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This paper is concerned with studying Lyapunov characterization of integral input-to-state stability (iISS) for switched nonlinear time-varying systems. Sufficient conditions are given to verify iISS for switched nonlinear time-varying systems under a time-varying state-dependent switching law designed, which allow all subsystems to be not integral input-to-state stable (iISS) and the time derivative of Lyapunov functions of individual subsystems to be indefinite. An indefinite multiple Lyapunov functions (iMLFs) method for analyzing the dynamic behavior of switched nonlinear time-varying systems is provided. Also, an iMLFs-based small-gain theorem for switched interconnected nonlinear time-varying systems is presented, where each lower dimensional subsystem is allowed to be not iISS, which extends the small-gain technique from its original nonswitched nonlinear time-invariant version to a switched nonlinear time-varying version. Finally, an illustrative example is used to demonstrate the feasibility of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper aims to solve the finite-time stabilization problem for stochastic high-order nonlinear systems with stochastic inverse dynamics. By characterizing unmeasured stochastic inverse dynamics with finite-time stochastic input-to-state stability (FT-SISS), borrowing from some ideas of the Lyapunov function, sign function, backstepping and FT-SISS methods, and using stochastic finite-time stability theory, a new design and analysis method is given to guarantee global finite-time stability in probability of the closed-loop stochastic systems. [ABSTRACT FROM AUTHOR]

This paper studies regional boundary stabilization of nonlinear Schrödinger equation with state delay and bounded internal disturbance. The boundary constrained control law is designed by using the backstepping method. Regional input-to-state stability of the perturbed system with time-delay is established by a Lyapunov function and a generalized Halanay's inequality. Estimates on the set of initial conditions are found starting from which the solutions are exponentially attracted to a bounded set. A numerical example demonstrates the efficiency of the results. [ABSTRACT FROM AUTHOR]

This paper develops a new sampling-based method for stability verification of piecewise continuous nonlinear systems via Lyapunov functions. Depending on the nonlinear system dynamics, the candidate Lyapunov function and the set of states of interest, verifying stability requires solving complex, possibly non-convex or infeasible optimization problems. To avoid such problems, the proposed approach firstly distributes the verification of Lyapunov's inequality on a finite sampling of a bounded set of states of interest. Secondly, it extends the validity of Lyapunov's inequality to an infinite, bounded set of states by automatically exploiting local continuity properties. A sampling-based method for estimating the domain of attraction (DOA) via level sets of a validated Lyapunov function is also presented. Efficient state-space exploration is achieved using multi-resolution sampling and hyper-rectangles as basic sampling blocks. The operations that need to be performed for each sampling point in the state-space can be carried out in parallel, which improves scalability. The proposed methodology is illustrated for various benchmark examples from the literature. [ABSTRACT FROM AUTHOR]

This paper considers the problem of synthesizing output feedback controllers subject to sparsity constraints. This problem is known to be generically NP-hard, unless the plant satisfies the quadratic invariance property. Our main results show that, even if this property does not hold, tractable convex relaxations with optimality certificates can be obtained by recasting the problem into a polynomial optimization through the use of polyhedral Lyapunov functions. Combining these ideas with rank minimization tools leads to a computationally attractive algorithm. As an alternative, we present a second relaxation, with lower computational complexity, based on finding the best sparse estimate of a desired control action. These results are illustrated with several examples. [ABSTRACT FROM AUTHOR]

The stability and stabilization problems for a class of switched discrete-time nonlinear systems are studied in this paper. Each nonlinear subsystem of the presented switched system is modeled as a piecewise affine (PWA) one by splitting the state space into polyhedron regions. With the aid of a simple searching strategy for active state transition pairs at a switching instant, i.e., the so-called $\mathbb {S}$ -arbitrary switching approach, the stability criteria are derived via the relaxed piecewise quadratic Lyapunov function technique. Then, using the descriptor system approach, a family of PWA stabilizing controllers are designed to guarantee exponential stability of the resulting closed-loop control system, and the corresponding PWA controller gains could be calculated using numerical software. The validity and potential of the developed techniques are verified through a numerical example. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the stabilizability of controlled discrete-time switched linear systems (CSLSs), where the discrete switching control input and the continuous control input coexist. For a given positive integer $h$ , we propose a quantitative metric of stabilizability, called $h$ -contraction rate ($h$ -CR), and prove that it serves as an indicator of the stabilizability in the following sense: the $h$ -CR is less than one for sufficiently large $h$ if and only if the switched linear system is stabilizable. In addition, we develop computational tools to estimate upper and lower bounds of the $h$ -CR. In particular, the upper bound estimation is expressed as a semidefinite programming problem. We derive its Lagrangian dual problem and prove that the dual problem with an additional rank constraint gives an exact characterization of the $h$ -CR. An algorithm is developed to solve the dual problem with the rank constraint. [ABSTRACT FROM AUTHOR]

Recently, an extension of the super-twisting algorithm for relative degrees $m\geq {}1$ has been proposed. However, as of yet, no Lyapunov functions for this algorithm exist. This paper discusses the construction of Lyapunov functions by means of the sum-of-squares technique for  $m=1$. Sign definiteness of both Lyapunov function and its time derivative is shown in spite of numerically obtained—and hence possibly inexact—sum-of-squares decompositions. By choosing the Lyapunov function to be a positive semidefinite, the finite time attractivity of the system's multiple equilibria is shown. A simple modification of this semidefinite function yields a positive definite Lyapunov function as well. Based on this approach, a set of the algorithm's tuning parameters ensuring finite-time convergence and stability in the presence of bounded uncertainties is proposed. Finally, a generalization of the approach for $m>1$ is outlined. [ABSTRACT FROM AUTHOR]

This paper studies the projected saddle-point dynamics associated to a convex–concave function, which we term saddle function. The dynamics consists of gradient descent of the saddle function in variables corresponding to convexity and (projected) gradient ascent in variables corresponding to concavity. We examine the role that the local and/or global nature of the convexity–concavity properties of the saddle function plays in guaranteeing convergence and robustness of the dynamics. Under the assumption that the saddle function is twice continuously differentiable, we provide a novel characterization of the omega-limit set of the trajectories of this dynamics in terms of the diagonal blocks of the Hessian. Using this characterization, we establish global asymptotic convergence of the dynamics under local strong convexity–concavity of the saddle function. When strong convexity–concavity holds globally, we establish three results. First, we identify a Lyapunov function (that decreases strictly along the trajectory) for the projected saddle-point dynamics when the saddle function corresponds to the Lagrangian of a general constrained convex optimization problem. Second, for the particular case when the saddle function is the Lagrangian of an equality-constrained optimization problem, we show input-to-state stability (ISS) of the saddle-point dynamics by providing an ISS Lyapunov function. Third, we use the latter result to design an opportunistic state-triggered implementation of the dynamics. Various examples illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an event-based boundary control for a $2\times 2$ coupled linear hyperbolic system. We use a well-established backstepping controller which stabilizes the system along with a dynamic triggering condition which determines when the controller must be updated. The main contributions rely on the definition of an event-based controller under which global exponential stability of the system is achieved and, furthermore, the existence of a minimal dwell-time between two triggering times is guaranteed. The well-posedness of the system under the event-based controller is stated. A simulation example is presented to illustrate the results. [ABSTRACT FROM AUTHOR]

This paper addresses integral input-to-state stability (iISS) for stochastic nonlinear dynamical systems and develops Lyapunov-type criteria. The developments offer an iISS small-gain framework for verifying robustness of interconnected systems with respect to disturbances. The framework not only includes input-to-state stability (ISS) as a special case, but also highlights fundamental differences between the stochastic and deterministic systems. In particular, dealing with nonlinearities in interconnection gives rise to a unique issue of degradation in the presence of stochastic noises. This paper addresses this issue by coping judiciously with the effect of the gradient of Lyapunov functions of subsystems through diffusion fields in composing a Lyapunov function of the entire system. This paper employs two types of robustness notions, and demonstrates that the influence of the stochastic degradation on one notion is considerably different from the influence on the other notion in dealing with interconnected systems. [ABSTRACT FROM PUBLISHER]

This paper presents a unification and a generalization of the small-gain theory subsuming a wide range of existing small-gain theorems. In particular, we introduce small-gain conditions that are necessary and sufficient to ensure input-to-state stability (ISS) with respect to closed sets. Toward this end, we first develop a Lyapunov characterization of $\omega$ ISS via finite-step $\omega$ ISS Lyapunov functions. Then, we provide the small-gain conditions to guarantee $\omega$ ISS of a network of systems. Finally, applications of our results to partial ISS, ISS of time-varying systems, synchronization problems, incremental stability, and distributed observers are given. [ABSTRACT FROM AUTHOR]

In this note, we address the problem of stabilization of port-Hamiltonian systems via the ubiquitous proportional-integral-derivative (PID) controller. The design is based on passivity theory, hence the first step is to identify all passive outputs of the system, which is the first contribution of the paper. Adding a PID around this signal ensures that the closed-loop system is \mathcal L_2-stable for all positive PID gains. Global stability (and/or global attractivity) of a desired constant equilibrium is also guaranteed for a new class of systems for which a Lyapunov function can be constructed. A second contribution is to prove that this class—that is identified via some easily verifiable integrability conditions—is strictly larger than the ones previously reported in the literature. Comparisons of the proposed PID controller with control-by-interconnection passivity-based control are also discussed. [ABSTRACT FROM PUBLISHER]

In this paper, we study the cooperative global robust output regulation problem for nonlinear multiagent systems in output feedback form with any relative degree subject to directed switching networks. Through the augmentation by a distributed internal model and a distributed observer of the given system, we first convert the original problem into the global robust stabilization problem of the so-called extended augmented system, which is a multi-input-coupled switched system. Then, we establish a few technical lemmas to lay the foundation for constructing a switching control law and the corresponding multiple Lyapunov functions for the switched system. Then, we present the solvability conditions of our problem via the multiple Lyapunov functions and average dwell time method. The main result will be applied to the leader-following synchronization problem for a group of hyperchaotic Lorenz multiagent systems with a directed switching network. [ABSTRACT FROM PUBLISHER]

In this paper, we study geometric properties of basins of attraction of monotone systems. Our results are based on a combination of monotone systems theory and spectral operator theory. We exploit the framework of the Koopman operator, which provides a linear infinite-dimensional description of nonlinear dynamical systems and spectral operator-theoretic notions such as eigenvalues and eigenfunctions. The sublevel sets of the dominant eigenfunction form a family of nested forward-invariant sets and the basin of attraction is the largest of these sets. The boundaries of these sets, called isostables, allow studying temporal properties of the system. Our first observation is that the dominant eigenfunction is increasing in every variable in the case of monotone systems. This is a strong geometric property which simplifies the computation of isostables. We also show how variations in basins of attraction can be bounded under parametric uncertainty in the vector field of monotone systems. Finally, we study the properties of the parameter set for which a monotone system is multistable. Our results are illustrated on several systems of two to four dimensions. [ABSTRACT FROM PUBLISHER]

In this paper we present a method for the addition of integral action to nonpassive outputs of a class of port-Hamiltonian (pH) systems. The proposed integral controller is a dynamic extension, constructed from the open-loop system, such that the closed loop preserves the pH form. It is shown that the controller is able to reject the effects of both matched and unmatched disturbances, preserving the regulation of the nonpassive outputs. Previous solutions to this problem have relied on a change of coordinates whereas the presented solution is developed using the original state vector and, therefore, retains its physical interpretation. In addition, the resulting closed-loop dynamics have a natural interpretation as a control by interconnection scheme. [ABSTRACT FROM AUTHOR]

For decades, scientists have studied opinion formation in social networks, where information travels via word of mouth. The particularly interesting case is when polar opinions—Democrats versus Republicans or iOS versus Android—compete in the network. The central problem is to design and analyze a model that captures how polar opinions evolve in the real world. In this paper, we propose a general nonlinear model of polar opinion dynamics, rooted in several theories of sociology and social psychology. The model's key distinguishing trait is that unlike in the existing linear models, such as DeGroot and Friedkin–Johnsen models, an individual's susceptibility to persuasion is a function of his or her current opinion. For example, a person holding a neutral opinion may be rather malleable, while “extremists” may be strongly committed to their current beliefs. We also study three specializations of our general model, whose susceptibility functions correspond to different sociopsychological theories. We provide a comprehensive theoretical analysis of our nonlinear models’ behavior using several tools from nonsmooth analysis of dynamical systems. To study convergence, we use nonsmooth max–min Lyapunov functions together with the generalized Invariance Principle. For our general model, we derive a general sufficient condition for the convergence to consensus. For the specialized models, we provide a full theoretical analysis of their convergence—whether to consensus or disagreement. Our results are rather general and easily apply to the analysis of other nonlinear models defined over directed networks, with Lyapunov functions constructed out of convex components. [ABSTRACT FROM PUBLISHER]

This paper addresses the output reachable set estimation problem for continuous-time switched linear systems consisting of Hurwtiz stable subsystems. Based on a common Lyapunov function approach, the output reachable set is estimated by a union of bounding ellipsoids. Then, multiple Lyapunov functions with time-scheduled structure are employed to estimate the output reachable set for switched systems under dwell-time constraint. Furthermore, the safety verification problem of uncertain switched systems is investigated based on the result of output reachable set estimation. First, a sufficient condition ensuring the existence of an approximate bisimulation relation between two switched linear systems with a prescribed precision is proposed. Then, the safety verification for an uncertain switched system can be performed through an alternative safety verification for a switched system with exact parameters. Numerical examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper introduces the notion of common zeroing-output systems (CZOS) to analyze the stability of switched systems. The concept of CZOS allows one to verify weak zero-state detectability. It characterizes a common behavior of any individual subsystem when the output signal for each subsystem is “approaching” zero. Heuristically speaking, it removes the effect of switching behavior, and thus enables one to analyze stability properties in systems with complex switching signals. With the help of CZOS, the Krasovskii–LaSalle theorem can be extended to switched nonlinear time-varying systems with both arbitrary switching and more general restricted switching cases. For switched nonlinear time-invariant systems, the needed detectability condition is further simplified, leading to several new stability results. Particularly, when a switched linear time-invariant system is considered, it is possible to generate a recursive method, which combines a Krasovskii–LaSalle result and a nested Matrosov result, to find a CZOS if it exists. The power of the proposed CZOS is demonstrated by consensus problems in literature to obtain a stronger convergence result with weaker conditions. [ABSTRACT FROM AUTHOR]

This paper addresses the design of output controllers of chosen order in a given set for ensuring a number of desired properties, such as robust D-stability and robust performance. It is shown that a sufficient condition for determining such controllers can be obtained by solving convex optimization problems with linear matrix inequality constraints through the introduction of a Lyapunov function depending polynomially on both uncertainty and controller. The proposed condition is also necessary for sufficiently large degrees of the polynomials involved under some assumptions. [ABSTRACT FROM AUTHOR]

This paper studies stochastic input-to-state stability and stochastic global stability for impulsive stochastic nonlinear systems. Using fixed dwell-time condition and Lyapunov-based approach, sufficient conditions are established for the stability properties. Two cases are studied: 1) the case that the continuous dynamics is stable and 2) the case that the impulsive effects are stable. Furthermore, the relations among different dwell-time conditions are studied. Finally, two examples are used to illustrate the developed theory. [ABSTRACT FROM AUTHOR]

This paper proposes an original interval-based method to compute an outer approximation of all invariant sets (such as limit cycles) of a continuous-time nonlinear dynamic system, which are included inside a prior set of the state space. Contrary to all other existing approaches, our method has the following properties: first, it is guaranteed (a solution cannot be lost); second, it is applicable to a large class of systems without any specific assumption such as the knowledge of a Lyapunov function or any partial linearity; and third, there is no need to integrate the system. [ABSTRACT FROM AUTHOR]

This paper investigates the state observer design of a class of unknown input switched linear systems via mode-dependent dwell time switchings. The distinguishing feature of the proposed method is that strong detectability condition of subsystems of the switched systems is unnecessarily required. First, a time-varying coordinate transformation is introduced to design a suitable reduced-order observer for each subsystem. Then, computable sufficient conditions on the synthesis of the observers are proposed in the framework of a mode-dependent dwell time technique. Since the observer of an individual subsystem cannot be designed due to unavailability of strong detectability condition of the subsystem, the state of the switched system is estimated under the condition of confining the dwell time by a pair of upper and lower bounds, restricting the growth of the Lyapunov function of the active subsystem and forcing “energy” of the overall switched system to decrease at switching instants. Finally, an example is presented to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Safety critical systems involve the tight coupling between potentially conflicting control objectives and safety constraints. As a means of creating a formal framework for controlling systems of this form, and with a view toward automotive applications, this paper develops a methodology that allows safety conditions—expressed as control barrier functions—to be unified with performance objectives—expressed as control Lyapunov functions—in the context of real-time optimization-based controllers. Safety conditions are specified in terms of forward invariance of a set, and are verified via two novel generalizations of barrier functions; in each case, the existence of a barrier function satisfying Lyapunov-like conditions implies forward invariance of the set, and the relationship between these two classes of barrier functions is characterized. In addition, each of these formulations yields a notion of control barrier function (CBF), providing inequality constraints in the control input that, when satisfied, again imply forward invariance of the set. Through these constructions, CBFs can naturally be unified with control Lyapunov functions (CLFs) in the context of a quadratic program (QP); this allows for the achievement of control objectives (represented by CLFs) subject to conditions on the admissible states of the system (represented by CBFs). The mediation of safety and performance through a QP is demonstrated on adaptive cruise control and lane keeping, two automotive control problems that present both safety and performance considerations coupled with actuator bounds. [ABSTRACT FROM AUTHOR]

This paper analyzes distributed control protocols for first- and second-order networked dynamical systems. We propose a class of nonlinear consensus controllers where the input of each agent can be written as a product of a nonlinear gain, and a sum of nonlinear interaction functions. By using integral Lyapunov functions, we prove the stability of the proposed control protocols, and explicitly characterize the equilibrium set. We also propose a distributed proportional-integral (PI) controller for networked dynamical systems. The PI controllers successfully attenuate constant disturbances in the network. We prove that agents with single-integrator dynamics are stable for any integral gain, and give an explicit tight upper bound on the integral gain for when the system is stable for agents with double-integrator dynamics. Throughout the paper we highlight some possible applications of the proposed controllers by realistic simulations of autonomous satellites, power systems and building temperature control. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of exponentially stabilizing periodic orbits in a special class of hybrid models—systems with impulse effects—through control Lyapunov functions. The periodic orbit is assumed to lie in a C^1 submanifold Z that is contained in the zero set of an output function and is invariant under both the continuous and discrete dynamics; the associated restriction dynamics are termed the hybrid zero dynamics. The orbit is furthermore assumed to be exponentially stable within the hybrid zero dynamics. Prior results on the stabilization of such periodic orbits with respect to the full-order dynamics of the system with impulse effects have relied on input-output linearization of the dynamics transverse to the zero dynamics manifold. The principal result of this paper demonstrates that a variant of control Lyapunov functions that enforce rapid exponential convergence to the zero dynamics surface, Z, can be used to achieve exponential stability of the periodic orbit in the full-order dynamics, thereby significantly extending the class of stabilizing controllers. The main result is illustrated on a hybrid model of a bipedal walking robot through simulations and is utilized to experimentally achieve bipedal locomotion via control Lyapunov functions. [ABSTRACT FROM PUBLISHER]

This paper proposes necessary and sufficient conditions for stability and performance analysis of 2-D mixed continuous-discrete-time systems that can be checked with convex optimization, in particular linear matrix inequalities (LMIs). Specifically, the first contribution of the paper is a condition for exponential stability based on the introduction of a complex Lyapunov function depending polynomially on a parameter and on the use of the Gram matrix method. It is shown that this condition is sufficient for any chosen degree of the complex Lyapunov function, and necessary for an a priori known degree. The second contribution is a non-Lyapunov condition for exponential stability based on eigenvalue products. This condition is necessary and sufficient, and has the advantage of requiring a significantly smaller computational burden for achieving necessity. Lastly, the third contribution is to show how upper bounds on the \cal H\infty norm of 2-D mixed continuous-discrete-time systems can be obtained through a semidefinite program based on complex Lyapunov functions. A necessary and sufficient condition is provided for establishing the tightness of the found upper bounds. [ABSTRACT FROM PUBLISHER]

This paper gives a solution to the problem of verifying stability of networks consisting of integral input-to-state stable (iISS) subsystems. The iISS small-gain theorem developed recently has been restricted to interconnections of two subsystems. For large-scale systems, stability criteria relying only on gain-type information that were previously developed address only input-to-state stable (ISS) subsystems. To address the stability problem involving iISS subsystems interconnected in general structure, this paper shows how to construct Lyapunov functions of the network by means of a sum of nonlinearly rescaled individual Lyapunov functions of subsystems under an appropriate small-gain condition. [ABSTRACT FROM AUTHOR]

Event-triggered control (ETC) is a control strategy that is especially suited for applications where communication resources are scarce. By updating and communicating sensor and actuator data only when needed for stability or performance purposes, ETC is capable of reducing the amount of communications, while still retaining a satisfactory closed-loop performance. In this paper, an ETC strategy is proposed by striking a balance between conventional periodic sampled-data control and ETC, leading to so-called periodic event-triggered control (PETC). In PETC, the event-triggering condition is verified periodically and at every sampling time it is decided whether or not to compute and to transmit new measurements and new control signals. The periodic character of the triggering conditions leads to various implementation benefits, including a minimum inter-event time of (at least) the sampling interval of the event-triggering condition. The PETC strategies developed in this paper apply to both static state-feedback and dynamical output-based controllers, as well as to both centralized and decentralized (periodic) event-triggering conditions. To analyze the stability and the \cal L2-gain properties of the resulting PETC systems, three different approaches will be presented based on 1) impulsive systems, 2) piecewise linear systems, and 3) perturbed linear systems. Moreover, the advantages and disadvantages of each of the three approaches will be discussed and the developed theory will be illustrated using a numerical example. [ABSTRACT FROM AUTHOR]

In this paper, we present a generalization of the super-twisting algorithm for perturbed chains of integrators of arbitrary order. This higher order super-twisting (HOST) controller is homogeneous with respect to a family of dilations and is continuous. It is built as a dynamic controller (with respect to the state variable of the chain of integrators) and the convergence analysis is performed by the use of a homogeneous strict Lyapunov function which is explicitly constructed. The effectiveness of the controller is finally illustrated with simulations for a chain of integrators of order four, first pure then perturbed, where we compare the performances of two HOST controllers. [ABSTRACT FROM AUTHOR]

In this paper, a strict linear Lyapunov function is developed in order to investigate the exponential stability of a linear hyperbolic partial differential equation with positive boundary conditions. Based on the method of characteristics, some properties of the positive solutions are derived for the hyperbolic initial boundary value problems. The dissipative boundary condition in terms of linear inequalities is proven to be not only sufficient but also necessary under an extra assumption on the velocities of the hyperbolic systems. An application to control of the freeway traffic modeled by the Aw–Rascle traffic flow equation illustrates and motivates the theoretical results. The boundary control strategies are designed by integrating the on-ramp metering with the mainline speed limit. Finally, the proposed feedback laws are tested under simulation, first in the free-flow case and then in the congestion mode, which show adequate performance to stabilize the local freeway traffic. [ABSTRACT FROM PUBLISHER]

The goal of this paper is to study the exponential stabilization problem for autonomous discrete-time switched linear systems (SLSs), where only the discrete mode can be controlled. Our approach is based on periodic control Lyapunov functions whose value decreases periodically instead of at each time step as in the classical control Lyapunov functions. Using periodic control Lyapunov functions, we develop stabilizability analysis and controller synthesis conditions that are less conservative than existing results in that they apply to a larger class of SLSs. Utilizing recent results on the switched optimal control problems, a constructive way to find periodic control Lyapunov functions is presented. [ABSTRACT FROM PUBLISHER]

This paper deals with the problem of asymptotically stabilizing the splay state configuration of a network of identical pulse coupled oscillators through the design of the their phase response function. The network of pulse coupled oscillators is modeled as a hybrid system. The design of the phase response function is performed to achieve almost global asymptotic stability of a set, wherein oscillators’ phases are evenly distributed on the unit circle. To establish such a result, a novel Lyapunov function is proposed. Robustness with respect to frequency perturbation is assessed. Finally, the effectiveness of the proposed methodology is shown in an example. [ABSTRACT FROM AUTHOR]

Given a static plant described by a differentiable input-output function, which is completely unknown, but whose Jacobian takes values in a known polytope in the matrix space, this paper considers the problem of tuning (i.e., driving to a desired value) the output, by suitably choosing the input. It is shown that, if the polytope is robustly nonsingular (or has full rank, in the nonsquare case), then a suitable tuning scheme drives the output to the desired point. The proof exploits a Lyapunov-like function and applies a well-known game-theoretic result, concerning the existence of a saddle point for a min-max zero-sum game. When the plant output is represented in an implicit form, it is shown that the same result can be obtained, resorting to a different Lyapunov-like function. The case in which proper input or output constraints must be enforced during the transient is considered as well. Some application examples are proposed to show the effectiveness of the approach. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of designing time-varying quantizers for the stabilization of switched linear systems with quantized output and switching delays. The detection delays of switches are assumed to be time-varying but bounded, and the dwell time of the switching signal is assumed to be larger than the maximum delay. Given a switching controller, we analyze reachable sets of the closed-loop state by using a common Lyapunov function and then construct a quantizer that guarantees asymptotic stability. A sufficient condition for the existence of such a quantizer is characterized by the maximum switching delay and the dwell time. [ABSTRACT FROM AUTHOR]

In this paper, we address robust static anti-windup compensator design and performance analysis for saturated linear closed loops in the presence of nonlinear probabilistic parameter uncertainties via randomized techniques. The proposed static anti-windup analysis and robust performance synthesis correspond to several optimization goals, ranging from minimization of the nonlinear input/output gain to maximization of the stability region or maximization of the domain of attraction. We also introduce a novel paradigm accounting for uncertainties in the energy of the disturbance inputs. Due to the special structure of linear static anti-windup design, wherein the design variables are decoupled from the Lyapunov certificates, we introduce a significant extension, called scenario with certificates (SwC), of the so-called scenario approach for uncertain optimization problems. This extension is of independent interest for similar robust synthesis problems involving parameter-dependent Lyapunov functions. We demonstrate that the scenario with certificates robust design formulation is appealing because it provides a way to implicitly design the parameter-dependent Lyapunov functions and to remove restrictive assumptions about convexity with respect to the uncertain parameters. Subsequently, to reduce the computational cost, we present a sequential randomized algorithm for iteratively solving this problem. The obtained results are illustrated by numerical examples. [ABSTRACT FROM PUBLISHER]

This paper investigates the weighted-averaging dynamic for unconstrained and constrained consensus problems. Through the use of a suitably defined adjoint dynamic, quadratic Lyapunov comparison functions are constructed to analyze the behavior of weighted-averaging dynamic. As a result, new convergence rate results are obtained that capture the graph structure in a novel way. In particular, the exponential convergence rate is established for unconstrained consensus with the exponent of the order of 1-O(1/(m\log2m)) for special tree-like regular graphs. Also, the exponential convergence rate is established for constrained consensus over time-varying graphs, which nontrivially extends the existing result limited to the static graph case and the use of uniform weight matrices. Our main results are developed for directed graphs that are weakly connected, and we also provide statements regarding the rates in case of joint connectivity. [ABSTRACT FROM PUBLISHER]

The design of robust controllers for linear systems with structured time-varying uncertainty is a fundamental problem in control systems. This problem is still open because the available methods for robust analysis lead to nonconvex optimization whenever design variables are present. This paper proposes a novel approach that overcomes this difficulty. Specifically, continuous-time linear systems with polytopic time-varying uncertainty are considered. The basic problem consists of designing a robust static output feedback controller ensuring robust asymptotical stability. A novel approach is proposed through the introduction of robust stabilizability functions (RSFs), i.e., functions able to establish whether a controller is robustly asymptotically stabilizing. In particular, RSFs based on homogeneous polynomial Lyapunov functions (HPLFs) are searched for. The proposed approach requires the solution of two convex optimization problems with constraints expressed as linear matrix inequalities (LMIs). For any size of the LMIs, the proposed approach provides a sufficient condition for the existence of a sought controller. Moreover, this condition is also necessary when the size of the LMIs is large enough. Several extensions of the proposed methodology are presented, which deal with the synthesis of controllers ensuring desired decay rate, fixed-order dynamic output controllers, and controllers with the minimum norm. [ABSTRACT FROM PUBLISHER]

The convergence properties of popular nonlinear attitude estimators can be traced to the choice of an attitude error function. This paper considers a nonlinear deterministic direction cosine matrix estimator whose form is derived from an alternate attitude error function. While the resulting estimator shares several properties with those previously presented in the literature, the careful selection of an attitude error function results in an estimator with superior convergence properties. The attitude estimate is propagated using a rate gyroscope measurement and corrected using two or more vector measurements. Simulation and experimental results are presented that highlight the desirable properties of the proposed estimator. [ABSTRACT FROM PUBLISHER]

In this paper, exponential stabilizability of continuous-time positive switched systems is investigated. For two-dimensional systems, exponential stabilizability by means of a switching control law can be achieved if and only if there exists a Hurwitz convex combination of the (Metzler) system matrices. In the higher dimensional case, it is shown by means of an example that the existence of a Hurwitz convex combination is only sufficient for exponential stabilizability, and that such a combination can be found if and only if there exists a smooth, positively homogeneous and co-positive control Lyapunov function for the system. In the general case, exponential stabilizability ensures the existence of a concave, positively homogeneous and co-positive control Lyapunov function, but this is not always smooth. The results obtained in the first part of the paper are exploited to characterize exponential stabilizability of positive switched systems with delays, and to provide a description of all the “switched equilibrium points” of an affine positive switched system. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel distributed method to stabilize a set of agents moving in a two dimensional environment to a desired rigid formation. In our approach, each agent computes its control input using the relative positions of a set of formation neighbors but, contrary to most existing works, this information is expressed in the agent's own independent local coordinate frame, without requiring any common reference. The controller is based on the minimization of a Lyapunov function that includes locally computed rotation matrices, which are required due to the absence of a common orientation. Our contribution is that the proposed distributed coordinate-free method achieves global stabilization to a rigid formation with the agents using only partial information of the team, does not require any leader units, and is applicable to both single-integrator or unicycle agents. To guarantee global stability, we require that the network induced by the agent interactions belongs to a certain class of undirected rigid graphs in two dimensions, which we explicitly characterize. The performance of the proposed method is illustrated with numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce continuous-time linear compartmental switched systems and investigate their stability and stabilizability properties. By their nature, these systems are always stable. Necessary and sufficient conditions for asymptotic stability for arbitrary switching functions, and sufficient conditions for asymptotic stability under certain dwell-time conditions on the switching functions are proposed. Finally, stabilizability is thoroughly investigated and proved to be equivalent to the existence of a Hurwitz convex combination of the subsystem matrices, a condition that, for positive switched systems, is only sufficient for stabilizability. [ABSTRACT FROM AUTHOR]

In this paper, we establish results similar in nature to the invariance principle for a class of stochastic hybrid systems modeled by set-valued mappings. In particular, we characterize the set to which bounded random solutions converge in terms of a new concept called weak total recurrence. A refinement of the result under the existence of a non-increasing on average Lyapunov-like function is also established. A comparison between weak total recurrence and the frequently studied invariance concepts are presented. Finally, application of the main results to establish weak sufficient conditions for certifying stochastic stability properties like uniform global asymptotic stability in probability for compact sets and uniform global recurrence for open, bounded sets are discussed. [ABSTRACT FROM PUBLISHER]

This paper is concerned with the stability analysis of a lossless Euler-Bernoulli beam that carries a tip payload which is coupled to a finite-dimensional nonlinear dynamic feedback system. The latter comprises dynamic systems satisfying the nonlinear KYP lemma, which may represent the closed-loop dynamics of subordinate controlled actuators, as well as the interaction with a nonlinear passive environment. Global-in-time wellposedness and asymptotic stability is rigorously proven for the resulting closed-loop partial differential equation–ordinary differential equation (PDE–ODE) system. The analysis is based on semigroup theory for the corresponding first order evolution problem. For the large-time analysis, precompactness of the trajectories is shown by deriving uniform-in-time bounds on the solution and its time derivatives. [ABSTRACT FROM AUTHOR]

This paper addresses the stability and L\infty-gain analysis problem for continuous-time positive fractional-order delay systems with incommensurate orders between zero and one. A necessary and sufficient condition is firstly given to characterize the positivity of continuous-time fractional-order systems with bounded time-varying delays. Moreover, by exploiting the monotonic and asymptotic property of the constant delay system by virtue of the positivity, and comparing the trajectory of the time-varying delay system with that of the constant delay system, it is proved that the asymptotic stability of positive fractional-order systems is not sensitive to the magnitude of delays. In addition, it is shown that the L\infty-gain of a positive fractional-order system is independent of the magnitude of delays and fully determined by the system matrices. Finally, a numerical example is given to show the validity of the theoretical results. [ABSTRACT FROM PUBLISHER]

This paper addresses the problem of state estimation in nonlinear Lagrangian dynamical systems with frictionless unilateral constraints using the position measurement as output. The discontinuous velocity variable in such systems is modeled as a function of bounded variation (so that Zeno phenomenon is not ruled out). Since the derivative of such functions is represented with the Lebesgue-Stieltjes measure, the framework of measure differential inclusions (MDIs) is used to describe the dynamics. A class of estimators is proposed, which also uses the framework of MDIs, and is shown to generate asymptotically converging state estimates. The existence and uniqueness of solutions for the proposed estimators is rigorously proven. The global stability of error dynamics is analyzed using the generalized Lyapunov methods for functions of bounded variation. As particular cases of our estimators, we provide an explicit construction of a full-order observer, and a reduced-order observer. [ABSTRACT FROM AUTHOR]

In this paper, we study the consensus problem for second-order multi-agent systems with heterogeneous unknown inertias and control gains under a general directed graph. Unlike the existing consensus algorithms for second-order multi-agent systems in which all agents are assumed to have common unit inertias or share common control gains, we allow the inertias and the control gains to be heterogeneous and time-varying for each agent. We propose fully distributed consensus algorithms over a general directed graph when there exist, respectively, absolute velocity damping and relative velocity damping. Novel integral-type Lyapunov functions are proposed to study the consensus convergence. Moreover, the adaptive $\sigma$- modification schemes for the gain adaptation are proposed, which renders smaller control gains and thus requires smaller amplitude on the control input without sacrificing consensus convergence. Furthermore, we show that one proposed algorithm also works for consensus of agents with intrinsic Lipschitz nonlinear dynamics. The control gains are varying and updated by distributed adaptive laws. As a result, the proposed algorithms require no global information and thus can be implemented in a fully distributed manner. [ABSTRACT FROM AUTHOR]

This paper is concerned with computational methods for Lyapunov-based stabilization of an attractor set of a nonlinear dynamical system. Based upon a stochastic representation of deterministic dynamics, a Lyapunov measure is used for these purposes. The paper poses and solves the co-design problem of jointly obtaining a control Lyapunov measure and a state feedback controller. The computational framework employs set-oriented numerical techniques. Using these techniques, the resulting co-design problem is shown to lead to a finite number of linear inequalities. These inequalities determine the feasible set of the solutions to the co-design problem. A particular solution can be efficiently obtained using methods of linear programming. [ABSTRACT FROM AUTHOR]

Many applications require the interpretation of the Euclidean coordinates of features of a 3-D object through 2-D images. In this paper, the relative range and the Euclidean coordinates of an object or camera undergoing general affine motion are determined for paracatadioptric imaging systems via a nonlinear observer. The nonlinear observer asymptotically determines the range information provided that the motion parameters are known. The observer is developed through a Lyapunov-based design and stability analysis, and simulation results are provided that illustrate the performance of the state estimator. The contributions of this paper are that the developed observer: 1) can be applied to both affine and nonaffine systems; 2) can be used for both the fixed and moving camera configurations; 3) is continuous; and 4) yields an asymptotic result. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the analysis of linear infinite-dimensional control systems that should be able to compensate and/or track signals that are periodic. Adopting the name given in the seminal paper by Hara et al., we call them repetitive controllers. We analyze the asymptotic stability in the Lyapunov sense of finite-dimensional positive real plants coupled with pure delays. For this class of systems, we initially prove convergence in the weak topology to later deduce convergence in the strong one. [ABSTRACT FROM AUTHOR]