Keywords: Feedback Control Systems
Abstract: Classical control theory does not scale well for large systems like traffic networks, power networks and chemical reaction networks. However, many of these applications can be handled efficiently using the concept of positive system, exploiting that the set of positive states is left invariant by the dynamics. Positive systems, and the nonlinear counterpart monotone systems, are common in many branches of science and engineering.

In this presentation, we will highlight several fundamental advantages of positive control systems: Verification and performance optimization can be done in with a complexity that scales linearly with the number of states and interconnections. Distributed controllers can be designed by convex optimization. Lyapunov functions and storage functions for nonlinear monotone systems can be built from scalar functions of the states, with dramatic simplifications as a result.

In spite of a rich set of existing results, several fundamental questions in control of positive systems remain open. For example, negative feedback can easily destroy positivity of the closed loop system. On the other hand, intuition tells us that something is wrong with a traffic control system where fewer cars leads to more congestion. Hence, we need to better understand how to design dynamic feedback controllers while preserving closed loop positivity.

As a case study, we study evolutionary dynamics of HIV virus in the human body. The dynamics can be described by a linear positive system, where the effects of drugs are visible in the diagonal elements. Thanks to the positive dynamics, drug therapies can be designed by convex optimization, but only after a nonlinear transformation of the state space.

Keywords: Nonlinear Systems and Control
Abstract:

Keywords: Optimal Control
Abstract: Riccati equations play a central role in a large number of areas associated with systems and control theory. A lot of work has been done by a large number of people about studying the type of solutions for the differential Riccati equation, the algebraic Riccati equation and the algebraic Riccati inequality. In this talk, I present yet another way to look at the Riccati equations (an approach which to the best of my knowledge has not been used before). Conceptually, this approach seems to lessen the complexity of understanding the structure of solutions and may perhaps lead to useful insights into iterative algorithms that are employed for Riccati equations.

Keywords: Delay Systems, Stability, Infinite Dimensional Systems Theory
Abstract: Numerical computation methods for the spectrum of monodromy operators are investigated. They are based on so-called fast-sample/fast-hold approximation. In previous work, the zero-th, 1st and 3rd order polynomials are used as the hold functions. Motivated by the fact that computational efficiency is improved as the order of the polynomial increases, we use higher-order polynomials as the hold functions to derive discrete-time approximations of the operator in matrix forms. Efficiency of the computation is compared through numerical examples.

Keywords: Delay Systems, Systems on Graphs, Nonlinear Systems and Control
Abstract: Sometimes a network of dynamical systems shows a form of incomplete synchronization characterized by synchronization of some but not all of its systems. This type of incomplete synchronization is called partial synchronization. Partial synchronization is associated with the existence of partial synchronization manifolds, which are linear invariant subspaces of C, the state space of the network of systems. We focus on partial synchronization manifolds in networks of systems that interact via linear coupling functions subject to time-delays. For such networks a number of necessary and sufficient conditions for the existence of partial synchronization manifolds are presented. A simple, illustrative example demonstrates the application of our results.

Keywords: Networked Control Systems, Delay Systems, Nonlinear Systems and Control
Abstract: The paper addresses the problem of consensus robustness against small heterogeneous input delays in the agents. The agents are assumed to have first-order dynamics and may be nonlinearly coupled; the couplings maps may be uncertain, assumed only to satisfy a sector (slope) restriction and a symmetry condition. The topology of the network may be switching and uncertain, however, it is supposed to be undirected and uniformly connected (which is known to be almost necessary for consensus). Under the mentioned assumptions, we obtain explicit analytical conditions for the consensus robustness.

Keywords: Stability, Delay Systems
Abstract: This manuscript is concerned with stability analysis of linear sampled-data systems with non-uniform sampling patterns. The stability problem is tackled from a continuous-time point of view, via the so-called "input delay approach". The system is viewed as feedback interconnection of a stable linear time-varying system and the "average delay-difference" operator, which is utilized to model the "aperiodic sampling operation". Characterization based on integral quadratic constrains (IQC) is identified for the "average delay-difference" operator, and the IQC theory is applied to derive convex stability criteria. The link between the IQC approach and the the Lyapunov-Krasovskii approach are made, and it is shown that one of the proposed criteria is equivalent to the main result of a recent article by A. Seurat, which generalizes many previous results. Moreover, a new criterion which is proven to be less conservative is proposed.

Keywords: Stability, Biological Systems
Abstract: Consider a chemical reaction network with general monotone rate functions. Conditions are given for the system to be reducible to a compartmental system. Consequently, every solution will approach an equilibrium point. Comparison is made with other approaches.

Keywords: Systems Biology, Optimal Control, Biological Systems
Abstract: Bacterial growth is a fundamental process in which cells sustain and reproduce themselves from available matter and energy. Optimisation principles have been widely used to explain and predict the growth behaviour of microorganisms, assumed to be optimized by evolution. This has given rise, among other things, to bacterial growth laws describing how the abundance of components of the gene expression machinery increases with the growth rate. These studies have mainly focused on the situation where the system is in balanced growth, a steady state in which all cell components grow at the same rate. Balanced-growth conditions, however, are far from natural growth conditions in which the environment is continually changing. We focus on the optimal allocation of resources between the gene expression machinery and other subsystems during growth-phase transitions. We describe an abstract model of the biochemical reaction processes occurring in the cell, based on first principles and articulated around two subsystems: the gene expression machinery and the uptake of nutrients from the environment. Using this so-called self-replicator model, we investigate the optimal dynamic reallocation of resources following a rapid change in the environment. We formulate our question as an optimal control problem that can be solved using Pontryagin’s maximum principle. Preliminary results have shown the predominance of bang-singular control of resource allocation following abrupt environmental transitions.

Keywords: Systems Biology, Stability, Systems on Graphs
Abstract: By making minimal assumptions on the kinetics of chemical reactions we study the stability of steady states in metabolic pathways in relation to the topology of the metabolic network. Here we report our results on metabolic pathways with irreversible reactions. We show that the steady states of linear pathways are always locally asymptotically stable. This is not necessarily true for branched and cyclic pathways, but stability in such networks is assured under mild conditions.

Keywords: Nonlinear Systems and Control, Process Control
Abstract: The properties MAL-CRN having equal number of species and complexes (called square CRNs) are studied in this paper. It is shown that linear weakly reversible MAL-CRNs have a unique realization, therefore all of their realization properties are system properties. Moreover, these CRNs have always zero deficiency, that implies the stability of their positive steady state point within the appropriate stoichiometric compatibility class. Additionally, it is shown that weakly reversible nonlinear square MAL-CRNs have also zero deficiency and a unique realization. Furthermore, a dynamically similar stable linear ODE model can be associated to a nonlinear square CRN by using translated X-factorable phase space transformations and nonlinear variable transformations. This way it is shown, that their unique positive steady state point within the appropriate stoichiometric compatibility class is also globally asymptotically stable under mild conditions.

Keywords: Systems on Graphs, Large Scale Systems, Mathematical Theory of Networks and Circuits
Abstract: We study individual agents with identical linear dynamics interconnected in a network, and in particular the lifetime of the transient behaviour, before the asymptotically dominant behaviour is reached. In application and main contribution, we give explicit expressions for the mixing time of non-Poissonian random walkers on a network, and show that it is typically either determined by the network topology alone, or by the waiting time distribution alone.

Keywords: Networked Control Systems, Systems on Graphs, Linear Systems
Abstract: This paper examines the strong structural controllability of networked systems. We use the notion of zero forcing set, a concept borrowed from combinatorial matrix theory, in order to solve the problem of determining a minimum-size input node set S so that a system with a self-damped/undamped tree structure is strongly S-controllable.

Keywords: Networked Control Systems, Systems on Graphs, Quantum Control
Abstract: We analyze controllability properties for two different classes of bilinear interconnected systems. The first class consists of networks of single-input single-output (SISO) linear systems, where coupling parameters act as control variables. We characterize the system Lie algebra of the resulting bilinear control system. Necessary and sufficient conditions for accessibility are derived in terms of the underlying interconnection graph. Similar results are stated for the multi-input multi-output (MIMO) case. Such problems are relevant for analyzing controllability of switching networks of linear systems. In the second part, motivated by applications in quantum control, we investigate parallel connections of bilinear systems. We deduce from a more general accessibility result on bilinear systems on semisimple Lie groups a necessary and sufficient accessibility/controllability condition for parallel connected bilinear systems.

Keywords: Networked Control Systems, Large Scale Systems
Abstract: A seminal result in decentralized control is the development of fixed modes by Wang and Davison in 1973 - that plant modes which cannot be moved with a static decentralized controller cannot be moved by a dynamic one either, and that the other modes which can be moved can be shifted to any chosen location with arbitrary precision. These results were developed for perfectly decentralized, or block diagonal, information structure, where each control input may only depend on a single corresponding measurement. Furthermore, the results were claimed after a preliminary step was demonstrated, omitting a rigorous induction for each of these results, and the remaining task is nontrivial.

In this paper, we consider fixed modes for arbitrary information structures, where certain control inputs may depend on some measurements but not others. We provide a comprehensive proof that the modes which cannot be altered by a static controller with the given structure cannot be moved by a dynamic one either, thus generalizing and solidifying the first part of Wang and Davison’s result. A follow-up paper discusses the second part.

Keywords: Coding Theory
Abstract: For static traitor tracing, the codes proposed by Tardos are the best known codes with regard to code length. Tardos codes are asymptotic optimal according to the low bound on code length proved by Peikert et al. The computational complexity of the tracing algorithm for Tardos codes is O(N), where N is the number of authorized users. Recently, Tardos codes have been adapted to dynamic traitor tracing. The adapted codes have the same tracing complexity as Tardos codes, that is, O(N).

In this extended abstract, we report our research of attempting to develop dynamic traitor-tracing schemes, using the near optimal codes obtained from the concatenation of Tardos codes and algebraic-geometric codes, which allow an efficient tracing procedure with complexity O(log(N)).

Keywords: Coding Theory, Information Theory
Abstract: Since 2003, Algebraic Attacks have received a lot of attention in the cryptography literature. In this context, algebraic immunity quantifies the resistance of a Boolean function to the standard algebraic attack of the pseudo-random generators using it as a nonlinear Boolean function. A high value of algebraic immunity is now an absolutely necessary cryptographic criterion for a resistance to algebraic attacks but is not sufficient, because of a more general kind of attacks so- called Fast Algebraic Attacks. In view of these attacks, the study of the set of annihilators of a Boolean function has become very important. We show that studying the annihilators of a Boolean function can be translated in studying the codewords of a linear code. We then explain how to exploit that connection to evaluate or estimate the algebraic immunity of a cryptographic function.

Keywords: Coding Theory, Information Theory
Abstract: Minimal linear codes are such that the support of every codeword does not contain the support of another linearly independent codeword. Such codes have applications in cryptography, e.g. to secret sharing and secure two-party computations. We pursue here the study of minimal codes and construct infinite families with asymptotically non-zero rates. We also introduce a relaxation to almost minimal codes, where a fraction of codewords is allowed to violate the minimality constraint. Finally, we construct new minimal codes based on hyperovals.

Keywords: Coding Theory
Abstract: We propose a secret sharing scheme based on rings or modules that admit a decomposition given by orthogonal idempotents of the ring. The secret is decomposed into partial secrets that belong to the projections induced by every idempotent. These projections are indeed ideal codes and thus the subsecrets are distributed among a group of users using the generator matrix of these ideal codes.

Keywords: Infinite Dimensional Systems Theory, Control of Distributed Parameter Systems, Robust and H-Infinity Control
Abstract: Starting from a very general formulation of the Internal Model Principle it is shown that a robust controller tracking/rejecting signals generated by an infinite-dimensional exosystem can be decomposed into a servocompensator and a stabilizing controller. The servocompensator contains an internal model of the exosystem generating the reference and disturbance signals and the stabilizing controller stabilizes the infinite-dimensional closed-loop system.

Keywords: Nonlinear Systems and Control, Stochastic Modeling and Stochastic Systems Theory, Optimal Control
Abstract: In the present work we prove existence and uniqueness of the solution for the stochastic nonlinear equation with Neumann boundary conditions, under general monotonicity assumptions, motivated by physical applications. To this purpose we shall use an optimal control approach based on the variational principle of Brezis and Ekeland.

Keywords: Control of Distributed Parameter Systems, Mechanical Systems, Feedback Control Systems
Abstract: Current controlled piezoelectric beams have been very popular recently due to the substantial reduction of hysteresis between the input and output. Even though there are many different PDE models for the voltage and charge controlled piezoelectric beams, there is no thoroughly derived PDE model in the literature for current controlled piezoelectric beams. A variational approach is utilized to derive the boundary value problem which models a single piezoelectric beam actuated by a current source at the electrodes. Magnetic effects are included. In addition to the Euler-Bernoulli displacement assumptions for the mechanical part, we assume that electrical and magnetic vector potential terms are quadratic-through thickness. Since current controller at the electrodes can control only the stretching equations, only the stretching equations are considered. Unlike the voltage or the charge actuation cases, a bounded control operator in the natural energy space is obtained.

Keywords: Control of Distributed Parameter Systems, Infinite Dimensional Systems Theory, Networked Control Systems
Abstract: This work considers the optimization of the synchronization gains for a class of networked systems governed by parabolic partial differential equations. The control and optimization objective is to ensure that (i) the state of each of the networked systems follows a reference model (leader) and (ii) all pairwise state differences of the networked systems converge to zero in an appropriate norm. A matching condition is assumed in order for a static output feedback controller to enable each of the networked systems to follow the state of the leader. A consensus protocol utilizing only the output signals from each of the networked systems is proposed to ensure that synchronization is attained. The synchronization gains utilized in the consensus protocol can be viewed as the strengths of the interconnections among the network nodes. Using the resulting aggregate systems, the choice of the synchronization gains is recast as an optimization problem in which the total energy of the aggregate dynamics is minimized. In this setting, the weighted version of the graph Laplacian matrix associated with an undirected connected graph, is used as an alternate parameterization of the synchronization gains. The resulting optimal value of the synchronization gains is thus found as the solution to a parameterized operator Lyapunov equation associated with the total energy of the aggregate system. The proposed results are demonstrated by numerical example for a 1D partial differential equation.

Keywords: Optimization : Theory and Algorithms, System Identification, Signal Processing
Abstract: Nonexistence of a best rank-R approximation of a three-way array hampers the practical use of the Canonical Polyadic Decomposition (CPD) for exploratory data analysis. We present theoretical results on nonexistence of a best rank-R approximation and propose a method to overcome this problem by augmenting the CPD with one or more interaction terms.

Keywords: Optimization : Theory and Algorithms, Signal Processing
Abstract: Nonnegative matrix factorization (NMF) under the separability assumption can provably be solved efficiently, even in the presence of noise, and has been shown to be a powerful technique in document classification and hyperspectral unmixing. This problem is referred to as near-separable NMF and requires that there exists a cone spanned by a small subset of the columns of the input nonnegative matrix approximately containing all columns. In this paper, we propose a preconditioning based on semidefinite programming making the input matrix well-conditioned. This in turn can improve significantly the performance of near-separable NMF algorithms which is illustrated on the popular successive projection algorithm (SPA). The new preconditioned SPA is provably more robust to noise, and outperforms SPA on several synthetic data sets. We also show how an active-set method allow us to apply the preconditioning on large-scale real-world hyperspectral images.

Keywords: Optimization : Theory and Algorithms
Abstract: In this paper, we investigate the complexity of the numerical construction of the Hankel structured low-rank approximation (HSLRA) problem, and develop a family of algorithms to solve this problem. Briefly, HSLRA is the problem of finding the closest (in some pre-defined norm) rank r approximation of a given Hankel matrix, which is also of Hankel structure. Unlike many other methods described in the literature the family of algorithms we propose has the property of guaranteed convergence.

Keywords: Optimization : Theory and Algorithms, Nonlinear Systems and Control, Large Scale Systems
Abstract: We propose a Riemannian optimization approach for computing low-rank solutions of the algebraic Riccati equation. The scheme alternates between fixed-rank optimization and rank-one updates. The fixed-rank optimization is on the set of fixed-rank symmetric positive definite matrices which is endowed with a particular Riemannian metric (and geometry) that is tuned to the structure of the objective function. We specifically discuss the implementation of a Riemannian trust-region algorithm that is potentially scalable to large-scale problems. The rank-one update is based on a descent direction that ensures a monotonic decrease of the cost function. Preliminary numerical results on standard small-scale benchmarks show that we obtain solutions to the Riccati equation at lower ranks than the standard approaches.

Keywords: Applications of Algebraic and Differential Geometry in Systems Theory
Abstract: We define and study preorderings and orderings on rings of the form M_n(R) where R is a commutative unital ring. We extend the Artin-Lang theorem and Krivine-Stengle Stellens" atze (both abstract and geometric) from R to M_n(R). While the orderings of M_n(R) are in one-to-one correspondence with the orderings of R, this is not true for preorderings. Therefore, our theory is not Morita equivalent to the classical real algebraic geometry.

Keywords: Applications of Algebraic and Differential Geometry in Systems Theory, Operator Theoretic Methods in Systems Theory, Optimization : Theory and Algorithms
Abstract: The talk concerns inequalities for functions having matrix variables. The functions are typically (noncommutative) polynomials or rational functions. A focus of much attention are the inequalities corresponding to convexity. Such mathematics is central to linear systems problems which are specified entirely by a signal flow diagram and L^2 performance specs on signals.

At this point we have:

A. versions of the classical real algebraic geometry description of when one polynomial p is nonnegative on the domain where another polynomial q is nonnegative.

B. classification of convex non-commutative polynomials, rational functions and varieties. Now we know that such matrix convexity typically forces the presence of some LMI.

C. some theory of matrix convex hulls.

D. some theory of changes of variables to achieve noncommutative convexity.

E. other.

The talk will select a recent topic from this. The work originates in trying to develop some theory for studying the matrix inequalities which are ubiquitous in linear engineering systems and control. Most of the work is done jointly by J. William Helton, Igor Klep and Scott A McCullough with parts having serious contributions by Jaka Cimpric, Chris Nelson and Markus Schweighofer.

The talks of Bill Helton and Igor Klep in this session are both on this topic and will be coordinated.

Keywords: Operator Theoretic Methods in Systems Theory
Abstract: The talk concerns inequalities for functions having matrix variables. The functions are typically (noncommutative) polynomials or rational functions. A focus of much attention are the inequalities corresponding to convexity. Such mathematics is central to linear systems problems which are specified entirely by a signal flow diagram and L^2 performance specs on signals.

At this point we have:

(A) versions of the classical real algebraic geometry description of when one polynomial p is nonnegative on the domain where another polynomial q is nonnegative;

(B) classification of convex non-commutative polynomials, rational functions and varieties. Now we know that such matrix convexity typically forces the presence of some LMI;

(C) some theory of matrix convex hulls;

(D) some theory of changes of variables to achieve noncommutative convexity;

(E) other.

The talk will select a recent topic from this. The work originates in trying to develop some theory for studying the matrix inequalities which are ubiquitous in linear engineering systems and control. Most of the work is done jointly by J. William Helton, Igor Klep and Scott A. McCullough with parts having serious contributions by Jaka Cimpric, Chris Nelson and Markus Schweighofer.

The talks of Bill Helton and Igor Klep in this session are both on this topic and will be coordinated. No paper will be submitted to the proceedings.

Keywords: Linear Systems, Operator Theoretic Methods in Systems Theory, Feedback Control Systems
Abstract: (Generalized) Positive Systems - an Overview -------------------------------------------- In scalar terminology (employed for simplicity) positive functions analytically map the right half plane to itself. We address matrix valued rational functions. In electrical networks theory, they were identified with the driving point immittance of passive circuit. More generally, they are associated with passivity. Generalized positive functions map (almost everywhere) the imaginary axis to the right half plane. We here address several structural properties of these sets: Factorization of generalized positive functions. Even-Odd partitioning. Closure under: (i) Positive scaling, (ii) Summation (iii) Inversion, (iv) Composition (v) Static and Dynamic output feedbacks. Whenever admit state-space realization, these sets may be characterized through a corresponding Kalman-Yakubovich-Popov Lemma. Properties of families of realization matrices will be introduced. In this case, simple operations lead to unexpected properties. If time permits, an open problem will be presented. Joint work with D. Alpay, Math. dept. Ben-Gurion University, Israel.

Keywords: Hybrid Systems, Delay Systems
Abstract: A hybrid static gain observer for systems comprized of a linear time-delayed continuous part and an impulsive feedback is suggested. The purpose of the observer is to asymptotically drive the state estimation error in the continuous states to zero and synchronize the sequence of pulse modulation instants estimated by the observer with that of the plant. Conditions on the observer gain matrix to locally stabilize the observer error along an arbitrary periodic plant solution are obtained and the observer performance is illustrated by numerical simulations.

Keywords: Hybrid Systems, Mathematical Theory of Networks and Circuits, Linear Systems
Abstract: In this paper, we use the switched linear differential systems framework cite{submitted} to model electrical devices with switching dynamics. Modularity, i.e. independent modelling and incremental combination of complex dynamics, is an important feature of this approach, since we can incorporate new dynamic modes to the bank without altering the existing ones. This makes our approach ideal for describing complex systems (e.g. energy distribution networks). Our modelling approach differs fundamentally from the traditional representation-based theory where the use of a global state space is required.

Keywords: Hybrid Systems, Networked Control Systems, Stability
Abstract: The objective is to present a new type of triggering conditions together with new proof concepts for the event-based coordination of multi-agents. As a first step, we focus on the rendez-vous of two identical systems modeled as double integrators with additional damping in the velocity dynamics. The reduced number of agents in the system allows us to illustrate the design of the new triggering rule while avoiding technical issues that arise in the full multi-agent system. We first propose an event-triggering policy which relies on a designed clock variable whose dynamics depend on the relative position of the agents. We then explain how to derive self-triggered and time-triggered rules. The overall system is modeled as a hybrid system in each case, and an invariance principle is used to conclude about the rendez-vous. Simulation results are provided to illustrate the efficiency of the proposed controllers.

Keywords: Hybrid Systems, Stability, Nonlinear Systems and Control
Abstract: We prove a novel Lyapunov-based small-gain theorem for interconnections of n hybrid systems, which are not necessarily input-to-state stable. This result unifies and extends several small-gain theorems for hybrid and impulsive systems, proposed in the last few years. Also we show how the average dwell-time (ADT) clocks and reverse ADT clocks can be used to modify the Lyapunov functions for subsystems and to enlarge the applicability of derived small-gain theorems.

Keywords: Stochastic Control and Estimation, Control of Distributed Parameter Systems, Adaptive Control
Abstract: A linear-quadratic control problem with an infinite time horizon for some infinite-dimensional controlled stochastic differential equations driven by a fractional Brownian motion is formulated and solved: The feedback form of the optimal control and the optimal cost are given explicitly. The optimal control is the sum of the well known linear feedback control for the associated deterministic linear-quadratic control problem and a suitable prediction of the adjoint optimal system response to the future noise. The noise covariance operator as well as the control operator in the system equation can in general be unbounded, so the results can also be applied where the noise or the control are on the boundary of the domain or at discrete points in the domain. A continuous dependence of the optimal cost on some parameters of the systems is also verified. Some examples of controlled stochastic partial differential equations are given.

Keywords: Stochastic Control and Estimation, Optimal Control, Nonlinear Systems and Control
Abstract: Some two person zero sum stochastic differential games are formulated and solved in the n-sphere for an arbitrary positive integer n. The payoff functional has a distance symmetry from the geometry of the n-sphere as a Riemannian symmetric space. Explicit solutions are obtained for the optimal control strategies of the two players in a direct way that does not require solutions of Hamilton-Jacobi-Isaacs equations. The optimal strategies are optimal for a large family of strategies.

Keywords: Stochastic Modeling and Stochastic Systems Theory, Stochastic Control and Estimation, Adaptive Control
Abstract: Assume that the state process is a controlled Markov process (X_t), which is partially observed with observation process (Y_t). We consider a game with two players: the maximizer who controls off-line he is choosing for each time a control function (control rule), which is a function of the current state of the process and filtering process plus eventually (in the case of MINMAX game) current value of control of the minimizer (the oponent). The minimizer knows control functions of the maximizer but gets only a partial observation (Y_t) of the state process plus, in the case of MAXMIN game, the current value of control of the maximizer. In the paper upper and lower values of the game are characterized and a simple counterexample for which upper value is strictly greater than the lower value of the game is shown.

Keywords: Nonlinear Filtering and Estimation, Stochastic Control and Estimation, Optimization : Theory and Algorithms
Abstract: The paper extends the results on the problem of change point detection for Markov processes generalizing the results contained in the publications Sarnowski & Szajowski(2011), Szajowski(2011) and Ochman-Gozdek & Szajowski(2013). The short description are as follows. A random sequence having segments being the homogeneous Markov processes is registered. Each segment has his own, known transition probability law and the length of the segment is unknown and random. Joint a priori distribution of the disorder moments is given. In this note the aim is to indicate the segment of given length between disorders with maximal probabilities. The case with various precision for over and under estimation of the middle point is analyzed including situation when the disorders do not appear with positive probability. The observed sequence, when the change point is known, has the Markov properties. The results explain the structure of optimal detector in various circumstances and shows new details of the solution construction as well insignificantly extends range of application. The motivation for this investigation is the modelling of selection the suspicious observations in the experiments. Such observation can be treated as outliers or disturbed. The objectives is to detect such inaccuracy immediately or in very short time before or after it appearance with highest probability. The problem is reformulated to optimal stopping of the observed sequences. The form of optimal decision function is presented. The application of the results to analysis of piecewise deterministic processes with change points appearing at the moment of jumps is shown (cf. Herberts & Jensen(2004), Poor & Hadjiliadis(2009), Ferenstein & Pasternak-Winiarski(2011)).

Keywords: Stability, Nonlinear Systems and Control, Robust and H-Infinity Control
Abstract: Input-to-State Stability (ISS) and the ISS-Lyapunov function have proved to be useful tools for the analysis and design of nonlinear systems in a variety of contexts. Motivated by the fact that many feedback control laws, such as model predictive control or event-based control, lead to discontinuous discrete-time dynamics, we investigate ISS-Lyapunov functions for such systems. ISS-Lyapunov functions were originally introduced in a so-called implication-form and, in many cases, this has been shown to be equivalent to an ISS-Lyapunov function of dissipation-form. However, for discontinuous dynamics, we demonstrate via an example that this equivalence no longer holds. We therefore propose a stronger implication-form ISS-Lyapunov which re-establishes the equivalence to dissipation-form ISS-Lyapunov functions and to the ISS property for discontinuous systems.

Keywords: Large Scale Systems, Stability, Nonlinear Systems and Control
Abstract: We present a nonlinear Lyapunov function based small-gain theorem for analyzing input-to-state stability of discrete-time large-scale systems. Motivated by the fact that many feedback control laws lead to discontinuous closed loop systems, we pose no continuity assumptions on the system dynamics. For characterizing input-to-state stability in this discontinuous setting, we utilize a recently introduced strong implication-form ISS-Lyapunov function.

Keywords: Stability, Hybrid Systems, Nonlinear Systems and Control
Abstract: We propose the notion of nondecreasing Lyapunov functions which can be used to prove stability or other properties of the system in question. This notion is in particular useful in studying switched or hybrid systems. We illustrate the concept by a general construction of such a nondecreasing Lyapunov function for a class of planar hybrid systems. It is noted that this class encompasses switched systems for which no piecewise-quadratic (classical) Lyapunov function exists.

Keywords: Nonlinear Systems and Control, Adaptive Control, Feedback Control Systems
Abstract: In this paper we investigate the application of funnel control to unconstrained path-following problems. While funnel control is a high-gain based time-varying feedback strategy applicable to minimum-phase systems with known relative degree, path following refers to the problem of tracking a reference path in an output space. A particular feature of path following is that the timing along the reference path is not determined a priori. The timing is adjusted online by the controller and may be exploited to improve the tracking performance. We show that the combination of funnel control and path following is applicable to fully-actuated rigid-link revolute-joint robots.

Keywords: Control of Distributed Parameter Systems, Stability, Infinite Dimensional Systems Theory
Abstract: This mini-course is devoted at illustrating the latest results on the stability analysis, and on the (boundary) stabilisation of distributed port-Hamiltonian systems. Several related topics will be discussed in five short lessons. More in detail, at the beginning, the problems of well-posedness of distributed port-Hamiltonian systems, existence of solutions for the associated systems of PDEs, and definition of a boundary control system in port-Hamiltonian form are addressed. These topics can be seen as the foundations of the remaining part of the course, that is more focused on stability criterions in case of static and dynamic boundary control, and on the energy-based control of this class of infinite dimensional systems. The emphasis is on linear systems with one-dimensional domain, but it is worth of mentioning that most of the proposed techniques can be in principal adapted to the nonlinear scenario. However, this extension is not trivial, and it is still an open problem.