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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]

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 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]

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]

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]

Periodic event-triggered control (PETC) is an appealing paradigm for the implementation of controllers on platforms with limited communication resources, a typical example being networked control systems. In PETC, transmissions over the communication channel are triggered by an event generator, which depends solely on the available plant and controller data and is only evaluated at given sampling instants to enable its digital implementation. In this paper, we consider the general scenario, where the controller communicates with the plant via multiple decoupled networks. Each network may contain multiple nodes, in which case a dedicated protocol is used to schedule transmissions among these nodes. The transmission instants over the networks are asynchronous and generated by local event generators. At given sampling instants, the local event generator evaluates a rule, which only involves the measurements and the control inputs available locally, to decide whether a transmission is needed over the considered network. Following the emulation approach, we show how to design local triggering generators to ensure input-to-state stability and $\mathcal {L}_p$ stability for the overall system based on a continuous-time output-feedback controller that robustly stabilizes the network-free system. The method is applied to a class of Lipschitz nonlinear systems, for which we formulate the design conditions as linear matrix inequalities. The effectiveness of the scheme is illustrated via simulations of a nonlinear example. [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]

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]

In this paper, the stability and $L_2$ -gain properties of linear impulsive delay systems with delayed impulses are studied. Commonly employed techniques, in which the delayed impulses are treated using Newton–Leibniz formula, may not be applicable to $L_2$ -gain analysis, since they make the disturbance input appear in the impulse part. In order to circumvent the difficulty, we first augment the considered system to a time-delay system with switching nondelayed impulses. Due to the absence of delayed impulses, this new approach has advantages in constructing Lyapunov functions and handling the effects of impulse delays on the system performance. Switching-based time-dependent Lyapunov functions are introduced to deal with the resultant switching impulses of the augmented system. Sufficient conditions for exponential stability and $L_2$ -gain properties are derived in terms of linear matrix inequalities. Numerical examples are provided to illustrate the efficiency of the new approach. [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]

It is proved in this paper that the existence of suitable multiple Lyapunov functions is a necessary and sufficient condition for a discrete-time nonlinear switching system, with given switches 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 switches digraph, the less is the number of inequalities that are involved in the provided necessary and sufficient Lyapunov conditions. [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]

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 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 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 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 addresses the problem of determining switching rules for affine switched systems such that the system state is driven to a desired point and a guaranteed cost is minimized. The switching rule is determined by solving an LMI problem and global asymptotic stability of the tracking error dynamics is guaranteed even if sliding motions occur on any switching surface of the system. The potential of the results is illustrated on a true refrigeration system, namely a domestic refrigerator, where the purpose is to control the temperature in the fresh food and the freezer compartments. [ABSTRACT FROM PUBLISHER]

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, the problem of frequency-limited \mmb H\infty model reduction for positive linear time-invariant systems is investigated. Specifically, our goal is to find a stable positive reduced-order model for a given positive system such that the \mmb H\infty norm of the error system is bounded over a frequency interval of interest. A new condition in terms of matrix inequality is developed for characterizing the frequency-limited \mmb H\infty performance. Then an equivalent parametrization of a positive reduced-order model is derived, based on which, an iterative algorithm is constructed for optimizing the reduced-order model. The algorithm utilizes coarse reduced-order models resulting from (generalized) balanced truncation as the initial value. Both continuous- and discrete-time systems are considered in the same framework. Numerical examples clearly show the effectiveness and advantages of the proposed model reduction method. [ABSTRACT FROM AUTHOR]

Although sum of squares programming has been used extensively over the past decade for the stability analysis of nonlinear systems, several fundamental questions remain unanswered. In this paper, we show that exponential stability of a polynomial vector field on a bounded set implies the existence of a Lyapunov function which is a sum of squares of polynomials. In particular, the main result states that if a system is exponentially stable on a bounded nonempty set, then there exists a sum of squares Lyapunov function which is exponentially decreasing on that bounded set. Furthermore, we derive a bound on the degree of this converse Lyapunov function as a function of the continuity and stability properties of the vector field. The proof is constructive and uses the Picard iteration. Our result implies that semidefinite programming can be used to answer the question of stability of a polynomial vector field with a bound on complexity. [ABSTRACT FROM AUTHOR]

This paper considers asymptotic stability and transient performance of discrete-time piecewise affine systems. We propose a procedure to construct a nested sequence of finite-state symbolic models, each of which abstracts the original piecewise affine system and leads to linear matrix inequalities for guaranteed stability and performance levels. This sequence is in the order of decreasing conservatism, and hence gives us the option to pay more computational cost and analyze a finer symbolic model within the sequence in return for less conservative results. Moreover, in the special case where this sequence is finite, an exact analysis of stability and performance is achieved via semidefinite programming. [ABSTRACT FROM PUBLISHER]