Control Theory Literature Research Articles

On scaling robust feedback control and state estimation problems in power networks

Many mainstream robust control/estimation algorithms for power networks are designed using the Lyapunov theory as it provides performance guarantees for linear/nonlinear models of uncertain power networks but comes at the expense of scalability and sensitivity. In particular, Lyapunov-based approaches rely on forming semi-definite programs (SDPs) that are (i) not scalable and (ii) extremely sensitive to the choice of the bounding scalar that ensures the strict feasibility of the linear matrix inequalities (LMIs). This paper addresses these two issues by employing a celebrated non-Lyapunov approach (NLA) from the control theory literature. In lieu of linearized models of power grids, we focus on (the more representative) nonlinear differential algebraic equation (DAE) models and showcase the simplicity, scalability, and parameter-resiliency of NLA. For some power systems, the approach is nearly fifty times faster than solving SDPs via standard solvers with almost no impact on the performance. The case studies also demonstrate that NLA can be applied to more realistic scenarios in which (i) only partial state data is available and (ii) sparsity structures are imposed on the feedback gain. The paper also showcases that virtually no degradation in state estimation quality is experienced when applying NLA.

Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints

<p style='text-indent:20px;'>A Pontryagin maximum principle for an optimal control problem in three dimensional linearized compressible viscous flows subject to state constraints is established using the Ekeland variational principle. Since the system considered here is of coupled parabolic-hyperbolic type, the well developed control theory literature using abstract semigroup approach to linear and semilinear partial differential equations does not seem to contain problems of the type studied in this paper. The controls are distributed over a bounded domain, while the state variables are subject to a set of constraints and governed by the compressible Navier-Stokes equations linearized around a suitably regular base state. The maximum principle is of integral-type and obtained for minimizers of a tracking-type integral cost functional.</p>

Neuro-based Canonical Transformation of Port Controlled Hamiltonian Systems

In the literature of control theory, tracking control of port controlled Hamiltonian systems is generally achieved using canonical transformation. Closed form evaluation of state-feedback for the canonical transformation requires the solution of certain partial differential equations which becomes very difficult for nonlinear systems. This paper presents the application of neural networks for the canonical transformation of port controlled Hamiltonian systems. Instead of solving the partial differential equations, neural networks are used to approximate the closed-form state-feedback required for canonical transformation. Ultimate boundedness of the tracking and neural network weight errors is guaranteed. The proposed approach is structure preserving. The application of neural networks is direct and off-line processing of neural networks is not needed. Efficacy of the proposed approach is demonstrated with the examples of a mass-spring system, a two-link robot arm and an Autonomous Underwater Vehicle (AUV).

Infinitesimal homeostasis in three-node input-output networks.

Homeostasis occurs in a system where an output variable is approximately constant on an interval on variation of an input variable [Formula: see text]. Homeostasis plays an important role in the regulation of biological systems, cf. Ferrell (Cell Syst 2:62-67, 2016), Tang and McMillen (J Theor Biol 408:274-289, 2016), Nijhout et al. (BMC Biol 13:79, 2015), and Nijhout et al. (Wiley Interdiscip Rev Syst Biol Med 11:e1440, 2018). A method for finding homeostasis in mathematical models is given in the control theory literature as points where the derivative of the output variable with respect to [Formula: see text] is identically zero. Such points are called perfect homeostasis or perfect adaptation. Alternatively, Golubitsky and Stewart (J Math Biol 74:387-407, 2017) use an infinitesimal notion of homeostasis (namely, the derivative of the input-output function is zero at an isolated point) to introduce singularity theory into the study of homeostasis. Reed et al. (Bull Math Biol 79(9):1-24, 2017) give two examples of infinitesimal homeostasis in three-node chemical reaction systems: feedforward excitation and substrate inhibition. In this paper we show that there are 13 different three-node networks leading to 78 three-node input-output network configurations, under the assumption that there is one input node, one output node, and they are distinct. The different configurations are based on which node is the input node and which node is the output node. We show nonetheless that there are only three basic mechanisms for three-node input-output networks that lead to infinitesimal homeostasis and we call them structural homeostasis, Haldane homeostasis, and null-degradation homeostasis. Substantial parts of this classification are given in Ma et al. (Cell 138:760-773, 2009) and Ferrell (2016) among others. Our contributions include giving a complete classification using general admissible systems (Golubitsky and Stewart in Bull Am Math Soc 43:305-364, 2006) rather than specific biochemical models, relating the types of infinitesimal homeostasis to the graph theoretic existence of simple paths, and providing the basis to use singularity theory to study higher codimension homeostasis singularities such as the chair singularities introduced in Nijhout and Reed (Integr Comp Biol 54(2):264-275, 2014. https://doi.org/10.1093/icb/icu010) and Nijhout et al. (Math Biosci 257:104-110, 2014). See Golubitsky and Stewart (2017). The first two of these mechanisms are illustrated by feedforward excitation and substrate inhibition. Structural homeostasis occurs only when the network has a feedforward loop as a subnetwork; that is, when there are two distinct simple paths connecting the input node to the output node. Moreover, when the network is just the feedforward loop motif itself, one of the paths must be excitatory and one inhibitory to support infinitesimal homeostasis. Haldane homeostasis occurs when there is a single simple path from the input node to the output node and then only when one of the couplings along this path has strength 0. Null-degradation homeostasis is illustrated by a biochemical example from Ma etal. (2009); this kind of homeostasis can occur only when the degradation constant of the third node is 0. The paper ends with an analysis of Haldane homeostasis infinitesimal chair singularities.

A Riccati solution for the ideal MHD plasma response with applications to real-time stability control

Active feedback control of ideal MHD stability in a tokamak requires rapid plasma stability analysis. Toward this end, we reformulate the δW stability method with a Hamilton-Jacobi theory, elucidating analytical and numerical features of the generic tokamak ideal MHD stability problem. The plasma response matrix is demonstrated to be the solution of an ideal MHD matrix Riccati differential equation. Since Riccati equations are prevalent in the control theory literature, such a shift in perspective brings to bear a range of numerical methods that are well-suited to the robust, fast solution of control problems. We discuss the usefulness of Riccati techniques in solving the stiff ordinary differential equations often encountered in ideal MHD stability analyses—for example, in tokamak edge and stellarator physics. We demonstrate the applicability of such methods to an existing 2D ideal MHD stability code—DCON [A. H. Glasser, Phys. Plasmas 23, 072505 (2016)]—enabling its parallel operation in near real-time, with wall-clock time ≪1s. Such speed may help enable active feedback ideal MHD stability control, especially in tokamak plasmas whose ideal MHD equilibria evolve with inductive timescale τ≳ 1s—as in ITER.

Multivariable State Feedback Control as a Foundation for Lithium-Ion Battery Pack Charge and Capacity Balancing

This article presents a framework for examining the impact of different balancing strategies on cell-to-cell variations in state of charge (SOC), maximum capacity, and resistance within a lithium-ion battery pack. The literature motivates this work by showing that imbalance in both SOC and maximum capacity places limitations on pack-level charge and discharge capabilities. The use of equalization circuits to mitigate differences in SOC and voltage between cells in a battery pack is well-established. The goal of this article, in contrast, is to make the case for the use of state feedback control as a tool for balancing both SOC and maximum charge capacity. Specifically, the article draws on the electrochemical modeling and control theory literature to examine two balancing strategies: voltage balancing and a proposed novel multivariable SOC/capacity balancing strategy. The latter strategy navigates the implicit short-term trade-offs between SOC and capacity balancing by inducing a temporary SOC imbalance between cells to increase the dissipation rate of capacity imbalance. Results indicate that voltage balancing fails to eliminate capacity and resistance imbalance between cells. In contrast, the proposed method is able to eliminate charge, capacity and resistance imbalance within the lifespan of the pack, and extends pack lifespan by 9.2%.

Dynamic of Distributed Parameter Systems; From Infinite to Finite Systems

This work focuses on describing the dynamicbehavior of distributed parameter systems by means ofspatial discretization. In control and system theory literature, distributed parameter systems are simply thesystems that are represented mathematically by partialdifferential equations. Thus, the key notion of this work isrepresenting the dynamic behavior of distributed parametersystems by means of spatial discretization of the dynamics ofthe system. Spatial discretization, or lumping, aims totransform a partial differential equation to a set of ordinarydifferential equations. The proposed concept approximatesthe dynamic behavior of the physical system by lumping itto small tranches and then using the finite differencesmethod to interconnect formally these lumps and simulatesthe dynamic of the overall system.

Generalized inverses of the vandermonde matrix: Applications in control theory

In the literature of control and system theory, several explicit formulae appeared for solving square Vandermonde systems and computing the inverse of it. In the present paper, we will discuss and present analytically the generalized inverses of the rectangular and square Vandermonde matrix. These matrices have been appeared recently in an interesting control and system theory problem, where the change of the initial state of a linear descriptor system in (almost) zero time is required.

Cascaded PID Controller Design for Heating Furnace Temperature Control

The paper evaluates the performance of cascaded PID controller designs for the temperature control of an industrial heating furnace. From the control theory literature it is clear that ideal PID controller is an obsolete for the control of non-linear processes like temperature. PID controller in cascaded architecture is the best choice compared to conventional single loop control system for controlling these nonlinear processes. However, it is constrained in choosing the better PID gains. Hence, this paper is such an approach to set the better values of PID gains in cascaded form by evaluating the performance with conventional tuning formulas. Performance analysis of various algorithms was carried out by finding the system's dynamic performance characteristics in each case. The entire system is modeled by using MATLAB/Simulink, The simulation results indicate that the proposed cascaded PID design could results to rapidity in response with robust dynamic performance.

Characterization of Information Channels for Asymptotic Mean Stationarity and Stochastic Stability of Nonstationary/Unstable Linear Systems

Stabilization of nonstationary linear systems over noisy communication channels is considered. Stochastically stable sources, and unstable but noise-free or bounded-noise systems have been extensively studied in the information theory and control theory literature since the 1970s, with a renewed interest in the past decade. There have also been studies on noncausal and causal coding of unstable/nonstationary linear Gaussian sources. In this paper, tight necessary and sufficient conditions for stochastic stabilizability of unstable (nonstationary) possibly multidimensional linear systems driven by Gaussian noise over discrete channels (possibly with memory and feedback) are presented. Stochastic stability notions include recurrence, asymptotic mean stationarity and sample path ergodicity, and the existence of finite second moments. Our constructive proof uses random-time state-dependent stochastic drift criteria for stabilization of Markov chains. For asymptotic mean stationarity (and thus sample path ergodicity), it is sufficient that the capacity of a channel is (strictly) greater than the sum of the logarithms of the unstable pole magnitudes for memoryless channels and a class of channels with memory. This condition is also necessary under a mild technical condition. Sufficient conditions for the existence of finite average second moments for such systems driven by unbounded noise are provided.

Social Bonds, Self-Control, and Adult Criminality

Recent modifications to self-control theory suggest that influential factors (bonds) equate to self-control in the calculation of whether or not to engage in deviant behavior. Hirschi argued that self-control should fare better as a theory when it is operationalized as the number and salience of an individual’s social bonds, rather than as a cognitive scale, or count of previous acts, as suggested by the original theory. This study extends the control theory literature by assessing the impact of redefined self-control, as well as attitudinal self-control, on adult criminal behavior. Data analyzed were from Waves 10 and 11 of the National Youth Survey Family Study. Findings suggest that both forms of self-control (new and old) are equivalently predictive of adult crime, yet it is unlikely that they are capturing the same phenomenon during adulthood. Implications for control theory are discussed.

On the Perturbation of Homogeneous LTI Descriptor Differential Systems

AbstractIn the literature of linear system and mathematical control theory, several different techniques have been developed for the solution of homogeneous LTI descriptor differential systems. In this paper, we apply the Weierstrass canonical form, and we investigate the conditions under which a system with specific structure and desirable properties is constructed. Consequently, we define the family of perturbed pencils such as the solutions of the initial and the relative perturbed system are r-close with respect to a Frobenius distance. A step-algorithm is also presented enhancing more the results of the paper.

Stabilization of the Inverted Pendulum on a Skate

In the classical control theory literature, the inverted pendulum stabilization problem has been addressed in two ways: as an application of the vibrational control principle, or by imagining that the pivot is on a cart moved according to a suitable feedback law. Several other stabilization methods have been recently proposed.In this paper we propose a new approach: the pendulum is assembled on an ice skate, subject to some constraints and equipped with a driving device. The upper equilibrium position is stabilized provided that the speed of the skate is sufficiently large.

Short-time asymptotics of heat kernels for a class of hypoelliptic operators

We compute the short-time asymptotics of heat kernels for a family of hypoelliptic operators, and we relate these to the value function of an associated variational problem previously investigated in the control theory literature. These heat kernels generalize to the noncontact case a heat kernel that has been previously obtained for the contact hypoelliptic Laplacian defined on the Heisenberg group.

Stabilization of the inverted pendulum on a skate

In the classical control theory literature, the inverted pendulum stabilization problem has been addressed in two ways: as an application of the vibrational control principle, or by imagining that the pivot is on a cart moved according to a suitable feedback law. Several other stabilization methods have been recently proposed. In this paper we propose a new approach: the pendulum is assembled on an ice skate, subject to some constraints and equipped with a driving device. The upper equilibrium position is stabilized provided that the speed of the skate is sufficiently large.

Frequency-Specific Effects of Stabilization Policies

In this paper, we describe a set of trade-offs that policymakers face with respect to fluctua? tions at different frequencies. A complete treat? ment is given in Brock, Durlauf, and Rondina (2008). The presence of frequency-specific effects of policy choices has generally not been a focus of macroeconomic policy analysis, important exceptions include Alexei Onatski and Noah M. Williams (2003). Our analysis dif? fers from previous studies in that our objective is to understand to what extent a policymaker is forced, when choosing a feedback rule to sta? bilize the economy, to exacerbate volatility at some frequencies in order to reduce them at oth? ers, or whether it is possible to reduce volatil? ity at all frequencies. These trade-offs, known as design limits in the control theory literature, are important in understanding what stabiliza? tion policies do, and also possess implications for policy design in the presence of policymaker ignorance.

Invariants of pseudo-random number generators

Pseudo-random number generators of the form Xn+i = P{xn), Vn = h{xn) are ubiquitous in applications ranging from cryptology to statistics. Such systems have been studied extensively in the control theory literature when Xn € K. In this paper we make a detailed study of the invariants of such systems when the underlying field is the Galois field of two elements. We consider various groups that act on such system.

Local asymptotic feedback stabilization to a submanifold: Topological conditions

We consider the problem of local asymptotic feedback stabilization of a control system x ˙ = f ( x , u ) defined in R n to a compact, connected, oriented, embedded codimension one submanifold P of R n using a continuous feedback law. This generalizes the problem of local asymptotic feedback stabilization to a point which has been previously considered in the control theory literature. It is natural to expect the topology of P to play a role in deciding whether or not local asymptotic stabilization to P of the system x ˙ = f ( x , u ) is feasible via continuous feedback, and our aim in this paper is precisely to outline necessary conditions on the topology of P for stabilization via continuous feedback to be achievable.

Economic Properties of the Risk Sensitive Criterion for Portfolio Management

The idea of using stochastic control methods for theoretical studies of portfolio management has long been standard, with maximum expected utility criteria commonly being used. But in recent years a new kind of criterion, the risk sensitive criterion, has emerged from the control theory literature and been applied to portfolio management. This paper studies various economic properties of this criterion for portfolio management, thereby providing justification for its theoretical and practical use. In particular, it is shown that the risk sensitive criterion amounts to maximizing a portfolio's risk adjusted growth rate. In other words, it is essentially the same as what is commonly done in practice: find the best trade‐off between a portfolio's average return and its average volatility.

Control Theory for Linear Systems

9R15. Control Theory for Linear Systems. - HL Trentelman (Dept of Math, Univ of Groningen, PO Box 800, Groningen, 9700 AV, Netherlands), AA Stoorvogel, M Hautus (Dept of Math and Comput Sci, Eindhoven Univ of Tech, PO Box 513, Eindhoven, 5600 MB, Netherlands). Springer-Verlag London Ltd, Surrey, UK. 2001. 389 pp. ISBN 1-85233-316-2. $84.95.Reviewed by L Dewell (Adv Tech Center, Lockheed Martin Advanced Tech Center, 3251 Hanover St, Palo Alto CA 94304).In the preface of this book, the authors stake out a strikingly broad boundary: “the theory of feedback control design for linear, finite-dimensional, time-invariant state space systems with inputs and outputs.” It is a testament to the recent growth of this field that had such a title appeared in the early 1970s, it would scarcely require 100 pages, let alone the nearly 400 pages of this book. Indeed, such a wide scope of a book today may either elicit suspicions of the authors of either being overly cursory in their treatment or overly parochial in their interpretation of the subject. In this book, however, the authors treat the subject with a coherence and an organization that made the large scope seem perfectly natural and pleasantly illuminating. The book appears to be adapted primarily to a reader interested in the theory of control for finite-dimensional linear systems, as opposed to applied engineers or scientists. The very restriction of the subject to such systems, however, allows for the applied control systems analyst to gain insights into recent developments in the field. The focus is certainly towards the development of a tight theoretical basis for the results, rather than applied examples, case studies or exercises. The authors assume that the reader is proficient in the linear algebra tools to analyze state space linear systems. In the opinion of this reviewer, it is probably best that the reader approach the book with an existing knowledge of the major results of the past few decades on the control of state space systems—most particularly linear quadratic regulators (LQR) and the linear quadratic Gaussian (LQG) problem. This book can quite readily stand on its own as an introduction to the recent results of H-infinity control theory from this foundation. The authors approach their subject from the point of view of geometric control theory, which requires a particular nomenclature and foundation with which the reader may not initially be familiar. The authors spend the first two chapters building such a foundation. In the next two chapters, the authors deal with the problem of disturbance decoupling and the notions of controlled and conditioned invariant subspaces. These are very fascinating chapters which alone make the book a worthwhile read. The following three chapters delve deeply into the structural properties of linear control systems, including system zeros and system invertibility. The next five chapters “deliver the goods” that the authors advertised in their preface—tracking and regulation, LQ optimal control, stochastic optimal control, and H-infinity control in finite- and infinite-time, with full-state and with measurement feedback. Each chapter was superbly organized into an introduction, a presentation of ideas, and a set of concluding notes. The notes were particularly pleasant to read, as they give the reader a clear connection between the material and the work of others over the last several decades. The index of the book is excellent (it includes a symbol index, which is frequently overlooked), and all results are painstakingly supported by proofs. If there is one area where the book seems inconsistent, it is in the paucity of numerical examples or case studies to illustrate the material of the book. This was doubtless a conscious decision of the authors, but the decision resulted in the work having a certain opacity and theoretical detachment that was distracting at times. The authors motivate the book in Chapter 1 by discussing, in detail, a practical problem of stationkeeping control of a geosynchronous satellite. After such a strong practical introduction, it would seem appropriate to apply the theory to a practical problem, perhaps even to the satellite problem suggested by the authors. Moreover, the authors stress in Chapters 4 and 6 that a disturbance decoupling controller can actually be numerically computed by algorithms they provide. An actual demonstration of such a controller synthesis would have been very powerful. Control Theory for Linear Systems is a fine contribution to the literature of control theory of linear systems. It offers a powerful organization and structure to the very broad subject of the control of finite-dimension linear systems and brings the torrent of activity of this field over the last several decades to a tight focus and perspective that has been lacking heretofore. This book is highly recommended for purchase by libraries and researchers in the field.