratic programming (QP) problems, such as those that arise in Model Predictive Control (MPC). The main idea is to rephrase the QP problem as a Least Distance Problem (LDP) that is solved via a NNLS reformulation. While the method is rather general for solving strictly convex QP's subject to linear inequality constraints, it is particularly useful for embedded MPC because (i) is very fast, compared to other existing state-of-the-art QP algorithms, (ii) is very simple to code, requiring only basic arithmetic operations for computing $LDL^{T}$ decompositions recursively to solve linear systems of equations, (iii) contrary to iterative methods, provides the solution or recognizes infeasibility in a finite number of steps.

Global optimization problems whose objective function is expensive to evaluate can be solved effectively by recursively fitting a surrogate function to function samples and minimizing an acquisition function to generate new samples. The acquisition step trades off between seeking for a new optimization vector where the surrogate is minimum (exploitation of the surrogate) and looking for regions of the feasible space that have not yet been visited and that may potentially contain better values of the objective function (exploration of the feasible space). This paper proposes a new global optimization algorithm that uses inverse distance weighting (IDW) and radial basis functions (RBF) to construct the acquisition function. Rather arbitrary constraints that are simple to evaluate can be easily taken into account. Compared to Bayesian optimization, the proposed algorithm, that we call GLIS (GLobal minimum using Inverse distance weighting and Surrogate radial basis functions), is competitive and computationally lighter, as we show in a set of benchmark global optimization and hyperparameter tuning problems. MATLAB and Python implementations of GLIS are available at http://cse.lab.imtlucca.it/~bemporad/glis.

In this paper we consider the problem of energy disaggregation, commonly referred in the literature as “non-intrusive load monitoring”. The problem is to estimate the end-use power consumption profiles of individual household appliance using only aggregated power measurements. We propose a two-stage supervised approach. At the first stage, dynamical models of individual appliances are estimated using disaggregated training data gathered over a short intrusive period. The consumption profiles of individual appliances are described by PieceWise Affine AutoRegressive (PWA-AR) models with multiple operating modes, which are estimated via a moving horizon PWA regression algorithm. Once the model of each appliance is identified, a binary quadratic programming problem is solved at the second stage to determine the set of active appliances which contribute to the instantaneous aggregated power, along with their operating modes. A benchmark dataset is used to assess the performance of the presented disaggregation approach.

The contribution of this paper is a novel tree-based stochastic model predictive control (SMPC) approach to solve the optimal exit-time control problem for stochastic systems, that is to maximize the expected value of the first time instant at which prescribed constraints are violated. A scenario tree with a specified number of tree nodes is used to encode the most likely system behavior, where each path on the tree corresponds to a distinct disturbance scenario. For linear discrete-time systems with an additive random disturbance, a mixed-integer linear program (MILP) obtains solutions arbitrarily close to the optimal solution for a sufficient number of tree nodes. In order to compensate for an incomplete scenario tree and/or unmodeled effects, feedback is provided by recomputing the MILP solution over a receding time horizon based on the current state and disturbance / scenario tree. Two numerical case studies, including an adaptive cruise control problem, demonstrate the effectiveness of the proposed SMPC scheme compared to dynamic programming solutions.

We present a novel branch-and-bound solver for mixed-integer quadratic programs (MIQPs) that efficiently exploits the first-order OSQP solver for the quadratic program (QP) sub-problems. Our algorithm is very robust, requires no dynamic memory allocation and is division-free once an initial factorization is computed. Thus, it suitable for embedded applications with low computing power. Moreover, it does not require any assumption on the problem data such as strict convexity of the objective function. We exploit factorization caching and warm-starting to reduce the computational cost of QP relaxations during branch-and-bound and over repeated solutions of parametric MIQPs such as those arising in embedded control, portfolio optimization, and machine learning. Numerical examples show that our method, using a simple high-level Python implementation interfaced with the OSQP solver, is competitive with established commercial solvers.

Model predictive control (MPC) is one of the most successful techniques adopted in industry to control multivariable systems under constraints on input and output variables. To circumvent the main drawback of MPC, i.e., the need to solve a Quadratic Program (QP) on line to compute the control action, explicit MPC was proposed in the past to precompute the control law off line using multiparametric QP (mpQP). The resulting form of the MPC law is piecewise affine, which is extremely easy to code, can be computed online by simple arithmetic operations, and requires a maximum number of iterations that can be exactly determined a priori. On the other hand, the offline computations to solve the mpQP problem require detecting emptiness, full-dimensionality, and minimal hyperplane representations of polyhedra, and other computational geometric operations. While most of the existing methods solve such operations via linear programming, the approach proposed in this paper relies on a nonnegative least squares (NNLS) solver that is very simple to code and fast to execute. In addition, the new approach exploits QP duality to identify and construct critical regions and to handle degeneracy issues.

In the consensus problem on multi-agent systems, in which the states of the agents represent opinions, the agents aim at reaching a common opinion (or consensus state) through local exchange of information. An important design problem is to choose the degree of interconnection of the subsystems to achieve a good trade-off between a small number of interconnections and a fast convergence to the consensus state, which is the average of the initial opinions under mild conditions. This paper addresses this problem through $l_1$ -norm and $l_0$ -“pseudo-norm” regularized versions of the well-known Fastest Mixing Markov-Chain (FMMC) problem. We show that such versions can be interpreted as robust forms of the FMMC problem and provide results to guide the choice of the regularization parameter.

This paper proposes a new algorithm for solving Mixed-Integer Quadratic Programming (MIQP) problems. The algorithm is particularly tailored to solving small-scale MIQPs such as those that arise in embedded hybrid Model Predictive Control (MPC) applications. The approach combines branch and bound (B&B) with nonnegative least squares (NNLS), that are used to solve Quadratic Programming (QP) relaxations. The QP algorithm extends a method recently proposed by the author for solving strictly convex QP's, by (i) handling equality and bilateral inequality constraints, (ii) warm starting, and (iii) exploiting easy-to-compute lower bounds on the optimal cost to reduce the number of QP iterations required to solve the relaxed problems. The proposed MIQP algorithm has a speed of execution that is comparable to state- of-the-art commercial MIQP solvers and is relatively simple to code, as it requires only basic arithmetic operations to solve least-square problems.

This paper presents a simple method for a posteriori (historical) multi-variate multi-stage optimal trading under transaction costs and a diversification constraint. Starting from a given amount of money in some currency, we analyze the stage-wise optimal allocation over a time horizon with potential investments in multiple currencies and various assets. Three variants are discussed, including unconstrained trading frequency, a fixed number of total admissable trades, and the waiting of a specific time-period after every executed trade until the next trade. The developed methods are based on efficient graph generation and consequent graph search, and are evaluated quantitatively on real-world data. The fundamental motivation of this work is preparatory labeling of financial time-series data for supervised machine learning., 25 pages, 4 figures, 6 tables

We seek a discussion about the most suitable feedback control structure for stock trading under the consideration of proportional transaction costs. Suitability refers to robustness and performance capability. Both are tested by considering different one-step ahead prediction qualities, including the ideal case, correct prediction of the direction of change in daily stock prices and the worst-case. Feedback control structures are partitioned into two general classes: stochastic model predictive control (SMPC) and genetic. For the former class three controllers are discussed, whereby it is distinguished between two Markowitz- and one dynamic hedging-inspired SMPC formulation. For the latter class five trading algorithms are disucssed, whereby it is distinguished between two different moving average (MA) based, two trading range (TR) based, and one strategy based on historical optimal (HistOpt) trajectories. This paper also gives a preliminary discussion about how modified dynamic hedging-inspired SMPC formulations may serve as alternatives to Markowitz portfolio optimization. The combinations of all of the eight controllers with five different one-step ahead prediction methods are backtested for daily trading of the 30 components of the German stock market index DAX for the time period between November 27, 2015 and November 25, 2016., 7 pages, 3 figures, 3 tables, presented as a poster at XVIII Workshop on Quantitative Finance (QFW2017) in Milano on January 25-27, 2017

This paper presents a novel two-stage regularized moving-horizon algorithm for PieceWise Affine (PWA) regression. At the first stage, the training samples are processed iteratively, and a Mixed-Integer Quadratic-Programming (MIQP) problem is solved to find the sequence of active modes and the model parameters which best match the training data, within a relatively short time window in the past. According to a moving-horizon strategy, only the last element of the optimal sequence of active modes is kept, and the next sample is processed by shifting forward the estimation horizon. A regularization term on the model parameters is included in the cost of the formulated MIQP problem, to partly take into account also the past training data outside the considered time horizon. At the second stage, linear multi-category discrimination techniques are used to compute a polyhedral partition of the regressor space based on the estimated sequence of active modes.

A hierarchical algorithm is presented for optimally and automatically combining various option investments to cost-efficiently realize a desired profit-vs.-underlying-price profile (profit profile). The algorithm assumes that a user-defined reference shape is defined and a set of plain vanilla options in which long and short investment positions can be taken are given. Within the presented framework, the desired profit profile can be of arbitrary piecewise-affine (PWA) shape. Depending on future underlying price predictions, it typically represents a bearish or bullish market outlook, or displays bi-modal shape for conditional market outlooks. The method provides a tool for portfolio optimization that is flexible enough to trade off different user-preferences such as exploiting on conditional market outlooks, realizing leverage, and most notably guaranteeing predictable worst-case losses for risk-minimization. The framework can easily be extended to account for different derivative contracts such as exotic options.

This paper presents new approaches based on linear programming (LP) and mixed-integer linear programming (MILP) to solve the optimal exit-time control problem, that is to maximize the first exit-time of a system from a prescribed set. For linear discrete-time systems with known disturbance inputs, we show that an optimal solution can be obtained by solving an MILP and suboptimal solutions are obtained via LP. For both the MILP and LP, an iterative scheme is introduced that improves robustness and computation time. In addition, feedback control strategies are formulated using model predictive control (MPC) techniques. Two numerical examples of a linearized van der Pol oscillator and of spacecraft attitude control demonstrate the efficiency of the proposed approaches in solving optimal exit-time control problems.

Despite the proven advantages of scenario-based stochastic model predictive control for the operational control of water networks, its applicability is limited by its considerable computational footprint. In this paper we fully exploit the structure of these problems and solve them using a proximal gradient algorithm parallelizing the involved operations. The proposed methodology is applied and validated on a case study: the water network of the city of Barcelona., Comment: 11 pages in double column, 7 figures

In this paper we study constrained stochastic optimal control problems for Markovian switching systems, an extension of Markovian jump linear systems (MJLS), where the subsystems are allowed to be nonlinear. We develop appropriate notions of invariance and stability for such systems and provide terminal conditions for stochastic model predictive control (SMPC) that guarantee mean-square stability and robust constraint fulfillment of the Markovian switching system in closed-loop with the SMPC law under very weak assumptions. In the special but important case of constrained MJLS we present an algorithm for computing explicitly the SMPC control law off-line, that combines dynamic programming with parametric piecewise quadratic optimization.

This note describes a model predictive control (MPC) formulation for discrete-time linear systems with hard constraints on control and state variables, under the assumption that the solution of the associated quadratic program is neither optimal nor satisfies the inequality constraints. This is common in embedded control applications, for which real-time constraints and limited computing resources dictate restrictions on the possible number of on-line iterations that can be performed within a sampling period. The proposed approach is rather general, in that it does not refer to a particular optimization algorithm, and is based on the definition of an alternative MPC problem that we assume can only be solved within bounded levels of suboptimality, and violation of the inequality constraints. By showing that the inexact solution is a feasible suboptimal one for the original problem, asymptotic or exponential stability is guaranteed for the closed-loop system. Based on the above general results, we focus on a specific dual accelerated gradient-projection method to obtain a stabilizing MPC law that only requires a predetermined maximum number of on-line iterations.

This paper proposes a dual fast gradient-projection method for solving quadratic programming problems that arise in linear model predictive control with general polyhedral constraints on inputs and states. The proposed algorithm is quite suitable for embedded control applications in that: (1) it is extremely simple and easy to code; (2) the number of iterations to reach a given accuracy in terms of optimality and feasibility of the primal solution can be estimated quite tightly; (3) the computational cost per iteration increases only linearly with the prediction horizon; and (4) the algorithm is also applicable to linear time-varying (LTV) model predictive control problems, with an extra on-line computational effort that is still linear with the prediction horizon.

This paper proposes a dynamic controller structure and a systematic design procedure for stabilizing discrete-time hybrid systems. The proposed approach is based on the concept of control Lyapunov functions (CLFs), which, when available, can be used to design a stabilizing state-feedback control law. In general, the construction of a CLF for hybrid dynamical systems involving both continuous and discrete states is extremely complicated, especially in the presence of non-trivial discrete dynamics. Therefore, we introduce the novel concept of a hybrid control Lyapunov function, which allows the compositional design of a discrete and a continuous part of the CLF, and we formally prove that the existence of a hybrid CLF guarantees the existence of a classical CLF. A constructive procedure is provided to synthesize a hybrid CLF, by expanding the dynamics of the hybrid system with a specific controller dynamics. We show that this synthesis procedure leads to a dynamic controller that can be implemented by a receding horizon control strategy, and that the associated optimization problem is numerically tractable for a fairly general class of hybrid systems, useful in real world applications. Compared to classical hybrid receding horizon control algorithms, the proposed approach typically requires a shorter prediction horizon to guarantee asymptotic stability of the closed-loop system, which yields a reduction of the computational burden, as illustrated through two examples.

In this paper we investigate the problem of optimal real-time power dispatch of an interconnection of conventional power generation plants, renewable resources and energy storage systems. The objective is to minimize imbalance costs and maximize profits whilst satisfying user demand. The managing company is able to trade energy on an electricity market. Energy prices, demand and renewable generation are considered stochastic processes. We show that under certain assumptions, the stochastic power dispatch problem over a finite horizon can be recast into a stochastic optimization formulation but with deterministic constraints. We carry out a systematic study of stochastic optimization methods to solve this problem. We also show that this problem can be approximated by a proper deterministic optimization problem using the sample average approximation method, which can then be solved by standard means.

Stochastic optimal control problems are typically of rather large scale involving millions of decision variables, but possess a certain structure which can be exploited by first-order methods such as forward-backward splitting and the alternating direction method of multipliers (ADMM). In this paper, we use the forward-backward envelope, a real-valued continuously differentiable penalty function, to recast the dual of the original nonsmooth problem as an unconstrained problem which we solve via the limited-memory BFGS algorithm. We show that the proposed method leads to a significant improvement of the convergence rate without increasing much the computational cost per iteration.

This paper presents a spatial-based trajectory planning method for automated vehicles under actuator, obstacle avoidance, and vehicle dimension constraints. Starting from a nonlinear kinematic bicycle model, vehicle dynamics are transformed to a road-aligned coordinate frame with path along the road centerline replacing time as the dependent variable. Space-varying vehicle dimension constraints are linearized around a reference path to pose convex optimization problems. Such constraints do not require to inflate obstacles by safety-margins and therefore maximize performance in very constrained environments. A sequential linear programming (SLP) algorithm is motivated. A linear program (LP) is solved at each SLP-iteration. The relation between LP formulation and maximum admissible traveling speeds within vehicle tire friction limits is discussed. The proposed method is evaluated in a roomy and in a tight maneuvering driving scenario, whereby a comparison to a semi-analytical clothoid-based path planner is given. Effectiveness is demonstrated particularly for very constrained environments, requiring to account for constraints and planning over the entire obstacle constellation space., Comment: - 7 pages, 13 figures, extended version of ITSC 2017 conference paper

SUMMARY Hybrid Petri nets represent a powerful modeling formalism that offers the possibility of integrating, in a natural way, continuous and discrete dynamics in a single net model. Usual control approaches for hybrid nets can be divided into discrete-time and continuous-time approaches. Continuous-time approaches are usually more precise, but can be computationally prohibitive. Discrete-time approaches are less complex, but can entail mode-mismatch errors due to fixed time discretization. This work proposes an optimization-based event-driven control approach that applies on continuous time models and where the control actions change when discrete events occur. Such an approach is computationally feasible for systems of interest in practice and avoids mode-mismatch errors. In order to handle modelling errors and exogenous disturbances, the proposed approach is implemented in a closed-loop strategy based on event-driven model predictive control. Copyright © 2013 John Wiley & Sons, Ltd.

This paper addresses the task of finding the shortest path to a target point on the boundary of an agricultural working area given a current position and heading of a ground vehicle within that area. Constraints such as admittance of turning on the spot, lane- and corridor-constraining motion as well as repressed area minimization are taken into account. We distinguish between orchard- and vineyard-like areas and agricultural fields growing, in particular, rapeseed and cereals. For the former application, dynamic programming and label-correcting algorithms are compared and, based on a coordinate transformation, a heuristic is motivated therefore. For the latter agricultural field application, we constrain the shortest path objective by allowing agricultural machinery to only use already existing tractor-lane traces and thus introduce a repressed area minimization constraint. Therefore, a customized and novel shortest path finding method is derived, before its optimality is proven. The outcome of this work is equally applicable for autonomous as well as manually driven agricultural ground vehicles.

Optimal control theory and machine learning techniques are combined to formulate and solve in closed form an optimal control formulation of online learning from supervised examples with regularization of the updates. The connections with the classical linear quadratic gaussian (LQG) optimal control problem, of which the proposed learning paradigm is a nontrivial variation as it involves random matrices, are investigated. The obtained optimal solutions are compared with the Kalman filter estimate of the parameter vector to be learned. It is shown that the proposed algorithm is less sensitive to outliers with respect to the Kalman estimate (thanks to the presence of the regularization term), thus providing smoother estimates with respect to time. The basic formulation of the proposed online learning framework refers to a discrete-time setting with a finite learning horizon and a linear model. Various extensions are investigated, including the infinite learning horizon and, via the so-called kernel trick, the case of nonlinear models.

Least Squares Support Vector Machine (LS-SVM) is a computationally efficient kernel-based regression approach which has been recently applied to nonparametric identification of Linear Parameter Varying (LPV) systems. In contrast to parametric LPV identification approaches, LS-SVM based methods obviate the need to parameterize the scheduling dependence of the LPV model coefficients in terms of a-priori specified basis functions. However, an accurate selection of the underlying model order (in terms of number of input lags, output lags and input delay) is still a critical issue in the identification of LPV systems in the LS-SVM setting. In this paper, we address this issue by extending the LS-SVM method to sparse LPV model identification, which, besides non-parametric estimation of the model coefficients, achieves datadriven model order selection via convex optimization. The main idea of the proposed method is to first estimate the coefficients of an over-parameterized LPV model through LS-SVM. The estimated coefficients are then scaled by polynomial weights, which are shrunk towards zero to enforce sparsity in the final LPV model estimate. The properties of the proposed approach are illustrated via simulation examples.

© 2016 EUCA. We introduce a new stochastic gradient algorithm, SAAGA, and investigate its employment for solving Stochastic MPC problems and multi-stage stochastic optimization programs in general. The method is particularly attractive for scenario-based formulations that involve a large number of scenarios, for which 'batch' formulations may become inefficient due to high computational costs. Benefits of the method include cheap computations per iteration and fast convergence due to the sparsity of the proposed problem decomposition. ispartof: pages:154-159 ispartof: Proc. of the 2016 European Control Conference pages:154-159 ispartof: ECC 2016 location:Aalborg, Denmark date:Jul - Jul 2016 status: published

Piecewise affine (PWA) regression is a supervised learning method which aims at estimating, from a set of training data, a PWA map approximating the relationship between a set of explanatory variables (commonly called regressors) and continuous-valued outputs. In this paper, we describe a recursive and numerically efficient PWA regression algorithm, and discuss its application to the identification of multi-input multi-output PWA dynamical models in autoregressive form and to the identification of linear parameter-varying models.

This paper investigates stochastic stabilization procedures based on quadratic and piecewise linear Lyapunov functions for discrete-time linear systems affected by multiplicative disturbances and subject to linear constraints on inputs and states. A stochastic model predictive control (SMPC) design approach is proposed to optimize closed-loop performance while enforcing constraints. Conditions for stochastic convergence and robust constraints fulfillment of the closed-loop system are enforced by solving linear matrix inequality problems off line. Performance is optimized on line using multistage stochastic optimization based on enumeration of scenarios, that amounts to solving a quadratic program subject to either quadratic or linear constraints. In the latter case, an explicit form is computable to ease the implementation of the proposed SMPC law. The approach can deal with a very general class of stochastic disturbance processes with discrete probability distribution. The effectiveness of the proposed SMPC formulation is shown on a numerical example and compared to traditional MPC schemes.

Derivative contracts require the replication of the product by means of a dynamic portfolio composed of simpler, more liquid securities. For a broad class of options encountered in financial engineering we propose a solution to the problem of finding a hedging portfolio using a discrete-time stochastic model predictive control and receding horizon optimization. By employing existing option pricing engines for estimating future option prices (possibly in an approximate way, to increase computation speed), in the absence of transaction costs the resulting stochastic optimization problem is easily solved at each trading date as a least-squares problem with as many variables as the number of traded assets and as many constraints as the number of predicted scenarios. As shown through numerical examples, the approach is particularly useful and numerically viable for exotic options where closed-form results are not available, as well as relatively long expiration dates where tree-based stochastic approaches are ...