# Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification

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It is shown that non-observations of the target improves the posterior target distribution, lowering the uncertainty when the target cannot be directly observed. In addition to presenting a filter that performs data association and tracking, a track-health monitoring scheme is proposed that monitors system performance.

Keywords: Estimation , Sensor networks , Randomized algorithms Abstract: Networked systems comprised of multiple nodes with sensing, processing, and communication capabilities are able to provide more accurate estimates of some state of a dynamic process through communication between the network nodes.

This paper considers the distributed estimation or tracking problem and focuses on a class of consensus normalized algorithms. A distributed algorithm consisting of two well-studied parts, namely, Simultaneous Perturbation Stochastic Approximation SPSA and the consensus approach is proposed for networked systems with uncertainties. Such combination allows us to relax the assumption regarding the strong convexity of the minimized mean-risk functional, which may not be fulfilled in the distributed optimization problems.

For the proposed algorithm we get a mean squared upper bound of residual between estimates and unknown states. The theoretically established properties of proposed algorithm are illustrated by simulation results. Keywords: Optimization algorithms , Agents-based systems , Distributed control Abstract: In this paper, the distributed optimization problem is investigated via input feedforward passivity.

First, an input-feedforward-passivity-based continuous-time distributed algorithm is proposed. It is shown that the error system of the proposed algorithm can be interpreted as output feedback interconnections of a group of Input Feedforward Passive IFP systems. Second, based on this IFP framework, the distributed algorithm is studied over directed and uniformly jointly strongly connected switching topologies. Specifically, the continuous-time distributed algorithm for uniformly jointly strongly connected digraphs has never been considered before.

Sufficient convergence conditions are derived for the design of a suitable coupling gain. Keywords: Robust control , Stability of linear systems , Uncertain systems Abstract: In this paper, we introduce a definition of phase response for a class of multi-input multi-output MIMO linear time-invariant LTI systems, the frequency responses of which are sectorial at all frequencies.

This phase concept generalizes the notions of positive realness and negative imaginariness. We also define the half-sectorial systems and provide a time-domain interpretation. As a starting point in an endeavour to develop a comprehensive phase theory for MIMO systems, we establish a small phase theorem for feedback stability, which complements the well-known small gain theorem.

In addition, we derive a sectored real lemma for phase-bounded systems as a natural counterpart of the bounded real lemma. Keywords: Network analysis and control , Large-scale systems , Linear systems Abstract: The relaxation systems are an important subclass of the passive systems that arise naturally in applications.

We exploit the fact that they have highly structured state-space realisations to derive analytical solutions to some simple H-infinity type optimal control problems. The resulting controllers are also relaxation systems, and often sparse. This makes them ideal candidates for applications in large-scale problems, which we demonstrate by designing simple, sparse, electrical circuits to optimally control large inductive networks and to solve linear regression problems.

Keywords: Compartmental and Positive systems , Stability of linear systems , Observers for Linear systems Abstract: This paper describes a generalized internally positive representation of a diagonalizable matrix and proves that its stability is equivalent to the fact that its eigenvalues belong to the zone described by the Karpelevich Theorem. This in turn implies the minimality of the generalized internally positive representation of complex numbers.

Keywords: Control of networks , Network analysis and control , Compartmental and Positive systems Abstract: Principal submatrices of the controllability Gramian and their inverses are examined, for a network-consensus model with inputs at a subset of network nodes. Several properties of the Gramian submatrices and their inverses -- including dominant eigenvalues and eigenvectors, diagonal entries, and sign patterns -- are determined by exploiting the special doubly-nonnegative structure of the matrices. In addition, majorizations for these properties are obtained in terms of cutsets in the network's graph.

The asymptotic long time horizon structure of the controllability Gramian is also analyzed. The results on the Gramian are used to study metrics for target control of the network-consensus model. The approach provides a numerical tool to study multi-stable systems, beyond Lyapunov analysis. The theory is illustrated on two examples: a consensus problem with some repulsive interactions and second order agent dynamics, and a controlled duffing oscillator.

Keywords: Uncertain systems , Fault detection , Estimation Abstract: Online change detection involves monitoring a stream of data for changes in the statistical properties of incoming observations. A good change detector will detect any changes shortly after they occur, while raising few false alarms. Although there are algorithms with confirmed optimality properties for this task, they rely on the exact specifications of the relevant probability distributions and this limits their practicality. In this work we describe a kernel-based variant of the Cumulative Sum CUSUM change detection algorithm that can detect changes under less restrictive assumptions.

The KCUSUM algorithm is applicable in settings where there is a large amount of background data available and it is desirable to detect a change away from this background setting. Exploiting the random-walk structure of the test statistic, we derive bounds on the performance of the algorithm, including the expected delay and the average time to false alarm. Keywords: Uncertain systems , Optimization , Smart grid Abstract: Many optimization problems incorporate uncertainty affecting their parameters and thus their objective functions and constraints.

As an example, in chance-constrained optimization the constraints need to be satisfied with a certain probability. To solve these problems, scenario optimization is a well established methodology that ensures feasibility of the solution by enforcing it to satisfy a given number of samples of the constraints. The main theoretical results in scenario optimization provide the methods to determine the necessary number of samples, or to compute the risk based on the number of so-called support constraints. In this paper, we propose a methodology to remove constraints after observing the number of support constraints and the consequent risk.

Additionally, we show the effectiveness of the approach with an illustrative example and an application to power distribution grid management when solving the optimal power flow problem.

In this problem, uncertainty in the loads converts the admissible voltage limits into chance-constraints. Keywords: Output regulation , Robust adaptive control , Optimization Abstract: This paper proposes an extremum-seeking control approach for the regulation of a class of minimum phase nonlinear systems to the optimum of a measured objective function.

The nonlinear systems are subject to the effects of exogenous disturbances driven by unknown dynamics. A Lie bracket averaging technique is used to design the extremum seeking regulation mechanism.

The internal model is estimated directly using a derivative action that exploits the convexity of the measured cost function. This mechanism avoids the need for an internal model estimation approach. A stability analysis shows that the system achieves a practical output regulation of the unknown optimum equilibrium. A simulation study demonstrates the effectiveness of the technique. Keywords: Output regulation , Switched systems , Uncertain systems Abstract: This paper addresses the problem of designing an Adaptive Feedforward Control AFC system for uncertain linear systems affected by a multi-sinusoidal disturbance with known frequencies.

A novel State-Norm Estimator-based SNE- based switching mechanism is proposed to remove the long- standing assumption that either the sign of the real part or the imaginary part of the transfer function of the plant at the frequencies of excitation are needed to be known. The distinctive feature of the proposed mechanism in comparison to previous solutions from the authors is a lower order of the controller. This feature is achieved with the use of suitable notch filters, allowing a decoupled design of the switching mechanism. Furthermore, the presence of the bounded noise is considered in the analysis.

The effectiveness and robustness of the proposed method are illustrated by means of numerical examples. Keywords: Uncertain systems , Information technology systems , Information theory and control Abstract: We consider privacy against hypothesis-testing adversaries within a non-stochastic framework. We develop a theory of non-stochastic hypothesis testing by borrowing the notion of uncertain variables from non-stochastic information theory.

We define tests as binary-valued mappings on uncertain variables and prove a fundamental bound on the performance of tests in non-stochastic hypothesis testing. We use this bound to develop a measure of privacy. We then construct reporting policies with prescribed privacy and utility guarantees. The utility of a reporting policy is measured by the distance between reported and original values. We illustrate the effects of using such privacy-preserving reporting polices on a publicly-available practical dataset of preferences and demographics of young individuals with Slovakian nationality.

Keywords: Uncertain systems , Game theory , Robust control Abstract: Variational inequalities are modeling tools used to capture a variety of decision-making problems arising in mathematical optimization, operations research, game theory. The scenario approach is a set of techniques developed to tackle stochastic optimization problems, take decisions based on historical data, and quantify their risk. The overarching goal of this manuscript is to bridge these two areas of research, and thus broaden the class of problems amenable to be studied under the lens of the scenario approach.

First and foremost, we provide out-of-samples feasibility guarantees for the solution of variational and quasi variational inequality problems. Second, we apply these results to two classes of uncertain games. In the first class, the uncertainty enters in the constraint sets, while in the second class the uncertainty enters in the cost functions.

Finally, we exemplify the quality and relevance of our bounds through numerical simulations on a demand-response model. Keywords: Stability of nonlinear systems , Robotics Abstract: The use of small, lightweight autonomous agents for tasks such as distributed environmental sampling and communication network emplacement presents an appealing potential technology that is low-cost with minimal effort for deployment.

A key challenge with such compact devices is the feasible control authority that can be realized. Here, we consider the design, modeling and stability of a negatively buoyant, shape-actuated autonomous agent operating in a fluid. We evaluate the viable equilibria states for cases of zero and non-zero rotation rates and assess the corresponding system stability in both cases. Results are demonstrated in simulation. The key in this combination is that the contraction metric is a linear matrix inequality with a special structure stemming from the configuration manifold SO 3.

We demonstrate our results through simulations. Keywords: Stability of nonlinear systems , Lyapunov methods Abstract: In this paper we study an alternative method for determining stability of dynamical systems by inspecting higher order derivatives of a Lyapunov function. The system can be time invariant or time varying; in both cases we define the higher order derivatives when there are inputs.

We then claim and prove that if there exists a linear combination of those higher order derivatives with non-negative coefficients except that the coefficient of the 0-th order term needs to be positive which is negative semi-definite, then the system is globally uniformly asymptotically stable.

The proof involves repeated applications of comparison principle for first order differential relations. We also show that a system with inputs whose auxiliary system admits a Lyapunov function satisfying the aforementioned conditions is input-to-state stable. Keywords: Stability of nonlinear systems , Uncertain systems Abstract: In this paper, we propose a state-feedback controller, designed with the help of the prescribed performance control methodology, to achieve prescribed performance attributes i. Besides being continuous, the proposed solution does not utilize either bounds of the system nonlinearities or high order derivatives of the desired output trajectory.

In addition, no hard calculations, analytic or numerical, are required; making its implementation straightforward. Simulation studies clarify and verify the approach. Keywords: Stability of nonlinear systems , Distributed parameter systems Abstract: In this paper, we consider the mean-field model of noisy bounded confidence opinion dynamics under exogenous influence of static radical opinions. The long-term behavior of the model is analyzed by providing a sufficient condition for exponential convergence of the dynamics to stationary state.

The stationary state is also characterized by a global estimate for a sufficiently large noise. Furthermore, we consider the order-disorder transition in the model in order to identify the effect of the relative mass of the radicals on the critical noise level at which this transition occurs. A numerical scheme for approximating the critical noise level is provided and validated through numerical simulations of the mean-field model and the corresponding agent-based model for a particular distribution of radical opinions. Keywords: Stability of nonlinear systems , Lyapunov methods Abstract: A class of generalized nonlinear Persidskii systems is considered in the paper.

The conditions of input-to-state and integral input-to-state stability are established, which can be checked using linear matrix inequalities. The issues of discretization of this class of dynamics are analyzed using the Euler methods. The proposed theory is applied to a Lotka—Volterra model. Keywords: Optimal control , Markov processes , Optimization algorithms Abstract: Active perception strategies enable an agent to selectively gather information in a way to improve its performance.

In applications in which the agent does not have prior knowledge about the available information sources, it is crucial to synthesize active perception strategies at runtime. We consider a setting in which at runtime an agent is capable of gathering information under a limited budget. We pose the problem in the context of partially observable Markov decision processes. We propose a generalized greedy strategy that selects a subset of information sources with near-optimality guarantees on uncertainty reduction. Our theoretical analysis establishes that the proposed active perception strategy achieves near-optimal performance in terms of expected cumulative reward.

We demonstrate the resulting strategies in simulations on a robotic navigation problem. Keywords: Optimal control Abstract: This article concerns the issues of the proper formulation of a certain class of bilevel optimal control problems with dynamics specified by sweeping processes. A typical instance of this class of problems arises in the motion control of a structured crowd in a confined space. By a structured crowd, it is meant that the population is organized in subsets of individuals that remain in a certain bounded set.

The problem formulation is discussed and a solution concept is provided. Then, conditions under which the problem is proper or well-posed are derived. Keywords: Optimal control , Smart grid , Computational methods Abstract: Demand dispatch is the science of extracting virtual energy storage through the automatic control of deferrable loads to provide balancing or regulation services to the grid, while maintaining consumer-end quality of service. The control of a large collection of heterogeneous loads is in part a resource allocation problem, since different classes of loads are more valuable for different services.

The goal of this paper is to unveil the structure of the optimal solution to the resource allocation problem, and investigate short term market implications. It is found that the marginal cost for each load class evolves in a two-dimensional subspace: spanned by a co-state process and its derivative.

The resource allocation problem is recast to construct a dynamic competitive equilibrium model, in which the consumer utility is the negative of the cost of deviation from ideal QoS. It is found that a competitive equilibrium exists with the equilibrium price equal to the negative of an optimal co-state process.

Keywords: Optimal control , Optimization algorithms , Constrained control Abstract: This work is concerned with a stabilizing adaptive dynamic programming ADP approach to approximate solution of a given infinite-horizon optimal control problem. Since the latter problem cannot, in general, be solved exactly, a parametrized function approximator for the infinite-horizon cost function is introduced in ADP so called "critic". This critic is used to adapt the parameters of the function approximator.

The so called "actor" in turn derives the optimal input of the system. It is a notoriously hard problem to guarantee closed-loop stability of ADP due to the use of approximation structures in the control scheme. Since at least stabilizability is always assumed in the analyses of ADP, it is justified to invoke a respective Lyapunov function. The proposed ADP scheme explicitly uses the said Lyapunov function to simultaneously optimize the critic and guarantee closed-loop stability.

A Hessian-free optimization routine is utilized for the actor and critic optimization problems. Convergence to prescribed vicinities of the optima is shown. A computational study showed significant performance improvement for the critic-based approach compared a nominal stabilizing controller for a range of initial conditions. Keywords: Optimal control , Systems biology , Delay systems Abstract: In this paper we analyze two optimal control problems for the scallop: a two-link swimmer that is able to self-propel changing dynamics between two fluids regimes.

We address and solve explicitly the minimum time problem and the minimum quadratic one, computing the cost needed to move the swimmer between two fixed positions using a periodic control. We focus on the case of only one switching in the dynamics and exploiting the structure of the equation of motion we are able to split the problem into simpler ones. We solve explicitly each sub-problem obtaining a discontinuous global solution. Then we approximate it through a suitable sequence of continuous functions. Keywords: Optimal control , Predictive control for linear systems Abstract: This article provides a solution for the continuous-time Linear Quadratic Regulator LQR subject to a scalar state constraint.

Using a dichotomy transformation, novel properties for the finite-horizon LQR are derived; the unknown boundary conditions are explicitly expressed as a function of the horizon length, the initial state, and the final state or, cost of the final state. Practical relevance of these novel properties are demonstrated with an algorithm to compute the continuous-time LQR subject to a scalar state constraint. The proposed algorithm uses the analytical conditions for optimality, without a priori discretization, to find only those sampling time instances that mark the start and end of a constrained interval.

Each subinterval consists of a finite-horizon LQR, hence, a solution can be efficiently computed and the computational complexity does not grow with the horizon length. In fact, an infinite horizon can be handled. The algorithm is demonstrated with a simulation example. Keywords: Optimization , Optimization algorithms , Robust control Abstract: Feedback-based online optimization algorithms have gained traction in recent years because of their simple implementation, their ability to reject disturbances in real time, and their increased robustness to model mismatch.

While the robustness properties have been observed both in simulation and experimental results, the theoretical analysis in the literature is mostly limited to nominal conditions. In this work, we propose a framework to systematically assess the robust stability of feedback-based online optimization algorithms. We leverage tools from monotone operator theory, variational inequalities and classical robust control to obtain tractable numerical tests that guarantee robust convergence properties of online algorithms in feedback with a physical system, even in the presence of disturbances and model uncertainty.

Keywords: Smart grid , Control of networks , Optimization algorithms Abstract: In this paper, a novel distributed control strategy achieving feasible current sharing and voltage regulation in Direct Current DC microgrids is proposed. Secondly, we design a controller, the unforced dynamics of which represent the continuous time primal-dual dynamics of the considered optimization problem.

Then, a passive interconnection between the physical plant and the controller is presented. Keywords: Optimization , Optimization algorithms , Machine learning Abstract: This paper leverages a framework based on averaged operators to tackle the problem of tracking fixed points associated with maps that evolve over time. In particular, the paper considers the Krasnosel'skii-Mann method in a settings where: i the underlying map may change at each step of the algorithm, thus leading to a "running" implementation of the Krasnosel'skii-Mann method; and, ii an imperfect information of the map may be available.

An imperfect knowledge of the maps can capture cases where processors feature a finite precision or quantization errors, or the case where part of the map is obtained from measurements. The analytical results are applicable to inexact running algorithms for solving optimization problems, whenever the algorithmic steps can be written in the form of a composition of averaged operators; examples are provided for inexact running gradient methods and the forward-backward splitting method. Weak convergence of the cumulative fixed-point residual is investigated for the non-expansive case; linear convergence to a unique fixed-point trajectory is showed in the case of inexact running algorithms emerging from contractive operators.

Keywords: Optimization , Power systems , Network analysis and control Abstract: This paper proves that in an unbalanced multiphase network with a tree topology, the semidefinite programming relaxation of optimal power flow problems is exact when critical buses are not adjacent to each other. Here a critical bus either contributes directly to the cost function or is where an injection constraint is tight at optimality.

Our result generalizes a sufficient condition for exact relaxation in single-phase tree networks to tree networks with arbitrary number of phases. Keywords: Optimization , Smart grid , Communication networks Abstract: We study distributed convex constrained optimization on a time-varying multi-agent network. Each agent has access to its own local cost function, its local constraints, and its instant number of out-neighbors.

The collective goal is to minimize the sum of the cost functions over the set of all constraints. We utilize the push-sum protocol to be able to solve this distributed optimization problem. We adapt the push-sum optimization algorithm, which has been studied in context of unconstrained optimization so far, to convex constrained optimization by introducing an appropriate choice of penalty functions and penalty parameters.

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Under some additional technical assumptions on the gradients we prove convergence of the distributed penalty-based push-sum algorithm to the optimal value of the global objective function. We apply the proposed penalty-based push-sum algorithm to the problem of distributed energy management in smart grid and discuss the advantages of this novel procedure in comparison with existing ones. Reviewed methods either do not allow for intuitive trade-off tuning between the objectives of synchronous state restoration, local reference tracking, and disturbance rejection, or do not consider all of these objectives.

In this paper, we address all of these objectives for voltage restoration in droop-controlled inverter-based islanded micro- grids. By using distributed model predictive control DMPC in series with an unscented Kalman Filter UKF , we design a secondary voltage controller to restore the voltage to the reference in finite time. The DMPC solves a reference tracking problem while rejecting reactive power disturbances in a noisy system.

The method we present accounts for non-zero mean disturbances by design of a random-walk estimator. We propose a novel algorithm that does not rely on iterative computations. Instead, controlled invariant sets are computed in two moves: 1 we lift the problem to a higher dimensional space where a controlled invariant set is computed in closed-form; 2 we project the resulting set back to the original domain to obtain the desired controlled invariant set.

One of the advantages of the proposed method is the ability to handle larger systems. For a monotone transition system the sets of states and inputs are equipped with partial orders, moreover, the transitions preserve the ordering on the states. We propose a lazy algorithm that exploits priorities on the states and inputs. To compute the maximal controlled invariant set, only inputs with the lowest priorities are used. Then, starting from the states with the highest priorities, transitions are computed on-the-fly and only when a particular region of the state space needs to be explored.

Once this set is computed, controller synthesis is straightforward by exploring different inputs and using their priorities. We prove the completeness of our algorithm w. Finally, we illustrate the advantages of the proposed approach on a vehicle platooning problem. A major drawback of existing ABCS techniques is the lack of flexibility against changes in the disturbance model; any change in the model results in a complete re-computation of the abstraction and the controller.

This flexibility is relevant to situations when disturbances are learned or estimated during operation in an environment which is previously not known precisely. As time passes, the disturbance model is progressively refined. The monolithic nature and high computational cost of existing algorithms make ABCS unsuited for such scenarios. In this paper, we present an incremental algorithm to locally adapt abstractions to changes in the disturbance model. Only the parts of the space which are affected by the changes are updated and the rest of the abstraction is reused.

Our new abstraction method allows to apply existing incremental techniques to update the discrete controller locally for the changed abstraction. This results in an incremental ABCS algorithm. We empirically show the benefit of dynamic abstraction adaptation on two large examples: a 5-dimensional vehicle model and a dimensional quadrotor model. In both cases, the speed-up over complete re-computation is significant. The challenge of getting high-quality formal specifications is well documented.

This challenge is further exacerbated in CPS with artificial-intelligence- or machine-learning-based components. This paper presents a problem called 'semantic inference', the goal of which is to automatically translate the behavior of a CPS to a formal specification written in signal temporal logic STL.

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To reduce the potential combinatorial explosion inherent to the problem, this paper adopts a search strategy called agenda-based computation, which is inspired by natural language processing. Based on such a strategy, the semantic inference problem can be formulated as a Markov decision process MDP and then solved using reinforcement learning RL. The obtained formal specification can be viewed as an interpretable classifier, which, on the one hand, can classify desirable and undesirable behaviors, and, on the other hand, is expressed in a human-understandable form.

The performance of the proposed method is demonstrated with a case study. We impose the surveillance requirement as a temporal logic constraint in a two-player player-information game. We then use a greedy approach to determine vantage points for map visibility at runtime by optimizing a notion of information gain, namely, the number of newly-seen states.

By using a convolutional neural network trained on a class of environments, we can efficiently approximate the amount of information gain at each potential vantage point and design paths to these points that do not violate the surveillance requirement on the moving target. Potential vantage points are chosen such that moving to that location will not violate the surveillance requirement regardless of the action chosen by the target for all time in the future. Our method combines formal guarantees of correctness from formal methods with the scalability of machine learning to provide an efficient approach to maximizing map visibility subject to correctness guarantees with respect to a surveillance requirement.

In this paper, we introduce a framework for planning in uncertain environments that yields guaranteed satisfaction probabilities. We show that point-based value iteration can be combined with probabilistic roadmaps to efficiently solve this planning problem over the belief space of the uncertain environment. The dynamical systems addressed exhibit a hybrid behavior which foresees a nonlinear continuous-time state evolution interrupted by abrupt discontinuities at isolated time instants.

The problem considered consists in finding a state feedback such that the system output is rendered totally insensitive to the disturbance. Both the case of static state feedback and that of dynamic state feedback are considered. A necessary and sufficient condition for the existence of a static state feedback that solves the problem in the multivariable case is proven by defining suitable tools in the context of the differential geometric approach.

This result is restated in the terms of the differential algebraic approach for systems with a single output and this paves the way to conjecturing a necessary and sufficient condition for the existence of a decoupling dynamic state feedback in the multivariable case. We find optimal controllers to transition between these types of orbits subject to PHZD constraints, along with finding optimal periodic orbits associated to different PHZD surfaces for different walking speeds. Additionally, optimal controllers that form a connecting surface between these distinct PHZD surfaces, along with transitions between them are synthesized.

The two methods are compared with performance metrics associated with the cost of transport. The results are illustrated on a 5 degree of freedom planar bipedal robot. Keywords: Hybrid systems , Optimal control , Linear systems Abstract: The problem of steering the state of a double inte- grator from a given initial condition to the origin in minimum time and in the presence of point-wise constraint on the control input has been thoroughly characterized in the case of purely continuous-time systems.

The objective of this paper consists in extending the theory to the hybrid setting, namely with the double integrator dynamics potentially undergoing state-driven jumps. In particular, the optimal solution is characterized and compared with the purely continuous one, namely assessing potential advantages of a hybrid scheme in controlling the state of a double integrator to the origin.

Moreover, we first construct the optimal solution in the case of a prescribed number of jumps before reaching the origin and then we provide sufficient conditions ensuring that the overall minimum-time solution undergoes a finite number of jumps. As the intuition suggests, it is given by the restriction of the dynamics onto the largest subspace over which the trajectories are constrained to ensure zero output. Such a dynamics is characterized by a subset of the flowing zeros and a subset of the zeros which can be fictitiously associated to the jumping dynamics.

Keywords: Hybrid systems , Linear parameter-varying systems , Robust control Abstract: In this paper, an impulsive observer--based controller is designed for a class of linear systems with parameter uncertainties. The impulsive observer uses sampled measurements of the system output. The controller is designed making use of the regulation theory, ensuring the stabilization property, both at sampling instants and in the intersampling.

Furthermore, the resulting controller results to be structurally robust with respect to parameter uncertainties. The dynamic controller is tested on an example to demonstrate the effectiveness of the proposed approach. Keywords: Hybrid systems , Lyapunov methods Abstract: Weakly forward pre- invariant sets guarantee the existence of at least one maximal solution, when starting from any point in the set, that stays in that set. As a continuation to prior works and using multiple barrier functions, this paper studies weak forward invariance in hybrid systems modeled by constrained inclusions.

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We propose sufficient conditions to guarantee weak forward invariance of a closed set generated by the intersection of the zero-sublevel sets of the different components of a vector function called multiple barrier function. Our sufficient conditions are in terms of the multiple barrier function generating the set. Moreover, along the flow part of the hybrid system, our conditions are of two types. The first type of flow conditions need to hold only at the boundary of the set and weak forward invariance is shown by imposing transversality conditions on the intersection between the zerosublevel sets of the components of the barrier function.

The second type of conditions require both transversality and flow conditions on an external complement of the boundary of the considered set. Examples throughout the paper illustrate the results. Keywords: Stochastic optimal control , Time-varying systems , Aerospace Abstract: The present paper extends the classically studied chance-constrained optimal control to incorporate continuous-time chance constraints. While the classical approaches provide risk guarantees only at discretized epochs, it is essential for most physical systems to have continuous-time risk guarantees; it is especially important for unstable systems.

The solution method finds a sequence of feedback control policies for LTV systems that minimizes the expected control cost subject to continuous-time chance constraints. We demonstrate the approach with a spacecraft orbit control scenario on an unstable orbit. Monte Carlo simulations with the optimized feedback policies show that our approach respects the continuous-time chance constraints whereas a classical approach results in the constraint violation between the discretized epochs.

Keywords: Stochastic optimal control , Machine learning , Optimization algorithms Abstract: We analyze a tree search problem with an underlying Markov decision process, in which the goal is to identify the best action at the root that achieves the highest cumulative reward. We present a new tree policy that optimally allocates a limited computing budget to maximize a lower bound on the probability of correctly selecting the best action at each node.

Compared to the widely used Upper Confidence Bound UCB type of tree policies, the new tree policy presents a more balanced approach to manage the exploration and exploitation trade-off when the sampling budget is limited. Furthermore, UCB assumes that the support of reward distribution is known, whereas our algorithm relaxes this assumption, and can be applied to game trees with mild modifications. A numerical experiment is conducted to demonstrate the efficiency of our algorithm in selecting the best action at the root.

Keywords: Stochastic optimal control , Adaptive control , Iterative learning control Abstract: Abstract: Under the Dynamic Resource Allocation DRA model, an administrator has the mission to allocate dynamically a limited budget of resources to the nodes of a network in order to reduce a diffusion process DP e.

The standard DRA assumes that the administrator has constantly full information and instantaneous access to the entire network. Towards bringing such strategies closer to real-life constraints, we first present the Restricted DRA model extension where, at each intervention round, the access is restricted to only a fraction of the network nodes, called sample. Our model introduces a sequential aspect in the decision process over the sample of each round, offering a completely new perspective to the dynamic DP control. Finally, we incorporate several sequential selection algorithms to SDRA control strategies and compare their performance in SIS epidemic simulations.

Keywords: Stochastic optimal control , Stochastic systems , Learning Abstract: We present RAEVL: Random Actions with Empirical Value Learning, one of the first practical algorithm for stochastic systems with continuous state and action spaces that finds near-optimal policies with high probability. It combines ideas of random search over action space with randomized function approximation and empirical value learning.

Theoretical analysis is done by viewing each iteration as application of a random operator, and doing probabilistic contraction analysis. This is combined with error concentration analysis for randomized function approximation and randomized optimization over actions. Preliminary numerical results indicate good performance. For the lower and upper value functions of the SZSDG, we establish the dynamic programming principle via the generalized stochastic backward semigroup associated with the BSDE. We then show that the lower and upper value functions are viscosity solutions to the associated Hamilton-Jacobi-Isaacs partial differential equations together with an algebraic equation.

This additional algebraic equation emerges due to dependence of the diffusion term on the solution of the BSDE. Keywords: Stochastic optimal control , Optimization Abstract: This paper constructs bounds on the expected value of a scalar function of a random vector.

The bounds are obtained using an optimization method, which can be computed efficiently using state-of-the-art solvers, and do not require integration or sampling the random vector. This optimization based approach is especially useful in stochastic programming, where the criteria to be minimized takes the form of an expected value. In particular, we minimize the bounds to solve problems of discrete time finite horizon open-loop control with stochastic perturbations and also uncertainty in the system's parameters.

We illustrate this application with two numerical examples. Keywords: Optimization algorithms , Distributed control , Large-scale systems Abstract: This paper introduces a novel distributed algorithm over static directed graphs for solving big data convex optimization problems in which the dimension of the decision variable can be extremely high and the objective function can be nonsmooth. In the proposed algorithm nodes in the network update and communicate only blocks of their current solution estimate rather than the entire vector.

The algorithm consists of two main steps: a consensus step and a subgradient update on a single block of the optimization variable which is then broadcast to neighbors. Agents are shown to asymptotically achieve consensus by studying a block-wise consensus protocol over random graphs. Then convergence to the optimal cost is proven in expected value by exploiting the consensus of agents estimates and randomness of the algorithm. Finally, as a numerical example, a distributed linear classification problem is solved by means of the proposed algorithm.

Keywords: Optimization algorithms , Large-scale systems , Distributed control Abstract: In this paper, we consider a network of processors that want to cooperatively solve a large-scale, convex optimization problem. Each processor has knowledge of a local cost function that depends only on a local variable. The goal is to minimize the sum of the local costs, while making the variables satisfy both local constraints and a global coupling constraint. We propose a simple, fully distributed algorithm, that works in a random, time-varying communication model, where at each iteration multiple edges are randomly drawn from an underlying graph.

The algorithm is interpreted as a primal decomposition scheme applied to an equivalent problem reformulation. Almost sure convergence to the optimal cost of the original problem is proven by resorting to approaches from block subgradient methods. Specifically, the communication structure is mapped to a block structure, where the blocks correspond to the graph edges and are randomly selected at each iteration.

Moreover, an almost sure asymptotic primal recovery property, with no averaging mechanisms, is shown. A numerical example corroborates the theoretical analysis. Keywords: Optimization algorithms , Networked control systems , Distributed control Abstract: We study a class of distributed optimization problems of minimizing the sum of potentially non-differentiable convex objective functions without requiring strong convexity. A novel approach to the analysis of asynchronous distributed optimization is developed. An iterative algorithm based on dual decomposition and block coordinate ascent is implemented in an edge based manner.

We extend available results in the literature by allowing multiple and potentially overlapping blocks to be updated at the same time with non-uniform probabilities assigned to different blocks.

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Sublinear convergence with probability one is proved for the algorithm under the aforementioned weak assumptions. A numerical example is provided to illustrate the effectiveness of the algorithm. This problem naturally calls for distributed optimization algorithms, in which the agents cooperatively solve the problem through local computations and communications with neighbors. While many of the existing distributed algorithms with constant stepsize can only converge to a neighborhood of optimal solution, some recent methods based on augmented Lagrangian and method of multipliers can achieve exact convergence with a fixed stepsize.

However, these methods either suffer from slow convergence speed or require minimization at each iteration. In this work, we develop a class of distributed first-order primal-dual methods, which allows for multiple primal steps per iteration.

### Introduction

This general framework makes it possible to control the trade-off between the performance and the execution complexity in primal-dual algorithms. We show that for strongly convex and Lipschitz gradient objective functions, this class of algorithms converges linearly to the optimal solution under appropriate constant stepsize choices. Simulation results confirm the superior performance of our algorithm compared to existing methods.

Keywords: Optimization algorithms , Distributed control , Predictive control for linear systems Abstract: In this paper, we propose to reduce the number of iterations required in the implementation of distributed Model Predictive Control dMPC based on dual decomposition. To this aim, a dynamic Lagrange multipliers fixation algorithm DLMFA is proposed by continually fixing the value of Lagrange multipliers, and a local optimization problems dynamic sizing algorithm LOPDSA is proposed by continually reducing the size of local optimization problem during the iteration through an original prediction horizon reduction.

The proposed algorithms are based on the Uzawa method, which is improved because of the specific nature of the constraints in a MPC context. The basics of these improvements stem from the particular behavior of the Lagrange multipliers and their fluctuations over the prediction horizon. Numerical experiments have shown that the iteration number, as well as the computation time of LOPDSA, are significantly reduced compared to Uzawa method.

Keywords: Networked control systems , Control over communications , LMIs Abstract: We address the problem of stabilizing a set of discrete-time systems over a communication network. We consider the case when the number of channels is limited and propose a dynamic scheduling scheme that, at a given time, determines which subset of systems get access to the channel for feedback control. This problem is addressed by considering two separate problemsscheduling systems over noiseless channels and stabilizing a system over an AWGN channel.

The scheduling problem is addressed in the switched system framework by making use of a min-type Lyapunov function. We provide a sufficient condition in the form of Linear Matrix Inequalities LMIs to schedule a subset of systems while achieving stability of all systems. We also explicitly determine the scheduling scheme. Next, we provide a novel LMI-based necessary and sufficient condition for stabilization of a discrete-time system over a discrete-time AWGN channel.

Finally, we appropriately combine the two results to obtain an LMI-based sufficient condition for the join scheduling-stabilization problem. Keywords: Networked control systems , Sensor networks , Optimization algorithms Abstract: In the optimal sampling problem, we are interested in selecting the optimal subset of times to sample a sensor such that the mean square estimation error MSE between the unobserved states and the estimated states is minimized.

In this problem, the states evolve according to a discrete, LTI system and the sensor takes measurements according to a discrete, LTI system. A Kalman Filter recursively estimates the evolving states based on the sensor measurements. Ideally, we would select all available times in the horizon of interest to sample the sensor for estimating the states.

However, there are communication and energy costs affiliated with sampling and therefore we aim to minimize the estimation error when the number of times we can sample is fixed. There have been multiple attempts to solve this problem by relaxing the original problem, which is NP-hard. Such relaxations allow for nice algorithms, but provide no guarantees on the gap between the solution of the relaxed problem and the solution of the original problem.

We leverage the idea of supermodularity in discrete optimization to show a greedy solution to the sampling problem will produce a near-optimal solution with an approximation factor. To prove the supermodularity property for the mean squared estimation error as a function of samples, we make a few assumptions: the covariance matrices for the system and measurement noise are diagonal matrices of the form constant times identity with certain restrictions on the constant; C and A matrices only have positive elements.

Keywords: Networked control systems , Distributed control , Agents-based systems Abstract: A distributed event-triggered controller is developed for approximate leader-follower consensus while being robust to Byzantine agents for a homogeneous multi-agent system MAS. The event-triggered strategy enables intermittent communication and sensing.

Moreover, each agent can detect Byzantine adversaries within their neighbor set and selectively disregard their transmission to achieve approximate leader-follower consensus. A non-smooth Lyapunov stability analysis is leveraged to prove consensus of the MAS.

Keywords: Networked control systems , Sampled-data control , Time-varying systems Abstract: The paper studies the H-inf optimal control in discrete-time systems under the constraint that the information exchange between the sensor- and actuator-side parts of the controller is intermittent, a priori unknown, and independent of the process. Khammash , Listening to the noise: random fluctuations reveal gene network parameters , Molecular Systems Biology , vol. Neuert, B. Munsky, R. Tan, L. Teytelman, M. Khammash et al. Papoulis , Probability, random variables, and stochastic processes.

McGraw-Hill series in electrical engineering , Paulsson , Models of stochastic gene expression , Physics of Life Reviews , vol. Sanft, S. Wu, M. Roh, J. Fu, R. Lim et al. Stefan, C. Pinel, S. Pinhal, E. Cinquemani, J. Geiselmann et al. Zechner, J. Ruess, P. Krenn, S. Pelet, M. Peter et al. Zechner, M. Unger, S. Peter, and H. Koeppl , Scalable inference of heterogeneous reaction kinetics from pooled single-cell recordings , Nature Methods , vol. Zulkower, M. Page, D. Ropers, J.

Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification
Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification
Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification
Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification
Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification
Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification
Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification
Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification
Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification

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