Keywords: Information Theory
Abstract:

Keywords: Feedback Control Systems
Abstract: The interplay between and mirroring of external features and internal properties of a system is a classical topic in systems and control theory. In this lecture I elaborate on this theme from an algebraic point of view, for the case of systems described by higher-order linear differential equations in the external variables. In this setting, a basic problem is how to construct the internal (state) variables; I will show how this can be done iusing polynomial algebra. Such construction is based upon an intrinsic link between the external system representation and the internal variable. The state variable obtained in this way makes it easy to translate external properties into properties of the internal (state) behavior. I will illustrate applications of this idea in realization theory, model reduction, and identification.

Keywords: Mathematical Theory of Networks and Circuits
Abstract:

Keywords: Mathematical Theory of Networks and Circuits, Systems Biology, Process Control
Abstract: A canonical representation of the set of feasible equilibrium solutions for MAL reaction networks is given in terms of a class of strictly stable Metzler matrices which define the so-called family of solutions. Main consequences of such representation are a simple characterization of uniqueness of complex balance solutions as well as a short proof of the deficiency one theorem. Future directions may involve detection or design of networks having multiple equilibria.

Keywords: Mathematical Theory of Networks and Circuits, Physical Systems Theory, Stability
Abstract: We present a new proof, using the log-sum inequality and graph-theoretic ideas, that the pseudo-Helmholtz free energy function is a Lyapunov function for complex-balanced mass-action systems.

Keywords: Physical Systems Theory, Nonlinear Systems and Control, Biological Systems
Abstract: We consider open chemical reaction networks, i.e. ones with inflows and outflows. We assume that all the inflows to the network are constant and all outflows obey the mass action kinetics rate law. A complex-balanced open reaction network is one that admits a complex-balanced steady state. We show how such a system can be modelled using graph-theoretic principles. It is well known that every steady state of a complex-balanced open reaction network is complex-balanced. Using graph-theoretic principles, we provide a more insightful proof of this fact as compared to the known proof.

Keywords: Mathematical Theory of Networks and Circuits, Systems Biology
Abstract: In systems biology there is a growing interest in the question, whether or not a given mathematical model can admit more than one steady state. As parameter values are often unknown or subject to a high uncertainty, one is often interested in the question, whether or not a given mathematical model can exhibit multistationarity at all. And, if this is the case, one is interested in those parts of the network that are responsible for multistationarity. For certain networks with mass action kinetics, multistationarity can be established by Feinberg’s Deficiency One Algorithm. This algorithm, however, is limited to so-called regular deficiency one networks and many realistic networks have a deficiency greater than one. In a previous publication it was suggested to analyze certain well defined subnetworks that are guaranteed to be of deficiency one. If these subnetworks are regular, then one can use the Deficiency One Algorithm to establish multistationarity and, if successful, extend it to the overall network. We call those subnetworks multistationarity causing structures. Unfortunately realistic reaction networks often lead to subnetworks that are irregular. The results presented here exploit the structure of such subnetworks to derive conditions for multistationarity that are independent of the Deficiency One Algorithm. Thus, in particular, these conditions are applicable to irregular subnetworks. As a proof of principle we apply these results to a model of textit{E. coli}'s metabolism and establish a multistationarity causing structure.

Keywords: Algebraic Systems Theory, Linear Systems, Numerical and Symbolic
Abstract: The first part of the talk will recall the results of [1] relating to the Ladenheim catalogue. In particular it will be seen that all but two of the 108 Ladenheim networks realise regular biquadratics. The remaining two ({#}70 and {#}95) are capable of realising some but not all non-regular biquadratics, as shown by considering a canonical form for biquadratics. Reichert's theorem shows that additional resistors do not expand the Ladenheim class [2]. Reichert's original proof contained a complex topological argument. The proof was reworked and amplified in [3]. This talk will discuss an alternative approach to proving Reichert's theorem which dispenses with the topological argument.

Keywords: Mathematical Theory of Networks and Circuits, Mechanical Systems, Systems on Graphs
Abstract: It is well known that the impedance of a passive circuit is necessarily positive-real [1]. The Bott-Duffin procedure shows that any positive-real function has an RLC realisation, possibly with the number of reactive elements (inductors and capacitors) greatly exceeding the McMillan degree. In [2] it was shown, for series-parallel circuits, that the Bott-Duffin procedure is minimal in the number of reactive elements (six) for the biquadratic minimum function. For general circuits, the best available result is the Reza-Pantell-Fialkow-Gerst simplification, published simultaneously in the 1954 papers [3– 5], which reduces the number of reactive elements to five for the general biquadratic minimum function. In this extended abstract, we present an additional class of equivalent circuits which have not appeared previously in the literature. In the accompanying talk, we will show the remarkable result that the Reza-Pantell-Fialkow-Gerst simplification produces circuits which contain the least possible number of reactive elements for the realisation of certain biquadratic minimum functions.

Keywords: Mathematical Theory of Networks and Circuits, Linear Systems
Abstract: A survey on the development of resistive network synthesis is presented. Tellegen's result is first introduced, which shows that paramountcy is the necessary and sufficient condition for synthesis if n<=3. Then, some more general cases are presented, including works by Cederbaum, Guillemin, Brown, Reed, Weinberg, and Boesch. These results range from necessary realizability conditions of general resistive n-port networks to realizability conditions of a subclass of such networks, which are mostly based on graph theory. Recently, there has been some new development on the realization of resistive n-port networks containing 2n terminals, motivated by mechanical network synthesis using the element extraction method.

Keywords: Large Scale Systems, Cellular Automata, Systems on Graphs
Abstract: Many cascading phenomena in social, economic and technological networks can be studied using epidemic models. We consider a threshold model which we analyze in large directed random networks with heterogeneous agents. The tree-like local structure of those networks has driven our approach, that describe the infection process with two recursive equations for the expected fractional infection on trees. Our main contribution is a concentration theorem, which guarantees that the infection process, on a generic instance of the network and initial condition, concentrates around its expectation on a tree. With the recursive equations it is possible to approximate the dynamic of the infection and recover the final extension of the contagion. In heterogeneous networks we observe selective percolation among the weaker nodes and we prove a condition for the uniqueness of the phase transition.

Keywords: Large Scale Systems, Linear Systems, Optimal Control
Abstract: A common approach to distributed control design is to impose sparsity constraints on the controller structure. Such constraints, however, may greatly complicate the control design procedure. This paper promotes an alternative structure, which is not sparse, yet might nevertheless be well suited for distributed control purposes. The structure appears as the optimal solution to a class of coordination problems arising in multi-agent applications. The controller comprises a diagonal (decentralized) part, complemented by a rank-one coordination term. Although this term relies on information about all subsystems, its implementation only requires a simple averaging operation.

Keywords: Networked Control Systems, Adaptive Control, Large Scale Systems
Abstract: In this paper, we deal with a leader following problem of heterogeneous uncertain multi-agent systems. We propose a cooperative adaptive control framework which is composed of two parts: a cooperative adaptive controller and a direct relative state feedback controller. Then we show that in this proposed multi-agent system, all the state variable errors and adaptive gain errors are Lyapunov stable and also all the signals used in the controller are bounded even if the leader is not asymptotically stable. From this property, we explain that the proposed controller can deal with an important application of multi-agent systems, a vehicle formation problem.

Keywords: Robust and H-Infinity Control, Feedback Control Systems, Networked Control Systems
Abstract: Recently we proposed a state-space solution of the structured H_infty-synthesis problem for nested interconnections. These are characterized by generalized plants with a block-triangular structure of the control channel and the search for optimal controllers whose transfer matrix shares this structure. The existing approach is based on eliminating the structured controller matrices and results in LMI existence conditions for H_infty-controllers. The present paper serves to develop an alternative approach with a novel structured convexifying controller parameter transformations which allows the extension of the design procedure to H_2- and mixed H_2/H_infty-cost criteria.

Keywords: Coding Theory
Abstract: We construct a new F_q-linear code over F_q^b replacing the elements in the large field F_q^b by elements in F_q[C], where C is the companion matrix of a primitive polynomial of degree b. Then, we propose a decoding algorithm for the F_q-linear code over the erasure channel, based on solving a linear system over F_q.

Keywords: Communication Systems, Information Theory
Abstract: In a wireless storage system, having to communicate over a fading channel makes repair transmissions prone to physical layer errors. The first approach to combat fading is to utilize the existing optimal space-time codes. However, it was recently pointed out that such codes are in general too complex to decode when the number of helper nodes is bigger than the number of antennas at the newcomer or data collector. In this paper, a novel protocol for wireless storage transmissions based on algebraic space-time codes is presented in order to improve the system reliability while enabling feasible decoding. The diversity-multiplexing gain tradeoff (DMT) of the system together with sphere-decodability even with low number of antennas are used as the main design criteria, thus naturally establishing a DMT-complexity tradeoff. It is shown that the proposed protocol outperforms the simpler time-division multiple access (TDMA) protocol, while still falling behind the optimal DMT.

Keywords: Coding Theory
Abstract: The decoding of the Interleaved Chinese Remainder (ICR) code is reformulated as finding a short vector in a Z-module and a decoding algorithm for ICR codes is proposed. The complexity of our algorithm is linear in the length of the code. We give a proof of the failure probability and derive the decoding radius.

Keywords: Coding Theory, Information Theory
Abstract: We focus on erasure codes for distributed storage. The distributed storage setting imposes locality requirements because of easy repair demands on the decoder. We first establish the characterization of various locality properties in terms of the generator matrix of the code. These lead to bounds on locality and notions of optimality. We then examine the locality properties of a family of nonbinary codes with simplex structure. We investigate their optimality and design several easy repair decoding methods. In particular, we show that any correctable erasure pattern can be solved by easy repair.

Keywords: Infinite Dimensional Systems Theory, Control of Distributed Parameter Systems, Operator Theoretic Methods in Systems Theory
Abstract: It is well-known that the algebraic Riccati equation for infinite-dimensional systems with very unbounded control operators in its classical form is not well-defined. Several alternatives have been proposed in the literature. We review some of these and give special emphasis to the recently proposed operator node Riccati equation, which is a generalization of the Lur'e form of the algebraic Riccati equation. We show how the other reviewed alternative Riccati equations can be obtained from this operator node Riccati equation.

Keywords: Infinite Dimensional Systems Theory, Control of Distributed Parameter Systems
Abstract: Let Sigma = (S;X,U,Y) be an i/s/o system in continuous time. Here the state space X, the input space U, and the output space Y are Hilbert spaces, and the generator S of Sigma is a closed operator from X x U to X x Y. A continuously differentiable X-valued function x is a classical trajectory of Sigma on some interval I with input function u and output function y (both of which are assumed to be continuous) if [x(t);u(t)] belongs to the domain of S and [x'(t);y(t)] = S [x(t);u(t)] for all t in I. For such a system Sigma we define the notions of the resolvent set rho(Sigma) and the i/s/o resolvent matrix S. The i/s/o resolvent matrix S of Sigma is an analytic 2 x 2 block operator matrix defined on rho(Sigma), and intuitively it maps the initial state x₀ at time zero and the restriction to rho(Sigma) of the formal Laplace transform of the input u into the restriction to rho(Sigma) of the formal Laplace transforms of the state x and the output y. The i/s/o resolvent matrix is a fundamental tool in the frequency domain analysis of Sigma, and it makes it possible to give natural extensions of many significant notions in the theory of well-posed i/s/o systems to the class of possibly non-well-posed i/s/o systems with a nonempty resolvent set. Examples of such notions which can be extended are controllability, observability, minimality, restrictions, projections, compressions, intertwinements, similarities, and pseudo-similarities.

Keywords: Control of Distributed Parameter Systems, Stability, Infinite Dimensional Systems Theory
Abstract: This paper is concerned with systems modelled by linear singularly perturbed partial differential equations. More precisely a class of linear systems of conservation laws with a small perturbation parameter is investigated. By setting the perturbation parameter to zero, the full system leads to two subsystems, the reduced system standing for the slow dynamics and the boundary-layer system representing the fast dynamics. The exponential stability for both subsystems are obtained by the stability of the overall system of conservation laws. However, the stability of the two subsystems does not imply the stability of the full system. The approximation of the solution for the overall system by the solution for the reduced system is validated via Lyapunov techniques.

Keywords: Control of Distributed Parameter Systems, Feedback Control Systems, Infinite Dimensional Systems Theory
Abstract: In this talk we consider forced infinitedimensional Lur’e systems. We investigate input-to-state stability (ISS) and convergent-input-convergentstate (CICS) properties. In particular, it is shown that, under the conditions of the classical circle criterion, the Lur’e system is ISS.

Keywords: Multidimensional Systems, Optimization : Theory and Algorithms, Large Scale Systems
Abstract: Numerical methods in higher dimensions using tensor factorizations I this talk I will collect recent advances in the solution of high-dimensional problems in different application areas: chemistry, biology, mathematics. The language of low-rank factorization gives a unified view on different algorithms for the solution of seemingly diverse and unconnected problems. Typical applications include quantum chemistry, chemical master equation in biology, latent variable models in data analyses.

Keywords: Optimization : Theory and Algorithms, System Identification, Signal Processing
Abstract: Decompositions of higher-order tensors are becoming more and more important in signal processing, data analysis, machine learning, scientific computing, optimization and many other fields. A new trend is the study of coupled matrix/tensor decompositions (i.e., decompositions of multiple matrices and/or tensors that are linked in one or several ways). Applications can be found in various fields and include recommender systems, advanced array processing systems, multimodal biomedical data analysis and data completion. We give a short overview and discuss the state-of-the-art in the generalization of results for tensor decompositions to coupled matrix/tensor decompositions. We briefly discuss the remarkable uniqueness properties, which make these decompositions important tools for signal separation. Factor properties (such as orthogonality and triangularity, but also nonnegativity, exponential structure, etc.) may be imposed when useful but are not required for uniqueness per se. Also remarkable, in the exact case the decompositions may under mild conditions be computed using only tools from standard linear algebra. We touch upon the computation of inexact decompositions via numerical optimization. We illustrate some of the ideas using Tensorlab, a Matlab toolbox for tensors and tensor computations that we have recently released, and of which version 2 provides a comprehensive framework for the computation of (possibly constrained) coupled matrix/tensor decompositions.

Keywords: System Identification
Abstract: This work investigates the use of low nuclear norm matrices in methods of recursive identification and adaptive modelling. The key idea is to use a Hankel matrix associated to a low-dimensional system, and to keep it as low-rank as possible during adaption. In this way, one can guarantee the power of the predictions made by this strategy, in case the true system can be approximated well as a low-dimensional system. This result is obtained by using the convex technique of the nuclear-norm projection. Proper formulation leads to a strategy with high predictive power, while guaranteeing a low-dimensional minimal realisation of the models which are built along the process. This work extends the work of Fazel et. al. and Liu et al. to a context of online learning. An example is given towards the task of fault detection.

Keywords: Multidimensional Systems, Operator Theoretic Methods in Systems Theory, Linear Systems
Abstract: Overdetermined multidimensional systems (also called vessels) are a class of multidimensional linear input/state/output systems with the evolution of the whole state vector specified by several separate update equations in several linearly independent directions. Such systems come equipped with compatibility conditions specifying the admissible input and output signals, and their frequency domain analysis leads to a transfer function that is a mapping of vector bundles on an algebraic curve embedded in a projective space. In this talk we will discuss overdetermined multidimensional systems that are impedance conservative rather than scattering conservative, so that their transfer function has a positive real part --- in an appropriate sense --- instead of being contractive.

Keywords: Operator Theoretic Methods in Systems Theory
Abstract: For the strictly positive case the maximum entropy solution X to the Leech problem G(z)X(z) = K(z) and ||X(z)||_inf ≤ 1, with G and K stable rational matrix functions, is proved to be a stable rational matrix function. An explicit state space realization for X is given, and ||X||_inf turns out to be strictly less than one. The matrices involved in this realization are computed from the matrices appearing in a state space realization of the data functions G and K. A formula for the entropy of X is also given. The approach is based on the central solution for the commutant lifting theorem.

Keywords: Linear Systems
Abstract: We describe the notion of hyperbolicity for subvarieties of P^d of codimension higher than 1 and tie this notion to a corresponding Grassmannian. We will also describe a certain class of determinantal representations, called Livsic-type determinantal representations. We will show how the notions are related in particular in the case of curves in P^d.

Keywords: Optimization : Theory and Algorithms
Abstract: Semidefinite programming (SDP) representations of convex sets are of great importance in optimization and applied mathematics. In this talk we discuss several techniques for producing lower bounds on the size of SDP representations. Based on joint work with Hamza Fawzi and James Saunderson (MIT).

Keywords: Computations in Systems Theory, Linear Systems
Abstract: The Non-symmetric Algebraic Riccati Equations are an extension of Algebraic Riccati Equations: they are equations which are quadratic in a rectangular matrix variable. Such equations may exhibit distinct solutions. Several among them have particular properties in terms of robustness and are useful in control system theory and in game theory. One can cite, if they exist: the strongly stabilizing solution, dichotomic solutions and reverse dichotomic solutions. This paper provides numerical efficient algorithms to compute them by using the Matrix Sign Function of appropriated characteristic matrices. The suitability and efficiency of these algorithms are illustrated by several academic examples.

Keywords: Computations in Systems Theory, Nonlinear Systems and Control
Abstract: In this paper we consider the problem of finding input-output representations of discrete-time subsystems which contain sufficient information to effectively prove safety properties in a closed-loop system. The main difficulty is that safety requires knowledge of the behaviour over unbounded time intervals. We show that regular expressions over discretised inputs and outputs provide sufficient information, and that these can be effectively computed from a state-space representation given natural conditions on the system and the input, state and output spaces.

Keywords: Computational Control, Delay Systems, Internet Control
Abstract: In this paper we discuss the notion of stability applied to computational systems with focus on Cloud com- puting systems. We consider the case study of a web service application hosted on a private Cloud environment. The effects of input delay on auto-scaling mechanism used for resource provisioning on a cloud is examined. The stability analysis is developed based on a dynamic model of the Cloud obtained by system identification. For the model obtained we apply LMI conditions necessary for estimating the upper bound on the time delay for stability. This upper bound is derived based on the assumption of a predefined maximum workload in connections/sec or equivalently the throughput parameter. This analysis of time delay helps in evaluating the performance of the Cloud, measured in terms of the response time of accessing the web service and the throughput. The experiments are performed on a Eucalyptus based cloud setup.

Keywords: Signal Processing, Optimization : Theory and Algorithms
Abstract: We present a framework to design an orthogonal wavelet with compact support and vanishing moments, tuned to a given application. This is achieved by optimizing a criterion, such that a prototype signal, which is characteristic for the application, becomes sparse in the wavelet domain. This approach is beneficial for compression and detection purposes. A parameterization is developed for which orthogonality and compact support are built in, and in terms of which we can express the vanishing moment conditions conveniently. The approach is developed for critically sampled wavelet transforms as well as for the stationary wavelet transform. Several examples illustrate the methods.

Keywords: Quantum Control, Nonlinear Systems and Control
Abstract: In this paper we study the so-called spin-boson system, namely a spin-1/2 particle in interaction with a distinguished mode of a quantized bosonic field. The system is controlled via an external field acting on the bosonic part. Applying geometric control techniques to the Galerkin approximation and using perturbation theory to guarantee non-resonance of the spectrum of the drift operator, we prove approximate controllability of the system, for almost every value of the interaction parameter.

Keywords: Quantum Control, Physical Systems Theory
Abstract: When dealing with the control of quantum systems in interaction with an external environment, one of the most commonly used models is the Lindblad-Kossakowski equation for the density matrix. Its derivation relies on various approximations, which are usually justified in terms of weak interaction between the system and its surroundings and special properties of the bath interacting with the system. This equation contains dissipative corrections, accounting for the interaction with the environment, whose expression strongly depends on the adopted Markov approximation. In the case of coherently controlled} Lindblad-Kossakowski equation, it is usually assumed that this correction is independent of the control Hamiltonian. However, this procedure is not consistent with the rigorous derivation of the Markov approximation in the standard case, the so-called weak coupling limit, in which the dissipative contribution depends on the coherent part of the dynamics.

In this paper we discuss the rigorous derivation of the Lindblad-Kossakowski equation in the weak coupling limit regime, following the original derivation of Davies, in the case where the control is present, and explicitly displaying the dependence of the model on the control. We then consider the special case of a 2-level system in a bosonic bath, and show explicitly how the model depends on the parameters in the Hamiltonian which contains the control. We also discuss the impact of this more rigorous modeling procedure in the current research on methods to find controls that decouple the system from the environment. We indicate directions and problems for further research.

Keywords: Quantum Control, Systems on Graphs
Abstract: In this paper we propose a continuous-time, dissipative Markov dynamics that asymptotically drives a network of n-dimensional quantum systems to the set of states that are invariant under the action of the subsystem permutation group. The Lindblad-type generator of the dynamics is built with two-body subsystem swap operators, thus satisfying locality constraints, and preserve symmetric observables. The potential of the proposed generator in combination with local control and measurement is illustrated with two applications: generation of a global pure state and estimation of the network size.

Keywords: Algebraic Systems Theory, Computations in Systems Theory, Numerical and Symbolic
Abstract: We consider polynomially nonlinear state-space systems and given algebraic varieties. A variety V is said to be controlled invariant w.r.t. a given system if we can find a polynomial state feedback law that causes the closed loop system to have V as an invariant set. If this task can be achieved by a polynomial output feedback law, V is called controlled and conditioned invariant. This concept leads to the problem of determining the intersection of a certain (affine) submodule of a free module over a polynomial ring with a free module over a subalgebra of this ring. We suggest various approaches to do this and to decide whether a variety is controlled and conditioned invariant, and if so, to compute all output feedback laws achieving the task.

Keywords: Delay Systems, Algebraic Systems Theory, Computations in Systems Theory
Abstract: A standard framework in analyzing Time-delay systems consists first, in identifying the associated crossing roots and secondly, then, in characterizing the local bifurcations of such roots with respect to small variations of the system parameters. Moreover, the dynamics of such spectral values are strongly related to their multiplicities (algebraic/geometric). This paper focuses on an interesting type of such singularities; that is when the zero spectral value is multiple. The simplest case, which is quite common in applications, is characterized by an algebraic multiplicity two and a geometric multiplicity one known as Bogdanov-Takens singularity. Unlike finite dimensional systems, the algebraic multiplicity of the zero spectral value may exceed the dimension of the delay-free system of differential equations. To the best of the authors’ knowledge, the bound of such a multiplicity for Time-delay systems was not deeply investigated in the literature. Our contribution is two fold. First, we emphasize the link between the multiplicity characterization and Birkhoff matrices. Secondly, we elaborate a constructive bound for the zero spectral value in the regular case; i.e. when the delay polynomials of a given quasipolynomial are complete, as well as in the singular case; i.e. when such polynomials are sparse. In the last case, the established bound is sharper than Polya-Szego ̈ generic bound.

Keywords: Algebraic Systems Theory, Multidimensional Systems
Abstract: In this communication, we discuss two estimation problems dealing with partial derivatives systems. Namely, estimating partial derivatives of a multivariate noisy signal and identifying parameters of partial differential equations. The multivariate noisy signal is expressed as a truncated Taylor expression in a small time interval. An algebraic method can be then used to estimate its partial derivatives in the operational domain. The same approach applies for the parameter identification problem. The main objects in the aforementioned algebraic method are particular differential operators called annihilators. Stafford's theorem can be applied to formalize the left ideal in the Weyl algebra formed by the annihilators. In the spatial domain, iterated integrals present in the estimates yield noise filtering.

Keywords: Nonlinear Filtering and Estimation, Algebraic Systems Theory, Nonlinear Systems and Control
Abstract: By numerical differentiation it is meant here approximation of the derivative at current time t of a real valued function of a single real variable (the time, t) which is available only through finitely many of its past samples which are possibly imprecisely known. Numerical differentiation is a major ill-posed inverse problem. It has not yet received a definite solution. But it is encountered in many engineering problems. Efficient numerical differentiation algorithms thus are key-enablers of engineering designs such as in control system theory, specifically estimation, control synthesis, and other practical issues. This communication presents theoretical and implementation details on several numerical differentiation algorithms for potential use in observation problems. In particular, these algorithms may be used as ingredients for alternative solutions to the longstanding problem of observer design for nonlinear systems.

Keywords: Linear Systems, Networked Control Systems
Abstract: This paper presents an output feedback controller for practical coordinated tracking of multi-agent systems. It is assumed that the agents under consideration have large parameter uncertainties, that they are affected by external disturbance, and that the agent dynamics can be different from one another. To handle the uncertainty, external disturbance, and heterogeneity effectively, a reduced-order version of disturbance observer has been employed. Compared to the recent result which relies on the conventional disturbance observer, the proposed controller has the following advantages. Firstly, the order of controller is reduced. Secondly, the stability proof as well as the design procedure becomes simpler. Simulation results are included to demonstrate that the proposed controller achieves practical coordinated tracking successfully.

Keywords: Networked Control Systems, Nonlinear Systems and Control
Abstract: This paper studies a chattering-free formation keeping control using very coarse information.

Keywords: Numerical and Symbolic, Nonlinear Systems and Control, Applications of Algebraic and Differential Geometry in Systems Theory
Abstract: The automated modeling of multi-physical dynamical systems is usually realized by coupling different subsystems together via certain interface or coupling conditions. This approach results in large-scale high-index differential-algebraic equations (DAEs). Since the direct numerical simulation of these kind of systems leads to instabilities and possibly non-convergence of the numerical methods a regularization or remodeling of the system is required. In many simulation environments a kind of structural analysis based on the sparsity pattern of the system is used to determine the index and a reduced system model. However, this approach is not reliable for certain problem classes, in particular not for coupled systems of DAEs. We will present a new approach for the remodeling of coupled dynamical systems that combines the structural analysis, in particular the signature method of Pryce, with classical algebraic regularization techniques and thus allows to handle so-called structurally singular systems and also enables a proper treatment of redundancies or inconsistencies in the system.

Keywords: Systems on Graphs, Networked Control Systems, Nonlinear Systems and Control
Abstract: This paper deals with the exact formation control of a group of point masses subject to ideal Coulomb friction. It is shown that the discontinuity of the Coulomb force prevents existing continuous controllers to achieve the formation objective. Based on previous work on course information exchange a discontinuous controller is proposed, which provides a solution to the exact formation problem in the presence of Coulomb friction.

Keywords: Feedback Control Systems, Nonlinear Systems and Control, Discrete Event Systems
Abstract: The control of asynchronous sequential machines by static state feedback is considered with the objective of matching a prescribed model. A necessary and sufficient condition for model matching by static state feedback is derived. This condition is based on counting the number of members of certain sets; it is expressed in terms of features of a matrix derived from the given recursion functions of the controlled machine and the desired model.

Keywords: Nonlinear Systems and Control, Linear Systems
Abstract: This paper deals with a class of dynamical systems obtained from interconnecting linear systems with static set-valued relations. We first show that such an interconnection can be described by a differential inclusions with a maximal monotone set-valued mappings when the underlying linear system is passive and the static relation is maximal monotone. Based on the classical results on such differential inclusions, we conclude that such interconnections are well-posed in the sense of existence and uniqueness of solutions.

Keywords: Nonlinear Systems and Control, Stability, Feedback Control Systems
Abstract: We present input-to-state stability results for forced Lur’e systems. The main result shows that conditions very similar to those in the complex Aizerman conjecture guarantee input-to-state stability of the Lur’e system. As a corollary of the main result, we obtain a sufficient condition for input-to-state stability which is reminiscent of the circle criterion.

Keywords: Stability, Nonlinear Systems and Control, Feedback Control Systems
Abstract: We identify a stability/instability trichotomy for a class of non-negative continuous-time Lur’e systems. Asymptotic as well as input-to-state stability concepts (ISS) are considered. The presented trichotomy rests on Perron- Frobenius theory, absolute stability theory and recent ISS results for Lur’e systems.

Keywords: Feedback Control Systems, Large Scale Systems, Optimal Control
Abstract: In this paper we propose a rigorous mathematical treatment of control problems for teams of controlled motions. Each member of the team is described by second-order dynamics. The proposed scheme is directed at solving problems of output feedback target control. For linear-convex systems such problems may be solved completely --- from theory to computation. In the nonlinear case this may be reached for fairly broad classes of systems. The solution methods rely on nonlinear analysis, set-valued calculus and Hamiltonian techniques for dynamic processes, including their matrix version applicable to large dimensions.

Keywords: Networked Control Systems, Systems on Graphs, Optimization : Theory and Algorithms
Abstract: In this paper the dynamics of signed networks is studied. Initially, we show how factions can be formed in a signed network and demonstrate that the factions formed in a weakly balanced network are consistent with what is expected. Later, we shift our focus to constructing signed networks that result in factions of known structure. In the end, we consider the case in which a group of influencer nodes manipulate their local connections in the network in order to steer to nodes to form any arbitrary faction structure. This allows the definition of a novel centrality measure for sets of nodes in a signed network.

Keywords: Optimization : Theory and Algorithms, Systems on Graphs
Abstract: This abstract deals with the problem of optimally placing unreliable sensors in a one-dimensional environment. We assume that sensors can fail with a certain probability and we minimize the expected maximum distance from any point in the environment to the closest active sensor. Under these assumptions, we provide a computational method for the optimal placement and we estimate the relative quality of equispaced and random placements, showing that the former achieves optimality when the number of sensors goes to infinity.

Keywords: Networked Control Systems, Systems on Graphs, Stability
Abstract: A sparse matrix space is a vector space of matrices with entries that are either fixed zeros or arbitrary real. Letting the pattern of zeros and arbitrary entries define an adjacency matrix (by setting the arbitrary entries to one), we can attach a graph to such vector spaces and think of it as describing the decentralization topology of a control system. We want to determine whether this topology can sustain stable dynamics, or equivalently whether the corresponding sparse matrix space contains stable (Hurwitz) matrices. We present in this extended abstract a necessary and sufficient condition for a symmetric sparse matrix space to contain stable matrices.

Keywords: Coding Theory
Abstract: In this work, from a controllable and observable input-state-output representation of two convolutional codes, we present an input-state-output representation of a convolutional product code. We also establish a lower bound on the free distance of the product code in terms of the free distance of the constituent codes.

Keywords: Coding Theory, Linear Systems, Communication Systems
Abstract: In this paper the authors introduce a model for evaluating the error correcting capabilities of convolutional codes. Until now, the developed models were limited to a binary field with only two elements. However, in this work we focus on the analysis of MDS, sMDS and not MDS codes defined over any finite field. We validate theoretical results of convolutional codes in a more realistic model to obtain the recovering error capability and ´we compare it with the results obtained simulating binary convolutional codes used currently in common communications systems as GSM or 802.11a.

Keywords: Coding Theory, Information Theory
Abstract: The introduction of Algebraic Geometry by Goppa as a tool for code construction resulted in a rich family of linear codes that also allowed to obtain already existing codes. This motivated the work of Pellikaan, Shen and van Wee about which linear codes could be obtained as algebraic geometric codes.

Similar tools have been used to develop the convolutional counterpart of Goppa codes yielding some optimal constructions and, as in the linear case, some codes which were already known. Therefore it seems a natural question to determine which convolutional codes can be constructed using algebraic geometric tools. It has been shown by the authors that in fact any convolutional code can be given an algebraic structure, i.e. they can be obtained by evaluation of a certain subspace of rational functions, and they have given conditions so that it is defined by a complete linear system.

We present here such structure with explicit expressions considering the projective line as the curve over which the construction is carried out. That is to say, we expose how any convolutional code can be obtained as a convolutional Goppa code over the projective line. In addition we give the explicit conditions so that a code is defined by a complete linear system, i.e. the subspace of functions is the subspace of global sections of a certain divisor. Further, we consider the particular case of memory 0 codes, i.e. linear codes, which implies that our conclusions are of application to linear codes as well. In particular, considering linear codes as a class of convolutional codes allows to overcome the limitation on the number of rational points when working over a finite field.

Keywords: Coding Theory, Multidimensional Systems, Linear Systems
Abstract: In this paper we consider a special class of 2D convolutional codes (composition codes) whose encoders G(d1,d2) can be decomposed as the product of two 1D encoders, i.e., G(d1,d2)=G2(d2)G1(d1). We prove that if G2 is a systematic encoder, then the composition code associated to G has a minimal 2D state-space realization (by means of a separable Roesser model) that can be obtained from minimal state space realizations of the 1D codes associated with G1 and G2.

Keywords: Hybrid Systems, Optimal Control, Numerical and Symbolic
Abstract: The application of Dynamic Programming techniques to hybrid systems leads to a Hamilton-Jacobi-Bellman equation in the form of a quasi-variational inequality, whose approximation may be carried out by an adaptation of classical monotone schemes. We present here some further improvements which aim at applying this numerical framework to problems of applicative interest, and in particular the use of policy iteration to speed up the iterative solution, and a practical construction of the (approximate) optimal control in feedback form.

Keywords: Optimization : Theory and Algorithms, Optimal Control, Large Scale Systems
Abstract: The Howard's algorithm (also called policy iteration algorithm) is a classical method for the resolution of a discrete Hamilton-Jacobi equation. This technique, developed by Bellman and Howard, is of large use in applications because of the high efficiency and performances. A special beneficial characteristic is that, in presence of a finite number of controls, the algorithm reaches the solution in finite time; this gives to the method a special interest for the resolution of problems with a big dimensionality, common issue in the sector. The performances of the method can be significantly improved by using parallel calculus but the parallelization of it is not an easy task. This is due to the strict relation between the value of the solution, even in two distant points of the domain. In this contribution we propose an original parallel version of the algorithm driven by a recent idea of domain decomposition. This permit to derive some important properties of the algorithm and proving the convergence. The key point about the conditions of connection between the sub domains will be pointed out and validated. Through some tests and examples will be shown the good features of the algorithm.

Keywords: Optimal Control, Feedback Control Systems, Numerical and Symbolic
Abstract: In this paper we consider a semi-Lagrangian scheme for minimum time problems with Lp-penalization. The minimum time function of the penalized control problem can be characterized as the solution of a Hamilton-Jacobi Bellman (HJB) equation. Furthermore, the minimum time converges with respect to the penalization parameter to the minimum time of the non-penalized problem. To solve the control problem we formulate the discrete dynamic programming principle and set up a semi-Lagrangian scheme. Various numerical examples are presented studying the effects of different choices of the penalization parameters.

Keywords: Stability, Optimal Control, Nonlinear Systems and Control
Abstract: Recent research has established the importance of dissipativity for proving stability of economic MPC in the case of a steady state. In many cases, though, steady state operation is not economically optimal and periodic operation of the system yields a better performance. In this paper, we propose three different ways of extending the notion of dissipativity for periodic systems and illustrate them with three examples.

Keywords: Optimization : Theory and Algorithms, System Identification, Signal Processing
Abstract: We present a new connection between higher-order tensors and affinely structured matrices, in the context of low-rank approximation. In particular, we show that the tensor low multilinear rank approximation problem can be reformulated as a structured matrix low-rank approximation, the latter being an extensively studied and well understood problem.

We first consider symmetric tensors. Although the symmetric tensor problem is at least as difficult as the general unstructured tensor problem, the symmetry allows us to simplify %the problem {em formulation} and to and clearly show the relation to the matrix structured low-rank approximation problem. By imposing linear equality constraints in the optimization problem, the proposed approach is applicable to unstructured tensors, as well as to affinely structured tensors. Therefore, it can be used to find (locally) optimal low multilinear rank approximation with a predefined structure.

An advantage of the proposed approach is that it can deal with more difficult variations of the main problem, including having missing and fixed elements in the given tensor or approximating with respect to a weighted norm. The drawback is its higher computational cost, compared to existing algorithms, partially due to the generality of the approach.

Keywords: Signal Processing, System Identification, Optimization : Theory and Algorithms
Abstract: Extensions of singular spectrum analysis (SSA) for processing of non-rectangular images, which can be also used for analysis of images and time series with gaps, are considered. A circular version is suggested, which allows application of the method to the data given on a circle or on a cylinder, e.g. cylindrical projection of a 3D ellipsoid. The constructed Shaped SSA method with planar or circular topology is able to produce low-rank approximations for images of complex shapes. Together with Shaped SSA, a shaped version of the subspace-based ESPRIT method for frequency estimation is developed. Examples of 2D circular SSA and 2D Shaped ESPRIT are presented.

Keywords: Optimization : Theory and Algorithms, System Identification, Signal Processing
Abstract: Recent research on statistical learning methods leveraged multilinear algebra and tensor-based models in reproducing kernel Hilbert spaces (RKHSs). We study implications of this framework for the identification of discrete-time dynamical systems that depend upon a (spatial) indexing variable. When this variable is discrete and the system of interest is linear, the proposed technique corresponds to learn a finite dimensional tensor under a constraint on the multilinear rank, hereby reducing the number of parameters needed in the estimation. More generally, the approach requires to learn a static mapping within a RKHS. The choice of reproducing kernel leads to different classes of models, which might depend nonlinearly on past inputs and outputs.

Keywords: Applications of Algebraic and Differential Geometry in Systems Theory, Optimization : Theory and Algorithms, System Identification
Abstract: We are focused on numerical methods for decomposing a multivariate polynomial as a sum of univariate polynomials in linear forms. The main tool is the recent result on correspondence between the Waring rank of a homogeneous polynomial and the rank of a partially known quasi-Hankel matrix constructed from the coefficients of the polynomial. Based on this correspondence, we show that the original decomposition problem can be reformulated as structured low-rank matrix completion (or as structured low-rank approximation in the case of approximate decomposition). We construct algorithms for the polynomial decomposition problem. In the case of bivariate polynomials, we provide an extension of the well-known Sylvester algorithm for binary forms.

Keywords: Operator Theoretic Methods in Systems Theory
Abstract: We show that multiplication by a one of the variables can lower the Agler norm of a multivariable function that is analytic on the open d-disk. This is in contrast to the H^infty-norm, where multiplication by a one of the variables leaves the norm unchanged, a consequence of the Schwarz lemma. To show this failure of the Schwarz lemma for the Schur-Agler class the theory of determinantal representations is used.

Keywords: Infinite Dimensional Systems Theory, Operator Theoretic Methods in Systems Theory, Control of Distributed Parameter Systems
Abstract: For an arbitrary given operator Schur function defined on the complex right-half plane, we give a controllable energy-preserving and an observable co-energy-preserving de Branges-Rovnyak functional model realization. Topics appearing only in the right-half-plane setting, such as the extrapolation space, are also discussed.

Keywords: Linear Systems, Operator Theoretic Methods in Systems Theory, Multidimensional Systems
Abstract: The Sz.-Nagy-Foias model theory for *-strongly stable contraction operators combined with the Beurling-Lax theorem establishes a correspondence between any two of four kinds of objects: shift-invariant subspaces of vector-valued Hardy space over the unit disk, operator-valued inner functions on the unit disk, conservative discrete-time input/state/output linear systems, and *-strongly stable Hilbert-space contraction operators. We discuss recent work exploring an extension of all these ideas where the Hardy space is replaced by a weighted Bergman space over the unit disk, the conservative linear system is replaced by a more general discrete-time time-varying linear system with system operators satisfying more elaborate metric constraints, and the result model theory is for the class of kappa-hypercontraction operators on a Hilbert space. We also discuss multivariate extensions of these ideas to the case where the linear system evolves over a free semigroup rather than the lattice of nonnegative integers, in which case weighted Bergman-Fock spaces of noncommutative formal power series come into play.

Keywords: Nonlinear Systems and Control, Computations in Systems Theory, Signal Processing
Abstract: Nonlinear Fourier transforms arise when nonlinear evolution equations are solved with the inverse scattering method. In this paper, the inverse nonlinear Fourier transform that arises during the solution of the nonlinear Schroedinger equation on the real line is investigated. The inverse problem consists in the synthesis of a potential for the Zakharov-Shabat differential operator such that the spectrum of the operator fulfills certain prescribed properties. Instead of discretizing an exact solution for the continuous case, a numerical discretization of the forward transform is inverted exactly in this paper. The focus is on the case of finitely many samples, and the problems at hand take the form of interpolation problems. A connection between analytic interpolation with degree constraint and fiber Braggs grating design is established, leading to a new synthesis method. A new method to compute reflectionless potentials is presented as well.

Keywords: Optimal Control, Stochastic Control and Estimation, Stochastic Modeling and Stochastic Systems Theory
Abstract: This paper combines linear programming and nonlinear optimization to demonstrate a new approach to the solution of impulse control problems. It examines the optimal process arising from the Bounded Follower Problem of Benes, Shepp and Witsenhausen (1980) in which one seeks to minimize the discounted second moment of the difference between a Brownian motion process W and its follower, which is limited to pursuing W using bounded velocity. Additional control of the optimal process is included in the model through the imposition of costly instantaneous shifts of the process. By first restricting the set of admissible impulse policies, this paper identifies two linear programs over spaces of measures into which the stochastic problem is imbedded, thus providing lower bounds on the optimal value. The first problem naturally leads to a nonlinear optimization problem which one is able to solve. The optimizers of this nonlinear function are then used to specify an impulse policy whose expected cost, for some initial positions, equals the lower bound and hence this policy is optimal. Solving the dual of the second linear program determines the optimal value for every initial position, again over the restricted set of policies. A stochastic argument then demonstrates that the expected cost of each admissible impulse control policy is bounded below by the optimal expected cost over the restricted class of policies.

Keywords: Stochastic Modeling and Stochastic Systems Theory, Stochastic Control and Estimation, Control issues in Finance
Abstract: This paper focuses on management of surplus and optimization the stream of dividend payment in a continuous-time financial market. The surplus process is modeled by a Markov regime-switching jump diffusion process subject to mixed regular and singular controls. The regular controls including reinsurance strategies and investment allocations are considered simultaneously to maximize the payoff function. Capital injections, considered as impulse controls, are mandatory to avoid solvency when cash reserve is insufficient. Our goal is to maximize the present value of the difference between the cumulative dividend payment and the possible capital injections. Using dynamic programming principle, the value function is a solution of coupled system of nonlinear integro-differential quasi-variational inequalities. Due to the complexity of the regime-switching process and randomness of capital injections, it is virtually impossible to obtain analytical solutions. Thus numerical schemes are a viable and in many cases only alternative. We construct a two component discrete-time controlled Markov chain to approximate the jump diffusion process and the hidden Markov chain. Convergence of the approximation algorithms is demonstrated; examples are presented to illustrate the applicability of the numerical methods.

Keywords: Nonlinear Systems and Control, Stochastic Modeling and Stochastic Systems Theory, System Identification
Abstract: The goal of this study is to classify magnetic resonance imaging (MRI) of brains of patients with Alzheimer's Disease (AD) and those without AD and then to identify changes in brain MRI in early stages of AD. A novel approach based on the diffusion map framework, which is considered to be one of the leading manifold learning methods, is used for this classification. Diffusion mapping provides dimensionality reduction of the data as well as pattern recognition that can be used to distinguish brains of patients with AD from brains of patients without AD. A new algorithm, which is an extension of diffusion maps, constructs coordinates that generate efficient geometric representations of the complex structures in the MRI. In addition, this method is adapted to the MRI and accounts for the variability in calibration of the MRI of different patients. The algorithm is tested on MRI data from patients who developed AD and those who did not.

Keywords: Optimal Control, Process Control, Stochastic Modeling and Stochastic Systems Theory
Abstract: The classical newsvendor problem is one of the simplest decision problems under uncertainty. Given the distribution function of the random [unknown] demand for a product, and assuming that the cost and price parameters are known, a vendor decides on the number of units to be ordered/produced at the beginning of a fixed sales period; his objective, when doing so, is to maximize his expected profit when selling this batch of units on the market. Among the many variations of the classical newsvendor problem, the extension of the model to the case where a vendor not only decides on the order size, but also chooses the selling price, is most important and has been extensively studied. We call a decision maker who faces such a situation a passive price-setting newsvendor; we call the decision maker an active vendor if he dynamically adjusts prices at any instant of time during the sales period. It is a nontrivial problem to quantify the gap between the optimal expected profit of an active vendor and the expected profit of a passive one. For the follwoing special case, we will present analytical formulas of both expressions and will derive an estimate of the differene of the two quantities: (i) the passive newsvendor faces a Poisson demand where the price elasticity of the intensity of the distribution function is constant, (ii) the active vendor deals with a similarly structured but controlled jump Markov process. In addition to the theoretical results, we will present efficient numerical schemes for computing the optimal policies, and we will discuss the results of some numerical studies.

Keywords: Quantum Control, Large Scale Systems, Stochastic Modeling and Stochastic Systems Theory
Abstract: This paper is concerned with translation invariant networks of linear quantum stochastic systems with nearest neighbour interaction mediated by boson fields. The systems are associated with sites of a one-dimensional chain or a multidimensional lattice and are governed by coupled linear quantum stochastic differential equations (QSDEs). Such interconnections of open quantum systems are relevant, for example, to the phonon theory of crystalline solids, atom trapping in optical lattices and quantum metamaterials. In order to represent a large-scale open quantum harmonic oscillator, the coefficients of the coupled QSDEs must satisfy certain physical realizability conditions. These are established in the form of matrix algebraic equations for the parameters of an individual building block of the network and its interaction with the neighbours and external fields. We also discuss the computation of mean square performance functionals with block Toeplitz weighting matrices for such systems in the thermodynamic limit per site for unboundedly increasing fragments of the lattice.

Keywords: Quantum Control, Infinite Dimensional Systems Theory
Abstract: In the present analysis, we consider the controllability problem of the abstract bilinear Schrodinger equation.

We are interested by approximation of this equation by finite dimensional systems.

Assuming that the uncontrolled problem has a pure discrete spectrum and the control potential is in some sense regular with respect to uncontrolled equation, we show that such an approximation is possible. More precisely the solutions are approximated by their projections on finite dimensional subspaces spanned by the eigenvectors of uncontrolled problem.

This approximation is uniform in time and in the control command, if this control has bounded variation with a priori bounded total variation. Hence if these finite dimensional systems are controllable with a fixed bound on the total variation of u then the system is approximatively controllable.

The main outcome of our analysis is that we can build solutions for low regular controls such as bounded variation ones and even Radon measures.

Keywords: Quantum Control
Abstract: We study the controllability of a closed control-affine quantum system driven by two or more external fields. We provide a sufficient condition for controllability in terms of existence of conical intersections between eigenvalues of the Hamiltonian in dependence of the controls seen as parameters. The spectral condition appears naturally in the adiabatic control framework and yields approximate controllability in the infinite-dimensional case. In the finite-dimensional case it implies that the system is Lie-bracket generating when lifted to the group of unitary transformations, and in particular that it is exactly controllable. Hence, Lie algebraic conditions are deduced from purely spectral properties.

Keywords: Quantum Control
Abstract: We present a constructive method to control the bilinear Schrödinger equation by means of three controlled external fields. The method is based on adiabatic techniques and works if the spectrum of the Hamiltonian admits eigenvalue intersections, with respect to variations of the controls, and if the latter are conical. We provide sharp estimates of the relation between the error and the controllability time, namely by following adiabatically special curves in the space of controls.

Keywords: Mathematical Theory of Networks and Circuits, Networked Control Systems, Systems on Graphs
Abstract: -

Keywords: Infinite Dimensional Systems Theory, Control of Distributed Parameter Systems, Operator Theoretic Methods in Systems Theory
Abstract: This minicourse is an introduction to well-posed linear time-invartiant (LTI) systems on Hilbert spaces for non-specialists, while also giving a glimse into nonlinear well-posed systems. In Part I (GW) we recall the more general concept of a system node, classical and generalized solutions of system equations, criteria for well-posedness, the subclass of regular linear systems. Motivated by physical examples, we recall the concepts of impedance passive and scattering passive systems, energy preserving and conservative systems. We illustrate this theory by examples governed by heat and wave equations. In Part II (HZ) we give well-posedness results that are derived from energy considerations. We present the main results about a special class of well-posed linear systems: those that are port-Hamiltonian systems in one space dimension, including 1D Schrodinger equations and the Euler Bernoulli beam equation. In Part III (GW) we present special classes of well- posed systems that occur frequently as models of physical systems. These include parabolic well-posed systems, conservative systems "from thin air" and their generalization that includes the Maxwell system. In Part IV (MT) we explore well-posed linear systems with feedback. After a brief review of the linear feedback theory, we concentrate on nonlinear static feedback. First we consider nonlinear static and Lipschitz feedback from output to input, and then we move to unbounded bilinear feedback from the state and the output to the input. Our results are based on small gain arguments. We apply them to the Burgers equation and the Navier Stokes equations.

Keywords: Control of Distributed Parameter Systems
Abstract: