An LMI approach to minimum sensitivity analysis with application to fault detection

This paper systematically studies the minimum input sensitivity analysis problem. The lowest level of sensitivity of system outputs to system inputs is defined as an H_- index. A full characterization of the H_- index is given, first, in terms of matrix equalities and inequalities, and then in terms of linear matrix inequalities (LMIs), as a dual of the Bounded Real Lemma. A related problem of input observability is also studied, with new necessary and sufficient conditions given, which are necessary for a fault detection system to have a nonzero worst-case fault sensitivity. The above results are applied to the problem of fault detection filter analysis, with numerical examples given to show the effectiveness of the proposed approaches.     

Linear matrix inequality tests for frequency domain inequalities with affine multipliers

This paper revisits an alternative formulation of the Kalman-Yakubovich-Popov (KYP) Lemma, relating an infinite dimensional Frequency Domain Inequality (FDI) to a pair of finite dimensional Linear Matrix Inequalities (LMI). It is shown that this alternative formulation allows a certain class of the coefficient matrix of the FDI to be frequency dependent without introducing conservatism. The construction provided in the present paper is helpful in other problems where system augmentation is commonly used, such as those involving rational or polynomial multipliers for stability and performance analysis.     

On coefficient inequalities of functions associated with conic domains

In this paper, the concepts of Janowski functions and the conic regions are combined to define a new domain which represents the conic type regions. Different views of this modified conic domain for specific values are shown graphically for better understanding of the behavior of this domain. The class of such functions which map the open unit disk E onto this modified conic domain is defined. Also the classes of k-uniformly Janowski convex and k-Janowski starlike functions are defined and their coefficient inequalities are formulated. The coefficient bound for a certain class of analytic functions, proved by Owa et al. (2006) in [16], has also been improved.     

Exponential stability of singular systems with multiple time-varying delays

This paper deals with the class of continuous-time singular linear systems with multiple time-varying delays in a range. The global exponential stability problem of this class of systems is addressed. Delay-range-dependent sufficient conditions such that the system is regular, impulse-free and α-stable are developed in the linear matrix inequality (LMI) setting. Moreover, an estimate of the convergence rate of such stable systems is presented. A numerical example is employed to show the usefulness of the proposed results.     

Application of a fault detection filter to structural health monitoring

A model-based fault detection filter is developed for structural health monitoring of a simply supported beam. The structural damage represented in the plant model is shown to decompose into a known fault direction vector maintaining a fixed direction, dependent on the damage location, and an arbitrary fault magnitude representing the extent of the damage. According to detection filter theory, if damage occurs, under certain circumstances the fault will be uniquely detected and identified through an associated invariance in the direction imposed on the fault detection filter residuals. The spectral algorithm used to design the detection filter is based on a left eigenstructure assignment approach which accommodates system sensitivities that are revealed as ill-conditioned matrices formed from the eigenvectors in the construction of the detection filter gains. The detection filter is applied to data from an aluminum simply supported beam with four piezoelectric sensors and one piezoelectric actuator. By exciting the structure at the first natural frequency, damage in the form of a 5 mm saw cut made to one side of the beam is detected and localized.     

Design of linear time-invariant controllers for multirate systems

In this paper we discuss a frequency domain approach to model multirate single-input single-output (SISO) systems which facilitates design of linear time-invariant (LTI) controllers operating at the fast rate. To illustrate the approach we consider a dual-rate system with slow output measurements and fast control actions. We obtain necessary and sufficient conditions for the existence of stabilizing linear time-invariant (LTI) controllers for which model matching is also achieved at the fast rate with a desired single-rate system. Moreover, a solution to the problem of parameterizing the set of such LTI controllers is also given.     

Vector transport for shape-from-shading

In this paper we describe a new shape-from-shading method. We show how the parallel transport of surface normals can be used to impose curvature consistency and also to iteratively update surface normal directions so as to improve the brightness error. We commence by showing how to make local estimates of the Hessian matrix from surface normal information. With the local Hessian matrix to hand, we develop an “EM-like” algorithm for updating the surface normal directions. At each image location, parallel transport is applied to the neighbouring surface normals to generate a sample of local surface orientation predictions. From this sample, a local weighted estimate of the image brightness is made. The transported surface normal which gives the brightness prediction which is closest to this value is selected as the revised estimate of surface orientation. The revised surface normals obtained in this way may in turn be used to re-estimate the Hessian matrix, and the process iterated until stability is reached. We experiment with the method on a variety of real world and synthetic data. Here we explore the properties of the fields of surface normals and the height data delivered by the method.     

Robust fault detection and isolation of time-delay systems using a geometric approach

This paper investigates the development of Fault Detection and Isolation (FDI) filters for both retarded and neutral time-delay systems with unknown time-varying delays. Using a geometric framework, the notion of a finite unobservability subspace is introduced for time-delay systems and an algorithm for its construction is presented. A bank of residual generators is then designed so that each residual is affected by one fault and is partially decoupled from the others while the H_∞ norm of the transfer function between the disturbances and the uncertainties in delays and the residuals are guaranteed to remain less than a prescribed value. Furthermore, it is shown that in the case of known delays it is possible to generate residuals that enjoy perfect decoupling properties among faults. Simulation results presented demonstrate the effectiveness of our proposed FDI algorithms.     

Gain-scheduling control of LFT systems using parameter-dependent Lyapunov functions

In this paper, we propose a new control design approach for linear fractional transformation (LFT) systems using parameter-dependent Lyapunov functions. Instead of assuming parameter dependency in LFT fashion, we consider general parameter-dependent controllers to achieve better closed-loop performance. Using full-block multipliers, new LPV synthesis conditions have been derived in terms of finite number of linear matrix inequalities (LMIs). Both continuous- and discrete-time cases are discussed. A ship steering example has been used to demonstrate advantages and benefits of the proposed approach.     

New bounded real lemma for discrete-time singular systems

In this paper, the problem of bounded real lemma for discrete-time singular system with strict matrix inequalities is investigated. It is proved that the bounded real lemma for discrete singular system can be described by a strict matrix inequality. The results lead to more tractable and reliable computation when applying them to design control systems.     

On using prototype reduction schemes to enhance the computation of volume-based inter-class overlap measures

In most pattern recognition (PR) applications, it is advantageous if the accuracy (or error rate) of the classifier can be evaluated or bounded prior to testing it in a real-life setting. It is also well known that if the two class-conditional distributions have a large overlapping volume (almost all the available work on “overlapping of classes” deals with the case when there are only two classes), the classification accuracy is poor. This is because if we intend to use the classification accuracy as a criterion for evaluating a PR system, the points within the overlapping volume tend to lead to maximal misclassification. Unfortunately, the computation of the indices which quantify the overlapping volume is expensive. In this vein, we propose a strategy of using a prototype reduction scheme (PRS) to approximately, but quickly, compute the latter. In this paper, we demonstrate, first of all, that this is an extremely expedient proposition. Indeed, we show that by completely discarding (we are not aware of any reported scheme which discards “irrelevant” sample (training) points, and which simultaneously attains to an almost-comparable accuracy) the points not included by the PRS, we can obtain a reduced set of sample points, using which, in turn, the measures for the overlapping volume can be computed. The value of the corresponding figures is comparable to those obtained with the original training set (i.e., the one which considers all the data points) even though the computations required to obtain the prototypes and the corresponding measures are significantly less. The proposed method has been rigorously tested on artificial and real-life datasets, and the results obtained are, in our opinion, quite impressive—sometimes faster by two orders of magnitude.     

Three results on frequency assignment in linear cellular networks

In the frequency assignment problem we are given a graph representing a wireless network and a sequence of requests, where each request is associated with a vertex. Each request has two more attributes: its arrival and departure times, and it is considered active from the time of arrival to the time of departure. We want to assign frequencies to all requests so that at each time step any two active requests associated with the same or adjacent vertices use different frequencies. The objective is to minimize the number of frequencies used.We focus exclusively on the special case of the problem when the underlying graph is a linear network (path). For this case, we consider both the offline and online versions of the problem, and we present three results. First, in the incremental online case, where the requests arrive over time, but never depart, we give an algorithm with an optimal (asymptotic) competitive ratio . Second, in the general online case, where the requests arrive and depart over time, we improve the current lower bound on the (asymptotic) competitive ratio to . Third, we prove that the offline version of this problem is NP-complete.     

Split Bregman method for large scale fused Lasso

Ordering of regression or classification coefficients occurs in many real-world applications. Fused Lasso exploits this ordering by explicitly regularizing the differences between neighboring coefficients through an ?₁ norm regularizer. However, due to nonseparability and nonsmoothness of the regularization term, solving the fused Lasso problem is computationally demanding. Existing solvers can only deal with problems of small or medium size, or a special case of the fused Lasso problem in which the predictor matrix is the identity matrix. In this paper, we propose an iterative algorithm based on the split Bregman method to solve a class of large-scale fused Lasso problems, including a generalized fused Lasso and a fused Lasso support vector classifier. We derive our algorithm using an augmented Lagrangian method and prove its convergence properties. The performance of our method is tested on both artificial data and real-world applications including proteomic data from mass spectrometry and genomic data from array comparative genomic hybridization (array CGH). We demonstrate that our method is many times faster than the existing solvers, and show that it is especially efficient for large p, small n problems, where p is the number of variables and n is the number of samples.     

Simultaneous LQ control of a set of LTI systems using constrained generalized sampled-data hold functions

In this paper, sampled-data control of a set of continuous-time LTI systems is considered. It is assumed that a predefined guaranteed continuous-time quadratic cost function, which is, in fact, the sum of the performance indices for all systems, is given. The main objective here is to design a decentralized periodic output feedback controller with a prespecified form, e.g., polynomial, piecewise constant, exponential, etc., which minimizes the above mentioned guaranteed cost function. This problem is first formulated as a set of matrix inequalities, and then by using a well-known technique, it is reformulated as a LMI problem. The set of linear matrix inequalities obtained provides necessary and sufficient conditions for the existence of a decentralized optimal simultaneous stabilizing controller with the prespecified form (rather than a general form). Moreover, an algorithm is presented to solve the resultant LMI problem. Finally, the efficiency of the proposed method is demonstrated in two numerical examples.     

Model-free Q-learning designs for linear discrete-time zero-sum games with application to H-infinity control

In this paper, the optimal strategies for discrete-time linear system quadratic zero-sum games related to the H-infinity optimal control problem are solved in forward time without knowing the system dynamical matrices. The idea is to solve for an action dependent value function Q(x,u,w) of the zero-sum game instead of solving for the state dependent value function V(x) which satisfies a corresponding game algebraic Riccati equation (GARE). Since the state and actions spaces are continuous, two action networks and one critic network are used that are adaptively tuned in forward time using adaptive critic methods. The result is a Q-learning approximate dynamic programming (ADP) model-free approach that solves the zero-sum game forward in time. It is shown that the critic converges to the game value function and the action networks converge to the Nash equilibrium of the game. Proofs of convergence of the algorithm are shown. It is proven that the algorithm ends up to be a model-free iterative algorithm to solve the GARE of the linear quadratic discrete-time zero-sum game. The effectiveness of this method is shown by performing an H-infinity control autopilot design for an F-16 aircraft.     

Controller certification

A modern problem from aerospace control involves the certification of a large set of potential controllers with either a single plant or a fleet of potential plant systems, with both plants and controllers being MIMO and, for the moment, linear. Experiments on a limited number of controller/plant pairs should establish the stability and a certain level of margin of the complete set. We consider this certification problem for a set of controllers and provide algorithms for selecting an efficient subset for testing. This is done for a finite set of candidate controllers and, at least for SISO plants, for compact infinite set. In doing this, the ν-gap metric will be the main tool. Computational examples are given, including one of certification of an aircraft engine controller. The overarching aim is to introduce truly MIMO margin calculations and to understand their efficacy in certifying stability over a set of controllers and in replacing legacy single-loop gain and phase margin calculations.     

H∞ filtering for discrete-time systems with randomly varying sensor delays

This paper investigates an H_∞ filtering problem for discrete-time systems with randomly varying sensor delays. The stochastic variable involved is a Bernoulli distributed white sequence appearing in measured outputs. This measurement mode can be used to characterize the effect of communication delays and/or data-loss in information transmissions across limited bandwidth communication channels over a wide area. H_∞ filtering of this class of systems is used to design a filter using the measurements with random delays to ensure the mean-square stochastic stability of the filtering error system and to guarantee a prescribed H_∞ filtering performance. A sufficient condition for the existence of such a filter is presented in terms of the feasibility of a linear matrix inequality (LMI). Finally, a numerical example is given to illustrate the effectiveness of the proposed approach.     

Modelling series of studies with a common structure

Consider the situation where the Structuration des Tableaux à Trois Indices de la Statistique (STATIS) methodology is applied to a series of studies, each study being represented by data and weight matrices. Relations between studies may be captured by the Hilbert-Schmidt product of these matrices. Specifically, the eigenvalues and eigenvectors of the Hilbert-Schmidt matrix S may be used to obtain a geometrical representation of the studies. The studies in a series may further be considered to have a common structure whenever their corresponding points lie along the first axis. The matrix S can be expressed as the sum of a rank 1 matrix λuu^T with an error matrix E. Therefore, the components of the vector are sufficient to locate the points associated to the studies. Former models for S where vec(E) are mathematically tractable and yet do not take into account the symmetry of the matrix S. Thus a new symmetric model is proposed as well as the corresponding tests for a common structure. It is further shown how to assess the goodness of fit of such models. An application to the human immunodeficiency virus (HIV) infection is used for assessing the proposed model.     

Detecting change-points in Markov chains

Markov chains provide a flexible model for dependent random variables with applications in such disciplines as physics, environmental science and economics. In the applied study of Markov chains, it may be of interest to assess whether the transition probability matrix changes during an observed realization of the process. If such changes occur, it would be of interest to estimate the transitions where the changes take place and the probability transition matrix before and after each change. For the case when the number of changes is known, standard likelihood theory is developed to address this problem. The bootstrap is used to aid in the computation of p-values. When the number of changes is unknown, the AIC and BIC measures are used for model selection. The proposed methods are studied empirically and are applied to example sets of data.