Keywords: Optimal Control, Optimization : Theory and Algorithms, Algebraic Systems Theory
Abstract: Optimal control problems with oscillation (chattering controls) and concentration (impulsive controls) can have integral performance criteria such that concentration of the control signal occurs at a discontinuity of the state signal. Techniques from functional analysis (anisotropic parametrized measures) are applied to give a precise meaning of the integral cost and to allow for the sound application of numerical methods. In the oral presentation of this work, we show how this can be combined with the Lasserre hierarchy of semidefinite programming relaxations. This includes in particular the use of compactification techniques allowing for unbounded time, state and control.

Keywords: Applications of Algebraic and Differential Geometry in Systems Theory, Operator Theoretic Methods in Systems Theory, Optimization : Theory and Algorithms
Abstract: The talk will be one of 3 coordinated talks (Helton, Klep, Volčič) concerning inequalities and equalities for functions having matrix variables. Matrix multiplication does not commute, so the functions we study are typically noncommutative (nc) polynomials or rational functions. Such mathematics is central to linear systems problems which are specified entirely by a signal flow diagram and L^2 performance specs on signals. The reason being: a system connection law amounts to an algebraic operation on nc quantities, while L^2 performance constraints, through use of quadratic "storage functions", convert to matrix inequalities.

Studies of commutative polynomial inequalities are central to classical mathematics and come under the heading of Real Algebraic Geometry (RAG). Hence linear systems problems cry out for development of a noncommutative RAG. This field has developed greatly over the last decade, pushed partly by systems motivation and greatly by unfolding the parallels to classical RAG.

Keywords: Operator Theoretic Methods in Systems Theory, Optimization : Theory and Algorithms, Multidimensional Systems
Abstract: Recent years saw a remarkable emergence of several general theories that are matrix or free noncommutative analogues of well established classical theories in the commutative setting. Since matrices are natural variables in many if not most problems in system theory, it is not surprising that these theories are often related to systems and control. In this talk (which is based on joint work with J. Ball and G. Marx) I will discuss a solution to the Nevanlinna-Pick interpolation problem for noncommutative functions on a matrix convex set. It is based on a general interpolation and realization theorem for noncommutative functions on generalized balls (balls defined by an arbitrary operator valued nc function) together with the description of a matrix convex set by an (in general infinite dimensional) linear matrix inequality.

Keywords: Optimization : Theory and Algorithms
Abstract: Hyperbolic polynomials are multivariate generalizations of characteristic polynomials of real sym- metric matrices. Hyperbolic programming is the problem of computing the infimum of a linear function when restricted to the hyperbolicity cone of a hyperbolic poly- nomial, a generalization of semidefinite programming. In the talk I will discuss an approach based on symbolic computation, relying on the multiplicity structure of the algebraic boundary of the cone, without the assumption of determinantal representability of the given polynomial. This allows us to design exact algorithms able to certify the multiplicity of the solution and the optimal value of the linear function. This is joint work with Daniel Plaumann, University of Dortmund. In the rest of the talk, I will discuss a work in progress, joint with Rainer Sinn, FU Berlin.

Keywords: Nonlinear Systems and Control, Networked Control Systems, Stability
Abstract: This paper studies the problem of constructing a zero-dynamics attack on “nonlinear and uncertain” cyber-physical systems being of non-minimum phase, particularly for the case of the quadruple-tank process. In most of the previous works, the zero-dynamics attack is usually designed by linearizing the nonlinear system at an operating point. As a consequence, the stealthiness of the attack may be easily violated whenever the plant has even small model uncertainty or the state trajectory under the attack moves too far from the operating point (so that the linearization is not accurate enough). Without relying on the linearization of the plant at all, in this paper we propose a nonlinear zero-dynamics attack based on the Byrnes-Isidori normal form representation. In particular, it is shown via the Lyapunov analysis that the proposed attack for the quadruple-tank process always remains stealthy until some of the tanks become empty or overflow even in the presence of small parametric uncertainty, which cannot be ensured by the existing methods. Simulation results are presented to verify the performance of the proposed attack.

Keywords: Linear Systems, Coding Theory
Abstract: We address the problem of attack detection and attack correction for multi-output discrete-time linear time-invariant systems under sensor attacks. More specifically, we focus on the situation where adversarial attack signals are added to some of the system’s output signals. A `security index' is defined to characterize the vulnerability of a system against such sensor attacks and methods are given to calculate this index for various system representations. Algorithms are presented to detect and correct for sensor attacks on a noise-free linear system.

Keywords: Stochastic Control and Estimation, Signal Processing
Abstract: Security of networked Bayesian source localization algorithms is analysed in this paper. The Bayesian estimators construct the probability density function of the location of the source from quantized measurements. Understanding fundamental limits in the performance of an adversary for manipulating the posterior of the Bayesian estimator is the main focus of the paper. The analysis is performed for cases where the estimator is not aware of the presence of the adversary. The results are then generalized to the case where the estimator is aware of the attacker.

Keywords: Signal Processing
Abstract: We are interested in the secure estimation problem of a linear time-varying Gaussian process. m sensors are deployed to measure the process state and p out of m sensors might undergo integrity attack, which means their measurements can be arbitrarily manipulated by attackers. We first show that the Kalman filter can be decomposed into m local estimators and then summed up to obtain the Kalman estimate. Then we show a least square interpretation to the fusion process and based on which a convex optimization based secure estimation scheme is proposed. The secure estimation algorithm guarantees that when all the sensors are benign, the secure estimate coincides with the Kalman filter. When less than half of the sensors are compromised, the secure estimation scheme can still generate an estimate with bounded error. In the end, numerical simulations are conducted to verify the effectiveness of the proposed algorithm.

Keywords: Nonlinear Filtering and Estimation, Information Theory
Abstract: This paper studies the design of an optimal privacy-aware estimator for a single sensor estimation problem. The sensor’s measurement is a (possibly non-linear) function of a private random variable, a public random variable and the measurement noise. Both public and private random variables are assumed to be discrete valued, and the measurement noise is arbitrarily distributed. The sensor provides an estimate of the public random variable for an untrusted entity, named the cloud. The objective is to design the estimator of the public random variable such that a level of privacy for the private random variable is guaranteed. The privacy metric is defined as the discrete conditional entropy of the private random variable given the output of the estimator. A binary loss function is considered for the estimation of the public random variable. The optimal estimator design problem is posed as the minimization of the average loss function subject to a constraint on the privacy level of the private random variable. It is shown that the objective function is linear and the privacy constraint is convex in the optimization variables. Thus, the optimal privacy-aware estimator can be designed by solving an infinite dimensional convex optimization problem.

Keywords: Artificial Intelligence
Abstract: Anomaly detection discovers regular patterns in unlabeled data and identifies the non-conforming data points, which in some cases are the result of malicious attacks by adversaries. Learners such as One-Class Support Vector Machines (OCSVMs) have been successfully used in anomaly detection, yet their performance may degrade significantly in adversarial conditions such as integrity attacks. This work focuses on integrity attacks, where the adversary distorts the training data in order to successfully avoid detection during evaluation. This paper presents a unique combination of anomaly detection using (1) OCSVMs in the presence of adversaries who distort training data in a targeted manner and (2) nonlinear randomized kernel methods, which facilitate computational and conceptual simplification through dimension reduction. We theoretically analyze the effects of adversarial distortions on the separating margin of OCSVMs and provide supporting empirical evidence. The proposed approach introduces a layer of uncertainty on top of the OCSVM learner, making it challenging for the adversary to guess the specific configuration of the learner.

Keywords: Nonlinear Systems and Control, Control of Distributed Parameter Systems, Stability
Abstract: We introduce a monotonicity-based method for studying input-to-state stability (ISS) of nonlinear parabolic equations with boundary inputs. We first show that a monotone control system is ISS if and only if it is ISS w.r.t. constant inputs. Then we show by means of classical maximum principles that nonlinear parabolic equations with boundary disturbances are monotone control systems.

With these two facts, we establish that ISS of the original nonlinear parabolic PDE with constant textit{boundary disturbances} is equivalent to ISS of a closely related nonlinear parabolic PDE with constant textit{distributed disturbances} and zero boundary condition. The last problem is conceptually much simpler and can be handled by means of various recently developed techniques.

Keywords: Control of Distributed Parameter Systems, Infinite Dimensional Systems Theory
Abstract: The guidance of mobile sensor-agents used in the estimation of spatially distributed processes often neglects the effects of the process itself on the health and reliability of the sensor. When the spatially distributed process negatively impacts the health status and functionality of the sensing devices then the standard gradient-based sensor guidance will accelerate the demise of the sensor and negatively impact the performance of the estimator. A mixed policy that takes into account the cumulative effects of the environment on the health status of the sensing devices will provide a compromise between sensor functionality and estimator performance. This work considers a mixed policy where initially an information-sensitive guidance policy is implemented. Using performance-based criteria, the sensor guidance switches to an information-neutral policy and when certain thresholds pertaining to the life expectancy of the sensing devices are exceeded, then the guidance policy switches to an information-averse policy. The proposed mixed-guidance policy is demonstrated with a 2D advection-diffusion partial differential equation.

Keywords: Infinite Dimensional Systems Theory, Hybrid Systems, Control of Distributed Parameter Systems
Abstract: In this paper we consider some issues in modeling and approximation of composite control systems defined by coupled partial differential equations and ordinary differential equations. Although these systems are motivated by applications to thermal management systems, the fundamental issues occur in many hybrid systems of subsystems. We establish a well-posedness result and provide a short discussion of how the problem formulation can impact the choice of a specific numerical approximation. In particular, the form of the coupling can impact the choice of finite volume, finite element or higher order schemes and their convergence properties. Examples are given to illustrate the ideas.

Keywords: Robust and H-Infinity Control, Infinite Dimensional Systems Theory, Large Scale Systems
Abstract: We address H-infinity control of a class of systems governed by parabolic partial differential equations (PDEs). Closed-form expressions for optimal state feedback with full information are stated. Unlike traditional methods for H-infinity-controller synthesis, no iteration is needed to obtain the optimal controller and a closed form expression for the controller is readily calculated. These results are used in improvements to general purpose algorithms for calculating H-infinity control for PDEs and large-scale systems.

Keywords: Optimal Control, Nonlinear Systems and Control, Control of Distributed Parameter Systems
Abstract: We consider an approximating control design for optimal mixing of a non-dissipative scalar field in unsteady Stokes flows. The objective of our approach is to achieve optimal mixing at a given final time, via an active control of the flow velocity through boundary inputs. Due to the zero diffusivity of the scalar field, establishing the well-posedness of its Gaeaux derivative requires high regularity on velocity field. This condition results in the need to penalize the time derivative of the boundary control in the cost functional. As a result, the optimality system becomes difficult to solve. Our current approximating approach will provide a more transparent optimality system, with the set of admissible controls being L2 in both time and space. This is achieved by first introducing a small diffusivity to the scalar equation and then establishing a rigorous analysis of convergence of the approximating control problem to the original one as the diffusivity approaches to zero. Uniqueness of the optimal solution is obtained for the two dimensional case.

Keywords: Infinite Dimensional Systems Theory, Stability, Control of Distributed Parameter Systems
Abstract: We study input-to-state stability of bilinear control system with possibly unbounded control operator and unbounded bilinearity. We show that every internally exponentially stable bilinear control system is integral input-to-state stable. An application to the bilinearly controlled Fokker-Planck equation is given.

Keywords: Delay Systems, Infinite Dimensional Systems Theory, Stability
Abstract: We consider distributed stabilization of 1-D Korteweg-de Vries-Burgers (KdVB) equation in the presence of constant input delay. The delay may be uncertain, but bounded by a known upper bound. On the basis of spatially distributed point measurements, we design a regionally stabilizing controller applied through distributed in space shape functions. The existing Lyapunov-Krasovskii functionals for heat equation that depend on the state derivative are not applicable to KdVB equation because of the third spatial derivative. We suggest a new Lyapunov-Krasovskii functional that leads to regional stability conditions of the closed-loop system in terms of linear matrix inequalities (LMIs). By solving these LMIs, an upper bound on delay that preserves regional stability can be found, together with an estimate on the set of initial conditions starting from which the state trajectories of the system are exponentially converging to zero. This estimate includes a priori Lyapunov- based bounds on the solutions of the open-loop system on the initial time interval of the length of delay. A numerical example illustrates the efficiency of the method.

Keywords: Control of Distributed Parameter Systems, Infinite Dimensional Systems Theory, Feedback Control Systems
Abstract: In this paper, a robust output regulation problem is considered for general infinite-dimensional systems. More precisely, for the design of the regulator, the controlled output y(t) and the reference signal y_r(t) are assumed to be available. One mode of the designed regulator is driven by the tracking error e_y(t) and is used for the reference signal tracking and disturbance rejection, while another mode driven by y(t) is used to address the plant stabilization. In particular, simpler and sufficient conditions are provided for injection and feedback gains. The robustness is achieved based on Internal Model Principle and verified for non-destabilizing system with model uncertainties and simultaneous satisfaction of {cal G}-conditions is guaranteed. An uncertain hyperbolic PDE system is used to demonstrate the paper results.

Keywords: Control of Distributed Parameter Systems
Abstract: This paper proposes a new method for the estimation of an in-domain source of a first order time-varying hyperbolic system. This system description is used to model the heat transfer in solar collectors where the source depends on the solar irradiance intensity and the collector's properties. The proposed method is based on the modulating functions approach. With this, the estimation problem is transformed into an algebraic system of equations. A detailed derivation of the resulting source estimation equations is given. Necessary conditions on the modulating functions for the solvability of this system of equations are also provided. Different test cases are presented to assess the performance of the proposed method.

Keywords: Operator Theoretic Methods in Systems Theory, Signal Processing, Infinite Dimensional Systems Theory
Abstract: The paper presents a new algorithm which ``differentiates'' noisy signals with values in a Hilbert space. The algorithm provides an iterative estimation procedure which converges to the weighted pseudoinverse of a compact operator with (possibly unbounded) linear weighting operator for a wide class of signals. The case of differentiation of 1D signals corrupted by a random noise with uncertain but bounded second moments is studied in details.

Keywords: Stability, Infinite Dimensional Systems Theory, Optimal Control
Abstract: We analyze the sensitivity of the extremal equations that arise when concluding first order optimality conditions for time dependent optimization problems. More specifically, we consider parabolic PDEs with a linear quadratic performance criterion. We prove the solutions boundedness with respect to the right-hand side of the first order optimality condition which includes initial data. As a consequence, it can be shown that the influence of a perturbation at a certain time decays exponentially in the temporal distance to the time of perturbation. Moreover, a quantitative turnpike theorem can be derived.

Keywords: Control of Distributed Parameter Systems, Robust and H-Infinity Control, Infinite Dimensional Systems Theory
Abstract: For many problems of estimation there are disturbances other than Gaussian noise. Also, for many applications only some aspect of the state needs to be estimated, not the whole state. In an H_infty estimation approach the aim is to reduce the estimation error over all disturbances. A derivation of H_infty-output estimation in the infinite-dimensional setting is provided. A framework for calculation using finite-dimensional approximations is described. Output estimation is compared with a Kalman filter for an example with disturbances.

Keywords: Networked Control Systems, Optimization : Theory and Algorithms, Nonlinear Systems and Control
Abstract: In this paper, we study an optimal steady-state regulation problem of a multi-agent network with agents of high-order dynamics. This problem can be taken as an extended version of the existing steady-state regulation problem for higher-order agents but with a distributed optimization requirement, where the state variables of agents are driven to the optimal solution of a global cost function in a cooperative way. To solve this problem, we first construct an optimal signal generator, and then employed an embedded control scheme by embedding the generator in the feedback loop to obtain the final distributed control laws. Several kinds of control laws are given with different information under standard assumptions to show the effectiveness of this embedded technique.

Keywords: Optimization : Theory and Algorithms
Abstract: This paper studies the distributed computation of a linear matrix equation over multi-agent networks from an optimization perspective. In the problem formulation, each agent only knows its own matrices and communicates with its neighbors. Then, a distributed continuous-time algorithm is proposed for solving its least squares solution from a distributed constrained optimization viewpoint. With help of the Lyapunov stability and semi-stability analysis, we prove the convergence of the algorithm for any initial condition.

Keywords: Large Scale Systems, Optimization : Theory and Algorithms, Linear Systems
Abstract: In this paper, a distributed resource allocation problem is investigated for disturbed continuous-time multi-agent systems, where the communication among agents is depicted by strongly connected and weight-balanced digraphs. A distributed continuous-time algorithm is developed based on internal model principle, and the algorithm can ensure the multi-agent systems exponentially converge to the optimal allocation even in the presence of external disturbance.

Keywords: Networked Control Systems, Mathematical Theory of Networks and Circuits, Linear Systems
Abstract: This paper considers containment control problem for second-order multi-agent systems with disturbances generated from linear exosystems and nonlinear exosystems, respectively, under independent position and velocity topologies. Firstly, for the case that the disturbances are generated from linear exosystems, linear disturbance observer and control protocol are proposed for each follower using the relative states among neighboring agents; Secondly, for the case that the disturbances are generated from nonlinear exosystems, based on dynamic-gain technique, nonlinear disturbance observer and control protocol are provided, under which all the followers are driven to the convex hull spanned by the leaders.

Keywords: Optimization : Theory and Algorithms, Nonlinear Systems and Control
Abstract: We presents a continuous-time multi-agent system for distributed saddle-point seeking subject to bounded constraints. In the system, two groups of agents are employed for computing the two state vectors in a saddle-point, respectively. Each agent seeks for consensus with the agents in the same group, and simultaneously optimize its local objective functions by competing with the agents in the opposite group. A projection operator is introduced into the dynamics of each agent for dealing with bounded constraints. Two types of local interactions are considered. First, we consider proportional consensus protocols only. In this case, the gradient term of each agent is equipped with a disminishing gain. Second, we consider proportional-integral consensus protocols but without diminishing gain. In both cases, it is shown that the proposed systems can converge to the saddle-point set of a convex-concave function provided the communication topology is undirected and connected.

Keywords: Communication Systems, Optimization : Theory and Algorithms
Abstract: To utilize the limited licensed frequency spectrum sufficiently, small cellular systems (SCY)s have been deployed to realize the spectrum reuse on licensed bands between the small base stations (SBS)s and macro base station (MBS). To further assist the transmission on licensed bands, the SBSs are also allowed to share the unlicensed bands with Wi-Fi systems, which is named as LTE-U based SBS. One of the critical challenges for the LTE-U system is how to deal with impact to the existing Wi-Fi users and ensure harmonious coexistence. In this paper, we propose one distributed and one decentralized resource management scheme for the LTE-U based SCYs to maximize the SE on licensed bands and guarantee harmonious coexistence with Wi-Fi systems on unlicensed bands, while satisfying SUEs' and MUEs' quality of service requirement. For the distributed scheme, there is not information exchange between the MBS and SBSs. On the contrary, for the decentralized scheme, there exist a limited interaction between the MBS and SBSs based on the backhaul resource constraint.

Keywords: Large Scale Systems, Systems on Graphs, Infinite Dimensional Systems Theory
Abstract: To achieve control objectives for extremely complex and very large scale networks using standard methods is essentially intractable. In this paper, we exploit our previously proposed graphon control methodology to approximately regulate complex network systems by the use of graphon theory and the theory of infinite dimensional systems. Conditions on the exact controllability and the approximate controllability on graphon dynamical systems are investigated. Approximation schemes to approximately regulate large network systems with linear quadratic cost are developed. The convergence properties of the approximation schemes are proved. Finally, two simulations of the application of graphon-LQR control to complex networks are presented.

Keywords: Large Scale Systems, Networked Control Systems, Linear Systems
Abstract: In this paper, we propose an index of the localizability for the distributed design of local controllers. We consider the case where a large-scale system is controlled by a retrofit controller. The retrofit controller is a local plug-in controller such that, rather than an entire system model, only a model of the subsystem of interest is required for the controller design. In addition, the retrofit controller guarantees robust stability in the sense that the control system is stable for any variations of subsystems other than the subsystem of interest, as long as the entire system before retrofit control is stable. The localizability index is defined as the Hcal_{infty} norm of an error system defined based on the isolation of the subsystem of interest from the entire system. The proposed index measures how much the control performance of the subsystem of interest is invariant with respect to the variation of subsystems other than the subsystem of interest. Then, we show that the localizability index is estimated from only the model parameters of the subsystem of interest. Finally, a retrofit controller placement problem for power systems is analyzed by the localizability index.

Keywords: Large Scale Systems, Stochastic Modeling and Stochastic Systems Theory, Systems on Graphs
Abstract: In this work, we focus on strategies to influence the opinion dynamics of a well-connected society. We propose a generalization of the popular voter model. This variant of the voter model can model a wide range of individuals including strong-willed individuals whose opinion evolution is independent of their neighbors as well as conformist individuals who tend to adopt the opinion of the majority. Motivated by political campaigns which aim to influence opinion dynamics by the end of a fixed deadline, we focus on influencing strategies for finite time horizons. We characterize the nature of the optimal influencing strategies as a function of the nature of individuals forming the society. Using this, we show that for a society consisting of strong-willed individuals, the optimal strategy is to influence towards the end of the finite time horizon, whereas, for a society consisting of individuals who are affected by their peers, it could be optimal to influence in the initial phase of the finite time horizon.

Keywords: Large Scale Systems, Mathematical Theory of Networks and Circuits, Mechanical Systems
Abstract: Electro-mechanical oscillations in power systems are typically controlled by simple decentralised controllers. We derive a formula for computing the delay margin of such controllers when the power system is represented by a simple mechanical network. This formula reveals a clear trade-off between system damping, inertia, and robustness to delays. In particular, it shows that reducing system inertia, which is a common consequence of increased renewable generation, can reduce robustness to unmodelled dynamics.

Keywords: Large Scale Systems
Abstract: This extended abstract presents an asynchronous algorithm to solve a class of distributed parametric learning problems in a multi-agent network. Each agent acquires its private streaming data to establish a local learning model at times determined by its own clock. It is not assumed that the agents’ clocks are synchronized or that the “event times” at which any one agent updates its variables are evenly spaced. The goal is for each agent to converge to a common global learning model, defined as the average of all local ones, by communicating only with its neighbors. It is shown that the algorithm solves the asynchronous distributed parametric learning problems almost surely, or at least with high probability, for any repeatedly jointly strongly connected sequence of neighbor graphs defined on the merged sequence of all agents’ event times.

Keywords: Operator Theoretic Methods in Systems Theory, Networked Control Systems, Large Scale Systems
Abstract: In this note we introduce a new approach for modeling and analyzing collective behavior of a group of agents using moments. We represent the group of agents by their moments and show how the dynamics of the moments can be modeled. Then approximate trajectories of the moments can be computed and an inverse problem is solved to recover macro-scale properties of the group of agents. To illustrate the theory, a numerical example with interactions between the agents is given.

Keywords: Systems on Graphs, Physical Systems Theory, Infinite Dimensional Systems Theory
Abstract: Port-Hamiltonian systems theory has proven to be a versatile and insightful framework for the modeling, simulation and control of general (finite- and infinite-dimensional) multi-physics systems. Its main strengths are its compositionality (power-conserving interconnections of port-Hamiltonian systems are again port-Hamiltonian), as well as the fact that the port-Hamiltonian formulation makes explicit the presence of balance equations (i.e., conservation laws with source terms) implying the preservation (or decrease) of the internal energy, contrary to other network modeling approaches. Its most important feature is to define a geometric structure (called Dirac structure) capturing the network interconnection structure. The identification of these structural properties has turned out to be of crucial importance in the development of modular dynamical models for complex multi-domain physical systems, the numerical simulation preserving the physical invariants, and in the design of control strategies, which in the nonlinear case do not always aim at compensating nonlinearities, but instead make use of beneficial nonlinearities. The aim of this mini-course is to present recent developments on the definition of Dirac structures and port-Hamiltonian systems associated with complex discrete topologies such as k-complexes (k-dimensional generalizations of graph topologies) and relate them with the port-Hamiltonian formulation of complex systems of balance equations defined on bounded spatial domains, their direct formulation on discrete spatial domains, and spatial semi-discretization using geometric structure preserving methods. The mini-course will consist of two blocks, each of 100 minutes.

Keywords: Operator Theoretic Methods in Systems Theory, Optimization : Theory and Algorithms
Abstract: The talk will be one of 3 coordinated talks (Helton, Klep, Volčič) concerning inequalities and equalities for functions having matrix variables. Matrix multiplication does not commute, so the functions we study are typically noncommutative (nc) polynomials or rational functions. Such mathematics is central to linear systems problems which are specified entirely by a signal flow diagram and L^2 performance specs on signals. The reason being: a system connection law amounts to an algebraic operation on nc quantities, while L^2 performance constraints, through use of quadratic "storage functions", convert to matrix inequalities.

Studies of commutative polynomial inequalities are central to classical mathematics and come under the heading of Real Algebraic Geometry (RAG). Hence linear systems problems cry out for development of a noncommutative RAG. This field has developed greatly over the last decade, pushed partly by systems motivation and greatly by unfolding the parallels to classical RAG.

Keywords: Operator Theoretic Methods in Systems Theory, Optimization : Theory and Algorithms
Abstract: The talk will be one of 3 coordinated talks (Helton, Klep, Volčič) concerning inequalities and equalities for functions having matrix variables. Matrix multiplication does not commute, so the functions we study are typically noncommutative (nc) polynomials or rational functions. Such mathematics is central to linear systems problems which are specified entirely by a signal flow diagram and L^2 performance specs on signals. The reason being: a system connection law amounts to an algebraic operation on nc quantities, while L^2 performance constraints, through use of quadratic "storage functions", convert to matrix inequalities.

Studies of commutative polynomial inequalities are central to classical mathematics and come under the heading of Real Algebraic Geometry (RAG). Hence linear systems problems cry out for development of a noncommutative RAG. This field has developed greatly over the last decade, pushed partly by systems motivation and greatly by unfolding the parallels to classical RAG.

Keywords: Optimization : Theory and Algorithms, Operator Theoretic Methods in Systems Theory, Optimal Control
Abstract: Semidefinite programming can be considered over any real closed field, including fields of Puiseux series equipped with their nonarchimedean valuation. Nonarchimedean semidefinite programs encode parametric families of classical semidefinite programs, for sufficiently large values of the parameter. Recently, a correspondence has been established between nonarchimedean semidefinite programs and stochastic mean payoff games with perfect information. This correspondence relies on tropical geometry. It allows one to solve generic nonarchimedean semidefinite feasibility problems, of large scale, by means of stochastic game algorithms. In this paper, we show that the mean payoff of these games can be interpreted as a condition number for the corresponding nonarchimedean feasibility problems. This number measures how close a feasible instance is from being infeasible, and vice versa. We show that it coincides with the maximal radius of a ball in Hilbert's projective metric, that is included in the feasible set. The geometric interpretation of the condition number relies in particular on a duality theorem for tropical semidefinite feasibility programs. Then, we bound the complexity of the feasibility problem in terms of the condition number. We finally give explicit bounds for this condition number, in terms of the characteristics of the stochastic game. As a consequence, we show that the simplest algorithm to decide whether a stochastic mean payoff game is winning, namely value iteration, has a pseudopolynomial complexity when the number of random positions is fixed.

Keywords: Optimization : Theory and Algorithms
Abstract: We present a new approach to certifying the nonnegativity of homogeneous multivariate polynomials that is based on the theory of hyperbolic polynomials. Moreover, the search for certificates of nonnegativity can be automated by solving a hyperbolic optimization problem. The main technical fact that enables these nonnegativity certificates is a polynomial parameterization (up to closure) of the dual cone of a hyperbolicity cone, a construction essentially due to Kummer, Plaumann, and Vinzant. This extended abstract presents the basic idea of such hyperbolic certificates of nonnegativity, and discusses what is known about the relationship between sums of squares and polynomials with hyperbolic certificates of nonnegativity.

Keywords: Stochastic Control and Estimation, Signal Processing, Networked Control Systems
Abstract: In this work, we consider an event-triggered risksensitive state estimation problem for hidden Markov models. The event-triggered condition considered is general, which involves the current measurement and past information. We use the reference probability measure approach in solving this problem. We derive the linear recursion in the unnormalized information state conditioned on the available information of the estimator, based on which risk-sensitive maximum a posterior estimates can be evaluated. The results are extended to the vector measurement scenario as well via a sequential event-triggered approach.

Keywords: Optimization : Theory and Algorithms
Abstract: How to evaluate the intrinsic quality of products and the user reputation becomes a significant yet difficult problem. In this paper, considering the trust relationship between users, we improve the traditional iterative ranking methods by combining the traditional iterative methods with the TrustRank method. The user’s reputation depends on the user’s rating behavior and the TrustRank value. Our simulations demonstrate that the modified methods posses a better robustness and performance than the existing methods.

Keywords: Networked Control Systems
Abstract: This paper studies semi-global containment control problem for a multi-agent system. The dynamics of each follower agent in the system is described by a discrete-time linear system in the presence of actuator position and rate saturation. For each follower agent we construct a linear state feedback control law by using low gain approach such that, the states of follower agents will converge to the convex hull formed by the leader agents asymptotically when the communication topology among follower agents is a connected undirected graph and each leader agent is a neighbor of at least one follower agent.

Keywords: Stochastic Control and Estimation, Networked Control Systems, Physical Systems Theory
Abstract: In this paper, we consider a consensus-based distributed filtering over wireless sensor networks under linear attacks. Suppose that in the network a malicious attacker injects false data into the data. First, we design an optimal estimator by minimizing the covariance of state estimation error. Then, we propose an effective detector for each sensor to resist the malicious data transmitted between the sensors in the network. Finally, the performances of the proposed estimator with the detector is demonstrated by comparing with the other typical attacking strategies.

Keywords: Networked Control Systems, Communication Systems, Stochastic Control and Estimation
Abstract: This paper proposes a design framework of sensor communication by using the stochastic event triggers, which aims at best saving the communication resources. The system to be considered is as follows: a sensor takes measurements of the states of a dynamic process and sends the information to a remote estimator, and the estimator computes the state estimates for the dynamic process. To save communication resources, a set of stochastic event triggers on the measurements are assigned to the sensor. At each sampled time, when no trigger is triggered, the sensor sends nothing to the estimator; when one of them is triggered, the sensor sends the identity code of the corresponding trigger. It is shown that once the estimator receives the identity of the trigger, it is equivalent for the estimator to receiving a measurement from a virtual sensor. Based on it, the system performance under the proposed communication is analyzed, and the design of specific models is considered.

Keywords: Infinite Dimensional Systems Theory, Networked Control Systems, Operator Theoretic Methods in Systems Theory
Abstract: The focus of this paper is on the finite or infinite dimensional class of spatially distributed linear systems with Hermitian and sparse state matrices. We show that exponential stability of this class of systems can be inferred in a decentralized and spatially localized manner, which is practically relevant to many real-world applications (e.g., systems with spatially discredited PDE models). Then, we obtain several sufficient conditions that allow us to adjust strength of existing couplings in a network in order to sparsify or grow a network, while ensuring global stability. Our proposed necessary and sufficient stability certificates are independent of the dimension of the entire system. Moreover, they only require localized knowledge about the state matrix of the system, which makes these verifiable conditions desirable for design of robust spatially distributed linear systems against subsystem failure and replacement.

Keywords: Control of Distributed Parameter Systems, Stability, Feedback Control Systems
Abstract: In this paper, we consider the stability of a hyperbolic system of the interconnected Schr"{o}dinger and wave equations, where the only distributed dissipative control is forced at the wave equation and there is no any control fixed at the Schr"{o}dinger. The Schr"{o}dinger can be exponentially stabilized by the inter-change transmission between the Schr"{o}dinger and wave at the interconnection boundary. We show that the whole system is well-posed and exponentially stable in the energy Hilbert space. A numerical computation is presented for the distributions of the spectrum of the whole system, and it is found that the spectrum of the Schr"{o}dinger depends largely on the interconnected transmission parameter and the decay of the wave equation.

Keywords: Infinite Dimensional Systems Theory, Control of Distributed Parameter Systems, Stochastic Control and Estimation
Abstract: This paper presents an ongoing research on the infinite horizon Linear Quadratic Gaussian (LQG for short) control problem for stochastic port-Hamiltonian systems on infinite-dimensional spaces with bounded input, output and noise operators. An adapted version of the separation principle is stated for this specific class of systems. Under suitable conditions, the LQG controller is shown to preserve the stochastic port-Hamiltonian structure. Finally, we propose some perspectives and open tracks to follow. The theory is illustrated on an example of vibrating string subject to a Hilbert space valued random forcing.

Keywords: Infinite Dimensional Systems Theory, Nonlinear Systems and Control
Abstract: Formulations of open physical systems within the framework of Non-Equilibrium Reversible/Irreversible Coupling (associated with the acronym “GENERIC") is related in this work with state-space realizations that are given as boundary port-Hamiltonian systems. This reformulation is carried out explicitly by splitting the dynamics of the system into a reversible contribution given by a Poisson bracket and an irreversible contribution given by a symmetric dissipation bracket, and is facilitated by the introduction of an exergy-like potential.

Keywords: Nonlinear Systems and Control, Infinite Dimensional Systems Theory, Optimal Control
Abstract: Semi-linear partial differential equations model a wide spectrum of physical systems with distributed parameters. It is shown that under sufficient conditions on nonlinearities in the systems and in cost functions, an optimal control input and optimal actuator design exist. First-order optimality conditions are obtained that characterize the optimizer. The results are used to address vibration suppression in a nonlinear railway track.

Keywords: Optimization : Theory and Algorithms, Signal Processing, Information Theory
Abstract: This paper deals with optimizing a transceiver, which consists of a linear time invariant (LTI) matrix filter at the transmitter and a generalized (nonlinear) decision feedback (DF) receiver, for {em single-carrier} data transmission over multi-input multi-output (MIMO) inter-symbol interference (ISI) channels. Considering a spatial multiplexing approach for which multiple scalar substreams are transmitted simultaneously, we adopt some cost function f_0({{rm MSE}_i}) to measure the overall system performance, where {rm MSE}_i's are the mean-squared errors (MSE) of the unquantized estimates at the receiver. Based on majorization theory and the generalized triangular decomposition (GTD), we derive the optimum DF based transceiver which minimizes f_0({{rm MSE}_i}) subject to the total input power constraint. It is proven that for {em any} cost function f_0 the optimum transmitter is of the same special structure and hence the original complicated matrix optimization problem can be significantly simplified to a problem with scalar variables. Furthermore, if the cost function is specialized to the cases where the composite function f_0 circ exp is Schur-convex, then the optimum nonlinear transceiver turns out to be a generalization of the uniform channel decomposition (UCD) scheme; when f_0 circ exp is Schur-concave, the optimum nonlinear design degenerates to diagonal transmission, which converts a MIMO ISI channel into multiple decoupled SISO ISI channels.

Keywords: Optimization : Theory and Algorithms
Abstract: In this paper, we analyze the existence conditions of a special class of constrained zero-one matrices with uncertainties. Classically, it asks how to characterize a class of zero-one matrices with given row sum and column sum vectors. The solution is given by a concise majorization inequality, widely known as the Gale-Ryser theorem. The authors further explore it in a more complicated case. Firstly, some fixed zeros or ones are given in a prescribed staircase pattern of positions. Secondly, each row or column sum is not exactly given but described by a boundary interval. It shows that such a constrained zero-one matrix exists if and only if two structure tensors are jointly nonnegative, which is consistent with the previous majorization condition.

Keywords: Networked Control Systems, Optimization : Theory and Algorithms
Abstract: This paper presents that the majorization theory plays an essential role in a class of sensor scheduling problems, whose solutions all have periodic or uniformly distributed patterns. This paper revisits the problem of communication time scheduling for a single sensor with local computation capability, and strengthens its original result by the majorization theory. The scheduling for a single normal sensor in a general-order system is also studied, and the optimal schedules for minimizing the upper bound of objective function is provided.

Keywords: Stochastic Control and Estimation
Abstract: We consider a sensor transmission power control problem for remote state estimation. In this problem, a sensor sends its local estimate to a remote estimator over a wireless packet-dropping communication channel. The transmission power is determined by a recently proposed algorithm which uses the innovative information contained in the measurement. In the current paper, we focus on parameter optimization arising from the selection of design parameters for this power controller. The existing work obtained a suboptimal solution to the parameter optimization problem, while by using a vector rearrangement inequality argument and the vector majorization, we now show that there exists an optimal solution within a subset of the whole feasible set. By leveraging this property, we obtain an optimal solution via solving a convex optimization problem.

Keywords: Systems on Graphs, Stability
Abstract: In this paper, we study the stability of a linear time-invariant dynamical network with both node and edge dynamics. It is demonstrated that at each frequency, the phases of the eigenvalues of the loop-gain transfer function are bounded from above (below, respectively) by the sum of the maximal (minimal, respectively) phases of the node and edge dynamics. Based on this property and the generalized Nyquist stability criterion, a sufficient condition for closed-loop stability is derived in terms of the phases of node and edge dynamics. This serves as an interesting starting point in developing a more general “small phase theorem”.

Keywords: Mechanical Systems, Networked Control Systems, Nonlinear Systems and Control
Abstract: In this paper, the attitude tracking problem is considered using the rotation matrices. Due to the inherent topological restriction, it is impossible to achieve global stability with any continuous attitude control system on SO(3). Hence in this work, we propose some control protocols that achieve almost global tracking asymptotically and in finite time, respectively. In these protocols, no world frame is needed and only relative state information are requested. For the closed-loop systems, Filippov solutions and non-smooth analysis techniques are adopted to handle the discontinuities.

Keywords: Stochastic Control and Estimation, Networked Control Systems
Abstract: We consider the problem of estimating the state of a linear time-invariant Gaussian system using N sensors, where a subset of the sensors can potentially be compromised by an adversary. In this case, localizing the compromised sensors is of crucial importance for obtaining an accurate state estimate. Inspired by the clustering algorithm in machine learning, we propose a Gaussian-mixture-model-based detection mechanism. It clusters the local state estimates autonomously and provides beliefs for the sensors, based on which measurements can be fused accordingly. When a subset of the sensors are under attack, we analyze the remote estimation error covariance for the proposed clustering-based detection algorithm and evaluate the performance of the proposed algorithm through the average belief. Moreover, the effectiveness of the proposed algorithm is verified under different attack scenarios through simulation examples.

Keywords: Systems on Graphs, Mathematical Theory of Networks and Circuits, Networked Control Systems
Abstract: In this extended abstract, we revisit the problem of Kalman filtering with intermittent observations. It is known that the definition of non-degeneracy plays a key role in determining the critical value of packet arrival probability for bounded estimate errors. In this extended abstract, we provide topological conditions for the existence of eigen-cycle, which can lead to degenerate systems. We prove that aperiodicity of the graph corresponding to the A matrix is a necessary condition for non-existence of eigen-cycle. We also prove that a special class of aperiodic graphs does not contain non-zero eigen-cycles.

Keywords: Applications of Algebraic and Differential Geometry in Systems Theory, Infinite Dimensional Systems Theory, Nonlinear Systems and Control
Abstract: Symmetry groups of PDEs allow to transform solutions continuously into other solutions. In this paper, we use this property for the observability analysis of nonlinear PDEs with input and output. Based on a differential-geometric representation of the nonlinear system, we derive conditions for the existence of special symmetry groups that do not change the trajectories of the input and the output. If such a symmetry group exists, every solution can be transformed into other solutions with the same input and output trajectories but different initial conditions, and this property can be used to prove that the system is not observable. We also put emphasis on showing how the approach simplifies for linear systems, and how it is related to the well-known observability concepts from infinite-dimensional linear systems theory.

Keywords: Algebraic Systems Theory, Applications of Algebraic and Differential Geometry in Systems Theory, Multidimensional Systems
Abstract: We study invariance properties with polynomially nonlinear and time-varying ODE systems (also termed one-dimensional or 1D systems due to the presence of one independent variable). The results are used to characterize time-varying controlled invariant varieties, that is, varieties that can be rendered invariant by state feedback. Analogous questions are also considered for a class of PDE systems in two independent variables (so-called two-dimensional or 2D systems), namely those which can be described by continuous Roesser models.

Keywords: Algebraic Systems Theory, Linear Systems, Mathematical Theory of Networks and Circuits
Abstract: A necessary condition for stability of a finite-dimensional linear time-invariant system is that all the coefficients of the characteristic equation are strictly positive. However, it is well-known that this condition is not sufficient, except for n less than 3. In this paper, we show that any polynomial that has positive coefficients cannot have roots on the negative real axis. Conversely, if a polynomial has no roots on the positive real axis, a polynomial with positive coefficients can be found so that the product of the two polynomials also has positive coefficients. A simple upper bound for the degree of this multiplier polynomial is given. One application of the main result is that under a strict condition, it is possible to find a non-minimal realization of a given transfer function using only positive multipliers (except for the "minus" in the standard feedback comparator).

Keywords: Linear Systems, Hybrid Systems, Algebraic Systems Theory
Abstract: This paper studies stabilizability problem with respect to an arbitrary stability domain for strict closed additive convex processes. The main results state necessary and sufficient conditions in terms of the eigenstructure of the dual process. The results presented in this paper are stronger than those of existing in the literature in two respects: (i) they are valid for arbitrary stability domains and (ii) they guarantee existence of Bohl-type stable trajectories. We also demonstrate the application of the main results by streamlining them for linear systems subject to conic input constraints.

Keywords: Hybrid Systems, Optimal Control, Feedback Control Systems
Abstract: A stabilization problem for a switched differential-algebraic system is investigated. We propose an approach for solving effectively the stabilization problem for an autonomous linear switched differential-algebraic system based on projector and flow matrices. In switched DAEs, the switches can induce jumps in certain state-variables, and it has been shown that the formulation as a switched DAE already implicitly defines these jumps, no additional jump map must be given. These jumps can be calculated in terms of the consistency projectors. The essence of this method is to design a stabilizing controller for switched differential-algebraic systems, i.e., the continuous dynamics of each subsystem are described by sets of differential-algebraic equations, using an averaging switched DAE model based on consistency projector and flow matrices to guarantee convergence towards an equilibrium point via fast switching.

Keywords: Delay Systems, Robust and H-Infinity Control, Control of Distributed Parameter Systems
Abstract: In this paper, we develop an SOS approach for design of observers for time-delay systems. The method is an extension of recently developed algorithms for control of infinite-dimensional systems. The observers we design are more general than the class of observers most commonly associated with time-delay systems in that they directly correct both the estimate of present state as well as the history of the state. The result is that the observer is itself a PDE. In this case the traditional notions of strong and weak observability do not apply and the resulting observer-based controllers can significantly outperform existing approaches.

Keywords: Discrete Event Systems
Abstract: The notion of switchable languages has been defined by Kumar, Takai, Fabian and Ushio in [Kumar-et-al.05]. It deals with switching supervisory control, where switching means switching between two specifications. In this paper, we first extend the notion of switchable languages to n languages, (n greater than 3). Then we consider a discrete-event system modeled with weighted automata. The use of weighted automata is justified by the fact that it allows us to synthesize a switching supervisory controller based on the cost associated to each event, like energy for example. Finally the proposed methodology is applied to a simple example.

Keywords: Delay Systems, Stability
Abstract: This paper proposes an analytic method to characterize the behavior of multiple critical roots of a retarded system with two delays. Expressing locally the related characteristic function, as a Weierstrass polynomial, we derive several results to analyze the stability behavior of such characteristic roots with respect to small variations on the delay parameters. The proposed results are illustrated by considering several numerical examples.

Keywords: Networked Control Systems, Feedback Control Systems, Robust and H-Infinity Control
Abstract: In this extended abstract, a stability certificate is presented for sampled-data implementations of static output feedback control with asynchronous sensor sampling and zero-order-hold actuator updates, given continuous-time two-input-single-output linear time-invariant system dynamics. The approach involves structured integral quadratic constraint (IQC) based robustness analysis, given individual bounds on the uncertain sample and two update intervals involved, and relationships between these. New IQCs for an operator that depends on multiple time-varying delays are also presented.

Keywords: Nonlinear Systems and Control, Delay Systems, Feedback Control Systems
Abstract: In this paper, we study the input-output decoupling and linearization of nonlinear two-input two-output time-varying delay systems. When working with delay systems, two problems may arise when constructing an invertible feedback transformation for which the input-output map of the feedback modified systems is linear. The first issue is the boundedness of the control and the second one is its causality. We develop an algorithm allowing the construction of a causal and bounded feedback which permits to solve the input-output decoupling and linearization problem. The idea of our algorithm is to introduce, at each step, when the input-output decoupling is not possible, an artificial delay for the input that appears "too early" in the system. To that end, we propose at each step a precise procedure for defining a simple feedback transformation.