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This paper considers the optimal control for hybrid systems whose trajectories transition between distinct subsystems when state-dependent constraints are satisfied. Though this class of systems is useful while modeling a variety of physical systems undergoing contact, the construction of a numerical method for their optimal control has proven challenging due to the combinatorial nature of the state-dependent switching and the potential discontinuities that arise during switches. This paper constructs a convex relaxation-based approach to solve this optimal control problem by formulating the problem in the space of relaxed controls, which gives rise to a linear program whose solution is proven to compute the globally optimal controller. This conceptual program is solved using a sequence of semidefinite programs whose solutions are proven to converge from below to the true solution of the original optimal control problem. Finally, a method to synthesize the optimal controller is developed. Using an array of examples, the performance of the proposed method is validated on problems with known solutions and also compared to a commercial solver. [ABSTRACT FROM AUTHOR]

This paper considers the quadratic control problem of discrete-time Markov jump linear systems with constraints on the norm of the state and control variables. We assume that the Markov chain parameter is not available, and instead, there is a detector, which emits signals providing information on this parameter. It is desired to derive a feedback linear control using the information provided by this detector in order to stochastically stabilize the closed-loop system, satisfy the constraints whenever the initial conditions belong to an invariant set, and minimize an upper bound for the quadratic cost. We show that a linear matrix inequality (LMI) optimization problem can be formulated in order to obtain a solution for this problem. Two other related problems, one for minimizing the guaranteed quadratic cost considering fixed initial conditions and the other for maximizing an estimate of the domain of an invariant set, can also be formulated using our LMI approach. This paper is concluded with some numerical simulations. [ABSTRACT FROM AUTHOR]

Recently, Thompson sampling has been shown to achieve good theoretical performance guarantees for stochastic control problems with parameter uncertainty when the state, control, and parameter spaces are all finite. Much less is known however about the performance of Thompson sampling when applied to continuous or more general spaces, which constitutes an important class of problems in practice. In this paper, we study Thompson sampling when applied to a broad class of average cost stochastic control problems where the state, control, and parameter spaces are all general measurable spaces. The main contributions of our paper are establishing theoretical performance guarantees for Thompson sampling as measured by: first, expected posterior sampling error; and second, average per period regret. [ABSTRACT FROM AUTHOR]

This paper presents reduction theorems for stability, attractivity, and asymptotic stability of compact subsets of the state space of a hybrid dynamical system. Given two closed sets $\Gamma _1 \subset \Gamma _2 \subset \mathbb {R}^n$ , with $\Gamma _1$ compact, the theorems presented in this paper give conditions under which a qualitative property of $\Gamma _1$ that holds relative to $\Gamma _2$ (stability, attractivity, or asymptotic stability) can be guaranteed to also hold relative to the state space of the hybrid system. As a consequence of these results, sufficient conditions are presented for the stability of compact sets in cascade-connected hybrid systems. We also present a result for hybrid systems with outputs that converge to zero along solutions. If such a system enjoys a detectability property with respect to a set $\Gamma _1$ , then $\Gamma _1$ is globally attractive. The theory of this paper is used to develop a hybrid estimator for the period of oscillation of a sinusoidal signal. [ABSTRACT FROM AUTHOR]

This paper considers the optimization problem for partially observable Markov decision processes (POMDPs) with the continuous state, observation, and action spaces. POMDPs with the discrete spaces have emerged as a promising approach to the decision systems with imperfect state information. However, in recent applications of POMDPs, there are many problems that have continuous states, observations, and actions. For such problems, due to the infinite dimensionality of the belief space, the existing studies usually discretize the continuous spaces with the sufficient or nonsufficient statistics, which may cause the curse of dimensionality and performance degradation. In this paper, based on the sensitivity analysis of the performance criteria, we have developed a simulation-based policy iteration algorithm to find the local optimal observation-based policy for POMDPs with the continuous spaces. The proposed algorithm needs none of the specific assumptions and prior information, and has a low computational complexity. One numerical example of the complicated multiple-input multiple-output beamforming problem shows that the algorithm has a significant performance improvement. [ABSTRACT FROM AUTHOR]

In this paper, we propose a control law for the almost-global asymptotic tracking (AGAT) of a smooth reference trajectory for a fully actuated simple mechanical system (SMS) evolving on a Riemannian manifold that can be embedded in a Euclidean space. The existing results on tracking for an SMS are either local, or almost global, only in the case the manifold is a Lie group. In the latter case, the notion of a configuration error is naturally defined by the group operation and facilitates a global analysis. However, such a notion is not intrinsic to a Riemannian manifold. In this paper, we define a configuration error followed by error dynamics on a Riemannian manifold, and then, prove the AGAT. The results are demonstrated for a spherical pendulum, which is an SMS on $S^2$ and for a particle moving on a Lissajous curve in $\mathbb {R}^3$. [ABSTRACT FROM AUTHOR]

The finite-time attitude synchronization problem is considered in this paper, where the rotation of each rigid body is expressed using the axis-angle representation. Two discontinuous and distributed controllers using the vectorized signum function are proposed, which guarantee almost global and local convergence, respectively. Filippov solutions and nonsmooth analysis techniques are adopted to handle the discontinuities. Sufficient conditions are provided to guarantee finite-time convergence and boundedness of the solutions. Simulation examples are provided to verify the performances of the control protocols designed in this paper. [ABSTRACT FROM AUTHOR]

This paper proposes a general approach to design convergent coordination control laws for multiagent systems subject to motion constraints. The main contribution of this paper is to prove in a constructive way that a gradient-descent coordination control law designed for single integrators can be easily modified to adapt for various motion constraints such as nonholonomic dynamics, linear/angular velocity saturation, and other path constraints while preserving the convergence of the entire multiagent system. The proposed approach is applicable to a wide range of coordination tasks such as rendezvous and formation control in two and three dimensions. As a special application, the proposed approach solves the problem of distance-based formation control subject to nonholonomic and velocity saturation constraints. [ABSTRACT FROM AUTHOR]

In this paper, we apply Thompson sampling to a class of average reward stochastic control problems with parameter uncertainty. Specifically, we study an average reward stochastic control problem over an infinite horizon in which both the reward and state transition distributions are parameterized by an unknown parameter taking values in a finite space. The main result of this paper is a proof showing that Thompson sampling achieves a worst case average per period regret of O(T^-1), which is asymptotically optimal. [ABSTRACT FROM PUBLISHER]

In this paper, we study mixed traffic flow in which some cars are under car-following control and others are under bilateral control. We also provide the necessary (modular string) stability condition for this type of mixed traffic, which can be viewed as an extension of the condition for pure car-following control-based traffic. This necessary stability condition provides some indication of how the introduction of self-driving cars (under bilateral control) will affect today's traffic. [ABSTRACT FROM AUTHOR]

In this paper, two important structural properties, i.e., strong observability and strong detectability, are introduced for linear hybrid systems with periodic jumps. These properties are characterized in terms of geometric and algebraic conditions over the system matrices. The concepts and characterizations of the weakly unobservable subspace and the hybrid invariant zeros are also introduced, respectively. An algorithm to compute the weakly unobservable subspace is provided. In addition, it is shown that there exists a close relationship between the hybrid invariant zeros and the properties ofstrong observability and strong detectability. Some examples illustrate the proposed properties. [ABSTRACT FROM AUTHOR]

We develop a novel lifting technique for nonlinear system identification based on the framework of the Koopman operator. The key idea is to identify the linear (infinite dimensional) Koopman operator in the lifted space of observables, instead of identifying the nonlinear system in the state space, a process which results in a linear method for nonlinear systems identification. The proposed lifting technique is an indirect method that does not require to compute time derivatives and is therefore well-suited to low-sampling rate data sets. Considering different finite-dimensional subspaces to approximate and identify the Koopman operator, we propose two numerical schemes: a main method and a dual method. The main method is a parametric identification technique that can accurately reconstruct the vector field of a broad class of systems. The dual method provides estimates of the vector field at the data points and is well-suited to identify high-dimensional systems with small datasets. This paper describes the two methods, provides theoretical convergence results, and illustrates the lifting techniques with several examples. [ABSTRACT FROM AUTHOR]

This paper develops a ratio-of-distance (RoD) rigidity theory to study when a framework can be uniquely determined by a set of RoD constraints up to similar transformations (translation, rotation, scaling, and reflection). In particular, a framework is specified by a set of RoD constraints (the RoD of a pair of edges joining a common vertex) instead of distance, bearing, and angle constraints assumed in existing literature. Its relations to three existing rigidity theories (distance, bearing, and angle rigidity theories) are established. The proposed RoD rigidity theory is further applied to the RoD-based similar formation stabilization problem, where the desired formation shape is expressed as a set of RoD constraints. Finally, numerical simulations are presented to illustrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

Modern applications of robotics typically involve a robot control system with an inner proportional–integral (PI) or proportional–integral–derivative (PID) control loop and an outer user-specified control loop for specifying the velocity (or position) command. Most existing outer loop controllers, however, do not take into consideration the dynamic effects of robots and their effectiveness relies on the ad hoc assumption that the inner PI or PID control loop is fast enough, and other torque-based controllers cannot be implemented in robots with closed architecture. This paper investigates the adaptive control of robotic systems with an inner/outer loop structure, taking into account the effects of the dynamics and uncertainties, and both the task-space control and joint-space control are considered. We propose a dynamic modularity approach to resolve this issue, and a class of adaptive outer loop control schemes is proposed and their role is to dynamically generate the joint velocity (or position) command for the low-level joint servoing loop. Without relying on the ad hoc assumption that the joint servoing loop is fast enough or modification of the low-level joint controller structure, we show that the proposed outer loop controllers can ensure stability and convergence of the closed-loop system. The formulated dynamic modularity approach is validated by the experimental results using different generations of the UR10 robots. [ABSTRACT FROM AUTHOR]

In this paper, we provide a general strategy based on Lyapunov functionals to analyze global asymptotic stability of linear infinite-dimensional systems subject to nonlinear dampings under the assumption that the origin of the system is globally asymptotically stable with a linear damping. To do so, we use the fact that for any linear infinite-dimensional system that is globally exponentially stable, there exists a Lyapunov functional. Then, we derive a Lyapunov functional for the nonlinear system, which is the sum of a Lyapunov functional coming from the linear system and another term that compensates the nonlinearity. Our results are then applied to the linearized Korteweg–de Vries equation and some wave equations. [ABSTRACT FROM AUTHOR]

In this paper, we present a hybrid control strategy that allows for global asymptotic tracking of reference trajectories evolving on smooth manifolds, with nominal robustness. Two different versions of the hybrid controller are presented: one which allows for discontinuities of the plant input and a second one that removes the discontinuities via dynamic extension. By taking an exosystem approach, we provide a general construction of a hybrid controller that guarantees global asymptotic stability of the zero tracking error set. The proposed construction relies on the existence of proper indicators and a transport map-like function for the given manifold. We provide a construction of these functions for the case where each chart in a smooth atlas for the manifold maps its domain onto the Euclidean space. We also provide conditions for exponential convergence to the zero tracking error set. To illustrate these properties, the proposed controller is exercised on three different compact manifolds—the two-dimensional sphere, the unit-quaternion group, and the special orthogonal group of order three—and further applied to the problems of obstacle avoidance in the plane and global synchronization on the circle. [ABSTRACT FROM AUTHOR]

This paper deals with the enforcement of generalized mutual exclusion constraints (GMECs) on time Petri nets (TPNs) with uncontrollable transitions by restricting the firing intervals of controllable transitions. Existing approaches do not exploit the timing information and consequently the system permissiveness is limited. The key idea behind the proposed approach is the online computation of a graph representing a reduced portion of the state space of a TPN system, and precisely the states that can be reached from the current one by firing only uncontrollable transitions. Such a graph is called partial modified state class graph (PMSCG) and is derived from another graph recently presented in the literature. Based on the PMSCG, a procedure to compute a supervisory control law enforcing a GMEC on a TPN system in a maximally permissive way is presented. [ABSTRACT FROM AUTHOR]

This paper investigates a formation control problem in which a group of kinematic unicycles is made to converge to a desired formation with parallel heading angles and come to a stop. A control law is presented, which solves this problem for almost all initial conditions in any given compact set. The proposed control law is local and distributed, meaning that each unicycle is only required to sense its relative displacement measured in its own body frame, and the relative heading angle with respect to each of its neighbors. No communication between the unicycles is required. The sensing graph is assumed to be connected, undirected, and time invariant. The idea used to solve the above-mentioned formation control problem is to rigidly attach to the body frame of each unicycle an appropriate fixed offset vector. Stabilizing the desired formation amounts to achieving consensus of the endpoints of the offset vectors, and simultaneously synchronizing the unicycles’ heading angles. A control law achieving this goal is constructed by combining a bounded translational consensus controller with an attitude synchronizer. As a special case, the proposed solution solves the full unicycle synchronization problem, in which the unicycle positions are made to converge to each other, while the unicycle headings are made to align. [ABSTRACT FROM AUTHOR]

This paper presents a dual-objective nonlinear model predictive control (NMPC) algorithm for stabilizing a target neighborhood of a state-space manifold of a nonlinear dynamical system and for concurrently optimizing an economic objective in this neighborhood. The control design is based on the transverse normal form description of the system dynamics. The NMPC scheme solves two optimal control problems (OCPs) in sequence, the first (transversal) OCP provides conditions for the convergence to the target neighborhood. The second (tangential) OCP optimizes the economic objective without compromising the convergence. The stability and performance properties of the resulting control scheme are discussed and its efficacy is illustrated in a tutorial example. [ABSTRACT FROM AUTHOR]

In this paper, we consider a broad class of infinite horizon discrete-time optimal control models that involve a nonnegative cost function and an affine mapping in their dynamic programming equation. They include as special cases several classical models, such as stochastic undiscounted nonnegative cost problems, stochastic multiplicative cost problems, and risk-sensitive problems with exponential cost. We focus on the case where the state space is finite and the control space has some compactness properties, and we emphasize shortest path-type models. We assume that the affine mapping has a semicontractive character, whereby for some policies it is a contraction, whereas for others it is not. In one line of analysis, we impose assumptions guaranteeing that the noncontractive policies cannot be optimal. Under these assumptions, we prove strong results that resemble those for discounted Markovian decision problems, such as the uniqueness of solution of Bellman's equation, and the validity of forms of value and policy iteration. In the absence of these assumptions, the results are weaker and unusual in character: the optimal cost function need not be a solution of Bellman's equation, and may not be found by value or policy iteration. Instead the optimal cost function over just the contractive policies is the largest solution of Bellman's equation, and can be computed by a variety of algorithms. [ABSTRACT FROM AUTHOR]

This paper addresses a distributed formation control problem of multiple mobile agents using relative positions and local bearings. First, a formation control method via distributed pattern matching is proposed, which is executed over each clique (i.e., complete induced subgraph) of a network. It is shown that this method achieves the best control performance for a given desired formation and network topology in the following sense: The closest formation to the desired one is achieved among all formations achievable by distributed and relative control over the network. Next, a necessary and sufficient network condition is derived under which the desired formation can be obtained. It turns out that a new concept of connectivity, called clique rigidity, plays a crucial role. Finally, the effectiveness of the proposed method is illustrated through simulations in both two and three dimensional spaces. [ABSTRACT FROM AUTHOR]

In this paper, we apply symmetry reduction techniques from geometric mechanics to sufficient conditions for local optimality in optimal control problems. After reinterpreting some previous results for left-invariant problems on Lie groups, we focus on optimal control problems with subgroup symmetry. For these problems, the necessary conditions for optimality can be simplified by exploiting symmetries so as to reduce the number of variables needed to describe trajectories of the system. We show that sufficient conditions for optimality, based on the non-existence of conjugate points, can be simplified in an analogous way to the necessary conditions. We demonstrate these simplifications by analyzing an optimal control problem that models a spinning top in a gravitational field, and we give particular attention to the example of an axisymmetric sleeping top. The results we derive in this paper allow us to determine which trajectories of a sleeping top are locally optimal solutions of the optimal control problem, which is a new result that has not appeared in previous literature. [ABSTRACT FROM PUBLISHER]

Decentralized optimization of distributed stochastic dynamical systems with two or more controls of the decision makers (DMs) has been an active area of research for over half a century. Although, such decentralized optimization problems are often formulated utilizing static team and person-by-person (PbP) optimality criteria, the corresponding static team theory results have not been extended to dynamical systems. In this first part of the two-part paper, we derive team and PbP optimality conditions for distributed stochastic differential systems, when the controls of the DMs generate actions based on different information structures. The necessary conditions are given by a Hamiltonian System described by coupled backward and forward stochastic differential equations (SDEs) and a conditional Hamiltonian, conditioned on the information structures available to the controls of the DMs. The sufficient conditions state that PbP optimality implies team optimality, if the Hamiltonian is convex in the state and/or actions spaces of the controls of the DMs. We show existence of relaxed team optimal strategies, when the information structures are not affected by the controls of the DMs. Throughout the paper we discuss similarities to analogous optimality conditions of centralized decision or control of stochastic systems, and we note a connections to mean field stochastic optimal control problems. [ABSTRACT FROM AUTHOR]

We are interested in understanding stability (almost sure boundedness) of stochastic approximation algorithms (SAs) driven by a “controlled Markov” process. Analyzing this class of algorithms is important, since many reinforcement learning (RL) algorithms can be cast as SAs driven by a “controlled Markov” process. In this paper, we present easily verifiable sufficient conditions for stability and convergence of SAs driven by a “controlled Markov” process. Many RL applications involve continuous state spaces. While our analysis readily ensures stability for such continuous state applications, traditional analyses do not. As compared to literature, our analysis presents a two-fold generalization: 1) the Markov process may evolve in a continuous state space and 2) the process need not be ergodic under any given stationary policy. Temporal difference (TD) learning is an important policy evaluation method in RL. The theory developed herein, is used to analyze generalized $\text{TD}(0)$ , an important variant of TD. Our theory is also used to analyze a TD formulation of supervised learning for forecasting problems. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the exact controllability properties of an advection–diffusion equation on a bounded domain, using time- and space-dependent velocity fields as the control parameters. This partial differential equation (PDE) is the Kolmogorov forward equation for a reflected diffusion process that models the spatiotemporal evolution of a swarm of agents. We prove that if a target probability density has bounded first-order weak derivatives and is uniformly bounded from below by a positive constant, then it can be reached in finite time using control inputs that are bounded in space and time. We then extend this controllability result to a class of advection–diffusion–reaction PDEs that corresponds to a hybrid switching diffusion process (HSDP), in which case the reaction parameters are additionally incorporated as the control inputs. For the HSDP, we first constructively prove controllability of the associated continuous-time Markov chain (CTMC) system in which the state space is finite. Then, we show that our controllability results for the advection–diffusion equation and the CTMC can be combined to establish controllability of the forward equation of the HSDP. Finally, we provide constructive solutions to the problem of asymptotically stabilizing an HSDP to a target nonnegative stationary distribution using time-independent state feedback laws, which correspond to spatially dependent coefficients of the associated system of PDEs. [ABSTRACT FROM AUTHOR]

This paper generalizes Thompson and Hilbert metrics to the space of spectral densities. The resulting complete metric space has the differentiable structure of a Finsler manifold with explicit geodesics. The corresponding distances are filtering invariant, can be computed efficiently, and admit geodesic paths that preserve rationality; these are properties of fundamental importance in many engineering applications. [ABSTRACT FROM AUTHOR]

This paper studies bearing-based formation control of a group of autonomous agents with the leader-first follower (LFF) structure in an arbitrary dimensional space. First, the bearing-based Henneberg construction and some properties of the LFF formation are introduced. Then, we propose and analyze bearing-only control laws that almost globally stabilize LFF formations to a desired formation. Further strategies to rotate and rescale the target formation are also discussed. Finally, simulation results are provided to support the analysis. [ABSTRACT FROM AUTHOR]

This paper is concerned with variational optimal control constructions whose solution yields a sampling algorithm. The particular form of the sampling algorithm considered here is a particle filter, designed to numerically approximate the solution to the global optimization problem. The theoretical significance of this study comes from its variational aspects. Specifically, the control input represents the solution of a mean-field-type optimal control problem. Its parametric counterpart, obtained when a parametric form of density is known a priori, is shown to be equivalent to the natural gradient algorithm. Explicit formulae for the filter are derived when the objective function is quadratic and the density is Gaussian. The optimal control construction of the particle filter is a significant departure from the classical importance sampling–resampling-based approaches. [ABSTRACT FROM AUTHOR]

This paper considers the stability of a single-input single-output (SISO) system with an input/output (I/O) delay, called SISO delay system hereafter. We first derive an algebraic criterion to solve the complete stability w.r.t. the delay $\tau$. Next, the stability analysis is extended to the case with additional free (plant and/or controller) parameters, denoted by a vector $X$. By adopting a mathematical tool for polynomial algebra, the discrimination system, the complete classification of the critical imaginary roots (CIRs) in the whole $X$ - $\tau$ parameter space may be obtained. Furthermore, with the above results, we present the stability crossing sets (SCSs) in a new form, called the colored SCSs. For each point on the SCS the asymptotic behavior information of the corresponding CIR can be coded by a certain color (red, green, or yellow). Hence, the stability in the $X$ - $\tau$ space can be analyzed easily and intuitively. Moreover, the geometry of the SCSs can be investigated from a new perspective, which will help not only to better understand the stability of the SISO delay system but also to considerably simplify the procedure for generating the colored SCSs. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamic formation control problem for multi-agent systems over directed networks, in which a desired spatial shape is time-varying instead of fixed one as usually assumed in the literature. Inspired by the fact that the existing approaches, including absolute positions based, relative positions based, inter-agent distances based, and inter-agent bearings based, for specifying a formation shape are not invariant under all three transformations: translations, rotations, and scalings, a novel specification for formation shapes is proposed that is invariant under translations, rotations, and scalings in two- and three-dimensional spaces, and thus more intrinsic. In doing so, a new notion, called matrix-valued Laplacian, for graphs is introduced in detail along with some useful properties. It is demonstrated that the matrix-valued Laplacian provides much flexibility. Subsequently, two controllers are designed for guaranteeing the achievement of a dynamic formation shape. It is proved that arbitrary dynamic geometric shape can be reached by the designed controllers. Finally, two numerical examples are provided for demonstrating the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

Several optimization-based control design techniques can be cast in the form of parametric optimization problems. The multiparametric quadratic programming (mpQP) represents a popular class often related to the control of constrained linear systems. The complete solution to mpQP takes the form of explicit feedback functions with a piecewise affine structure, valid in polyhedral partitions of the feasible parameter space known as critical regions. The recently proposed combinatorial approach for solving mpQP has shown better efficiency than geometric approaches in finding the complete solution to problems with high dimensions of the parameter vectors. The drawback of this method, on the other hand, is that it tends to become very slow as the number of constraints increases in the problem. This paper presents an alternative method for enumerating all optimal active sets in an mpQP based on theoretical properties of adjacent critical regions and their corresponding optimal active sets. Consequently, it results in excluding a noticeable number of feasible but not optimal candidate active sets from investigation. Therefore, the number of linear programs that should be solved decreases noticeably and the algorithm becomes faster. Simulation results confirm the reliability of the suggested method in finding the complete solution to the mpQPs while decreasing the computational time compared favorably with the best alternative approaches. [ABSTRACT FROM AUTHOR]

For a system matrix $M$ , this paper explores the smallest (Frobenius) norm additive structured perturbation $\delta M$ for which a system property $\mathcal {P}$ (e.g., controllability, observability, stability, etc.) fails to hold, i.e., $\delta M$ is the structured perturbation with smallest Frobenius norm such that there exists a property matrix $R\in \mathcal {P}$ for which $M{-}\delta M{-}R$ drops rank. The Frobenius norm is used because of its direct dependence on the magnitude of each entry in the perturbation matrix. Necessary conditions on a locally minimum norm structured rank-reducing perturbation $\delta M$ and associated property matrix $R$ are set forth and proven. An iterative algorithm is also set forth that computes a locally minimum norm structured perturbation and associated property matrix satisfying the necessary conditions. Algorithm convergence is proven using a discrete Lyapunov function. [ABSTRACT FROM AUTHOR]

Given a convex potential in a space with convex obstacles, an artificial potential is used to navigate to the minimum of the natural potential while avoiding collisions. The artificial potential combines the natural potential with potentials that repel the agent from the border of the obstacles. This is a popular approach to navigation problems because it can be implemented with spatially local information that is acquired during operation time. Artificial potentials can, however, have local minima that prevent navigation to the minimum of the natural potential. This paper derives conditions that guarantee artificial potentials to have a single minimum that is arbitrarily close to the minimum of the natural potential. The qualitative implication is that artificial potentials succeed when either the condition number—the ratio of the maximum and the minimum eigenvalue—of the Hessian of the natural potential is not large and the obstacles are not too flat, or when the destination is not close to the border of an obstacle. Numerical analyses explore the practical value of these theoretical conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we describe a possible generalization of the Wasserstein-2 metric, originally defined on the space of scalar probability densities, to the space of Hermitian matrices with trace one and to the space of matrix-valued probability densities. Our approach follows a control-theoretic optimization formulation of the Wasserstein-2 metric, having its roots in fluid dynamics, and utilizes certain results from the quantum mechanics of open systems, in particular the Lindblad equation. It allows determining the gradient flow for the quantum entropy relative to this matricial Wasserstein metric. [ABSTRACT FROM AUTHOR]

A fundamental problem that the bearing rigidity theory studies is to determine when a framework can be uniquely determined up to a translation and a scaling factor by its inter-neighbor bearings. While many previous works focused on the bearing rigidity of two-dimensional frameworks, a first contribution of this paper is to extend these results to arbitrary dimensions. It is shown that a framework in an arbitrary dimension can be uniquely determined up to a translation and a scaling factor by the bearings if and only if the framework is infinitesimally bearing rigid. In this paper, the proposed bearing rigidity theory is further applied to the bearing-only formation stabilization problem where the target formation is defined by inter-neighbor bearings and the feedback control uses only bearing measurements. Nonlinear distributed bearing-only formation control laws are proposed for the cases with and without a global orientation. It is proved that the control laws can almost globally stabilize infinitesimally bearing rigid formations. Numerical simulations are provided to support the analysis. [ABSTRACT FROM AUTHOR]

In this paper, we develop a compositional approach to abstraction and safety synthesis for a general class of discrete-time nonlinear systems. Our approach makes it possible to define a symbolic abstraction by composing a set of symbolic subsystems that are overlapping in the sense that they can share some common state variables. We develop compositional safety synthesis techniques using such overlapping symbolic subsystems. Comparisons, in terms of conservativeness and of computational complexity, between abstractions and controllers obtained from different system decompositions are provided. Numerical experiments show that the proposed approach for symbolic control synthesis enables a significant complexity reduction with respect to the centralized approach, while reducing the conservatism with respect to compositional approaches using nonoverlapping subsystems. [ABSTRACT FROM AUTHOR]

The classical notion of rigidity (that a formation of agents in $\mathbb {R}^2$ or $\mathbb {R}^3$ is rigidly constrained by interagent distances up to rigid-body transformations of space) is inherently dependent on the nature of Euclidean space and the nature of distance measurements. In this paper, we present a generalized formulation of rigidity, where agent states may lie in heterogeneous and non-Euclidean state spaces with arbitrary differentiable measurement constraints. A key aspect of our approach is to recognize the crucial role that the symmetry action of rigid-body transformations plays in classical rigidity theory. We consider a general symmetry given by a Lie-group action on a heterogeneous state space and define global rigidity of a formation to be the case where the interagent measurements fully constrain the agent locations up to the invariance encoded by the group action. In this framework, we develop general definitions of local rigidity and infinitesimal rigidity and introduce a new notion of robust rigidity that we believe will be important for control applications. To motivate the development, we show how the proposed theory can be applied to generalizations of the established problems of network localization and formation control. The provided results are directly applicable to networks of robotic vehicles involving a mixture of bearing and distance sensors, as well as cases where a collection of ground, submersible, and aerial vehicles operate in a single formation. [ABSTRACT FROM AUTHOR]

Control allocation (CA) is part of a hierarchical control architecture and distributes the desired control effort of a controller among a set of redundant actuators as it is used, for example, to meet high reliability requirements. Before applying CA, a factorization of a linear plant's input matrix must be carried out. This paper investigates the influence of this factorization on two common algorithms for constrained CA: direct allocation and redistributed pseudoinverse (RPINV). It is shown why the factorization does not affect the first method, whereas it influences the latter one as soon as the number of actuator saturations exceeds a certain threshold. On this basis a modification of RPINV is proposed, which allows the prioritization of virtual control vector components. If the actuator saturations prevent an exact solution, the errors of those components with high priority are preferentially forced to zero. Finally simulations demonstrate the main results. [ABSTRACT FROM PUBLISHER]

In this paper, we demonstrate that the dynamics of an $n$ -dimensional Lindblad control system can be separated into its inter- and intraorbit dynamics when there is fast controllability. This can be viewed as a control system on the simplex of density operator spectra, where projectors representing the eigenspaces are viewed as control variables. The local controllability properties of this control system can be analyzed when the control set of projectors is limited to a finite subset. In particular, there is a natural finite subset of $n!$ projector tuples that are effective for low-purity orbits. [ABSTRACT FROM PUBLISHER]

We study formation control problems. Our approach is to let a group of systems maximize their pairwise distances while bringing them all to a given submanifold, determining the shape of the formation. The algorithm we propose allows us to initialize the positions of the individual systems in the ambient space of the given submanifold but brings them to the desired formation asymptotically in a stable fashion. Our control inherently consists of a distributed component, maximizing the pairwise distances, and a decentralized component, asymptotically stabilizing the submanifold. We establish a graph-theoretical interpretation of the equilibria that our control enforces and extends our approach to systems living on the special Euclidean group. Throughout this paper, we illustrate our approach on different examples. [ABSTRACT FROM AUTHOR]

We study three existing issues associated with optimization of long-run average rewards of time-nonhomogeneous Markov processes in continuous time with continuous state spaces: 1) the underselectivity, i.e., the optimal policies do not depend on their actions in any finite time period; 2) its related issue, the bias optimality, i.e., policies that optimize both long-run average and transient total rewards, and 3) the effects of a nonsmooth point of a value function on performance optimization. These issues require considerations of the performance in the entire period with an infinite horizon, and therefore are not easily solvable by dynamic programming, which works backwards in time and takes a local view at a particular time instant. In this paper, we take a different approach called the relative optimization theory, which is based on a direct comparison of the performance measures of any two policies. We derive tight necessary and sufficient optimality conditions that take the underselectivity into consideration; we derive bias optimality conditions for both long-run average and transient rewards; and we show that the effect of a wide class of nonsmooth points, called semismooth points, of a value function on the long-run average performance is zero and can be ignored. [ABSTRACT FROM AUTHOR]

This paper introduces a large class of differential-geometric distances between finite-dimensional linear dynamical systems, collectively called the alignment distance. Contrary to the existing distances, the alignment distance is based on the state-space description of dynamical systems, and is defined on the manifolds of systems of fixed order and fixed input–output dimension under a matrix rank constraint (e.g., minimality, controllability, or observability). While the quotient topology and principal fiber bundle structure associated with such manifolds have been known since the early days of modern control theory, distances natural to this structure have not been studied. The starting point for defining such a distance is to identify a linear system of order $n$ with its equivalence class of state-space realizations, all related by the so-called similarity action, i.e., state-space change of basis under $GL(n)$, the Lie group of nonsingular $n \times n$ matrices. The main idea of the alignment distance is to first find the best state-space change of basis that brings a realization of a system “as close as possible” to a realization of another system (the alignment step), and then compare the aligned realizations. A direct implementation of this idea, due to noncompactness of $GL(n)$ , is complicated. However, using the notion of “reduction of the structure group” of a principal bundle, we show that the change of basis can be restricted to an orthogonal change of basis, provided one uses realizations in a reduced subbundle. This key observation brings about significant computational benefits. As a technical contribution (possibly of independent interest), we show that several forms of realization balancing available in the control literature have differential-geometric significance, and are, indeed, examples of reducing the structure group from $GL(n)$ to its subgroup of orthogonal matrices $O(n)$ . The alignment distance can be defined for stable and unstable systems, discrete or continuous-time, and stochastic systems. [ABSTRACT FROM AUTHOR]

In this paper, a compact representation of the reachability graph of a Petri net is proposed. The transition set of a Petri net is partitioned into the subsets of explicit and implicit transitions in such a way that the subnet induced by implicit transitions does not contain directed cycles. The firing of implicit transitions can be abstracted so that the reachability set of the net can be completely characterized by a subset of reachable markings called basis makings. We show that to determine a max-cardinality- TI basis partition is an NP-hard problem, but a max-set- TI basis partition can be determined in polynomial time. The generalized version of the marking reachability problem in a Petri net can be solved by a practically efficient algorithm based on the basis reachability graph. Finally, this approach is further extended to unbounded nets. [ABSTRACT FROM AUTHOR]

In this paper, an explicit nonlinear model predictive controller (ENMPC) for the stabilization of nonlinear systems is investigated. The proposed ENMPC is constructed using tensored polynomial basis functions and samples drawn from low-discrepancy sequences. Solutions of a finite-horizon optimal control problem at the sampled nodes are used 1) to learn an inner and outer approximation of the feasible region of the ENMPC using support vector machines, and 2) to construct the ENMPC control surface on the computed feasible region using regression or sparse-grid interpolation, depending on the shape of the feasible region. The attractiveness of the proposed control scheme lies in its tractability to higher-dimensional systems with feasibility and stability guarantees, significantly small online computation times, and ease of implementation. [ABSTRACT FROM PUBLISHER]

This paper addresses stationary dynamic output-feedback control of discrete-time Markovian jumping linear systems (MJLS). A rather general setup is adopted, with indirect and noisy output measurements, infinite time horizon, and not necessarily ergodic Markov chains. The class of admissible controllers consists of all stabilizing systems, dynamic or memoryless, with arbitrary dimension. The quality of stabilization achieved by an admissible controller is measured by a performance criterion described by a long run average cost, under some standard conditions. The optimal controller can be computed off-line resting on two sets of Riccati equations and it only requires the storage of $4N$ matrices where $N$ is the cardinality of the Markov state space. We present an example of a quad-rotor system to illustrate the results and compare the performance of three different control schemes with the proposed one, indicating that it is an interesting, simple alternative for controlling MJLS. Although the systems under consideration are subject to random perturbations, the proof of the main result is based on arguments typical of deterministic systems. [ABSTRACT FROM AUTHOR]

This paper deals with a wave equation with a one-dimensional space variable, which describes the dynamics of string deflection. Two kinds of control are considered: a distributed action and a boundary control. It is supposed that the control signal is subject to a cone-bounded nonlinearity. This kind of feedback laws includes (but is not restricted to) saturating inputs. By closing the loop with such a nonlinear control, it is thus obtained a nonlinear partial differential equation, which is the generalization of the classical 1D wave equation. The well-posedness is proven by using nonlinear semigroups techniques. Considering a sector condition to tackle the control nonlinearity and assuming that a tuning parameter has a suitable sign, the asymptotic stability of the closed-loop system is proven by Lyapunov techniques. Some numerical simulations illustrate the asymptotic stability of the closed-loop nonlinear partial differential equations. [ABSTRACT FROM PUBLISHER]

Impulsive solutions in LTI dynamical systems have received ample attention, but primarily for descriptor systems, i.e., first order Differential Algebraic Equations (DAEs). This paper focuses on the impulsive behavior of higher order dynamical systems and analyzes the causes of impulses in the context of interconnection of one or more dynamical systems. We extend the definition of impulse-controllability to the higher order case. Amongst the various nonequivalent notions of impulse-controllability for first order systems available in the literature, which mostly rely on the input/output structure of the system, our definition, based on a so-called state-map obtained directly from the system equations, generalizes many key first order results to the higher order case. In particular, we show that our higher-order-extension of the definition of impulse controllability generalizes the equivalence between impulse controllability and the ability to eliminate impulses in the closed loop by interconnecting with a suitable controller. This requires an extension of the definition of regularity of interconnection from behaviors involving only smooth trajectories to behaviors on the positive half line involving impulsive-smooth trajectories. [ABSTRACT FROM PUBLISHER]

The paper introduces and studies differentially positive systems, that is, systems whose linearization along an arbitrary trajectory is positive. A generalization of Perron–Frobenius theory is developed in this differential framework to show that the property induces a conal order that strongly constrains the asymptotic behavior of solutions. The results illustrate that behaviors constrained by local order properties extend beyond the well-studied class of linear positive systems and monotone systems, which both require a constant cone field and a linear state space. [ABSTRACT FROM PUBLISHER]

Information relaxation and duality in Markov decision processes have been studied recently by several researchers with the goal to derive dual bounds on the value function. In this paper we extend this dual formulation to controlled Markov diffusions: in a similar way we relax the constraint that the decision should be made based on the current information and impose a penalty to punish the access to the information in advance. We establish the weak duality, strong duality and complementary slackness results in a parallel way as those in Markov decision processes. We further explore the structure of the optimal penalties and expose the connection between the optimal penalties for Markov decision processes and controlled Markov diffusions. We demonstrate the use of this dual representation in a classic dynamic portfolio choice problem through a new class of penalties, which require little extra computation and produce small duality gap on the optimal value. [ABSTRACT FROM PUBLISHER]

This paper focuses on the task of safety controller synthesis, that is, designing a controller that will take a system from any point within a compact set of initial states to a point inside a set of acceptable goal states, while never entering any state that is deemed unsafe. To do this we use a human generated trajectory-based approach. We introduce the control autobisimulation function, which is the analog of the control Lyapunov function for approximate bisimulation. We consider a class of hybrid systems and use this function to determine a set of admissible feedback control laws that guarantee trajectory robustness for underlying dynamics that are linear affine, feedback linearizable, and differentially flat. This property ensures that any trajectory of the closed-loop system that is initialized within some neighborhood of a nominal trajectory will stay within some tube of the nominal trajectory when given the same input. This feedback control and input can be used as the controller for some subset of the initial states. We demonstrate how to combine multiple trajectories into a synthesized controller that satisfies the safety problem. [ABSTRACT FROM AUTHOR]