Keywords:Feedback Control Systems Abstract: Models of the brain, the climate, or the power grid, call for multi-resolution modeling principles because they are about behaviors that span a broad range of scales described by heterogenous data of vastly different resolution.

The talk will present a feedback principle for localization in range, time, and space and explores how it suggests a generic architecture for multi resolution behaviors that can be tightly and robustly controlled at different scales.

The proposed theory is entirely inspired from manifestations of this multi-resolution loop shaping principle in the regulation of neuronal behaviors. The theory will be illustrated with the control of bursting, a two-resolution behavior in time, and the cellular control of network oscillations, a two-resolution behavior in time and space.

The essence of the proposed principle is that localization in a given window is achieved when positive feedback suitably balances negative feedback. We will argue that studying systems near this balance calls for novel analysis tools and also suggests novel design tools. We will report on recent work on singularity theory and differential positivity to address those challenges.

Keywords:Linear Systems, Networked Control Systems Abstract: In this work we consider the problem of obtaining a bound on the error of the H2-norm of a linear time invariant system when using structured controllability and/or observability Gramians. In particular we consider dynamical systems whose drift matrices are lower block triangular and Gramians that have a block diagonal structure. We motivate the problem by showing that autonomous triangular systems always admit a diagonal Lyapunov function. We then show how the search for block diagonal Gramians can be interpreted in a decentralised manner and provide error bounds on the norm estimation.

Keywords:Networked Control Systems, Communication Systems Abstract: This paper introduces a novel dynamic event-based scheduling mechanism for networked control systems (NCSs) composed of multiple linear heterogeneous stochastic plants whose feedback loops are closed over a shared constrained communication channel. Each subsystem competes for the channel access in order to update its own controller with true local state values. Employing an emulation-based control policy, a probabilistic scheduler allocates the communication resource according to a prioritized error-based (PEB) measure. Based on this policy, a higher chance of transmission is assigned to the subsystems with higher errors, while the other requests are blocked when the channel capacity is reached. Under some mild assumptions, the probabilistic nature of PEB scheduling scheme facilitates an approximative decentralized implementation. We evaluate the stochastic stability of the overall NCS scheduled by PEB policy in terms of networked-induced error ergodicity, by applying the drift criterion over a multi time-step horizon. Moreover, we derive uniform performance bounds for the networked-induced error variance, which demonstrates a significant reduction in comparison with static and random access scheduling schemes such as TDMA and CSMA.

Keywords:Networked Control Systems, Nonlinear Systems and Control, Hybrid Systems Abstract: We design output-based event-triggered controllers to stabilize a class of nonlinear systems. We start from an output feedback law which stabilizes the plant in the absence of sampling and we then synthesize the event-triggering condition. The proposed event-triggering condition combines event-triggered and time-triggered techniques. The idea is to turn on the event-triggering mechanism only after a fixed amount of time has elapsed since the last transmission. This time is computed based on results on the stabilization of time-driven sampled-data systems. The overall strategy ensures an asymptotic stability property for the closed-loop system. Moreover, it has the advantage to enforce a (uniform) minimum amount of time between two transmissions which can be directly tuned. We show that the results are applicable to linear time-invariant systems as a particular case and we illustrate the approach for the stabilization of a nonlinear single-link robot arm model.

We apply Girsanov's theorem to transformed the initial stochastic dynamic team problem to an equivalent team problem, under a reference probability space, with state process and information structures independent of any of the team decisions. Subsequently, we derive team and Person-by-Person (PbP) optimality conditions, via the stochastic Pontryagin's maximum principle, consisting of forward and backward stochastic differential equations, and a set of conditional variational Hamiltonians with respect to the information structures of the team members. Finally, we relate the backward stochastic differential equation to the value process of the stochastic team problem.

Keywords:Nonlinear Systems and Control, Networked Control Systems, Optimal Control Abstract: We propose a split version of the optimality principle and an associated split optimal policy iteration in order to solve the dynamic programming problem for medium sized quantized nonlinear systems which consist of weakly coupled subsystems of small size. We show convergence of the scheme in special cases and present numerical experiments for discrete and continuous time systems.

Keywords:Nonlinear Systems and Control, Algebraic Systems Theory Abstract: This paper describes a feedback transformation group for the class of nonlinear single-input, single-output systems that can be represented in terms of Chen-Fliess functional expansions. There is no a priori requirement that these input-output systems have a state space realization, so the results presented here are independent of any particular state space coordinate system or state space embedding when a realization is available. The group is referred to as affine since it can always represent the input-output feedback linearization law of any control affine state space realization having a well defined relative degree in the usual sense. It is further shown in a coordinate free fashion that the invariants of this transformation group correspond exactly with transfer functions of Brunovsky forms when the group acts on a generating series having a well defined relative degree (defined purely from an input-output point of view). This represents a generalization of the author's previous work where a significantly smaller subgroup of this transformation group was considered.

Keywords:Applications of Algebraic and Differential Geometry in Systems Theory, Nonlinear Systems and Control Abstract: In this paper, we introduce the concept of x-maximal flatness. A control system is x-maximally flat if the number of new states gained by each successive derivation of the flat output is the largest possible. Firstly, we show that the only control-linear systems that are x-maximally flat are those that are static feedback equivalent to the m-chained form. Secondly, we generalize that result from control-linear systems to control-affine systems whose control-linear subsystem is static feedback equivalent to the m-chained form. We prove that they are x-maximally flat if and only if the drift exhibits a triangular form compatible with the m-chained form (and recently characterized by Silveira et al. and by the authors). We also show that if we skip the assumption of the x-maximal flatness,the latter condition is not necessary for x-flatness of control-affine system whose associated control-linear subsystem is static feedback equivalent to the m-chained form.

Keywords:Nonlinear Systems and Control, Algebraic Systems Theory, Feedback Control Systems Abstract: This paper addresses the input-output decoupling problem for discrete-time nonlinear systems by measurement feedback. Necessary and sufficient conditions are given to solve the problem by static or dynamic measurement feedback, respectively. Since the dynamic measurement solution presented here depends on the solution of the input-output linearization problem, a sufficient condition for linearizability of certain functions is also given. Finally, the derived conditions are specified to solve the problem by output feedback.

Keywords:Stability, Linear Systems, Nonlinear Systems and Control Abstract: We show that a sub-homogeneous positive monotone system with bounded heterogeneous time-varying delays is globally asymptotically stable if and only if the corresponding delay-free system is globally asymptotically stable. The proof is based on an extension of a delay-independent stability result for monotone systems under constant delays by Smith to systems with bounded heterogeneous time-varying delays. Under the additional assumption of positivity and sub-homogeneous vector fields, we establish the aforementioned delay insensitivity property and derive a novel test for global asymptotic stability. If the system has a unique equilibrium point in the positive orthant, we prove that our stability test is necessary and sufficient. Specialized to positive linear systems, our results extend and sharpen existing results from the literature.

Keywords:System Identification, Nonlinear Systems and Control Abstract: The paper deals with the observer design problem for a wide class of triangular nonlinear time-varying systems. The results of the present work generalize previous results in the literature dealing with the observer design problem for triangular systems and particularly, constitute generalizations of those obtained in a recent authors work concerning a class of triangular systems, whose dynamics have “p-normal form”. The only hypotheses we make, are forward completeness and a monotonicity assumption which guarantees that the systems under consideration are globally observable. For the case where it is a priori known that the initial state of the system belongs to a given bounded subset of the state space, a Luenberger-type observer is constructed and for the general case, the state estimation is exhibited by means of a switching sequence of time-varying dynamics.

Keywords:Nonlinear Systems and Control, Stability, Stochastic Modeling and Stochastic Systems Theory Abstract: In this paper, we improve the notion of stochastic stability by clarifying convergence rates of sample paths and intensity rates of diffusion terms, based on the shapes of stochastic Lyapunov functions. Using this notion, we clarify conditions that scaling operations for stochastic Lyapunov functions are available. Then, we create a measure of the distance between stochastic and deterministic stability properties by considering the inclusive relation between various stochastic stability properties.

Keywords:Nonlinear Systems and Control, Stability, Stochastic Modeling and Stochastic Systems Theory Abstract: By adopting on a notion of stochastic differential contraction, the paper presents new results on the incremental mean squared gain (IMSG) analysis of nonlinear systems with stochastic inputs. The relative power between two stochastic processes is defined as the asymptotic average (over time) of the second moment of the point-wise distance (in some metric) between the two processes. The IMSG of a system is then defined as the relative power between two output trajectories driven by two independent instances of i.i.d. inputs with unit relative power. While contracting metrics have been previously used for analysis of nonlinear systems, the formulation and analysis method in this paper lead to new conditions that yield a more accurate upper bound on the system gain. The idea is to introduce a notion of stochastic differential contraction which does not explicitly embed an exponential rate of contraction. This approach is more suitable for analysis of systems with stochastic inputs. In particular, and unlike previous approaches, the standard H_2-norm analysis results for linear systems can be recovered as a special case in this setting.

Keywords:Signal Processing, Linear Systems, Infinite Dimensional Systems Theory Abstract: The purpose of this paper is to elucidate a dichotomy between past and future in prediction of multivariate time-series. More specifically, vector-valued gaussian stochastic processes may be deterministic in one time-direction and not the other. This fact, which is absent in scalar-valued processes, is deeply rooted in the geometry of the shift-operator. The exposition and the examples we discuss are based on the work of Douglas, Shapiro and Shields on cyclic vectors of the backward shift and relate to classical ideas going back to Wiener and Komogoroff. The paper builds on examples and the goal is to provide insight to a control engineering audience.

Keywords:Coding Theory Abstract: We decode Reed-Solomon codes using soft information provided at the receiver. The Extended Euclidean Algorithm (EEA) is considered as an initial step to obtain an intermediate result. The final decoding result is obtained by interpolating the output of the EEA through the least reliable positions of the received word. We refer to this decoding method as Reduced list decoding, since not all positions in the received word are used in the interpolation as in other list decoding methods, such as the Guruswami-Sudan and Wu algorithms. As a result, the complexity of the interpolation step will be reduced considerably. The probability of failure can be minimized by adjusting certain parameters, making it comparable with the Koetter-Vardy algorithm but having a much lower complexity.

Keywords:Coding Theory Abstract: Hermitian codes belong to a powerful class of algebraic geometry codes, which allow to correct many independent errors. We show that Hermitian codes can correct many bursts of errors as well. Decoding of a Hermitian (N,K) code over F_{q^2} can be reduced to decoding of interleaved extended Reed-Solomon codes. Using this fact, we propose an efficient unique decoding algorithm correcting up to (N-K)/(q+1) bursts. Decoding failure probability is upper bounded by 1/q^2 and exponentially decreases with the number of bursts. It is also shown that low rate Hermitian codes can correct even more bursts of errors using "power" and "mixed" decoding. Time complexity of the algorithms is O(N^{5/3}) field operations.

Keywords:Coding Theory Abstract: We present two new methods for estimating the minimum distance of affine variety codes. Namely one for dual codes and one for primary codes. Our bound for dual codes improves previous results by Salazar et al., whereas our bound for primary codes is completely new. As becomes clear from the bounds, affine variety codes can be very good, also in the cases where they are not one-point algebraic geometric codes in disguise.

Keywords:Coding Theory Abstract: Coding theory and the theory of finite projective spaces, also called Galois geometries, are closely linked to each other. A large variety of problems in coding theory can be retranslated into equivalent problems on specific substructures of finite projective spaces. These latter links include functional codes and projective Reed-Muller codes. This talk presents recent results on functional codes and projective Reed-Muller codes, and presents the main ideas and techniques which led to these new results.

Keywords:Coding Theory, Communication Systems Abstract: Considering a linear code C and an ML-decoder, we use a quadratic function for measuring (evaluating) errors occurred during the encoding-decoding process, and show (with some restrictions also prove) that an encoding of C (that can actually be seen as a permutation of C) determined by a lexicographic order is sub-optimal.

Keywords:Control of Distributed Parameter Systems, Nonlinear Systems and Control, Feedback Control Systems Abstract: This note deals with the stabilization problem of a rotating disk with a flexible beam attached to its center. Assuming that a torque control is applied on the disk and a shear force control is exerted at the free end of the beam, we end up with a nondissipative closed loop system. We prove that the system can be nonuniformly exponentially stabilized provided that the angular velocity of the disk is less than the square root of the first eigenvalue of the related self-adjoint positive definite operator. This result is illustrated by numerical examples.

Keywords:Control of Distributed Parameter Systems, Infinite Dimensional Systems Theory, Feedback Control Systems Abstract: This paper deals with the stabilization of infinite dimensional linear systems by state feedback with a positivity constraint on the state components. It is assumed that the open loop system is positive and unstable. The positive stabilization problem consists mainly of a) analyzing necessary and/or sufficient conditions on the system parameters that guarantee its positive stabilizability, i.e. the existence of a stabilizing state feedback operator preserving positivity, and b) developing useful methods for getting such a state feedback law. The main part of this contribution is the description and the analysis of a specific method for computing a positively stabilizing state feedback, under suitable conditions on the nominal system. The main feature of this method is the fact that it guarantees that the closed loop dynamics are nonnegative by designing a control law such that the resulting input trajectory is nonnegative.

Inst. De Mathématiques De Toulouse, UMR CNRS 5219, Univ

Keywords:Control of Distributed Parameter Systems, Nonlinear Systems and Control, Infinite Dimensional Systems Theory Abstract: We study a system coupling the incompressible Navier-Stokes equations in a 3D parallelepiped type domain with a damped plate equation. The plate is located in a part of the upper boundary of the fluid domain. The fluid domain depends on the deformation of the plate, and therefore it depends on time.

We are interested in the stabilization, with a prescribed decay rate, of such a system in a neighborhood of a stationary solution, by a Dirichlet control acting at the boundary of the fluid domain.

For that, we first study the stabilizability of the corresponding linearized system and we determine a finite-dimensional feedback control able to stabilize the linearized model.

A crucial step in the analysis consists in showing that this linearized system can be rewritten thanks to an analytic semigroup, the infinitesimal generator of which has a compact resolvent.

A fixed-point argument is used to prove the local stabilization of the original nonlinear system. The main difficulties comes from the coupling between the fluid and plate equations, and the fact that the fluid domain varies with time, giving rise to geometric nonlinearities.

The results of the paper may be adapted to other more complex geometrical configurations for the same type of system. Ongoing research concerns the numerics of the control problem.

Keywords:Infinite Dimensional Systems Theory, Stability, Feedback Control Systems Abstract: We consider the indirect stabilization of systems of hyperbolic-type equations, such as wave and plate equations with different boundary conditions. By energy method, we show that a single feedback allows to stabilize the full system at a polynomial rate. Furthermore, we exploit refined resolvent estimates deducing the optimal decay rate of the energy of the system. Numerical simulations show resonance effect between the components of the system and confirm optimality of the proven decay rate.

Acad. of Mathematics and Systems Science, Acad. Sinica

Keywords:Control of Distributed Parameter Systems, Infinite Dimensional Systems Theory, Stability Abstract: In this paper, we consider boundary stabilization for a one-dimensional Euler-Bernoulli equation with boundary moment control and disturbance. The active disturbance rejection control (ADRC) and sliding mode control (SMC) approaches are adopted in investigation. By the ADRC approach, an extended state observer with time varying gain is designed to estimate the disturbance. It is shown that the closed-loop system is asymptotically stable after canceling the disturbance in the feedback loop. In the second part, the SMC is applied to reject the disturbance. The well-posedness of the closed-loop system via SMC is proven and the monotonicity of the ``reaching condition'' is presented without differentiation for the sliding mode function which may not always exist for the weak solution. The numerical experiments are presented to illustrate the convergence and the peaking value reduction caused by the constant high gain in literature. The control energy is compared numerically for two approaches.

Keywords:Mathematical Theory of Networks and Circuits, Linear Systems Abstract: We consider positive real rational matrix-valued rational functions. We show that the pointwise kernel of functions as well as the pointwise kernel of the Hermitian part is constant in the right complex half plane. These results are the basis for a decomposition for positive real matrices under orthogonal similarity transformation. We further consider positive real matrices which have a certain symmetry property that is known as "reciprocity". A decomposition for reciprocal and positive real matrices under block orthogonal transformation is derived. We illustrate our results by applying them to transfer functions arising in electrical circuit theory.

Keywords:Mathematical Theory of Networks and Circuits, Linear Systems Abstract: We examine RLC networks described by the impedance, or admittance matrices, which have a natural topology associated with the R, L, C structural matrices of the network. The McMillan degree of the integral- differential description introduced by the impedance and admittance matrices is related to the rank properties of the R, L, C structural matrices. We prove that the maximum possible value of the McMillan degree is given as the sum of the ranks of the capacitance and inductance matrices.We also provide necessary and sufficient conditions, of determinantal or of rank type, for this degree to be achieved.

Keywords:Mathematical Theory of Networks and Circuits, Optimal Control, Robust and H-Infinity Control Abstract: We consider multi-port RLC circuits and study the problems of charging the circuit to a specified state with the minimum supply of energy and that of discharging the circuit from a specified state with maximum energy extraction. These are known to be respectively the anti-stabilizing and stabilizing solutions of the associated Algebraic Riccati Equation (ARE). Using Hamiltonian matrix arguments, we prove that for multi-port RLC circuits, the state space realizations of the impedance Z(s) and the admittance Y(s) are related such that they both admit the same stabilizing solution of the ARE. The same holds for the anti-stabilizing solution too. We next show that the corresponding `closed loop state transition matrices' computed using Z(s) or Y(s) are equal too.

We next consider single-port RLC circuits, for which we provide capacitor/inductor loop/cut-set conditions with respect to the port that result in a pole at the origin or a pair of purely imaginary poles. In this case we show, using network topological arguments, that all the ARE solutions and the corresponding closed loop transition matrices share common eigenvectors. We give physical RLC-circuit based insights for these results. These results have potential implications for port-controlled Hamiltonian matrices.

Keywords:Mathematical Theory of Networks and Circuits, Physical Systems Theory, Nonlinear Systems and Control Abstract: In May 2008, HP Lab engineers announced their physical realization of the `missing' fourth basic circuit element in electronics: the memristor. Not often a technological discovery attracted so much attention from the media. Apart from the wildest possible speculations on future applications in new non-volatile memory devices with human brain synthesizing properties and suggestions to rewrite the existing textbooks on circuit theory, the discovery met with much skepticism as well. In this talk, we review a collection of scientific statements, arguments, and counter examples that critically address the existence of memristors.

Keywords:Mathematical Theory of Networks and Circuits, Physical Systems Theory, Signal Processing Abstract: In this paper, we present an alternative method to determine the active, reactive, and scattered power in an electrical network driven by an ideal non-sinusoidal AC voltage source. The method is based on orthogonally projecting the current demanded by the load onto the space of all (anti)derivatives of the source voltage of odd order. It is applicable in both the time-domain and frequency-domain. The obtained projection is the reactive current that can be compensated by a finite-dimensional lossless linear time-invariant compensator placed in parallel to the load. The parameters of the compensator can also be obtained from the projection. With the compensator, the apparent power delivered by the source can be reduced, while the active power demanded by the load remains unchanged. In this way, the power factor of the energy transport from source to load can be improved.

Keywords:Optimal Control, Stochastic Control and Estimation, Nonlinear Systems and Control Abstract: This work is concerned with a general class of stochastic optimal control problems in presence of state-constraints. When state-constraints are taken into account and in absence of quite restrictive controllability assumptions on the dynamics, the continuity of the value function cannot be guaranteed and some well-known problems arise in its characterization as a viscosity solution of a Hamilton-Jacobi-Bellman equation. The approach proposed in this work leads to a characterization of the epigraph of the value function translating, at a first stage, the optimal control problem into a state-constrained stochastic target problem with unbounded controls. This new formulation of the problem has the advantage to allow to solve it by a level set approach, where the state-constraints can be managed by an exact penalization technique.

Keywords:Optimal Control, Robust and H-Infinity Control, Stability Abstract: We consider a finite-horizon optimal control problem for a system subject to perturbations. We compare the performance of the nominal optimal control sequence under perturbations with a shrinking horizon strategy in which a re-optimization for the nominal model is performed in each sampling instant using the current perturbed system state as new initial value. We analyze the potential performance improvement using suitable moduli of continuity as well as stability and controllability properties and illustrate our findings by numerical simulations.

Keywords:Optimal Control, Nonlinear Systems and Control Abstract: We show that exact penalization techniques can be applied to optimal control problems with state constraints under a hard to verify hypothesis. Investigating conditions implying our hypothetical hypothesis we discuss some recent theoretical results on regularity of multipliers for optimal control problem involving first order state constraints. We show by an example that known conditions asserting regularity of the multipliers do not prevent the appearance of atoms in the multiplier measure. Our accompanying example is treated both numerically and analytically. Extension to cover problems with additional mixed state constraints is also discussed.

Keywords:Stability, Linear Systems, Optimal Control Abstract: This paper is concerned with stability and recursive feasibility of constrained linear receding horizon control schemes without terminal constraints and costs. Particular attention is paid to characterize the basin of attraction S of the asymptotically stable equilibrium. For stabilizable linear systems with quadratic costs and convex constraints we show that any compact subset of the interior of the viability kernel is contained in S for sufficiently large optimization horizon N. An analysis at the boundary of the viability kernel provides a connection between the growth of the infinite horizon optimal value function and stationarity of the feasible sets. Several examples are provided which illustrate the results obtained.

Keywords:Optimal Control, Computations in Systems Theory, Optimization : Theory and Algorithms Abstract: Converging hierarchies of finite-dimensional convex semidefinite relaxations have been proposed for state-constrained optimal control problems featuring oscillation phenonema, using the notion of relaxed control or Young measure, interpreting the control as the conditional of an occupation measure for the systems trajectories. These semidefinite relaxations were later on extended to optimal control problems depending linearly on the control input and typically featuring concentration phenomena, interpreting the control as a measure of time with a discrete singular component modeling discontinuities or jumps of the state trajectories. In this contribution, we use measures introduced originally by DiPerna and Majda in the partial differential equations literature to model simultaneously, and in a unified framework, possible oscillation and concentration effects of the optimal control policy. We show that hierarchies of semidefinite relaxations can also be constructed to deal numerically with optimal control problems with nonconvex polynomial extended vector field (i.e. Lagrangian and dynamics) and nonconvex semi-algebraic state constraints.

Keywords:Systems on Graphs, Large Scale Systems, Linear Systems Abstract: Upper bounds are derived on the total variation distance between the invariant distributions of two stochastic matrices differing on a subset W of rows. Such bounds depend on three parameters: the mixing time and the minimal expected hitting time on W for the Markov chain associated to one of the matrices; and the escape time from W for the Markov chain associated to the other matrix. These results, obtained through coupling techniques, prove particularly useful in scenarios where W is a small subset of the state space, even if the difference between the two matrices is not small in any norm. Several applications to large-scale network problems are discussed, including robustness of Google's PageRank algorithm, distributed averaging and consensus algorithms, and interacting particle systems.

Keywords:Systems on Graphs, Large Scale Systems, Robust and H-Infinity Control Abstract: We consider network flow over directed graphs between a single origin-destination pair, where the network state consists of flows and activation status of the links. The evolution of the activation status of a link depends on its saturation status and the activation status of the downstream links. The flow dynamics is determined by activation status of the links and node-wise routing policies under flow balance constraints at the nodes. Within this framework, we formulate discrete time dynamics to model cascading failure under disturbances that reduce link-wise flow capacities, where the time epochs correspond to a change in the activation status of the links. A salient feature of this model is that, unlike (bond) percolation models, the links to become inactive successively are not necessarily adjacent to each other. The margin of resilience is defined as the minimum, among all possible disturbances, of the link-wise sum of reductions in flow capacities, under which the links outgoing from the origin node become inactive in finite time. We propose a backward propagation algorithm to compute an upper bound on the margin of resilience, and describe scenarios under which this bound is tight within the class of oblivious routing policies, i.e., policies that have no information about the disturbance.

Keywords:Systems on Graphs, Large Scale Systems Abstract: A central issue in game theory is that of providing sound dynamical foundations for Nash equilibria, whose original notion is a purely static one. Several efforts have been made, generating new research fields such evolutionary game theory or the theory of learning in games. However, satisfactory answers to the aforementioned problem have been found only for special classes of games.

Another dimension of complexity is added when one considers games in networks. Here, the network can be meant to encode both constraints in the reward dependencies structure, as well as in the information structure determining the learning process. Often in the literature, the two networks have been assumed to coincide. This is the case, e.g., in population games where rewards depend only on empirical frequencies of the actions chosen by the population of agents and the population is totally mixed so that no information constraints are present, or in evolutionary coordination games over networks where agents update their actions according to (noisy) best response to their neighbors states.

In this paper we study the imitative noisy best response dynamics with small spontaneous mutations for population games under the assumption that pairwise learning interactions can only take place along the edges of a preassigned graph. Our main result shows that if the game is a potential one and the graphs are large-scale expanders, Nash equilibria can be recovered in the double limit of large population size and zero mutation noise. More specifically, we show that the invariant probability measure of the Markov process describing the learning dynamics concentrates in a neighborhood of the set of Nash equilibria.

Keywords:Computer Networks, Networked Control Systems, Systems on Graphs Abstract: In this paper, we study finite-time convergence of gossip algorithms. We show that there exists a symmetric gossip algorithm that converges in finite time if and only if the number of network nodes is a power of two, while there always exists a globally finite-time convergent gossip algorithm despite the number of nodes if asymmetric gossiping is allowed. For n=2^m nodes, we prove that a fastest convergence can be reached in mn node updates via symmetric gossiping. On the other hand, for n=2^m+r nodes with 0leq r<2^m, it requires at least mn+2r node updates for achieving a finite-time convergence in cooperation with asymmetric interactions.

Keywords:Mathematical Theory of Networks and Circuits, Stochastic Modeling and Stochastic Systems Theory, Transportation Systems Abstract: A distributed routing control algorithm for dynamical networks has recently been presented in the literature cite{CSADF1,CSADF2}. The model describes the time evolution of the density at the edges of a network and describes a distributed routing control that allows the density to converge to a Wardrop equilibrium. This is characterized by an equal traffic density on all used paths. We borrow the idea and rearrange the density model recasting this within the framework of mean-field games cite{HCM07,LL07}. The problem we analyze in this paper has striking similarities with the optimal planning problem cite{ACC12} which in turn can be linked back to mean-field games. Essentially, in optimal planning problems the idea is to drive the density of players from a given initial configuration to a target one at time T by an appropriate design of the optimal decisions of the agents.

The problem setup involves a population of individuals or players traversing the edges of a network in the attempt to reach a destination node starting from a source node. From a microscopic standpoint, each player jumps from one edge to an adjacent one according to a continuous-time Markov model. Players select the transition rates, which represent the control. From a macroscopic perspective, each edge is then characterized by dynamics describing the time evolution of the density of individuals on that edge. These dynamics take the form of a classical forward Kolmogorov ordinary differential equation (ODE). In the second part of the paper, we extend our analysis to the case where the Kolmogorov equation turns into a stochastic differential equation (SDE) driven by a Brownian motion.

For the problem at hand we highlight three main results. First, we provide a mean-field game formulation of the

Keywords:Infinite Dimensional Systems Theory, Physical Systems Theory, Applications of Algebraic and Differential Geometry in Systems Theory Abstract: This mini-course is dealing with geometric structures that arise in the modelling of infinite-dimensional port-Hamiltonian systems. In four short lessons different geometric structures as well as their system theoretic consequences will be analyzed. This includes symplectic, polysymplectic as well as Stokes-Dirac structures, where the focus is put on the latter class, especially also with regard to diffusion systems and to higher order differential operators. Additionally, the structure preserving discretization by using simplicial Dirac structures will be discussed by applying tools from discrete exterior calculus. Finally, the connection to functional analytic concepts on Hilbert spaces will be presented.

Keywords:Biological Systems, Nonlinear Systems and Control, Systems Biology Abstract: The foundation of today’s computational neuroscience is the 1952 paper of Hodgkin and Huxley. This paper is a model of behavioral modeling. It uses an exquisite combination of parsimonious experiments, biophysical principles, and curve fitting, to reduce the mechanism of the action potential to an elementary switching RC circuit. Yet the default textbook presentation of Hodgkin Huxley model is an obscure set of four nonlinear differential equations that produce oscillations for a well chosen set of parameters. The talk will review the basics of this historical model and emphasizes the importance of regarding this model as a behavior, that is, an open system, regulated by elementary but fundamental feedback principles.

Keywords:Systems Biology, Multidimensional Systems, Biological Systems Abstract: Experimental data on gene regulation and protein interaction is often very qualitative, with the only information available about pairwise interactions is the presence of either up-or down- regulation. Since majority of the parameters for any model in such a situation are not constrained by data, it is important to understand how different choices of parameters affect the dynamics and, therefore, the predictions of such a model. Continuous time Boolean networks, or Glass networks, represent an attractive platform for qualitative studies of gene regulation, since the dynamics at fixed parameters is relatively easily to compute. However, it is quite difficult to analytically understand how changes of parameters affect dynamics. On the other hand, the Database for Dynamics is an numerical approach to study global dynamics over a parameter space. The results obtained by this method provably capture the dynamics a predetermined spatial scale. We combine these two approaches to present a method to study Glass networks over parameter spaces. We apply our method to experimental data for cell cycle dynamics.

Keywords:Biological Systems, Numerical and Symbolic, Systems Biology Abstract: Gene regulatory networks control the response of living cells to changes in their environment. A class of piecewise-linear (PWL) models, which capture the switch-like interactions between genes by means of step functions, has been found useful for describing the dynamics of gene regulatory networks. The step functions lead to discontinuities in the right-hand side of the differential equations. This has motivated extensions of the PWL models based on differential inclusions and Filippov solutions, whose analysis requires sophisticated mathematical tools. We present a method for the numerical analysis of one proposed extension, called Aizermann-Pyatnitskii (AP)-extension, by reformulating the PWL models as a mixed complementarity system. This allows the application of powerful methods developed for this class of nonsmooth dynamical systems, in particular those implemented in the SICONOS platform. We also show that under a set of reasonable biological assumptions, putting constraints on the right-hand side of the PWL models, AP-extensions and classical Filippov (F)-extensions are equivalent. This means that the proposed numerical method is valid for a range of different solution concepts. We illustrate the practical interest of our approach through the numerical analysis of three well-known networks developed in the field of synthetic biology.

Keywords:Biological Systems, Mathematical Theory of Networks and Circuits, Networked Control Systems Abstract: Existing general purpose software tools for qualitative analysis of gene regulatory networks can handle most types of dynamical behaviour, but one remaining problematic case is that of transient damped oscillations, which certainly can occur. Current methods allow a solution trajectory to be followed by explicit computation of switching points, where the regulation of one or more gene is turned on or off, or more generally, allow parameter ranges to be determined for which a qualitative solution is followed. However, during intervals on which two or more gene products undergo damped oscillations, an arbitrary or even an infinite number of switching points can occur before another switch occurs to disrupt the damped oscillations. Here we resolve the resulting computational difficulties by explicitly calculating the end point of the transient oscillation. For a specific example, we compute regions corresponding to all possible itineraries, which can be used to find periodic solutions with a ‘bursting’ phase. We assume equal degradation rates, though the method should be generalizable to a wider class of networks.

Keywords:Systems Biology, Mathematical Theory of Networks and Circuits Abstract: Many biochemical network models include quantities that are conserved in molarity. While important theoretical results rely on such conservation relations, they often pose a problem for numerical analyses. Several analysis routines that are very relevant for biochemical models, for example sensitivity or stability analysis, intrinsically cannot deal with networks where conservation relations are present. To apply these routines, one generally needs to construct an equivalent reduced model without the conservation relations. Here, a method to remove conservation relations from biochemical reaction network models is proposed. The method is based on an orthogonal decomposition for the stoichiometric matrix, which makes the approach numerically efficient even for very large networks. Based on this decomposition, a reduced differential equation which describes the dynamics within a specified stoichiometric class is derived. Finally, applications of the reduction approach for steady state computation and stability analysis are discussed.

Keywords:Systems Biology, Hybrid Systems, Nonlinear Systems and Control Abstract: This paper will discuss the coupling of two oscillators in cyanobacteria. The circadian rhythm of cyanobacteria is driven by an oscillator based on an ordered sequence of phosphorylations of a protein (KaiC). As the cyanobacterial cell grows and divides, the amount of protein KaiC itself is regulated by a transcription/translation cycle. Our goals are to study: (i) the robustness of the circadian rhythm with respect to the perturbations inherent to the noisy environment of the cell, including cell growth and division; and (ii) to what extent the growth curves of cyanobacterial colonies, under different external light conditions, are modulated by the circadian cycle. A combination of hybrid and continuous approaches will be used to identify basic mechanisms and analyse the dynamics of the coupled system.

Keywords:Feedback Control Systems Abstract: Switched systems and control with limited information are two research areas that have evolved rapidly but separately over the last two decades. In this talk we undertake their unified study by considering a stabilization problem for a switched linear system with sampled and quantized state measurements. In our setting, at the sampling times the active mode of the switched system is known, but between the sampling times the switching signal is unknown and is only subject to mild slow-switching assumptions. An important ingredient in our stabilizing quantized feedback control strategy is the propagation of reachable set over-approximations for switched systems, a problem that has received a lot of attention in the hybrid systems literature and for which we propose a novel algorithm. This work is based on the paper: D. Liberzon, Finite data-rate feedback stabilization of switched and hybrid linear systems, Automatica, vol. 50, no. 2, pp. 409-420, Feb 2014.

Keywords:Biological Systems Abstract: The mathematical analysis of models for biological systems covers a wide range of challenging problems that include model assembly or model reduction, parameter estimation, or control of a system towards a desired state. Many different formalisms and methodologies can be used to study these problems, but here the focus will be on Boolean and hybrid modeling frameworks, which facilitate the development of intuitive, computationally amenable, and mathematically rigorous, methods of analysis.

The first part of this presentation will address the problem of predicting the dynamical properties of large biological networks, which are often obtained by assembling several smaller modules. Each module will be represented by a Boolean network, the corresponding asynchronous state transition graph and its attractors. Two new objects are introduced, the asymptotic and the cross- graphs, constructed from the strongly connected components of the modules' transition graphs. By using the notion of feedback interconnection, it is shown that the attractors of the large Boolean network can be fully identified in terms of cross-products of the attractors of each module, a method which implies a large computational cost reduction.

In the second part some examples will be presented, including a model for cellular growth of bacteria E.coli and a model for gene pattern formation in Drosophila embryo, to illustrate how a combination of different mathematical formalisms leads to gaining quantitative knowledge and predictive power for biological systems.

Keywords: Abstract: In this talk we discuss algorithms and present a computational framework for solving a large class of dynamic clustering, coverage control and aggregation problems, ranging from those that arise in deployment of mobile sensor networks, to identification of ensemble spike trains in neuronal data, to reduction of graphs and Markov chains. This framework provides the ability to identify natural clusters in an underlying dynamic data set and connectivity structures in large graphs, and allows us to address inherent trade-offs such as those between cluster resolution and computational cost. We can further define the problem of minimizing an instantaneous coverage metric as an optimization problem using a maximum entropy formulation constructed specifically for the dynamic setting, where locating cluster centers and tracking their associated dynamics is cast as a control design problem.

Keywords:Networked Control Systems Abstract: In this paper we consider networks of identical linear time-invariant single-input-single-output systems. We consider the situation in which the couplings between the individual systems depend on a parameter varying over compact real interval. We examine a family of circularly interconnected harmonic oscillators. Also, we consider a family of systems that are arranged in a row. We show that in both cases there exists a broadcasted parameter-independent open-loop control input steering all individual system towards a predefined parameter-dependent family of terminal states. The investigation of this problem leads naturally to the analysis of one-parameter dependent linear systems and, in particular, uniform ensemble controllability. To this end, we generalize existing results on uniform ensemble controllability of parameter-dependent single-input linear systems so that the parameter space can be any compact subset of the real line.

Keywords:Mathematical Theory of Networks and Circuits, Biological Systems, Networked Control Systems Abstract: We study synchronization of nonlinear systems that satisfy an incremental passivity property. We consider the case where the control input is subject to a class of disturbances, including constant and sinusoidal disturbances with unknown phases and magnitudes and known frequencies. We design a distributed control law that recovers the synchronization of the nonlinear systems in the presence of the disturbances. Simulation results of Goodwin oscillators illustrate the effectiveness of the control law.

Keywords:Networked Control Systems, Nonlinear Systems and Control, Large Scale Systems Abstract: We consider consensus algorithms for multi-agent networks with discrete-time linear identical MIMO agents. The agents may be of arbitrary order, the interaction topology may be time-varying and the couplings may be nonlinear and uncertain, however assumed to satisfy a slope restriction or, more generally, quadratic constraint. Using the discrete-time version of the KYP Lemma (referred to as the Kalman-Szegő Lemma), we derive a criterion which provide consensus in such a network for any uncertain couplings from the mentioned class. This criterion is close in spirit to the celebrated Tsypkin criterion for discrete time Lurie system.

Keywords:Networked Control Systems, Large Scale Systems, Stochastic Control and Estimation Abstract: In this paper, we analyze convergence of the consensus problems in multi-agent systems. The system considered here has a directed graph topology and the communication among the agents includes noise. A stochastic approximation method is applied to a consensus algorithm and the relation between the closeness of the agreement and the number of iterations is clarified. A required number of iterations for the desired closeness can be estimated by a probabilistic guarantee.

Keywords:Linear Systems, Stability, Feedback Control Systems Abstract: We study consensus problem over a network of dynamic agents with time-dependent communication links which may disconnect for long intervals of time. We assume that the nodes are interconnected in a ring topology. The originality of our results lays, in part, in our method of proof: we leave behind graph theory for linear time-invariant systems and use instead, stability theory. In particular, we employ the small-gain theorem to establish simple checkable conditions on the network interconnections, to guarantee the achievement of consensus. Simulations on an academic example are provided to demonstrate the effectiveness of the theoretical results.

Keywords:Networked Control Systems, Optimal Control, Systems on Graphs Abstract: In this paper, decentralized feedback control strategies are derived for min-max time consensus tracking of multi-agent systems with bounded inputs that are commu- nicating over directed graphs. Each agent is a linear time invariant system with distinct and rational eigenvalues. The graph contains a directed spanning tree rooted at an agent which generates the reference trajectory for the other agents. This spanning tree is locally identified and the tail agent of each edge tries to match its states with that of the head of the edge in min-max possible time. Using recent Gröbner basis based methods by the authors, these local min-max time problems are solved by deriving Nash equilibrium feedback strategies for time optimal pursuit-evasion games. The local min-max strategies lead to global consensus tracking in min-max time under some conditions.

Keywords:System Identification Abstract: In this paper, we will develop an online statistical change detection method for a multivariable system in closed-loop. The proposed method is based on online hypotheses testing by observing a test signal generated by a recursive least squares algorithm for estimation of a bunch of selected matrix-valued Markov parameters of a high-order ARX model. A test signal generated sequentially by the recursive algorithm is studied and its asymptotic whiteness is proved explicitly. The proposed online change detection method is based on a likelihood ratio test to examine the change in the covariance of the aforementioned test signal.

Keywords:System Identification Abstract: Explicit formulae of dominant parts of the estimation errors of A and C matrices in PO-MOESP are derived based on a proposing lemma on perturbations to the singular subspaces. The formulae are derived directly from a slightly modifed PO-MOESP algorithm. Based on the formulae, the effect of the weighting matrices in the singular value decomposition is analyzed. It is also shown that the original PO-MOESP method is optimal in a sense that singular values of some matrix in the Frobenius norm of the estimation error are minimized.

Keywords:System Identification, Linear Systems, Mechanical Systems Abstract: This paper deals with the identification of errors-in-variables (EIV) models corrupted by additive and uncorrelated white noises when the noise-free input is an arbitrary signal, not necessarily periodic. In particular, a frequency domain method is proposed, under the assumption that the ratio of the noise variances is known.

Keywords:System Identification, Linear Systems, Networked Control Systems Abstract: Identification of particular dynamical transfers (modules) in a general dynamic network is addressed in the context of prediction error identification. Depending on the availability of measured signals, and the possible presence of (external) excitation signals, conditions are derived for the consistent estimation of prespecified modules. For this purpose existing closed-loop identification methods have been generalized to the situation of general dynamic networks with complex interconnection structures, including feedback. Topological conditions on the network can be verified by graph theoretical tools. The achieved results can be used as a basis for sensor and actuator placement schemes as well as for network structure identification problems.

Keywords:System Identification, Signal Processing, Stochastic Modeling and Stochastic Systems Theory Abstract: In many practical applications second order moments are used for estimation of the power spectrum. The moments are estimated from the available data and then the inverse problem to seek a spectrum consistent with the moments is solved. The focus here is on the second step in this procedure. Usually, the second order moments considered are the covariances of the process, and well known methods are available for efficiently determining the autoregressive part of a generating model from them. The autoregressive part adds poles to the model and in order to also have zeros additional information needs to be determined from the data. Here, we consider the case where second order moments of the inverse of the power spectrum, so called inverse covariances, also are given. The exact inverse problem corresponding to autoregressive moving average models with one pole and one zero is studied explicitly, and the solvability of the inverse problem is considered. Then different exact and approximative interpolation problems are considered from a global optimization approach. The exact interpolation result generalizes a result by T.T. Georgiou on a maximally random process from a smoothing perspective. For the approximation a quadratic function of the interpolation error is minimized in one approach and the distance between interpolants of the covariances and the inverse covariances are minimized in the second approach.

Keywords:Optimization : Theory and Algorithms, Systems on Graphs, Nonlinear Systems and Control Abstract: A dual decomposition approach for the distributed control of both reactive power and storage control in microgrids is presented. This approach considers physical constraints inherent to storage systems, as well as voltage regulation constraints. The objective is to minimize a cost function consisting on a linear combination of the overall active power cost over a finite time horizon, and the transmission losses over the microgrid. A distributed algorithm is introduced for which we present the corresponding convergence result. In addition, we introduce a novel approximate distributed voltage prediction algorithm. Simulations demonstrate the dual decomposition algorithm performance on a particular microgrid case.

Keywords:Large Scale Systems, Feedback Control Systems, Stability Abstract: This paper considers the robustness of a recently proposed distributed distant-downstream control architecture for irrigation channels. The components of this can be designed in a scalable fashion (i.e. pool-by pool) to nominally achieve an L_infty string-stability property that concerns the spatial propagation of transient flow peaks. The robustness of this property is investigated via known LMI based analysis conditions for bounding the L_infty induced norm of systems with uncertain transfer functions. Application of the conditions, which are only sufficient, does not confirm string-stability robustness for the channel example presented. However, the robust induced-norm bounds obtained for substantial pool-delay parameter uncertainty are such that the degree to which transient flow peaks could be amplified remains reasonable at worst. Illustrative simulations are presented.

Keywords:Nonlinear Systems and Control, Networked Control Systems, Large Scale Systems Abstract: This paper studies the problem of frequency regulation in power grids under unknown and possible time-varying load changes, while minimizing the generation costs in some particular cases. We formulate this problem as an optimal output agreement problem for distribution networks. This problem has been recently addressed by some of the authors using incremental passivity and dynamic internal-model-based controllers. We believe that this framework is general enough to allow for more complex control scenarios in future extensions.

Keywords:Transportation Systems, Optimization : Theory and Algorithms, Networked Control Systems Abstract: This paper studies the problem of optimal flow control in dynamic inventory systems. A dynamic optimal distribution problem, including time-varying supply and demand, capacity constraints on the transportation lines, and convex flow cost functions of Legendre-type, is formalized and solved. The time-varying optimal flow is characterized in terms of the time-varying dual variables of a corresponding network optimization problem. A dynamic feedback controller is proposed that regulates the flows asymptotically to the optimal flows and achieves in addition a balancing of all storage levels.

Keywords:Optimal Control, Networked Control Systems, Linear Systems Abstract: This paper studies the optimal control of a micro grid of biogas prosumers equipped with local storage devices. Excess biogas can be upgraded and injected into the low-pressure gas grid or, alternatively, shipped per lorry to be used elsewhere in an effort to create revenue. The aim of the control process is to maximize the prosumers' profit from the biogas they produce, with a size restriction on the local biogas storage. The problem is solved in centralized and distributed MPC schemes in order to compare their capabilities to control a micro grid. We perform simulations with a realistic average gas usage pattern over a year to study the economic feasibility of local biogas storage.

Keywords:Nonlinear Systems and Control, Systems on Graphs Abstract: This paper considers the attitude stabilization and synchronization of rigid bodies. We use the logarithm map to exponentially stabilize the attitude of a rigid body with an almost global region of attraction. Interestingly, we are able to obtain a closed form solution for the kinematics. We extend this in a discontinuous manner and analyze global stability. Further, we present consensus protocols for exponential attitude and angular velocity synchronization under directed information flow. Simulation results are presented to validate the control laws.

Keywords:Artificial Intelligence, Robotics, Linear Systems Abstract: This paper takes a step towards defining what it means for a dynamical system, such as a robot, to be able to learn through the notion of learnability. It takes a system-theoretic view to define what learning is and establishes when a system can and can not learn. Equipped with this definition of learnability, we provide a learnability result for linear systems with quadratic costs that is then applied to two different types of mobile robots, namely a simulated two-wheel inverted pendulum robot and a real, locomoting robotic platform.

Keywords:Delay Systems, Robotics, Optimal Control Abstract: Control of cooperative bilateral teleoperation systems over delayed communication is considered. A recently proposed quadratically invariant control architecture is adopted. It is shown that within this architecture, the H2 optimal controller can be found analytically in spite of the decentralized nature of the problem. As a first step, we show that the problem can be split into a number of independent centralized control problems with delays. This, in turn, allows us to apply the recent loop shifting techniques for finding an efficient solution and revealing the optimal controller structure.

Keywords:Discrete Event Systems, Robotics, Nonlinear Systems and Control Abstract: In this paper, the authors present a new kind of cubic mobile robot whose locomotion is dominated by rolling constraints. We discuss the controllability on the robot in a discrete state space. The behaviors of the robot are described as motions of a rolling cube with special constraints originated in their actuation structure. Considering some cases of input patterns, we concluded that the robot has the global reachability on its position.

Keywords:Networked Control Systems, Robotics Abstract: In this paper, we investigate whether we can qualitatively recover the appropriate group sizes for a team of predators by varying environmental and operational conditions. The result is a combination of biologically inspired analytical and algorithmic tools that not only establish guaranteed capture conditions, but also identify the number of predators needed for a successful capture. We implement the parameterized model on a team of mobile robots to validate that it is possible to generate a cooperative strategy that achieves capture with this model and these tools.

Keywords:Optimal Control, Robotics Abstract: This paper considers the optimal control problem of connecting two periodic trajectories with maximal persistence. A maximally persistent trajectory is close to the periodic type in the sense that the norm of the image of this trajectory under the operator defining the periodic type is minimal among all trajectories. A solution is obtained in this paper for the case when the two trajectories have the same period but it turns out to be only piecewise continuous, and so an alternate norm is employed to obtain a continuous connection. The problem of connecting periodic trajectories is of interest because the operating characteristics of many biological and artificial systems are limit cycles, and so there is a need for a unified optimal framework of connections between different operating regimes. This paper is a first step towards that goal.

Inst. Tecnologico Y De Estudios Superiores De Monterrey

Keywords:Robotics, Nonlinear Systems and Control, Mechanical Systems Abstract: This paper proposes a surveillance and defense system based on a 3SPS-1S parallel manipulator. The manipulator has three pure rotation degrees of freedom thanks to the central leg, which also increases the stiffness of the robot. The inverse and forward dynamics of the manipulator are solved. Two different control methods are presented, one is to control each leg separately, and the other to control the platform’s position directly. Some control examples are present for each method. A prototype of the manipulator as a sentry gun is shown, and a test scenario is presented in order to show a way the robot can be used.

Keywords:Nonlinear Systems and Control, Stability Abstract: Under normal operating conditions, the electrical signals in a power system are sinusoidal waveforms with the same constant frequency, also known as synchronous frequency. Since a deviation from the synchronous frequency may cause serious damage in the components of a power system, one of the most important problems in power systems, also known as the transient stability problem, is the synchronization of all the signals on the synchronous frequency.

Traditional models used for the study of transient stability problem are based on several assumptions: 1) all waveforms in the model are sinusoidal; 2) the fre- quencies of these waveforms are very close to the synchronous frequency; 3) the reactive power is neglected in the power balance equation of each generator. These assumptions are not compatible with transients during which waveforms are known not to be sinusoidal. Moreover, the models based on these restrictive assumptions prevent us from having a clear understanding of how energy moves between different components of a power system. Using the framework of port-Hamiltonian systems it is possible to derive a more general model from rst principles. Such model is still tractable and it does not require any of the stated assumptions. Furthermore, we show, using the aforementioned model, how to infer transient stability of the interconnected system by checking a simple condition for each individual generator.Since we reason about the interconnected system by applying a condition to the individual generators, the presented results are also compositional. This energy-based framework is already used in the recent works [3, 4], where a single machine connected to a load is analyzed in the restrictive scenario that there are no losses. The transient stability conditions presented in this extended abstract are applicable for a larger class of power systems which are composed of lossy transmission lines, constant impedance and constant current loads.

Keywords:Feedback Control Systems, Networked Control Systems, Optimization : Theory and Algorithms Abstract: We consider the problem of minimizing the power generation cost by exploiting the microgenerators dispersed in the power distribution network. The proposed strategy requires that the intelligent agents, located at the microgenerator buses, measure their voltage and then actuate the physical layer by adjusting the amount of injected power, according to a feedback control law derived from a projected gradient descent strategy. Simulations are provided in order to illustrate the algorithm behavior.

Keywords:Mathematical Theory of Networks and Circuits, Nonlinear Systems and Control, Systems on Graphs Abstract: Motivated by the growing interest in smart grid architectures, we consider the problem of voltage stability and reactive power balancing in microgrids. It is generally believed that high-voltage solutions of the polynomial reactive power flow equations are stable, but the locations of the associated equilibria are unknown, as is the critical network load where stability is lost. Inspired by the "control by interconnection" paradigm developed for port-Hamiltonian systems, we propose a novel droop-like inverter controller which is quadratic in the local voltage magnitude. Remarkably, under this controller the closed-loop network is again a well-posed electrical circuit. We find that the equilibria of the quadratic droop-controlled network are in exact correspondence with the solutions of a reduced power flow equation. We consider general network topologies and study some simple yet insightful solutions of this equation. As first result, for the frequently-encountered case of a parallel microgrid, we present a concise, closed-form, necessary and sufficient condition for the existence of an exponentially stable high-voltage network equilibrium. Our condition establishes the existence of a critical inductive load for the network, which depends only on the network topology, admittances, and controller gains. Going further, as a second result we show that in microgrids of arbitrary topology, for sufficiently-high reference voltages, the existence and local exponential stability of a high-voltage equilibrium point is assured. We provide an approximate expression for this fixed point and find the limiting value of the approximation error for high reference voltages.

Keywords:Transportation Systems, Physical Systems Theory, Large Scale Systems Abstract: With this contribution to the study of electric power system behavior we bring into focus the role of eigenvalue sensitivity under incremental changes of stationary power flow solutions.This analysis is motivated by the ubiquity of high spectral sensitivity in a large class of physical systems that are distributed in space, where the underlying mechanism is related to the nonnormality of the linearized dynamics. To this end we relate high sensitivity in stressed power grids to elements in the analysis of nonnormal systems, namely the ill-conditioning due to spatial transport processes. In doing so we present two recently, and independently derived novel formulas for eigenvalue deviations under changes in operating point and relate them to each other. It turns out that these eigenvalue sensitivity formulas play a fundamental role for power system behavior and we establish relations to recently proposed phase-coupled oscillator models for power systems. To conclude we discuss the use of the proposed eigenvalue sensitivity formulas for more flexible operation architectures in which PMU data may be incorporated for real-time coordination of local controls.

Keywords:Optimal Control, Optimization : Theory and Algorithms, Networked Control Systems Abstract: Conventional analysis and control approaches to inter-area oscillations in bulk power systems are based on a modal perspective. Typically, inter-area oscillations are identified from spatial profiles of poorly damped modes, and they are damped using carefully tuned decentralized controllers. To improve upon the limitations of conventional decentralized strategies, recent efforts aim at distributed wide-area control which involves the communication of remote signals. Here, we introduce a novel approach to the analysis and control of inter-area oscillations. Our framework is based on a stochastically driven system with performance outputs chosen such that the H2 norm is associated with incoherent inter-area oscillations. We show that an analysis of the output covariance matrix offers new insights relative to modal approaches. Next, we leverage the recently proposed sparsity-promoting optimal control approach to design controllers that use relative angle measurements and simultaneously optimize the closed-loop performance and the control architecture. For the IEEE 39 New England model, we investigate performance trade-offs of different control architectures and show that optimal retuning of decentralized control strategies can effectively guard against inter-areas oscillations.

Keywords:Multidimensional Systems, Large Scale Systems, Optimization : Theory and Algorithms Abstract: Present paper proposes a method to provide necessary series compensation by formulating transient stability problem as a synchronization problem, providing an edge control to keep the line flows within permissible limits. With the aim to satisfy relative rotor angle criteria in addition to line flow control having multiple constraints on a set of state variables as well as on control variables makes it necessary to call for an efficient optimization tool. Knowing the strengths and effectiveness of convex optimization present paper proposes an application of the same in finding necessary series compensation using TCSC to address this multidimensional problem in remarkably shortest possible time which facilitates on-line synchronization. In the process, post fault conditions are mapped to pre-fault healthy conditions so as to guarantee stable and secure power flow even after credible line contingency without shedding any load (i.e. maintaining demand supply balance), which if not followed may lead to cascade tripping.

Keywords:Optimal Control, Adaptive Control, Linear Systems Abstract: In embedded computers, there are delays due to computation time. Unless they are considered, a controlled system may be unstable. If the system is unknown, Q-learning-based optimal control is one of the useful approaches. Applying it to a system, we can obtain the optimal feedback gain for the unknown system. In this paper, we propose Q-learning-based optimal feedback control taking the delay into consideration. First, we assume that all states can be observed and consider a state feedback controller. The input at the next time is estimated by the current state and the input. Then an optimal feedback gain is provided by sequences of pairs of the state and the input. Next, we show an output feedback controller. If the system is observable, the state can be reconstructed by the last L input and output data, where L is an integer equal to or larger than the observability index of the system. An optimal feedback gain is provided by these sequences. Finally, we apply the proposed adaptive state feedback controller to a quadrotor and show its efficiency by simulation.

Keywords:Optimal Control, Computational Control, Linear Systems Abstract: The recently developed max-plus primal space fundamental solution provides a new explicit representation of solutions to the difference Riccati equation (DRE) that is ubiquitous in system and optimal control / filtering theory. This representation provides a new tool to explore various key properties of the DRE. This paper presents new results on these properties for a class of DREs via the associated max-plus primal space fundamental solution.

Keywords:Optimal Control, Nonlinear Systems and Control Abstract: The aim of this note is to compare the averaged optimal coplanar transfer with one input towards circular orbits when the costs are transfer time and energy consumption. While the energy case leads to the analysis of a 2-D Riemannian metric usingthe standard tools of Riemannian geometry (curvature computations, geodesic convexity), the time-minimal case is associated to a Finsler metric which is not differentiable. Nevertheless a qualitative analysis of the singularites and the geodesic flow is given in this article to describe the optimal transfers and contrast them with those of the energy case.

Keywords:Optimal Control, Nonlinear Systems and Control, Optimization : Theory and Algorithms Abstract: Some optimal control problems do not have solution in the class of classic controls. This suggests the need of a relaxation or extension of the control problem ensuring the existence of a solution in some enlarged class of controls. This work aimes at the development of an extension for optimal control problems with a nonlinear control dynamics and the control function which takes values in some closed, but not necessarily bounded, set. To achieve this goal, we exploit the approach of R.V. Gamkrelidze based on the generalized controls, but related to discontinuous arcs. This leads to the notion of generalized impulsive control.

Keywords:Optimal Control, Stochastic Control and Estimation, Feedback Control Systems Abstract: In this paper, we investigate the existence of a mean-square stabilizing solution to a continuous-time modified algebraic Riccati equation (MARE). Such an MARE comes up in the linear-quadratic optimal control of a class of stochastic systems. In most existing research works, only sufficient conditions are given for the existence of a mean-square stabilizing solution in terms of stabilizability and observability (or detectability) of the underlying stochastic system. In some papers, numerical conditions are provided. However, such conditions do not have explicit interpretations with respect to the dynamical properties of the underlying stochastic system. Here, a necessary and sufficient condition is given directly in terms of the system parameters and has explicit interpretations with respect to the dynamical properties of the underlying stochastic system. It is shown that the common assumption or condition of observability or detectability of certain stochastic system is not necessary.

Keywords:Adaptive Control, Optimal Control, Neural Networks Abstract: In this paper a partially model-free reinforcement learning (RL) algorithm based on experience replay is developed for finding online the Nash equilibrium solution of the multi-player nonzero-sum (NZS) differential games. In order to avoid the performance degradation or even system instability, the amplitude limitation on the control inputs is considered in the design procedure. The proposed algorithm is implemented on actor-critic structure for every player in the game, where both actor and critic networks are tuned at the same time. The game players learn online the solution of the constrained coupled Hamilton-Jacobi (HJ) equations, without using any knowledge on the internal system dynamics. The idea of experience replay is used to relax the requirement for checking the restrictive persistence of excitation (PE) condition which is difficult to verify or implement online. The closed-loop stability is analyzed and the convergence to the Nash equilibrium of the game is shown. A simulation study example is provided showing the effectiveness of the proposed approach.

Keywords:Applications of Algebraic and Differential Geometry in Systems Theory, Mathematical Theory of Networks and Circuits, Nonlinear Systems and Control Abstract: The variational framework for linear electric circuits introduced in [OTCOM13] is extended to general nonlinear circuits. Based on a constrained Lagrangian formulation that takes the basic circuit laws into account, the equations of motion of a nonlinear electric circuit consisting of inductors, capacitors, resistors and voltage sources are derived. The resulting differential-algebraic system can be reduced by performing the variational principle on a reduced space. It is shown under which conditions the corresponding reduced Lagrangian is regular and thus the equations of motion transform to an ordinary differential equation system. Based on a discrete variational formulation, a variational integrator for the structure-preserving simulation of nonlinear electric circuits is derived and demonstrated by numerical examples.

Keywords:Control of Distributed Parameter Systems, Computational Control, Numerical and Symbolic Abstract: A geometric spatial reduction method is presented in this paper. It applies to port Hamiltonian models for open systems of balance equations. It is based on projections which make use of the symmetries and on the preservation of the "natural" power pairing for the considered system. The method is applied to a system of two coupled parabolic equations describing the poloidal magnetic flux diffusion and heat radial transport in tokamak reactors. There are two reduction steps first to reduce the model from 3D to 1D, then to reduce the model from 1D to 0D. The assumptions of axial symmetry and quasi-static equilibrium of the plasma are used to perform the reduction from 3D to 1D by using simple integration formulas on toroidal coordinates’ surfaces. A Galerkin-type pseudo-spectral spatial reduction method is used to reduce the 1D model to a 0D one. Both reductions are made symplectic with respect to the power-pairings in the magnetic and thermal domains. Finally, the 0D plasma control model is obtained by reduction of the multi-domains couplings between the two original PDEs. These couplings are the Lorentz force (with non-uniform resistivity) and the bootstrap current.

Keywords:Infinite Dimensional Systems Theory, Computations in Systems Theory, Linear Systems Abstract: This paper presents a new port-Hamiltonian formulation for the transmission line. By eliminating the dual complex from the underlying Dirac structure, it enables structure-preserving spatial semi-discretization using mixed finite elements. The proposed discretization scheme preserves not only the port-Hamiltonian structure but also important conservation laws. A numerical example using the p version of the finite element method demonstrates the improvements in convergence attainable by higher-order discretization schemes.

Keywords:Control of Distributed Parameter Systems, Nonlinear Systems and Control, Computational Control Abstract: The paper deals with the feedforward control problem for a class of hyperbolic distributed parameter systems in one spatial dimension (generalized transmission systems). Based on the models obtained from a structure preserving port-Hamiltonian discretization, a modular approach is presented to determine input trajectories which correspond to prespecified trajectories of the non-collocated outputs. The presence of a feedthrough term in the discretized models allows to immediately express the inverse dynamics which must be integrated to solve the feedforward control problem. In general, the inverse dynamics for non-collocated pairs of in- and outputs is unstable, hence, techniques from the dynamic inversion of non-minimum phase systems can be applied. The resulting modular approach is presented for linear systems, and, as a sketch, for the nonlinear case. The proposed procedure is a building block to solve the feedforward control problem in networks of nonlinear transmission systems.

Keywords:Numerical and Symbolic, Physical Systems Theory, Computational Control Abstract: This paper describes the mathematical setting of continuum models such as elasticity and fluid flow. From this continuous description we derive a discrete formulation which satisfies the conservation laws from the outset. The only approximation required is for the material-dependent constitutive equations.

Keywords:Applications of Algebraic and Differential Geometry in Systems Theory, Mechanical Systems, Numerical and Symbolic Abstract: Many gauge field theories can be described using a multisymplectic Lagrangian formulation, where the Lagrangian density involves space-time differential forms. While there has been much work on finite-element exterior calculus for spatial and tensor product space-time domains, there has been less done from the perspective of space-time simplicial complexes. One critical aspect is that the Hodge star is now taken with respect to a pseudo-Riemannian metric, and this is most naturally expressed in space-time adapted coordinates, as opposed to the barycentric coordinates that the Whitney forms (and their higher-degree generalizations) are typically expressed in terms of.

We introduce a novel characterization of Whitney forms and their Hodge dual with respect to a pseudo-Riemannian metric that is independent of the choice of coordinates, and then apply it to a variational discretization of the covariant formulation of Maxwell's equations. Since the Lagrangian density for this is expressed in terms of the exterior derivative of the four-potential, the use of finite-dimensional function spaces that respects the de Rham cohomology results in a discretization that inherits the gauge symmetries of the continuous problem. This then yields a variational discretization that exhibits a discrete Noether's theorem, which implies that an associated multi-momentum is automatically conserved by the discretization.

Keywords:Hybrid Systems, Linear Systems Abstract: The paper studies the reach control problem (RCP) to make trajectories of an affine system defined on a simplex reach and exit a prescribed facet of the simplex in finite time without first leaving the simplex. Affine feedbacks are generally used to solve RCP, and there is an emerging belief that affine feedback and continuous state feedback are equivalent with respect to solvability of RCP on simplices. This equivalence has been proved under an assumption on the triangulation of the state space. There remains the question of whether this result can also be proved under arbitrary triangulations. In this paper, we show that the answer is negative by constructing an example for which no solution based on affine feedback exists, yet a continuous state feedback solves the problem. Then for single-input affine systems we provide a constructive method for synthesis of multi-affine feedbacks to solve RCP on simplices for the case where affine feedbacks fail to solve the problem.

Keywords:Hybrid Systems, Linear Systems Abstract: This paper studies reachability problem for discrete-time bimodal piecewise linear systems. Lack of convexity in discrete-time reachability problem is a major issue that prevents generalizations of the existing results for such systems in continuous-time. In Yurtseven et. al., "Controllability of a class of bimodal discrete-time piecewise linear systems", Systems and Control Letters (62), 338-344, 2013, necessary and sufficient conditions for controllability of discrete-time bimodal systems were presented only under a quite restrictive condition on the dimension of the underlying zero dynamics. In this paper, we take a completely different approach that is based on the recent results for the reachability of linear systems with conical output constraints and provide necessary and sufficient conditions without any assumptions on the zero dynamics.

Keywords:Hybrid Systems, Linear Systems, Feedback Control Systems Abstract: This paper studies the problem of stabilizing a continuous-time switched linear system by quantized output feedback. We assume that the quantized output and the switching signal are available to the controller at all time. We develop an encoding strategy by using multiple Lyapunov functions and an average dwell-time property. The encoding strategy is based on the results in the case of a single mode. An additional adjustment of the “zoom” parameter is required at every switching time.

Keywords:Hybrid Systems, Nonlinear Systems and Control, Linear Systems Abstract: In this paper we tackle the disturbance decoupling problem for continuous bimodal piecewise linear systems. After establishing necessary and sufficient geometric conditions for such a system to be disturbance decoupled, we study state feedback and dynamic feedback controllers, both mode-dependent and mode-independent. For these feedback schemes, we provide necessary and sufficient conditions for the solvability of the disturbance decoupling problem. Also, we provide subspace algorithms in order to verify the presented conditions.

Keywords:Linear Systems, Hybrid Systems, Feedback Control Systems Abstract: This paper concerns robust controlled invariant subspaces and robust conditioned invariant subspaces for a family of vertex systems for polytopic uncertain switched linear systems. Further, disturbance decoupling problem via static output feedback under arbitrary switching for polytopic uncertain switched linear systems is also investigated.

Keywords:Linear Systems, Hybrid Systems, Stability Abstract: In this paper, a class of linear dynamical systems, called linear reset systems, is studied from a geometric point of view. Their state satisfies a system of linear differential equations (with constant coefficients) but they are provided with a mechanism which resets the state when a certain condition is met.

In particular, when the dynamical system without reset is stable, sufficient conditions on the reset structure are given, which guarantee asymptotic stability of the corresponding reset system.

Keywords:Large Scale Systems, Optimization : Theory and Algorithms Abstract: In elastic shape analysis, a representation of a shape is invariant to translation, scaling, rotation and re-parameterization and important problems (such as computing the distance and geodesic between two curves, the mean of a set of curves, and other statistical analyses) require finding a best rotation and re-parameterization between two curves. In this paper, we focus on this key subproblem and study different tools for optimizations on the joint group of rotations and re-parameterizations. In this conference paper, we give a first account of a novel Riemannian optimization approach and evaluate its use in computing the distance between two curves and classification using two public data sets. Experiments show significant advantages in computational time and reliability in performance compared to the current state-of-the-art method. Further information will become available in a forthcoming full version of this conference paper.

Keywords:Optimization : Theory and Algorithms Abstract: In this paper, we characterize a monotonicity of the solution of interval quadratic programming that involves an interval-valued parameter in the objective function and linear constraints. To give a characterization of monotonicity, we first analyze the derivative of a solution candidate for quadratic programming, on the basis of the Karush-Kuhn- Tucker condition. Then, deriving a condition for the objective function to assure the monotonicity, we propose a method to exactly find the upper and lower limits of optimal solutions by a finite number of operations. Finally, we provide an illustrative example of power supply scheduling to demonstrate the efficiency of our solution method.

Keywords:Optimization : Theory and Algorithms, Mathematical Theory of Networks and Circuits, Communication Systems Abstract: Given a set of nodes in the plane, the min-power centre is a point that minimises the cost of the star centred at this point and spanning all nodes. The cost of the star is defined as the sum of the costs of its nodes, where the cost of a node is an increasing function of the length of its longest incident edge. The min-power centre problem provides a model for optimally locating a cluster-head amongst a set of radio transmitters. We provide upper bounds for the performance of the centroid (centre of mass) of the given nodes as an approximation to the quadratic min-power centre.

Keywords:Optimization : Theory and Algorithms, Optimal Control Abstract: What influence can be exerted by one or a few nodes in the consensus or stationary distribution reached by a global network? We address this general question regarding the sensitivity of the invariant probability distribution of an irreducible row-stochastic matrix to the perturbation of a few of its rows, and provide tight bounds on the infinity-norm of the perturbation that can be computed in polynomial time.

Keywords:Optimization : Theory and Algorithms, Optimal Control, Process Control Abstract: In this paper we propose an approach for solving convex quadratic programs (QPs) with linear equalities and general linear inequalities by the alternating direction method of multipliers (ADMM). ADMM has attracted considerable interest in recent years in different application fields, especially due to the simplicity of the iteration. We focus on the application of ADMM to the QPs that are solved in Model Predictive Control (MPC) algorithms, where the inequalities represent limits on the states and controls. After introducing our ADMM iteration, we provide a proof of convergence based on the theory of maximal monotone operators. The proving approach allows us to identify a more general measure to monitor the rate of convergence than those previously used and to characterize the optimal step size for the ADMM iterations for the considered class of QPs. While the mathematical result has a similar structure to previous contributions, it allows us to relax some of the previously required assumptions that currently limit the applicability to the QP of model predictive control. The results are validated through numerical simulations on a large number of publicly available QPs, which are generated from an MPC for controlling of a four tank process.

Keywords:Optimization : Theory and Algorithms, Stability, Large Scale Systems Abstract: Modern power networks often have partitioned structures, with disjoint parts of the network being operated by competing agents. It is therefore of significant interest to be able to solve the Optimal Power Flow (OPF) problem for the entire network in a distributed manner without requiring significant exchange of sensitive operational information between rival operators. For networks whose regions are interconnected over a tree structure, we consider a comparison of two potential distributed schemes for solving the OPF problem. For the augmented algorithm, with incorporated higher-order dynamics, we prove a result of guaranteed convergence to the set of solutions of the global problem. By contrast, for classical unregularized dual decompositions, we demonstrate that there exist network scenarios in which convergence breaks down and the algorithm fails to give a solution of the OPF problem.

Keywords:Control of Distributed Parameter Systems, Infinite Dimensional Systems Theory, Mathematical Theory of Networks and Circuits Abstract: In this paper we present control of infinite- dimensional systems by power shaping methods, which have been used extensively for control of finite dimensional systems. Towards achieving the results we work within the Brayton Moser framework, by using the system of transmission line as an example and derive passivity of the system with respect to the boundary voltages and derivatives of current at the boundary. We then solve the stabilization problem by interconnecting the system through a finite-dimensional controller and generating Casimirs for the closed-loop system. Finally we explore possi- bility of generating other alternate passive maps.

Keywords:Control of Distributed Parameter Systems, Infinite Dimensional Systems Theory, Operator Theoretic Methods in Systems Theory Abstract: In this paper we address the state feedback regulator problem for regular boundary control systems (a special class of infinite-dimensional linear systems). The plant is assumed to be exponentially stable and is driven by a linear (possibly infinite-dimensional) exosystem via a disturbance signal. The exosystem has its spectrum in the closed right half-plane and also generates the reference signal for the plant output. The regulator problem is to design a controller that, while guaranteeing the stability of the closed-loop system without the exosystem, drives the tracking error to zero. A particular version of this problem is the state feedback regulator problem in which the states of the exosystem and the plant are known to the controller. Under suitable assumptions, we show that the latter problem is solvable if and only if a system of three algebraic equations, called the regulator equations, is solvable. We derive conditions, in terms of the transfer function of the plant and eigenvalues of the exosystem, for the solvability of the regulator equations. An example illustrating our theory is presented.

Keywords:Control of Distributed Parameter Systems, Nonlinear Systems and Control, Stability Abstract: This paper presents a systematic way of constructing control Lyapunov functions for a class of partial differential equations in terms of a geometric viewpoint, a symplectic structure. The equations are assumed to include up to second order derivatives with respect to a time coordinate and higher order derivatives with respect to spatial coordinates.

Keywords:Control of Distributed Parameter Systems, Optimal Control, Numerical and Symbolic Abstract: We consider the null-controllability problem for rotating Timoshenko beams. We present a short survey of the results on different forms of reachability conditions and determining the minimal exact controllability time. The main result concerns the problem of optimal control in the sense of minimal energy. We give a new numerical approach of finding the optimal control. The method is based on the analysis of the specific properties of the moment problem corresponding to the problem of rest-to-rest controllability.

Keywords:Stability, Infinite Dimensional Systems Theory, Nonlinear Systems and Control Abstract: Lyapunov's indirect method is an attractive method for analyzing stability of non-linear systems since only the stability of the corresponding linearized system needs to be determined. Unfortunately, the proof for finite-dimensional systems does not generalize to infinite-dimensions. In this paper a unified approach to Lyapunov's indirect method for infinite-dimensional system is described. It is shown how existing sufficient conditions fit this framework and a new sufficient condition is presented.

Keywords:Linear Systems, Control of Distributed Parameter Systems Abstract: We propose to use the observer-based algorithm of Ramdani, Tucsnak and Weiss (Automatica, 2010) for the initial state recovery of the wave equation involved in thermoacoustic tomography. We proved the rate of convergence of the iterative algorithm to the observable part of the initial state. We performed 3D numerical test in the relevant case where the measurement is performed on a grid of transducers on a half-sphere.

Keywords:Signal Processing, Robust and H-Infinity Control, Linear Systems Abstract: There has been remarkable progress in sampled-data control theory in the last two decades. The main achievement here is that there exists a digital (discrete-time) control law that takes the intersample behavior into account and makes the overall analog (continuous-time) performance optimal, in the sense of H-infinity norm. This naturally suggests its application to digital signal processing where the same hybrid nature of analog and digital is always prevalent. A crucial observation here is that the perfect band-limiting hypothesis, widely accepted in signal processing, is often inadequate for many practical situations. In practice, the original analog signals (sounds, images, etc.) are neither fully band-limited nor even close to be band-limited in the current processing standards. Sampled-data control theory provides an ideal platform for dealing with these problems, and our new design method has become highly successful in commercial applications of sound processing chips (45 million chips produced to date) and AAC/MP3 sound processing audio players applications (iPhone/iPod App).

This mini-course intends to provide

1. Introductory overview of modern sampled-data control theory; in particular, we carefully review a. the difficulties in dealing with sampled-data systems, b. how lifting reduces a sampled-data control system into a linear, time-invariant, discrete-time system, c. how this reduction recovers such classical notions of transfer operators, steady-state response, and frequency response, etc., and d. how an H-infinity sampled-data control problem can be reduced to a discrete-time H-infinity problem.

After covering such fundamental issues, we review a basic problem of signal reconstruction in digital signal processing. It is noted that t