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The intrinsic energy minimization in dynamical systems offers a valuable tool for minimizing the objective functions of computationally challenging problems in combinatorial optimization. However, most prior works have focused on mapping such dynamics to combinatorial optimization problems whose objective functions have quadratic degree (e.g., MaxCut); such problems can be represented and analyzed using graphs. However, the work on developing such models for problems that need objective functions with degree greater than two, and subsequently, entail the use of hypergraph data structures, is relatively sparse. In this work, we develop dynamical system-inspired computational models for several such problems. Specifically, we define the 'energy function' for hypergraph-based combinatorial problems ranging from Boolean SAT and its variants to integer factorization, and subsequently, define the resulting system dynamics. We also show that the design approach is applicable to optimization problems with quadratic degree, and use it develop a new dynamical system formulation for minimizing the Ising Hamiltonian. Our work not only expands on the scope of problems that can be directly mapped to, and solved using physics-inspired models, but also creates new opportunities to design high-performance accelerators for solving combinatorial optimization.

The massive increase of data traffic, the widespread proliferation of wireless applications and the full-scale deployment of 5G and the IoT, imply a steep increase in cellular networks energy use, resulting in a significant carbon footprint. This paper presents a comprehensive model to show the interaction between the networking and energy features of the problem and study the economical and technical viability of green networking. Solar equipment, cell zooming, energy management and dynamic user allocation are considered in the upgrading network planning process. We propose a mixed-integer optimization model to minimize long-term capital costs and operational energy expenditures in a heterogeneous on-grid cellular network with different types of base station, including solar. Based on eight scenarios where realistic costs of solar panels, batteries, and inverters were considered, we first found that solar base stations are currently not economically interesting for cellular operators. We next studied the impact of a significant and progressive carbon tax on reducing greenhouse gas emissions (GHG). We found that, at current energy and equipment prices, a carbon tax ten-fold the current value is the only element that could make green base stations economically viable., Comment: 25 pages, 12 figures, 51 references, journal paper to ieee transaciton green communications and networks

In this paper, a novel control strategy namely disturbance observer-based control is first applied to stabilization and disturbance rejection for an anti-stable stochastic heat equation with Neumann boundary actuation and unknown boundary external disturbance generated by an exogenous system. A disturbance observer-based boundary control is designed based on the backstepping approach and estimation/cancellation strategy, where the unknown disturbance is estimated in real time by a disturbance observer and rejected in the closed-loop, while the in-domain multiplicative noise whose intensity is within a known finite interval is attenuated. It is shown that the resulting closed-loop system is exponentially stable in the sense of both mean square and almost surely. A numerical example is demonstrated to validate the effectiveness of the proposed control approach.

We consider distributed optimization methods for problems where forming the Hessian is computationally challenging and communication is a significant bottleneck. We leverage randomized sketches for reducing the problem dimensions as well as preserving privacy and improving straggler resilience in asynchronous distributed systems. We derive novel approximation guarantees for classical sketching methods and establish tight concentration results that serve as both upper and lower bounds on the error. We then extend our analysis to the accuracy of parameter averaging for distributed sketches. Furthermore, we develop unbiased parameter averaging methods for randomized second order optimization for regularized problems that employ sketching of the Hessian. Existing works do not take the bias of the estimators into consideration, which limits their application to massively parallel computation. We provide closed-form formulas for regularization parameters and step sizes that provably minimize the bias for sketched Newton directions. Additionally, we demonstrate the implications of our theoretical findings via large scale experiments on a serverless cloud computing platform., arXiv admin note: text overlap with arXiv:2002.06540

We illustrate the potential of PDE-based traffic flow control in cardiovascular flow analysis, monitoring, and control, presenting a PDE-based control-oriented formulation, for 1-D blood flow dynamics in the presence of stenosis. This is achieved adopting an approach for modeling and analysis that relies on the potential correspondence of 1-D blood flow dynamics in the presence of stenosis, with 1-D traffic flow dynamics in the presence of bottleneck. We reveal such correspondence in relation to the respective (for the two flow types), speed dynamics and a (consistent with them) fundamental diagram-based reduction; bottleneck dynamic effects description and resulting boundary conditions; and free-flow/congested regimes characterization., Comment: Submitted to IEEE Transactions on Automatic Control on October 22, 2021

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.

In this paper, we address the efficient implementation of moving horizon state estimation of constrained discrete-time linear systems. We propose a novel iteration scheme which employs a proximity-based formulation of the underlying optimization algorithm and reduces computational effort by performing only a limited number of optimization iterations each time a new measurement is received. We outline conditions under which global exponential stability of the underlying estimation errors is ensured. Performance guarantees of the iteration scheme in terms of regret upper bounds are also established. A combined result shows that both exponential stability and a sublinear regret which can be rendered smaller by increasing the number of optimization iterations can be guaranteed. The stability and regret results of the proposed estimator are showcased through numerical simulations.

The rate-regulation tradeoff, defined between two objective functions, one penalizing the packet rate and one the regulation cost, can express the fundamental performance bound of networked control systems. However, the characterization of the set of globally optimal solutions in this tradeoff for multi-dimensional Gauss-Markov processes has been an open problem. In the present article, we characterize a policy profile that belongs to this set without imposing any restrictions on the information structure or the policy structure. We prove that such a policy profile consists of a symmetric threshold triggering policy based on the value of information and a certainty-equivalent control policy based on a non-Gaussian linear estimator. These policies are deterministic and can be designed separately. Besides, we provide a global optimality analysis for the value of information $\text{VoI}_k$, a semantic metric that emerges from the rate-regulation tradeoff as the difference between the benefit and the cost of a data packet. We prove that it is globally optimal that a data packet containing sensory information at time $k$ be transmitted to the controller only if $\text{VoI}_k$ becomes nonnegative. These results have important implications in the areas of communication and control., Comment: arXiv admin note: text overlap with arXiv:1812.07534

This paper investigates the joint problems of dynamic state estimation of algebraic variables (voltage and phase angle) and generator states (rotor angle and frequency) of nonlinear differential algebraic equation (NDAE) power network models, under uncertainty. Traditionally, these two problems have been decoupled due to complexity of handling NDAE models. In particular, this paper offers the first attempt to solve the aforementioned problem in a coupled approach where the algebraic and generator states estimates are simultaneously computed. The proposed estimation algorithm herein is endowed with the following properties: (i) it is fairly simple to implement and based on well-understood Lyapunov theory; (ii) considers various sources of uncertainty from generator control inputs, loads, renewables, process and measurement noise; (iii) models phasor measurement unit installations at arbitrary buses; and (iv) is computationally less intensive than the decoupled approach in the literature., Comment: IEEE Transactions on Power Systems, In Press, June 2022

Electric vehicles (EVs) are being rapidly adopted due to their economic and societal benefits. Autonomous mobility-on-demand (AMoD) systems also embrace this trend. However, the long charging time and high recharging frequency of EVs pose challenges to efficiently managing EV AMoD systems. The complicated dynamic charging and mobility process of EV AMoD systems makes the demand and supply uncertainties significant when designing vehicle balancing algorithms. In this work, we design a data-driven distributionally robust optimization (DRO) approach to balance EVs for both the mobility service and the charging process. The optimization goal is to minimize the worst-case expected cost under both passenger mobility demand uncertainties and EV supply uncertainties. We then propose a novel distributional uncertainty sets construction algorithm that guarantees the produced parameters are contained in desired confidence regions with a given probability. To solve the proposed DRO AMoD EV balancing problem, we derive an equivalent computationally tractable convex optimization problem. Based on real-world EV data of a taxi system, we show that with our solution the average total balancing cost is reduced by 14.49%, and the average mobility fairness and charging fairness are improved by 15.78% and 34.51%, respectively, compared to solutions that do not consider uncertainties., 16 pages

A dual adaptive model predictive control (MPC) algorithm is presented for linear, time-invariant systems subject to bounded disturbances and parametric uncertainty in the state-space matrices. Online set-membership identification is performed to reduce the uncertainty and thus control affects both the informativity of identification and the system’s performance. The main contribution of the paper is to include this dual effect in the MPC optimization problem using a predicted worst-case cost in the objective function. This allows the controller to perform active exploration, that is, the control input reduces the uncertainty in the regions of the parameter space that have most influence on the performance. Additionally, the MPC algorithm ensures robust constraint satisfaction of state and input constraints. Advantages of the proposed algorithm are shown by comparing it to a passive adaptive MPC algorithm from the literature., IEEE Transactions on Automatic Control, 68 (5), ISSN:0018-9286, ISSN:1558-2523

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 paper, 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 considers the distributed online convex optimization problem with time-varying constraints over a network of agents. This is a sequential decision making problem with two sequences of arbitrarily varying convex loss and constraint functions. At each round, each agent selects a decision from the decision set, and then only a portion of the loss function and a coordinate block of the constraint function at this round are privately revealed to this agent. The goal of the network is to minimize the network-wide loss accumulated over time. Two distributed online algorithms with full-information and bandit feedback are proposed. Both dynamic and static network regret bounds are analyzed for the proposed algorithms, and network cumulative constraint violation is used to measure constraint violation, which excludes the situation that strictly feasible constraints can compensate the effects of violated constraints. In particular, we show that the proposed algorithms achieve $\mathcal{O}(T^{\max\{\kappa,1-\kappa\}})$ static network regret and $\mathcal{O}(T^{1-\kappa/2})$ network cumulative constraint violation, where $T$ is the time horizon and $\kappa\in(0,1)$ is a user-defined trade-off parameter. Moreover, if the loss functions are strongly convex, then the static network regret bound can be reduced to $\mathcal{O}(T^{\kappa})$. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.

The non-convex complementarity constraints present a fundamental computational challenge in energy constrained optimization problems. In this work, we present a new, linear, and robust battery optimization formulation that sidesteps the need for battery complementarity constraints and integers and prove analytically that the formulation guarantees that all energy constraints are satisfied which ensures that the optimized battery dispatch is physically realizable. In addition, we bound the worst-case model mismatch and discuss conservativeness. Simulation results further illustrate the effectiveness of this approach.

Mirror descent (MD) is a powerful first-order optimization technique that subsumes several optimization algorithms including gradient descent (GD). In this work, we develop a semi-definite programming (SDP) framework to analyze the convergence rate of MD in centralized and distributed settings under both strongly convex and non-strongly convex assumptions. We view MD with a dynamical system lens and leverage quadratic constraints (QCs) to provide explicit convergence rates based on Lyapunov stability. For centralized MD under strongly convex assumption, we develop a SDP that certifies exponential convergence rates. We prove that the SDP always has a feasible solution that recovers the optimal GD rate as a special case. We complement our analysis by providing the $O(1/k)$ convergence rate for convex problems. Next, we analyze the convergence of distributed MD and characterize the rate using SDP. To the best of our knowledge, the numerical rate of distributed MD has not been previously reported in the literature. We further prove an $O(1/k)$ convergence rate for distributed MD in the convex setting. Our numerical experiments on strongly convex problems indicate that our framework certifies superior convergence rates compared to the existing rates for distributed GD.

Determining the reachable set for a given nonlinear control system is crucial for system control and planning. However, computing such a set is impossible if the system's dynamics are not fully known. This paper is motivated by a scenario where a system suffers an adverse event mid-operation, resulting in a substantial change to the system's dynamics, rendering them largely unknown. Our objective is to conservatively approximate the system's reachable set solely from its local dynamics at a single point and the bounds on the rate of change of its dynamics. We translate this knowledge about the system dynamics into an ordinary differential inclusion. We then derive a conservative approximation of the velocities available to the system at every system state. An inclusion using this approximation can be interpreted as a control system; the trajectories of the derived control system are guaranteed to be trajectories of the unknown system. To illustrate the practical implementation and consequences of our work, we apply our algorithm to a simplified model of an unmanned aerial vehicle.

We develop a probabilistic control algorithm, $\texttt{GTLProCo}$, for swarms of agents with heterogeneous dynamics and objectives, subject to high-level task specifications. The resulting algorithm not only achieves decentralized control of the swarm but also significantly improves scalability over state-of-the-art existing algorithms. Specifically, we study a setting in which the agents move along the nodes of a graph, and the high-level task specifications for the swarm are expressed in a recently-proposed language called graph temporal logic (GTL). By constraining the distribution of the swarm over the nodes of the graph, GTL can specify a wide range of properties, including safety, progress, and response. $\texttt{GTLProCo}$, agnostic to the number of agents comprising the swarm, controls the density distribution of the swarm in a decentralized and probabilistic manner. To this end, it synthesizes a time-varying Markov chain modeling the time evolution of the density distribution under the GTL constraints. We first identify a subset of GTL, namely reach-avoid specifications, for which we can reduce the synthesis of such a Markov chain to either linear or semi-definite programs. Then, in the general case, we formulate the synthesis of the Markov chain as a mixed-integer nonlinear program (MINLP). We exploit the structure of the problem to provide an efficient sequential mixed-integer linear programming scheme with trust regions to solve the MINLP. We empirically demonstrate that our sequential scheme is at least three orders of magnitude faster than off-the-shelf MINLP solvers and illustrate the effectiveness of $\texttt{GTLProCo}$ in several swarm scenarios., Comment: Initial submission to TAC

The problem of finding the unique low dimensional decomposition of a given matrix has been a fundamental and recurrent problem in many areas. In this paper, we study the problem of seeking a unique decomposition of a low rank matrix $Y\in \mathbb{R}^{p\times n}$ that admits a sparse representation. Specifically, we consider $Y = A X\in \mathbb{R}^{p\times n}$ where the matrix $A\in \mathbb{R}^{p\times r}$ has full column rank, with $r < \min\{n,p\}$, and the matrix $X\in \mathbb{R}^{r\times n}$ is element-wise sparse. We prove that this sparse decomposition of $Y$ can be uniquely identified, up to some intrinsic signed permutation. Our approach relies on solving a nonconvex optimization problem constrained over the unit sphere. Our geometric analysis for the nonconvex optimization landscape shows that any {\em strict} local solution is close to the ground truth solution, and can be recovered by a simple data-driven initialization followed with any second order descent algorithm. At last, we corroborate these theoretical results with numerical experiments., Accepted by 2021 Neurips, in IEEE Transactions on Information Theory, 2022

This work contributes to the field of optimal control of bilinear systems. It concerns a continuous time, finite dimensional, bilinear state equation with a quadratic performance index to be minimized. The state equation is non-autonomous and comprises a deterministic, a-priori known excitation. The control trajectory is constrained to an admissible set without a specific structure. The performance index is a functional, quadratic in the state variables and control signals. Krotov's method is used for solving this problem by means of an improving sequence. To this end, the required sequence of an improving functions is formulated. Finally, the solution is encapsulated in an algorithm form and a numerical example of structural control problem is provided., \copyright 2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works

In this work, we study decentralized convex constrained optimization problems in networks. We focus on the dual averaging-based algorithmic framework that is well-documented to be superior in handling constraints and complex communication environments simultaneously. Two new decentralized dual averaging (DDA) algorithms are proposed. In the first one, a second-order dynamic average consensus protocol is tailored for DDA-type algorithms, which equips each agent with a provably more accurate estimate of the global dual variable than conventional schemes. We rigorously prove that the proposed algorithm attains $\mathcal{O}(1/t)$ convergence for general convex and smooth problems, for which existing DDA methods were only known to converge at $\mathcal{O}(1/\sqrt{t})$ prior to our work. In the second one, we use the extrapolation technique to accelerate the convergence of DDA. Compared to existing accelerated algorithms, where typically two different variables are exchanged among agents at each time, the proposed algorithm only seeks consensus on local gradients. Then, the extrapolation is performed based on two sequences of primal variables which are determined by the accumulations of gradients at two consecutive time instants, respectively. The algorithm is proved to converge at $\mathcal{O}(1)\left(\frac{1}{t^2}+\frac{1}{t(1-\beta)^2}\right)$, where $\beta$ denotes the second largest singular value of the mixing matrix. We remark that the condition for the algorithmic parameter to guarantee convergence does not rely on the spectrum of the mixing matrix, making itself easy to satisfy in practice. Finally, numerical results are presented to demonstrate the efficiency of the proposed methods., Comment: 15 pages, 4 figures

Distributed linear algebraic equation over networks, where nodes hold a part of problem data and cooperatively solve the equation via node-to-node communications, is a basic distributed computation task receiving an increasing research attention. Communications over a network have a stochastic nature, with both temporal and spatial dependence due to link failures, packet dropouts or node recreation, etc. In this paper, we study the convergence and convergence rate of distributed linear equation protocols over a $\ast$-mixing random network, where the temporal and spatial dependencies between the node-to-node communications are allowed. When the network linear equation admits exact solutions, we prove the mean-squared exponential convergence rate of the distributed projection consensus algorithm, while the lower and upper bound estimations of the convergence rate are also given for independent and identically distributed (i.i.d.) random graphs. Motivated by the randomized Kaczmarz algorithm, we also propose a distributed randomized projection consensus algorithm, where each node randomly selects one row of local linear equations for projection per iteration, and establish an exponential convergence rate. When the network linear equation admits no exact solution, we prove that a distributed gradient-descent-like algorithm with diminishing step-sizes can drive all nodes' states to a least-squares solution at a sublinear rate. These results collectively illustrate that distributed computations may overcome communication correlations if the prototype algorithms enjoy certain contractive properties or are designed with suitable parameters., 17 pages

This paper studies the adaptive optimal stationary control of continuous-time linear stochastic systems with both additive and multiplicative noises, using reinforcement learning techniques. Based on policy iteration, a novel off-policy reinforcement learning algorithm, named optimistic least-squares-based policy iteration, is proposed which is able to find iteratively near-optimal policies of the adaptive optimal stationary control problem directly from input/state data without explicitly identifying any system matrices, starting from an initial admissible control policy. The solutions given by the proposed optimistic least-squares-based policy iteration are proved to converge to a small neighborhood of the optimal solution with probability one, under mild conditions. The application of the proposed algorithm to a triple inverted pendulum example validates its feasibility and effectiveness., Comment: 10 pages, 3 figures

We present a parallelized primal-dual algorithm for solving constrained convex optimization problems. The algorithm is "block-based," in that vectors of primal and dual variables are partitioned into blocks, each of which is updated only by a single processor. We consider four possible forms of asynchrony: in updates to primal variables, updates to dual variables, communications of primal variables, and communications of dual variables. We construct a family of explicit counterexamples to show the need to eliminate asynchronous communication of dual variables, though the other forms of asynchrony are permitted, all without requiring bounds on delays. A first-order primal-dual update law is developed and shown to be robust to asynchrony. We then derive convergence rates to a Lagrangian saddle point in terms of the operations agents execute, without specifying any timing or pattern with which they must be executed. These convergence rates include an "asynchrony penalty" that we quantify and present ways to mitigate. Numerical results illustrate these developments., Comment: arXiv admin note: text overlap with arXiv:2004.05142

This study presents a vehicle-level distributed coordination strategy to control a mixed traffic stream of connected automated vehicles (CAVs) and connected human-driven vehicles (CHVs) through signalized intersections. We use CAVs as mobile traffic controllers during a newly introduced white phase, during which CAVs will negotiate the right-of-way to lead a group of CHVs while CHVs must follow their immediate front vehicle. The white phase will not be activated under low CAV penetration rates, where vehicles must wait for green signals. We have formulated this problem as a distributed mixed-integer non-linear program and developed a methodology to form an agreement among all vehicles on their trajectories and signal timing parameters. The agreement on trajectories is reached through an iterative process, where CAVs update their trajectory based on shared trajectory of other vehicles to avoid collisions and share their trajectory with other vehicles. Additionally, the agreement on signal timing parameters is formed through a voting process where the most voted feasible signal timing parameters are selected. The numerical experiments indicate that the proposed methodology can efficiently control vehicle movements at signalized intersections under various CAV market shares. The introduced white phase reduces the total delay by 3.2% to 94.06% compared to cooperative trajectory and signal optimization under different CAV market shares in our tests. In addition, our numerical results show that the proposed technique yields reductions in total delay, ranging from 40.2% - 98.9%, compared to those of a fully-actuated signal control obtained from a state-of-practice traffic signal optimization software., 15 pages. 20 figures, 4 tables

We present a Model Predictive Control (MPC) algorithm for energy management in aircraft with hybrid electric propulsion systems consisting of gas turbine and electric motor components. Series and parallel configurations are considered. By combining a point-mass aircraft dynamical model with models of electrical losses and losses in the gas turbine, the fuel consumed over a given future flight path is minimised subject to constraints on the battery, electric motor and gas turbine. The optimization is formulated as a convex problem under mild assumptions and its solution is used to define a predictive energy management control law that takes into account the variation in aircraft mass during flight. We investigate the performance of algorithms for solving this problem. An Alternating Direction Method of Multipliers (ADMM) algorithm is proposed and compared with a general purpose convex interior point solver. We also show that the ADMM implementation reduces the required computation time by orders of magnitude in comparison with a general purpose nonlinear programming solver, making it suitable for real-time supervisory energy management control., Comment: arXiv admin note: text overlap with arXiv:2004.02582

In this paper, we study accelerating a Laplacian-based dynamic average consensus algorithm by splitting the conventional delay-free disagreement feedback into weighted summation of a current and an outdated term. We determine for what weighted sum there exists a range of time delay that results in the higher rate of convergence for the algorithm. For such weights, using the Lambert W function, we obtain the rate increasing range of the time delay, the maximum reachable rate and comment on the value of the corresponding maximizer delay. We also study the effect of use of outdated feedback on the control effort of the agents and show that only for some specific affine combination of the immediate and outdated feedback the control effort of the agents does not go beyond that of the delay-free algorithm. Additionally, we demonstrate that using outdated feedback does not increase the steady state tracking error of the average consensus algorithm. Lastly, we determine the optimum combination of the current and the outdated feedback weights to achieve the maximum increase in the rate of convergence without increasing the control effort of the agents. We demonstrate our results through a numerical example.

Existing frameworks for transfer learning are incomplete from a systems theoretic perspective. They place emphasis on notions of domain and task, and neglect notions of structure and behavior. In doing so, they limit the extent to which formalism can be carried through into the elaboration of their frameworks. Herein, we use Mesarovician systems theory to define transfer learning as a relation on sets and subsequently characterize the general nature of transfer learning as a mathematical construct. We interpret existing frameworks in terms of ours and go beyond existing frameworks to define notions of transferability, transfer roughness, and transfer distance. Importantly, despite its formalism, our framework avoids the detailed mathematics of learning theory or machine learning solution methods without excluding their consideration. As such, we provide a formal, general systems framework for modeling transfer learning that offers a rigorous foundation for system design and analysis.

Convergence (virtual) bidding is an important part of two-settlement electric power markets as it can effectively reduce discrepancies between the day-ahead and real-time markets. Consequently, there is extensive research into the bidding strategies of virtual participants aiming to obtain optimal bids to submit to the day-ahead market. In this paper, we introduce a price-based general stochastic optimization framework to obtain optimal convergence bid curves. Within this framework, we develop a computationally tractable linear programming-based optimization model, which produces bid prices and volumes simultaneously. We also show that different approximations and simplifications in the general model lead naturally to state-of-the-art convergence bidding approaches, such as self-scheduling and opportunistic approaches. Our general framework also provides a straightforward way to compare the performance of these models, which is demonstrated by numerical experiments on the California (CAISO) market.

This paper considers distributed stochastic optimization, in which a number of agents cooperate to optimize a global objective function through local computations and information exchanges with neighbors over a network. Stochastic optimization problems are usually tackled by variants of projected stochastic gradient descent. However, projecting a point onto a feasible set is often expensive. The Frank-Wolfe (FW) method has well-documented merits in handling convex constraints, but existing stochastic FW algorithms are basically developed for centralized settings. In this context, the present work puts forth a distributed stochastic Frank-Wolfe solver, by judiciously combining Nesterov's momentum and gradient tracking techniques for stochastic convex and nonconvex optimization over networks. It is shown that the convergence rate of the proposed algorithm is $\mathcal{O}(k^{-\frac{1}{2}})$ for convex optimization, and $\mathcal{O}(1/\mathrm{log}_2(k))$ for nonconvex optimization. The efficacy of the algorithm is demonstrated by numerical simulations against a number of competing alternatives., Comment: 15 pages, 11 figures, 2 tables

Motivated by various distributed control applications, we consider a linear system with Gaussian noise observed by multiple sensors which transmit measurements over a dynamic lossy network. We characterize the stationary optimal sensor scheduling policy for the finite horizon, discounted, and long-term average cost problems and show that the value iteration algorithm converges to a solution of the average cost problem. We further show that the suboptimal policies provided by the rolling horizon truncation of the value iteration also guarantee stability and provide near-optimal average cost. Lastly, we provide qualitative characterizations of the multidimensional set of measurement loss rates for which the system is stabilizable for a static network, significantly extending earlier results on intermittent observations., 25 pages

Robustness is a basic property of any control system. In the context of linear output regulation, it was proved that embedding an internal model of the exogenous signals is necessary and sufficient to achieve tracking of the desired reference signals in spite of external disturbances and parametric uncertainties. This result is commonly known as "internal model principle". A complete extension of such linear result to general nonlinear systems is still an open problem, which is exacerbated by the large number of alternative definitions of uncertainty and desired control goals that are possible in a nonlinear setting. In this paper, we develop a general framework in which all these different notions can be formally characterized in a unifying way. Classical results are reinterpreted in the proposed setting, and new results and insights are presented with a focus on robust rejection/tracking of arbitrary harmonic content. Moreover, we show by counter-example that, in the relevant case of continuous unstructured uncertainties, there are problems for which no smooth finite-dimensional robust regulator exists ensuring exact regulation., updated version 04.2021

Semidefinite and sum-of-squares (SOS) optimization are fundamental computational tools in many areas, including linear and nonlinear systems theory. However, the scale of problems that can be addressed reliably and efficiently is still limited. In this paper, we introduce a new notion of block factor-width-two matrices and build a new hierarchy of inner and outer approximations of the cone of positive semidefinite (PSD) matrices. This notion is a block extension of the standard factor-width-two matrices, and allows for an improved inner-approximation of the PSD cone. In the context of SOS optimization, this leads to a block extension of the scaled diagonally dominant sum-of-squares (SDSOS) polynomials. By varying a matrix partition, the notion of block factor-width-two matrices can balance a trade-off between the computation scalability and solution quality for solving semidefinite and SOS optimization problems. Numerical experiments on a range of large-scale instances confirm our theoretical findings., Accepted for publication as a regular paper at IEEE TAC. Code is available through https://github.com/zhengy09/SDPfw

The persistence of excitation (PE) condition is sufficient to ensure parameter convergence in adaptive estimation problems. Recent results on adaptive estimation in reproducing kernel Hilbert spaces (RKHS) introduce PE conditions for RKHS. This paper presents sufficient conditions for PE for the particular class of uniformly embedded reproducing kernel Hilbert spaces (RKHS) defined over smooth Riemannian manifolds. This paper also studies the implications of the sufficient condition in the case when the RKHS is finite or infinite-dimensional. When the RKHS is finite-dimensional, the sufficient condition implies parameter convergence as in the conventional analysis. On the other hand, when the RKHS is infinite-dimensional, the same condition implies that the function estimate error is ultimately bounded by a constant that depends on the approximation error in the infinite-dimensional RKHS. We illustrate the effectiveness of the sufficient condition in a practical example., Comment: 11 pages, 2 figures

In this paper, we investigate the architecture of an optimal controller that maximizes the convergence speed of a consensus protocol with single-integrator dynamics. Under the assumption that communication delays increase with the number of hops from which information is allowed to reach each agent, we address the optimal control design under delayed feedback and show that the optimal controller features, in general, a sparsely connected architecture., 6 pages, 7 figures; published on IEEE Control Systems Letters

In this paper, we present and study a robust unit commitment model and some variants that consider complex temperature and demand uncertainties. Since there is a strong relationship among the efficiency of gas generators, demand, and temperature in practical systems, our robust models have both left- and right-hand-side (LHS and RHS, respectively) uncertainties. Unlike many existing robust models with RHS uncertainty only, the introduction of LHS uncertainty imposes a huge challenge in computing robust solutions. For those complex formulations, we analyze their structures, derive important properties, and design exact and fast approximation solution strategies under the column-and-constraint generation framework. Numerical experiments are conducted on typical IEEE test systems, which showcase the great performance of our solution methods and demonstrate a clear impact of complex and correlated uncertainties in system operations.

This paper presents a family of algorithms for decentralized convex composite problems. We consider the setting of a network of agents that cooperatively minimize a global objective function composed of a sum of local functions plus a regularizer. Through the use of intermediate consensus variables, we remove the need for inner communication loops between agents when computing curvature-guided updates. A general scheme is presented which unifies the analysis for a plethora of computing choices, including gradient descent, Newton updates, and BFGS updates. Our analysis establishes sublinear convergence rates under convex objective functions with Lipschitz continuous gradients, as well as linear convergence rates when the local functions are further assumed to be strongly convex. Moreover, we explicitly characterize the acceleration due to curvature information. Last but not the least, we present an asynchronous implementation for the proposed algorithms, which removes the need for a central clock, with linear convergence rates established in expectation under strongly convex objectives. We ascertain the effectiveness of the proposed methods with numerical experiments on benchmark datasets.

Urban air mobility is a concept that promotes aerial modes of transport in urban areas. In these areas, the location and capacity of the vertiports--where the travelers embark and disembark the aircraft--not only affect the flight delays of the aircraft, but can also aggravate the congestion of ground vehicles by creating extra ground travel demands. We introduce a mathematical model for selecting the location and capacity of the vertiports that minimizes the traffic congestion in hybrid air-ground transportation networks. Our model is based on a mathematical program with bilinear equilibrium constraints. Furthermore, we show how to compute a global optimal solution of this mathematical program by solving a mixed integer linear program. We demonstrate our results via the Anaheim transportation network model, which contains more than 400 nodes and 900 links.

This paper builds Wasserstein ambiguity sets for the unknown probability distribution of dynamic random variables leveraging noisy partial-state observations. The constructed ambiguity sets contain the true distribution of the data with quantifiable probability and can be exploited to formulate robust stochastic optimization problems with out-of-sample guarantees. We assume the random variable evolves in discrete time under uncertain initial conditions and dynamics, and that noisy partial measurements are available. All random elements have unknown probability distributions and we make inferences about the distribution of the state vector using several output samples from multiple realizations of the process. To this end, we leverage an observer to estimate the state of each independent realization and exploit the outcome to construct the ambiguity sets. We illustrate our results in an economic dispatch problem involving distributed energy resources over which the scheduler has no direct control.

In this paper, we introduce a fast row-stochastic decentralized algorithm, referred to as FRSD, to solve consensus optimization problems over directed communication graphs. The proposed algorithm only utilizes row-stochastic weights, leading to certain practical advantages in broadcast communication settings over those requiring column-stochastic weights. Under the assumption that each node-specific function is smooth and strongly convex, we show that the FRSD iterate sequence converges with a linear rate to the optimal consensus solution. In contrast to the existing methods for directed networks, FRSD enjoys linear convergence without employing a gradient tracking (GT) technique explicitly, rather it implements GT implicitly with the use of a novel momentum term, which leads to a significant reduction in communication and storage overhead for each node when FRSD is implemented for solving high-dimensional problems over small-to-medium scale networks. In the numerical tests, we compare FRSD with other state-of-the-art methods, which use row-stochastic and/or column-stochastic weights.

We develop multi-stage linear decision rules (LDRs) for dynamic power system generation and energy storage investment planning under uncertainty and propose their chance-constrained optimization with performance guarantees. First, the optimized LDRs guarantee operational and carbon policy feasibility of the resulting dynamic investment plan even when the planning uncertainty distribution is ambiguous. Second, the optimized LDRs internalize the tolerance of the system planner towards the stochasticity (variance) of uncertain investment outcomes. They can eventually produce a quasi-deterministic investment plan, which is insensitive to uncertainty (as in deterministic planning) but robust to its realizations (as in stochastic planning). Last, we certify the performance of the optimized LDRs with the bound on their sub-optimality due to their linear functional form. Using this bound, we guarantee that the preference of LDRs over less restrictive -- yet poorly scalable -- scenario-based optimization does not lead to financial losses exceeding this bound. We use a testbed of the U.S. Southeast power system to reveal the trade-offs between the cost, stochasticity, and feasibility of LDR-based investments. We also conclude that the LDR sub-optimality depends on the amount of uncertainty and the tightness of chance constraints on operational, investment and policy variables.

Communication has been seen as a significant bottleneck in industrial applications over large-scale networks. To alleviate the communication burden, sign-based optimization algorithms have gained popularity recently in both industrial and academic communities, which is shown to be closely related to adaptive gradient methods, such as Adam. Along this line, this paper investigates faster convergence for a variant of sign-based gradient descent, called scaled signGD, in three cases: 1) the objective function is strongly convex; 2) the objective function is nonconvex but satisfies the Polyak-Lojasiewicz (PL) inequality; 3) the gradient is stochastic, called scaled signGD in this case. For the first two cases, it can be shown that the scaled signGD converges at a linear rate. For case 3), the algorithm is shown to converge linearly to a neighborhood of the optimal value when a constant learning rate is employed, and the algorithm converges at a rate of $O(1/k)$ when using a diminishing learning rate, where $k$ is the iteration number. The results are also extended to the distributed setting by majority vote in a parameter-server framework. Finally, numerical experiments on logistic regression are performed to corroborate the theoretical findings.

This paper proposes a novel CTA (Combine-Then-Adapt)-based decentralized algorithm for solving convex composite optimization problems over undirected and connected networks. The local loss function in these problems contains both smooth and nonsmooth terms. The proposed algorithm uses uncoordinated network-independent constant stepsizes and only needs to approximately solve a sequence of proximal mappings, which is advantageous for solving decentralized composite optimization problems where the proximal mappings of the nonsmooth loss functions may not have analytical solutions. For the general convex case, we prove an O(1/k) convergence rate of the proposed algorithm, which can be improved to o(1/k) if the proximal mappings are solved exactly. Furthermore, with metric subregularity, we establish a linear convergence rate for the proposed algorithm. Numerical experiments demonstrate the efficiency of the algorithm.

A trading system is said to be {robust} if it generates a robust return regardless of market direction. To this end, a consistently positive expected trading gain is often used as a robustness metric for a trading system. In this paper, we propose a new class of trading policies called the {double linear policy} in an asset trading scenario when the transaction costs are involved. Unlike many existing papers, we first show that the desired robust positive expected gain may disappear when transaction costs are involved. Then we quantify under what conditions the desired positivity can still be preserved. In addition, we conduct heavy Monte-Carlo simulations for an underlying asset whose prices are governed by a geometric Brownian motion with jumps to validate our theory. A more realistic backtesting example involving historical data for cryptocurrency Bitcoin-USD is also studied., Submitted to for possible publication. arXiv admin note: substantial text overlap with arXiv:2202.02300

This work considered an online distributed optimization problem, with a group of agents whose local objective functions vary with time. Moreover, the value of the objective function is revealed to the corresponding agent after the decision is executed per time-step. Thus, each agent can only update the decision variable based on the revealed value and information collected from the neighbors, without the knowledge on the explicit expression of the objective function. To solve this problem, an online gradient-free distributed projected gradient descent (DPGD) algorithm is presented, where each agent locally approximates the gradient based on two point values. With some standard assumptions on the communication graph and the objective functions, we provide the bound for the dynamic regret as a function of the minimizer path length, step-size and smoothing parameter. Under appropriate selections of the step-size and smoothing parameter, we prove that the dynamic regret is sublinear with respect to the time duration $T$ if the minimizer path length also grows sublinearly. Finally, the effectiveness of the proposed algorithm is illustrated through numerical simulations.

We study the spread of a novel state in a network, in the presence of an exogenous control. The considered controlled evolutionary dynamics is a non-homogeneous Markov process that describes the evolution of the states of all nodes in the network. Through a rigorous analysis, we estimate the performance of the system by establishing upper and lower bounds on the expected time needed for the novel state to replace the original one. Such bounds are expressed in terms of the support and intensity of the control policy (specifically, the set of nodes that can be controlled and its energy) and of the network topology and establish fundamental limitations on the system's performance. Leveraging these results, we are able to classify network structures depending on the possibility to control the system using simple open-loop control policies. Finally, we propose a feedback control policy that, using little knowledge of the network topology and of the system's evolution at a macroscopic level, allows for a substantial speed up of the spreading process with respect to simple open-loop control policies. All these theoretical results are presented together with explanatory examples, for which Monte Carlo simulations corroborate our analytical findings., Comment: 15 pages; 7 Figures

We present a new neural network to approximate convex functions. This network has the particularity to approximate the function with cuts and can be easily adapted to partial convexity. We give an universal approximation theorem in the full convex case and give many numerical results proving it efficiency., 10 pages 16 figures

This paper presents a data-learned linear Koopman embedding of nonlinear networked dynamics and uses it to enable real-time model predictive emergency voltage control in a power network. The approach involves a novel data-driven ``basis-dictionary free" lifting of the system dynamics into a higher dimensional linear space over which an MPC (model predictive control) is exercised, making it both scalable and rapid for practical real-time implementation. A Koopman-inspired deep neural network (KDNN) encoder-decoder architecture for the linear embedding of the underlying dynamics under distributed controls is presented, in which the end-to-end components of the KDNN comprising of a triple of transforms is learned from the system trajectory data in one go: A Neural Network (NN)-based lifting to a higher dimension, a linear dynamics within that higher dimension, and an NN-based projection to the original space. This data-learned approach relieves the burden of the ad-hoc selection of the nonlinear basis functions (e.g., polynomial or radial) used in conventional approaches for lifting to higher dimensional linear space. We validate the efficacy and robustness of the approach via application to the standard IEEE 39-bus system.

Adaptive networks (ANs) are effective real time techniques to process and track events observed by sensor networks and, more recently, to equip Internet of Things (IoT) applications. ANs operate over nodes equipped with collaborative adaptive filters that solve distributively an estimation problem common to the whole network. However, they do not guarantee that nodes do not lose from cooperation, as compared to its non-cooperative operation; that poor nodes are rejected and exceptional nodes estimates reach the entire network; and that performance is uniform over all nodes. In order to enforce such properties, this work introduces the concept of distributed universal estimation, which encompasses the new concepts of local universality, global universality and universality with respect to the non-cooperative operation. We then construct a new cooperation protocol that is proven to be distributively universal, outperforming direct competitors from the literature, as shown by several simulations. Mean and mean-square analytical models are developed, with good agreement between theory and simulations.

Many robotic tasks, such as human-robot interactions or the handling of fragile objects, require tight control and limitation of appearing forces and moments alongside sensible motion control to achieve safe yet high-performance operation. We propose a learning-supported model predictive force and motion control scheme that provides stochastic safety guarantees while adapting to changing situations. Gaussian processes are used to learn the uncertain relations that map the robot's states to the forces and moments. The model predictive controller uses these Gaussian process models to achieve precise motion and force control under stochastic constraint satisfaction. As the uncertainty only occurs in the static model parts -- the output equations -- a computationally efficient stochastic MPC formulation is used. Analysis of recursive feasibility of the optimal control problem and convergence of the closed loop system for the static uncertainty case are given. Chance constraint formulation and back-offs are constructed based on the variance of the Gaussian process to guarantee safe operation. The approach is illustrated on a lightweight robot in simulations and experiments.

A control system is called self-powered if the only energy it requires for operation is that which it absorbs from the plant. For a linear feedback law to be feasible for a self-powered control system, its feedback signal must be colocated with the control inputs, and its input-output mapping must satisfy an associated passivity constraint. The imposition of such a feedback law can be viewed equivalently as the imposition of a linear passive shunt admittance at the actuation ports of the plant. In this paper we consider the use of actively-controlled electronics to impose a self-powered linear feedback law. To be feasible, it is insufficient that the imposed admittance be passive, because parasitic losses must additionally be overcome. We derive sufficient feasibility conditions which explicitly account for these losses. In the finite-dimensional, time-invariant case, the feasibility condition distills to a more conservative version of the Positive Real Lemma, which is parametrized by various loss parameters. Three examples are given, in which this condition is used to determine the least-efficient loss parameters necessary to realize a desired feedback law.