Publications and preprints

Working papers

• "Newton and interior-point methods for (constrained) nonconvex-nonconcave minmax optimization with stability guarantees" (with R. Chinchilla and J. P. Hespanha), submitted for journal publication
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  We address the problem of finding a local solution to a nonconvex-nonconcave minmax optimization using Newton-type methods, including interior-point ones. We modify the Hessian matrix of these methods such that, at each step, the modified Newton update direction can be seen as the solution to a quadratic program that locally approximates the minmax problem. Moreover, we show that by selecting the modification in an appropriate way, the only stable points of the algorithm's iterations are local minmax points. Using numerical examples, we show that the computation time of our algorithm scales roughly linearly with the number of nonzero elements in the Hessian. For minmax control problems with per-stage costs, this generally leads to computation times that scale linearly with the horizon length.

• "Adaptive learning in two-player Stackelberg games with application to network security" (with R. Poovendran and J. P. Hespanha), submitted for journal publication
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  An adaptive learning approach is proposed for solving two-player Stackelberg games with incomplete information. Specifically, the follower’s cost is unknown to the leader, who knows only that the follower’s response to its own action belongs to some parametric family of functions, but not the actual parameter value. The proposed approach is capable of simultaneously estimating the unknown parameter and optimizing the leader’s action. It ensures that the estimated follower’s action and leader’s cost become indistinguishable from their actual values in finite time, up to a preselected, arbitrarily small error bound; moreover, the first-order necessary condition for optimality holds asymptotically in time for the estimated leader’s cost. Under a persistent excitation condition, the parameter estimation error can be bounded by a preselected, arbitrarily small constant in finite time as well. When the parametric function known to the leader does not match the follower’s response function perfectly, the same convergence results can be ensured for preselected error bounds proportional to the size of the mismatch. The approach and convergence results are also extended to the case where the follower’s actions cannot be observed, and are illustrated by simulation examples in the domain of network security.

Journals

• "Topological entropy of switched nonlinear and interconnected systems" (with D. Liberzon and J. P. Hespanha), Mathematics of Control, Signals, and Systems, vol. 35, no. 3, pp. 641–683, Sep. 2023
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  A general upper bound for topological entropy of switched nonlinear systems is constructed, using an asymptotic average of upper limits of the matrix measures of Jacobian matrices of strongly persistent individual modes, weighted by their active rates. A general lower bound is constructed as well, using a similar weighted average of lower limits of the traces of these Jacobian matrices. In a case of interconnected structure, the general upper bound is readily applied to derive upper bounds for entropy that depend only on “network-level” information. In a case of block-diagonal structure, less conservative upper and lower bounds for entropy are constructed. In each case, upper bounds for entropy that require less information about the switching signal are also derived. The upper bounds for entropy and their relations are illustrated by numerical examples of a switched Lotka–Volterra ecosystem model.

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• "Forecasting COVID-19 cases based on a parameter-varying stochastic SIR model" (with J. P. Hespanha, R. Chinchilla, R. R. Costa, and M. K. Erdal), Annual Reviews in Control, vol. 51, pp. 460–476, Apr. 2021
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  We address the prediction of the number of new cases and deaths for the coronavirus disease 2019 (COVID-19) over a future horizon from historical data (forecasting). We use a model-based approach based on a stochastic Susceptible-Infections-Removed (SIR) model with time-varying parameters, which capture the evolution of the disease dynamics in response to changes in social behavior, non-pharmaceutical interventions, and testing rates. We show that, in the presence of asymptomatic cases, such model includes internal parameters and states that cannot be uniquely identified solely on the basis of measurements of new cases and deaths, but this does not preclude the construction of reliable forecasts for future values of these measurements. Such forecasts and associated confidence intervals can be computed using an iterative algorithm based on nonlinear optimization solvers, without the need for Monte Carlo sampling. Our results have been validated on an extensive COVID-19 dataset covering the period from March through December 2020 on 144 regions around the globe.

• "Topological entropy of switched linear systems: General matrices and matrices with commutation relations" (with A. J. Schmidt, D. Liberzon, and J. P. Hespanha), Mathematics of Control, Signals, and Systems, vol. 32, no. 3, pp. 411–453, Sep. 2020
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  This paper studies a notion of topological entropy for switched systems, formulated in terms of the minimal number of trajectories needed to approximate all trajectories with a finite precision. For general switched linear systems, we prove that the topological entropy is independent of the set of initial states. We construct an upper bound for the topological entropy in terms of an average of the measures of system matrices of individual modes, weighted by their corresponding active times, and a lower bound in terms of an active-time-weighted average of their traces. For switched linear systems with scalar-valued state and those with pairwise commuting matrices, we establish formulae for the topological entropy in terms of active-time-weighted averages of the eigenvalues of system matrices of individual modes. For the more general case with simultaneously triangularizable matrices, we construct upper bounds for the topological entropy that only depend on the eigenvalues, their order in a simultaneous triangularization, and the active times. In each case above, we also establish upper bounds that are more conservative but require less information on the system matrices or on the switching, with their relations illustrated by numerical examples. Stability conditions inspired by the upper bounds for the topological entropy are presented as well.

• "Feedback stabilization of switched linear systems with unknown disturbances under data-rate constraints" (with D. Liberzon), IEEE Transactions on Automatic Control, vol. 63, no. 7, pp. 2107–2122, Jul. 2018
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  We study the problem of stabilizing a switched linear system with a completely unknown disturbance using sampled and quantized state feedback. The switching is assumed to be slow enough in the sense of combined dwell-time and average dwell-time, each individual mode is assumed to be stabilizable, and the data rate is assumed to be large enough but finite. By extending the approach of reachable-set approximation and propagation from an earlier result on the disturbance-free case, we develop a communication and control strategy that achieves a variant of input-to-state stability with exponential decay. An estimate of the disturbance bound is introduced to counteract the unknown disturbance, and a novel algorithm is designed to adjust the estimate and recover the state when it escapes the range of quantization.

• "Stabilization of networked control systems under clock offsets and quantization" (with K. Okano, M. Wakaiki, and J. P. Hespanha), IEEE Transactions on Automatic Control, vol. 63, no. 6, pp. 1708–1723, Jun. 2018
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  This paper studies the impact of clock mismatches and quantization on networked control systems. We consider a scenario where the plant's state is measured by a sensor that communicates with the controller through a network. Variable communication delays and clock jitter do not permit a perfect synchronization between the clocks of the sensor and controller. We investigate limitations on the clock offset tolerable for stabilization of the feedback system. For a process with a scalar-valued state, we show that there exists a tight bound on the offset above which the closed-loop system cannot be stabilized with any causal controllers. For higher dimensional plants, if the plant has two distinct poles, then the effect of clock mismatches can be canceled with a finite number of measurements, and hence there is no fundamental limitation. We also consider the case where the measurements are subject to quantization in addition to clock mismatches. For first-order plants, we present necessary conditions and sufficient conditions for stabilizability, which show that a larger clock offset requires a finer quantization.

• "Lyapunov small-gain theorems for networks of not necessarily ISS hybrid systems" (with A. Mironchenko and D. Liberzon), Automatica, vol. 88, pp. 10–20, Feb. 2018
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  We prove a novel Lyapunov-based small-gain theorem for networks composed of n ≥ 2 hybrid subsystems which are not necessarily input-to-state stable. This result unifies and extends several small-gain theorems for hybrid and impulsive systems proposed in the last few years. We also show how average dwell-time (ADT) clocks and reverse ADT clocks can be used to modify the ISS Lyapunov functions for subsystems and to enlarge the applicability of the derived small-gain theorems.

• "A Lyapunov-based small-gain theorem for interconnected switched systems" (with D. Liberzon), Systems & Control Letters, vol. 78, pp. 47–54, Apr. 2015
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  Stability of an interconnected system consisting of two switched systems is investigated in the scenario where in both switched systems there may exist some subsystems that are not input-to-state stable (non-ISS). We show that, providing the switching signals neither switch too frequently nor activate non-ISS subsystems for too long, a small-gain theorem can be used to conclude global asymptotic stability (GAS) of the interconnected system. For each switched system, with the constraints on the switching signal being modeled by an auxiliary timer, a correspondent hybrid system is defined to enable the construction of a hybrid ISS Lyapunov function. Apart from justifying the ISS property of their corresponding switched systems, these hybrid ISS Lyapunov functions are then combined to establish a Lyapunov-type small-gain condition which guarantees that the interconnected system is globally asymptotically stable.

Book chapter

• "Modeling and mitigating link-flooding distributed denial-of-service attacks via learning in Stackelberg games" (with J. P. Hespanha), in Handbook of Reinforcement Learning and Control open_in_new, K. G. Vamvoudakis, Y. Wan, F. L. Lewis, and D. Cansever, Eds.   Springer, 2021, pp. 433–463
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  This work formulates the mitigation of link-flooding distributed denial-of-service attacks as a routing problem among parallel links. In order to address the challenge that the adversary can observe the routing strategy before assigning attack traffic, we model the conflict between routing and attack as a Stackelberg game. For a general class of adversaries, we establish a characterization of an optimal attack strategy that reduces the search space to a finite set, and construct explicit formulae for Stackelberg equilibria and costs for a special class of networks. When the attack objective and capacity are unknown, we propose a learning-based approach that predicts the routing cost using a neural network and minimizes the predicted cost via projected gradient descent. A simulation study is provided to demonstrate the effectiveness of our approach.

Conferences

• "State estimation for asynchronously switched sampled-data systems" (with S. C. Shankar and J. P. Hespanha), to be published in Proc. 61st IEEE Conference on Decision and Control (CDC 2022)
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  Asynchronously switched sampled-data systems can help model power systems and vehicles that evolve in continuous-time with switching behavior and discrete time measurements. We address the problem of jointly estimating a switching signal, with uncertainty in the exact switching times, as well as the continuous states of the system. We prove stability of the standard Kalman filter under uncertainty in the switching time, with statistical bounds relating to the sampling period. We then propose a method for estimation of switching times as well as a method for efficient joint estimation of the state and switching signal inspired by the interacting multiple-model extended-Viterbi algorithm. We validate our algorithms in simulation for a power converter and maneuvering vehicle.

• "A tale of two doses: Model identification and optimal vaccination for COVID-19" (with R. Chinchilla, M. K. Erdal, R. R. Costa, and J. P. Hespanha), in Proc. 60th IEEE Conference on Decision and Control (CDC 2021), 2021, pp. 3544–3550 (invited paper)
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  We address the model identification and the computation of optimal vaccination policies for the coronavirus disease 2019 (COVID-19). We consider a stochastic Susceptible–Infected–Removed (SIR) model that captures the effect of multiple vaccine treatments, each requiring a different number of doses and providing different levels of protection against the disease. We show that the inclusion of vaccination data enables the estimation of the state of the model and key model parameters that are otherwise not identifiable. This estimates can, in turn, be used to design strategic approaches to vaccination that aim at minimizing the number of deaths and the economic cost of the disease. We illustrate these results with numerical examples.

  See the slides of the talk
• "Topological entropy of switched nonlinear systems" (with D. Liberzon and J. P. Hespanha), in Proc. 24th ACM International Conference on Hybrid Systems: Computation and Control (HSCC 2021), 2021, 11 pages
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  This paper studies topological entropy of switched nonlinear systems. We construct a general upper bound for the topological entropy in terms of an average of the asymptotic suprema of the measures of Jacobian matrices of individual modes, weighted by the corresponding active rates. A general lower bound is also established in terms of an active-rate-weighted average of the asymptotic infima of the traces of these Jacobian matrices. For switched systems with diagonal modes, we establish upper and lower bounds that only depend on the eigenvalues of Jacobian matrices, their relative order among individual modes, and the active rates. For both cases, we also establish upper bounds that are more conservative but require less information on the switching, with their relations illustrated by numerical examples of a switched Lotka–Volterra ecosystem model.

  See the slides and watch the videoopen_in_new of the talk
• "Adaptive learning in two-player Stackelberg games with continuous action sets" (with R. Poovendran and J. P. Hespanha), in Proc. 58th IEEE Conference on Decision and Control (CDC 2019), 2019, pp. 6905–6911
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  We study a two-player Stackelberg game in which the follower’s strategy depends on a parameter vector that is unknown to the leader. An adaptive learning algorithm is designed to simultaneously estimate the unknown parameter and minimize the leader’s cost, based on adaptive control techniques and hysteresis switching. The algorithm guarantees that the leader’s cost predicted using the parameter estimate becomes indistinguishable from its actual cost in finite time, up to a preselected, arbitrarily small error threshold, and that the first-order necessary condition for optimality holds asymptotically for the predicted cost. If an additional persistent excitation condition holds, then the parameter estimation error can also be bounded by a preselected, arbitrarily small threshold in finite time. The algorithm and convergence results are illustrated via a simple simulation example in the domain of network security.

  See the slidesfile_download of the talk
• "On topological entropy and stability of switched linear systems" (with J. P. Hespanha and D. Liberzon), in Proc. 22nd ACM International Conference on Hybrid Systems: Computation and Control (HSCC 2019), 2019, pp. 119–127 (Best Paper Award winner)
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  This paper studies topological entropy and stability properties of switched linear systems. First, we show that the exponential growth rates of solutions of a switched linear system are essentially upper bounded by its topological entropy. Second, we estimate the topological entropy of a switched linear system by decomposing it into a part that is generated by scalar multiples of the identity matrix and a part that has zero entropy, and proving that the overall topological entropy is upper bounded by that of the former. Third, we prove that a switched linear system is globally exponentially stable if its topological entropy remains zero under a destabilizing perturbation. Finally, the entropy estimation via decomposition and the entropy-based stability condition are applied to three classes of switched linear systems to construct novel upper bounds for topological entropy and novel sufficient conditions for global exponential stability.

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• "On topological entropy of switched linear systems with pairwise commuting matrices" (with J. P. Hespanha), in Proc. 56th Annual Allerton Conference on Communication, Control, and Computing (Allerton 2018), 2018, pp. 429–436 (invited paper)
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  We study a notion of topological entropy for switched systems, formulated in terms of the minimal number of initial states needed to approximate all initial states within a finite precision. This paper focuses on the topological entropy of switched linear systems with pairwise commuting matrices. First, we prove there exists a simultaneous change of basis under which each of the matrices can be decomposed into a diagonal part and a nilpotent part, and all the diagonal and nilpotent parts are pairwise commuting. Then a formula for the topological entropy is established in terms of the component-wise averages of the eigenvalues, weighted by the active time of each mode, which indicates that the topological entropy is independent of the nilpotent parts above. We also present how the formula generalizes known results for the non-switched case and the case with simultaneously diagonalizable matrices, and construct more general but more conservative upper bounds for the entropy. A numerical example is provided to demonstrate properties of the formula and the upper and lower bounds for the topological entropy.

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• "On topological entropy of switched linear systems with diagonal, triangular, and general matrices" (with A. J. Schmidt and D. Liberzon), in Proc. 57th IEEE Conference on Decision and Control (CDC 2018), 2018, pp. 5682–5687
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  This paper introduces a notion of topological entropy for switched systems, formulated using the minimal number of initial states needed to approximate all initial states within a finite precision. We show that it can be equivalently defined using the maximal number of initial states separable within a finite precision, and introduce switching-related quantities such as the active time of each mode, which prove to be useful in calculating the topological entropy of switched linear systems. For general switched linear systems, we show that the topological entropy is independent of the set of initial states, and establish upper and lower bounds using the active-time-weighted averages of the norms and traces of system matrices in individual modes, respectively. For switched linear systems with scalar-valued state or simultaneously diagonalizable matrices, we derive formulae for the topological entropy using active-time-weighted averages of eigenvalues, which can be extended to the case with simultaneously triangularizable matrices to obtain an upper bound. In these three cases with special matrix structure, we also provide more general but more conservative upper bounds for the topological entropy.

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• "Modeling and mitigating the Coremelt attack" (with H. Hosseini, D. Sahabandu, A. Clark, J. P. Hespanha, and R. Poovendran), in Proc. 2018 American Control Conference (ACC 2018), 2018, pp. 3410–3416
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  This paper studies the Coremelt attack, a link-flooding Distributed Denial of Service attack that exhausts the bandwidth at a core network link using low-intensity traffic flows between subverted machines. A dynamical system model is formulated for analyzing the effect of Coremelt attack on a single-link Transmission Control Protocol (TCP) network and developing mitigation methods. For the case with a limited number of subverted sources, a modified TCP algorithm is developed for the attackers to achieve a desired congestion level. A mitigation method is proposed to improve the link usage of legitimate users when the link is under attack. The network performance under different attack and mitigation scenarios is illustrated through simulation results.

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• "Stabilization of interconnected switched control-affine systems via a Lyapunov-based small-gain approach" (with D. Liberzon and Z.-P. Jiang), in Proc. 2017 American Control Conference (ACC 2017), 2017, pp. 5182–5187
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  We study the feedback stabilization of interconnected switched control-affine systems with both input-to-state stable (ISS) and non-ISS modes. Provided that the switching is slow in the sense of average dwell-time and the active time of non-ISS modes is short in proportion, suitable feedback controls are designed to achieve input-to-state practical stability (ISpS) with an arbitrarily small constant. We devise such feedback controls by extending a previous small-gain theorem on stability of interconnected switched systems to the ISpS context, and proposing a novel Lyapunov-based gain-assignment scheme.

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• "Analysis of different Lyapunov function constructions for interconnected hybrid systems" (with A. Mironchenko and D. Liberzon), in Proc. 55th IEEE Conference on Decision and Control (CDC 2016), 2016, pp. 465–470 (invited paper)
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  This paper studies stability of interconnections of hybrid dynamical systems, in the general scenario that the continuous or discrete dynamics of subsystems may have destabilizing effects. We analyze two existing methods of constructing Lyapunov functions for the interconnection based on candidate ISS Lyapunov functions for subsystems, small-gain conditions, and auxiliary timers modeling restrictions on jump frequency in terms of average dwell-time and reverse average dwell-time. We compare their feasibility and limitations for different types of subsystem dynamics, and examine a case that the combination of them is needed to establish global asymptotic stability.

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• "Finite data-rate stabilization of a switched linear system with unknown disturbance" (with D. Liberzon), in Proc. 10th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2016), 2016, pp. 1103–1108
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  We study the stabilization of a switched linear system with unknown disturbance using sampled and quantized state feedback. The switching is slow in the sense of combined dwell-time and average dwell-time, while the active mode is unknown except at sampling times. Each mode of the switched system is stabilizable, and the disturbance admits an unknown bound. A communication and control strategy is designed to achieve practical stability and exponential convergence w.r.t. the initial state with a nonlinear gain on the disturbance, provided the data-rate meets given lower bounds. Compared with previous results, a more involved algorithm is developed to handle effects of the unknown disturbance based on employing an iteratively updated estimate of the disturbance bound and expanding the over-approximations of reachable sets over sampling intervals from the case without disturbance.

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• "Stabilizing a switched linear system with disturbance by sampled-data quantized feedback" (with D. Liberzon), in Proc. 2015 American Control Conference (ACC 2015), 2015, pp. 2193–2198
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  We study the problem of stabilizing a switched linear system with disturbance using sampled and quantized measurements of its state. The switching is assumed to be slow in the sense of combined dwell-time and average dwell-time, while the active mode is unknown except at sampling times. Each mode of the switched linear system is assumed to be stabilizable, and the magnitude of the disturbance is constrained by a known bound. A communication and control strategy is designed to guarantee bounded-input-bounded-state (BIBS) stability of the switched linear system and an exponential convergence rate with respect to the initial state, providing the data rate satisfies certain lower bounds. Such lower bounds are established by expanding the over-approximation bounds of reachable sets over sampling intervals derived in a previous paper to accommodate effects of the disturbance.

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• "Input-to-state stability for switched systems with unstable subsystems: a hybrid Lyapunov construction" (with D. Liberzon), in Proc. 53rd IEEE Conference on Decision and Control (CDC 2014), 2014, pp. 6240–6245
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  The input-to-state stability (ISS) of a nonlinear switched system is investigated in the scenario where there may exist some subsystems that are not input-to-state stable (non-ISS). We show that, providing the switching signal neither switches too frequently nor activates non-ISS subsystems for too long, a hybrid ISS Lyapunov function can be constructed to guarantee ISS of the switched system. With the constraints on the switching signal being modeled by a novel auxiliary timer, a hybrid system is defined so that the solutions to the two systems are correspondent. After the construction and verification of an ISS Lyapunov function, ISS of all complete solutions to the hybrid system, and therefore all solutions to the switched system, is conveniently proved.

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• "Lyapunov small-gain theorems for not necessarily ISS hybrid systems" (with A. Mironchenko and D. Liberzon), presented at the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014), 2014, pp. 1001–1008
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  We prove a novel Lyapunov-based small-gain theorem for interconnections of n hybrid systems, which are not necessarily input-to-state stable. This result unifies and extends several small-gain theorems for hybrid and impulsive systems, proposed in the last few years. Also we show how the average dwell-time (ADT) clocks and reverse ADT clocks can be used to modify the Lyapunov functions for subsystems and to enlarge the applicability of derived small-gain theorems.

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Ph.D. Dissertation

• "Switched and hybrid systems with inputs: Small-gain theorems, control with limited information, and topological entropy", Ph.D. dissertation, University of Illinois at Urbana-Champaign, 2017
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  In this thesis, we study stability and stabilization of switched and hybrid systems with inputs. We consider primarily two topics in this area: small-gain theorems for interconnected switched and hybrid systems, and control of switched linear systems with limited information. Read moreexpand_more Read lessexpand_less

  First, we study input-to-state practical stability (ISpS) of interconnections of two switched nonlinear subsystems with independent switchings and possibly non-ISpS modes. Provided that for each subsystem, the switching is slow in the sense of an average dwell-time (ADT), and the total active time of non-ISpS modes is short in proportion, Lyapunov-based small-gain theorems are established via hybrid system techniques. By augmenting each subsystem with a hybrid auxiliary timer that models the constraints on switching, we enable a construction of hybrid ISpS-Lyapunov functions, and consequently, a convenient formulation of a small-gain condition for ISpS of the interconnection. Based on our small-gain theorem, we demonstrate the stabilization of interconnected switched control-affine systems using gain-assignment techniques.

  Second, we investigate input-to-state stability (ISS) of networks composed of n ≥ 2 hybrid subsystems with possibly non-ISS dynamics. Lyapunov-based small-gain theorems are established based on the notion of candidate ISS-Lyapunov functions, which unifies and extends several previous results for interconnected hybrid and impulsive systems. In order to apply our small-gain theorem to different combinations of non-ISS dynamics, we adopt the method of modifying candidate exponential ISS-Lyapunov functions using ADT and reverse ADT timers. The effect of such modiffcations on the Lyapunov feedback gains between two interconnected hybrid systems is discussed in detail through a case-by-case study.

  Third, we consider the problem of stabilizing a switched linear system with a completely unknown disturbance using sampled and quantized state feediiback. The switching is assumed to be slow enough in the sense of combined dwell-time and average dwell-time, each individual mode is assumed to be stabilizable, and the data rate is assumed to be large enough but finite. By extending the approach of reachable-set approximation and propagation from an earlier result on the disturbance-free case, we develop a communication and control strategy that achieves a variant of input-to-state stability with exponential decay. An estimate of the disturbance bound is introduced to compensate for the unknown disturbance, and a novel algorithm is designed to adjust the estimate and recover the state when it escapes the range of quantization.

  Last, motivated by the connection between the minimum data rate needed to stabilize a linear time-invariant system and its topological entropy, we examine a notion of topological entropy for switched systems with a known switching signal. This notion is formulated in terms of the number of initial points such that the corresponding trajectories approximate all trajectories within a certain error, and can be equivalently defined using the number of initial points that are separable up to a certain precision. We first calculate the topological entropy of a switched scalar system based on the active rates of its modes. This approach is then generalized to nonscalar switched linear systems with certain Lie structures to establish entropy bounds in terms of the active rate and eigenvalues of each mode.

Technical reports