Search

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 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 proposes a less conservative stability criterion for discrete-time Lur'e systems with sector and slope restrictions by constructing a novel Lyapunov functional. Compared with the Lyapunov functional in the literature, this paper fully utilizes the sector and slope restrictions in our Lyapunov functional, which includes integral terms involved with the sector restriction of the nonlinearities $\phi (y_{i})$ and $\phi (y_{i+1})$ , integral terms involved with the slope restriction between $\phi (y_{i})$ and $\phi (y_{i+1})$ , and a quadratic term with an augmented vector related to available vectors for representing upper and lower bounds of all integral terms. The positive definiteness of a matrix appearing in the quadratic term can be relaxed by utilizing the lower bounds of all integral terms. Based on the Lyapunov functional, an improved stability criterion is derived in terms of linear matrix inequalities. Numerical examples show the effectiveness of the proposed criterion. [ABSTRACT FROM AUTHOR]

This paper deals with the estimation of convergence rate and domain of attraction of control-Lyapunov functions in Lyapunov-based control. This pair of estimation problems has been considered only for input-affine systems with constraints on the input norm. In this paper, we propose a novel optimization framework to address the estimation of convergence rate and domain of attraction. Specifically, we formulate the estimation problems as min–max bilevel programs for the decay rate of the Lyapunov function, where the inner problem can be resolved using Karush–Kuhn–Tucker optimality conditions, and the resulting single-level programs can be transformed into and solved as mixed-integer nonlinear programs. The proposed approach is applicable to systems with input-nonaffinity or more general forms of input constraints under an input-convexity assumption. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of determining the \cal H\infty and \cal H2 norms of 2-D mixed continuous-discrete-time systems. The first contribution is to propose a novel approach based on the use of complex Lyapunov functions with even rational parametric dependence, which searches for upper bounds on the sought norms via linear matrix inequalities (LMIs). The second contribution is to show that the upper bounds provided are nonconservative by using Lyapunov functions in the chosen class with sufficiently large degree. The third contribution is to provide conditions for establishing the tightness of the upper bounds. The fourth contribution is to show how the numerical complexity of the proposed approach can be significantly reduced by proposing a new necessary and sufficient LMI condition for establishing positive semidefiniteness of even Hermitian matrix polynomials. This result is also exploited to derive an improved necessary and sufficient LMI condition for establishing exponential stability of 2-D mixed continuous-discrete-time systems. Some numerical examples illustrate the proposed approach. It is worth remarking that nonconservative LMI methods for determining the \cal H\infty and \cal H2 norms of 2-D mixed continuous-discrete-time systems have not been proposed yet in the literature. [ABSTRACT FROM PUBLISHER]

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 presents a small-gain result for large-scale interconnected systems composed of hybrid subsystems. It is assumed that the subsystems are input-to-output practically stable (IOpS) with offset and unboundedness observability (UO), and the interconnected system can be proved to be IOpS with offset and UO as long as the proposed small-gain condition is satisfied. If, moreover, the assumption of IOpS on the subsystems is reduced to input-to-output stability (IOS), then the interconnected system is IOS. [ABSTRACT FROM AUTHOR]

This paper deals with the certification problem for robust quadratic stability, robust state convergence, and robust quadratic performance of linear systems that exhibit bounded rates of variation in their parameters. We consider both continuous-time (CT) and discrete-time (DT) parameter-varying systems. In this paper, we provide a uniform method for this certification problem in both cases and we show that, contrary to what was claimed previously, the DT case requires a significantly different treatment compared to the existing CT results. In the established uniform approach, quadratic Lyapunov functions, which are affine in the parameter, are used to certify robust stability, robust convergence rates, and robust performance in terms of linear matrix inequality feasibility tests. To exemplify the procedure, we solve the certification problem for $\mathscr {L}_2$ -gain performance both in the CT and the DT cases. A numerical example is given to show that the proposed approach is less conservative than a method with slack variables. [ABSTRACT FROM AUTHOR]

Lyapunov's second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle. The Lyapunov function endows the state-space with a Finsler structure. Incremental stability is inferred from infinitesimal contraction of the Finsler metrics through integration along solutions curves. [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 considers a standard consensus algorithm under output saturations. In this case, the global consensus cannot be realized due to the existence of stable equilibrium points that do not belong to the consensus manifold. Therefore, this paper investigates necessary and sufficient initial conditions for the achievement of consensus that characterize an exact domain of attraction. Specifically, this paper considers single-integrator agents with both fixed and time-varying undirected graphs. Then, we show that the consensus will be achieved if and only if the average of the initial states is within the minimum saturation level. [ABSTRACT FROM AUTHOR]

This paper addresses the problems of determining the \mathcal H_2 norm and the root mean square (RMS) gain of continuous-time switched linear systems. A novel class of Lyapunov functions is proposed for reaching this goal, called homogeneous rational Lyapunov functions (HRLFs). It is shown that sufficient conditions for establishing upper bounds of the sought performance indexes in the case of arbitrary switching can be given in terms of linear matrix inequality (LMI) feasibility tests by searching for an HRLF of chosen degree. Moreover, it is shown that these conditions are also necessary by searching for an HRLF of degree sufficiently large. It is worth mentioning that necessary and sufficient LMI conditions have not been proposed yet in the literature for the considered problems. Hence, the paper continues by considering the case of switching with dwell time constraints, showing that analogous LMI conditions can be obtained for this case by searching for a family of HRLFs mutually constrained by the dwell time specification. Some numerical examples illustrate the proposed methodology and highlight the advantages with respect to the existing works. [ABSTRACT FROM AUTHOR]

In this paper, two control schemes producing a continuous control signal are presented for a perturbed triple integrator: The continuous twisting algorithm (3-CTA) and the output feedback continuous twisting algorithm (3-OFCTA). The first one is a state feedback controller that ensures global finite-time stability of the origin, despite of Lipschitz, nonvanishing, and matched disturbances. It provides steady-state precision of fourth order of the output w.r.t. sampling step. The second one is an output feedback controller, which uses a third-order robust and exact differentiator as an observer. By requiring only information of the measurable output, the 3-OFCTA preserves all features of robustness, convergence, and precision of the state feedback 3-CTA. Moreover, it is proven that a separation principle applies, so that the gains of the controller and of the observer can be selected independently to assure stability. [ABSTRACT FROM AUTHOR]

This paper considers the problem of designing adaptive feedforward control (AFC) systems for uncertain single-input single-output linear systems perturbed by multisinusoidal disturbances of known frequencies. The proposed approach removes the longstanding assumption that either the sign of the real part or the imaginary part of the transfer function of a stable plant at the frequency of excitation is known for AFC to be applicable, which is referred to in this paper as a strictly positive real (SPR)-like condition. Notable features of the solution are that persistence of excitation is not required, and stability analysis tools based on averaging are avoided; hence, the requirement of an exponentially stable equilibrium for the closed-loop system is circumvented. [ABSTRACT FROM AUTHOR]

This paper employs detectability ideas to decide uniform global asymptotic stability (UGAS) of the trivial solution for a class of switched nonlinear time-varying systems when the trivial solution is uniformly globally stable. Using the notion of limiting behaviors of the state, output, and switching signals, the concept of a limiting zeroing-output solution is introduced. This leads to a definition of weak zero-state detectability (WZSD) that can be used to check UGAS, (uniformly for a set of switched signals). En route to establish this, a number of new stability results are derived. For example, under appropriate conditions, it is feasible to decide UGAS even when the switching signal does not satisfy an averaged dwell-time condition. It is also shown that WZSD of the original switched system can be verified by detectability conditions of much simpler auxiliary systems. Moreover, UGAS can be guaranteed without requiring that in each allowable system (without switching), the trivial solution is attractive. The effectiveness of the proposed concept is illustrated by a few examples including a switched semi-quasi-Z-source inverter. [ABSTRACT FROM AUTHOR]

In this paper, we present a method to robustify asymptotically stable nonlinear systems by adding an integral action that rejects unknown additive disturbances. The proposed approach uses a port-Hamiltonian (pH) representation of the open-loop dynamics, which, relying on the asymptotic stability property, is guaranteed to exist. The integral action controller preserves the pH structure, and, by adding a suitable cross term between the plant and the controller states to the closed-loop energy function, it avoids the unnatural coordinate transformation used in the past. The controller is shown to be robust against some common types of modeling uncertainty, including unknown friction dynamics in mechanical systems. [ABSTRACT FROM AUTHOR]

It is proved in this paper that the existence of a delay-dependent suitable Lyapunov function is a necessary and sufficient condition for a discrete-time fully nonlinear time-delay system, with given delays digraph, to be globally asymptotically stable. The same result is provided for the input-to-state stability. The less is the number of edges in the delays digraph, the less is the number of inequalities that are involved in the provided necessary and sufficient Lyapunov conditions. The case of arbitrary time-varying time delays, with no constraints as long as bounded, is covered as a special case. [ABSTRACT FROM AUTHOR]

In this paper, we study connections between structured storage or Lyapunov functions of a class of interconnected systems (dynamical networks) and dissipativity properties of the individual systems. We prove that if a dynamical network, composed as a set of linear time invariant systems interconnected over an acyclic graph, admits an additive quadratic Lyapunov function, then the individual systems in the network are dissipative with respect to a (nonempty) set of interconnection neutral supply functions. Each supply function from this set is defined on a single interconnection link in the network. Specific characterizations of neutral supply functions are presented, which imply robustness of network stability/dissipativity to removal of interconnection links. [ABSTRACT FROM AUTHOR]

We prove that if a homogeneous, continuously differentiable vector field is asymptotically stable, then it admits a Lyapunov function, which is the ratio of two polynomials (i.e., a rational function). We further show that when the vector field is polynomial, the Lyapunov inequalities on both the rational function and its derivative have sum of squares certificates and, hence, such a Lyapunov function can always be found by semidefinite programming. This generalizes the classical fact that an asymptotically stable linear system admits a quadratic Lyapunov function, which satisfies a certain linear matrix inequality. In addition to homogeneous vector fields, the result can be useful for showing local asymptotic stability of nonhomogeneous systems by proving asymptotic stability of their lowest order homogeneous component. This paper also includes some negative results: We show that in absence of homogeneity, globally asymptotically stable polynomial vector fields may fail to admit a global rational Lyapunov function, and in presence of homogeneity, the degree of the numerator of a rational Lyapunov function may need to be arbitrarily high (even for vector fields of fixed degree and dimension). On the other hand, we also give a family of homogeneous polynomial vector fields that admit a low-degree rational Lyapunov function but necessitate polynomial Lyapunov functions of arbitrarily high degree. This shows the potential benefits of working with rational Lyapunov functions, particularly as the ones whose existence we guarantee have structured denominators and are not more expensive to search for than polynomial ones. [ABSTRACT FROM AUTHOR]

In this paper, we study attitude synchronization for elements in the unit sphere in $\mathbb {R}^{\scriptscriptstyle {\scriptscriptstyle {3}}}$ and for elements in the three-dimensional (3-D) rotation group, for a network with switching topology. The agents’ angular velocities are assumed to be the control inputs, and a switching control law for each agent is devised that guarantees synchronization, provided that all elements are initially contained in a region, which we identify later in the paper. The control law is decentralized and it does not require a common orientation frame among all agents. We refer to synchronization of unit vectors in $\mathbb {R}^{\scriptscriptstyle {3}}$ as incomplete synchronization, and of 3-D rotation matrices as complete synchronization. Our main contribution lies in showing that these two problems can be analyzed under a common framework, where all agents’ dynamics are transformed into unit vectors dynamics on a sphere of appropriate dimension. [ABSTRACT FROM AUTHOR]

The explicit reference governor (ERG) is a simple and systematic approach that provides constraint handling capabilities to prestabilized systems. The basic idea behind this approach is to translate state and input constraints into an upper-bound on the value of the Lyapunov function, which is then enforced by suitably manipulating the derivative of the applied reference. When designing the ERG, one of the main challenges is the determination of an upper-bound on the value of the Lyapunov function that ensures constraints satisfaction. This paper proposes a systematic approach for estimating online the optimal upper-bound for systems subject to the intersection of concave constraints. To do this, the Barrier function method is used. The effect of the estimation error caused by the time-varying nature of the auxiliary reference on the constraint satisfaction capability of the ERG is studied analytically. A procedure is proposed to modify the estimated upper-bound to avoid constraints violation in the presence of estimation errors. The effectiveness of the proposed scheme is demonstrated through a simulation study on an overhead gantry crane system. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a method to obtain necessary and sufficient stability conditions for systems governed by one-dimensional nonlinear hyperbolic partial-differential equations with closed-loop integral controllers, when the linear frequency analysis cannot be used anymore. We study the stability of a general nonlinear transport equation where the control input and the measured output are both located on the boundaries. The principle of the method is to extract the limiting part of the stability from the solution using a projector on a finite-dimensional space and then use a Lyapunov approach. This paper improves a result of Trinh, Andrieu, and Xu, and gives an optimal condition for the design of the controller. The results are illustrated with numerical simulations where the predicted stable and unstable regions can be clearly identified. [ABSTRACT FROM AUTHOR]

An essential problem in the coordination of multiple agents is formation control. Significant challenges to the theoretical design may arise when the multiagent system is subject to uncertainty. This paper considers the robust multitask formation control problem for multiple agents, whose communication and measurements are disturbed by uncertain parameters. The control objectives include achieving the desired configuration, avoiding collisions, and preserving the connectivity of the uncertain topology. To achieve these objectives, we first provide conditions in terms of linear matrix inequalities for checking the connectivity of uncertain topologies. Then, we propose a new type of Lyapunov-like barrier function, called parametric Lyapunov-like barrier function, that is applicable to multiagent systems with uncertainties in communication and measurements. It is shown that this new type of Lyapunov-like barrier function guarantees the robust multitask formation and displays advantages over parameter-independent Lyapunov-like barrier functions. The efficacy of the proposed method is demonstrated via simulation results. [ABSTRACT FROM AUTHOR]

Previous small-gain results for interconnected integral input-to-state (iISS) systems have been restricted to systems that are strongly iISS. This paper removes this restriction by allowing cross terms between external inputs and states in the Lyapunov decrease of component systems, and subsequently constructing a nonseparable Lyapunov function. An example demonstrates the use of this new small-gain formulation. [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 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]

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]

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]

A geometric homogeneity is introduced for evolution equations in a Banach space. Scalability property of solutions of homogeneous evolution equations is proven. Some qualitative characteristics of stability of trivial solution are also provided. In particular, finite-time stability of homogeneous evolution equations is studied. Theoretical results are illustrated on important classes of partial differential equations. [ABSTRACT FROM PUBLISHER]

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]

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]

Zeno behavior is a dynamic phenomenon unique to hybrid systems in which an infinite number of discrete transitions occurs in a finite amount of time. This behavior commonly arises in mechanical systems undergoing impacts and optimal control problems, but its characterization for general hybrid systems is not completely understood. The goal of this paper is to develop a stability theory for Zeno hybrid systems that parallels classical Lyapunov theory; that is, we present Lyapunov-like sufficient conditions for Zeno behavior obtained by mapping solutions of complex hybrid systems to solutions of simpler Zeno hybrid systems defined on the first quadrant of the plane. These conditions are applied to Lagrangian hybrid systems, which model mechanical systems undergoing impacts, yielding simple sufficient conditions for Zeno behavior. Finally, the results are applied to robotic bipedal walking. [ABSTRACT FROM AUTHOR]

In this paper, the stability, stabilization, and $L_{2}$ -gain problems are investigated for periodic piecewise systems with time-varying subsystems. Continuous Lyapunov function with time-varying Lyapunov matrix is adopted. A condition guaranteeing the negative definiteness of a matrix polynomial, deriving from the Lyapunov derivative, is first obtained. Based on such a condition, an exponential stability condition is provided. Moreover, a state-feedback controller with time-varying gain is developed to stabilize the unstable periodic piecewise time-varying system. The $L_{2}$ -gain criterion for periodic piecewise time-varying system is also studied. Numerical examples are given to show the validity of the proposed techniques. [ABSTRACT FROM AUTHOR]

This paper investigates standard $H_\infty$ control of switched systems via dwell-time switchings without posing any internal stability requirements on subsystems of the switched systems. First, a sufficient condition is formed by specifying lower and upper bounds of the dwell time, constraining upper bound of derivative of a Lyapunov function of the active subsystem, and forcing the Lyapunov function values of the overall switched system to decrease at switching times to achieve standard $H_\infty$ control of unforced switched linear systems. Then, in the same framework of the dwell time, sufficient conditions are given for that of the corresponding forced switched linear systems by further designing state feedback controllers. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]

In this paper, tools to study forward invariance properties with robustness to disturbances, referred to as robust forward invariance, are proposed for hybrid dynamical systems modeled as hybrid inclusions. Hybrid inclusions are given in terms of differential and difference inclusions with state and disturbance constraints, for whose definition only four objects are required. The proposed robust forward invariance notions allow for the diverse type of solutions to such systems (with and without disturbances), including solutions that have persistent flows and jumps, that are Zeno, and that stop to exist after finite amount of (hybrid) time. Sufficient conditions for sets to enjoy such properties are presented. These conditions are given in terms of the objects defining the hybrid inclusions and the set to be rendered robust forward invariant. In addition, as special cases, these conditions are exploited to state results on nominal forward invariance for hybrid systems without disturbances. Furthermore, results that provide conditions to render the sublevel sets of Lyapunov-like functions forward invariant are established. Analysis of a controlled inverter system is presented as an application of our results. Academic examples are given throughout this paper to illustrate the main ideas. [ABSTRACT FROM AUTHOR]

Although the finite-time control theory and application have been studied very well in the literature, few results are concerned with the effect of sampled-data control. In this paper, first, the stability analysis of a chain of integrators under a discretized finite-time controller is given. Then, in the presence of disturbance, the robustness of the closed-loop system is also investigated. Based on a Lyapunov method, the ultimate bound for the steady states of the considered closed-loop system is estimated. Some simulation and examples are provided to verify the proposed theoretical results. [ABSTRACT FROM AUTHOR]

Compositional control methods are well-suited for the design of large-scale control systems. QSR-dissipativity and passivity are closely related to stability and may be utilized to show Lyapunov and finite-gain stability for dynamical systems under certain conditions. Additionally, passivity is preserved under feedback and parallel interconnections of passive systems and may be utilized to build larger stable systems. In this paper, we present a QSR-dissipativity and passivity based analysis for event-triggered networked control systems (NCSs). We characterize QSR-dissipativity, passivity, and stability conditions for an event-triggered NCS in cases where an input–output event-triggering sampler is located on the plant's side, controller's side, or both sides leading to a considerable decrease in the communication load among subunits. We calculate passivity levels for the entire NCS. We show that the general passivity and stability conditions of the NCS depend on passivity properties of the plant and controller. Our results illustrate the tradeoffs amongst passivity levels, stability, and the rates of communication between subsystems. The motivation behind our work is that by designing and connecting individual stable passive event-triggered NCSs with desired passivity levels, one can design larger stable passive compositional networked systems that are suitable for the design and control of large-scale systems, such as cyber-physical systems. [ABSTRACT FROM AUTHOR]

This paper considers the input-to-state stability (ISS) of time-varying switched systems with time delays, where the upper bound estimation for the operator of Lyapunov function (UBEOL) is assumed to be time varying and mode dependent. The ISS and integral ISS are investigated for time-varying switched systems with time delays by using the Lyapunov–Razumikhin and comparison theorem methods. Since the coefficient in the UBEOL is time varying and takes a positive/negative value, the subsystems consist of both ISS and non-ISS subsystems, simultaneously. It is shown that our presented results have wider applications than some existing works. Two examples, including one of the consensus for time-varying multiagent systems with cooperative and competitive protocols, are presented to demonstrate the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]

Stability analysis of nonlinear singular systems, arising from differential algebraic equations, is addressed in this paper, exploiting the fact that it belongs to the class of Positvstellensatz problems. By means of the direct Lyapunov method and a variety of convex descriptor forms, sufficient conditions in terms of linear matrix inequalities are developed. Issues concerning proper initialization and simulation are also considered. Illustrative examples are provided. [ABSTRACT FROM AUTHOR]

The goal of this paper is to design a stabilizing feedback boundary control for a reaction–diffusion partial differential equation (PDE), where the boundary control is subject to a constant delay while the equation may be unstable without any control. For this system, which is equivalent to a parabolic equation coupled with a transport equation, a prediction-based control is explicitly computed by splitting the infinite-dimensional system into two parts: a finite-dimensional unstable part and a stable infinite-dimensional part. A finite-dimensional delayed controller is computed for the unstable part, and it is shown that this controller stabilizes the whole PDE. The proof is based on an explicit expression of the classical Artstein transformation combined with an adequately designed Lyapunov function. A numerical simulation illustrates the constructive feedback design method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the regional stabilization problem for discrete time-delay systems with actuator saturations. Different from some existing delay-independent techniques handling the saturation nonlinearity, a delay-dependent polytopic approach is first proposed in a discrete-time framework. Then, by combining with an augmented Lyapunov function, a discrete Wirtinger-based inequality and some novel analysis techniques, improved delay-dependent regional stabilization conditions are established by means of linear matrix inequalities. Finally, it is shown via two numerical examples that the obtained results are less conservative than some existing ones yet providing a larger estimate of an initial condition set. [ABSTRACT FROM AUTHOR]

This paper generalizes the LaSalle–Yoshizawa Theorem to switched nonsmooth systems. The Filippov and Krasovskii regularizations of a switched system are shown to be contained within the convex hull of the Filippov and Krasovskii regularizations of the subsystems, respectively. A common candidate Lyapunov function that has a negative semidefinite generalized time derivative along the trajectories of the subsystems is shown to be sufficient to establish LaSalle–Yoshizawa-like results for the switched system. Of independent interest, are the results on approximate continuity and Filippov regularization of set-valued maps, reduction of differential inclusions using Lipschitz continuous regular functions, and comparative remarks on different generalizations of the time derivative along the trajectories of a nonsmooth system. [ABSTRACT FROM AUTHOR]

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 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]

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]

The paper deals with the design of sampled-data controllers which preserve the stabilizing performance of a continuous-time backstepping control strategy. This is achieved through matching of the control Lyapunov functions evolutions at the sampling instants. The method is developed for systems in strict-feedback form. The results are discussed and compared with similar strategies through simulated examples. [ABSTRACT FROM AUTHOR]

In this paper we deal with the stabilizability property for discrete-time switched linear systems. A recent necessary and sufficient characterization of stabilizability, based on set theory, is considered as the reference for comparing the computation-oriented sufficient conditions. The classical BMI conditions based on Lyapunov-Metzler inequalities are considered and extended. Novel LMI conditions for stabilizability, derived from the geometric ones, are presented that permit to combine generality with computational affordability. For the different conditions, the geometrical interpretations are provided and the induced stabilizing switching laws are given. The relations and the implications between the stabilizability conditions are analyzed to infer and compare their conservatism and their complexity. [ABSTRACT FROM AUTHOR]