Interest In Control Theory Research Articles

Student Feedback on Educational Innovation in Control Engineering: Active Learning in Practice

<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Contribution:</i> An education innovation in control engineering using practical setups and its evaluation based on a three-year student feedback study and examination grades. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Background:</i> Based on extensive research, education’s transition toward active learning and more practical experience has been shown to increase learning outcomes. Contrary to virtual and remote labs, a practical session with an individual setup for each student provides the most practical experience. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Intended Outcomes:</i> To show a positive effect on learning performance by integrating practical sessions in basic control engineering. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Application Design:</i> Presenting low cost setups that can be mass produced and adapt to the course’s growing complexity. These setups are evaluated during a three-year feedback study. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Findings:</i> The developed setups increased understanding of theoretical concepts. The new methodology significantly improved students’ average grades. The students’ interest in control theory is triggered. This case study could guide other institutions toward successfully implementing highly individual practical sessions for large groups.

Disturbance decoupling and design of unknown input observers for hybrid systems with state-driven jumps

In this paper two classical problems of interest in control theory, namely disturbance decoupling and the design of Unknown Input Observers (UIO), are revisited in the context of linear hybrid systems in the presence of state-driven jumps, induced by multi-affine sets. Interestingly, the latter feature renders the overall problem intrinsically nonlinear, hence classical (geometric) conditions cannot be straightforwardly applied. Firstly, it is shown that the definition of the disturbance decoupling control objective needs to be adapted to the hybrid setting, by distinguishing between weak and strong decoupling. In the former, the desired requirement consists in ensuring that the output of the plant is independent of the disturbance on each flow subinterval and across the successive jump, while the latter requires that both the output trajectory and the corresponding time domain, which is potentially perturbed in the uncontrolled case, are not affected by the disturbance. Easily verifiable geometric necessary and sufficient conditions to achieve weak or strong decoupling are presented and discussed. Then, it is shown that the solution to an auxiliary strong decoupling control problem is instrumental for the design of unknown input observers, whose existence conditions mimic and extend those for purely continuous-time systems. The paper is concluded by numerical simulations that corroborate the theoretical analysis.

G-stability one-leg hybrid methods for solving DAEs

This paper introduces the solution of differential algebraic equations using two hybrid classes and their twin one-leg with improved stability properties. Physical systems of interest in control theory are sometimes described by systems of differential algebraic equations (DAEs) and ordinary differential equations (ODEs) which are zero index DAEs. The study of the first hybrid class includes the order of convergence, A(α)-stability, stability regions, and G-stability for its one-leg twin in two cases: for step (k=1) and steps (k=2). For the second class, G-stability of its one-leg twin is studied in two cases: for steps (k=2) and steps (k=3). Test problems are introduced with different step size at different end points.

Diagrammatic algebra: from linear to concurrent systems

We introduce the resource calculus, a string diagrammatic language for concurrent systems. Significantly, it uses the same syntax and operational semantics as the signal flow calculus --- an algebraic formalism for signal flow graphs, which is a combinatorial model of computation of interest in control theory. Indeed, our approach stems from the simple but fruitful observation that, by replacing real numbers (modelling signals) with natural numbers (modelling resources) in the operational semantics, concurrent behaviour patterns emerge. The resource calculus is canonical: we equip it and its stateful extension with equational theories that characterise the underlying space of definable behaviours---a convex algebraic universe of additive relations---via isomorphisms of categories. Finally, we demonstrate that our calculus is sufficiently expressive to capture behaviour definable by classical Petri nets.

Parrondo’s Paradox: Gambling games from noise induced transport - a new study

Noise is known to disrupt a preferred act or motion. Yet, noise and broken symmetry in asymmetric potential, together, can make a directed motion possible, in fact essential in some cases. In Biology, in order to understand the motions of molecular motors in cells ratchets are imagined as useful models. Besides, there are optical ratchets and quantum ratchets considered in order to understand corresponding transport phenomena in physics. The discretized actions of ratchets can be well understood in terms of probabilistic gambling games which are of immense interest in Control theory – a powerful tool in physics, engineering, economics, biology and social sciences.

Discrete-Time Resonant Observer Based Control for Periodic Signal Rejection

Rejection of periodic signals has been a subject of great interest in control theory and engineering applications. Periodic disturbances are usually present in mechatronic, hydraulic and electronics systems, between others. This paper proposes a resonant observer-based control for periodic signal rejection. The observer includes, in an embedded way, an internal model of the disturbance that is based on its harmonic decomposition. The resulting disturbance estimation is used by a control law that cancels the periodic disturbance term while controlling a specified tracking task. The proposed scheme lets the control designer to address the disturbance estimation and tracking problems separately. Experimental results, on a mechatronic test bed, show that the proposed resonant observer-based control successfully rejects periodic disturbances.

A quiver approach to studying orbit spaces of linear systems

Orbit spaces associated to linear actions are of particular interest in control theory. Their geometrical properties can be naturally investigated by using the representations of quivers as an abstract framework. The aim of the paper is to bring into attention an application of this approach and to show how the use of quivers makes it easy handling concepts arising in control theory. Specifically, the natural duality between controllable and observable systems, as well as the construction of compactifications for the associated orbit spaces is interpreted in terms of opposite quivers.

Cueing an unresolved personal goal causes persistent ruminative self-focus: An experimental evaluation of control theories of rumination

Background and ObjectivesControl theory predicts that the detection of goal discrepancies initiates ruminative self-focus (Martin & Tesser, 1996). Despite the breadth of applications and interest in control theory, there is a lack of experimental evidence evaluating this prediction. The present study provided the first experimental test of this prediction. MethodsWe examined uninstructed state rumination in response to the cueing of resolved and unresolved goals in a non-clinical population using a novel measure of online rumination. ResultsConsistent with control theory, cueing an unresolved goal resulted in significantly greater recurrent intrusive ruminative thoughts than cueing a resolved goal. Individual differences in trait rumination moderated the impact of the goal cueing task on the extent of state rumination: individuals who had a stronger tendency to habitually ruminate were more susceptible to the effects of cueing goal discrepancies. LimitationsThe findings await replication in a clinically depressed sample where there is greater variability and higher levels of trait rumination. ConclusionsThese results indicate that control theories of goal pursuit provide a valuable framework for understanding the circumstances that trigger state rumination. Additionally, our measure of uninstructed online state rumination was found to be a valid and sensitive index of the extent and temporal course of state rumination, indicating its value for further investigating the proximal causes of state rumination.

On the time-optimal problem for three- and four-dimensional control systems

The paper is devoted to the time-optimal problem for three- and four-dimensional nonlinear control systems with one-dimensional control. We obtain sufficient conditions for a time-optimal control to be equivalent (in the Lebesgue sense) to a piecewise constant control that is also optimal, has a finite number of discontinuity points, and takes only extreme values. Such optimal controls are called bang-bang solutions and are of considerable interest in control theory and its applications.

Positivity, Complete Positivity, and SOS Representation

The gap between positive multivariate polynomials and the sum-of-squares (SOS) representation has resurfaced as a problem of importance and interest in control theory. A representation theorem for a bihermitian form is proved. A linear map and its dual are associated with this unique form. Complete positivity and not just positivity of the linear maps is necessary as well as sufficient for SOS representation of the form.

Extension of the Barzilai–Borwein Method for Quadratic Forms in Finite Euclidean Spaces

In this paper, we present an extension of the Barzilai–Borwein method for the minimization of a quadratic form defined in a finite-dimensional real Euclidean space. We present a convergence analysis of the proposed method. We also present some applications in the resolution of linear matrix equations such as the Sylvester equation, which are of great interest in control theory and other engineering disciplines. Our numerical results indicate that the new method competes satisfactorily in the resolution of linear matrix equations with the function lyap of the Toolbox of Control of MATLAB.

Boundary control, quiet boundaries, super-stability and super-instability

This paper deals with the boundary feedback control of a bar undergoing axial vibrations. The feedback force is applied to one end of the bar and it is proportional to its velocity. The problem is of fundamental interest in control theory, structural dynamics, and the development of ‘quiet boundaries’ in fields like earthquake engineering and computational mechanics. The system is not self-adjoint and exhibits a variety of interesting behaviors which are explained through a combination of several inter-twined strands of thought using mathematics, physical interpretations, and numerical simulations. Besides providing a rigorous mathematical solution to the problem, the paper explains the physical origin of super-stable behavior, introduces the new concept of super-unstable behavior, and points out that these regimes of behavior are intricately connected with the continuum model that is used. It is shown that any finite-dimensional approximation of the system, no matter how finely discretized, can not qualitatively depict superstable behavior.

Some new directions in control theory inspired by systems biology

This paper, addressed primarily to engineers and mathematicians with an interest in control theory, argues that entirely new theoretical problems arise naturally when addressing questions in the field of systems biology. Examples from the author's recent work are used to illustrate this point.

SINGULAR CHARACTERISTICS OF FIRST ORDER PDEs IN OPTIMAL CONTROL AND DIFFERENTIAL GAMES

The interconnection between the maximum principle [13] and dynamic programming [3], the two main methods in the theory of optimal control and differential games, is now understood much better due to the theory of viscosity solutions of firstand second-order nonlinear PDEs [2, 8, 14]. The value functions in optimal control and differential games have been proved to be generalized (viscosity) solutions of the corresponding Hamilton–Jacobi–Bellman–Isaacs (HJBI) equations. The classical method of characteristics (MC), which reduces the solution of a PDE to the integration of an ODE (characteristic) system, is one of the attractive and powerful tools for solving nonlinear first-order PDEs arising in control theory and mathematical physics [5]. Characteristics are of special interest in control theory, since they represent optimal paths [10]. This is clear in regarding the regular characteristics. The nonsmoothness of the generalized (e.g., viscosity) solution and/or of the Hamiltonian (left-hand-side function of the PDE), i.e., the presence of singular surfaces, is often referred to as an obstacle to the implementation of the MC. On the other hand, the construction of singular surfaces and lines is an essential and interesting part of the solution of a problem in control theory. The method of singular characteristics (MSC) shows that one can overcome this obstacle by using the same notion of characteristics suitably modified [11,12]. In this paper, a new notion of singular characteristics (SC) is suggested, which, together with the classical (regular) ones, form generalized characteristics. SC are effective for the construction of singular lines, surfaces, and manifolds of a nonsmooth solution (to firstor second-order PDEs), which carry essential information on the corresponding control or physical problem or phenomenon. Singular characteristics were found owing to investigation of singular paths in differential games and optimal control [4,9,10]. Regular paths in these domains are known to be governed by a Hamiltonian ODE system, the characteristic system for the HJBI-equation. In many cases, singular paths are described by similar equations by using the so-called singular controls. The attempt to eliminate singular controls from these equations has led to the discovery of singular characteristics, which appear to be inherent not only to game or control problems but also to the general nonlinear first-order PDEs. Owing to the notion of SC, many singular lines and surfaces known in control theory have received their invariant description in terms of general (abstract) PDEs. The possible interconnection between singular paths and characteristics (in some generalized sense) was mentioned in [7]. This general mathematical insight into the nature of a singularity, as a rule, simplifies the solution procedure of a game or control problem. Correspondingly, the experience accumulated during the construction of solutions to the latter problems is highly useful for the understanding the structure of singularities of the viscosity solutions to general PDEs. The considerations of the present paper are given in general mathematical terms rather than in terms specific for control theory. The ODE systems of singular characteristics are derived for several types of well-known singular surfaces in optimal control and differential games. The solutions of particular problems are presented.

Basic problems in stability and design of switched systems

By a switched system, we mean a hybrid dynamical system consisting of a family of continuous-time subsystems and a rule that orchestrates the switching between them. The article surveys developments in three basic problems regarding stability and design of switched systems. These problems are: stability for arbitrary switching sequences, stability for certain useful classes of switching sequences, and construction of stabilizing switching sequences. We also provide motivation for studying these problems by discussing how they arise in connection with various questions of interest in control theory and applications.

Orders of Input/Output Differential Equations and State-Space Dimensions

This paper deals with the orders of input/output equations satisfied by nonlinear systems. Such equations represent differential (or difference, in the discrete-time case) relations between high-order derivatives (or shifts, respectively) of input and output signals. It is shown that, under analyticity assumptions, there cannot exist equations of order less than the minimal dimension of any observable realization; this generalizes the known situation in the classical linear case. The results depend on new facts, themselves of considerable interest in control theory, regarding universal inputs for observability in the discrete case, and observation spaces in both the discrete and continuous cases. Included in the paper is also a new and simple self-contained proof of Sussmann's universal input theorem for continuous-time analytic systems.

Vertices and segments of interval plants are not sufficient for step response analyses

Interval plants are of interest in control theory as models of uncertain systems. They are useful because many worst-case analyses of these models are simple to carry out. For example, robust stability of an interval plant can be determined by investigating only the four Kharitonov vertices of the denominator polynomial. Also, the maximum peak of the Bode magnitude plot can be found using just 16 special plant vertices. These 16 vertices are connected by 32 special line segments. Most stability and frequency domain analyses that cannot be done using only the special vertices can be carried out using just the segments. From these results, it is tempting to conjecture that the 16 vertices or at least the 32 segments are adequate for step response analyses. This paper presents examples showing that these conjectures are not true.

Structural aspects of controllability and Sobservability—II. digraph decomposition

The graph theoretic aspects of controllability and observability are examined and related to the tensorial formulation of Part I of the paper. Particular emphasis is given to the significance of the system digraph decomposition and the relevance of this to certain system algebraic properties of interest in control theory.