Keywords: Infinite Dimensional Systems Theory
Abstract:

Keywords: Networked Control Systems, Robust and H-Infinity Control, Linear Systems
Abstract: This paper deals with robust synchronization of directed and undirected multi-agent networks with uncertain agent dynamics. Given a network with identical nominal dynamics, we allow uncertainty in the form of coprime factor perturbations of the transfer matrix of the agent dynamics. These perturbations are assumed to be stable and have H-infinity norm that is bounded by an a priori given desired tolerance. We derive state space equations for dynamic observer based protocols that achieve robust synchronization in the presence of such uncertainty. We show that this robust synchronization of the network by the dynamic protocol is equivalent to robust stabilization of a single linear system by all controllers from a given set of feedback controllers. The synchronizing protocols are expressed in terms of real symmetric solutions to algebraic Riccati equations related to the nominal agent dynamics, and contain a weighting factor depending on the eigenvalues of the graph Laplacian. We obtain an achievable interval, i.e. an interval such that for each value of the tolerance contained in this interval there exists a robustly synchronizing protocol. For undirected networks, the supremum of this interval is proportional to the square root of the quotient of the smallest and the largest eigenvalue of the graph Laplacian.

Keywords: Systems on Graphs, Nonlinear Systems and Control, Robust and H-Infinity Control
Abstract: In this paper, we deal with robust synchronization problems for uncertain dynamical networks of identical Lur'e systems diffusively interconnected by means of measurement outputs. In contrast to stabilization of one single Lur'e system with a passive static nonlinearity in the negative feedback loop, in our paper the feedback nonlinearities are assumed to be incrementally passive. We assume that the interconnection topologies among these Lur'e agents are undirected and connected throughout this paper. A distributed dynamical protocol is proposed. We establish sufficient conditions for the existence of such protocol that robustly synchronizes the Lur'e dynamical network. The protocol parameter matrices are computed in terms of the system matrices of the individual agent, but also the second smallest and the largest eigenvalues of the Laplacian matrix associated with the interconnection topology.

Keywords: Large Scale Systems, Linear Systems
Abstract: This note proposes a unified approach to analyse linear time-invariant consensus problems via the use of integral quadratic constraints (IQCs) without recourse to loop transformations, which may cloud the inherent structural properties of the multi-agent networked systems. The main technical hindrance to using IQCs lies in the presence of the marginally stable integral action in consensus setups. It is shown that by working with conditions defined on modified signal spaces of interests and exploiting the graph structure underlying the connections between the dynamic systems, IQC methods can be applied directly to consensus analysis. A decentralised and scalable condition for consensus is proposed in this setting, which generalises some of the existing results in the literature.

Keywords: Systems on Graphs, Networked Control Systems, Linear Systems
Abstract: The paper addresses the synchronization problem for a network of identical, linear time-invariant state-space models. The notion of synchronizability is investigated and a set of sufficient and necessary conditions relating synchronizability to the dynamical properties of the subunits are provided. The paper also extends recent results about synchronization of passive linear systems by proving that networks of linear, detectable, passive systems can be synchronized by any (possibly directed) connected interconnection topology. The theory is illustrated with several examples.

Keywords: Networked Control Systems, Large Scale Systems, Systems on Graphs
Abstract: This paper is concerned with state synchronization of linear agents subject to input saturation over a fixed undirected communication graph. We first derive a sufficient condition for achieving the synchronization via relative state feedback control law for any initial condition. Based on this analysis result, we present a linear matrix inequality (LMI) condition for designing the synchronizing state feedback gain. The present LMI condition is scalable as long as we can calculate the eigenvalues of the Laplacian of the communication graph, and is readily solved by an existing convex programming algorithm.

Keywords: Applications of Algebraic and Differential Geometry in Systems Theory, Linear Systems, Feedback Control Systems
Abstract: This paper proposes a functional controller for motion control of the Robotino. The proposed controller takes under consideration a functional decoupling control strategy realized using a geometric approach and the inversability property of the DC-drives with which the Robotino is equipped. For a given control structure the functional controllability is proven for motion trajectories of class C3, continuous functions with third derivative also continuous. Horizon, Vertical and Angular motions are considered and once the decoupling between these motions is obtained, a Model Predictive Control (MPC) strategy is used in combination with an inverse drive model. Simulation results using real data of Robotino are shown.

Keywords: Nonlinear Systems and Control, Algebraic Systems Theory
Abstract: Given two nonlinear input-output systems written in terms of Chen-Fliess functional expansions, the feedback interconnected system is in the same class with a generating series that can be computed explicitly via the antipode of a certain Faa di Bruno type Hopf algebra. This defines the feedback product of two generating series. Existing methods to compute this antipode are based on matrix inversion and are known to be inefficient. This paper has two objectives. The first is to use some recent advances in the area to produce a completely recursive algorithm for computing the antipode of the Hopf algebra for the SISO feedback group. The second objective is to provide a Mathematica implementation of the algorithm and evaluate its performance against the existing method using matrix inversion.

Keywords: Nonlinear Systems and Control, Applications of Algebraic and Differential Geometry in Systems Theory, Hybrid Systems
Abstract: Systems driven by impulsive inputs are readily modeled for linear systems. Extension of the results to nonlinear systems has run into many problems. Consequently impulsive nonlinear systems are usually described by the effect of impulses, thus giving rise to jumps in the state space. However, it is not clear a priori how these jumps are to be related to the effective impulsive inputs (the causes) to the system. We derive such results for bilinear systems of arbitrary order, and discuss further extensions to other classes of systems. The method is justifiable by taking an approach touching on non-standard analysis. It is based on first principles considering singular functions as sequences of regular functions, pretty much as they are thought in an undergraduate signal course, and leads to a definition of insensible times and functions. The latter provide the fine structure extension.

Keywords: Nonlinear Systems and Control, Hybrid Systems, Transportation Systems
Abstract: In this paper, we propose a new observer design for nonlinear systems that can be linearized using a change of coordinates and a singular time re-scaling. Our observer is a switched system and the observer error dynamics are described, after time re-scaling, by a switched linear system that is uniformly exponentially stable. We also give necessary and sufficient conditions for linearizability via a change of coordinates and a singular time re-scaling. Our methods are illustrated by an example coming from the ABS literature.

Keywords: Optimal Control, Biological Systems, Robotics
Abstract: This paper solves the optimization problem for a simplified one-dimensional worm model when the friction force depends on the direction of the motion. The motion of the worm is controlled by the actuator force f(t) which is assumed to be piecewise continuous and always generates the same force in the opposite directions. The paper derives the necessary condition for the force which maximizes the average velocity or minimizes the power over a unit distance. The maximum excursion of the worm body and the force are bounded. A simulation is given at the end of the paper.

Keywords: Mathematical Theory of Networks and Circuits, Linear Systems
Abstract: The birth of theoretical electrical engineering could be dated to Heaviside’s brilliant idea (early 1900’s) of replacing linear differential equations by an algebraic symbolism. This allowed Ohm’s Law to be applied not only to resistors but also to the reactive elements (capacitors, inductors) thereby creating a general notion of impedance. The problem of realization (this was not the terminology used a hundred years ago) arose immediately: “Given an impedance, is there a passive RCL network which has that impedance?” (The easy question, “Given a resistance, is there a resistor with that resistance?” has already given rise to an industry.) Surprisingly quickly, a nice answer to such questions was given in 1924 by a young “math- ematician”, Ronald M. Foster (1896-1998), (B.S. in Mathematics 1917 from Harvard), whom Bell Laboratories was converting into an “engineer”. His answer was YES, with reservations. In his work surroundings, Foster was concerned mainly with lossless (CL) networks, and, for that case, the answer was unequivocably “yes”, with classical mathe- matics (polynomials and rational functions) providing precise, necesssary and sufficient, conditions. Foster created a family of thitherto unknown networks for the purpose and gave explicit formulas for the capacitors and inductors of those networks to realize the desired impedance. When (and only when) the computed component values turned out to be positive, the impedance was realizable.

Keywords: Mathematical Theory of Networks and Circuits, Linear Systems
Abstract: Since the data in the Foster-Ladenheim catalog are more than seventy-five (75) years old (see first lecture), some of you may regard talking about it as scientific archeology. But no, our forefathers, the old boys, surprisingly, knew much more about things than we thought they did. Actually, the Ladenheim catalog is a rich source of information, some on the surface, some to be made precise. Foster’s contributions: (1) The idea of a “generic” network. (“Generic” means: no further simplification possible without affecting basic properties; nowhere explicitly defined by Foster, even though the celebrated “Foster and Cauer canonical forms”, discovered in the 1920’s (see Guillemin’s book) are, of course, generic networks and duly appear in the catalog.) (2) That straightforward algebraic manipulation can lead to explicit solution of the real- ization problem. A rediscovery of some tricks of classical invariant theory. Foster’s omissions: (3) Calling “discriminant” the “resultant” in the 1962 survey paper, in spite of accurate use of the traditional Sylvester definition of the resultant. (Cultural obstacles between engineering and mathematics, but also sloppy teaching of mathematics to engineering students in the Boston area in the early twentieth century.) (4) Not recognizing that the network realization problem, as reformulated by himself, shows many affinities (use of resultants and related invariants) to the classical theory of projective invariants involving a pair of polynomials, well developed before 1900. (5) Getting actual criteria for the positive realizability of a fixed impedance. (Very difficult problem, still largely unsolved; perhaps already solved, for a large class of networks, by “regularity” in the sense of Malcolm Smith and Jason Jiang, ...)

Keywords: Mathematical Theory of Networks and Circuits, Mechanical Systems
Abstract: The motivation provided by mechanical network synthesis to make a fresh attack on certain questions in circuit synthesis will be briefly recalled. The classical early work on RLC synthesis, beginning with the works of Foster and Cauer and culminating in the Bott-Duffin construction, will be explained in a tutorial manner. Recent work on RLC synthesis by Jiang and Smith and Hughes and Smith will be introduced. The proof in T.H. Hughes and M.C. Smith, 2014, “On the minimality and uniqueness of the Bott-Duffin procedure”, IEEE Trans. Aut. Contr., (to appear), showing the surprising result that the Bott-Duffin construction for a biquadratic minimum function is the simplest possible among series-parallel circuits, will be explained.

Keywords: Mathematical Theory of Networks and Circuits, Algebraic Systems Theory, Mechanical Systems
Abstract: A fundamental result in circuit synthesis states that the McMillan degree of a passive circuit’s impedance is less than or equal to the number of reactive elements in the circuit. More recently, Hughes and Smith [1] connected the individual numbers of inductors and capacitors in a circuit to a generalisation of the Cauchy index for the circuit’s impedance, which was named the extended Cauchy index. There is a close connection between the Cauchy index of a real-rational function and many classical algebraic results relating to pairs of polynomial functions [2]. Using this connection, it is possible to derive algebraic constraints on circuit impedance functions relating to the precise numbers of inductors and capacitors in that circuit. In this paper, we first present these algebraic constraints. We will then show a relationship between the extended Cauchy index and properties of continued fraction expansions of real-rational functions, which we use to provide insight into circuit synthesis procedures.

Keywords: Algebraic Systems Theory, Linear Systems, Mathematical Theory of Networks and Circuits
Abstract: The paper addresses some synthesis problems that arise in the control of large scale networks of linear systems. For homogeneous networks of identical SISO systems we characterize the rational transfer functions of the networks in terms of associated Galois groups. We then study the problem of synthesizing broadcast open loop controls for networks from knowledge of local controls for the node systems. The difficulty in doing so is both a function of the size of the network and the complexity of the interconnection patterns. We shall focus on two points: The characterization of controllability and the computation of open loop controls that steer the system from rest to an arbitrary prescribed state. Explicit formulas for the open loop controls are derived for parallel connection of N linear state space systems.

Keywords: Coding Theory
Abstract: We transpose the theory of rank metric and Gabidulin codes to the case of fields which are not finite fields. The Frobenius automorphism is replaced by any element of the Galois group of a cyclic algebraic extension of a base field. We use our framework to define Gabidulin codes over the field of rational functions using algebraic function fields with a cyclic Galois group. This gives a linear subspace of matrices whose coefficients are rational function, such that the rank of each of this matrix is lower bounded, where the rank is comprised in term of linear combination with rational functions. We provide two examples based on Kummer and Artin-Schreier extensions.The matrices that we obtain may be interpreted as generating matrices of convolutional codes.

Keywords: Coding Theory, Communication Systems
Abstract: Ideal group codes are defined as principal ideals in the group algebra of a finite group G over a finite field F. An overview of their properties along with their encoding and syndrome decoding are presented. The correction of a single error, using syndromes, is described in detail.

Keywords: Coding Theory
Abstract: We provide a generalized version of the Grey-Rankin bound for self-complementary codes over finite Frobenius rings and present some examples where the bound can be applied and where the bound is sharp.

Keywords: Coding Theory
Abstract: The problem of constructing Insertion/Deletion codes for the editing distance is reduced to constructing codes over the integers for the Manhattan distance by run length coding. These latter codes are constructed by truncation of translates of lattices. These lattices in turn are obtained from Construction A applied to binary codes and Z4􀀀codes. Complete weight enumerators of these codes allow to compute the generalized theta series of the corresponding lattices. Asymptotic Gilbert and Hamming type of bounds are derived in large dimensions. Constructive bounds better than the Gilbert type bound are derived by use of geometric codes.

Keywords: Coding Theory, Information Theory
Abstract: One-point Goppa codes are among the most studied algebraic geometric codes. They are obtained evaluating vector spaces of rational functions on curves defined over finite fields at appropriate sets of points. In some relevant cases (e.g., for codes from norm-trace curves) the dual minimum-weight codewords of such codes are characterized by the intersections of the underlying curve with lines and conics in the plane. We describe a geometric method to determine the dual minimum distance of such codes and explicitly count their dual minimum-weight codewords.

Keywords: Infinite Dimensional Systems Theory, Linear Systems, Operator Theoretic Methods in Systems Theory
Abstract: For the wave equation on a bounded Lipschitz domain we characterize all homogeneous boundary conditions for which this partial differential equation generates a contraction semigroup in the energy space. The proof uses the Hille-Phillips characterization of generators of contraction semigroups. To apply this characterization we first have to determine the dual operator of the grad and div operator, seen as unbounded operators on the space of square integrable functions.

Keywords: Control of Distributed Parameter Systems, Infinite Dimensional Systems Theory
Abstract: This paper aims at providing some synthesis between two alternative representations of systems of two conservation laws and to make a link between the notions of dissipativity and the Small-Gain theorem. The first one, based on the invariance of its coordinates, is the representation in Riemann coordinates which has been applied successfully for the stabilization of linear and non-linear hyperbolic systems of conservation laws. The second representation is based on physical modeling and leads to port Hamiltonian systems which are extensions of infinite-dimensional Hamiltonian systems defined on Dirac structure encompassing pairs of conjugated boundary variables. We propose in this paper to link the passivity of the Hamiltonian functional, expressed in Riemann coordinates, with the Small Gain Theorem.

Keywords: Infinite Dimensional Systems Theory
Abstract: We study robustness of well-posedness in L² with maximal regularity under non-autonomous multiplicative perturbation for evolution equations governed by non-autonomous form.

Keywords: Infinite Dimensional Systems Theory, Operator Theoretic Methods in Systems Theory, Linear Systems
Abstract: The weighted Weiss conjecture states that the system theoretic property of weighted admissibility can be characterised by a resolvent growth condition. For positive weights, it is known that the conjecture is true if the system is governed by a normal operator; however, the conjecture fails if the system operator is the unilateral shift on the Hardy space (discrete time) or the right-shift semigroup (continuous time). To contrast and complement these counterexamples, in this talk positive results are presented characterising weighted admissibility of linear systems governed by shift operators and shift semigroups. These results are shown to be equivalent to the question of whether certain generalized Hankel operators satisfy a reproducing kernel thesis.

Keywords: Nonlinear Systems and Control, Stability, Optimization : Theory and Algorithms
Abstract: The existence of a control Lyapunov function with the weak infinitesimal decrease via the Dini or the proximal subdifferential and the lower Hamiltonian characterizes asymptotic controllability of nonlinear control systems and differential inclusions. We study the class of nonlinear differential inclusions with a right-hand side formed by the convex hull of active C² functions which are defined on subregions of the domain. For a simplicial triangulation we parametrize a control Lyapunov function (clf) for nonlinear control systems by a continuous, piecewise affine (CPA) function via its values at the nodes and demand a suitable negative upper bound in the weak decrease condition on all vertices of all simplices.

Applying estimates of the proximal subdifferential via active gradients we can set up a mixed integer linear problem (MILP) with inequalities at the nodes of the triangulation which can be solved to obtain a CPA function. The computed function is a clf for the nonlinear control system.

We compare this novel approach with the one applied to compute Lyapunov functions for strongly asymptotically stable differential inclusions and give a first numerical example.

Keywords: Nonlinear Systems and Control, Stability
Abstract: Conley showed that the state-space of a dynamical system can be decomposed into a gradient-like part and a chain-recurrent part, and that this decomposition is characterized by a so-called complete Lyapunov function for the system. Kalies, Mischaikow, and VanderVorst proposed a combinatorial method to compute discrete approximations to such complete Lyapunov functions. Their approach uses a finite subdivision of a compact subset of the state-space and a combinatorial multivalued map to approximate the dynamics. They proved that as the diameter of the elements of the subdivision approaches zero the resulting approximations to complete Lyapunov functions converge to a true complete Lyapunov function for the system.

Ban and Kalies implemented this algorithm and used it to compute an approximation to a complete Lyapunov function for a time-T mapping of the van der Pol oscillator.

The CPA method to compute continuous and piecewise affine Lyapunov functions uses linear programming to parameterize true Lyapunov functions for continuous dynamical systems.

In this paper we propose using the CPA method to evaluate approximations to complete Lyapunov functions computed by the combinatorial method. Especially, we can explicitly compute the region of the state-space where the orbital derivative of the approximation is negative. Further, we use the RBF method to solve a Zubov equation and compare the solution V with the complete Lyapunov function computed by the combinatorial method for the van der Pol oscillator.

Keywords: Nonlinear Systems and Control, Stability, Optimal Control
Abstract: In this paper, we present a numerical algorithm for computing a local ISS Lyapunov function for systems which are locally input-to-state stable (ISS) on compact subsets of the state space. The algorithm relies on a linear programming problem and computes a continuous, piecewise affine ISS Lyapunov function on a simplicial grid covering the given compact set excluding a small neighborhood of the origin. We show that the ISS Lyapunov function delivered by the algorithm is a viscosity subsolution of a partial differential equation.

Keywords: Optimal Control, Large Scale Systems, Control of Distributed Parameter Systems
Abstract: We present a numerical approach for the timeoptimal feedback control of an advection-reaction-diffusion model. Our approach is composed by three main building blocks: approximation of the abstract system dynamics, feedback computation based on dynamic programming and state observation. For the approximation of the abstract dynamics, we consider a finite element semi-discretization in space, leading to a large-scale dynamical system, whose dimension is reduced by means of a Balanced Truncation algorithm. Next, we apply the dynamic programming principle over the reduced model, and characterize the value function of the optimal control as a viscosity solution of a Hamilton-Jacobi-Bellman equation, which is numerically approximated with a semi-Lagrangian scheme. Finally, the computation of the corresponding feedback controls and its insertion into the control loop is performed by implementing a Luenberger observer.

Keywords: Optimal Control, Feedback Control Systems, Stochastic Control and Estimation
Abstract: A Model Predictive Control scheme is applied to track the solution of a Fokker-Planck equation over a fixed time horizon. We analyse the dependence of the total cost functional on several parameters of the algorithm, in particular on the prediction horizon, on the regularization parameter, and on the sampling time. Comparison among different numerical simulations show valuable improvements by properly tuning the scheme's parameters. Our numerical study is complemented by a theoretical controllability analysis explaining the superior performance of controls with time and space dependence.

Keywords: Operator Theoretic Methods in Systems Theory
Abstract: We establish a fixed point theorem for mappings of square matrices of all sizes which respect the matrix sizes and direct sums of matrices. The conclusions are stronger if such a mapping also respects matrix similarities, i.e., is a noncommutative function. As a special case, we prove the corresponding contractive mapping theorem which can be viewed as a new version of the Banach Fixed Point Theorem. This result is then applied to prove the existence and uniqueness of a solution of the initial value problem for ODEs in noncommutative spaces. As a by-product of the ideas developed in this paper, we establish a noncommutative version of the principle of nested closed sets.

Keywords: Multidimensional Systems, Optimization : Theory and Algorithms, Operator Theoretic Methods in Systems Theory
Abstract: Noncommutative rational power series have been of importance in system theory since their first appearance in the work on formal languages and finite automata in the late 1950s. They form an important tool of studying noncommutative rational functions that are regular at 0, or more generally at a scalar point. Using the new ideas stemming from the general theory of noncommutative functions, we consider power series expansions of a noncommutative rational function around a general matrix point.

Keywords: Optimization : Theory and Algorithms, Operator Theoretic Methods in Systems Theory
Abstract: In this talk, we intend to present the main results of our work ``Application of Jacobi's Representation Theorem to locally multiplicatively convex topological real Algebras''. For a commutative unital algebra A, we determine the closure of a module S in A over the sum of 2d-th powers in A, for any integer d, with respect to any locally multiplicatively convex topology. The motivation comes from the quest, going back to Hilbert, of relating nonnegative polynomials to sums of squares.

Keywords: Robust and H-Infinity Control, Operator Theoretic Methods in Systems Theory, Multidimensional Systems
Abstract: The structured singular value (often referred to simply as mu) was introduced independently by Doyle and Safanov as a tool for analyzing robustness of system stability and performance in the presence of structured uncertainty in the system parameters. While the structured singular value provides a necessary and sufficient criterion for robustness with respect to a structured ball of uncertainty, it is notoriously difficult to actually compute. The method of diagonal (or simply "D") scaling, on the other hand, provides an easily computable upper bound for the structured singular value, but provides an exact evaluation of mu (or even a useful upper bound for mu) only in special cases. However it was discovered in the 1990s that enhancement of the uncertainly structure to allow what can be interpreted as time-varying uncertainty (equivalently, letting the uncertainty parameters be freely noncommuting operators on an infinite-dimensional separable Hilbert space) resulted in the D-scaling procedure leading to an exact evaluation of mu. This report discusses recent refinements of these results to allow repetitions of full blocks as well as nonsquare blocks in the uncertainty structure.

Keywords: Mechanical Systems, Stability, Optimization : Theory and Algorithms
Abstract: We consider the optimal damping problem for a linear vibrational system Mddot{x}+Ddot{x}+Kx=0, where M and K are positive definite matrices. For the damping optimization we use a criterion based on minimization of the integral of the solution's amplitude over a given time interval. Finding the optimal damping D is a very demanding problem, and using this approach the computational cost comes mainly from a large number of matrix exponential computations. We propose an efficient numerical scheme to accelerate these computations. The performance of our approach is illustrated by numerical results for an n-mass oscillator.

Keywords: Nonlinear Filtering and Estimation, Nonlinear Systems and Control, Mechanical Systems
Abstract: In this note, we propose a new approach for nonlinear observer design based on Sylvester equations that depend on states and inputs. Using a state-dependent linear representation and generalizing the Luenberger observer formulation, a class of nonlinear observer with analytic expression is proposed. The proposed design method does not rely on numerical computation unlike existing methods such as the state-dependent Riccati equation approach, or extended Kalman filter. A numerical simulation for 2-dimensional pendulum control is illustrated to verify the effectiveness of the proposed observer.

Keywords: Nonlinear Systems and Control, Mechanical Systems, Stability
Abstract: The control of 4-DOF underactuated overhead crane system poses a challenging control problem as it has extra degree of freedom (DOF) compared to its popular 3-DOF variant. The extra DOF represents strong state coupling and hence more complicated system dynamics. We propose Geometric - Passivity Based Control (PBC) methodology for synthesis of nonlinear stabilising feedback control law. The structure of the split tangent space is modified along the actuated direction in such a way that the power flow is established between the controller and the un-actuated subsystems. The passivating outputs of the modified system are identified which are utilized for energy shaping of the system. The nonlinear control law thus obtained achieves the control objective of precise payload positioning with elimination of payload swings. The main advantage of proposed nonlinear design methodology is that obviates the need of solving PDE's or obtaining nonlinear transformations to synthesize the control law. The simulation results are presented to validate the nonlinear control law. The system parameters and constraints on input forces are considered to represent the experimental setup of 4-DOF overhead crane system.

Keywords: Nonlinear Systems and Control, Mechanical Systems, Stability
Abstract: In this paper, a nonlinear control law is proposed for the curved ball and beam system with the ball at the upright position. The stabilization of such a system is challenging, as compared to conventional ball and (straight) beam system, due to the presence of two unstable equilibrium points and the gyroscopic forces appearing in its dynamics. Further, the system belongs to the interesting class of underactuated mechanical systems. We propose Geometric Passivity Based Control (PBC) methodology for synthesizing stabilizing control law for the system under consideration. The power flow between the controller and the unactuated dynamics is established by manipulating the mechanical connection. An additional dissipation term is included in the control law to deal with the gyroscopic forces. The main advantage of the proposed controller design strategy is that it does not involve solving any Partial Differential Equations (PDEs) or nonlinear transformations. The simulation results are presented to validate the control law. The system parameters used in simulation adheres to the physical model constraints.

Keywords: Nonlinear Systems and Control, Multidimensional Systems, Mechanical Systems
Abstract: For multidimensional and infinite-dimensional control systems with periodic differentiable nonlinearities and denumerable sets of equilibria the problem of cycle-slipping is investigated. By means of Lyapunov periodic functions, Kalman-Yakubovich-Popov lemma, and Popov functionals new frequency-algebraic estimates for a number of slipped cycles are obtained.

Keywords: Algebraic Systems Theory, Multidimensional Systems, Linear Systems
Abstract: In this talk we define and characterize the property of controlled invariance in the framework of the behavioral appproach to systems and control.

Keywords: Multidimensional Systems, Algebraic Systems Theory, Linear Systems
Abstract: In this paper we study a generalized tracking and disturbance rejection problem for multidimensional linear behaviours. Given a multidimensional plant, our first goal is to design a compensator to be connected to the plant through regular partial interconnection, in such a way that the overall controlled system is autonomous and stable, when no exogenous signal acts on the system. On the other hand, when exogenous signals affect the controlled system evolution, we want to impose that a suitable linear combination of the overall system trajectories is "negligibile" in a sense we will clarify within the paper. This problem set-up formalizes a number of classical control problems, first of all tracking of some (reference) signal together with rejection of another (disturbance) signal. The adopted approach is extremely general and it is based on the idea of describing all behaviour trajectories as the sum of a "transient signal" and a "steady state" component, a decomposition that relies on Gabriel's localization theory. Necessary and sufficient conditions for the problem solvability are provided, and the compensators that satisfy the control goal are characterized in terms of an internal model condition.

Keywords: Algebraic Systems Theory, Infinite Dimensional Systems Theory, Delay Systems
Abstract: We investigate one-dimensional convolution behaviors that comprise differential and delay-differential behaviors in particular. We thus continue work of, for instance, Fliess, Gluesing-Luerssen, Mounier, Rocha, Vettori, Willems, Yamamoto, Zampieri of the last twenty-five years. The signal space of these behaviors is the space E of complex-valued smooth functions on the real line with its standard Frechet topology on which the dual space E' with the strong topology of distributions with compact support acts by a variant of the convolution product. Also via convolution E' is a commutative domain and topological algebra and E is an E'-module. By definition generalized behaviors (gen. beh.) are closed, translation-invariant subspaces or, equivalently, closed E'-submodules B of some finite power of E. A gen. beh. is called a behavior if it can be described by finitely many convolution equations as usual. The torsion elements of E, i.e., the functions which are annihilated by some nonzero distribution in E', are called mean-periodic functions and were studied by well-known analysts, for instance by Delsarte (1935), Schwartz (1947), Kahane (1953), Ehrenpreis (1955, 1960) and Berenstein and Taylor (1980). Their results are significant for the study of convolution behaviors. Typical new results are the following: (i) The E'-submodule PE of E of polynomial-exponential functions is injective and thus permits (Willems') elimination procedure for the polynomial-exponential part of gen. beh. whereas E itself is not injective. (ii) We construct all autonomous gen. beh.. (iii) We show that many gen. beh., in particular all autonomous ones, are indeed behaviors, but cannot presently decide whether even all gen. beh. are behaviors. Schwartz' seminal result (1947) in this context was that every closed, translation-invariant subspace of E is a behavior and indeed described by two equations. (iv) We construct and investigate input/output decompositions of generalized behaviors.

Keywords: Algebraic Systems Theory, Multidimensional Systems, Computations in Systems Theory
Abstract: Within the algebraic analysis approach to multidimensional systems, the behavioural approach can be understood as a dual theory to the module-theoretic approach. This duality is exact when the signal space is an injective cogenerator left module over the ring of functional operators. In this paper, we consider the case of a general signal space and investigate the connection between the properties of the module M defining the system and the obstruction to the existence of parametrizations of this system. To do so, we study the Cartan-Eilenberg resolution of a certain complex and a Grothendieck spectral sequence connecting the obstructions to the existence of parametrizations to the obstructions for M to be projective.

Keywords: Algebraic Systems Theory, Multidimensional Systems, Computations in Systems Theory
Abstract: The problem of characterizing the restriction of the solutions of an n-D system to a subvector space of Rⁿ has recently been investigated in the literature of multidimensional systems theory. In this paper, we characterize the restriction of an n-D behaviour to an algebraic or analytic submanifold of Rⁿ. To do that, we first use the algebraic analysis approach to multidimensional systems. We then show that the restriction of an n-D behaviour to an algebraic or analytic submanifold can be characterized in terms of the inverse image of the differential module defining the behaviour. Characterization of inverse images of differential modules is investigated. Finally, using the above results, we explain Kashiwara's extension of the Cauchy-Kowalevski theorem for general n-D behaviours and non-characteristic algebraic or analytic submanifolds.

Keywords: Linear Systems, Stability
Abstract: New sequences of monotonically increasing sets are introduced, for linear discrete-time systems subject to input and state constraints. The elements of the set sequences are controlled invariant and admissible regions of stabilizability. They are generated from the iterative application of the inverse reachability mapping, its geometric generalization, called the inverse directional reachability mapping, and mappings constructed by parts of the one-step inverse reachability and the one-step inverse directional reachability set.The four proposed set sequences converge to the maximal region of stabilizability.

Keywords: Linear Systems, Stability
Abstract: This note studies stabilization problems arising in discrete estimation (or preview tracking) problems when unstable exosystems are used to impose steady-state behavior constraints. The main theme is stabilizability conditions under FIR (finite impulse response) constraints. Necessary and sufficient stabilizability conditions are derived and all solutions are parametrized. Unlike the continuous-time case, where the problem is solvable under the standard detectability condition, stabilizability in the discrete-time case depends on the length of the impulse response and, sometimes, also on the smoothing lag (length of preview).

Keywords: Stability, Algebraic Systems Theory
Abstract: In this paper, we present the design of a simultaneous compensator for a segment of systems based on an interpolation method with stable polynomial interpolants. This problem leads to formulate conditions of polynomial divisibility in the case of the simultaneous control as a polynomial interpolation issue. The feasibility and infeasibility of the approach is also analyzed. Finally, an algorithm permitting to compute a simultaneous controller stabilizing a segment of systems is given.

Keywords: Stability, Computations in Systems Theory
Abstract: This paper presents an alternative approach for obtaining a converse Lyapunov theorem for discrete-time sys- tems. The proposed approach is constructive, as it provides an explicit Lyapunov function. The developed converse Lyapunov theorem establishes existence of global Lyapunov functions for globally exponentially stable (GES) systems and for globally asymptotically stable systems, Lyapunov functions on a subset of the positive orthant are derived. In particular, we discuss our result for both continuous and discontinuous dynamics.

Keywords: Stability, Linear Systems, Computations in Systems Theory
Abstract: In this paper we make use of the alternative converse Lyapunov theorem presented in [1] for specific classes of systems. We show that the developed converse Lyapunov theorem can be used to establish non–conservatism of a par- ticular type of Lyapunov functions. Most notably, a proof that the existence of conewise linear Lyapunov functions are non– conservative for globally exponentially stable (GES) conewise linear systems is given and, as a by–product, tractable con- struction of polyhedral Lyapunov functions for linear systems is attained.

Keywords: Optimization : Theory and Algorithms, Nonlinear Systems and Control, Optimal Control
Abstract:

Keywords:
Abstract: Quantum feedback has to take into account the intrinsic invasive character of measurements. We present here two feedback schemes for the stabilization of quantum states and their protection from decoherence: measurement-based feedback scheme where an open quantum system is stabilized by a classical controller; coherent or autonomous feedback scheme where a quantum system is stabilized by a quantum controller with decoherence (reservoir/dissipation engineering). The differences and complementary features of these stabilization schemes are illustrated with the photon box experiments realized in the group of Serge Haroche (Nobel Prize 2012) and Jean-Michel Raimond.

Keywords: Coding Theory
Abstract: Linear system theory and the theory of linear error control codes have several fundamental results in common. In this talk I will present some of these that can be formulated in terms of key results regarding minimal bases of polynomial vector modules.

Forney's "predictable degree" property is well-established in algebraic system theory. In this talk I will present a refinement of this property by defining minimality not only in terms of "degree" but also in terms of "position". Bases that are minimal in this sense have the "Predictable Leading Monomial (PLM)" property. Thus this refines the system-theoretic notion of "row-reducedness".

Minimal bases play a fundamental role in several types of algebraic decoding. In this talk we focus on two types of codes, the Reed-Solomon codes and the Gabidulin codes. Reed-Solomon codes are ubiquitous in applications, ranging from storage systems such as CD and DVD to bar codes, such as QR code and to space image transmission. Despite the fact that these codes were introduced as early as 1960, the last 15 years saw significant improvements in Reed-Solomon decoding performance. These developments were set off by a 1997 breakthrough paper by Sudan which presented a computationally feasible method to perform Reed-Solomon list decoding.

The last decade saw some of this attention shift to a punctured subcode of the Reed-Solomon code, namely the Gabidulin code, named after 1985 results by Gabidulin. Because of their linear evaluation property, Gabidulin codes are instrumental in for example Random Linear Network Coding where linear subspaces are sent rather than single vectors.

In the talk I present the algebraic decoding problem for both Reed-Solomon codes and Gabidulin codes. I will discuss differences and similarities and show how the abovementioned PLM property features prominently in the design of efficient iterative algorithms. In the terminology of the late Jan C. Willems, these algorithms essentially construct a Most Powerful Unfalsified Model (MPUM) for the data which is the smallest unfalsified model, the true heart of the matter. This talk is dedicated to Jan.

Keywords:
Abstract: Modeling of complex physical and technological processes frequently leads to dynamical systems of very large dimension. Numerical simulation, structural optimization and real-time control design of such complex systems becomes difficult due to rapidly growing computational time and storage requirements. Model order reduction aims to approximate large-scale dynamical systems by models of lower dimension that accurate capture the dynamical behavior of the original problems and preserve important system properties. In many industrial applications such as micro- and nanoelectronics, structural mechanics and fluid dynamics, parametrized models are often used in the early design stages, where the parameters describe geometric properties and material variations. In parametric model reduction, it is important to preserve the parametric dependence in the reduced-order models enabling fast simulations for many different parameter values. In this talk, we will review the recent development of projection-based model reduction methods for parameterized systems and highlight their advantages and disadvantages. We will also discuss the computational issues and present several examples from different application areas to demonstrate the performance of the various parametric model reduction approaches.

Keywords: Networked Control Systems, Nonlinear Systems and Control, Robust and H-Infinity Control
Abstract: This paper addresses stability issues of sampled data controllers. Considering a continuous-time linear plant and a linear discrete-time dynamic output feedback control law designed from a classical periodic sampling paradigm, the main goal is to assess the effects of aperiodic sampling on the closed-loop stability. This aperiodic sampling models for instance the communication delays and package losses through a network. In addition, the effects of control signal saturation on the stability and the maximal admissible sampling interval are also taken into account . In this context, based on the use of a looped functional, linear matrix inequalities (LMI) are derived to ensure the global asymptotic stability of the origin for the aperiodic sampled-data closed-loop system, provided a bound on the maximal sampling interval is given. An optimization problem in order to evaluate the maximal admissible value for the interval between two sampling instants is then associated to the LMI conditions.

Keywords: Networked Control Systems, Robust and H-Infinity Control, Optimization : Theory and Algorithms
Abstract: This paper presents a novel algorithm for the synthesis of robust distributed controllers for interconnected linear discrete time systems. For a network of interconnected and uncertain systems, the distributed controller achieves robust stability and a guaranteed level of robust performance in a well defined H_infty sense. The theory is developed for networks of linear discrete time systems where model uncertainty is described in the setting of linear fractional representations. A computationally tractable synthesis algorithm is proposed that employs the use of linear matrix inequalities in an iterative D-K type of algorithm. Convergence properties of the algorithm are inferred.

Keywords: Stability, Biological Systems, Large Scale Systems
Abstract: In this paper, we study instability analysis of a feedback system composed of a nominal linear system and a positive real uncertainty. We propose a new definition of generalized strictly positive realness to show that it is sufficient for preserving the same instability property in the feedback system as that in the nominal system for all uncertainties. Then, an illustrative example for robustness analysis of a genetic oscillator model with positive real uncertainties is presented. Finally, we discuss an implication of the derived instability condition from the viewpoint of stability and instability analysis of a positive real network system, i.e., a network system composed of positive real subsystems that are internally connected in a specific network topology. We show merits and demerits of such network construction strategy for global stability to be guaranteed.

Keywords: Networked Control Systems, Systems on Graphs, Mathematical Theory of Networks and Circuits
Abstract: We consider a basic model of a dynamical distribution network, modeled as a directed graph with storage variables corresponding to every vertex and flow inputs corresponding to every edge, subject to unknown but constant inflows and outflows. In cite{wei2013} we showed how a distributed proportional-integral controller structure, associating with every edge of the graph a controller state, regulates the state variables of the vertices, irrespective of the unknown constant inflows and outflows, in the sense that the storage variables converge to the same value (load balancing or consensus). In many practical cases, the flows on the edges are constrained. The main result of cite{wei2013} is a sufficient and necessary condition, which only depend on the structure of the network, for load balancing for arbitrary constraint intervals of which the intersection has nonempty interior. In this paper, we will consider the stability of the same model as in cite{wei2013} with given network structure and constraint intervals. We will derive a graphic condition, which is sufficient and necessary, for load balancing. This will be proved by a Lyapunov function and the analysis the kernel of incidence matrix of the network. Furthermore, we will show that by modified PI controller, the storage variable on the nodes can be driven to an arbitrary point of admissible set.

Keywords: Stability, Systems on Graphs, Networked Control Systems
Abstract: This work is part of a larger research effort investigating the role of symmetries in control of large-scale cyber-physical-systems (CPS). Most prior efforts have considered discrete symmetries, and the work in this paper reflects our initial efforts directed toward investigating continuous symmetries. Specifically, we consider a formation control problem, limited for now to only two agents, and determine all the point transformation symmetries associated with that system. We also determine a reduced description of the dynamics of the system particularly important for formation control and show stability of the system in those coordinates.

Keywords: Nonlinear Filtering and Estimation, Stochastic Control and Estimation
Abstract: This paper considers estimation theory for partially observed stochastic dynamical systems whose state equations are McKean-Vlasov type stochastic differential equations and hence contain a measure term corresponding to the distribution of the solution of the state process. Nonlinear filtering equations are provided based on the classification that either the measure term is stochastic or deterministic and that either the state or the measure term is estimated. When the measure term is deterministic the standard theory holds without any modification. In the situation where the measure term is random, the induced functions in the dynamics of the state become random and a similar recursion for the optimal filter is obtained. The joint estimation of state and the measure term is next considered. The extended state in this setup is shown to be a Polish space valued stochastic process with random functions in its state dynamics and a nonlinear filtering equation for this setup is provided. The main motivation for the development of estimation theory for McKean-Vlasov type dynamical systems arises from recent progress in mean field game theory and estimation problems in a mean field game framework are discussed. In particular, a mean field game setting with one major and many minor players is considered and nonlinear filtering formulations for the optimal estimation of both the major agent's state and the stochastic measure induced by the minor agents are provided.

Keywords: Optimization : Theory and Algorithms, Stochastic Control and Estimation, Control of Distributed Parameter Systems
Abstract: This paper addresses an initial study of a model-free approach based on Simultaneous Perturbation Stochastic Approximation (SPSA) for wind farm control. The SPSA based method is used to optimize the control action of each turbine such that the overall power production of wind farm is maximized. In order to validate our model-free design, a wind farm model with dynamic characterization of wake interaction amongst turbines as presented in [4] is considered. For simplicity, the proposed method is tested on a single row wind farm. Simulation results illustrate that the SPSA based method achieves a maximum total power production with faster convergence as compared to the other existing model-free methods.

Keywords: Stochastic Control and Estimation, Stochastic Modeling and Stochastic Systems Theory, Nonlinear Systems and Control
Abstract: The main purpose of this paper is to investigate stochastic input-to-state stability (SISS) property of a class of stochastic port-Hamiltonian systems (SPHSs). By using particular coordinate transformations and integral action, and feedback compensators derived from the stochastic generalized canonical transformations, we have clarified robustness against matched deterministic disturbances, and matched and unmatched stochastic noises for SPHSs. Thanks to stochastic calculus, we provide sufficient conditions of the noise port of the plant system as well as some design parameters of the controller for achieving SISS property.

Keywords: Stochastic Control and Estimation, System Identification
Abstract: This paper is devoted to moderate deviations for characterizing parameter estimation errors in system identification. Moderate deviations for system identification provide probabilistic error bounds that are beyond laws of large numbers and central limit theorems, and cannot be expressed in terms of large deviations bounds. Such error bounds are crucial in complexity analysis for system identification. Explicit error bounds are derived and their relations to identification complexity analysis are explored. Examples are included to illustrate the ideas and results of this paper.

Keywords: Stochastic Modeling and Stochastic Systems Theory, Linear Systems, Stochastic Control and Estimation
Abstract: A theory of stochastic modeling for stationary processes by general acausal realizations which involve a possibly unstable A matrix leads to a unified derivation of all known results on spectral factorization, Linear Matrix Inequalities and the Riccati inequality/equation in a coherent framework. In particular linear passive systems theory corresponds to stochastic modeling by causal realizations.

Keywords: Biological Systems, Hybrid Systems, Stability
Abstract: The efficiency of biological control as crop protection methodology is deeply impacted by inter- and intra-specific interactions. As such, Allee effects among biocontrol agents such as pest predators should be taken into account to design biological control schemes. We investigate this problem through two models of increasing complexity: the first one concentrates on the predator dynamics whose establishment is hampered by a strong Allee effect and the second one models both predator and prey dynamics. In the latter case, pest eradication is targeted despite diminished predation efficiency when predator density is low. Considering augmentative biological control through periodic impulsive predator releases we show that, when it succeeds, pest control is accelerated in both models by taking rare and large releases over frequent and smaller ones.

Keywords: Biological Systems, Nonlinear Systems and Control
Abstract: The aim of this paper is to analyse the dynamical behaviour of models of gene transcription-translation for the synthesis of RNA polymerase in a cell, with a closed loop from the produced RNA polymerase (end-product) to the transcription step (RNA polymerase is needed to transcribe its own gene). Using monotone system theory we study a reduced version of this model with two variables (mRNA and protein), and show that it has either a single stable trivial equilibrium in (0,0), or has an unstable zero equilibrium and a stable positive one.

Keywords: Biological Systems, Physical Systems Theory, Computations in Systems Theory
Abstract: Controllability of componentwise nonnegative discrete-time linear systems is considered. The key difference here from the well-established positive systems theory is that we permit the case where the input takes negative values, provided that the state remains nonnegative. Such a framework is very natural, moreover necessary, in situations such as population ecology to describe the control actions of harvesting or culling. The present contribution summarises recently published material by the authors and considers a novel application in low-gain PI control.

Keywords: Networked Control Systems, Biological Systems, Stability
Abstract: This paper addresses the stability (the existence and uniqueness of an point attractor) of Boolean networks under the assumption that partial information is available. By considering that a Boolean network is described by the network structure (connection rule among the nodes) and the Boolean function (transition rule of the state), we propose a new stability notion, called the structural stability, where the Boolean function is assumed to be unknown. For the structural stability, we present a necessary and sufficient condition, which is a simple condition characterized by the graph topology representing the network structure. Several examples are given to demonstrate our result.

Keywords: Networked Control Systems, Coding Theory
Abstract: Projective space codes have received an increasing amount of attention since they were shown to be applicable for error correction in the Network setting on Koetter and Kschichang. Since then, many optimally efficient codes have been discovered, some with good decoding algorithms. As in the case of classical coding theory, we can also ask if it is possible to have a list-decoding algorithm. For the most widely studied classes of codes, none so far exists. The problem of list decoding projective space codes was shown to be a problem of intersecting Schubert varieties with varieties defining the code. We examine this more closely and obtain a upper bound for list-decodability for lifted rank-metric codes.

Keywords: Coding Theory, Networked Control Systems
Abstract: An S_q[t,k,v] q-Steiner system is a collection of k-dimensional subspaces of the v-dimensional vector space F_q^v over the finite field F_q with q elements, called blocks, such that each t-dimensional subspace of F_q^v is contained in exactly one block. The smallest admissible parameters for which a q-Steiner system could exist is S_2[2,3,7]. Up to now the issue whether q-Steiner systems with these parameters exist or not is still unsolved. In this paper we investigate the automorphism group of a putative S_2[2,3,7] q-Steiner system. We conclude that in case of existence the automorphism group is cyclic and of order at most 4.

Keywords: Coding Theory, Information Theory, Communication Systems
Abstract: Cyclic orbit codes form a specific class of subspace codes as they play a crucial role for random network coding. We will define the stabilizer subfield of a cyclic orbit code and derive results about the cardinality and distance of the code in terms of the stabilizer subfield.

Keywords: Coding Theory, Networked Control Systems
Abstract: Consider a network G = (V, E), where V denotes the set of vertices in G, and E denotes the set of edges in G. A vertex in G is said to be a source if it is only incident with outgoing edges, and a sink if it is only incident with incoming edges. Often, a source or sink is referred to as a terminal vertex. A non-terminal vertex is said to be a hub if its degree is greater than or equal to 3. In this paper, we are primarily concerned with the minimum number of hubs needed when certain constraints on the flow demand between multiple pairs of sources and sinks are imposed. The flow demand constraints considered in this paper will be in terms of the vertex-cuts between pairs of sources and sinks. This can be justified by a vertex version of the max-flow min-cut theorem [1], which states that for a network with infinite edge-capacity and unit vertex-capacity, the maximum flow between one source and one sink is equal to the minimum vertex-cut between them. Here, we remark that with appropriately modified setup, our results can be stated in terms of edge-cuts as well.

Keywords: Systems on Graphs, Optimization : Theory and Algorithms, Mathematical Theory of Networks and Circuits
Abstract: Computing meaningful clusters of nodes is crucial to analyze large networks. In this paper, we present a pairwise node similarity measure that allows to extract roles, i.e. group of nodes sharing similar flow patterns within a network. We propose a low rank iterative scheme to approximate the similarity measure for very large networks. Finally, we show that our low rank similarity score successfully extracts the different roles in random graphs and that its performances are similar to the full rank pairwise similarity measure.

Keywords: Systems on Graphs, Physical Systems Theory, Mathematical Theory of Networks and Circuits
Abstract: A method for model order reduction of a large class of physical network systems is discussed. The method is based on clustering of the vertices of the underlying graph, and yields a reduced order model within the same class. This is illustrated on a number of physical examples, including mass-spring-damper systems, chemical reaction networks, and hydraulic networks.

Keywords: Networked Control Systems, Large Scale Systems, Systems on Graphs
Abstract: The paper presents a system theoretic analysis of network-of-networks which are formed from smaller factor networks via graph Kronecker products. We provide a compositional framework for extending trajectories, stabilizability, controllability and observability of the factor networks to that of the composite network-of-networks. We proceed to delve into the effectiveness of applying these composite features to approximate Kronecker product networks. Examples are provided throughout to illustrate the results.

Keywords: Systems on Graphs, Networked Control Systems, Nonlinear Systems and Control
Abstract: We introduce a new class of (dynamical) systems that inherently capture cascading effects (viewed as consequential effects) and are naturally amenable to combinations. We develop an axiomatic general theory around those systems, and guide the endeavor towards an understanding of cascading failure. The theory evolves as an interplay of lattices and fixed points, and its results may be instantiated to commonly studied models of cascade effects.

We characterize the systems through their fixed points, and equip them with two operators. We uncover properties of the operators, and express global systems through combinations of local systems. We enhance the theory with a notion of failure, and understand the class of shocks inducing a system to failure. We develop a notion of mu-rank to capture the energy of a system, and understand the minimal amount of effort required to fail a system, termed resilience. We deduce a dual notion of fragility and show that the combination of systems sets a limit on the amount of fragility inherited.

Keywords: Optimal Control, Operator Theoretic Methods in Systems Theory
Abstract: We survey two recent results by the author cite{Vigeral,SoVi2} (one in collaboration with Sylvain Sorin) that give examples of zero-sum stochastic games with four states, compact action sets for each player, and continuous payoff and transition functions, such that the discounted value does not converge as the discount factor tends to 0, and the value of the n-stage game does not converge as n goes to infinity. In particular we explain how the oscillations of transition functions can lead to oscillations of the values.

Keywords: Operator Theoretic Methods in Systems Theory, Optimization : Theory and Algorithms, Nonlinear Systems and Control
Abstract: Shapley operators are the dynamic programming operators of zero-sum stochastic games, they can be characterized as order preserving maps commuting with the addition of a constant. We study a subclass of Shapley operators which are characterized by the property of commuting with the multiplication by a positive constant. We call them payment-free, as they arise in the study of recursive games, in which the payment only occurs when the game stops. They also arise in the study of structural properties of parametric mean payoff games (the transition probabilities are fixed, not the transition payoffs) with finite action spaces and perfect information: their fixed point set can be shown to be all the possible mean payoff vectors of such games. A basic problem is to check whether the fixed point set of such an operator is trivial (reduced to the multiples of the unit vector), and more precisely to determine its characteristics, for instance decide whether there is a fixed point with a prescribed argmin. Yang and Zhao showed (in Systems and Control Letters, 2004) that the former problem is co-NP-complete for deterministic games. We show that the latter problem is polynomial, and deduce that the former problem remains in co-NP for stochastic games. The proof relies on the construction of a Galois connection between faces of the hypercube that are invariant by the operator, and on a reduction to a reachability problem in a directed hypergraph.

Keywords: Nonlinear Systems and Control, Physical Systems Theory, Optimal Control
Abstract: Two-point boundary-value problems for conservative systems are studied in the context of the stationary action principle. In particular, we consider the case where the initial boundary condition is the system position, and the terminal boundary condition may be a combination of position and velocity data. The emphasis is on the N-body problem under gravitation. When the duration is sufficiently short, one may use a differential game formulation to obtain a fundamental solution, where for specific initial position and terminal data, one obtains the particular solution via a min-plus convolution of a function related to the terminal data and another function associated with the fundamental solution. That latter function is obtained by minimization of a parameterized linear functional over a convex set. This convex set is the fundamental solution. For longer duration problems, one takes a stationary point rather than a minimum.

Keywords: Optimal Control, Computations in Systems Theory, Computational Control
Abstract: Recently, a max-plus dual space fundamental solution for a class of difference Riccati equations (DREs) has been developed. This fundamental solution is represented in terms of the kernel of a specific max-plus linear operator that plays the role of the dynamic programming evolution operator in a max-plus dual space. In order to fully understand connections between this dual space fundamental solution and evolution of the value function of the underlying optimal control problem, a new max-plus primal space fundamental solution for the same class of difference Riccati equations is presented. Connections and commutation results between this new primal space fundamental solution and the recently developed dual space fundamental solution are developed.

Keywords: Optimization : Theory and Algorithms
Abstract: A combinatorial simplex algorithm is an instance of the simplex method in which the pivoting depends on combinatorial data only. We show that any algorithm of this kind admits a tropical analogue which can be used to solve mean payoff games. Moreover, any combinatorial simplex algorithm with a strongly polynomial complexity (the existence of such an algorithm is open) would provide in this way a strongly polynomial algorithm solving mean payoff games (all the arithmetic operations being performed on data polynomially bounded in the size of the input, in particular). Mean payoff games are known to be in NP and co-NP; whether they can be solved in polynomial time is an open problem. Our algorithm relies on a tropical implementation of the simplex method over a real closed field of Hahn series. One of the key ingredients is a new scheme for symbolic perturbation which allows us to lift an arbitrary mean payoff game instance into a non-degenerate linear program over Hahn series.

Keywords: Robust and H-Infinity Control
Abstract: In a recent series of papers concrete procedures were derived to compute a stable rational matrix solution to the Leech equation with rational matrix data. In one of these papers the procedure involved a specific Douglas-Shapiro-Shields(DSS) factorization. In the present paper it is shown that one can take a class of DSS factorizations as the starting point of the procedure, leading to solutions which may have a lower McMillan degree as the one obtained in the original procedure.

Keywords: Robust and H-Infinity Control, Infinite Dimensional Systems Theory, Linear Systems
Abstract: The need for measuring distances between control systems is basic in control theory. A desirable metric for measuring closeness of control systems was introduced by Glenn Vinnicombe in 1993, and is called the nu-metric. Vinnicombe only considered linear control systems described by constant coefficient ODEs, but left open the question of handling PDEs or delay-differential ODEs. This issue was addressed in several recent articles. We give a survey of this new nu-metric in control theory (from an article by Ball and Sasane, called "Extension of the nu-metric", Complex Anal. Oper. Theory, 2012), and related developments which also covers such partial/delay-differential equations.

Keywords: Operator Theoretic Methods in Systems Theory, Multidimensional Systems
Abstract: We discuss transfer-function realization for multivariable holomorphic functions mapping the unit polydisk or the right polyhalfplane into the operator analogue of either the unit disk or the right halfplane (Schur/Herglotz functions over either the unit polydisk or the right polyhalfplane) which satisfy the appropriate stronger contractive/positive real part condition for the values of these functions on commutative tuples of strict contractions/strictly accretive operators (Schur--Agler/Herglotz--Agler functions over either the unit polydisk or the right polyhalfplane). As originally shown by Agler, the first case (polydisk to disk) can be solved via unitary extensions of a partially defined isometry constructed in a canonical way from a kernel decomposition for the function (the lurking-isometry method). We show how a geometric reformulation of the lurking-isometry method (embedding of a given isotropic subspace of a Krein space into a Lagrangian subspace---the lurking isotropic-subspace method) can be used to handle the second two cases (polydisk to halfplane and polyhalfplane to disk), as well as the last case (polyhalfplane to halfplane) if an additional growth condition at infinity is imposed. For the general fourth case, we show how a linear-fractional-transformation change of variable can be used to arrive at the appropriate symmetrized nonhomogeneous Bessmertnyu{i} long-resolvent realization. We also indicate how this last result recovers the classical integral representation formula for scalar-valued holomorphic functions mapping the right halfplane into itself.

Keywords: Stability, Delay Systems, Computations in Systems Theory
Abstract: The stability of equilibria is a question of fundamental importance in the analysis of dynamical systems. In many practical situations, time-delays are an unavoidable part of a system description. Notwithstanding the substantial body of work done on the analysis of linear time-invariant (LTI) delay systems many authors continue to work on developing implementable tests for the stability of such systems. Some of these approaches rely on Lyapunov functionals, while others proceed by analysing the spectrum of the system: namely checking whether all solutions of the characteristic equation lie in the open left half of the complex plane. In this talk, we will describe an approach to this latter question that uses a recently developed generalised Budan Fourier algorithm. We will also compare our method to recent approaches proposed in the literature and highlight some possible directions for future research.

Keywords: Operator Theoretic Methods in Systems Theory
Abstract: Positive and strongly positive completions of partial matrices with entries in a finite dimensional real C*-algebras are studied. Some well known results on completions of partial real and complex matrices are extended to this setting. As a particular case, existence of positive and strictly positive Toeplitz completions of partial Toeplitz matrices is proved, under natural hypotheses. Open problems pertaining to positive completions of partial matrices over infinite dimensional real C*-algebras are stated.

Keywords: Signal Processing, Multidimensional Systems, Optimization : Theory and Algorithms
Abstract: This paper considers the problem of low rank approximations of multi-linear functionals. Multilinear functionals are generally referred to as tensors and provide a natural object of study in multi-dimensional signal and system analysis. We propose a decomposition of an arbitrary order N tensor by means of mutual orthogonal rank 1 tensors. In a sense, this decomposition generalizes the standard singular value decomposition of matrices to multi-variable data objects. It is shown how this decomposition can be used for approximating arbitrary tensors.

Keywords: Algebraic Systems Theory, Computational Control, Nonlinear Systems and Control
Abstract: The motivation for model order reduction (MOR) stems from an inevitable fact about modern computing. The need for accurate modeling of physical phenomena often leads to large-scale dynamical systems that require long simulation times and large data storage. For instance, one such example is provided by the discretization of partial differential equations over fine grids, which leads to large-scale systems of ordinary differential equations. In these settings, MOR seeks models of low dimension that accurately capture the input-output behavior of the large-scale system while requiring only a fraction of the large-scale simulation time and storage.

A powerful and versatile approach to MOR is provided by the Loewner framework for rational interpolation. This approach was introduced by Antoulas and Anderson in 1986 and a major advance was made by Mayo and Antoulas in 2007. It has since been successfully applied to two main categories of systems: (a) linear systems with multiple inputs and multiple outputs, and (b) linear parametric systems. It is the purpose of this talk to present the most recent extension of the Loewner framework to classes of non-linear differential-algebraic systems, namely bilinear and quadratic non-linear differential algebraic systems.

Keywords: Computations in Systems Theory, Linear Systems, Delay Systems
Abstract: In this paper we address the problem of model order reduction of coupled systems. Using moment matching techniques, we compute families of parameterized reduced order models that achieve moment matching and preserve the block structure of the to-be-reduced model. In particular we adapt the theory to the particular case of the implementation of a specific control law. The result is a reduced order controller with a reduced number of blocks thatexhibit properties similar to the given control law.

Keywords: Linear Systems, Stochastic Modeling and Stochastic Systems Theory, Computations in Systems Theory
Abstract: We consider model order reduction of stochastic linear systems by balanced truncation. Two types of Gramians are suggested, both satisfying generalized Lyapunov equations. The first is motivated by energy functionals, the second is taylored to yield an error bound for the truncated system.

Keywords: Nonlinear Systems and Control
Abstract: This paper is concerned with model order reduc- tion based on balanced realization for symmetric nonlinear systems. A new notion of symmetry for nonlinear systems was characterized recently. It plays an important role in linear systems theory and is expected to provide new insights to nonlinear systems. In this paper, we provide a novel framework of balanced realization for this class of systems and apply it for model order reduction preserving symmetry.

Keywords: Multidimensional Systems, Linear Systems, Applications of Algebraic and Differential Geometry in Systems Theory
Abstract: We investigate the fault detection and isolation (FDI) problem for discrete-time linear repetitive processes (DLRP) using a geometric approach. We propose a 2-D model for these systems that incorporates the failure description. Based on this model, we formulate the FDI problem in geometric language and state sufficient conditions for solvability of the problem. We also develop a FDI procedure based on an asymptotic observer of the state.

Keywords: Linear Systems
Abstract: Positive state-space models describe large classes of processes in econometry, epidemiology, biology, ecology, chemistry, hydraulics and logistics. These models satisfy strict algebraic conditions that can be directly fulfilled when the models are obtained by means of traditional modeling techniques by selecting the state variables in a "natural" way i.e. by associating a well-defined physical meaning to every variable. The situation is more critical when positive state-space models must be obtained by means of realization techniques from transfer functions since, in this case, the fulfillment of positivity conditions could call for the introduction of spurious dynamics and non minimal parameterizations. A possible alternative consists in using quasi-positive models; this paper discusses the pros and cons of these solutions.

Keywords: Linear Systems
Abstract: This paper is concerned with the parameterisation of basis matrices and the simultaneous computation of friends of the output nulling subspaces Vstar, Vstargood and Rstar with the assignment of the corresponding inner and outer closed-loop free eigenstructure. Differently from the classical techniques presented in the literature so far on this topic, which are based on the standard pole assignment algorithms and are therefore applicable only in the non-defective case, the method presented in this paper can be applied in the case of closed-loop eigenvalues with arbitrary multiplicity.

Keywords: Linear Systems, Multidimensional Systems, Algebraic Systems Theory
Abstract: In this paper, we consider losslessness and dissipativeness for two-dimensional (2-D) behaviors using quadratic difference forms (QDFs) based on the behavioral approach. We derive necessary and sufficient conditions for the losslessness and dissipativeness in terms of dissipation equality and a certain frequency domain inequality as a main result.

Keywords: Linear Systems, Algebraic Systems Theory
Abstract: Control and observation are in general not dual problems. In this talk I provide examples of situations where the two problems are not dual as well as a discussion of the parts of the two problems that are dual. I propose two new general notions of duality, one internal to an observer-based feedback control loop, and one external duality concept for plant/controller resp. observer/plant pairs.

Keywords: Linear Systems
Abstract: This paper focuses on the preservation of the period in the periodic Kalman canonical decomposition with complex-valued coefficients. This clarifies essential difference between linear periodic systems with real-valued coefficients and with complex-valued coefficients.

Keywords: Quantum Control, Stability, Feedback Control Systems
Abstract: The goal of this mini-course is to present some of the particular features of the quantum systems that one needs to study with extra care while discussing system theory concepts such as state and parameter estimation, feedback control, active and passive stabilization. More precisely we will overview the following subjects: - Two-level quantum system, Quantum harmonic oscillator, composite spin-spring system, Jaynes-Cummings model. - Discrete-time dynamical models and generalized measurements: hidden Markov chain and Kraus maps, state estimation via quantum filtering and Bayes law. - Stabilization scheme relying on active measurement-based feedback and control Lyapunov techniques. - Dissipation and continuous-time Lindblad master equation. - Stabilization scheme relying on passive reservoir (dissipation) engineering.

Keywords: System Identification, Optimization : Theory and Algorithms, Computations in Systems Theory
Abstract: