Kronecker Power Research Articles

Matrix Factorization and Some Fast Discrete Transforms

In this paper, three discrete transforms related to vector spaces over finite fields are studied. For our purposes, and according to the properties of the finite fields, the most suitable transforms are as follows: for binary fields, this is the Walsh–Hadamard transform; for odd prime fields, the Vilenkin–Chrestenson transform; and for composite fields, the trace transform. A factorization of the transform matrices using Kronecker power is given so that the considered discrete transforms are reduced to the fast discrete transforms. Examples and applications are also presented of the considered transforms in coding theory for calculating the weight distribution of a linear code.

Moment propagation of polynomial systems through Carleman linearization for probabilistic safety analysis

We develop a method to approximate the moments of a discrete-time stochastic polynomial system. Our method is built upon Carleman linearization with truncation. Specifically, we take a stochastic polynomial system with finitely many states and transform it into an infinite-dimensional system with linear deterministic dynamics, which describe the exact evolution of the moments of the original polynomial system. We then truncate this deterministic system to obtain a finite-dimensional linear system, and use it for moment approximation by iteratively propagating the moments along the finite-dimensional linear dynamics across time. We provide efficient online computation methods for this propagation scheme with several error bounds for the approximation. Our results also show that precise values of certain moments at a given time step can be obtained when the truncated system is sufficiently large. Furthermore, we investigate techniques to reduce the offline computation load using reduced Kronecker power. Based on the obtained approximate moments and their errors, we also provide hyperellipsoidal regions that are safe for some given probability bound. Those bounds allow us to conduct probabilistic safety analysis online through convex optimization. We demonstrate our results on a logistic map with stochastic dynamics and a vehicle dynamics subject to stochastic disturbance.

Generalization of the Fast Fourier Transform with a Constant Structure

The widely popular famous fast Cooley–Tukey algorithms for the discrete Fourier transform of mixed radix are presented in two forms: classical and with a constant structure. A matrix representation of these algorithms is proposed in terms of two types of tensor product of matrices: the Kronecker product and the b-product. This matrix representation shows that the structure of these algorithms is identical to two fast Good algorithms for a Kronecker power of a matrix. A technique for constructing matrix-form fast algorithms for the discrete Fourier and Chrestenson transforms with mixed radix and for the discrete Vilenkin transform is demonstrated. It is shown that the constant-structured algorithm is preferable in the case of more sophisticated constructions

Distributed estimation for multi‐agent systems with relative measurements and quantized communication: A feedback quadratic framework

SummaryIn this paper, the feedback quadratic distributed state estimation problem is investigated for a class of linear discrete time‐varying multi‐agent systems subject to non‐Gaussian noises. For the multi‐agent systems, each agent has access to not only the local measurements but the relative states to its adjacent agents as well. Due to the bandwidth constraints of the digital communication networks, the signals are quantized and then exchanged between two adjacent agents, where the probabilistic uniform quantizations are taken into account. The purpose of the addressed problem is to design a novel quadratic distributed estimator for each agent based on the local information and the predictions received from the neighbors. In particular, an output injection term is introduced to handle the unstable systems. By means of the state/measurement augmentation approach, the underlying system is transformed into an augmented one, which aggregates the original vector and its second‐order Kronecker power. Accordingly, a distributed estimator is constructed such that an upper bound is ensured for the estimation error covariance and the suboptimal estimator parameters, which minimize such an upper bound, are subsequently calculated in terms of the solutions to certain matrix difference equations. Finally, an illustrative example is provided to verify the effectiveness and superiority of the proposed quadratic estimation scheme.

Quadratic estimation for discrete time-varying non-Gaussian systems with multiplicative noises and quantization effects

This paper is concerned with the remote state estimation problem for a class of linear discrete time-varying non-Gaussian systems with multiplicative noises. Due to bandwidth constraints in digital communication networks, the measured outputs are quantized before transmission by a probabilistic uniform quantizer. Our attention is focused on the design of a recursive quadratic estimator that exploits the quadratic functions of the measurements. By introducing a proper augmented system which aggregates the original state vector and its second-order Kronecker power, we are able to transfer the quadratic estimation problem into a corresponding linear estimation problem of the augmented state vector. An upper bound is first established for the covariance of the estimation error that is expressed in terms of the solutions to certain matrix difference equations, and such an upper bound is then minimized by designing the filter parameters in an iterative manner. Subsequently, we discuss the monotonicity of the optimized upper bound with respect to the quantization accuracy. A numerical example is provided to verify the effectiveness of the proposed filtering algorithm.

Derivatives of fourth order Kronecker power systems with applications in nonlinear elasticity

A natural way to describe systems with polynomial nonlinearities is using the Kronecker product. Particularly, third-order Kronecker power systems can express a wide range of systems from electronic engineering to nonlinear elasticity. But such development (e.g. equations of motions of elastic structures from nonlinear strain energy) requires standard formulation for the derivative of the Kronecker power of vectors with respect to the same vector. Such standard way cannot be found in literature. This paper presents a method to obtain the derivatives of Kronecker powers of vectors with respect to itself up to a power of four and also third-order Kronecker Power systems containing those terms in a concise matrix form. The matrix expression of these systems provides new approach for efficient numerical implementation, organized analysis and linearization. To demonstrate the strength of this method, an example of application for a finite element nonlinear Euler beam is also presented.

LQ Non-Gaussian Regulator With Markovian Control

This letter concerns the linear quadratic non-Gaussian (LQnG) sub-optimal control problem when the input signal travels through an unreliable network, namely a Gilbert-Elliot channel. In particular, the control input packet losses are modeled by a two-state Markov chain with known transition probability matrix, and we assume that the moments of the non-Gaussian noise sequences up to the fourth order are known. By mean of a suitable rewriting of the system through an output injection term, and by considering an augmented system with the second-order Kronecker power of the measurements, a simple solution is provided by substituting the Kalman predictor of the LQG control law with a quadratic optimal predictor. Numerical simulations show the effective ness of the proposed method.

Kalman-like filtering with intermittent observations and non-Gaussian noise

The paper concerns the sub-optimal filtering problem when the measurement signal is sent through an unreliable channel and the noise signals are not necessarily Gaussian. In particular, we assume that the measurement packet losses are modeled by an i.i.d. Bernoulli sequence with known probability mass function, and the moments of the (generally) non-Gaussian noise sequences up to the fourth order are known. By mean of a suitable rewriting of the system through an output injection term, and by considering an augmented system with the second-order Kronecker power of the measurements, an optimal solution among the quadratic transformations of the output is provided. Numerical simulations show the effectiveness of the proposed method.

Development of a decentralized nonlinear controller for a class of uncertain polynomial interconnected systems: Application for a large scale power system

This paper presents a new robust decentralized control of nonlinear interconnected systems, which is applied and validated on a large scale power system. Our work is performed in three steps. Firstly, we have developed the polynomial description of the nonlinear uncertain and interconnected system using odd Kronecker power of state vectors, which is an easy-manipulation model for such complex systems. Then we applied Lyapunov’s direct method of stability analysis, associated with a quadratic function, in order to determine a sufficient condition for global asymptotic stability by applying a nonlinear, decentralized and optimal polynomial control. Finally, we carried out a simulation study on a nonlinear uncertain power system with three interconnected machines. We considered different cases of perturbations on its state variables as well as different cases of fault locations. We prove via advanced simulations, the effectiveness of the proposed control technique which is able to mitigate the successive amplitudes of the oscillations, to limit the control actions and to enhance the power system transient stability.

New Good’s Type Kronecker Power Expansions

We propose a new version of the proof of Good’s theorem stating that the Kronecker power of an arbitrary square matrix can be represented as a matrix power of a sparse matrix Z. We propose new variants of sparse matrices Z. We observe that for another version of the tensor power of a matrix, the b-power, there exists an analog of another Good’s expansion but no analog of this theorem.

Nonlinear Identification and Control of Coupled Mass-Spring-Damper System using Polynomial Structures

The paper aims to identify and control the coupled mass-spring-damper system. A nonlinear discrete polynomial structure is elaborated. Its parameters are estimated using Recursive Least Squares (RLS) algorithm. Moreover, a feedback stabilizing control law based on Kronecker power is designed. Finally, simulations are presented to illustrate the effectiveness of the proposed structure.

Generalization of a stability domain estimation method for nonlinear discrete systems

In this paper, we present a generalization approach of an algebraic existing method to estimate stability regions for discrete nonlinear polynomial systems of degree 3. The existing method is based on the enlargement of a guaranteed stability region by applying various steps of a proposed algorithm. Its main limitation is that the initial result has only been subsequently developed in a particular case of a single iteration. The stability domain obtained is consequently not the widest one. Our main contribution in this paper is to develop generalized functions that allow the enlargement of the guaranteed stability region after k iterations, for any value of k. A required fundamental tool is developed and consists in a general formula allowing to give the result of the Kronecker power calculation of two matrices sum. The advantages of this generalization are to reach a larger region of asymptotic stability and to improve the existing methods results. Two application examples illustrate the proposed method.

Probabilistic evolution theory for the solution of explicit autonomous ordinary differential equations: squarified telescope matrices

Probabilistic evolution theory (PREVTH) is used for the solution of initial value problems of first order explicit autonomous ordinary differential equation sets with second degree multinomial right hand side functions. It is an approximation method based on Kronecker power series: a rewriting of multivariate Taylor series using matrices having certain flexible parameters. Kronecker power series have matrices which are called telescope matrices: \(n \times n^{j+1}\) matrices where j is the index of summation. The additive terms of each telescope matrix is formed through Kronecker product from both sides by Kronecker powers of identity matrices. Recently, squarification is proposed in order to avoid the growing of the matrices in size at each additive term of the series. This paper explains the squarification procedure: the procedure used in order to avoid Kronecker multiplications within PREVTH so that the sizes of the matrices do not grow and so that the amount of necessary computation is reduced. The recursion between squarified matrices is also given. As a numerical application, the solution of a Henon–Heiles system is provided.

Generating operator for discrete Chrestenson functions

We study the structure and properties of and a construction algorithm for a cyclic operator whose orbit consists of discrete Chrestenson functions arranged as columns of the matrix of a Kronecker power of the discrete Fourier transform. We analyze possibilities for extending the results to the case of discrete Vilenkin functions.

Probabilistic evolution approach to the expectation value dynamics of quantum mechanical operators, part I: integral representation of Kronecker power series and multivariate Hausdorff moment problems

This is the first one of two companion papers focusing on the establishment of a new path for the expectation value dynamics of the quantum mechanical operators. The main goal of these studies is to do quantum mechanics without explicitly solving Schrodinger wave equation, in other words, without using wave functions except their initially given forms. This goal is achieved by using Ehrenfest theorem and utilizing probabilistic evolution approach (PEA). PEA, first introduced by Metin Demiralp, is a method providing solutions to the nonlinear ordinary differential equations by transforming them to a set of linear ODEs at the cost of denumerably infinite dimensionality. It is recently shown that this method produces analytic solutions, if the initial conditions are given appropriately at some special cases. However, generalization of these conditions to the quantum mechanical applications is not straightforward due to the dispersion of the quantum mechanical systems. For this purpose, multivariate moment problems for the integral representation of the Kronecker power series are introduced and then solved yielding to more specific and precise convergence analysis for the quantum mechanical applications.

On the properties of a new tensor product of matrices

Previously, the author introduced a new tensor product of matrices according to which the matrix of the discrete Walsh-Paley transform can be represented as a power of the second-order discrete Walsh transform matrix H with respect to this product. This power is an analogue of the representation of the Sylvester-Hadamard matrix in the form of a Kronecker power of H. The properties of the new tensor product of matrices are examined and compared with those of the Kronecker product. An algebraic structure with the matrix H used as a generator element and with these two tensor products of matrices is constructed and analyzed. It is shown that the new tensor product operation proposed can be treated as a convenient mathematical language for describing the foundations of discrete Fourier analysis.

Quadratic Filtering of non-Gaussian Linear Systems with Random Observation Matrices

In this paper we consider the problem of state estimation for linear discrete-time non-Gaussian systems with random observation matrices. This is the model for systems with observation losses due to propagation through unreliable communication channels. Losses may result from intermittent failures that cause packet dropouts, as in the case of networks, or fading phenomena in propagation channel, as in the case of wireless networks. These are common problems in wireless sensor network, or networked control systems. In this paper, we do not make any assumption about the distribution of the observation matrix, thus encompassing a great variety of possible scenarios. We derive the quadratic estimate of the state by means of a recursive algorithm. The solution is obtained by applying the Kalman filter to a suitably augmented system, which is fully observable. The augmented system is constructed as the aggregate of the actual system and the observable part of a system having as state the second Kronecker power of the original state, namely the quadratic system. To extract the observable part of the quadratic system we exploit the knowledge of the rank of the corresponding observability matrix. This approach guarantees the internal stability of the estimation filter.

Probabilistic evolution approach for the solution of explicit autonomous ordinary differential equations. Part 2: Kernel separability, space extension, and, series solution via telescopic matrices

This work focuses on the Kronecker power series solution of the explicit conical ODEs. This means that the Kronecker power series of the descriptive function vector of the ODEs has only zeroth, first and second Kronecker powers of the unknowns hence the only nonvanishing matrix coefficients are \({\mathbf {F}}_0, {\mathbf {F}}_1\) and \({\mathbf {F}}_2\). We focus on the cases where \({\mathbf {F}}_0\) also vanishes. These enable us to get and solve a two block term recursive ODE and the accompanying initial conditions. The resulting Kronecker power series’ kernel can be expressed as a binary product whose first factor which in square matrix type and a second factor which is in purely rectangular matrix algebraic structure. The constancy adding space extension separates the temporal behavior of the kernel in a scalar first factor while the second factor is again in rectangular matrix structure. We also show that the definition and use of rectangular eigenvalue problem takes us to constant solution of the original ODEs.

Probabilistic evolution approach for the solution of explicit autonomous ordinary differential equations. Part 1: Arbitrariness and equipartition theorem in Kronecker power series

Probabilistic evolution approach is a newly developed theory which may be utilized for the solution of ordinary differential equations. The approach may directly be applied for initial value problems of explicit first order autonomous ordinary differential equation sets with analytic right hand side functions. Analyticity plays an important role since it facilitates the expansion into direct power series which is the key element of the approach. Direct power series appear not only in all applications of probabilistic evolution but also show themselves as a promising tool for novel approximation methods. In this work, similarities and differences between Taylor series and direct power series are rigorously studied. Arbitrariness in transposed vector coefficients of direct power series is detailed. Equipartition theorem of direct power series is conjectured and proven in order to obtain unique transposed vector coefficients.

Robust Reliable H∞ Control for a Class of Nonlinear Perturbed Systems with Electrical Application: An LMI Approach

AbstractThis paper discusses the robust feedback control for a class of nonlinear polynomial systems with H∞ robustness performance. The studied systems are described by perturbed polynomial models based on the Kronecker product and the Kronecker power of the state vector. The proposed nonlinear feedback control scheme is developed to provide the practical stability of the closed loop system using the well-known second Lyapunov method with quadratic function. A sufficient condition for robust stabilization and H∞ control performance is given such that ultimately bounds on the controlled system trajectories are ensured and at the same time the effect of external disturbance on control performance is attenuated to a minimum level. The developed control approach is formulated as a minimization problem and derived in the form of Linear Matrix Inequalities (LMIs). To asses the performances of the proposed control method through numerical simulations, a power electrical application is chosen to provide the effectiveness of the proposed methodology.