769 research outputs found

    Invariance Principles and Observability in Switched Systems with an Application in Consensus

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    Using any nonnegative function with a nonpositive derivative along trajectories to define a virtual output, the classic LaSalle invariance principle can be extended to switched nonlinear time-varying (NLTV) systems, by considering the weak observability (WO) associated with this output. WO is what the output informs about the limiting behavior of state trajectories (hidden in the zero locus of the output). In the context of switched NLTV systems, WO can be explored using the recently established framework of limiting zeroing-output solutions. Adding to this, an extension of the integral invariance principle for switched NLTV systems with a new method to guarantee uniform global attractivity of a closed set (without assuming uniform Lyapunov stability or dwell-time conditions) is proposed. By way of illustrating the proposed method, a leaderless consensus problem for nonholonomic mobile robots with a switching communication topology is addressed, yielding a new control strategy and a new convergence result

    Robust output stabilization: improving performance via supervisory control

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    We analyze robust stability, in an input-output sense, of switched stable systems. The primary goal (and contribution) of this paper is to design switching strategies to guarantee that input-output stable systems remain so under switching. We propose two types of {\em supervisors}: dwell-time and hysteresis based. While our results are stated as tools of analysis they serve a clear purpose in design: to improve performance. In that respect, we illustrate the utility of our findings by concisely addressing a problem of observer design for Lur'e-type systems; in particular, we design a hybrid observer that ensures ``fast'' convergence with ``low'' overshoots. As a second application of our main results we use hybrid control in the context of synchronization of chaotic oscillators with the goal of reducing control effort; an originality of the hybrid control in this context with respect to other contributions in the area is that it exploits the structure and chaotic behavior (boundedness of solutions) of Lorenz oscillators.Comment: Short version submitted to IEEE TA

    Joint Spectral Radius and Path-Complete Graph Lyapunov Functions

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    We introduce the framework of path-complete graph Lyapunov functions for approximation of the joint spectral radius. The approach is based on the analysis of the underlying switched system via inequalities imposed among multiple Lyapunov functions associated to a labeled directed graph. Inspired by concepts in automata theory and symbolic dynamics, we define a class of graphs called path-complete graphs, and show that any such graph gives rise to a method for proving stability of the switched system. This enables us to derive several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic, common sum of squares, and maximum/minimum-of-quadratics Lyapunov functions. We compare the quality of approximation obtained by certain classes of path-complete graphs including a family of dual graphs and all path-complete graphs with two nodes on an alphabet of two matrices. We provide approximation guarantees for several families of path-complete graphs, such as the De Bruijn graphs, establishing as a byproduct a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.Comment: To appear in SIAM Journal on Control and Optimization. Version 2 has gone through two major rounds of revision. In particular, a section on the performance of our algorithm on application-motivated problems has been added and a more comprehensive literature review is presente

    Robust Observer Design for Hybrid Dynamical Systems with Linear Maps and Approximately Known Jump Times

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    This paper proposes a general framework for the state estimation of plants given by hybrid systems with linear flow and jump maps, in the favorable case where their jump events can be detected (almost) instantaneously. A candidate observer consists of a copy of the plant's hybrid dynamics with continuous-time and/or discrete-time correction terms multiplied by two constant gains, and with jumps triggered by those of the plant. Assuming that the time between successive jumps is known to belong to a given closed set allows us to formulate an augmented system with a timer which keeps track of the time elapsed between successive jumps and facilitates the analysis. Then, since the jumps of the plant and of the observer are synchronized, the error system has time-invariant linear flow and jump maps, and a Lyapunov analysis leads to sufficient conditions for the design of the observer gains for uniform asymptotic stability in three different settings: continuous and discrete updates, only discrete updates, and only continuous updates. These conditions take the form of matrix inequalities, which we solve in examples including cases where the time between successive jumps is unbounded or tends to zero (Zeno behavior), and cases where either both the continuous and discrete dynamics, only one of them, or neither of them are detectable. Finally, we study the robustness of this approach when the jumps of the observer are delayed with respect to those of the plant. We show that if the plant's trajectories are bounded and the time between successive jumps is lower-bounded away from zero, the estimation error is bounded, and arbitrarily small outside the delay intervals between the plant's and the observer's jumps

    Preview Tracking Control of Linear Periodic Switched Systems with Dwell Time

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    This paper studies the preview tracking control problem for linear discrete-time periodic switched systems. Firstly, an augmented error system is constructed for each subsystem by stabilizing the augmented error systems through the method of optimal preview control, and the tracking problem of the switched system is transformed into the switched stability problem of closed-loop augmented error systems. Secondly, a switched Lyapunov function method is applied to search the minimal dwell time satisfying the switched stability of the closed-loop augmented error systems. Thirdly, the switched preview control input is solved from the controller of the individual augmented error system, and then the sufficient conditions and the preview controller can be obtained to guarantee the solvability of the original periodic switched preview tracking problem. Finally, numerical simulations show the effectiveness of the stabilization design method

    Modeling and Control of High-Voltage Direct-Current Transmission Systems: From Theory to Practice and Back

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    The problem of modeling and control of multi-terminal high-voltage direct-current transmission systems is addressed in this paper, which contains five main contributions. First, to propose a unified, physically motivated, modeling framework - based on port-Hamiltonian representations - of the various network topologies used in this application. Second, to prove that the system can be globally asymptotically stabilized with a decentralized PI control, that exploits its passivity properties. Close connections between the proposed PI and the popular Akagi's PQ instantaneous power method are also established. Third, to reveal the transient performance limitations of the proposed controller that, interestingly, is shown to be intrinsic to PI passivity-based control. Fourth, motivated by the latter, an outer-loop that overcomes the aforementioned limitations is proposed. The performance limitation of the PI, and its drastic improvement using outer-loop controls, are verified via simulations on a three-terminals benchmark example. A final contribution is a novel formulation of the power flow equations for the centralized references calculation

    A stability-theory perspective to synchronisation of heterogeneous networks

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    Dans ce mémoire, nous faisons une présentation de nos recherches dans le domaine de la synchronisation des systèmes dynamiques interconnectés en réseau. Une des originalités de nos travaux est qu'ils portent sur les réseaux hétérogènes, c'est à dire, des systèmes à dynamiques diverses. Au centre du cadre d'analyse que nous proposons, nous introduisons le concept de dynamique émergente. Il s'agit d'une dynamique "moyennée'' propre au réseau lui-même. Sous l'hypothèse qu'il existe un attracteur pour cette dynamique, nous montrons que le problème de synchronisation se divise en deux problèmes duaux : la stabilité de l'attracteur et la convergence des trajectoires de chaque système vers celles générées par la dynamique émergente. Nous étudions aussi le cas particulier des oscillateurs de Stuart-Landau

    Linear quadratic regulation control for falling liquid films

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    We propose and analyse a new methodology based on linear-quadratic regulation (LQR) for stabilising falling liquid films via blowing and suction at the base. LQR methods enable rapidly responding feedback control by precomputing a gain matrix, but are only suitable for systems of linear ordinary differential equations (ODEs). By contrast, the Navier-Stokes equations that describe the dynamics of a thin liquid film flowing down an inclined plane are too complex to stabilise with standard control-theoretical techniques. To bridge this gap we use reduced-order models - the Benney equation and a weighted-residual integral boundary layer model - obtained via asymptotic analysis to derive a multi-level control framework. This framework consists of an LQR feedback control designed for a linearised and discretised system of ODEs approximating the reduced-order system, which is then applied to the full Navier-Stokes system. The control scheme is tested via direct numerical simulation (DNS), and compared to analytical predictions of linear stability thresholds and minimum required actuator numbers. Comparing the strategy between the two reduced-order models we show that in both cases we can successfully stabilise towards a uniform flat film across their respective ranges of valid parameters, with the more accurate weighted-residual model outperforming the Benney-derived controls. The weighted-residual controls are also found to work successfully far beyond their anticipated range of applicability. The proposed methodology increases the feasibility of transferring robust control techniques towards real-world systems, and is also generalisable to other forms of actuation.Comment: 21 pages, 9 figures, 1 tabl
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