Linear System Theory Research Articles

The general theory of linear equation systems over semi-simple rings

The general theory of linear equation systems over semi-simple rings

Positive Consensus of Directed Multiagent Systems

This technical note investigates the positive consensus problem of multiagent systems with directed communication topologies, where all the agents have identical continuous-time positive linear dynamics. Existing works of such a problem mainly focus on the case, where networked communication topologies are of either undirected and incomplete graphs, or strongly connected directed graphs. In contrast to them, we study this problem in which the communication topologies of the multiagent system are described by directed graphs each containing a spanning tree, which is a more general and new scenario due to the interplay between the eigenvalues of the Laplacian matrix and the controller gains. Based on the existing results in spectral graph theory and positive linear systems theory, several necessary, and sufficient conditions on positive consensus of the directed multiagent system are derived through using linear matrix inequality techniques. A primal-dual iterative algorithm is developed for the computation of solutions. Finally, several numerical simulations are provided to illustrate the effectiveness of the proposed theoretical results.

Power Oscillation Analysis of PMSG Wind Power Generation System Considering Power Control Nonlinearity

High proportion of renewable energy generation, especially wind power generation, has changed the dynamic behavior of power systems and leads to emerging stability issues such as oscillation accidents. The commonly used stability analysis methods based on linear system theory cannot tackle the influence of nonlinear elements in the control loop. To fill this gap, the describing function method is used to analyze the stability and power oscillations of a grid-connected permanent magnet synchronous generator (PMSG) wind power generation system in this paper. A complete PMSG system model is taken into consideration, including a wind turbine, a generator, a machine-side converter, a grid-side converter, and a weak grid. Particularly, the nonlinear element introduced by maximum power point tracking control in the power control loop is modeled and analyzed, and the influences of critical system parameters are studied. With the high accuracy of the describing function method, it is revealed that the machine-side converter mainly influences the oscillation amplitude, while the grid-side converter mainly influences the oscillation frequency. Furthermore, the following results can be obtained: 1) High bandwidth of maximum power point tracking control and rotation speed control may enlarge the oscillation amplitude, while a fast current loop of the machine-side converter is beneficial to the stability; 2) Grid-side voltage loop can significantly affect the frequency of the oscillation, and improper high bandwidth of a phase-locked loop (PLL) is harmful to system stability; 3) A stronger grid is not always beneficial to the system stability, and the influence of grid strength should be carefully analyzed and examined when nonlinear elements are considered. All analyses are verified by simulations based on MATLAB/Simulink.

The Dual Characterization of Structured and Skewed Structured Singular Values

The structured singular values and skewed structured singular values are the well-known mathematical quantities and bridge the gap between linear algebra and system theory. It is well-known fact that an exact computation of these quantities is NP-hard. The NP-hard nature of structured singular values and skewed structured singular values allow us to provide an estimations of lower and upper bounds which guarantee the stability and instability of feedback systems in control. In this paper, we present new results on the dual characterization of structured singular values and skewed structured singular values. The results on the estimation of upper bounds for these two quantities are also computed.

인공지능 머신러닝을 이용한 카트-펜듈럼 이동제어 시스템 성형입력 설계

The tower crane is widely used in construction and transportation engineering. To improve working efficiency and safety, input shaping methods have been applied. Input shaping is a method of reducing residual vibration of flexible systems by convolving a sequence of impulses with unit step command. However, input shaping is based on the linear system theory in which its control performances are degraded, in case of nonlinearity and unmatched dynamics of the control systems. In this paper, a new optimal reference input shape design method based on minimizing cost function is suggested and applied, to a simple cart-pendulum system which is a simplified model of tower cranes. Since pendulum dynamics is nonlinear, analytic solution does not exist. To overcome this problem, in this paper, a machine learning approach is suggested to find optimal reference input shape for the cart position control. The feasibility of the proposed design method is verified through some simulation examples by using MatLab.

Identifiability analysis of linear ordinary differential equation systems with a single trajectory

Ordinary differential equations (ODEs) are widely used to model dynamical behavior of systems. It is important to perform identifiability analysis prior to estimating unknown parameters in ODEs (a.k.a. inverse problem), because if a system is unidentifiable, the estimation procedure may fail or produce erroneous and misleading results. Although several qualitative identifiability measures have been proposed, much less effort has been given to developing quantitative (continuous) scores that are robust to uncertainties in the data, especially for those cases in which the data are presented as a single trajectory beginning with one initial value. In this paper, we first derived a closed-form representation of linear ODE systems that are not identifiable based on a single trajectory. This representation helps researchers design practical systems and choose the right prior structural information in practice. Next, we proposed several quantitative scores for identifiability analysis in practice. In simulation studies, the proposed measures outperformed the main competing method significantly, especially when noise was presented in the data. We also discussed the asymptotic properties of practical identifiability for high-dimensional ODE systems and conclude that, without additional prior information, many random ODE systems are practically unidentifiable when the dimension approaches infinity.

Renormalized oscillation theory for singular linear Hamiltonian systems

Working with a general class of linear Hamiltonian systems on intervals with at least one singular endpoint which can be limit-point, limit-circle, or limit-intermediate, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of C2n. In the first part of the analysis we associate our linear Hamiltonian systems with families of well-defined self-adjoint operators, and in the latter part we employ the renormalized oscillation approach to count the number of eigenvalues these operators have on fixed intervals (λ1,λ2) whose closures do not intersect the essential spectrum of the operators. We conclude the analysis with two illustrative examples, indicating how the theory can be implemented in practice. This extends previous work by the authors for regular linear Hamiltonian systems.

Convolutional Code Theory based Steganography Technique

Steganography, as it is known, is a technique to hide a secret message within a message or collection of data that is not secret, and a problem in mathematics is to decipher the secret included in the message, to solve this problem a good tool It is the theory of codes. Unlike the existing works that use block codes to hide information using the steganographic process, in this work, we propose the use of convolutional coding theory in steganography to encrypt and decrypt messages methods to decrypt messages. Here, we suggest a steganographic protocol based on convolutional codes in which they are defined as discrete linear dynamical systems with which the properties on controllability and observability characteristic of linear systems theory can be applied, in particular the properties of output observability character which can be easily described using matrix language. The proposed decoding algorithm used for dissimulation is a linear decoding method, which has decreased both the time and space complexity, compared to the Viterbi decoding algorithm, sometimes used in other cases; indeed, we go from 2h.n to 2h=2.n, in memory space (with h: constraint height, and n: length of cover object). Moreover, the time complexity is better, while we can also denote that with the convolutional approach, we intend to take advantage of the time-depending transaction.

Feedback theory approach to positivity and stability of evolution equations

In this paper, we study the positivity and (uniform) exponential stability of a large class of perturbed semigroups. Our approach is essentially based on feedback theory of infinite-dimensional linear systems. The obtained results are applied to the stability of hyperbolic systems including those with a delay at the boundary conditions. The efficiency of our theoretical results is illustrated by some numerical simulations.

Computing Natural Frequencies and Mode Shapes of an Axially Moving Nonuniform Beam

Abstract The partial differential equation of motion of an axially moving beam with spatially varying geometric, mass, and material properties has been derived. Using the theory of linear time-varying systems and numerical optimization, a general algorithm has been developed to compute complex eigenvalues/natural frequencies, mode shapes, and the critical speed for stability. Numerical results from the new method are presented for beams with spatially varying rectangular cross sections with sinusoidal variation in thickness and sine-squared variation in width. They are also compared to those from the Galerkin method. It has been found that critical speed of the beam can be significantly reduced by nonuniformity in a beam's cross section.

Optimal guidance and estimation of a 2D diffusion–advection process by a team of mobile sensors

This paper describes an optimization framework to design guidance for a possibly heterogeneous team of multiple mobile sensors to estimate a spatiotemporal process modeled by a 2D diffusion–advection process. Owing to the abstract linear system representation of the process, we apply the Kalman–Bucy filter for estimation, where the sensors provide linear outputs. We propose an optimization problem that minimizes the sum of the trace of the covariance operator of the Kalman–Bucy filter and a generic mobility cost of the mobile sensors, subject to the sensors’ motion modeled by linear dynamics. We establish the existence of a solution to this problem. Moreover, we prove convergence to the exact optimal solution of the approximate optimal solution. That is, when evaluating these two solutions using the original cost function, the difference becomes arbitrarily small as the approximation gets finer. To compute the approximate solution, we use Pontryagin’s minimum principle after approximating the infinite-dimensional terms originating from the diffusion–advection process. The approximate solution is applied in simulation to analyze how a single mobile sensor’s performance depends on two important parameters: sensor noise variance and mobility penalty. We also illustrate the application of the framework to multiple sensors, in particular the performance of a heterogeneous team of sensors.

Generic Delay-L Left Invertibility of Structured Systems

This paper studies structured systems, namely linear systems where the state-space matrices have zeros in some fixed positions and free parameters in all other entries. This paper focuses on time-invariant systems in discrete time affected by an unknown input, and their delay-L left invertibility, namely the possibility to reconstruct the input sequence from the output sequence, assuming that the initial state is known, and requiring that the inputs can be reconstructed up to L time steps before the current output. Building upon classical results on linear systems theory and on structured systems, a graphical characterization is obtained of the integers L for which a structured system is generically delay-L left invertible.

Stabilization of 2 × 2 linear hyperbolic systems with delayed feedback boundary

This paper is concerned with the stabilization of linear hyperbolic systems with time lags in the boundary feedback. The well-posedness of such system is established. Moreover, we derived necessary and sufficient conditions on stabilization of hyperbolic systems with delayed feedback boundary. Our approach is mainly based on the feedback theory of infinite dimensional linear systems and the theory of positive semigroups.

Renormalized oscillation theory for regular linear non-Hamiltonian systems

<p style='text-indent:20px;'>In recent work, Baird et al. have generalized the definition of the Maslov index to paths of Grassmannian subspaces that are not necessarily contained in the Lagrangian Grassmannian [<xref ref-type="bibr" rid="b2">2</xref>]. Such an extension opens up the possibility of applications to non-Hamiltonian systems of ODE, and Baird and his collaborators have taken advantage of this observation to establish oscillation-type results for obtaining lower bounds on eigenvalue counts in this generalized setting. In the current analysis, the author shows that renormalized oscillation theory, appropriately defined in this generalized setting, can be applied in a natural way, and that it has the advantage, as in the traditional setting of linear Hamiltonian systems, of ensuring monotonicity of crossing points as the independent variable increases for a wide range of system/boundary-condition combinations. This seems to mark the first effort to extend the renormalized oscillation approach to the non-Hamiltonian setting.</p>

A spatially and size-structured population model with unbounded birth process

<p style='text-indent:20px;'>In this paper, we consider a spatially and size structured population model with unbounded birth process. Firstly, the model is transformed into a closed-loop system, and hence the well-posedness is established by using the feedback theory of regular linear systems. Moreover, the solution to the resulting closed-loop system is given by a perturbed semigroup. Secondly, we give a condition on birth and death rates in such a way that the solution decays exponentially. To do this, we show that the semigroup solution is positive and hence we derive a characterization of exponential stability due to the technique tools of positive semigroups. We mention that our results extend a previous work in [D. Yan and X. Fu, Comm. Pure Appl. Anal. 15 (2016), 637–655] to the unbounded situation.</p>

A Fuzzy-PIE Representation of T-S Fuzzy Systems with Delays and Stability Analysis via LPI method

Inspired by the recently proposed Partial Integral Equality(PIE) representation for linear delay systems, this paper proposes a fuzzy-PIE representation for T-S fuzzy systems with delays for the first time. Inspired by the free-weighting matrix technique, this paper introduces the free-weighting Partial Integral (PI) operators. Based on the novel representation and free-weighting PI operators, the stability issue is investigated for the T-S fuzzy systems with delays. The corresponding conditions are given as Linear Partial Inequality (LPI) and can be solved by the MATLAB toolbox PIETOOLS. Compared with the existing results, our method has no need of the bounding technique and a large amount of matrix operation. The numerical examples show the superiority of our method. This paper adds to the expanding field of LPI approach to fuzzy systems with delays.

Transforming interconnected systems and ladder circuits into implicit 2D models

In this paper, we consider two classes of spatially interconnected systems (including ladder circuits) modelled as mixed discrete-continuous linear 2D systems. Within the algebraic analysis approach to linear systems theory, we prove that these systems can always be transformed into equivalent implicit Roesser models. Moreover we show that ladder circuits can also be transformed into equivalent implicit Fornasini-Marchesini models. An advantage of our results compared to previous ones obtained through the zero coprime system equivalence approach is that the dimension of the state vector of the equivalent Roesser (or Fornasini-Marchesini) models is significantly smaller.

A scheduling algorithm for networked control systems

This paper deals with the design of scheduling logics for Networked Control Systems (NCSs) whose communication networks have limited capacity. We assume that only a subset of the plants can communicate with their controllers at any time instant. Our contributions are twofold. First, we present a probabilistic algorithm to design scheduling logics that, under certain conditions on the plant and controller dynamics and the capacity of the network, ensure stochastic stability of each plant in an NCS. Second, given the plant dynamics and the capacity of the network, we design static state-feedback controllers such that the conditions for stability under our scheduling logics are satisfied. The main apparatus for our analysis is a Markovian jump linear system representation of the individual plants in an NCS. Our stability conditions involve sets of matrix inequalities. We present numerical experiments to demonstrate our results.

A single-step recursive representation of foundation flexibility functions to soil-structure interaction using first-order IIR filters

The rational approximation of the dynamic impedance function of the foundation plays a vital role in the soil-structure interaction (SSI) analysis in the time domain. The filter representation of the rational approximation proposed by Safak provides a multi-step recursive parameter model facilitating incorporation into the time history analysis. However, it encounters both the uncertainty of the parameter identification and numerical instability. According to the linear time-invariant system theory in signal processing, this paper presents an improved single-step recursive representation of the foundation flexibility function using first-order infinite impulse response (IIR) filters. Based on the discrete responses of the foundation to the triangular unit impulse, the well-developed technologies of the digital filter can efficiently identify the stable filter parameters. It is easy to incorporate the filters into the motion equations of the structure in the forms of additional stiffness and single-step recursive variables. A computational scheme founded on the Newmark integration method is proposed to obtain the structural responses. According to the spectral radius theory, combining the filters with the Newmark integration method produces only a single operation matrix, thus effectively making the numerical stability guaranteed. Finally, a multi-degree-of-freedom model subjected to earthquake ground motion is employed to approve the proposed scheme.

Stochastic spatio-temporal optimization for control and co-design of systems in robotics and applied physics

Correlated with the trend of increasing degrees of freedom in robotic systems is a similar trend of rising interest in spatio-temporal systems described by partial differential equations (PDEs) among the robotics and control communities. These systems often exhibit dramatic under-actuation, high dimensionality, bifurcations, and multimodal instabilities. Their control represents many of the current-day challenges facing the robotics and automation communities. Not only are these systems challenging to control, but the design of their actuation is an NP-hard problem on its own. Recent methods either discretize the space before optimization, or apply tools from linear systems theory under restrictive linearity assumptions in order to arrive at a control solution. This manuscript provides a novel sampling-based stochastic optimization framework based entirely in Hilbert spaces suitable for the general class of semi-linear SPDEs which describes many systems in robotics and applied physics. This framework is utilized for simultaneous policy optimization and actuator co-design optimization. The resulting algorithm is based on variational optimization, and performs joint episodic optimization of the feedback control law and the actuation design over episodes. We study first and second order systems, and in doing so, extend several results to the case of second order SPDEs. Finally, we demonstrate the efficacy of the proposed approach with several simulated experiments on a variety of SPDEs in robotics and applied physics including an infinite degree-of-freedom soft robotic manipulator.