A “mixed” small gain and passivity theorem in the frequency domain

We show that the negative feedback interconnection of two causal, stable, linear time-invariant systems, with a “mixed” small gain and passivity property, is guaranteed to be finite-gain stable. This “mixed” small gain and passivity property refers to the characteristic that, at a particular frequency, systems in the feedback interconnection are either both “input and output strictly passive”; or both have “gain less than one”; or are both “input and output strictly passive” and simultaneously both have “gain less than one”. The “mixed” small gain and passivity property is described mathematically using the notion of dissipativity of systems, and finite-gain stability of the interconnection is proven via a stability result for dissipative interconnected systems.     

On ?∞, robust stability and performance with ?2 and ?∞ perturbations

In this paper we will consider systems with linear time-invariant perturbations. We will analyze robust performance in the ?₂ and ?_∞ settings. The ?₂ setting gives rise to the familiar case of structured singular values, and a stability criterion is given by the “small μ” theorem. We show that although the necessary and sufficient criterion of robust stability for the ?_∞ case (?_∞ stability with structured ?_∞-gain bounded perturbations) is the same “small μ” criterion, a system with ?₂-gain bounded perturbations is never ?_∞ stable.     

Detection filter design for LPV systems—a geometric approach

This paper investigates fault detection and isolation of linear parameter-varying (LPV) systems by using parameter-varying (C,A)-invariant subspace and parameter-varying unobservability subspaces. The so called “detection filter” approach, formulated as the fundamental problem of residual generation (FPRG) for linear time-invariant (LTI) systems, is extended for a class of LPV systems. The question of stability is addressed in the terms of Lyapunov quadratic stability by using linear matrix inequalities. The results are applied to the model of a generic small commercial aircraft.     

Realization and identification of autonomous linear periodically time-varying systems

The subsampling of a linear periodically time-varying system results in a collection of linear time-invariant systems with common poles. This key fact, known as “lifting”, is used in a two-step realization method. The first step is the realization of the time-invariant dynamics (the lifted system). Computationally, this step is a rank-revealing factorization of a block-Hankel matrix. The second step derives a state space representation of the periodic time-varying system. It is shown that no extra computations are required in the second step. The computational complexity of the overall method is therefore equal to the complexity for the realization of the lifted system. A modification of the realization method is proposed, which makes the complexity independent of the parameter variation period. Replacing the rank-revealing factorization in the realization algorithm by structured low-rank approximation yields a maximum likelihood identification method. Existing methods for structured low-rank approximation are used to identify efficiently a linear periodically time-varying system. These methods can deal with missing data.     

On a performance-robustness trade-off intrinsic to the natural anti-windup problem

Any natural definition of the anti-windup (AW) control problem requires the design of an add-on compensator which, connected to a saturating closed loop system (which would be well-behaved in the absence of saturation), guarantees stability and, as long as the saturation limits are never exceeded, preservation of the original linear behaviour. Three main contributions are given in this paper. First, it is shown that the “model-matching” requirement implied by the preservation of the linear response can be incompatible with the achievement of robust stability in the presence of large uncertainties, even if the controlled plant is robustly open loop stable. Then, a reasonable “weakened” AW problem is introduced, in which the “model-matching” requirement is considered just as a performance requirement (instead of a hard constraint) whose relaxation can be traded off with robustness to larger uncertainties. Finally, the approach proposed in Teel and Kapoor [(1997a). In Proceedings of the fourth European control conference] is extended to deal with the new problem, leading to a family of state feedback compensators parameterized in terms of a state feedback gain (already present in the original approach) and a stable linear time invariant filter. A detailed design procedure for determining suitable values of the parameters is also described.     

Decentralized switching control for hierarchical systems

In this paper, a decentralized switching scheme is introduced for uncertain interconnected systems with a structure which is “approximately hierarchical”, and where the switching controller acts on the control agents independently of each other. In this problem, it is assumed that the plant at any point of time can be described by a finite set of linear time-invariant (LTI) finite-dimensional models, but that the switching controller does not require any knowledge of these plant models; the only requirement made is that there exists a known finite set of decentralized controllers, containing at least one controller which can stabilize and regulate the actual physical plant at any time. Simulation results obtained for the proposed decentralized switching controller are given.     

Fixed-architecture controller synthesis for systems with input–output time-varying nonlinearities

In this paper we develop a fixed-architecture controller analysis and synthesis framework that addresses the problem of multivariable linear time-invariant systems subject to plant input and plant output time-varying nonlinearities while accounting for robust stability and robust performance over the allowable class of nonlinearities. The proposed framework is based on the classical Luré problem and the related Aizerman conjecture concerning the stability of a feedback loop involving a sector-bounded nonlinearity. Specifically, we extend the classical notions of absolute stability theory to guarantee closed-loop stability of multivariable systems in the presence of input nonlinearities. In order to capture closed-loop system performance we also consider the minimization of a quadratic performance criterion over the allowable class of input nonlinearities. Our approach is directly applicable to systems with saturating actuators and provides full and reduced-order dynamic compensators with a guaranteed domain of attraction. The principal result is a set of constructive sufficient conditions for absolute stabilization characterized via a coupled system of algebraic Riccati and Lyapunov equations. The effectiveness of design approach is illustrated by several numerical examples. © 1997 John Wiley & Sons, Ltd.     

Decomposition principle in model predictive control for linear systems with bounded disturbances

Considering a constrained linear system with bounded disturbances, this paper proposes a novel approach which aims at enlarging the domain of attraction by combining a set-based MPC approach with a decomposition principle. The idea of the paper is to extend the “pre-stabilizing” MPC, where the MPC control sequence is parameterized as perturbations to a given pre-stabilizing feedback gain, to the case where the pre-stabilizing feedback law is given as the linear combination of a set of feedback gains. This procedure leads to a relatively large terminal set and consequently a large domain of attraction even when using short prediction horizons. As time evolves, by minimizing the nominal performance index, the resulting controller reaches the desired optimal controller with a good asymptotic performance. Compared to the standard “pre-stabilizing” MPC, it combines the advantages of having a flexible choice of feedback gains, a large domain of attraction and a good asymptotic behavior.     

Periodically time-varying memory state-feedback controller synthesis for discrete-time linear systems

In this paper, we deal with discrete-time linear periodic/time-invariant systems with polytopic-type uncertainties and propose a new linear matrix inequality (LMI)-based method for robust state-feedback controller synthesis. In stark contrast with existing approaches that are confined to memoryless static controller synthesis, we explore dynamical controller synthesis and reveal a particular periodically time-varying memory state-feedback controller (PTVMSFC) structure that allows LMI-based synthesis. In the context of robust controller synthesis, we prove rigorously that the proposed design method encompasses the well-known extended-LMI-based static controller synthesis methods as particular cases. Through numerical experiments, we demonstrate that the suggested design method is indeed effective in achieving less conservative results, under both periodic and time-invariant settings. We finally derive a viable test to verify that the designed robust PTVMSFC is “exact” in the sense that it attains the best achievable robust performance. This exactness verification test works fine in practice, and we will show via a numerical example that exact robust control is indeed attained by designing PTVMSFCs, even for such a problem where the standard memoryless static state-feedback fails.     

Output-feedback stabilization of an unstable wave equation

We consider the problem of stabilization of a one-dimensional wave equation that contains instability at its free end and control on the opposite end. In contrast to classical collocated “boundary damper” feedbacks for the neutrally stable wave equations with one end satisfying a homogeneous boundary condition, the controllers and the associated observers designed in the paper are more complex due to the open-loop instability of the plant. The controller and observer gains are designed using the method of “backstepping,” which results in explicit formulae for the gain functions. We prove exponential stability and the existence and uniqueness of classical solutions for the closed-loop system. We also derive the explicit compensators in frequency domain. The results are illustrated with simulations.     

Solving macroeconomic models with “off-the-shelf” software: An example of potential pitfalls

When working with large-scale models or numerous small models, there can be a temptation to rely on default settings in proprietary software to derive solutions to the model. In this paper we show that, for the solution of non-linear dynamic models, this approach can be inappropriate. Alternative linear and non-linear specifications of a particular model are examined. One version of the model, expressed in levels, is highly non-linear. A second version of the model, expressed in logarithms, is linear. The dynamic solution of each model version has a combination of stable and unstable eigenvalues so that any dynamic solution requires the calculation of appropriate “jumps” in endogenous variables. We can derive a closed-form solution of the model, which we use as our “true” benchmark, for comparison with computational solutions of both linear and non-linear models. Our approach is to compare the “goodness of fit” of reverse-shooting solutions for both the linear and non-linear model, by comparing the computational solutions with the benchmark solution. Under the basic solution method with default settings, we show that there is significant difference between the computational solution for the non-linear model and the benchmark closed-form solution. We show that this result can be substantially improved using modifications to the solver and to parameter settings.     

ℋ∞ filtering for continuous-time linearsystems with delay

The problem of ℋ_∞ filtering for continuous-time linear systems with time-delayed measurement is investigated. The authors develop a methodology for designing linear filters which ensure a prescribed bound on the ℒ₂-induced gain from the noise signals to the estimation error. Filtering problems for time varying systems over a finite-horizon, as well as stationary infinite-horizon filtering for time-invariant systems, are tackled. In the finite-horizon case, our estimation procedure entails an overdesign that stems from the last d seconds of the time interval [0,T], where d is the delay length. This overdesign becomes smaller as T increases, and it vanishes in the infinite-horizon case     

Solving a non-linear model: The importance of model specification for deriving a suitable solution

In this paper, we consider a macroeconomic model with alternative linear and non-linear specifications. One version of the model, expressed in levels, is highly non-linear and has at least two steady-state equilibria. One of these equilibria has an economically meaningful interpretation, while the other does not have a sensible economic interpretation. A second version of the model, expressed in logarithms, is linear and has a unique steady-state equilibrium, which corresponds to the economically meaningful equilibrium of the non-linear version of the model. The dynamic solution of each model version has a combination of stable and unstable eigenvalues so that any dynamic solution requires the calculation of appropriate “jumps” in endogenous variables. Attempts to solve these models, using forward-shooting and reverse-shooting algorithms, show that the forward-shooting algorithm chooses the “wrong” solution for the non-linear model, but the “right” solution for the linear model. The reverse-shooting algorithm chooses the “right” solution in both cases. We demonstrate how this result is driven by particular properties of the two versions of the model.     

Robust compensation of a Cart–Inverted Pendulum system using a periodic controller: Experimental results

Based on Das and Dey (2007), this paper designs and implements a periodic controller to achieve, via zero-placement, robustness of a physical Cart–Inverted Pendulum system with respect to differential gain variations in the output sensors. Experimental results that verify the superiority of this controller over linear time-invariant (LTI) ones are also presented.     

Standard ML–NJ Weak Polymorphism and Imperative Constructs

Standard ML of New Jersey (SML–NJ) uses “weak type variables” to restrict the polymorphic use of functions that may allocate reference cells, manipulate continuations, or use exceptions. However, the type system used in the SML–NJ compiler has not previously been presented in a form other than source code nor proved correct. We present a set of typing rules, based on analysis of the concepts underlying “weak polymorphism”, that appears to subsume the implemented algorithm and uses type variables of only a slightly more general nature than the compiler. One insight in the analysis is that allowing a variable to occur both “ordinarily” and “weakly” in a type permits a simpler and more flexible formulation of the typing rules. In particular, we are able to treat applications of polymorphic functions to imperative arguments with greater flexibility than SML–NJ. The soundness of the type system is proved for imperative code using operational semantics, by showing that evaluation preserves typability. By incorporating assumptions about memory addresses in the type system, we avoid proofs by co-induction.     

Measuring the premium on common knowledge in computer-mediated coordination problems

Common knowledge, whereby everybody knows something, and everybody knows that everybody knows it, and so on ad infinitum, is claimed to be central to coordination in organizations. However, this claim has so far not received empirical support, as a method to empirically compare common knowledge with other forms of knowledge has not been available. In this article, we present a novel method through which we empirically estimate the common knowledge premium—the level of users’ preference of common knowledge over “knowledge by all” (where everybody knows something, but not necessarily everybody knows that everybody knows it). Using the method we show that a “premium” of common knowledge over “knowledge by all” does exist consistently, across populations and measuring variations. The findings provide empirical support for the centrality of common knowledge. Implications of the study are discussed.     

On ultimate boundedness around non-assignable equilibria of linear time-invariant systems

In this note we investigate the following questions: given a (finite-dimensional) linear time-invariant (LTI) multivariable system and a constant desired value for its output, say y_?. Assume there is no assignable equilibrium point corresponding to y_?. How “close” to y_? can we ultimately keep the output using LTI static state-feedback stabilizing controllers? Can this neighborhood of y_? be reduced with dynamic, nonlinear, time-varying controllers? Our main contributions are the proof that the optimal ultimate boundedness neighborhood is achieved with LTI static state-feedback, the explicit computation of the neighborhood's size and the proof, under some reasonable rank assumptions, that the system has non-assignable values for the output if and only if it has a transmission zero at zero. Interestingly, there is no connection between this problem and the more familiar concepts of controllability and observability.     

A convex parametrization of risk-adjusted stabilizing controllers

The focal point of this paper is the synthesis of controllers under risk-specifications. In recent years there has been a growing interest in the development of techniques for controller design where, instead of requiring that the performance specifications are met for every possible value of admissible uncertainty, it is required that the risk of performance violation is below a small well-defined risk level. In contrast to previous work, where the search for the controller gains is done using randomized algorithms, the results in this paper show that for a class of uncertain linear time invariant systems, the search for the “risk-adjusted” controller can be done efficiently using deterministic algorithms. More precisely, for the case when the characteristic polynomial of the closed loop system depends affinely on the uncertainty, we provide a convex parametrization of “risk-adjusted” stabilizing controllers.     

Sufficient bootstrapping

In this paper, we introduce an idea we refer to as sufficient bootstrapping, which is based on retaining only distinct individual responses, and also develop a theoretical framework for the techniques. We demonstrate through numerical illustrations that the proposed sufficient bootstrapping may be better than the conventional bootstrapping in certain situations. The expected gain by the sufficient bootstrapping has been computed for small and large sample sizes. The relative efficiency shows that there could be significant gain by the sufficient bootstrapping and it could reduce computational burden. Variance expressions for both the conventional and sufficient bootstrapping sample means are derived. Here the word “sufficient” is being used in the sense that it is “sufficient to take just one of any duplicated items in the bootstrap sample” and is not tightly connected to sufficiency in terms of any likelihood perspective. R code for comparing bootstrapping and sufficient bootstrapping are provided. A huge scope of further studies is suggested.     

Feminism asks the “Who” questions in HCI

In this brief personal essay, I describe some of the ways that feminism has influenced my life as a researcher and practitioner in HCI and CSCW - in the creation of work to be read by others, in the critical reading of works that were created by others, and in the planning and framing of practical work in enterprise workplaces. I discuss three variations of “Who” questions that feminism helps us to ask in HCI: The “who” of the identity of the user; the “who” of the identity of organizational actors; and the “who” of the practitioner or researcher.