arXiv subjects

Alberto Bemporad

Publications and source records attributed to Alberto Bemporad.

At least 73 records · Page 4Linked to original sources

arXiv2022

A construction-free coordinate-descent augmented-Lagrangian method for embedded linear MPC based on ARX models

This paper proposes a construction-free algorithm for solving linear MPC problems based on autoregressive with exogenous terms (ARX) input-output models. The solution algorithm relies on a coordinate-descent augmented Lagrangian (CDAL) method previously proposed by the authors, which we adapt here to exploit the special structure of ARX-based MPC. The CDAL-ARX algorithm enjoys the construction-free feature, in that it avoids explicitly constructing the quadratic programming (QP) problem associated with MPC, which would eliminate construction cost when the ARX model changes/adapts online. For example, the ARX model parameters are dependent on linear parameter-varying (LPV) scheduling signals, or recursively adapted from streaming input-output data with cheap computation cost, which make the ARX model widely used in adaptive control. Moreover, the implementation of the resulting CDAL-ARX algorithm is matrix-free and library-free, and hence amenable for deployment in industrial embedded platforms. We show the efficiency of CDAL-ARX in two numerical examples, also in comparison with MPC implementations based on other general-purpose quadratic programming solvers.

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arXiv2021

C-GLISp: Preference-Based Global Optimization under Unknown Constraints with Applications to Controller Calibration

Preference-based global optimization algorithms minimize an unknown objective function only based on whether the function is better, worse, or similar for given pairs of candidate optimization vectors. Such optimization problems arise in many real-life examples, such as finding the optimal calibration of the parameters of a control law. The calibrator can judge whether a particular combination of parameters leads to a better, worse, or similar closed-loop performance. Often, the search for the optimal parameters is also subject to unknown constraints. For example, the vector of calibration parameters must not lead to closed-loop instability. This paper extends an active preference learning algorithm introduced recently by the authors to handle unknown constraints. The proposed method, called C-GLISp, looks for an optimizer of the problem only based on preferences expressed on pairs of candidate vectors, and on whether a given vector is reported feasible and/or satisfactory. C-GLISp learns a surrogate of the underlying objective function based on the expressed preferences, and a surrogate of the probability that a sample is feasible and/or satisfactory based on whether each of the tested vectors was judged as such. The surrogate functions are used iteratively to propose a new candidate vector to test and judge. Numerical benchmarks and a semi-automated control calibration task demonstrate the effectiveness of C-GLISp, showing that it can reach near-optimal solutions within a small number of iterations.

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arXiv2021

Model Predictive Control with Environment Adaptation for Legged Locomotion

Re-planning in legged locomotion is crucial to track the desired user velocity while adapting to the terrain and rejecting external disturbances. In this work, we propose and test in experiments a real-time Nonlinear Model Predictive Control (NMPC) tailored to a legged robot for achieving dynamic locomotion on a variety of terrains. We introduce a mobility-based criterion to define an NMPC cost that enhances the locomotion of quadruped robots while maximizing leg mobility and improves adaptation to the terrain features. Our NMPC is based on the real-time iteration scheme that allows us to re-plan online at $25\,\mathrm{Hz}$ with a prediction horizon of $2$ seconds. We use the single rigid body dynamic model defined in the center of mass frame in order to increase the computational efficiency. In simulations, the NMPC is tested to traverse a set of pallets of different sizes, to walk into a V-shaped chimney,and to locomote over rough terrain. In real experiments, we demonstrate the effectiveness of our NMPC with the mobility feature that allowed IIT's $87\, \mathrm{kg}$ quadruped robot HyQ to achieve an omni-directional walk on flat terrain, to traverse a static pallet, and to adapt to a repositioned pallet during a walk.

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arXiv2021

Data-driven synthesis of Robust Invariant Sets and Controllers

This paper presents a method to identify an uncertain linear time-invariant (LTI) prediction model for tube-based Robust Model Predictive Control (RMPC). The uncertain model is determined from a given state-input dataset by formulating and solving a Semidefinite Programming problem (SDP), that also determines a static linear feedback gain and corresponding invariant sets satisfying the inclusions required to guarantee recursive feasibility and stability of the RMPC scheme, while minimizing an identification criterion. As demonstrated through an example, the proposed concurrent approach provides less conservative invariant sets than a sequential approach.

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arXiv2021

Computation of Input Disturbance Sets for Constrained Output Reachability

Linear models with additive unknown-but-bounded input disturbances are extensively used to model uncertainty in robust control systems design. Typically, the disturbance set is either assumed to be known a priori or estimated from data through set-membership identification. However, the problem of computing a suitable input disturbance set in case the set of possible output values is assigned a priori has received relatively little attention. This problem arises in many contexts, such as in supervisory control, actuator design, decentralized control, and others. In this paper, we propose a method to compute input disturbance sets (and the corresponding set of states) such that the resulting set of outputs matches as closely as possible a given set of outputs, while additionally satisfying strict (inner or outer) inclusion constraints. We formulate the problem as an optimization problem by relying on the concept of robust invariance. The effectiveness of the approach is demonstrated in numerical examples that illustrate how to solve safe reference set and input-constraint set computation problems.

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arXiv2021

Constrained Controller and Observer Design by Inverse Optimality

Model Predictive Control (MPC) is often tuned by trial and error. When a baseline linear controller exists that is already well tuned in the absence of constraints and MPC is introduced to enforce them, one would like to avoid altering the original linear feedback law whenever they are not active. We formulate this problem as a controller matching similar to [1]-[3], which we extend to a more general framework. We prove that a positive-definite stage cost matrix yielding this matching property can be computed for all stabilizing linear controllers. Additionally, we prove that the constrained estimation problem can also be solved similarly, by matching a linear observer with a Moving Horizon Estimator (MHE). Finally, we discuss various aspects of the practical implementation of the proposed technique in some examples.

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arXiv2021

A Dual Active-Set Solver for Embedded Quadratic Programming Using Recursive LDL' Updates

In this paper we present a dual active-set solver for quadratic programming which has properties suitable for use in embedded model predictive control applications. In particular, the solver is efficient, can easily be warm-started, and is simple to code. Moreover, the exact worst-case computational complexity of the solver can be determined offline and, by using outer proximal-point iterations, ill-conditioned problems can be handled in a robust manner.

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arXiv2021

Massively parallelizable proximal algorithms for large-scale stochastic optimal control problems

Scenario-based stochastic optimal control problems suffer from the curse of dimensionality as they can easily grow to six and seven figure sizes. First-order methods are suitable as they can deal with such large-scale problems, but may fail to achieve accurate solutions within a reasonable number of iterations. To achieve solutions of higher accuracy and high speed, in this paper we propose two proximal quasi-Newtonian limited-memory algorithms - MinFBE applied to the dual problem and the Newton-type alternating minimization algorithm (NAMA) - which can be massively parallelized on lockstep hardware such as graphics processing units (GPUs). We demonstrate the performance of these methods, in terms of convergence speed and parallelizability, on large-scale problems involving millions of variables.

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arXiv2021

Preference-based MPC calibration

Automating the calibration of the parameters of a control policy by means of global optimization requires quantifying a closed-loop performance function. As this can be impractical in many situations, in this paper we suggest a semi-automated calibration approach that requires instead a human calibrator to express a preference on whether a certain control policy is "better" than another one, therefore eliminating the need of an explicit performance index. In particular, we focus our attention on semi-automated calibration of Model Predictive Controllers (MPCs), for which we attempt computing the set of best calibration parameters by employing the recently-developed active preference-based optimization algorithm GLISp. Based on the preferences expressed by the human operator, GLISp learns a surrogate of the underlying closed-loop performance index that the calibrator (unconsciously) uses and proposes, iteratively, a new set of calibration parameters to him or her for testing and for comparison against previous experimental results. The resulting semi-automated calibration procedure is tested on two case studies, showing the capabilities of the approach in achieving near-optimal performance within a limited number of experiments.

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arXiv2021

An efficient bounded-variable nonlinear least-squares algorithm for embedded MPC

This paper presents a new approach to solve linear and nonlinear model predictive control (MPC) problems that requires small memory footprint and throughput and is particularly suitable when the model and/or controller parameters change at runtime. Typically MPC requires two phases: 1) construct an optimization problem based on the given MPC parameters (prediction model, tuning weights, prediction horizon, and constraints), which results in a quadratic or nonlinear programming problem, and then 2) call an optimization algorithm to solve the resulting problem. In the proposed approach the problem construction step is systematically eliminated, as in the optimization algorithm problem matrices are expressed in terms of abstract functions of the MPC parameters. We present a unifying algorithmic framework based on active-set methods with bounded variables that can cope with linear, nonlinear, and adaptive MPC variants based on a broad class of prediction models and a sum-of-squares cost function. The theoretical and numerical results demonstrate the potential, applicability, and efficiency of the proposed framework for practical real-time embedded MPC.

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arXiv2021

A machine-learning approach to synthesize virtual sensors for parameter-varying systems

This paper introduces a novel model-free approach to synthesize virtual sensors for the estimation of dynamical quantities that are unmeasurable at runtime but are available for design purposes on test benches. After collecting a dataset of measurements of such quantities, together with other variables that are also available during on-line operations, the virtual sensor is obtained using machine learning techniques by training a predictor whose inputs are the measured variables and the features extracted by a bank of linear observers fed with the same measures. The approach is applicable to infer the value of quantities such as physical states and other time-varying parameters that affect the dynamics of the system. The proposed virtual sensor architecture - whose structure can be related to the Multiple Model Adaptive Estimation framework - is conceived to keep computational and memory requirements as low as possible, so that it can be efficiently implemented in embedded hardware platforms. The effectiveness of the approach is shown in different numerical examples, involving the estimation of the scheduling parameter of a nonlinear parameter-varying system, the reconstruction of the mode of a switching linear system, and the estimation of the state of charge (SoC) of a lithium-ion battery.

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arXiv2021

Piecewise linear regression and classification

This paper proposes a method for solving multivariate regression and classification problems using piecewise linear predictors over a polyhedral partition of the feature space. The resulting algorithm that we call PARC (Piecewise Affine Regression and Classification) alternates between (i) solving ridge regression problems for numeric targets, softmax regression problems for categorical targets, and either softmax regression or cluster centroid computation for piecewise linear separation, and (ii) assigning the training points to different clusters on the basis of a criterion that balances prediction accuracy and piecewise-linear separability. We prove that PARC is a block-coordinate descent algorithm that optimizes a suitably constructed objective function, and that it converges in a finite number of steps to a local minimum of that function. The accuracy of the algorithm is extensively tested numerically on synthetic and real-world datasets, showing that the approach provides an extension of linear regression/classification that is particularly useful when the obtained predictor is used as part of an optimization model. A Python implementation of the algorithm described in this paper is available at http://cse.lab.imtlucca.it/~bemporad/parc .

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arXiv2021

Exact and Heuristic Methods with Warm-start for Embedded Mixed-Integer Quadratic Programming Based on Accelerated Dual Gradient Projection

Small-scale Mixed-Integer Quadratic Programming (MIQP) problems often arise in embedded control and estimation applications. Driven by the need for algorithmic simplicity to target computing platforms with limited memory and computing resources, this paper proposes a few approaches to solving MIQPs, either to optimality or suboptimally. We specialize an existing Accelerated Dual Gradient Projection (GPAD) algorithm to effectively solve the Quadratic Programming (QP) relaxation that arise during Branch and Bound (B&B) and propose a generic framework to warm-start the binary variables which reduces the number of QP relaxations. Moreover, in order to find an integer feasible combination of the binary variables upfront, two heuristic approaches are presented: ($i$) without using B&B, and ($ii$) using B&B with a significantly reduced number of QP relaxations. Both heuristic approaches return an integer feasible solution that may be suboptimal but involve a much reduced computation effort. Such a feasible solution can be either implemented directly or used to set an initial upper bound on the optimal cost in B&B. Through different hybrid control and estimation examples involving binary decision variables, we show that the performance of the proposed methods, although very simple to code, is comparable to that of state-of-the-art MIQP solvers.

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arXiv2021

Efficient Calibration of Embedded MPC

Model Predictive Control (MPC) is a powerful and flexible design tool of high-performance controllers for physical systems in the presence of input and output constraints. A challenge for the practitioner applying MPC is the need of tuning a large number of parameters such as prediction and control horizons, weight matrices of the MPC cost function, and observer gains, according to different trade-offs. The MPC design task is even more involved when the control law has to be deployed to an embedded hardware unit endowed with limited computational resources. In this case, real-time system requirements limit the complexity of the applicable MPC configuration, engendering additional design tradeoffs and requiring to tune further parameters, such as the sampling time and the tolerances used in the on-line numerical solver. To take into account closed-loop performance and real-time requirements, in this paper we tackle the embedded MPC design problem using a global, data-driven, optimization approach We showcase the potential of this approach by tuning an MPC controller on two hardware platforms characterized by largely different computational capabilities.

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arXiv2020

Reduction of the Number of Variables in Parametric Constrained Least-Squares Problems

For linearly constrained least-squares problems that depend on a vector of parameters, this paper proposes techniques for reducing the number of involved optimization variables. After first eliminating equality constraints in a numerically robust way by QR factorization, we propose a technique based on singular value decomposition (SVD) and unsupervised learning, that we call $K$-SVD, and neural classifiers to automatically partition the set of parameter vectors in $K$ nonlinear regions in which the original problem is approximated by using a smaller set of variables. For the special case of parametric constrained least-squares problems that arise from model predictive control (MPC) formulations, we propose a novel and very efficient QR factorization method for equality constraint elimination. Together with SVD or $K$-SVD, the method provides a numerically robust alternative to standard condensing and move blocking, and to other complexity reduction methods for MPC based on basis functions. We show the good performance of the proposed techniques in numerical tests and in a linearized MPC problem of a nonlinear benchmark process.

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arXiv2020

Safe Reinforcement Learning via Projection on a Safe Set: How to Achieve Optimality?

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.

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arXiv2020

OSQP: An Operator Splitting Solver for Quadratic Programs

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.

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arXiv2020

Global optimization via inverse distance weighting and radial basis functions

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 a combination of 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 \url{http://cse.lab.imtlucca.it/~bemporad/glis}.

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