Ground fault detection in inverter-based microgrid (IBM) systems is challenging, particularly in a real-time setting, as the fault current deviates slightly from the nominal value. This difficulty is reinforced when there are partially decoupled disturbances and modeling uncertainties. The conventional solution of installing more relays to obtain additional measurements is costly and also increases the complexity of the system. In this brief, we propose a data-assisted diagnosis scheme based on an optimization-based fault detection filter with the output current as the only measurement. Modeling the microgrid dynamics and the diagnosis filter, we formulate the filter design as a quadratic programming (QP) problem that accounts for decoupling partial disturbances, robustness to nondecoupled disturbances and modeling uncertainties by training with data, and ensuring fault sensitivity simultaneously. To ease the computational effort, we also provide an approximate but analytical solution to this QP. Additionally, we use classical statistical results to provide a thresholding mechanism that enjoys probabilistic false-alarm guarantees. Finally, we implement the IBM system with Simulink and real-time digital simulator (RTDS) to verify the effectiveness of the proposed method through simulations.

This paper proposes a nonlinear estimator for the robust reconstruction of process and sensor faults for a class of uncertain nonlinear systems. The proposed fault estimation method augments the system dynamics with an ultra-local (in time) internal state–space representation (a finite chain of integrators) of the fault vector. Next, a nonlinear state observer is designed based on the known parts of the augmented dynamics. This nonlinear filter (observer) reconstructs the fault signal as well as the states of the augmented system. We provide sufficient conditions that guarantee stability of the estimation error dynamics: firstly, asymptotic stability (i.e., exact fault estimation) in the absence of perturbations induced by the fault model mismatch (mismatch between internal ultra-local model for the fault and the actual fault dynamics), uncertainty, external disturbances, and measurement noise and, secondly, Input-to-State Stability (ISS) of the estimation error dynamics is guaranteed in the presence of these perturbations. In addition, to support performance-based estimator design, we provide Linear Matrix Inequality (LMI) conditions for L₂-gain and L₂−L_∞ induced norm and cast the synthesis of the estimator gains as a semi-definite program where the effect of model mismatch and external disturbances on the fault estimation error is minimized in the sense of L₂-gain, for an acceptable L₂−L_∞ induced norm with respect to measurement noise. The latter result facilitates a design that explicitly addresses the performance trade-off between noise sensitivity and robustness against model mismatch and external disturbances. Finally, numerical results for a benchmark system illustrate the performance of the proposed methodologies.

This paper studies the problem of fault detection and estimation (FDE) for linear time-invariant (LTI) systems with a particular focus on frequency content information of faults, possibly as multiple disjoint continuum ranges, and under both disturbances and stochastic noise. To ensure the worst-case fault sensitivity in the considered frequency ranges and mitigate the effects of disturbances and noise, an optimization framework incorporating a mixed H_{_}/H₂ performance index is developed to compute the optimal detection filter. Moreover, a thresholding rule is proposed to guarantee both the false alarm rate (FAR) and the fault detection rate (FDR). Next, shifting attention to fault estimation in specific frequency ranges, an exact reformulation of the optimal estimation filter design using the restricted H_∞ performance index is derived, which is inherently non-convex. However, focusing on finite frequency samples and fixed poles, a lower bound is established via a highly tractable quadratic programming (QP) problem. This lower bound together with an alternating optimization (AO) approach to the original estimation problem leads to a suboptimality gap for the overall estimation filter design. The effectiveness of the proposed approaches is validated through applications of a non-minimum phase hydraulic turbine system and a multi-area power system.

We establish a collection of closed-loop guarantees and propose a scalable optimization algorithm for distributionally robust model predictive control (DRMPC) applied to linear systems, convex constraints, and quadratic costs. Via standard assumptions for the terminal cost and constraint, we establish distribtionally robust long-term and stage-wise performance guarantees for the closed-loop system. We further demonstrate that a common choice of the terminal cost, i.e., via the discrete-algebraic Riccati equation, renders the origin input-to-state stable for the closed-loop system. This choice also ensures that the exact long-term performance of the closed-loop system is independent of the choice of ambiguity set for the DRMPC formulation. Thus, we establish conditions under which DRMPC does not provide a long-term performance benefit relative to stochastic MPC. To solve the DRMPC optimization problem, we propose a Newton-type algorithm that empirically achieves superlinear convergence and guarantees the feasibility of each iterate. We demonstrate the implications of the closed-loop guarantees and the scalability of the proposed algorithm via two examples. To facilitate the reproducibility of the results, we also provide open-source code to implement the proposed algorithm and generate the figures.

We study the data-driven finite-horizon linear quadratic regularization (LQR) problem reformulated as a semidefinite program (SDP). Our contribution is to propose two novel accelerated first-order methods for solving the resulting SDP. Our methods enjoy adaptive stepsize and adaptive smoothing parameters that speed up convergence and in turn, enhance scalability. Finally, we compare our accelerated first-order methods and show their benefits via numerical simulations on a benchmark LQR example.

This paper considers the problem of fault estimation in linear time-invariant systems when actuators are subject to unknown additive faults. A data-driven approach is proposed to design an inverse-system-based filter for reconstructing fault signals when the underlying fault subsystem can be either a minimum phase or non-minimum phase system. Unlike traditional two-step data-driven methods in the literature, the proposed method directly computes the filter parameters from input-output data to avoid the propagation of identification errors through an inverse operation into the fault estimates, which is the case in state-of-the-art filter designs. Furthermore, regarding out-of-sample performance of the filter, a kernel-based regularization is exploited to not only reduce the model complexity but also enable the design scheme to take advantage of available prior knowledge on the underlying system behavior. This knowledge can be incorporated into basis functions, promoting the desired solution to the optimization problem. To validate the effectiveness of the proposed method, a simulation study is conducted, demonstrating a notable reduction in estimation error compared to state-of-the-art methods.

We present a framework for learning of modeling uncertainties in Linear Time Invariant (LTI) systems to improve the predictive capacity of system models in the input-output sense. First, we propose a methodology to extend the LTI model with an uncertainty model. The proposed framework guarantees stability of the extended model. To achieve this, two semi-definite programs are provided that allow obtaining optimal uncertainty model parameters, given state and uncertainty data. Second, to obtain this data from available input-output trajectory data, we introduce a filter in which an internal model of the uncertainty is proposed. This filter is also designed via a semi-definite program with guaranteed robustness with respect to uncertainty model mismatches, disturbances, and noise. Numerical simulations are presented to illustrate the effectiveness and practicality of the proposed methodology in improving model accuracy, while guaranteeing model stability.

We study the problem of identifying a linear time-varying output map from measurements and linear time-varying system states, which are perturbed with Gaussian observation noise and process uncertainty, respectively. Employing a stochastic model as prior knowledge for the parameters of the unknown output map, we reconstruct their estimates from input/output pairs via a Bayesian approach to optimize the posterior probability density of the output map parameters. The resulting problem is a non-convex optimization, for which we propose a tractable linear matrix inequalities approximation to warm-start a first-order subsequent method. The efficacy of our algorithm is shown experimentally against classical Expectation Maximization and Dual Kalman Smoother approaches.

This article focuses on a class of distributionally robust optimization (DRO) problems where, unlike the growing body of the literature, the objective function is potentially nonlinear in the distribution. Existing methods to optimize nonlinear functions in probability space use the Frechet derivatives, which present both theoretical and computational challenges. Motivated by this, we propose an alternative notion for the derivative and corresponding smoothness based on Gateaux (G)-derivative for generic risk measures. These concepts are explained via three running risk measure examples of variance, entropic risk, and risk on finite support sets. We then propose a G-derivative based Frank–Wolfe (FW) algorithm for generic nonlinear optimization problems in probability spaces and establish its convergence under the proposed notion of smoothness in a completely norm-independent manner. We use the set-up of the FW algorithm to devise a methodology to compute a saddle point of the nonlinear DRO problem. Finally, we validate our theoretical results on two cases of the entropic and variance risk measures in the context of portfolio selection problems. In particular, we analyze their regularity conditions and “sufficient statistic”, compute the respective FW-oracle in various settings, and confirm the theoretical outcomes through numerical validation.

We study a diagnosis scheme to reliably detect the active mode of discrete-time, switched affine systems in the presence of measurement noise and asynchronous switching. The proposed scheme consists of two parts: (i) the construction of a bank of filters, and (ii) the introduction of a residual/threshold-based diagnosis rule. We develop an exact finite optimization-based framework to numerically solve an optimal bank of filters in which the contribution of measurement noise to the residual is minimized. The design problem is safely approximated through linear matrix inequalities and thus becomes tractable. We further propose a thresholding policy along with probabilistic false-alarm guarantees to estimate the active system mode in real-time. In comparison with the existing results, the guarantees improve from a polynomial dependency in the probability of false alarm to a logarithmic form. This improvement is achieved under the additional assumption of sub-Gaussianity, which is expected in many applications. The performance of the proposed approach is validated through a numerical example and an application of the building radiant system.

We introduce a distributionally robust minimium mean square error estimation model with a Wasserstein ambiguity set to recover an unknown signal from a noisy observation. The proposed model can be viewed as a zero-sum game between a statistician choosing an estimator—that is, a measurable function of the observation—and a fictitious adversary choosing a prior—that is, a pair of signal and noise distributions ranging over independent Wasserstein balls—with the goal to minimize and maximize the expected squared estimation error, respectively. We show that, if the Wasserstein balls are centered at normal distributions, then the zero-sum game admits a Nash equilibrium, by which the players’ optimal strategies are given by an affine estimator and a normal prior, respectively. We further prove that this Nash equilibrium can be computed by solving a tractable convex program. Finally, we develop a Frank–Wolfe algorithm that can solve this convex program orders of magnitude faster than state-of-the-art general-purpose solvers. We show that this algorithm enjoys a linear convergence rate and that its direction-finding subproblems can be solved in quasi-closed form.

In the past few years, online convex optimization (OCO) has received notable attention in the control literature thanks to its flexible real-time nature and powerful performance guarantees. In this article, we propose new step-size rules and OCO algorithms that simultaneously exploit gradient predictions, function predictions and dynamics, features particularly pertinent to control applications. The proposed algorithms enjoy static and dynamic regret bounds in terms of the dynamics of the reference action sequence, gradient prediction error, and function prediction error, which are generalizations of known regularity measures from the literature. We present results for both convex and strongly convex costs. We validate the performance of the proposed algorithms in a trajectory tracking case study, as well as portfolio optimization using real-world datasets.

This paper addresses the problem of robust process and sensor fault reconstruction for nonlinear systems. The proposed method augments the system dynamics with an approximated internal linear model of the combined contribution of known nonlinearities and unknown faults - leading to an approximated linear model in the augmented state. We exploit the broad modeling power of ultra-local models to characterize this internal dynamics. We use a linear filter to reconstruct the augmented state (simultaneously estimating the state of the original system and the sum of nonlinearities and faults). Having this combined estimate, we can simply subtract the analytic expression of nonlinearities from that of the corresponding estimate to reconstruct the fault vector. Because the nonlinearity does not play a role in the filter dynamics (it is only used as a static nonlinear output to estimate the fault), we can avoid standard restrictive assumptions like globally (one-sided) Lipschitz nonlinearities and/or the need for Lipschitz constants to carry out the filter design. The filter synthesis is posed as a mixed H₂/H_∞ optimization problem where the effect of disturbances and model mismatches is minimized in the H_∞ sense, for an acceptable H₂ performance with respect to measurement noise.

In this article, we investigate the problem of practical output regulation, i.e., to design a controller that brings the system output in the vicinity of a desired target value while keeping the other variables bounded. We consider uncertain systems that are possibly nonlinear and the uncertainty of their linear parts is modeled element wise through a parametric family of matrix boxes. An optimization-based design procedure is proposed that delivers a continuous-time control and estimates the maximal regulation error. We also analyze an event-triggered emulation of this controller, which can be implemented on a digital platform, along with an explicit estimate of the regulation error.

Estimating and detecting faults is crucial in ensuring safe and efficient automated systems. In the presence of disturbances, noise, or varying system dynamics, such estimation is even more challenging. To address this challenge, this letter proposes a novel filter to estimate multiple fault signals for a class of discrete-time linear parameter-varying (LPV) systems. The design of such a filter is formulated as an optimization problem and is solved recursively, while the system dynamics may vary over time. Conditions for the existence and detectability of the fault are introduced and the problem is formulated and solved using the quadratic programming framework. We further propose an approximate scheme that can be arbitrarily precise while it enjoys an analytical solution, which supports real-time implementation. The method is illustrated and validated on an automated vehicle's lateral dynamics, which is a practically relevant example for LPV systems. The results show that the estimation filter can decouple unknown disturbances and known or measurable parameter variations in the dynamics while estimating the unknown fault.

We introduce a distributionally robust maximum likelihood estimation model with a Wasserstein ambiguity set to infer the inverse covariance matrix of a p-dimensional Gaussian random vector from n independent samples. The proposed model minimizes the worst case (maximum) of Stein’s loss across all normal reference distributions within a prescribed Wasserstein distance from the normal distribution characterized by the sample mean and the sample covariance matrix. We prove that this estimation problem is equivalent to a semidefinite program that is tractable in theory but beyond the reach of general-purpose solvers for practically relevant problem dimensions p. In the absence of any prior structural information, the estimation problem has an analytical solution that is naturally interpreted as a nonlinear shrinkage estimator. Besides being invertible and well conditioned even for p > n, the new shrinkage estimator is rotation equivariant and preserves the order of the eigenvalues of the sample covariance matrix. These desirable properties are not imposed ad hoc but emerge naturally from the underlying distributionally robust optimization model. Finally, we develop a sequential quadratic approximation algorithm for efficiently solving the general estimation problem subject to conditional independence constraints typically encountered in Gaussian graphical models.

We present a learning method to learn the mapping from an input space to an action space, which is particularly suitable when the action is an optimal decision with respect to a certain unknown cost function. We use an inverse optimization approach to retrieve the cost function by introducing a new loss function and a new hypothesis class of mappings. A tractable convex reformulation of the learning problem is also presented. The method is effective for learning input-action mapping in continuous input-action space with input-output constraints, typically present in control systems. The learning approach can be effectively transformed to learn a Model Predictive Control (MPC) behaviour and a case study to mimic an MPC is presented, which is a rather computationally heavy control strategy. Simulation and experimental results show the effectiveness of the proposed approach.