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Recently there has been an increasing interest in primal-dual methods for model predictive control (MPC), which require minimizing the (augmented) Lagrangian at each iteration. We propose a novel first order primal-dual method, termed \emph{proportional-integral projected gradient method}, for MPC where the underlying finite horizon optimal control problem has both state and input constraints. Instead of minimizing the (augmented) Lagrangian, each iteration of our method only computes a single projection onto the state and input constraint set. Our method ensures that, along a sequence of averaged iterates, both the distance to optimum and the constraint violation converge to zero at a rate of \(O(1/k)\) if the objective function is convex, where \(k\) is the iteration number. If the objective function is strongly convex, this rate can be improved to \(O(1/k^2)\) for the distance to optimum and \(O(1/k^3)\) for the constraint violation. We compare our method against existing methods via a trajectory-planning example with convexified keep-out-zone constraints., Comment: Julia code available at: https://github.com/purnanandelango/pi-projgrad-demo

We present a method to compute the stochastic reachability safety probabilities for high-dimensional stochastic dynamical systems. Our approach takes advantage of a nonparametric learning technique known as conditional distribution embeddings to model the stochastic kernel using a data-driven approach. By embedding the dynamics and uncertainty within a reproducing kernel Hilbert space, it becomes possible to compute the safety probabilities for stochastic reachability problems as simple matrix operations and inner products. We employ a convergent approximation technique, random Fourier features, in order to alleviate the increased computational requirements for high-dimensional systems. This technique avoids the curse of dimensionality, and enables the computation of safety probabilities for high-dimensional systems without prior knowledge of the structure of the dynamics or uncertainty. We validate this approach on a double integrator system, and demonstrate its capabilities on a million-dimensional, nonlinear, non-Gaussian, repeated planar quadrotor system.

Classical linear quadratic (LQ) control centers around linear time-invariant (LTI) systems, where the control-state pairs introduce a quadratic cost with time-invariant parameters. Recent advancement in online optimization and control has provided novel tools to study LQ problems that are robust to time-varying cost parameters. Inspired by this line of research, we study the distributed online LQ problem for identical LTI systems. Consider a multi-agent network where each agent is modeled as an LTI system. The LTI systems are associated with decoupled, time-varying quadratic costs that are revealed sequentially. The goal of the network is to make the control sequence of all agents competitive to that of the best centralized policy in hindsight, captured by the notion of regret. We develop a distributed variant of the online LQ algorithm, which runs distributed online gradient descent with a projection to a semi-definite programming (SDP) to generate controllers. We establish a regret bound scaling as the square root of the finite time-horizon, implying that agents reach consensus as time grows. We further provide numerical experiments verifying our theoretical result.

In this paper, we discuss the methodology of generalizing the optimal control law from learned component tasks to unlearned composite tasks on Multi-Agent Systems (MASs), by using the linearity composition principle of linearly solvable optimal control (LSOC) problems. The proposed approach achieves both the compositionality and optimality of control actions simultaneously within the cooperative MAS framework in both discrete- and continuous-time in a sample-efficient manner, which reduces the burden of re-computation of the optimal control solutions for the new task on the MASs. We investigate the application of the proposed approach on the MAS with coordination between agents. The experiments show feasible results in investigated scenarios, including both discrete and continuous dynamical systems for task generalization without resampling., Accepted to the 2021 American Control Conference (ACC)

In this paper, we aim to explore optimal regional trajectory tracking control problems of the anomalous subdiffusion processes governed by time-fractional diffusion systems under the Neumann boundary conditions. Using eigenvalue theory of the system operator and the semigroup theory, we explore the existence and some estimates of the mild solution to the considered system. An approach on finding solution to the optimal problem that minimizes the regional trajectory tracking error and the corresponding control cost over a finite space and time domain is then explored via the Hilbert uniqueness method (HUM). The obtained results not only can be directly used to investigate the systems that are not controllable on the whole domain, but also yield an explicit expression of the control signal in terms of the desired trajectory. Most importantly, it is worth noting that our results in this paper are still novel even for the special case when the order of fractional derivative is equal to one. Finally, we provide a numerical example to illustrate our theoretical results., Comment: 6 pages, 5figures, Accepted to appear in the 2021 Online American Control Conference (ACC2021)

Security related questions for Cyber Physical Systems (CPS) have attracted much research attention in searching for novel methods for attack-resilient control and/or estimation. Specifically, false data injection attacks (FDIAs) have been shown to be capable of bypassing bad data detection (BDD), while arbitrarily compromising the integrity of state estimators and robust controller even with very sparse measurements corruption. Moreover, based on the inherent sparsity of pragmatic attack signals, $\ell_1$-minimization scheme has been used extensively to improve the design of attack-resilient estimators. For this, the theoretical maximum for the percentage of compromised nodes that can be accommodated has been shown to be $50\%$. In order to guarantee correct state recoveries for larger percentage of attacked nodes, researchers have begun to incorporate prior information into the underlying resilient observer design framework. For the most pragmatic cases, this prior information is often obtained through some data-driven machine learning process. Existing results have shown strong positive correlation between the tolerated attack percentages and the precision of the prior information. In this paper, we present a pruning method to improve the precision of the prior information, given corresponding stochastic uncertainty characteristics of the underlying machine learning model. Then a weighted $\ell_1$-minimization is proposed based on the pruned prior. The theoretical and simulation results show that the pruning method significantly improves the observer performance for much larger attack percentages, even when moderately accurate machine learning model used., 6

Motivated by applications arising from sensor networks and machine learning, we consider the problem of minimizing a finite sum of nondifferentiable convex functions where each component function is associated with an agent and a hard-to-project constraint set. Among well-known avenues to address finite sum problems is the class of incremental gradient (IG) methods where a single component function is selected at each iteration in a cyclic or randomized manner. When the problem is constrained, the existing IG schemes (including projected IG, proximal IAG, and SAGA) require a projection step onto the feasible set at each iteration. Consequently, the performance of these schemes is afflicted with costly projections when the problem includes: (1) nonlinear constraints, or (2) a large number of linear constraints. Our focus in this paper lies in addressing both of these challenges. We develop an algorithm called averaged iteratively regularized incremental gradient (aIR-IG) that does not involve any hard-to-project computation. Under mild assumptions, we derive non-asymptotic rates of convergence for both suboptimality and infeasibility metrics. Numerically, we show that the proposed scheme outperforms the standard projected IG methods on distributed soft-margin support vector machine problems.

The tuned liquid dampers (TLD) technology is a feasible and cost-effective seismic design. In order to improve its efficiency it is fundamental to find accurate models describing their dynamic. A TLD system can be modeled through the Housner model and its parameters can be estimated by solving a nonlinear state estimation problem. We propose a robust extended Kalman filter which alleviates the model discretization and the fact that the noise process is not known. We test the effectiveness of the proposed approach by using some experimental data corresponding to two classical seismic waves, namely the El Centro wave and the Hachinohe wave.

Selective laser melting (SLM) is one of emerging processes for effective metal additive manufacturing. Due to complex heat exchange and material phase changes, it is challenging to accurately model the SLM dynamics and design robust control of SLM process. In this paper, we first present a data-driven Gaussian process based dynamic model for SLM process and then design a model predictive control to regulate the melt pool size. Physical and process constraints are considered in the controller design. The learning model and control design are tested and validated with high-fidelity finite element simulation. The comparison results with other control design demonstrate the efficacy of the control design.

Recent trends express the impact of prosumers and small energy resources and storages in distribution systems, due to the increasing uptake of renewable resources. This research studies the effect of coordination of distributed resources with the utility grid and the role of prosumers in the operation of renewable microgrids. We formulated this problem as a social welfare maximization problem followed by employing the dual decomposition method to decompose it into sub-problems of the microgrid, distributed generators, prosumers, and consumers. Moreover, the corresponding power balance mechanism via price adjustment can be viewed as a Walrasian tatonnement process. Specifically, prosumers and consumers compete to adjust their energy exchange with other agents to maximize their profit gained by renewable emission reduction benefits while minimizing the associated cost of energy. To this end, we have adopted a peer-to-peer energy trading mechanism based on continuous double auction that can be viewed as a multi-parametric quadratic problem. Finally, we proposed a distributed adaptive algorithm that determines strategies as well as payment and assignment rules. The numerical result suggests that the proposed method can incentivize peer-to-peer energy trading while improving the cost fairness problem and the peak-to-average ratio., Comment: IEEE International Conference, ICEE2021

This paper investigates the impact of the changes in the demand of power systems on the quality of the solution procured by the convex relaxation methods for the AC optimal power flow (ACOPF) problem. This investigation needs various measures to evaluate the tightness of the solution procured by the convex relaxation approaches. Therefore, three tightness measures are leveraged to illustrate the performance of convex relaxation methods under different demand scenarios. The main issue of convex relaxation methods is recovering an optimal solution which is not necessarily feasible for the original non-convex problem in networks with cycles. Thus, a cycle measure is introduced to evaluate the performance of relaxation schemes. The presented case study investigates the merit of using various tightness measures to evaluate the performance of various relaxation methods under different circumstances., 6 Pages, 16 Figures, This paper is accepted for the 52nd North American Power Symposium

Learning-based control algorithms require data collection with abundant supervision for training. Safe exploration algorithms ensure the safety of this data collection process even when only partial knowledge is available. We present a new approach for optimal motion planning with safe exploration that integrates chance-constrained stochastic optimal control with dynamics learning and feedback control. We derive an iterative convex optimization algorithm that solves an \underline{Info}rmation-cost \underline{S}tochastic \underline{N}onlinear \underline{O}ptimal \underline{C}ontrol problem (Info-SNOC). The optimization objective encodes control cost for performance and exploration cost for learning, and the safety is incorporated as distributionally robust chance constraints. The dynamics are predicted from a robust regression model that is learned from data. The Info-SNOC algorithm is used to compute a sub-optimal pool of safe motion plans that aid in exploration for learning unknown residual dynamics under safety constraints. A stable feedback controller is used to execute the motion plan and collect data for model learning. We prove the safety of rollout from our exploration method and reduction in uncertainty over epochs, thereby guaranteeing the consistency of our learning method. We validate the effectiveness of Info-SNOC by designing and implementing a pool of safe trajectories for a planar robot. We demonstrate that our approach has higher success rate in ensuring safety when compared to a deterministic trajectory optimization approach., Accepted IEEE Robotics and Automation Letters 2020

Chance constrained program where one seeks to minimize an objective over decisions which satisfy randomly disturbed constraints with a given probability is computationally intractable. This paper proposes an approximate approach to address chance constrained program. Firstly, a single layer neural-network is used to approximate the function from decision domain to violation probability domain. The algorithm for updating parameters in single layer neural-network adopts sequential extreme learning machine. Based on the neural violation probability approximate model, a randomized algorithm is then proposed to approach the optimizer in the probabilistic feasible domain of decision. In the randomized algorithm, samples are extracted from decision domain uniformly at first. Then, violation probabilities of all samples are calculated according to neural violation probability approximate model. The ones with violation probability higher than the required level are discarded. The minimizer in the remained feasible decision samples is used to update sampling policy. The policy converges to the optimal feasible decision. Numerical simulations are implemented to validate the proposed method for non-convex problems comparing with scenario approach and parallel randomized algorithm. The result shows that proposed method have improved performance., arXiv admin note: substantial text overlap with arXiv:1911.00192

Intentional controlled islanding (ICI) is a final resort for preventing a cascading failure and catastrophic power system blackouts. This paper proposes a controlled islanding algorithm that uses spectral clustering over multi-layer graphs to find a suitable islanding solution. The multi-criteria objective function used in this controlled islanding algorithm involves the correlation coefficients between bus frequency components and minimal active and reactive power flow disruption. Similar to the previous studies, the algorithm is applied in two stages. In the first stage, groups of coherent buses are identified with the help of modularity clustering using correlation coefficients between bus frequency components. In the second stage, the ICI solution with minimum active and reactive power flow disruption and satisfying bus coherency is determined by grouping all nodes using spectral clustering on the multi-layer graph. Simulation studies on the IEEE 39-bus test system demonstrate the effectiveness of the method in determining an islanding solution in real time while addressing the generator coherency problem., Accepted in Journal of Modern Power System and Clean Energy on 28 Apr 2019

This paper proposes an adaptive primal-dual dynamics for distributed optimization in multi-agent systems. The proposed dynamics incorporates an adaptive synchronization law that reinforces the interconnection strength between the coupled agents. By strengthening the synchronization between the primal variables of the coupled agents, the given law accelerates the convergence of the proposed dynamics to the saddle-point solution. The resulting dynamics is represented as a feedback-interconnected networked system that proves to be passive. The passivity properties of the proposed dynamics are exploited along with the LaSalle's invariance principle for hybrid systems, to establish asymptotic convergence and stability of the saddle-point solution. Further, the primal dynamics is analyzed for the rate of convergence and stronger convergence bounds are established, it is proved that the primal dynamics achieve accelerated convergence under the adaptive synchronization. The robustness of the proposed dynamics is quantified using $L_{2}$ -gain analysis and the correlation between the rate of convergence and robustness of the proposed dynamics are presented. The effectiveness of the proposed dynamics is demonstrated by applying it to solve distributed least squares and distributed support vector machines problems.

In this work, a novel optimal scheduling approach is proposed for isolated microgrids (MGs) with renewable generations by incorporating demand response of electric vehicles (EVs). First, a bi-level programming-based MG scheduling model is proposed under real-time pricing environments, where the upper- and lower- levels seek to minimize the MG net operating cost and the EV charging cost. Second, a hybrid solution algorithm called JAYA-interior point method is put forward to solve the model. And finally, the simulation results demonstrate that incorporating demand response of electric vehicles is able to guide EV users to actively participate in MG scheduling and achieve the peak load shaving, which offers a fundamental way to balance the interests between MG and EV users., Comment: Accepted by IEEE Access

This paper proposes two novel nonlinear attitude filters evolved directly on the Special Orthogonal Group SO(3) able to ensure prescribed measures of transient and steady-state performance. The tracking performance of the normalized Euclidean distance of attitude error is trapped to initially start within a large set and converge systematically and asymptotically to the origin from almost any initial condition. The convergence rate is guaranteed to be less than the prescribed value and the steady-state error does not exceed a predefined small value. The first filter uses a set of vectorial measurements with the need for attitude reconstruction. The second filter instead uses only a rate gyroscope measurement and two or more vectorial measurements. These filters provide good attitude estimates with superior convergence properties and can be applied to measurements obtained from low cost inertial measurement units (IMUs). Simulation results illustrate the robustness and effectiveness of the proposed attitude filters with guaranteed performance considering high level of uncertainty in angular velocity along with body-frame vector measurements. Keywords: Attitude, estimate, estimator, observer, filter, nonlinear deterministic attitude filter, special orthogonal group, Euler angles, angle-axis, Rodrigues vector, mapping, parameterization, prescribed performance, representation, robust, Multiplicative Extended Kalman Filter, KF, EKF, MEKF, asymptotic stability, almost global asymptotic, noise, rotational matrix, identity, origin, orientation, body frame, inertial frame, rigid body, three dimensional, 3D, space, micro electromechanical systems, sensor, MEMS, roll, pitch, yaw, UAVs, QUAV, SVD, fixed, moving, vehicles, robot, robotic system, spacecraft, submarine, underwater vehicle, passive complementary filter, explicit complementary filter, autonomous, comparative study, SO(3)., 2018, IEEE Access

The traditional demand-responsive paratransit system plays an important role in connecting people to health care, particularly those who are carless, low-income, senior, underinsured/uninsured, or who have a disability. However, the existing paratransit system usually has low service quality due to long waiting times, low operation frequencies, and high costs. In order to improve the service quality, we propose to design a new demand-responsive paratransit system that offers public, Uber-like options for non-emergency medical transportation. Mixed integer programming models are thus developed to optimize the system operations with the objectives of minimizing user waiting times for riders as well as operating costs for operators. The results produced in this paper will assist local departments of transportation and transit agencies as they consider operational strategies to meet non-emergency medical transport needs.

In this paper, we consider an unconstrained distributed optimization problem over a network of agents, in which some agents are adversarial. We solve the problem via gradient-based distributed optimization algorithm and characterize the effect of the adversarial agents on the convergence of the algorithm to the optimal solution. The attack model considered is such that agents locally perturb their iterates before broadcasting it to neighbors; and we analyze the case in which the adversarial agents cooperate in perturbing their estimates and the case where each adversarial agent acts independently. Based on the attack model adopted in the paper, we show that the solution converges to the neighborhood of the optimal solution and depends on the magnitude of the attack (perturbation) term. The analyses presented establishes conditions under which the malicious agents have enough information to obstruct convergence to the optimal solution by the non-adversarial agents.

Data Assimilation (DA) is a computational tool that uses value from the model and the real measurement to arrive to an optimally acceptable value. Rather, this technique relies on the idea of Kalman gain. We point out that DA has two different perspectives based on the type of problem. In this paper, we look into two problem types: one that does not rely on the initial condition, and the other that is initial condition dependent. Data Assimilation is demonstrated on two examples: runoff monitoring and forecasting in the city of Dhaka (initial condition independent) and convection in the atmosphere (initial condition dependent). We show that standard DA works well for problems with no initial condition dependence and piecewise DA is to be utilized when the problem has initial condition dependence. In the first example, we exploited standard Data Assimilation to arrive at values that are more realistic than the ones from the model and the observations. The second example is where we devised a method to find the dynamics of the system in a piecewise manner in Data Assimilation framework and noticed that the data assimilated dynamics (even due to noisy initial condition) is in good agreement with the true dynamics for a reasonable extent of time in future., 5 pages

Convex regression (CR) is the problem of fitting a convex function to a finite number of noisy observations of an underlying convex function. CR is important in many domains and one of its workhorses is the non-parametric least square estimator (LSE). Currently, LSE delivers only non-smooth non-strongly convex function estimates. In this paper, leveraging recent results in convex interpolation, we generalize LSE to smooth strongly convex regression problems. The resulting algorithm relies on a convex quadratically constrained quadratic program. We also propose a parallel implementation, which leverages ADMM, that lessens the overall computational complexity to a tight $O(n^2)$ for $n$ observations. Numerical results support our findings., Comment: 6 pages, 3 figures

We address the problem of scheduling water resources in a power system via approximate dynamic programming. To this goal, we model a finite horizon economic dispatch problem with convex stage cost and affine dynamics, and consider a quadratic approximation of the value functions. Evaluating the achieved policy entails solving a quadratic program at each time step, while value function fitting can be cast as a semidefinite program. We test our proposed algorithm on a simplified version of the Uruguayan power system, achieving a four percent cost reduction with respect to the myopic policy. Agencia Nacional de Investigación e Innovación

Finding optimal parameter configurations for tunable GPU kernels is a non-trivial exercise for large search spaces, even when automated. This poses an optimization task on a non-convex search space, using an expensive to evaluate function with unknown derivative. These characteristics make a good candidate for Bayesian Optimization, which has not been applied to this problem before. However, the application of Bayesian Optimization to this problem is challenging. We demonstrate how to deal with the rough, discrete, constrained search spaces, containing invalid configurations. We introduce a novel contextual variance exploration factor, as well as new acquisition functions with improved scalability, combined with an informed acquisition function selection mechanism. By comparing the performance of our Bayesian Optimization implementation on various test cases to the existing search strategies in Kernel Tuner, as well as other Bayesian Optimization implementations, we demonstrate that our search strategies generalize well and consistently outperform other search strategies by a wide margin., In context of the 2021 International Workshop on Performance Modeling, Benchmarking and Simulation of High Performance Computer Systems (PMBS) at SuperComputing 2021. Pending publication in IEEE Computer Society Technical Consortium on High Performance Computing (TCHPC)

When the shunt elements in the Pi circuit line model are assumed zero, it has been proved that branch flow models are equivalent to bus injection models and that the second-order cone relaxation of optimal power flow problems on a radial network is exact under certain conditions. In this note we propose a branch flow model that includes nonzero line shunts and prove that the equivalence and the exactness of relaxation continue to hold under essentially the same conditions as for zero shunt elements., Comment: 6 pages, 1 figure

Distributed algorithms enable private Optimal Power Flow (OPF) computations by avoiding the need in sharing sensitive information localized in algorithms sub-problems. However, adversaries can still infer this information from the coordination signals exchanged across iterations. This paper seeks formal privacy guarantees for distributed OPF computations and provides differentially private algorithms for OPF computations based on the consensus Alternating Direction Method of Multipliers (ADMM). The proposed algorithms attain differential privacy by introducing static and dynamic random perturbations of OPF sub-problem solutions at each iteration. These perturbations are Laplacian and designed to prevent the inference of sensitive information, as well as to provide theoretical privacy guarantees for ADMM subproblems. Using a standard IEEE 118-node test case, the paper explores the fundamental trade-offs among privacy, algorithmic.convergence, and optimality losses.

This paper presents a new deep learning-based framework for robust nonlinear estimation and control using the concept of a Neural Contraction Metric (NCM). The NCM uses a deep long short-term memory recurrent neural network for a global approximation of an optimal contraction metric, the existence of which is a necessary and sufficient condition for exponential stability of nonlinear systems. The optimality stems from the fact that the contraction metrics sampled offline are the solutions of a convex optimization problem to minimize an upper bound of the steady-state Euclidean distance between perturbed and unperturbed system trajectories. We demonstrate how to exploit NCMs to design an online optimal estimator and controller for nonlinear systems with bounded disturbances utilizing their duality. The performance of our framework is illustrated through Lorenz oscillator state estimation and spacecraft optimal motion planning problems., IEEE Control Systems Letters (L-CSS). Preprint version, accepted June 2020

This paper proposes a hierarchical adaptive sampling scheme for passivity characterization of large-scale linear lumped macromodels. Here, large-scale is intended both in terms of dynamic order and especially number of input/output ports. Standard passivity characterization approaches based on spectral properties of associated Hamiltonian matrices are either inefficient or non-applicable for large-scale models, due to an excessive computational cost. This paper builds on existing adaptive sampling methods and proposes a hybrid multi-stage algorithm that is able to detect the passivity violations with limited computing resources. Results from extensive testing demonstrate a major reduction in computational requirements with respect to competing approaches., Submitted to the IEEE Transactions on Components, Packaging and Manufacturing Technology

We consider the problem of convergence to a saddle point of a concave-convex function via gradient dynamics. Since first introduced by Arrow, Hurwicz and Uzawa in [1] such dynamics have been extensively used in diverse areas, there are, however, features that render their analysis non trivial. These include the lack of convergence guarantees when the function considered is not strictly concave-convex and also the non-smoothness of subgradient dynamics. Our aim in this two part paper is to provide an explicit characterization to the asymptotic behaviour of general gradient and subgradient dynamics applied to a general concave-convex function. We show that despite the nonlinearity and non-smoothness of these dynamics their $\omega$-limit set is comprised of trajectories that solve only explicit linear ODEs that are characterized within the paper. More precisely, in Part I an exact characterization is provided to the asymptotic behaviour of unconstrained gradient dynamics. We also show that when convergence to a saddle point is not guaranteed then the system behaviour can be problematic, with arbitrarily small noise leading to an unbounded variance. In Part II we consider a general class of subgradient dynamics that restrict trajectories in an arbitrary convex domain, and show that when an equilibrium point exists their limiting trajectories are solutions of subgradient dynamics on only affine subspaces. The latter is a smooth class of dynamics with an asymptotic behaviour exactly characterized in Part I, as solutions to explicit linear ODEs. These results are used to formulate corresponding convergence criteria and are demonstrated with several examples and applications presented in Part II.

In this paper, we consider the problem of distributed consensus optimization over multi-agent networks with directed network topology. Assuming each agent has a local cost function that is smooth and strongly convex, the global objective is to minimize the average of all the local cost functions. To solve the problem, we introduce a robust gradient tracking method (R-Push-Pull) adapted from the recently proposed Push-Pull/AB algorithm [1], [2]. R-Push-Pull inherits the advantages of Push-Pull and enjoys linear convergence to the optimal solution with exact communication. Under noisy information exchange, R-Push-Pull is more robust than the existing gradient tracking based algorithms; the solutions obtained by each agent reach a neighborhood of the optimum in expectation exponentially fast under a constant stepsize policy. We provide a numerical example that demonstrate the effectiveness of R-Push-Pull.

This paper presents a novel data-driven, direct filtering approach for unknown linear time-invariant systems affected by unknown-but-bounded measurement noise. The proposed technique combines independent multistep prediction models, identified resorting to the Set Membership framework, to refine a set that is guaranteed to contain the true system output. The filtered output is then computed as the central value in such a set. By doing so, the method achieves an accurate output filtering and provides tight and minimal error bounds with respect to the true system output. To attain these results, the online solution of linear programs is required. A modified filtering approach with lower online computational cost is also presented, obtained by moving the solution of the optimization problems to an offline preliminary phase, at the cost of larger accuracy bounds. The performance of the proposed approaches are evaluated and compared with those of standard model-based filtering techniques in a numerical example.

In this paper, zero-sum mean-field type games (ZSMFTG) with linear dynamics and quadratic utility are studied under infinite-horizon discounted utility function. ZSMFTG are a class of games in which two decision makers whose utilities sum to zero, compete to influence a large population of agents. In particular, the case in which the transition and utility functions depend on the state, the action of the controllers, and the mean of the state and the actions, is investigated. The game is analyzed and explicit expressions for the Nash equilibrium strategies are derived. Moreover, two policy optimization methods that rely on policy gradient are proposed for both model-based and sample-based frameworks. In the first case, the gradients are computed exactly using the model whereas they are estimated using Monte-Carlo simulations in the second case. Numerical experiments show the convergence of the two players' controls as well as the utility function when the two algorithms are used in different scenarios., Comment: arXiv admin note: text overlap with arXiv:2009.00578

The present work deals with quantitative two-phase reach-avoid problems on nonlinear control systems. This class of optimal control problem requires the plant's state to visit two (rather than one) target sets in succession while minimizing a prescribed cost functional. As we illustrate, the naive approach, which subdivides the problem into the two evident classical reach-avoid tasks, usually does not result in an optimal solution. In contrast, we prove that an optimal controller is obtained by consecutively solving two special quantitative reach-avoid problems. In addition, we present a fully-automated method based on Symbolic Optimal Control to practically synthesize for the considered problem class approximately optimal controllers for sampled-data nonlinear plants. Experimental results on parcel delivery and on an aircraft routing mission confirm the practicality of our method., Comment: 14 pages, 7 figures

The problem of computing the reachable set for a given system is a quintessential question in nonlinear control theory. While previous work has yielded a plethora of approximate and analytical methods for determining such a set, these methods naturally require the knowledge of the controlled system dynamics throughout the state space. In contrast to such classical methods, this paper considers the question of estimating the reachable set of a control system using only the knowledge of local system dynamics at a single point and a bound on the rate of change of dynamics. Namely, motivated by the need for safety-critical planning for systems with unknown dynamics, we consider the problem of describing the guaranteed reachability set: the set of all states that are guaranteed to be reachable regardless of the true system dynamics, given the current knowledge about the system. We show that such a set can be underapproximated by a reachable set of a related known system whose dynamics at every state depend on the velocity vectors that are guaranteed to all control systems consistent with the assumed knowledge. Complementing the theory, numerical examples of a single-dimensional control system and a simple model of an aircraft in distress verify that such an underapproximation is meaningful in practice, and may indeed equal the desired guaranteed reachability set., v3: fixed error in Figures 1, 3, 4; all consequential material remains unchanged

We study the strong structural controllability (SSC) of diffusively coupled networks, where the external control inputs are injected to only some nodes, namely the leaders. For such systems, one measure of controllability is the dimension of strong structurally controllable subspace, which is equal to the smallest possible rank of controllability matrix under admissible (positive) coupling weights. In this paper, we compare two tight lower bounds on the dimension of strong structurally controllable subspace: one based on the distances of followers to leaders, and the other based on the graph coloring process known as zero forcing. We show that the distance-based lower bound is usually better than the zero-forcing-based bound when the leaders do not constitute a zero-forcing set. On the other hand, we also show that any set of leaders that can be shown to achieve complete SSC via the distance-based bound is necessarily a zero-forcing set. These results indicate that while the zero-forcing based approach may be preferable when the focus is only on verifying complete SSC, the distance-based approach is usually more informative when partial SSC is also of interest. Furthermore, we also present a novel bound based on the combination of these two approaches, which is always at least as good as, and in some cases strictly greater than, the maximum of the two bounds. We support our analysis with numerical results for various graphs and leader sets., Accepted to the 59th IEEE Conference on Decision and Control

Swing equations are an integral part of a large class of power system dynamical models used in rotor angle stability assessment. Despite intensive studies, some fundamental properties of lossy swing equations are still not fully understood. In this paper, we develop a sufficient condition for certifying the stability of equilibrium points (EPs) of these equations, and illustrate the effects of damping, inertia, and network topology on the stability properties of such EPs. The proposed certificate is suitable for real-time monitoring and fast stability assessment, as it is purely algebraic and can be evaluated in a parallel manner. Moreover, we provide a novel approach to quantitatively measure the degree of stability in power grids using the proposed certificate. Extensive computational experiments are conducted, demonstrating the practicality and effectiveness of the proposal.

Control of continuous time dynamics with multiplicative noise is a classic topic in stochastic optimal control. This work addresses the problem of designing infinite horizon optimal controls with stability guarantees for \textit{a single agent or large population systems} of identical, non-cooperative and non-networked agents, with multi-dimensional and nonlinear stochastic dynamics excited by multiplicative noise. For agent dynamics belonging to the class of reversible diffusion processes, we provide constraints on the state and control cost functions which guarantee stability of the closed-loop system under the action of the individual optimal controls. A condition relating the state-dependent control cost and volatility is introduced to prove the stability of the equilibrium density. This condition is a special case of the constraint required to use the path integral Feynman-Kac formula for computing the control. We investigate the connection between the stabilizing optimal control and the path integral formalism, leading us to a control law formulation expressed exclusively in terms of the desired equilibrium density., 6 pages, 0 figures, IEEE Conference on Decision and Control 2020 (accepted)

Vehicle-to-vehicle (V2V) communications have a great potential to improve traffic system performance. Most existing work of connected and autonomous vehicles (CAVs) focused on adaptation to downstream traffic conditions, neglecting the impact of CAVs' behaviors on upstream traffic flow. In this paper, we introduce a notion of Leading Cruise Control (LCC) that retains the basic car-following operation and explicitly considers the influence of the CAV's actions on the vehicles behind. We first present a detailed modeling process for LCC. Then, rigorous controllability analysis verifies the feasibility of exploiting the CAV as a leader to actively lead the motion of its following vehicles. Besides, the head-to-tail transfer function is derived for LCC under adequate employment of V2V connectivity. Numerical studies confirm the potential of LCC to strengthen the capability of CAVs in suppressing traffic instabilities and smoothing traffic flow., Comment: Accepted by the 59th IEEE Conference on Decision and Control (CDC 2020); 7 pages, 5 figures

The prolific rise in autonomous systems has led to questions regarding their safe instantiation in real-world scenarios. Failures in safety-critical contexts such as human-robot interactions or even autonomous driving can ultimately lead to loss of life. In this context, this paper aims to provide a method by which one can algorithmically test and evaluate an autonomous system. Given a black-box autonomous system with some operational specifications, we construct a minimax problem based on control barrier functions to generate a family of test parameters designed to optimally evaluate whether the system can satisfy the specifications. To illustrate our results, we utilize the Robotarium as a case study for an autonomous system that claims to satisfy waypoint navigation and obstacle avoidance simultaneously. We demonstrate that the proposed test synthesis framework systematically finds those sequences of events (tests) that identify points of system failure.

An explicit Model Predictive Control algorithm for large-scale structured linear systems is presented. We base our results on Distributed and Localized Model Predictive Control (DLMPC), a closed loop model predictive control scheme based on the System Level Synthesis (SLS) framework wherein only local state and model information needs to be exchanged between subsystems for the computation and implementation of control actions. We provide an explicit solution for each of the subproblems resulting from the distributed MPC scheme. We show that given the separability of the problem, the explicit solution is only divided into three regions per state and input instantiation, making the point location problem very efficient. Moreover, given the locality constraints, the subproblems are of much smaller dimension than the full problem, which consequently significantly reduces the computational overhead of explicit solutions. We conclude with numerical simulations to demonstrate the computational advantages of our method, in which we show a large improvement in runtime per MPC iteration as compared with the results of computing the optimization with a solver online.

The state estimation of continuous-time nonlinear systems in which a subset of sensor outputs can be maliciously controlled through injecting a potentially unbounded additive signal is considered in this paper. Analogous to our earlier work for continuous-time linear systems in \cite{chong2015observability}, we term the convergence of the estimates to the true states in the presence of sensor attacks as `observability under $M$ attacks', where $M$ refers to the number of sensors which the attacker has access to. Unlike the linear case, we only provide a sufficient condition such that a nonlinear system is observable under $M$ attacks. The condition requires the existence of asymptotic observers which are robust with respect to the attack signals in an input-to-state stable sense. We show that an algorithm to choose a compatible state estimate from the state estimates generated by the bank of observers achieves asymptotic state reconstruction. We also provide a constructive method for a class of nonlinear systems to design state observers which have the desirable robustness property. The relevance of this study is illustrated on monitoring the safe operation of a power distribution network., This paper has been accepted for publication at the 59th IEEE Conference on Decision and Control, 2020

We consider the distributed optimization problem, where a group of agents work together to optimize a common objective by communicating with neighboring agents and performing local computations. For a given algorithm, we use tools from robust control to systematically analyze the performance in the case where the communication network is time-varying. In particular, we assume only that the network is jointly connected over a finite time horizon (commonly referred to as B-connectivity), which does not require connectivity at each time instant. When applied to the distributed algorithm DIGing, our bounds are orders of magnitude tighter than those available in the literature., Comment: 6 pages, CDC 2020

Many control applications require that a system be constrained to a particular set of states, often termed as safe set. A practical and flexible method for rendering safe sets forward-invariant involves computing control input using Control Barrier Functions and Quadratic Programming methods. Many prior results however require the resulting control input to be continuous, which requires strong assumptions or can be difficult to demonstrate theoretically. In this paper we use differential inclusion methods to show that simultaneously rendering multiple sets invariant can be accomplished using a discontinuous control input. We present an optimization formulation which computes such control inputs and which can be posed in multiple forms, including a feasibility problem, a linear program, or a quadratic program. In addition, we discuss conditions under which the optimization problem is feasible and show that any feasible solution of the considered optimization problem which is measurable renders the multiple safe sets forward invariant.

We address the problem of assessing the robustness of the equilibria in uncertain, multi-agent games. Specifically, we focus on generalized Nash equilibrium problems in aggregative form subject to linear coupling constraints affected by uncertainty with a possibly unknown probability distribution. Within a data-driven context, we apply the scenario approach paradigm to provide a-posteriori feasibility certificates for the entire set of generalized Nash equilibria of the game. Then, we show that assessing the violation probability of such set merely requires to enumerate the constraints that ``shape'' it. For the class of aggregative games, this results in solving a feasibility problem on each active facet of the feasibility region, for which we propose a semi-decentralized algorithm. We demonstrate our theoretical results by means of an academic example.

Scenario optimization is by now a well established technique to perform designs in the presence of uncertainty. It relies on domain knowledge integrated with first-hand information that comes from data and generates solutions that are also accompanied by precise statements of reliability. In this paper, following recent developments in (Garatti and Campi, 2019), we venture beyond the traditional set-up of scenario optimization by analyzing the concept of constraints relaxation. By a solid theoretical underpinning, this new paradigm furnishes fundamental tools to perform designs that meet a proper compromise between robustness and performance. After suitably expanding the scope of constraints relaxation as proposed in (Garatti and Campi, 2019), we focus on various classical Support Vector methods in machine learning - including SVM (Support Vector Machine), SVR (Support Vector Regression) and SVDD (Support Vector Data Description) - and derive new results for the ability of these methods to generalize.

We consider the problem of estimating the dynamic latent states of an intracellular multiscale stochastic reaction network from time-course measurements of fluorescent reporters. We first prove that accurate solutions to the filtering problem can be constructed by solving the filtering problem for a reduced model that represents the dynamics as a hybrid process. The model reduction is based on exploiting the time-scale separations in the original network, and it can greatly reduce the computational effort required to simulate the dynamics. This enables us to develop efficient particle filters to solve the filtering problem for the original model by applying particle filters to the reduced model. We illustrate the accuracy and the computational efficiency of our approach using a numerical example., 6 pages, 1 figure. Accepted to CDC 2020

This paper studies selecting a subset of the system's output to minimize the state estimation mean square error (MSE). This results in the maximization problem of a set function defined on possible sensor selections subject to a cardinality constraint. We consider to solve it approximately by a greedy search. Since the MSE function is not submodular nor supermodular, the well-known performance guarantees for the greedy solutions do not hold in the present case. Thus, we use the quantities---the submodularity ratio and the curvature---to evaluate the degrees of submodularity and supermodularity of the objective function. By using the properties of the MSE function, we approximately compute these quantities and derive a performance guarantee for the greedy solutions. It is shown that the guarantee is less conservative than those in the existing results.

This paper considers the general problem of transitioning theoretically safe controllers to hardware. Concretely, we explore the application of control barrier functions (CBFs) to sampled-data systems: systems that evolve continuously but whose control actions are computed in discrete time-steps. While this model formulation is less commonly used than its continuous counterpart, it more accurately models the reality of most control systems in practice, making the safety guarantees more impactful. In this context, we prove robust set invariance with respect to zero-order hold controllers as well as state uncertainty, without the need to explicitly compute any control invariant sets. It is then shown that this formulation can be exploited to address input delays in this system, with the result being CBF constraints that are affine in the input. The results are demonstrated in a high-fidelity simulation of an unstable Segway robotic system in real-time., 6 pages, submitted to CDC 2020

Policy Iteration (PI) is a classical family of algorithms to compute an optimal policy for any given Markov Decision Problem (MDP). The basic idea in PI is to begin with some initial policy and to repeatedly update the policy to one from an improving set, until an optimal policy is reached. Different variants of PI result from the (switching) rule used for improvement. An important theoretical question is how many iterations a specified PI variant will take to terminate as a function of the number of states $n$ and the number of actions $k$ in the input MDP. While there has been considerable progress towards upper-bounding this number, there are fewer results on lower bounds. In particular, existing lower bounds primarily focus on the special case of $k = 2$ actions. We devise lower bounds for $k \geq 3$. Our main result is that a particular variant of PI can take $\Omega(k^{n/2})$ iterations to terminate. We also generalise existing constructions on $2$-action MDPs to scale lower bounds by a factor of $k$ for some common deterministic variants of PI, and by $\log(k)$ for corresponding randomised variants., Comment: 8 pages, 3 diagrams, 2 tables. Paper in IEEE CDC 2020

The design of navigation observers able to simultaneously estimate the position, linear velocity and orientation of a vehicle in a three-dimensional space is crucial in many robotics and aerospace applications. This problem was mainly dealt with using the extended Kalman filter and its variants which proved to be instrumental in many practical applications. Although practically efficient, the lack of strong stability guarantees of these algorithms motivated the emergence of a new class of geometric navigation observers relying on Riemannian geometry tools, leading to provable strong stability properties. The objective of this brief tutorial is to provide an overview of the existing estimation schemes, as well as some recently developed geometric nonlinear observers, for autonomous navigation systems relying on inertial measurement unit (IMU) and landmark measurements., 8 pages, accepted by the IEEE 59th Conference on Decision and Control (CDC 2020)

This paper studies the control of safety-critical dynamical systems in the presence of adversarial disturbances. We seek to synthesize state-feedback controllers to minimize a cost incurred due to the disturbance, while respecting a safety constraint. The safety constraint is given by a bound on an H-inf norm, while the cost is specified as an upper bound on the H-2 norm of the system. We consider an online setting where costs at each time are revealed only after the controller at that time is chosen. We propose an iterative approach to the synthesis of the controller by solving a modified discrete-time Riccati equation. Solutions of this equation enforce the safety constraint. We compare the cost of this controller with that of the optimal controller when one has complete knowledge of disturbances and costs in hindsight. We show that the regret function, which is defined as the difference between these costs, varies logarithmically with the time horizon. We validate our approach on a process control setup that is subject to two kinds of adversarial attacks., Paper accepted to the Conference on Decision and Control (CDC), 2020