Linear Descriptor Systems Research Articles

Application of the Algebraic Extension Method to the Construction of Orthogonal Bases for Partial Digital Convolutions

Mathematical tools have been developed that are analogous to the tool that allows one to reduce the description of linear systems in terms of convolution operations to a description in terms of amplitude-frequency characteristics. These tools are intended for use in cases where the system under consideration is described by partial digital convolutions. The basis of the proposed approach is the Fourier–Galois transform using orthogonal bases in corresponding fields. As applied to partial convolutions, the Fourier–Galois transform is decomposed into a set of such transforms, each of which corresponds to operations in a certain Galois field. It is shown that for adequate application of the Fourier–Galois transform to systems described by partial convolutions, it is necessary to ensure the same number of cycles in each of the transforms from the set specified above. To solve this problem, the method of algebraic extensions was used, a special case of which is the transition from real numbers to complex numbers. In this case, the number of cycles varies from p to pn/k, where p is a prime number, n and k are integers, and an arbitrary number divisor of pn can be chosen as k. This allows us to produce partial Fourier–Galois transforms corresponding to different Galois fields, for the same number of cycles. A specific example is presented demonstrating the constructiveness of the proposed approach.

Analyzing asymmetry in brain hierarchies with a linear state-space model of resting-state fMRI data.

This study challenges the traditional focus on zero-lag statistics in resting-state functional magnetic resonance imaging (rsfMRI) research. Instead, it advocates for considering time-lag interactions to unveil the directionality and asymmetries of the brain hierarchy. Effective connectivity (EC), the state matrix in dynamical causal modeling (DCM), is a commonly used metric for studying dynamical properties and causal interactions within a linear state-space system description. Here, we focused on how time-lag statistics are incorporated within the framework of DCM resulting in an asymmetric EC matrix. Our approach involves decomposing the EC matrix, revealing a steady-state differential cross-covariance matrix that is responsible for modeling information flow and introducing time-irreversibility. Specifically, the system's dynamics, influenced by the off-diagonal part of the differential covariance, exhibit a curl steady-state flow component that breaks detailed balance and diverges the dynamics from equilibrium. Our empirical findings indicate that the EC matrix's outgoing strengths correlate with the flow described by the differential cross covariance, while incoming strengths are primarily driven by zero-lag covariance, emphasizing conditional independence over directionality.

Transfer matrices with positive coefficients of descriptor linear systems

Transfer matrices with positive coefficients of descriptor positive linear continuous-time systems are addressed. Two methods of checking of the positivity of descriptor linear systems are proposed. It is shown that if the positive descriptor system is asymptotically stable then all coefficients of its transfer matrix are positive.

State estimators for discrete‐time descriptor linear systems with mixed uncertainties and state constraints

AbstractThis article presents two novel mixed‐uncertainty state estimators for discrete‐time descriptor linear systems, namely linear time‐varying mixed‐uncertainty filter (LTVMF) and linear time‐invariant mixed‐uncertainty filter (LTIMF). The former is based on the minimum‐variance approach, from which quadratic and explicit (special case of quadratic) formulations are derived and addressed to LTV systems. Both formulations incorporate the knowledge of state linear constraints, such as equality (in the descriptor form) and inequality, to mitigate precision and accuracy issues related to initialization and evolution of the state estimates. The LTIMF algorithm is based on the mixed criterion and addressed to LTI systems, with the low‐cost computation being its motivation. Both LTVMF and LTIMF algorithms solve state‐estimation problems in which the uncertainties are combined to yield the so‐called mixed‐uncertainty vector, which is composed by set‐bounded uncertainties, characterized by constrained zonotopes, and stochastic uncertainties, characterized by Gaussian random vectors. As mixed‐uncertainty vectors imply biobjective optimization problems, we innovatively present multiobjective arguments to justify the choice of the solution on the Pareto‐optimal front. In order to discuss the advantages and drawbacks, the state estimators are tested in two numerical examples.

Controllability and Observability of the Descriptor Linear Systems Reduced to the Standard Ones by Feedbacks

Abstract In this paper, a new method for the reduction of the descriptor linear systems to standards ones is presented and verified. The method uses a state and/or state derivative feedback of output and output derivative feedback in order to transform the descriptor system into a standard one. The controllability and observability properties of the original descriptor as well as transformed standard systems are proved. Simple numerical examples illustrate the theorems introduced.

Tight interval state estimate for discrete‐time descriptor linear systems

SummaryIn this work, two state estimation methods are proposed for a class of discrete‐time descriptor linear systems subject to bounded uncertainties. First, we propose set‐valued estimator algorithm using symmetric boxes to compute rigorous bounds of the system states. The observer gains are calculated using norm to attenuate the effects of the uncertainties and to improve the accuracy of the proposed estimator. Second, to obtain tighter state enclosures, zonotopic set computation are developed instead of interval analysis to design a new set‐valued state estimation algorithm. The performances of the proposed state estimation approaches are highlighted on different illustrative examples.

Impulse analysis in Kronecker form and impulse elimination for rectangular descriptor systems

Contrary to regular descriptor systems, there are various notions for impulse freeness and impulse controllability for rectangular linear descriptor systems. Moreover, rectangular descriptor systems generally do not have solutions for arbitrary initial conditions. This paper studies the conditions for free of impulse, and no impulsive modes admitting arbitrary initial conditions in the form of Kronecker canonical form (KCF) along with other equivalent and geometric conditions. The knowledge of KCF blocks significantly simplifies the analysis of the system solutions. Further, conditions for impulse controllability are investigated, and a proportional state feedback is designed such that the closed-loop system has no impulsive modes and admits arbitrary initial conditions. This makes the system solutions smooth without any restrictions on initial conditions. Physical and academic instances are provided to fortify the theory.

СТОХАСТИЧНА ДЕСКРИПТОРНА ГРА ПЕРЕСЛІДУВАННЯ

A differential pursuit game in a stochastic descriptor linear system is analyzed. The system dynamics is described by Ito’s stochastic differential algebraic equation. Solutions of the equation are presented by the formula of variation of constants in terms of the initial data and control unit. Constraints on the support functionals of two sets defined by the behaviors of the pursuer and evader are used to obtain the game completion conditions. The method of resolving functions is applied to construct a pursuer control bringing the dynamic vector of the system to a terminal set. The results are illustrated by an example of a stochastic descriptor system that describes transient states in a radio technical filter with random perturbations in the form of white noise. Keywords: stochastic differential algebraic equation, Wiener process, descriptor system, differential game, radio technical filter, white noise.

Robust model reference tracking control for high‐order descriptor linear systems subject to parameter uncertainties

AbstractThis paper investigates the robust model reference tracking control problem for high‐order descriptor linear systems (HODLS) subject to norm‐bounded parameter uncertainties. The problem is divided into two subproblems: a robust proportional plus derivative state feedback stabilization (RPPDSFS) problem and a robust feed‐forward compensation (RFFC) problem. In the presence of uncertainties, by using the controller consisting of RPPDSFS part and RFFC part, all the signals of the closed‐loop system are bounded. The latter is equivalent to seeking three coefficient matrices such that a series of linear matrix equations are satisfied, and simultaneously the effect caused by uncertainties is minimized. Based on the right coprime factorization, the RFFC problem is further converted into a minimization problem with a quadratic performance index and some linear constraints. Thereafter, a linear matrix equation that determines the optimal solution is achieved. The restrictive condition that the coefficient matrix of the augmented reference system is non‐defective is relaxed, which reduces the conservativeness. A numerical example and the application of a three‐axis dynamic flight motion simulator manifest the effectiveness and the practicability of the proposed algorithm.

New stability tests for fractional positive descriptor linear systems

New stability tests for fractional positive descriptor linear systems

Addressing stability challenges in linear descriptor systems: A unified approach to robust control

This paper focuses on studying the robust stability of linear descriptor systems with perturbations in the state matrix under rank-deficient and unmatched conditions. In fact, there are three main steps in this study. The first is constructing an equivalent optimal control to design a robust control. The second step is to solve the Riccati equation to find the optimal control solution. The third step is to implement the cost function of the equivalent optimal control to define the candidate Lyapunov function in order to verify the stability conditions. The proposed control method is verified through simulations. Our results demonstrate that the proposed approach can guarantee robust stability under uncertain and perturbed conditions.

Analysis of the pointwise completeness and the pointwise degeneracy of the standard and fractional descriptor linear systems and electrical circuits

The Drazin inverse of matrices is applied to analysis of the pointwise completeness and the pointwise degeneracy of the descriptor standard and fractional linear continuous-time and discrete-time systems. It is shown that: 1) The descriptor linear continuous-time system is pointwise complete if and only if the initial and final states belong to the same subspace. 2) The descriptor linear discrete-time system is not pointwise complete if its system matrix is singular. 3) System obtained by discretization of continuous-time system is always not pointwise complete. 4) The descriptor linear continuous-time system is not pointwise degenerated in any nonzero direction for all nonzero initial conditions. 5) The descriptor fractional system is pointwise complete if the matrix defined by (36) is invertible. 6) The descriptor fractional system is pointwise degenerated if and only if the condition (41) is satisfied. Considerations are illustrated by examples of descriptor linear electrical circuits.

A new bounded real lemma for discrete‐time linear descriptor systems

AbstractThis paper presents a new bounded real lemma for discrete‐time linear descriptor systems. We first present a necessary and sufficient condition for the admissibility of discrete‐time descriptor systems expressed in terms of strict linear matrix inequalities (LMIs). We then prove that a discrete‐time descriptor system is ‐stable if and only if a symmetric matrix satisfying two strict LMIs can be found. Two illustrative examples are provided for verification.

Laplace Distribution Based Online Identification of Linear Systems With Robust Recursive Expectation–Maximization Algorithm

The robust online identification problem of linear systems is considered in this paper using a faster robust recursive expectation-maximization (RREM) framework. To improve the convergence rate, the outliers, which would deteriorate the identified models, are accommodated with a Laplace distribution instead of Student's t-distribution. Then, the recursive transformation of the maximum likelihood function is realized with a recursive Q-function. The extensively recognized autoregressive exogenous (ARX) models are used for the description of general linear systems. As a result, the unknown parameters, including the regression coefficient vector of the ARX models, the variance of the noise without outliers, and the scale parameter of the Laplace distribution, are determined in a recursive manner. The performance of the proposed approach is tested with a simulated continuous fermentation reactor (CFR) system example and a coupled-tank experiment.

Adaptive observer design for a class of nonlinear descriptor systems

This paper concerns the adaptive observer design for a class of nonlinear descriptor systems. It allows the simultaneous estimation of the system states and its parameters. The presented observer is in a general form, generalizing the design for the full order and reduced order observers. The existence conditions for the observer are given and the parameterization of all the observer parameters is provided. The design is obtained in the LMI formulation. The case of the linear descriptor system can be directly obtained from the presented results as a particular case. A numerical example is provided to illustrate the design procedure.

Robust Stability Analysis of a Simple Data-Driven Model Predictive Control Approach

In this paper, we provide a theoretical analysis of closed-loop properties of a simple data-driven model predictive control (MPC) scheme. The formulation does not involve any terminal ingredients, thus allowing for a simple implementation without (potential) feasibility issues. The proposed approach relies on an implicit description of linear time-invariant systems based on behavioral systems theory, which only requires one input-output trajectory of an unknown system. For the nominal case with noise-free data, we prove that the data-driven MPC scheme ensures exponential stability for the closed loop if the prediction horizon is sufficiently long. Moreover, we analyze the robust data-driven MPC scheme for noisy output measurements for which we prove closed-loop practical exponential stability. The advantages of the presented approach are illustrated with a numerical example.

Eigenvalues assignment in descriptor linear systems by state and its derivative feedbacks

Eigenvalues assignment in descriptor linear systems by state and its derivative feedbacks

Analysis of distinguishability of linear descriptor control systems using Drazin inverse

This article aims to provide the equivalent criteria for the distinguishability of linear descriptor systems (LDS). Regularity of the matrix pencil, which, loosely speaking, guarantees the existence, and uniqueness of the solution of LDS for any inhomogeneity, is required in this article. A characterization of observability for LDS in terms of distinguishability is given. The Laplace transform together with the Cayley-Hamilton theorem exploited to derive Hautus-type criteria for the distinguishability. In addition, we present examples of distinguishable systems.

Stability analysis of robust control problems in linear descriptor systems with matching conditions

We investigate the optimal control equivalent and state feedback approach to linear time-invariant descriptor systems with norm-bounded uncertainty in matrix A(q). The goal is to find a controller that can stabilize the system even in the presence of random perturbations and to study the effect of initial conditions on the system’s stability. The systems with matching conditions and also linear algebraic equations with the full rank of the algebraic coefficient matrix are investigated. The solution to the optimal control problem (OCP) for the linear descriptor system (LDS) is the solution to the robust linear descriptor control problem provided. We will show that with theorems and examples.

Determining the response of optical systems in both time and harmonic domains with the singularity expansion method

Physical systems are characterized by their transfer operators in the harmonic domain. These operators are usually locally approximated as rational functions or pole expansions. We generalize this result and introduce the Multiple-Order Singularity Expansion Method (MOSEM) which offers an exact description of linear systems in terms of their singularities and Laurent series coefficients or zeros. The interest of this approach is first illustrated by the simple but fundamental case of a dispersive Fabry-Perot cavity, where it provides an analytical expression of the reflected field in both the time and harmonic domains. In a second step, we show that this method must be applied for defining the complex expression of the dielectric permittivity that describes the physical response of a system (the material) to an excitation field. This rigorous expression of the permittivity is shown to provide highly accurate results for a broad range of materials.