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This article investigates observability/controllability of a networked dynamic system (NDS) in which system matrices of its subsystems are expressed through linear fractional transformations (LFTs). Some relations have been obtained between this NDS and descriptor systems about their observability/controllability. A necessary and sufficient condition is established with the associated matrices depending affinely on subsystem parameters/connections. An attractive property of this condition is that all the required calculations are performed independently on each individual subsystem. Except well posedness, not any other conditions are asked for subsystem parameters/connections. This is in sharp contrast to recent results on structural observability/controllability, which is proven to be NP-hard. Some characteristics are established for a subsystem that are helpful in constructing an observable/controllable NDS. It has been made clear that subsystems with an input matrix of full column rank are helpful in constructing an observable NDS while subsystems with an output matrix of full row rank are helpful in constructing a controllable NDS. These results are extended to an NDS with descriptor form subsystems. As a byproduct, the full normal rank condition of previous works on network observability/controllability has been completely removed. On the other hand, satisfaction of this condition is shown to be appreciative in building an observable/controllability NDS.

This article studies the controllability degree via analyzing the condition number of Gramian matrix. Our aim is to explore how the network characteristics affect the controllability degree. Specifically, we prove that a large time parameter would worsen the controllability degree. The time parameter could be understood as the network coupling strength. For directed path networks, we derive how edge weights and time parameter jointly determine the best controllability degree. Furthermore, we prove that either adding a new edge or enhancing an existing edge weight appropriately would worsen the controllability degree. Moreover, through the numerical simulation of external inputs deployment, we find a significant statistical relationship between the controllability index and the controllability degree. In this article, the Gramian matrix reveals the importance of network characteristics that cannot be captured by classic Kalman rank condition or Popov–Belevitch–Hautus (PBH) test.

Nonnegative tensor decomposition is a versatile tool for multiway data analysis, by which the extracted components are nonnegative and usually sparse. Nevertheless, the sparsity is only a side effect and cannot be explicitly controlled without additional regularization. In this paper, we investigated the nonnegative CANDECOMP/PARAFAC (NCP) decomposition with the sparse regularization item using $$l_1$$ l 1 -norm (sparse NCP). When high sparsity is imposed, the factor matrices will contain more zero components and will not be of full column rank. Thus, the sparse NCP is prone to rank deficiency, and the algorithms of sparse NCP may not converge. In this paper, we proposed a novel model of sparse NCP with the proximal algorithm. The subproblems in the new model are strongly convex in the block coordinate descent (BCD) framework. Therefore, the new sparse NCP provides a full column rank condition and guarantees to converge to a stationary point. In addition, we proposed an inexact BCD scheme for sparse NCP, where each subproblem is updated multiple times to speed up the computation. In order to prove the effectiveness and efficiency of the sparse NCP with the proximal algorithm, we employed two optimization algorithms to solve the model, including inexact alternating nonnegative quadratic programming and inexact hierarchical alternating least squares. We evaluated the proposed sparse NCP methods by experiments on synthetic, real-world, small-scale, and large-scale tensor data. The experimental results demonstrate that our proposed algorithms can efficiently impose sparsity on factor matrices, extract meaningful sparse components, and outperform state-of-the-art methods.

In this paper, we consider families of linear systems (linear ensembles) defined by matrix pairs ( A ( θ ) , B ( θ ) ) depending on a parameter θ ∈ P that is varying over a compact subset P of the complex plane. In particular, we investigate the existence of open-loop controls which are independent of the parameter θ ∈ P and steer a given family of initial states x 0 ( θ ) arbitrarily close to a desired family of terminal states f ( θ ) in finite time. Here, the maps θ ↦ x 0 ( θ ) and θ ↦ f ( θ ) are assumed to lie in a common appropriately chosen Banach space X n ( P ) of C n -valued functions. If this task is solvable for all initial and terminal states, the pair ( A ( θ ) , B ( θ ) ) is called (completely) ensemble controllable with respect to X n ( P ) . Using a well-known infinite-dimensional version of the Kalman rank condition for systems on Banach spaces, we derive sufficient conditions for cascade and parallel connections of linear ensembles. Moreover, we prove an abstract decomposition theorem which results from a spectral splitting of the matrix family A ( θ ) . Based on these findings as well as on cyclicity conditions for multiplication operators and approximation theory, we obtain necessary and sufficient conditions for ensemble controllability (reachability) with respect to the space of continuous functions and the space of L q -functions. In the last section, results on averaged controllability (reachability) for linear families ( A ( θ ) , B ( θ ) , C ( θ ) ) are presented.

Among the existing estimators of factor-augmented regressions, the CCE approach is the most popular. A major reason for this popularity is the simplicity and good small-sample performance of the approach, making it very attractive from an empirical point of view. The main drawback is that most of the available asymptotic theory is based on quite restrictive assumptions, such as that the common factor component should be independent of the regressors. The present paper can be seen as a reaction to this. The purpose is to study the asymptotic properties of the pooled CCE estimator under more realistic conditions. In particular, the common factor component may be correlated with the regressors, and the true number of common factors, r , can be larger than the number of estimated factors, which in CCE is given by k + 1 , where k is the number of regressors. The main conclusion is that while the estimator is generally consistent, asymptotic normality can fail when r > k + 1 .

We study joint nonlinear state estimation with multi-period measurement vectors that are potentially corrupted by sparse gross errors. We consider a nonlinear sparse optimization formulation for joint sparse error correction and robust state estimation in a nonlinear sensing system wherein we exploit the sparsity and short-term invariance properties of gross error locations. We introduce a sequential convex approximation approach that can be used to solve the nonlinear sparse optimization problem with a convergence guarantee. We derive a necessary rank condition that identifiable gross error matrix should satisfy. Then, we present an identifiability-aware version of the proposed algorithm wherein we exploit the aforementioned identifiability condition to improve the accuracy of gross error localization. We demonstrate the efficacy of the proposed approach by applying it to power system nonlinear state estimation of IEEE 14-bus and 118-bus networks.

This paper introduces a version of the interdependent value model of Milgrom and Weber (1982), where the signals are given by an index gathering signal shifters observed by the econometrician and private ones specific to each bidders. The model primitives are shown to be nonparametrically identified from first-price auction bids under a testable mild rank condition. Identification holds for all possible signal values. This allows to consider a wide range of counterfactuals where this is important, as expected revenue in second-price auction. An estimation procedure is briefly discussed.

Controllability of a complex network system is related to finding a set of minimum number of nodes, known as drivers, controlling which allows having a full control on the dynamics of the network. For some applications, only a portion of the network is required to be controlled, for which target control has been proposed. Often, along the controlling route from driver nodes to target nodes, some mediators (intermediate nodes) are also unwillingly controlled, which might cause various side effects. In controlling cancerous cells, unwillingly controlling healthy cells, might result in weakening them, thus affecting the immune system against cancer. This manuscript proposes a suitable candidate solution to the problem of finding minimum number of driver nodes under minimal mediators. Although many others have attempted to develop algorithms to find minimum number of drivers for target control, the newly proposed algorithm is the first one that is capable of achieving this goal and at the same time, keeping the number of the mediators to a minimum. The proposed controllability condition, based on path lengths between node pairs, meets Kalman's controllability rank condition and can be applied on directed networks. Our results show that the path length is a major determinant of in properties of the target control under minimal mediators. As the average path length becomes larger, the ratio of drivers to target nodes decreases and the ratio of mediators to targets increases. The proposed methodology has potential applications in biological networks. The source code of the algorithm and the networks that have been used are available from the following link: https://github.com/LBBSoft/Target-Control-with-Minimal-Mediators.git

We study the stabilizability of a class of abstract parabolic equations of the form $$\begin{aligned} u'(t)+Au(t)+p(t)Bu(t)=0,\qquad t\ge 0 \end{aligned}$$ u ′ ( t ) + A u ( t ) + p ( t ) B u ( t ) = 0 , t ≥ 0 where the control $$p(\cdot )$$ p ( · ) is a scalar function, A is a self-adjoint operator on a Hilbert space X that satisfies $$A\ge -\sigma I$$ A ≥ - σ I , with $$\sigma >0$$ σ > 0 , and B is a bounded linear operator on X. Denoting by $$\{\lambda _k\}_{k\in {\mathbb {N}}^*}$$ { λ k } k ∈ N ∗ and $$\{\varphi _k\}_{j\in {\mathbb {N}}^*}$$ { φ k } j ∈ N ∗ the eigenvalues and the eigenfunctions of A, we show that the above system is locally stabilizable to the eigensolutions $$\psi _j={\mathrm{e}}^{-\lambda _j t}\varphi _j$$ ψ j = e - λ j t φ j with doubly exponential rate of convergence, provided that the associated linearized system is null controllable. Moreover, we give sufficient conditions for the pair $$\{A,B\}$$ { A , B } to satisfy such a property, namely a gap condition for A and a rank condition for B in the direction $$\varphi _j$$ φ j . We give several applications of our result to different kinds of parabolic equations.

Made available in DSpace on 2021-06-25T11:06:21Z (GMT). No. of bitstreams: 0 Previous issue date: 2020-09-01 Necessary optimality conditions for optimal control problems with mixed state-control equality constraints are obtained. The necessary conditions are given in the form of a weak maximum principle and are obtained under (i) a new regularity condition for problems with mixed linear equality constraints and (ii) a constant rank type condition for the general non-linear case. Some instances of problems with equality and inequality constraints are also covered. Illustrative examples are presented. Department of Applied Mathematics State University of Campinas (Unicamp), Rua Sérgio Buarque de Holanda, N. 651 Department of Applied Mathematics São Paulo State University (Unesp), Rua Cristóvão Colombo, N. 2265 Department of Applied Mathematics São Paulo State University (Unesp), Rua Cristóvão Colombo, N. 2265

We study linear groups G for which for every g in G there exists a subgroup E of G satisfying some sort of rank condition and such that for every x in G there is a positive integer m such that for all n ≥ m the repeated commutator [x, ng] lies in E. If E can always be chosen to be a Chernikov group (resp. a polycyclic-by-finite group) such G can be completely described (Wehrfritz in Adv Group Theory Appl 7:143–157, 2019; Wehrfritz in Boll Unione Mat Ital, to appear). For more general rank conditions our analyses below are complete for positive characteristics but are less so for characteristic zero.

For a discrete-time linear system, we use data from an open-loop experiment to design directly a linear feedback controller enforcing that a given (polyhedral) set of the state is invariant and given (polyhedral) constraints on the control are satisfied. By building on classical results from model-based set invariance and a fundamental result from Willems et al., the controller designed from data has the following desirable features. The satisfaction of the above properties is guaranteed only from data, it can be assessed by solving a numerically-efficient linear program, and, under a certain rank condition, a data-based solution is feasible if and only if a model-based solution is feasible.

In this paper we study the null controllability of some non diagonalizable degenerate parabolic systems of PDEs, we assume that the diffusion, coupling and controls matrices are constant and we characterize the null controllability by an algebraic condition so called \textit{Kalman's rank} condition.

This paper investigates the assessment of Granger causality (GC) between jointly Gaussian signals based on noisy or filtered measurements. To do so, a recent rank condition for inferring GC between jointly Gaussian stochastic processes is exploited. Sufficient conditions are derived under which GC can be reliably inferred from the second order moments of the noisy or filtered measurements. This approach does not require a model of the underlying Gaussian system to be identified. The noise signals are not required to be Gaussian or independent, and the filters may be noncausal or nonminimum-phase, as long as they are stable.

Let $f(x_1,\ldots,x_s)$ be a translation invariant indefinite quadratic form of integer coefficients with $s\ge 10$. Let $\mathcal{A}\subseteq \mathcal{P}\cap \{1,2,\ldots,X\}$. Let $X$ be sufficiently large. Subject to a rank condition, we prove that there exist distinct primes $p_1,\ldots,p_s\in \mathcal{A}$ such that $f(p_1,\ldots,p_s)=0$ as soon as $|\mathcal{A}|\ge \frac{X}{\log X} (\log\log X)^{-\frac{1}{80}}.$

This paper presents the results of structure–activity relationship (SAR) studies of 140 3,3’-(α,ω-dioxaalkan)bis(1-alkylimidazolium) chlorides. In the SAR analysis, the dominance-based rough set approach (DRSA) was used. For analyzed compounds, minimum inhibitory concentration (MIC) against strains of Staphylococcus aureus and Pseudomonas aeruginosa was determined. In order to perform the SAR analysis, a tabular information system was formed, in which tested compounds were described by means of condition attributes, characterizing the structure (substructure parameters and molecular descriptors) and their surface properties, and a decision attribute, classifying compounds with respect to values of MIC. DRSA allows to induce decision rules from data describing the compounds in terms of condition and decision attributes, and to rank condition attributes with respect to relevance using a Bayesian confirmation measure. Decision rules present the most important relationships between structure and surface properties of the compounds on one hand, and their antibacterial activity on the other hand. They also indicate directions of synthesizing more efficient antibacterial compounds. Moreover, the analysis showed differences in the application of various parameters for Gram-positive and Gram-negative strains, respectively.

For a semisimple Lie group G satisfying the equal rank condition, the most basic family of unitary irreducible representations is the discrete series found by Harish-Chandra. In this paper, we study some of the branching laws for these when restricted to a subgroup H of the same type by combining the classical results with the recent work of T. Kobayashi. We analyze aspects of having differential operators being symmetry-breaking operators; in particular, we prove in the so-called admissible case that every symmetry breaking (H-map) operator is a differential operator. We prove discrete decomposability under Harish-Chandra's condition of cusp form on the reproducing kernels. Our techniques are based on realizing discrete series representations as kernels of elliptic invariant differential operators.

We study proper holomorphic maps between bounded symmetric domains $D$ and $\Omega$. In particular, when $D$ and $\Omega$ are of the same rank $\ge 2$ such that all irreducible factors of $D$ are of rank $\ge 2$, we prove that any proper holomorphic map from $D$ to $\Omega$ is a totally geodesic holomorphic isometric embedding with respect to certain canonical K\"ahler metrics of $D$ and $\Omega$. We also obtain some results regarding holomorphic maps $F:D\to \Omega$ which map minimal disks of $D$ properly into rank-$1$ characteristic symmetric subspaces of $\Omega$. On the other hand, we obtain new rigidity results regarding semi-product proper holomorphic maps between $D$ and $\Omega$ under a certain rank condition on $D$ and $\Omega$., Comment: To appear in Proceedings of the American Mathematical Society

Two objectives are accomplished in this paper: First we establish the comparison between Wolff and Riesz potentials on space of homogeneous type in the sense of Coifman and Weiss ( Theorem 1.2 ), followed by a Hardy–Littlewood–Sobolev type inequality for Wolff potentials ( Theorem 1.3 ). Then applying this inequality, we consider a Lane–Emden type integral system and use regularity lifting to derive integrability estimates of positive solutions to the system, by which we also imply L ∞ estimates (see Theorem 1.4 , Theorem 1.5 ). Furthermore, we use a modified regularity lifting method to prove that the positive solutions are also Lipschitz continuous ( Theorem 1.6 ). Our results imply Wolff potentials and regularity of solutions to the Lane–Emden type integral systems on stratified groups and the Carnot–Carathedory metric spaces defined by a family of vector fields satisfying Hormander's finite rank condition as special cases.

A problem of characterizing conditions under which a topological change in a network of differential–algebraic equations (DAEs) can go undetected is considered. It is shown that initial conditions for which topological changes are indiscernible belong to a generalized eigenspace shared by the nominal system and the system resulting from a topological change. A condition in terms of eigenvectors of the nominal system is derived to check for existence of possibly indiscernible topological changes. For homogeneous networks this condition simplifies to the existence of an eigenvector of the Laplacian of network having equal components. Lastly, a rank condition is derived which can be used to check if a topological change preserves regularity of the nominal network.

Bilinear systems form models of wide-ranging applications in diverse areas of engineering and natural sciences. Investigating fundamental properties of such systems has been a prosperous subject of interest and remains essential toward the advancement of systems science and engineering. In this paper, we introduce an algebraic framework utilizing the theory of symmetric group to characterize controllability of bilinear systems evolving on special orthogonal and Euclidean groups. Our development is based on the most notable Lie algebra rank condition and offers an alternative to controllability analysis. The main idea of the developed approach lies in identifying the mapping of Lie brackets of vector fields governing the system dynamics to permutation multiplications on a symmetric group. Then, by leveraging the actions of the resulting permutations on a finite set, controllability and controllable submanifolds for bilinear systems evolving on the special Euclidean group can be explicitly characterized.

In a landmark contribution to the structural vector autoregression (SVARs) literature, Rubio-Ramirez, Waggoner, and Zha (2010, `Structural Vector Autoregressions: Theory of Identification and Algorithms for Inference,' Review of Economic Studies) shows a necessary and sufficient condition for equality restrictions to globally identify the structural parameters of a SVAR. The simplest form of the necessary and sufficient condition shown in Theorem 7 of Rubio-Ramirez et al (2010) checks the number of zero restrictions and the ranks of particular matrices without requiring knowledge of the true value of the structural or reduced-form parameters. However, this note shows by counterexample that this condition is not sufficient for global identification. Analytical investigation of the counterexample clarifies why their sufficiency claim breaks down. The problem with the rank condition is that it allows for the possibility that restrictions are redundant, in the sense that one or more restrictions may be implied by other restrictions, in which case the implied restriction contains no identifying information. We derive a modified necessary and sufficient condition for SVAR global identification and clarify how it can be assessed in practice.

In this paper, we adapt the fractional derivative approach to formulate the flow-conservation transportation networks, which consider the propagation dynamics and the users’ behaviors in terms of route choices. We then investigate the controllability of the fractional-order transportation networks by employing the Popov-Belevitch-Hautus rank condition and the QR decomposition algorithm. Furthermore, we provide the exact solutions for the full controllability pricing controller location problem, which includes where to locate the controllers and how many controllers are required at the location positions. Finally, we illustrate two numerical examples to validate the theoretical analysis.

In this paper, we study the concept of approximate controllability of retarded network systems of neutral type. On one hand, we reformulate such systems as free-delay boundary control systems on product spaces. On the other hand, we use the rich theory of infinite-dimensional linear systems to derive necessary and sufficient conditions for the approximate controllability. Moreover, we propose a rank condition for which we can easily verify the conditions of controllability. Our approach is mainly based on the feedback theory of regular linear systems in the Salamon-Weiss sense.

For homogeneous bilinear control systems, the control sets are characterized using a Lie algebra rank condition for the induced systems on projective space. This is based on a classical Diophantine approximation result. For affine control systems, the control sets around the equilibria for constant controls are characterized with particular attention to the question when the control sets are unbounded., Comment: 28 pages

Modally reduced order models are commonly adopted in system inversion. Their observability requires specific attention, since these models only accurately describe the dynamic behavior of the underlying system in a limited frequency range. This paper elaborates a methodology to investigate the observability of modally reduced order models with unknown parameters. The focus is on a particular type of model where the quasi-static contribution of the out-of-band modes is accounted for using so-called dummy modes. The observability test is performed by means of the commonly used Observability Rank Condition (ORC). The proposed methodology is illustrated by multiple examples from structural engineering. It is found that modally reduced order models serve as a valuable alternative for full order models when applied in system inversion. Not only are they computationally much less demanding, but due to their strong link with the underlying full order model, they also allow for the identification of physical parameters, such as mass or stiffness.

We study the set of intrinsic singularities of flat affine systems with $n-1$ controls and $n$ states using the notion of Lie-B��cklund atlas, previously introduced by the authors. For this purpose, we prove two easily computable sufficient conditions to construct flat outputs as a set of independent first integrals of distributions of vector fields, the first one in a generic case, namely in a neighborhood of a point where the $n-1$ control vector fields are independent, and the second one at a degenerate point where $p-1$ control vector fields are dependent of the $n-p$ others, with $p>1$. We show that the set of intrinsic singularities includes the set of points where the system does not satisfy the strong accessibility rank condition and is included in the set where the distribution of vector fields, introduced in the generic case, is singular. We conclude this analysis by three examples of apparent singularites of flat systems in generic and non generic degenerate cases., 24 pages

This paper is devoted to the state and input estimation of a linear time varying system in the presence of an unknown input (UI) in both state and measurement equations, and affected by Gaussian noises. The classical rank condition used in this kind of approach is relaxed in order to be able to be used in a wider range of systems. A state observer, that is an unbiased estimator with minimum error variance, is proposed. Then a UI observer is constructed, in order to be a best linear unbiased estimator, it follows a unique construction whether the direct feedthrough matrix is null or not. In a sense the proposed approach, generalizes and unifies the existing ones. Besides, a stability result is given for linear time invariant systems, which is a novelty for unbiased minimum variance observers relaxing the classical rank condition.

In this article, we propose a simple algorithm to cluster nonnegative data lying in disjoint subspaces. We analyze its performance in relation to a certain measure of correlation between said subspaces. We use our clustering algorithm to develop a matrix completion algorithm which can outperform standard matrix completion algorithms on data matrices satisfying a certain natural low rank condition.

We consider a generalization of the elliptic $L^p$-estimate suited for linear operators with non-trivial kernels. A classical result of Schulenberger and Wilcox (Ann. Mat. Pura Appl. (4) 88: 229-305, 1971) shows that if the operator has constant rank then the estimate holds. We prove necessity of the constant rank condition for such an estimate., Comment: 5 pages; weaker versions of the results here had previously appeared in arXiv:1909.03923

This work proposes an efficient algorithm to examine the observability and identifiability of rational nonlinear systems in the presence of unmeasured and unknown inputs. The proposed algorithm allows for determining whether the dynamic states, unknown parameters and unmeasured inputs of a dynamical system can be, in theory, successfully identified from a given set of input–output measurements. The underlying theory of the algorithm is based on a further extension of the recently suggested extended Observability Rank Condition while focusing on rational instead of analytic nonlinearities. For the robust development of the algorithm, a power series based framework is established for computing the observability matrix efficiently. The occurring framework substantially alleviates the computational burden of the standard implementations of the extended Observability Rank Condition approaches, which allows for applications to real-world engineering systems that are often large and complex. Several examples of large-scale and high-complexity engineering structures are used to demonstrate the performance and capability of the algorithm. Furthermore, the proposed algorithm is used to investigate the feasibility of monitoring a sub-system that is independently separated from a full system under the introduction of additional unmeasured inputs. The first author would like to gratefully acknowledge the financial support from the China Scholarship Council (CSC) for his Ph.D. studies.

In this paper, we study some non-negative integers related to a linear time varying system and to some Krylov sub-spaces associated to this system. Such integers are similar to the controllability indices and have been used in the literature to derive results on the controllability of linear systems. The purpose of this paper goes in the same direction by studying the local behavior of these integers especially nearby instants in the time interval with some maximal rank condition and then apply them to get some results which generalize the mentioned existing results.

In this Note, we consider a system of wave equations coupled by a nilpotent matrix with homogeneous Dirichlet boundary condition. We establish the uniqueness of the solution when partial Neumann observation satisfies Kalman's rank condition.

For the class of matrices over a field, the notion of `rank of a matrix' as defined by `the dimension of subspace generated by columns of that matrix' is folklore and cannot be generalized to the class of matrices over an arbitrary commutative ring. The `determinantal rank' defined by the size of largest submatrix having nonzero determinant, which is same as the column rank of given matrix when the commutative ring under consideration is a field, was considered to be the best alternative for the `rank' in the class of matrices over a commutative ring. Even this determinantal rank and the McCoy rank are not so efficient in describing several characteristics of matrices like in the case of discussing solvability of linear system. In the present article, the `rank--function' associated with the matrix as defined in [{\it Solvability of linear equations and rank--function}, K. Manjunatha Prasad, \url{http://dx.doi.org/10.1080/00927879708825854}] is discussed and the same is used to provide a necessary and sufficient condition for the existence of an outer inverse with specific column space and row space. Also, a rank condition is presented for the existence of Drazin inverse, as a special case of an outer inverse, and an iterative procedure to verify the same in terms of sum of principal minors of the given square matrix over a commutative ring is discussed.

In the tensor completion problem, one seeks to estimate a low-rank tensor based on a random sample of revealed entries. In terms of the required sample size, earlier work revealed a large gap between estimation with unbounded computational resources (using, for instance, tensor nuclear norm minimization) and polynomial-time algorithms. Among the latter, the best statistical guarantees have been proved, for third-order tensors, using the sixth level of the sum-of-squares (SOS) semidefinite programming hierarchy (Barak and Moitra, 2014). However, the SOS approach does not scale well to large problem instances. By contrast, spectral methods --- based on unfolding or matricizing the tensor --- are attractive for their low complexity, but have been believed to require a much larger sample size. This paper presents two main contributions. First, we propose a new unfolding-based method, which outperforms naive ones for symmetric $k$-th order tensors of rank $r$. For this result we make a study of singular space estimation for partially revealed matrices of large aspect ratio, which may be of independent interest. For third-order tensors, our algorithm matches the SOS method in terms of sample size (requiring about $rd^{3/2}$ revealed entries), subject to a worse rank condition ($r\ll d^{3/4}$ rather than $r\ll d^{3/2}$). We complement this result with a different spectral algorithm for third-order tensors in the overcomplete ($r\ge d$) regime. Under a random model, this second approach succeeds in estimating tensors of rank $d\le r \ll d^{3/2}$ from about $rd^{3/2}$ revealed entries.

In this paper, a necessary and sufficient condition for the existence of a proportional semistate feedback is established such that the closed loop system is regular and of index at most two. The condition is characterized by a rank condition that involves only the original system coefficient matrices. Also, a new rank condition ensuring that a given linear time-invariant descriptor system is regular and of index at most some specific value is also derived. The developed theory is illustrated through physical and numerical examples.

International audience; This study addresses the problem of guaranteed state and unknown input (UI) estimation for a class of linearparameter varying systems affected by UIs and other bounded perturbations. The context of interval observer in the cooperativeframework is adopted. Thus, lower and upper bounds of the state and UI are provided. No assumption is made on the UI. Theapproach is based on a rank condition allowing to decouple the UI from the state estimation errors (so that the state estimationis guaranteed, regardless of the variation of the UI). The convergence of the proposed interval observer is studied by Lyapunovtheory and stability conditions are expressed in terms of linear matrix inequalities which ease the design of the gain matrices ofthe observer. Finally, illustrative examples are given for discussion and comparison with respect to the existing results.

This work proposes a robust algorithm to examine the observability of linear systems whose dynamic states and parameters are to be identified. The observability of a dynamical system plays a fundamental role in predicting whether it would be successful to use system identification methods to estimate the dynamic states and parameters of the system from a given set of input–output measurements. The motivation of the development of the suggested algorithm arises from the need to address the significant physical memory requirements of the standard implementation of the Observability Rank Condition (ORC). The high computational cost of the ORC substantially limits its applicability to real-world engineering systems, even when the underlying dynamics can be reasonably approximated as linear. The framework of the algorithm is obtained through the derivation of a recursive formula for the computation of the observability matrix of linear systems with unknown parameters. To further improve the efficiency, robust numerical implementations of the algorithm are achieved through random realizations of the dynamic states and parameters and the use of modular operations. The superior performance of the algorithm is demonstrated using several examples of large linear dynamical models of engineering systems containing up to thousands of dynamic states and parameters.

The main contribution of this paper is to study the applicability of the bootstrap to estimating the distribution of the standard test of overidentifying restrictions of Hansen (1982) when the model is globally identified but the rank condition fails to hold (lack of first-order local identification). An important example for which these conditions are verified is the popular test of common conditionally heteroskedastic features proposed by Engle and Kozicki (1993) . As Dovonon and Renault (2013b) show, the Jacobian matrix for this model is identically zero at the true parameter value, resulting in a highly nonstandard limiting distribution that complicates the computation of critical values. We first show that the standard GMM bootstrap fails to consistently estimate the distribution of the overidentification restrictions test under lack of first-order identification. We then propose a new bootstrap method that is asymptotically valid in this context. The modification consists of adding an additional term that recenters the bootstrap moment conditions in a way as to ensure that the bootstrap Jacobian matrix is zero when evaluated at the GMM estimate.

We consider the problem of computing parameters of player cost functions in discrete-time nonzero-sum noncooperative dynamic games from open-loop Nash equilibria. Although similar inverse problems have been investigated in the optimal control literature where there is a single player (or decision maker), there has been limited attention given to the inverse dynamic game problem with multiple (competing) players. By exploiting the minimum principle of optimal control, we propose a method of inverse dynamic games for when the information structure of the game is open-loop. Our method involves solving a system of linear equations and is able to recover the true unknown parameters (up to an unknown scaling factor) whenever a testable rank condition holds. We illustrate our method in an example two-player game.