Search

Various efforts have been devoted to developing stabilizing distributed model predictive control (MPC) schemes for tracking piecewise constant references. In these schemes, terminal sets are usually computed offline and used in the MPC online phase to guarantee recursive feasibility and asymptotic stability. Maximal invariant terminal sets do not necessarily respect the distributed structure of the network, hindering the distributed implementation of the controller. On the other hand, ellipsoidal terminal sets respect the distributed structure, but may lead to conservative schemes. In this article, a novel distributed MPC scheme is proposed for reference tracking of networked dynamical systems, where the terminal ingredients are reconfigured online depending on the closed-loop states to alleviate the aforementioned issues. The resulting nonconvex infinite-dimensional problem is approximated using a quadratic program. The proposed scheme is tested in simulation, where the proposed MPC problem is solved using distributed optimization. [ABSTRACT FROM AUTHOR]

This article addresses the problem of consistently identifying a single transfer function in a network of dynamic systems using only observational data. It is assumed that the topology is partially known, the forcing inputs are not measured, and that only a subset of the nodes outputs is accessible. The developed technique is applicable to scenarios encompassing confounding variables and feedback loops, which are complicating factors potentially introducing bias in the estimate of the transfer function. The results are based on the prediction of the output node using the input node along with a set of additional auxiliary variables which are selected only from the observed nodes. Similar prediction error methods provide only sufficient conditions for the appropriate choice of auxiliary variables and assume a priori information about the location of strictly causal operators in the network. In this article, such an a priori knowledge is not required. A most remarkable feature of our approach is that the conditions for the selection of the auxiliary variables are purely graphical. Furthermore, within single-output prediction methods, such conditions are proven to be necessary and sufficient to consistently identify all networks with a given topology. A fundamental consequence of this characterization is to enable the search of a set of auxiliary variables minimizing a suitable cost function for single-output prediction error identification. In the article, we suggest possible approaches to tackle such optimal identification problems. [ABSTRACT FROM AUTHOR]

This article develops the sliding mode control (SMC) strategy for linear impulsive systems with matched disturbances. Such systems describe dynamical systems that evolve according to continuous process most of the time with matched disturbances, but instantaneously exhibit discontinuities, i.e., impulsive disturbances. A suitable sliding surface function is proposed and the resulting sliding mode dynamics are a class of linear impulsive systems. Some sufficient conditions coupled with impulse time sequences are proposed to guarantee the reachability of the predesigned sliding surface and the stability of the reduced-order impulsive systems under sliding motion, respectively. We show how restrictions on the structure of the systems and impulses should be imposed to ensure the feasibility of the proposed SMC strategy in comparison with general SMC in continuous space. To show the effectiveness of the proposed approach, two simulation examples are presented at the end. [ABSTRACT FROM AUTHOR]

Checking uniform attractivity of a time-varying dynamic system without a strict Lyapunov function is challenging as it requires the characterization of the limiting behavior of a set of trajectories. In the context of hybrid nonlinear time-varying systems, characterizing such limiting or convergent behaviors is even harder due to the complexity stemming from both continuous-time variations as well as discrete-time jumps. In this article, an extension of the standard hybrid time domain is introduced to define limiting behaviors, using set convergence, when time approaches either positive infinity or negative infinity. In particular, it is shown how to characterize limiting behaviors under the condition that an output signal approaches zero. Such limiting behaviors and their associated limiting systems can be used to verify uniform global attractivity. Particularly, a generalization of the classic Krasovskii–LaSalle theorem is obtained for hybrid time-varying systems. Two examples are used to demonstrate the effectiveness of the results. [ABSTRACT FROM AUTHOR]

Identification of dynamical systems from low-resolution quantized observations presents several challenges because of the limited amount of information available in the data and since proper algorithms have to be designed to handle the error due to quantization. In this article, we consider identification of infinite impulse response models from quantized outputs. Algorithms both for maximum-likelihood estimation and Bayesian inference are developed. Finally, a particle-filter approach is presented for recursive reconstruction of the latent nonquantized outputs from past quantized observations. [ABSTRACT FROM AUTHOR]

The deployment of autonomous systems in uncertain and dynamic environments has raised fundamental questions. Addressing these is pivotal to build fully autonomous systems and requires a systematic integration of planning and control. We first propose reactive risk signal interval temporal logic (ReRiSITL) as an extension of signal temporal logic (STL) to formulate complex spatiotemporal specifications. Unlike STL, ReRiSITL allows to consider uncontrollable propositions that may model humans as well as random environmental events such as sensor failures. Additionally, ReRiSITL allows to incorporate risk measures, such as (but not limited to) the conditional value-at-risk, to measure the risk of violating certain spatial specifications. Second, we propose an algorithm to check if an ReRiSITL specification is satisfiable. For this purpose, we abstract the ReRiSITL specification into a timed signal transducer and devise a game-based approach. Third, we propose a reactive planning and control framework for dynamical control systems under ReRiSITL specifications. [ABSTRACT FROM AUTHOR]

This article is devoted to the existence and the global stability of stationary solutions for stochastic Hopfield neural networks with time delays and additive white noises. Using the method of random dynamical systems, we present a new approach to guarantee that the infinite-dimensional stochastic flow generated by stochastic delay differential equations admits a globally attracting random equilibrium in the state-space of continuous functions. An example is given to illustrate the effectiveness of our results, and the forward trajectory synchronization will occur. [ABSTRACT FROM AUTHOR]

The problem of filter design is considered for continuous-time linear stochastic systems using distributed sensors. Each sensor unit, represented by a node in an undirected and connected graph, collects some information about the state and communicates its own estimate with the neighbors. It is stipulated that this communication between sensor nodes connected by an edge is time-sampled randomly and for each edge, the sampling process is an independent Poisson counter. Our proposed filtering algorithm for each sensor node is a stochastic hybrid system: It comprises a continuous-time differential equation, and at random time instants when communication takes place, each sensor node updates its state estimate based on the information received by its neighbors. In this setting, we compute the expectation of the error covariance matrix for each unit which is governed by a matrix differential equation. To study the asymptotic behavior of these covariance matrices, we show that if the gain matrices are appropriately chosen and the mean sampling rate is large enough, then the error covariances practically converge to a constant matrix. [ABSTRACT FROM AUTHOR]

Safety for dynamical systems is often posed as an invariance constraint, requiring the system trajectory to remain in some safe subset of the state-space for all time. This note presents new tools for studying reachability and set invariance for nondeterministic systems subject to a disturbance input using the theory of mixed-monotone dynamical systems. The vector field of a mixed-monotone system is characterized as being decomposable into increasing and decreasing components, which allows the dynamics to be embedded in a higher dimensional embedding system. Even though the original system is nondeterministic due to the unknown disturbance input, the embedding system has no disturbance and a single simulation of the embedding system provides bounds for reachable sets of the original dynamics. In this article, we present an efficient method for identifying robustly forward invariant and attractive sets for mixed-monotone systems by studying equilibria and their stability properties of the corresponding embedding system. We show how this approach can be applied to either the backward-time dynamics or a set of linearly transformed dynamics to establish different robustly forward invariant sets for the original dynamics, and we show also how periodic solutions to the embedding system establish invariant regions for the original dynamics as well. The findings of this work are demonstrated through two numerical examples and two case studies, including a five-dimensional planar quadrotor system. [ABSTRACT FROM AUTHOR]

We examine the problem of testing a parametric form of the nonlinearity of Hammerstein systems against a nonparametric alternative. This article proposes test statistics relying on nonparametric orthogonal series regression function estimates. An asymptotic theory of the proposed tests is established including the asymptotic normality under the null hypothesis that the parametric form of the system nonlinearity is correct. Moreover, it is shown that under nonparametric alternatives, the probability of detecting that the parametric model is incorrect tends to one. The rate of detecting local alternatives is also established. [ABSTRACT FROM AUTHOR]

We study the problem of remote state estimation in the presence of an eavesdropper. A sensor transmits state information over a packet-dropping link to a legitimate user. This information is randomly overheard by an eavesdropper. To reduce information leakage to the eavesdropper, previous studies have shown that by encoding the estimate with the acknowledgments (Acks), perfect secrecy can be achieved. However, this strategy greatly relies on the accuracy of the Acks and may easily fail if the Acks are compromised by cyberattacks. In this article, we tackle this issue by proposing to switch between sending an encoded state and sending the plain state to stay resilient against fake Acks. Specifically, we assume the Acks to be randomly attacked and derive recursive expressions for the minimum-mean-squared error estimates and error covariance matrices at the legitimate user and at the eavesdropper. Based upon this, we propose a transmission policy that depends on the probability of synchronization. We formulate a partially observable Markov decision process to model the evolution of the synchronization status and derive associated optimal transmission policies. Numerical examples are provided to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we propose a tractable nonlinear fault estimation filter along with explicit performance bounds for a class of linear dynamical systems in the presence of both additive and nonlinear multiplicative faults. We consider the case, where both faults may occur simultaneously and through an identical dynamical relationship, a setting that is relevant to several application domains, including automotive driving, aviation, and chemical plants. The proposed filter architecture combines tools from model-based approaches in the control literature and regression techniques from machine learning. To this end, we view the regression operator through a system-theoretic perspective to develop operator bounds that are then utilized to derive performance bounds for the proposed estimation filter. In the case of constant, simultaneously, and identically acting additive and multiplicative faults, it can be shown that the estimation error converges to zero with an exponential rate. The performance of the proposed estimation filter in the presence of incipient faults is validated through an application on the lateral safety systems of SAE level 4 automated vehicles. The numerical results show that the theoretical bounds of this study are indeed close to the actual estimation error. [ABSTRACT FROM AUTHOR]

Optimal control problems are prevalent in model-based control, state and parameter estimation, and experimental design for complex dynamical systems. An approach for obtaining solutions to these problems is based on the notion of parsimonious input parameterization and comprises two tasks: the enumeration of arc sequences followed by the computation of optimal values of a small number of decision variables for each sequence. This article proposes an efficient global solution method for single-input optimal control problems for nonlinear dynamical systems with a potentially large number of states or complex dynamics via sum-of-squares polynomials and parallel computing. The method approximates the problem for a given arc sequence as a polynomial optimization problem that can be efficiently solved to global optimality via semidefinite programming. It is established that the difference between the cost obtained by the proposed method and the globally optimal cost of the original problem is bounded and depends on the polynomial approximation error. The method is illustrated by simulation examples of a reaction system and a rocket. [ABSTRACT FROM AUTHOR]

This article is about the data-driven computation of optimal control for a class of control affine deterministic nonlinear systems. We assume that the control dynamical system model is not available, and the only information about the system dynamics is available in the form of time-series data. We provide a convex formulation for the optimal control problem (OCP) of the nonlinear system. The convex formulation relies on the duality result in the dynamical system’s stability theory involving density function and Perron–Frobenius operator. We formulate the OCP as an infinite-dimensional convex optimization program. The finite-dimensional approximation of the optimization problem relies on the recent advances made in the Koopman operator’s data-driven computation, which is dual to the Perron–Frobenius operator. Simulation results are presented to demonstrate the application of the developed framework. [ABSTRACT FROM AUTHOR]

A directed acyclic graph (DAG) is a common topology in biological, engineering, and social networks. A network topology is critical in determining a collective behavior of a network dynamic system. For example, the convergence rate of a consensus behavior in a multiagent system relies on the eigenvalues of the Laplacian associated with the network topology. This article aims to analyze the influence of adding a reverse edge into a DAG on convergence rate. It reveals the existence of the so-called interfering reverse edges; adding one single edge in this category can reduce the so-called dominant convergence rate even for a large network. More specifically, a necessary and sufficient condition of an interfering reverse edge is explicitly constructed. According to the condition, a computationally efficient method is proposed to assess an interfering reverse edge. [ABSTRACT FROM AUTHOR]

Second-order dynamical systems are important tools for solving optimization problems, and most of the existing works in this field have focused on unconstrained optimization problems. In this article, we propose an inertial primal–dual dynamical system with constant viscous damping and time scaling for the linear equality constrained convex optimization problem, which consists of a second-order ordinary differential equation (ODE) for the primal variable and a first-order ODE for the dual variable. When the scaling satisfies certain conditions, we prove its convergence property without assuming strong convexity. Even the convergence rate can become exponential when the scaling grows exponentially. We also show that the obtained convergence property of the dynamical system is preserved under a small perturbation. [ABSTRACT FROM AUTHOR]

We present order reduction results for linear time invariant descriptor systems. Results are given for both forced and unforced systems as well methods for constructing the reduced order systems. Our results establish a precise connection between classical and new results on this topic, and lead to an elementary construction of quasi-Weierstrass forms for a descriptor system. Examples are given to illustrate the usefulness of our results. [ABSTRACT FROM AUTHOR]

This article concerns robustness analysis for interconnections of two dynamical systems (described by upper semicontinuous differential inclusions) using a generalized notion of derivatives associated with locally Lipschitz Lyapunov functions obtained from a finite family of differentiable functions. We first provide sufficient conditions for input-to-state stability for differential inclusions, using a class of nonsmooth (but locally Lipschitz) candidate Lyapunov functions and the concept of Lie generalized derivative. In general our conditions are less conservative than the more common Clarke derivative-based conditions. We apply our result to state-dependent switched systems, and to the interconnection of two differential inclusions. As an example, we propose an observer-based controller for certain nonlinear two-mode state-dependent switched systems. [ABSTRACT FROM AUTHOR]

This article addresses the problem of learning the optimal control policy for a nonlinear stochastic dynamical. This problem is subject to the “curse of dimensionality” associated with the dynamic programming method. This article proposes a novel decoupled data-based control (D2C) algorithm that addresses this problem using a decoupled, “open-loop–closed-loop,” approach. First, an open-loop deterministic trajectory optimization problem is solved using a black-box simulation model of the dynamical system. Then, closed-loop control is developed around this open-loop trajectory by linearization of the dynamics about this nominal trajectory. By virtue of linearization, a linear quadratic regulator based algorithm can be used for this closed-loop control. We show that the performance of D2C algorithm is approximately optimal. Moreover, simulation performance suggests a significant reduction in training time compared to other state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

This article develops data-driven methods to identify eigenfunctions of the Koopman operator associated with a dynamical system and subspaces that are invariant under the operator. We build on Extended Dynamic Mode Decomposition (EDMD), a data-driven method that finds a finite-dimensional approximation of the Koopman operator on the span of a predefined dictionary of functions. We propose a necessary and sufficient condition to identify Koopman eigenfunctions based on the application of EDMD forward and backward in time. Moreover, we propose the Symmetric Subspace Decomposition (SSD) algorithm, an iterative method that provably identifies the maximal Koopman-invariant subspace and the Koopman eigenfunctions in the span of the dictionary. We also introduce the Streaming SSD algorithm, an online extension of SSD that only requires a small fixed memory and incorporates new data as is received. Finally, we propose an extension of SSD that approximates Koopman eigenfunctions and invariant subspaces when the dictionary does not contain sufficient informative eigenfunctions. [ABSTRACT FROM AUTHOR]

In this article, we propose a novel technique for the stability analysis of discrete-time switched systems under constrained switching based on the semitensor product (STP) of matrices and vector-representation of logic. We assume that the admissible switching signals are generated by a logic dynamical system, referred to as the logic dynamical generator (LDG) of the admissible switching. We demonstrate that switching with minimum dwell-time and restricted successors represent two special cases of this type, by constructing their respective LDGs. First, we transfer the LDG into algebraic form, based on the vector-representation of logic. Next, we merge the logic state of the LDG and the continuous state of the original switched system using the STP and derive the dynamics of the merged state; a switched system without constraints on switching. The stability of the switched system under constrained switching is equivalent to the stability of the merged system under arbitrary switching. Based on this, sufficient conditions for the asymptotical stability of nonlinear switched systems with analytic subsystems under constrained switching are obtained. Specifically, in the case of all subsystems being linear, necessary and sufficient conditions for asymptotical stability under constrained switching are proposed. [ABSTRACT FROM AUTHOR]

Semilinear discrete-time control systems with periodic coefficients are considered. The problem of uniform global asymptotic stabilization of the zero equilibrium of the closed-loop system by state feedback is studied. It is assumed that the free dynamic system has a Lyapunov stable zero equilibrium. The method for constructing a damping control is extended from time-invariant systems to time varying periodic semilinear discrete-time systems. By using this approach, sufficient conditions for uniform global asymptotic stabilization for those systems are obtained. Moreover, the converse Lyapunov theorem on Lyapunov (nonasymptotic) stability is proved for complex and real linear periodic discrete-time systems. Finally, examples of using the obtained results are presented. [ABSTRACT FROM AUTHOR]

LaSalle’s invariance principle is a commonly used extension of Lyapunov’s second method to study asymptotic stability of nonlinear systems. If the system can be written in polynomial form, the examination can be automated using algebraic geometry and quantifier elimination. This article addresses this automated examination using a method relying on polynomial ideals and applies it on some example systems. In addition, some properties of these special ideals are derived that allow to reduce the computational effort significantly. [ABSTRACT FROM AUTHOR]

The Lyapunov equation for polytopic linear inclusions over the space of Minkowski functions of nonempty compact and convex sets that contain the origin as an interior point is studied. In particular, necessary and sufficient conditions for the characterization, existence, and uniqueness of its fundamental solution are derived. [ABSTRACT FROM AUTHOR]

In this article, we investigate the distributed state estimation problem for a nonlinear dynamical system. Relying upon the assumption of the existence of a block-triangular “uniform observability” canonical form, local high-gain observers are used to stabilize the estimation error for the observable portion of the state, and exchange of information between neighbors is used to achieve consensus in the estimation of the state of the unobservable part. This article addresses, first, the case of an autonomous system and, then, the case of a controlled dynamical system under the assumption that the exogenous input is accessible by each node in the network. Based on the designed state estimator, each distributed sensor node can achieve asymptotic estimation of the full state, with an arbitrarily fixed decay rate of the observation error. [ABSTRACT FROM AUTHOR]

In this article, we investigate the behavior of agents with bounded rationality, attacking a set of stochastic sensors measuring the state of a binary event. The coordination problem between the attackers is formulated as a multiplayer non-zero-sum one-shot game. The objective of each attacker is to maximize the probability that a certain detector will produce an erroneous estimate of the true event, while they remain stealthy. To better predict the outcome of this game, we categorize the players based on the number of strategic thinking steps they will take. Each level- $k$ attacker behaves based on subjective beliefs of the others’ behaviors, which are quantified via a Poisson distribution over the lower levels. The expected best responses for the type of each attacker are derived. The limiting conditions as the cognitive levels increase, as well as when the attackers fully coordinate, are shown to converge to the Nash equilibrium. [ABSTRACT FROM AUTHOR]

In this article we study the problem of finite-time constrained optimal control of unknown stochastic linear time-invariant (LTI) systems, which is the key ingredient of a predictive control algorithm—albeit typically having access to a model. We propose a novel distributionally robust data-enabled predictive control (DeePC) algorithm which uses noise-corrupted input/output data to predict future trajectories and compute optimal control inputs while satisfying output chance constraints. The algorithm is based on 1) a nonparametric representation of the subspace spanning the system behavior, where past trajectories are sorted in Page or Hankel matrices; and 2) a distributionally robust optimization formulation which gives rise to strong probabilistic performance guarantees. We show that for certain objective functions, DeePC exhibits strong out-of-sample performance, and at the same time respects constraints with high probability. The algorithm provides an end-to-end approach to control design for unknown stochastic LTI systems. We illustrate the closed-loop performance of the DeePC in an aerial robotics case study. [ABSTRACT FROM AUTHOR]

Complex networked systems can be modeled and represented as graphs, with nodes representing the agents and the links describing the dynamic coupling between them. The fundamental objective of network identification for dynamic systems is to identify causal influence pathways. However, dynamically related data streams that originate from different sources are prone to corruption caused by asynchronous time-stamps, packet drops, and noise. In this article, we show that identifying causal structure using corrupt measurements results in the inference of spurious links. A necessary and sufficient condition that delineates the effects of corruption on a set of nodes is obtained. Our theory applies to nonlinear systems, and systems with feedback loops. Our results are obtained by the analysis of conditional directed information (DI) in dynamic Bayesian networks. We provide consistency results for the conditional DI estimator that we use by showing almost-sure convergence. [ABSTRACT FROM AUTHOR]

Fault detection and isolation is an area of engineering dealing with designing online protocols for systems that allow one to identify the existence of faults, pinpoint their exact location, and overcome them. We consider multi-agent systems, where faults correspond to the disappearance of links in the underlying graph, simulating communication failures between agents. We study the case in which the agents and controllers are maximal equilibrium-independent passive, and use the known connection between steady states of these multi-agent systems and network optimization theory. We first study asymptotic methods of differentiating the faultless system from its faulty versions by studying their steady-state outputs. We explain how to apply the asymptotic differentiation to detect and isolate communication faults, with graph-theoretic guarantees on the number of faults that can be isolated, assuming the existence of a “convergence assertion protocol,” a data-driven method of asserting that a multi-agent system converges to a conjectured limit. We then construct two data-based convergence assertion protocols. We demonstrate our results by a case study. [ABSTRACT FROM AUTHOR]

In this article, we propose a notion of high-order (zeroing) barrier functions (HOBFs) that generalizes the concept of zeroing barrier functions and guarantees set forward invariance by checking their higher order derivatives. The proposed formulation guarantees asymptotic stability of the forward invariant set, which is highly favorable for robustness with respect to model perturbations. No forward completeness assumption is needed in our setting in contrast to existing HOBF methods. For the case of controlled dynamical systems, we relax the requirement of uniform relative degree and propose a singularity-free control scheme that yields a locally Lipschitz control signal and guarantees safety. Furthermore, the proposed formulation accounts for “performance-critical” control: it guarantees that a subset of the forward invariant set will admit any existing, bounded control law while still ensuring forward invariance of the set. Finally, a nontrivial case study with rigid-body attitude dynamics and interconnected cell regions as the safe region is investigated. [ABSTRACT FROM AUTHOR]

This article is concerned with the state estimation of linear dynamic systems when some sensors are corrupted by attackers. This problem is known as resilient state estimation (RSE), and aims to achieve, under some conditions, the estimation of the true state despite the malicious attacks on sensors. The state-of-art RSE methods provides a bound on estimation errors when external disturbance exists. However, it is shown in this article that the effect of the disturbance on estimation error may be larger than that for conventional observers, or even worse, resiliency may be lost for the disturbance that exceeds the bound. To resolve this issue, unknown input observer (UIO) mechanism is adopted in RSE for the purpose of estimating true plant state under both sensor attacks and external disturbance. Also achieved in this work is the method of partial state UIO synthesis, which relaxes the design requirement for full state UIO. In relation to resiliency, it is shown that 2 $q$ redundant detectability is a necessary condition for robust and resilient state estimator in order to tolerate up to $q$ sensor attacks. Numerical examples are given to validate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Model-free reinforcement learning attempts to find an optimal control action for an unknown dynamical system by directly searching over the parameter space of controllers. The convergence behavior and statistical properties of these approaches are often poorly understood because of the nonconvex nature of the underlying optimization problems and the lack of exact gradient computation. In this article, we take a step toward demystifying the performance and efficiency of such methods by focusing on the standard infinite-horizon linear–quadratic regulator problem for continuous-time systems with unknown state-space parameters. We establish exponential stability for the ordinary differential equation (ODE) that governs the gradient-flow dynamics over the set of stabilizing feedback gains and show that a similar result holds for the gradient descent method that arises from the forward Euler discretization of the corresponding ODE. We also provide theoretical bounds on the convergence rate and sample complexity of the random search method with two-point gradient estimates. We prove that the required simulation time for achieving $\epsilon$ -accuracy in the model-free setup and the total number of function evaluations both scale as $\log \, (1/\epsilon)$. [ABSTRACT FROM AUTHOR]

In this article, we deal with infinite-dimensional nonlinear forward complete dynamical systems which are subject to uncertainties. We first extend the well-known Datko lemma to the framework of the considered class of systems. Thanks to this generalization, we provide characterizations of the uniform (with respect to uncertainties) local, semi-global, and global exponential stability, through the existence of coercive and non-coercive Lyapunov functionals. The importance of the obtained results is underlined through some applications concerning 1) exponential stability of nonlinear retarded systems with piecewise constant delays, 2) exponential stability preservation under sampling for semilinear control switching systems, and 3) the link between input-to-state stability and exponential stability of semilinear switching systems. [ABSTRACT FROM AUTHOR]

Weconsider the problem of learning a realization for a linear time-invariant (LTI) dynamical system from input/output data. Given a single input/output trajectory, we provide finite time analysis for learning the system’s Markov parameters, from which a balanced realization is estimated using the classical Ho–Kalman algorithm. By proving a robustness result for the Ho–Kalman algorithm and combining it with the sample complexity results for Markov parameters, we show how much data are needed to approximate the balanced realization of the system up to a desired accuracy with high probability. [ABSTRACT FROM AUTHOR]

We develop two generalizations of contraction theory, namely, semi-contraction and weak-contraction theory. First, using the notion of seminorm, we propose a geometric framework for semi-contraction theory. We introduce matrix semimeasures and characterize their properties. We show that the spectral abscissa of a matrix is the infimum over weighted semimeasures. For dynamical systems, we use the semimeasure of their Jacobian to characterize the contractivity properties of their trajectories. Second, for weakly contracting systems, we prove a dichotomy for the asymptotic behavior of their trajectories and novel sufficient conditions for convergence to an equilibrium. Third, we show that every trajectory of a doubly contracting system, i.e., a system that is both weakly and semi-contracting, converges to an equilibrium point. Finally, we apply our results to various important network systems, including affine averaging and affine flow systems, continuous-time distributed primal-dual algorithms, and networks of diffusively coupled dynamical systems. For diffusively coupled systems, the semi-contraction theory leads to a sufficient condition for synchronization that is sharper, in general, than previously known tests. [ABSTRACT FROM AUTHOR]

Control scheduling refers to the problem of assigning agents or actuators to act upon a dynamical system at specific times so as to minimize a quadratic control cost, such as the objectives of the linear-quadratic-Gaussian (LQG) or linear quadratic regulator problems. When budget or operational constraints are imposed on the schedule, this problem is in general NP-hard and its solution can therefore only be approximated even for moderately sized systems. The quality of this approximation depends on the structure of both the constraints and the objective. This article shows that greedy control scheduling is near-optimal when the constraints can be written as an intersection of matroids, algebraic structures that encode requirements such as limits on the number of agents deployed per time slot, total number of actuator uses, and duty cycle restrictions. To do so, it proves that the LQG cost function is $\alpha$ -supermodular and provides new $\alpha /(\alpha + P)$ -optimality certificates for the greedy minimization of such functions over an intersection of $P$ matroids. These certificates are shown to approach the $1/(1+P)$ guarantee of supermodular functions in relevant settings. These results support the use of greedy algorithms in nonsupermodular quadratic control problems as opposed to typical heuristics such as convex relaxations and surrogate figures of merit, e.g., the $\log \det$ of the controllability Gramian. [ABSTRACT FROM AUTHOR]

This article establishes the existence of Lyapunov functions for analyzing the stability of a class of state-constrained systems, and it describes algorithms for their numerical computation. The system model consists of a differential equation coupled with a set-valued relation that introduces discontinuities in the vector field at the boundaries of the constraint set. In particular, the set-valued relation is described by the subdifferential of the indicator function of a closed convex cone, which results in a cone-complementarity system. The question of analyzing the stability of such systems is addressed by constructing cone-copositive Lyapunov functions. As a first analytical result, we show that exponentially stable complementarity systems always admit a continuously differentiable cone-copositive Lyapunov function. Putting some more structure on the system vector field, such as homogeneity, we can show that the aforementioned functions can be approximated by a rational function of cone-copositive homogeneous polynomials. This latter class of functions is seen to be particularly amenable for numerical computation as we provide two types of algorithms for precisely that purpose. These algorithms consist of a hierarchy of either linear or semidefinite optimization problems for computing the desired cone-copositive Lyapunov function. Some examples are given to illustrate our approach. [ABSTRACT FROM AUTHOR]

In this article, we investigate a balancing-based model order reduction method for dynamical systems with a linear dynamical equation and a quadratic output function. With this aim, we propose a new algebraic observability Gramian for the system based on the Hilbert space adjoint theory. We then show that the proposed Gramian satisfies a particular type of generalized Lyapunov equation and we investigate its connection to energy functionals. It allows one to find states that are hard to control and hard to observe via an appropriate balancing transformation. Truncation of such states yields reduced-order models. Finally, based on $\mathscr {H}_2$ energy considerations, we derive error bounds, depending on the neglected singular values. We demonstrate the efficiency of the proposed methodology using a numerical example. [ABSTRACT FROM AUTHOR]

Given a plant whose trajectories of interest remain in a known compact set and an associated observer, we propose a general framework to modify this observer so that its state remains in a given convex set for all times, without altering the observer guaranteed performances in terms of convergence and robustness to external disturbances. The methodology can be applied to any time-varying continuous-time, discrete-time, and hybrid system/observer, for which a quadratic Lyapunov function is used for the analysis. The proposed approach is relevant, for instance, to remove the peaking phenomenon, to attenuate the effect of impulsive outliers in the measurement, to avoid aberrant estimates during transients, or to guarantee a given range for variables in embedded systems. [ABSTRACT FROM AUTHOR]

In multivehicle autonomous systems that operate under unknown or adversarial environments, it is a challenging task to simultaneously achieve source seeking and obstacle avoidance. Indeed, even for single-vehicle systems, smooth time-invariant feedback controllers based on navigation or barrier functions have been shown to be highly susceptible to arbitrarily small jamming signals that can induce instability in the closed-loop system, or that are able to stabilize spurious equilibria in the operational space. When the location of the source is further unknown, adaptive smooth source seeking dynamics based on averaging theory may suffer from similar limitations. In this article, we address this problem by introducing a class of novel distributed hybrid model-free controllers, that achieve robust source seeking and obstacle avoidance in multivehicle autonomous systems, with vehicles characterized by nonlinear continuous-time dynamics stabilizable by hybrid feedback. The hybrid source seeking law switches between a family of cooperative gradient-free controllers, derived from potential fields that satisfy mild invexity assumptions. The stability and robustness properties of the closed-loop system are analyzed using Lyapunov tools and singular perturbation theory for set-valued hybrid dynamical systems. The theoretical results are validated via numerical and experimental tests. [ABSTRACT FROM AUTHOR]

In this article, we develop a data-based controller design framework for diffusively coupled systems with guaranteed convergence to an $\epsilon$ -neighborhood of the desired formation. The controller is composed of a fixed controller with an adjustable gain on each edge. Via passivity theory and network optimization, we not only prove that there exists a gain attaining the desired formation control goal, but we present a data-based method to find an upper bound on this gain. Furthermore, by allowing for additional experiments, the conservatism of the upper bound can be reduced via iterative sampling schemes. The introduced scheme is based on the assumption of passive systems, which we relax by discussing different methods for estimating the systems’ passivity shortage, as well as applying transformations passivizing them. Finally, we illustrate the developed model-free cooperative control scheme with a case study. [ABSTRACT FROM AUTHOR]

This article studies the evolution of social power in influence networks with stubborn individuals. We formulate two models grounded on the Friedkin–Johnsen opinion dynamics and the reflected appraisal mechanism; the models are defined over issue sequences and over a single issue, respectively. The key advance over the original DeGroot–Friedkin dynamics is that the proposed models include the empirically observed phenomenon of stubbornness and are, therefore, more realistic. We obtain various results about the existence of equilibria, their uniqueness, and their global attractivity. For example, we show that the single-issue and issue-sequence models have the same equilibrium social power. We also show that the equilibrium social power depends only upon interpersonal accorded influence and individuals’ stubbornness in several cases. Roughly speaking, a more stubborn individual has more equilibrium social power. In comparison with the original DeGroot–Friedkin model, the introduction of stubbornness leads to fundamental sociological differences: Full autocracy can never be achieved, whereas democracy can be achieved under all network topologies. [ABSTRACT FROM AUTHOR]

We prove a robust converse barrier function theorem via the converse Lyapunov theory. While the use of a Lyapunov function as a barrier function is straightforward, the existence of a converse Lyapunov function as a barrier function for a given safety set is not. We establish this link by a robustness argument. We show that the closure of the forward reachable set of a robustly safe set must be robustly asymptotically stable under mild technical assumptions. As a result, all robustly safe dynamical systems must admit a robust barrier function in the form of a Lyapunov function for set stability. We present the results in both continuous-time and discrete-time settings and remark on connections with various barrier function conditions. [ABSTRACT FROM AUTHOR]

This article studies controllability of a discrete-time linear dynamical system using nonnegative and sparse inputs. These constraints on the control input arise naturally in many real-life systems, where the external influence on the system is unidirectional, and activating each input node adds to the cost of control. We derive the necessary and sufficient conditions for the controllability of the system, without imposing any constraints on the system matrices. Unlike the straightforward extension of the well-known Kalman-rank-based controllability criteria, the conditions presented in this article can be verified in polynomial time. The verification complexity of our conditions is independent of the sparsity level. Our results also provide a closed-form expression for the minimum number of control nodes to be activated at every time instant to ensure controllability of the system using nonnegative controls. Furthermore, for the systems satisfying the derived controllability conditions, we provide approximate algorithms that solve for the nonnegative inputs. [ABSTRACT FROM AUTHOR]

This article proposes a formal framework based on discrete event systems in order to analyze the democratic progress and regression in a society controlled by networked agents. For this purpose, we construct a simple model using a finite state automaton that describes the dynamic behavior of progress and regression in a democracy. We represent a network of agents as a directed graph where each agent has its own objective. Each agent may be a citizen or a group of people sharing a common objective, and it makes decisions on enabling or disabling events upon the observation of states of a system. Agents may have different decisions on the same event, and the final decision follows the majority rule. Upon this framework, we derive the necessary and sufficient conditions for a democratic system controlled by networked agents to be progressive or regressive, where a progressive one implies that it reaches a more equal state at which a larger number of agents meet their objectives. Finally, we obtain some convergence results for special graph topologies. [ABSTRACT FROM AUTHOR]

In this article, we propose a new method to obtain feedback controllers of an unknown dynamical system directly from noisy input/state data. The key ingredient of our design is a new matrix S-lemma that will be proven in this article. We provide both strict and nonstrict versions of this S-lemma, which are of interest in their own right. Thereafter, we will apply these results to data-driven control. In particular, we will derive nonconservative design methods for quadratic stabilization, $\mathcal {H}_2$ and $\mathcal {H}_{\infty }$ control, all in terms of data-based linear matrix inequalities. In contrast to previous work, the dimensions of our decision variables are independent of the time horizon of the experiment. Our approach, thus, enables control design from large datasets. [ABSTRACT FROM AUTHOR]

This article presents new mathematical results for exact and approximate relaxations of nonconvex optimal control problems defined on disconnected control sets. The exact relaxations are based upon extreme point relaxations wherein the relaxed control set is constructed so that its extreme points belong to the original set. A system property required for exactness is normality. In the absence of normality, control set approximations and system dynamic perturbations are introduced so that normality holds. The results hold for linear dynamical systems and nonlinear systems affine in the control. It is shown that the convex relaxations enable the efficient solution of problems in aerospace engineering, sparsity-promoting optimization, overactuated systems, and multiplexing systems. The computational guarantees associated with convex optimization make the relaxation techniques suitable for real-time control. [ABSTRACT FROM AUTHOR]

In this article, we present a Lyapunov framework for establishing stability with respect to a compact set for a nested interconnection of nonlinear dynamical systems ordered from slow to fast according to their convergence rates, where each of the dynamics is influenced only by the slower dynamics and the successive fastest one. The proposed approach explicitly considers more than two time scales, it does not require modeling multiple time scales via scalar time constants, and provides analytic bounds that make ad-hoc time-scale separation arguments rigorous. Motivated by the technical results, we develop a novel control strategy for a grid-forming power converter that consists of an inner cascaded two-degree-of-freedom controller and dispatchable virtual oscillator control as a reference model. The resulting closed-loop converter-based ac power system is in the form of a nested system with multiple time scales. We apply our technical results to obtain explicit bounds on the controller set points, branch powers, and control gains that guarantee almost global asymptotic stability of the multiconverter ac power system with respect to a prespecified solution of the ac power flow equations. Finally, we validate the performance of the proposed control structure in a case study using a high-fidelity simulation with detailed hardware validated converter models. [ABSTRACT FROM AUTHOR]

Network reconstruction of dynamical continuous-time (CT) systems is motivated by applications in many fields. Due to experimental limitations, especially in biology, data can be sampled at low frequencies, leading to significant challenges in network inference. We introduce the concept of “system aliasing” and characterize the minimal sampling frequency that allows reconstruction of CT systems from low sampled data. A test criterion is also proposed to detect the presence of system aliasing. With no system aliasing, this article provides an algorithm to reconstruct dynamic networks from full-state measurements in the presence of noise. With system aliasing, we add additional prior information such as sparsity to overcome the lack of identifiability. This article opens new directions in modeling of network systems where samples have significant costs. Such tools are essential to process available data in applications subject to experimental limitations. [ABSTRACT FROM AUTHOR]