Keywords: Systems on Graphs
Abstract: While they were first used to model networks of resistors, the Laplacian matrices of graphs are now used in many fields including Machine Learning, Computer Vision, Optimization and Network Analysis. The field of Spetcral Graph Theory is devoted to the study of their eigenvalues and eigenvectors.

We will survey many of the areas in which Laplacian matrices arise and will try to provide intuition about why they are so useful.

We will then describe recently developed algorithms that allow us to solve linear equations in these matrices in time that is nearly linear in their number of non-zero entries. These algorithms exploit approximations of graphs by sparser graphs and by trees. We explain what these approximations are and sketch how they lead to fast algorithms for solving linear equations.

Keywords: Systems on Graphs, Networked Control Systems, Optimization : Theory and Algorithms
Abstract: We investigate the control of evolutionary games on tree networks, presenting an algorithm that computes a set of agents that, when their strategies are fixed, will drive all other agents in the network to the desired strategy. The nodes in the network represent agents engaged in 2-player evolutionary games and the edges define who plays with whom. After each round, each agent updates its strategy to that of its neighbor who received the highest total payoff. We are interested in finding the minimum number of control agents needed to drive the entire network to a desired strategy. The proposed algorithms compute upper and lower bounds on the solution to this problem for arbitrary trees and payoff matrices. Simulations demonstrate that the control set corresponding to the upper bound is minimal in the majority of cases, and otherwise within a small number of agents from minimal.

Keywords: Systems on Graphs, Optimization : Theory and Algorithms, Networked Control Systems
Abstract: In this paper, we propose a systematic strategy for distributedly maximizing the connectivity of a network. A graph process is introduced to formulate the problem in which edges are added to an existing graph in a step-wise manner to maximize the algebraic connectivity of underlying network topology. This problem turns into an intractable combinatorial optimization and hence we explore a heuristic rule to approximately solve it. In dependence upon global network structure information, we first provide a local criterion of near-optimization by involving the elements of the Fiedler eigenvector of graph Laplacian. To conduct the graph process in a distributed fashion, a decentralized estimation algorithm is devised for locally attaining the knowledge of the Fiedler eigenvector. The proofs of convergence and convergence rate are provided. Synthetically, a novel heuristic strategy is proposed to deal with the network optimization problem in a distributed and generalized fashion. The effectiveness of proposed methods are demonstrated via some examples.

Keywords: Optimization : Theory and Algorithms, Networked Control Systems, Systems on Graphs
Abstract: This paper studies the regret of a family of distributed algorithms for online convex unconstrained optimization. A team of agents cooperate in a decision making process enabled by local interactions and the knowledge of each agent about the local cost functions associated with its decisions in previous rounds. We propose a class of online, second-order distributed coordination algorithms that combine subgradient descent on the local objectives revealed in the previous round and proportional-integral feedback on the disagreement among neighboring agents. The communication network is given by a time-varying sequence of connected graphs, and the local objectives can be adversarially adaptive to the agents’ behavior. The goal of each agent is to incur a cumulative cost over time with respect to the sum of local objectives across the network that is competitive with the best fixed and centralized decision in hindsight. For this, we establish the classical logarithmic regret bound under strong convexity of the local objectives.

Keywords: Networked Control Systems, Large Scale Systems
Abstract: This paper proposes protocols for a distributed optimization problem to minimize the average of objective functions of the agents in the network with satisfying local constraints of each agent. To decide whether the agents update their decision variables toward the direction of the gradient of objective or constraint functions, the agents receive information on fulfillment of constraints from neighbor agents as well as sharing the decision variables. Then the agents can know whether all the constraints of the agents were satisfied some steps ago. Convergence of subgradient-method based protocols is proved even with depending on past information of constraints. Numerical examples are presented to illustrate the protocols.

Keywords: Networked Control Systems, Optimization : Theory and Algorithms, Optimal Control
Abstract: The problem of finite-horizon linear quadratic control in a network of interconnected systems is studied. A gradient based algorithm is considered to solve the problem. Network structures are identified where the iterations can be calculated on a subsystem-by-subsystem basis in terms of model information from only a local subset of nodes in the network.

Keywords: Nonlinear Filtering and Estimation, Nonlinear Systems and Control, Stability
Abstract: The design of semiglobal and practical discrete-time reduced-order observers for nonlinear sampled-data systems of strict-feedback-like form is given. Robustness analysis of the designed observer and the comparison to the two-step deadbeat reduced-order observer are also discussed. A numerical example is given to show an efficiency of the designed observer.

Keywords: Large Scale Systems, Computations in Systems Theory, Computational Control
Abstract: The need for solving parameter-dependent Lyapunov matrix equations arises when applying model reduction to parameterized control systems. Although reduced basis methods have been shown to deal quite effectively with parameter dependencies, the dimensionality of the solution space for Lyapunov equations provides an additional challenge. We show how this challenge can be addressed by incorporating low-rank techniques into the reduced basis method.

Keywords: Nonlinear Systems and Control, Adaptive Control
Abstract: We consider tracking control for a class of nonlinear functional differential-algebraic systems. Funnel control, that is a static nonlinear proportional output error feedback, is applied to achieve tracking of a reference signal by the output signal with prescribed transient behavior.

Keywords: Feedback Control Systems, Nonlinear Systems and Control, Process Control
Abstract: This paper is devoted to the feedforward feedback control of sampled data control system. The starting point is a nonlinear system described by a system of explicit ordinary differential equations. The continuous-time system is converted to a time discretized one with the help of the implicit Euler method, which allows to use significantly larger sampling times as in the case of the explicit Euler method. The input to state linearizability by static feedback is checked, the sampled data Brunovsky normal form is the starting point for both trajectory planning and its stabilization. The design of the feedforward and feedback part is performed analogous to the time continuous case. Finally, the proposed approach is applied to a nonlinear hydraulic system, where its efficiency is shown by numerical simulations, where also Coulomb friction is taken into account.

Keywords: Nonlinear Systems and Control, Process Control, Mathematical Theory of Networks and Circuits
Abstract: Reaction networks refers to multiple reactions, like chemical, biochemical, biological etc., occurring simultaneously in a reactor. Traditional studies consider mainly isothermal operation conditions, thus focussing on the dynamics of the interconnected mass balance laws. This is however an unrealistic scenario in most real life applications since thermal gradients are often one of the main driving forces of any kind of reaction. In a recent work the set of dynamic equations governing the evolution of a general single chemical reaction has been modelled as irreversible port-Hamiltonian control systems. These systems express, just like standard port-Hamiltonian systems, the conservation of energy as a structural property but in addition they also express (as a structural property) the second law of Thermodynamics: the irreversible production of entropy. In this work the irreversible port-Hamiltonian formulation is extended to general reaction networks, allowing to propose an energy based model for non-isothermal reaction networks. The developments are performed for chemical reaction networks, but the model is general and can be applied to any kind of non-isothermal reaction. The results are illustrated on a simple 2 step chemical reaction network.

Keywords: Biological Systems
Abstract: A discontinuous Galerkin finite element method is used to simulate solutions to a nonlinear PDE used to model the biological process of transcription; the process by which RNA polymerase transcribe the genetic information from DNA and transfer it to mRNA. The transcription process is punctuated by short, frequent RNAP pauses, and these pauses cause a delay in the total time duration of the transcription process. The DG solution to the nonlinear model is used to calculate the delay and to quantify the effects of the pauses on the overall transcription time. Preliminary parameter studies indicate that in a system with multiple pauses both the location and time duration of the pauses can significantly effect the average delay experienced by an RNAP.

Keywords: Biological Systems, Delay Systems, Systems Biology
Abstract: In this paper we propose a coupled model for healthy and cancer cell dynamics in Acute Myeloid Leukemia consisting of two stages of maturation for cancer cells and three stages of maturation for healthy cells. The cell dynamics are modelled by nonlinear partial differential equations (PDE). Applying the method of characteristics enable us to reduce the PDE model to a nonlinear distributed delay system. For an equilibrium point of interest, necessary and sufficient conditions of local asymptotic stability are given. The results are illustrated with numerical examples and simulations.

Keywords: Biological Systems, Numerical and Symbolic, Systems Biology
Abstract: Numerical simulation of muscle cells and tissue is an established tool in cardiac electrophysiology, where the electrical behavior of excitable heart muscle cells is commonly modeled as a stiff, non-linear system of ordinary differential equations. A common feature of this system’s right-hand side is the heavy use of computationally expensive univariate functions of the membrane potential. In this article, we investigate the performance benefits of replacing these functions with cubic spline approximations in an automated model simplification process. Clear performance gains were found when evaluating the right-hand side in isolation and when performing multi-cellular simulations using a simple forward Euler method. Single cell simulations run with an adaptive method saw smaller gains due to a higher overhead from the solver. A parallel multi-cellular simulation was also investigated, but the overhead of the implementation overshadowed the evaluation time of the right-hand side.

Keywords: Systems Biology, Biological Systems
Abstract: One of the canonical problems in systems biology is to determine the topology of a genetic regulatory network, given some gene expression data. The data are naturally discrete, e.g. they can be obtained from sampling the (continuous time) gene expression levels.

In this work, we want to find conditions on the sampling period such that the monotonicity of the continuous-time system is preserved by the resulting discrete-time model. Since validation of the discrete-time model using experimental data has been fully characterized in our prior work, results from this paper can be used to inform the experimentalists about suitable sampling frequency for the purpose of network identification.

Keywords: Stochastic Modeling and Stochastic Systems Theory, Stability, Systems Biology
Abstract: The goal of this paper is to first review existing results on ergodicity analysis of stochastic reaction networks. These results are then extended to the analysis of interconnections of networks. We show that these tests can be cast as linear programs and that they can be solved in a distributed way. Several examples are given for illustration.

Keywords: Nonlinear Systems and Control, Stability, Large Scale Systems
Abstract: It is well known that input-to-state stability admits an astonishing number of equivalent characterizations. Here it is shown that for monotone systems on Rⁿ₊ there are some additional characterizations that are useful for network stability analysis. These characterizations include system theoretic properties, algebraic properties, as well as the problem of finding simultaneous bounds on solutions to a collection of inequalities.

Keywords: Linear Systems, Networked Control Systems, Stability
Abstract: The convergence to consensus of all products of a given set of matrices is known to be algorithmically decidable when all matrices in the set are stochastic. We formulate this question as a stability problem for switched systems, and show that the decidability result remains valid for more general classes than stochastic matrices. Our results make use of a general theorem of Lagarias and Wang on the convergence of switched systems, and allow showing as a byproduct that the bound provided by this theorem is tight.

Keywords: Transportation Systems, Networked Control Systems, Large Scale Systems
Abstract: We study dynamical transportation networks modeled as monotone systems of ODEs describing the time-evolution of traffic densities on different links of a directed graph. We consider the problem of designing link flow capacities in order to optimize a global objective function. We show that, via a suitable change of variables, the optimization problem can be transformed into a convex one, allowing for a computationally efficient, distributed solution.

Keywords: Linear Systems, Operator Theoretic Methods in Systems Theory, Optimal Control
Abstract: We consider the case where the input to a system is restricted to be in the positive cone of l∞, denoted by l∞+, and seek to characterize the system’s induced norm from l∞+ to l∞. We obtain an exact characterization of this norm which is particularly easy to calculate in the case of LTI systems. Furthermore, we consider the model matching problem to show that time-varying linear or nonlinear control or filtering does not improve the performance with respect to this norm for LTI systems. We also consider the case when the output is forced to be in the positive l∞ cone when the input is in this cone. Such a constraint is called external positivity. We show that synthesizing a stabilizing controller to ensure external positivity can be done by imposing linear constraints on the Youla parameter. Furthermore, for LTI plants, time-varying linear or nonlinear controllers cannot outperform LTI controllers. If internal positivity is sought, we show that a dynamic optimal controller offers no advantage over a static one.

Keywords: Control of Distributed Parameter Systems, System Identification, Optimization : Theory and Algorithms
Abstract: A method is developed to solve an optimal node activation problem in sensor networks whose measurements are supposed to be used to estimate unknown parameters of the underlying process model in the form of a PDE. Given a partition of the observation horizon into a finite number of consecutive intervals, the problem is set up to select gaged sites on each interval so that the determinant of the Fisher information matrix associated with the estimated parameters be maximal. The approach adopted here to circumvent the combinatorial nature of the sensor selection problem consists in operating on the spatial density of sensors, rather than on the sensor locations. The original problem then reduces to maximizing the determinant of the sum of finite convex combinations of some nonnegative definite matrices subject to additional box constraints on the weights of these combinations. Some separability characterizations of optimal solutions are indicated first, and then the block coordinate ascent method combined with simplicial decomposition is applied to obtain numerical solutions. As a result, a simple computational scheme is obtained which can be implemented without resorting to sophisticated numerical software.

Keywords: Control of Distributed Parameter Systems, Infinite Dimensional Systems Theory
Abstract: We consider environmental and meteorological applications which undergo significant parametric changes that alter the behavior of the system. For the transport process under study, which depends on the Pe ́clet number, the behavior changes from an advection dominated to a diffusion dominated process. For each value of the Péclet parameter, a set of optimal sensor locations and number is found and the associated state estimator is subsequently designed. A supervisory scheme is then utilized to schedule the sensors corresponding to the current value of the Péclet number by first turning off all the sensors associated with a different value of the Péclet number and then activate the sensors that are optimal for the current value of the Pe ́clet number. Simulation results are included to provide further insight on parameter-dependent sensor and observer scheduling for a representative 1D environmental process.

Keywords: Optimal Control, Linear Systems
Abstract: Observations of the chemical states of the atmosphere typically have low temporal and spatial density. Based on the insufficient density of observations, one possibility to optimise the states estimation is to target the locations of observations which can potentially result in the largest forecast improvement. The objective of this work is to study the optimal locations of observations of atmospheric chemical transport model (CTM) with emissions by two approaches. One approach is the singular vector analysis, which can identify the sensitivity of observations by determining the directions of maximum perturbation growth per finite interval. Another approach is to transform the problem of optimal placement of observations to the duality of the linear-quadratic optimal problem of the locations of the control hardware, which, based on linear-quadratic optimal, can provide a global optimal solution for observation locations.

Keywords: Control of Distributed Parameter Systems, Large Scale Systems, Computations in Systems Theory
Abstract: We present two challenging benchmark problems for creating reduced-order models for the unsteady Boussinesq equations. The first is a differentially-heated cavity flow in the quasi-periodic regime and the second is an unsteady Marsigli flow problem exhibiting the Kelvin-Helmholtz instability. The first can be modeled accurately using the principle interval decomposition (PID), which optimizes the length of time windows over which to perform the POD procedure and is highly effective in convective problems. The second benchmark problem requires the introduction of stabilization terms that capture the effect of energy dissipation that is missing in a truncated POD basis. We provide two eddy viscosity stabilization models and optimize the parameter selection to best represent simulation data. These benchmark problems demonstrate the feasibility of low-order nonlinear models for capturing the dynamics of complex thermal fluid behavior and can be used to validate alternative reduced-order modeling methodologies.

Keywords: Control of Distributed Parameter Systems, Nonlinear Systems and Control, Process Control
Abstract: The problem of output feedback control for fast evolving distributed parameter systems is investigated using adaptive proper orthogonal decomposition (APOD). The novelty lies in employing a modified data ensemble construction approach for APOD to construct models with enlarged region of accuracy. Modifying the construction procedure of ensemble snapshots in APOD allows the derivation of local valid low-dimensional reduced order dynamic models (ROMs) with enlarged region of accuracy for controller synthesis thus resulting in a computationally-efficient alternative to using large-dimensional models with global validity. This advantage is utilized for the synthesis of a robust state controller combined with a dynamic observer of the system states with an enlarged region of attraction. The proposed approach is successfully used to stabilize the Kuramoto-Sivashinksy equation at a spatially invariant steady state profile in the absence and presence of uncertainty when the open loop process exhibits highly nonlinear behavior. The original and the modified ensemble construction approaches for APOD are compared in different conditions and the effectiveness of the modified approach is presented.

Keywords: Infinite Dimensional Systems Theory, Optimal Control, Computations in Systems Theory
Abstract: Recent work concerning the development of fundamental solution semigroups for specific classes of integro-differential equations is generalized via the consideration of a class of operator differential Riccati equations. In particular, a max-plus dual space representation for the fundamental solution semigroup for this more general class of equations is constructed by exploiting max-plus linearity and semiconvexity properties of an associated optimal control problem. A class of time-indexed max-plus integral operators is shown to be instrumental in this construction, and in the evolution of the fundamental solution semigroup obtained.

Keywords: Discrete Event Systems, Computer Networks, Transportation Systems
Abstract: Linear max-plus systems describe the behavior of a large variety of complex systems. It is known that these systems show a periodic behavior after an initial transient phase. Assessment of the length of this transient phase provides important information on the performance of such systems, and is thus crucial in system design. We identify relevant parameters in a graph representation of these systems and propose a modular strategy to derive new upper bounds on the length of the transient phase. By that we are the first to give asymptotically tight and potentially subquadratic transience bounds. We use our bounds to derive new complexity results, in particular in distributed computing.

Keywords: Control issues in Finance, Robust and H-Infinity Control
Abstract: Dynamics of assets in financial markets are often modeled by stochastic differential equations driven by Brownian motion, which is uncertain in the sense of probability theory. On the other hand, robust control treats model uncertainty by regarding noises as unknown deterministic functions. In this paper, motivated by risk-averse/small noise limit, we consider an H^¥infty-control model for an optimal investment problem with logarithmic utility under unobserved economic factor. We point out this model can be a max-plus stochastic control formulation of the investment problem. By using so-called information state in H^¥infty-control theory, we reduce the problem to a completely observed differential game in infinite-dimensional space. Due to a special affine/quadratic structure of the problem, we can give an optimal investment strategy by solving a linear Isaacs partial differential equation in finite-dimensional space.

Keywords: Optimal Control, Large Scale Systems, Quantum Control
Abstract: Recently a new class of techniques termed the max-plus curse of dimensionality-free methods have been developed to solve nonlinear optimal control problems. In these methods the discretization in state space is avoided by using a max-plus basis expansion of the value function. This requires storing only the coefficients of the basis functions used for the representation. However, the number of basis functions grows exponentially with respect to the number of time steps of propagation to the time horizon of the control problem. This so called ``curse of complexity'' can be managed by applying a pruning procedure which selects the subset of basis functions that contribute most to the approximation of the value function. The pruning procedures described thus far in the literature rely on the solution of a sequence of high dimensional optimization problems which can become computationally expensive.

In this paper we show that if the max-plus basis functions are linear and the region of interest in state space is convex, the pruning problem can be efficiently solved by the bundle method. This approach combining the bundle method and semidefinite formulations is applied to the quantum gate synthesis problem, in which the state space is the special unitary group (which is non-convex). This is based on the observation that the convexification of the unitary group leads to an exact relaxation. The results are studied and validated via examples.

Keywords: Mathematical Theory of Networks and Circuits, Discrete Event Systems
Abstract: A max-plus matrix A is called weakly stable if the orbit of A does not reach an eigenvector of A for any starting vector x unless x is an eigenvector itself. This is in contrast to strongly stable (robust) matrices for which the orbit reaches an eigenvector with any nontrivial starting vector. Max-plus matrices are used to describe multiprocessor interactive systems for which reachability of a steady regime is equivalent to reachability of an eigenvector by a matrix orbit. We prove that an irreducible matrix is weakly stable if and only if its critical graph is a Hamiltonian cycle in the associated graph. We extend this condition to reducible matrices. These criteria can be checked in polynomial time. They complement the known criteria for strong stability which will also be presented.

Keywords: Robust and H-Infinity Control, Computations in Systems Theory
Abstract: In this note, the problem of computation of the stabilizing solution of a class of periodic discrete-time Riccati equation is addressed. Such an equation is closely related to the so called full information H_infty control problem of discrete-time periodic systems. A globally convergent iterative algorithm is proposed for this purpose. The performances of the proposed algorithm are illustrated on some numerical examples.

Keywords: Robust and H-Infinity Control, Linear Systems, Stability
Abstract: In this paper, we study robustness problems of inhomogeneous Markov chains in dynamic environments. First, we define the natural distributions of nonstationary Markov chains as a sequence of probability distributions, which represents a non-autonomous dynamical system on a suitable state space. Then, we study the robustness and sensitivity analysis of these distributions under the assumption of geometric ergodicity of the underlying Markov chain.

Keywords: Robust and H-Infinity Control, Nonlinear Systems and Control, Feedback Control Systems
Abstract: This paper presents a somewhat new perspective on the stability problem for uncertain LTI feedback systems with actuator input amplitude saturation. The solution is obtained using the QFT theory and a 3 Dof non-interfering control structure. Describing function analysis is used as a criterion for closed loop stability and limit cycle avoidance, but the Circle or Popov criteria could also be employed. The novelty is the parametrization of the three degrees of freedom. Two examples are given. The first is a benchmark problem and a comparison is made with other proposed solutions. The second is an example which was implemented and tested on an X-Y linear stage used for nano-positioning applications. Design and implementation considerations are given.

Keywords: Robust and H-Infinity Control, Optimal Control, Linear Systems
Abstract: In this paper we reconsider the existence of worstcase Nash equilibria in a noncooperative multi-player setting, where the dynamics are described by a linear differential equation and players’ preferences are modeled by quadratic cost functions. We consider an openloop information structure. We show that these equilibria can be obtained by determining the open-loop Nash equilibria of an associated linear quadratic differential game with an additional initial state constraint. We derive both necessary and sufficient conditions for solvability of the finite planning horizon problem. In particular we demonstrate that unlike in the standard linear quadratic differential game setting uniqueness of equilibria may fail to hold. A both necessary and sufficient condition under which there is a unique equilibrium is provided.

Keywords: Robust and H-Infinity Control, Stability, Stochastic Control and Estimation
Abstract: One of the problems for the anisotropy-based analysis with non-zero mean of input sequence is considered. The closed-loop system robust stability conditions on anisotropic norm of the system transfer function and the uncertainty transfer function are obtained for linear discrete time-invariant systems. For the systems with anisotropic norm bounded uncertainty, the robust stability criterion is derived.

Keywords: Hybrid Systems, Linear Systems, Operator Theoretic Methods in Systems Theory
Abstract: This paper studies a mathematical basis of piecewise linear approximation in the L_1 optimal control problem of sampled-data systems, which aims at minimizing the L_infty-induced norm of sampled-data systems. To take into account of intersample behavior, we first consider the lifted representation of sampled-data systems. The sampling interval [0,h) is then divided into M subintervals with an equal width by fast-lifting. Finally, the signals on each subinterval are approximated by linear functions through the introduction of two types of `linearizing operators.' This method is called piecewise linear approximation of sampled-data systems. By using the arguments of preadjoint operators, we verify that this approximation techniques gives a theoretical basis for tackling the analysis and synthesis problems on the L_1 optimal control of sampled-data systems.

Keywords: Infinite Dimensional Systems Theory, Linear Systems, Operator Theoretic Methods in Systems Theory
Abstract: For a large class of linear neutral type systems which include distributed delays we give the duality relation between exact controllability and exact observability. The characterization of exact observability is given.

Keywords: Infinite Dimensional Systems Theory, Operator Theoretic Methods in Systems Theory, Linear Systems
Abstract: The goal of the following is to use (infinite-dimensional) linear systems theory to approach the H^{infty}-functional calculus. This extends recent work by the authors using the notion of admissible observation operators to define f(A) where f is bounded, analytic in the left half-plane and A generates an exponentially stable strongly continuous semigroup. Among other things it is shown that f(A) is always weakly admissible and indeed coincides with classical approaches to H^{infty}-calculus.

Keywords: Infinite Dimensional Systems Theory, Stability, Operator Theoretic Methods in Systems Theory
Abstract: In this paper we discuss and compare different definitions for stability of an infinite-dimensional linear system. In particular, we concentrate on a situation where the semigroup generated by the system operator is polynomially stable. We derive conditions for strong input-output stability of the system. In addition, we introduce a weaker concept of polynomial input output stability for linear systems, and show that it corresponds to the recently introduced notion of P-stability in the frequency domain.

Keywords: Mathematical Theory of Networks and Circuits, Linear Systems, Operator Theoretic Methods in Systems Theory
Abstract: In this work we study a generalized Nevanlinna Pick interpolation problem, where transmission zero locations are imposed. Unlike in other variant of this problem considered by T.T Georgiou et al. the interpolation points are chosen on the boundary of the analyticity domain: that is, in our framework, on the real axis. This problem is motivated by important questions in electronic and microwave system design, and it relates to the broadband matching theories of Youla and Helton. An existence and uniqueness theorem is proved. The constructive proof is based on continuation techniques.

Keywords: Algebraic Systems Theory, Applications of Algebraic and Differential Geometry in Systems Theory, Multidimensional Systems
Abstract: In this presentation, we will develop a new approach based on numerical linear algebra to compute all the roots of a set of multivariate real polynomials.

We explore several properties of the Macaulay matrix that is constructed with the coefficients of the monomials of these real polynomials, including the determination of the number of solutions via the calculation of a rank, and the computation of the roots via generalized eigenvalue problems, that derive from an nD-realization problem in the null space of the Macaulay matrix.

We show how therefore polynomial optimization problems (such as e.g. Prediction Error Methods in system identification) boil down to solving a (large) eigenvalue problem.

Keywords: Algebraic Systems Theory, Mathematical Theory of Networks and Circuits
Abstract: We present an algebraic approach to the circuit synthesis problem for coupled-resonators filters. The latter yields a rigorous understanding of the relationship between the coupling geometry and the corresponding set of realizable filtering characteristics. In order to tackle the problem in practice, algorithms based on Groebner bases computations and continuation techniques are considered. Numerical results are presented for a 10th order filter.

Keywords: Optimization : Theory and Algorithms, Computations in Systems Theory, Multidimensional Systems
Abstract: In many applications of mathematics, including areas such as financial mathematics and control theory, optimization plays an important role. Often the criterion function is a polynomial or a rational function of the unknowns, which we will take to be real numbers here. In such a case the first order conditions for optimality can be written in the form of a polynomial system of equations and we speak of an algebraic optimization problem. The solutions of the polynomial system are called the critical points of the criterion function. The corresponding values of the criterion function are called critical values. There are only a finite number of such critical values and one can construct a univariate nonzero polynomial which is zero on the critical value set. Such a polynomial will be called a critical value polynomial (CVP).

In the talk we will explain methods to obtain such a critical value polynomial and its usage for determining the (global!) optimal value of the criterion.

The talk is based on joint research project with Marketa Adamova and Andrei Mustata(UCC)

Keywords: Nonlinear Systems and Control, Applications of Algebraic and Differential Geometry in Systems Theory, Systems Biology
Abstract: In this paper we investigate realization theory of a class of non-linear systems, called Nash systems, both in continuous and discrete time. To unify the approach for continuous and discrete time we will consider Nash systems on time scales. Nash systems are non-linear systems whose vector fields and readout maps are analytic semi-algebraic functions. We will present necessary and sufficient conditions for a given response map to have a realization as a Nash system on the time scale, and we will relate observability and reachability of Nash systems to minimality. The main contribution of the paper is twofold: 1) considering time scales framework for the class of semi-algebraic systems and solving realization problem, 2) results concerning discrete-time counterpart of the obtained characterization of the existence of Nash realizations on time scales.

Keywords: Systems on Graphs, Networked Control Systems, Stochastic Control and Estimation
Abstract: Discrete harmonic functions have long proved to be a fundamental analytical tool in applications as diverse as electrical networks, random walks, and mechanical systems of springs. Recently, they have emerged as models for the stationary state of some ergodic opinion dynamics over social networks. This mini-course will provide an introduction to such models and an overview of some recent results in the field. We illustrate emerging phenomena in large-scale social networks such as homogeneous influence in highly fluid networks, and persistent disagreement and fluctuations. Other current applications are presented in problems of distributed control and estimation, such as leader-follower models in robotics and estimation from relative measurements.

Keywords: Hybrid Systems, Stochastic Modeling and Stochastic Systems Theory, Discrete Event Systems
Abstract: The objectives of this mini-course is to introduce the participants to the topics of correct-by-design synthesis and verification of embedded control software. The main objectives are:

1- Make participants new to the topic familiar with relevant concepts and techniques in computer science such as: temporal logics, automata, simulation relations, safety/reachability games, etc.

2- Introduce abstraction techniques to construct finite-state models of infinite (stochastic) hybrid systems.

3- Introduce algorithmic techniques to automatically synthesize hybrid controllers satisfying some specifications of interest.

4- Illustrate the introduced concepts on the MATLAB-based toolbox Pessoa and on a newly developed toolbox for stochastic hybrid systems.

5- Exhibit the audience to new directions and areas in formal methods with applications in control.

6- Suggest novel applications of the discussed formal techniques.

Additionally, we hope that the active discussions with the participants will lead to fruitful interactions.

Keywords: Networked Control Systems
Abstract: We will present a compositional approach that uncovers the dynamical properties of a networked system by exploiting input-output properties of the subsystems and the structure of their interconnection. This approach overcomes the complexity of a large-scale, nonlinear network model by dividing the analysis and design tasks into two layers: At the network layer, we represent the subsystems (“nodes”) with input-output properties as abstractions of their detailed models and exploit these properties together with the interconnection structure to ascertain desirable behaviors, such as stability, synchronization, and spatial pattern formation. At the node layer, we verify or assign the relevant input-output properties using analytical and numerical techniques that do not rely on knowledge of global network properties. The proposed methodology is neither a “top-down” approach that ignores inherent features of the components, nor a “bottom-up” strategy that fails to reveal emergent network behaviors. Instead, it identifies key structural properties of the components and their interconnection that dictate the ensemble behavior. We will illustrate this approach with examples from multi-agent systems and biochemical reaction networks by first exhibiting inherent interconnection structures. We will further present a large-scale optimization technique that aims to certify desirable network properties by searching for subsystem input-output properties that are compatible with the interconnection structure.

Keywords: Infinite Dimensional Systems Theory
Abstract: Many physical systems can be formulated using a Hamiltonian framework. This class contains ordinary as well as partial differential equations. Each system in this class has a Hamiltonian, generally given by the energy function. In the study of Hamiltonian systems it is usually assumed that the system does not interact with its environment. However, for the purpose of control and for the interconnection of two or more Hamiltonian systems it is essential to take this interaction with the environment into account. This led to the class of port-Hamiltonian systems. The Hamiltonian/energy can be used to control a port-Hamiltonian system. For port-Hamiltonian systems described by ordinary differential equations this approach is very successful. Port-Hamiltonian systems described by partial differential equation is a subject of current research.

In this presentation, we show how that the class of linear infinite-dimensional port-Hamiltonian systems has many nice properties. Among others it is easy to show well-posedness of the homogeneous equation, and there are easy checkable conditions for exponential stability.

Keywords: Numerical and Symbolic
Abstract: In this talk, we describe a control theoretic approach towards the so-called PageRank algorithm, which has been employed at Google to assist the rankings in search results. This algorithm assigns a measure of popularity and importance to each page in the web; higher values are given to pages that are visited more often. We first introduce a distributed randomized approach for this algorithm, where web pages behave as agents computing their own PageRank by communicating with linked neighbor pages. This approach is meaningful in view of the computational difficulties due to the size of the problem, especially in the web environment where resources are available. Then, we extend this approach to systematically aggregate the web pages into groups by exploiting the sparsity structure in the web. For each group, an aggregated PageRank value is computed, which can then be distributed among the group members. The aggregation method is able to reduce the computation and communication loads in the distributed algorithm. We highlight the connection with singular perturbation techniques for large-scale Markov chains and multi-agent consensus. Finally, numerical simulations are presented based on a web obtained from real data to illustrate the results.