Linear Dynamical Systems Research Articles

Performance enhancement of large-scale linear dynamic MIMO systems using GWO-PID controller

The multi-input multi-output (MIMO) technique is becoming grown and integrated into wireless wideband communication. MIMO techniques suffer from a large-scale linear dynamic problem, it will be easy to adjust the proportional-integral-derivative (PID) of a continuous system, unlike the nonlinear model. This work displays the tuning of the PID controller for MIMO systems utilizing a statistical grey wolf optimization (GWO) and evaluated by objective function as integral time absolute error (ITAE). The instantaneous adjusting characteristic GWO approach is the criterion that distinguishes such a combination-proposed strategy from that existing in the traditional PID approach. The GWO algorithm searching-based methodology is used to determine the adequate gain factors of the PID controller. The suggested approach guarantees stability as the initial scheme for a steady state condition. A combination of ITAE combined with the GWO reduction method is adopted to reduce the steady-state transient time responses between the higher-order initial scheme and the unit amplitude response. Simulation outcomes are illustrated using MATLAB software to show the capability of adopting the GWO scheme for PID controlling.

Composition of Behavioural Assume–Guarantee Contracts

The growing complexity of modern engineering systems necessitates a method for design and analysis that is inherently modular. Methods based on using contracts for system design have successfully tackled this issue for a variety of system classes, but mostly in the context of discrete software systems. Motivated by this, we present assume-guarantee contracts for continuous linear dynamical systems with inputs and outputs. Such contracts serve as system specifications through two aspects. The assumptions specify the dynamic behaviour of the environment of the system, which provides inputs for it, while the guarantees specify the desired dynamic behaviour of the output of the system when interconnected with a relevant environment. This is formalized by utilizing the behavioural approach to system theory. We define and characterize notions of contract implementation and contract refinement, where the latter is used to compare contracts. We also define and characterize two notions of contract composition that allow one to reason about two types of system interconnections: series and feedback. The properties of refinement and composition allow contracts to be used for modular design and analysis.

Modelling and model-based tracking control of soft twisted and coiled actuators

Soft twisted and coiled actuators (STCAs) have promising prospects for application in the field of soft-bodied robots. However, the STCA has a complex energy conversion mechanism and a rate-dependent hysteresis nonlinearity, which brings great difficulties to its modelling and control. To solve these difficulties, this paper proposes modelling and model-based tracking control methods to realize the high-precision output-force tracking control of the STCA. First, a dynamics model framework is constructed, which consists of a thermo-electric model and a thermo-mechanical model, to describe the current-to-force process of the STCA. Among which, the thermo-mechanical model is established by combining the Bouc–Wen model and the linear dynamic system in series, which can accurately describe the rate-dependent hysteresis nonlinearity of the STCA. Then, a double closed-loop control architecture with a hysteresis inverse compensation controller is proposed, which can realize the timely control of the temperature and the compensation for the rate-dependent hysteresis nonlinearity of the STCA. Next, controller parameters are adjusted in the simulation environment to reduce the risk of the damage of the STCA and to obtain a satisfactory control effect. Finally, practical experiments are conducted, and root mean square errors of all experiment results are all less than 5.01%, which verifies the effectiveness of the proposed modelling and control methods.

Approximate Solutions for Optimal Control of Fixed Boundary Value Problems Using Variational and Minimum Approaches

The optimal control is the process of finding a control strategy that extreme some performance index for a dynamic system (partial differential equation) over the class of admissibility. The present work deals with a problem of fixed boundary with a control manipulated in the structure of the partial differential equation. An attractive computational method for determining the optimal control of unconstrained linear dynamic system with a quadratic performance index is presented. In the proposed method the difference between every state variable and its initial condition is represented by a finite - term polynomial series, this representation leads to a system of linear algebraic equations which represents the necessary condition of optimality. The linear algebraic system is solved by using two approaches namely the variational iteration method and the minimization approach for unconstrained optimization problem with estimation of gradient and Hessian matrix. These approaches are illustrated by two application examples.

The application of extended kalman filtering based on SLAM

EKF (Extended Kalman Filter) is a non-linear state estimation algorithm based on the development of Kalman filtering technology. In SLAM (Simultaneous Localization and Mapping), EKF technology is widely used to realize robots' independent positioning and map construction. EKF technology mainly estimates the state of the robot by using the information provided by the sensor, and then realizes the robot's self-positioning and the environment's map. This paper uses EKF algorithm for testing, and analyses the simulation results, which compared with Kalman filtering technology, the main difference between the two is that EKF can handle non -linear dynamic systems. In SLAM, robots often detect environmental detection through sensors such as laser radar, camera, and then update the position and posture of the robot by the receiving sensor data. In this process, the EKF technology can perform non -linearity by sensor data by sensor data Transformation makes the state of the robot more accurately estimate. Specifically, Extended Kalman Filtering technology can be divided into two steps: prediction and update. In the predicted phase, EKF uses the current robotic status and motion equation to predict the robot status of the next moment; in the update stage, EKF uses sensors to measure data to update the current state of the robot. By repeating the prediction and updating two steps, you can get the trajectory of the robot movement and the map of the environment where the robot is located.

Dynamics of Linear Control Systems on the Group of Proper Motions

Dynamics of Linear Control Systems on the Group of Proper Motions

A greedy Galerkin method to efficiently select sensors for linear dynamical systems

A key challenge in inverse problems is the selection of sensors to gather the most effective data. In this paper, we consider the problem of inferring the initial condition to a linear dynamical system and develop an efficient control-theoretical approach for greedily selecting sensors. Our method employs a Galerkin projection to reduce the size of the inverse problem, resulting in a computationally efficient algorithm for sensor selection. As a byproduct of our algorithm, we obtain a preconditioner for the inverse problem that enables the rapid recovery of the initial condition. We analyze the theoretical performance of our greedy sensor selection algorithm as well as the performance of the associated preconditioner. Finally, we verify our theoretical results on various inverse problems involving partial differential equations.

Energy‐based model order reduction for linear stochastic Galerkin systems of second order

AbstractWe consider a second‐order linear system of ordinary differential equations (ODEs) including random variables. A stochastic Galerkin method yields a larger deterministic linear system of ODEs. We apply a model order reduction (MOR) of this high‐dimensional linear dynamical system, where its internal energy represents a quadratic quantity of interest. We investigate the properties of this MOR with respect to stability, passivity and energy dissipation. Numerical results are shown for a system modelling a mass‐spring‐damper configuration.

Stochastic Galerkin method and port-Hamiltonian form for linear dynamical systems of second order

We investigate linear dynamical systems of second order. Uncertainty quantification is applied, where physical parameters are substituted by random variables. A stochastic Galerkin method yields a linear dynamical system of second order with high dimensionality. A structure-preserving model order reduction (MOR) produces a small linear dynamical system of second order again. We arrange an associated port-Hamiltonian (pH) formulation of first order for the second-order systems. Each pH system implies a Hamiltonian function describing an internal energy. We examine the properties of the Hamiltonian function for the stochastic Galerkin systems. We show numerical results using a test example, where both the stochastic Galerkin method and structure-preserving MOR are applied.

Stability analysis of the Robust model order reduction algorithm for parametric linear dynamical systems

AbstractWe study stability of a interpolatory model order reduction (MOR) algorithm for first‐order and second‐order parametric linear dynamical systems, with respect to inexact linear solves. This analysis is easily extendible to other MOR algorithms for such systems. Besides deriving the two conditions for stability, and subsequent experimentation, our most novel contribution here is achieving a backward stable algorithm.

A Novel relaxation of the static output feedback problem for a class of plants

The static output feedback control problem is important, as it is concerned with the case when one cannot measure all state variables. It seeks to obtain a stabilising control under these conditions and is, often, difficult to solve. For a certain class of linear dynamical systems, we provide a novel approach to obtain a stabilising control gain matrix. The class of plants considered is of those, for which we can measure at least half of all state variables and where those measurements affect at least half of all state variables. Moreover, when we write the (linearly transformed) system matrix in block form, we require that either the two lower block matrices are nonsingular, or, if the inputs affect the measured state variables directly, at least the lower left one is nonsingular. We then show that we can determine the stabilising control gain matrix from the solution of a linear matrix inequality. Finally, we apply the approach to different benchmark problems, where it performs well, and confirm its good performance through additional numerical experiments.

A dynamic-mode-decomposition-based acceleration method for unsteady adjoint equations at low Reynolds numbers

The computational cost of unsteady adjoint equations remains high in adjoint-based unsteady aerodynamic optimization. In this letter, the solution of unsteady adjoint equations is accelerated by dynamic mode decomposition (DMD). The pseudo-time marching of every real-time step is approximated as an infinite-dimensional linear dynamical system. Thereafter, DMD is utilized to analyze the adjoint vectors sampled from these pseudo-time marching. First-order zero frequency mode is selected to accelerate the pseudo-time marching of unsteady adjoint equations in every real-time step. Through flow past a stationary circular cylinder and an unsteady aerodynamic shape optimization example, the efficiency of solving unsteady adjoint equations is significantly improved. Results show that one hundred adjoint vectors contains enough information about the pseudo-time dynamics, and the adjoint dominant mode can be precisely predicted only by five snapshots produced from the adjoint vectors, which indicates DMD analysis for pseudo-time marching of unsteady adjoint equations is efficient.

Interpretable Design of Reservoir Computing Networks Using Realization Theory.

The reservoir computing networks (RCNs) have been successfully employed as a tool in learning and complex decision-making tasks. Despite their efficiency and low training cost, practical applications of RCNs rely heavily on empirical design. In this article, we develop an algorithm to design RCNs using the realization theory of linear dynamical systems. In particular, we introduce the notion of α -stable realization and provide an efficient approach to prune the size of a linear RCN without deteriorating the training accuracy. Furthermore, we derive a necessary and sufficient condition on the irreducibility of the number of hidden nodes in linear RCNs based on the concepts of controllability and observability from systems theory. Leveraging the linear RCN design, we provide a tractable procedure to realize RCNs with nonlinear activation functions. We present numerical experiments on forecasting time-delay systems and chaotic systems to validate the proposed RCN design methods and demonstrate their efficacy.

A Note on the Hausdorff Distance Between Norm Balls and Their Linear Maps

We consider the problem of computing the (two-sided) Hausdorff distance between the unit ℓp1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{p_{1}}$\\end{document} and ℓp2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{p_{2}}$\\end{document} norm balls in finite dimensional Euclidean space for 1≤p1<p2≤∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1 \\leq p_{1} < p_{2} \\leq \\infty $\\end{document}, and derive a closed-form formula for the same. We also derive a closed-form formula for the Hausdorff distance between the k1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$k_{1}$\\end{document} and k2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$k_{2}$\\end{document} unit D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$D$\\end{document}-norm balls, which are certain polyhedral norm balls in d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$d$\\end{document} dimensions for 1≤k1<k2≤d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1 \\leq k_{1} < k_{2} \\leq d$\\end{document}. When two different ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{p}$\\end{document} norm balls are transformed via a common linear map, we obtain several estimates for the Hausdorff distance between the resulting convex sets. These estimates upper bound the Hausdorff distance or its expectation, depending on whether the linear map is arbitrary or random. We then generalize the developments for the Hausdorff distance between two set-valued integrals obtained by applying a parametric family of linear maps to different ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{p}$\\end{document} unit norm balls, and then taking the Minkowski sums of the resulting sets in a limiting sense. To illustrate an application, we show that the problem of computing the Hausdorff distance between the reach sets of a linear dynamical system with different unit norm ball-valued input uncertainties, reduces to this set-valued integral setting.

Формування математичної моделі турбовального двигуна

The subject of the study is the process of forming a mathematical model (MM) of a turboshaft gas turbine engine and a twin-engine helicopter power plant, which provides the determination of parameters of the working process in steady and transient operating modes for use in the estimation of dynamic characteristics, in the analysis and synthesis of engine and helicopter automatic control systems. The goal is to substantiate the structure and methodology of MM formation intended for use in real and accelerated time scale systems. Tasks: implementation of the previously proposed MM structure taking into account the turboshaft engine performances, development of a methodology for determining the MM coefficients based on known information about the static and dynamic properties of the engine, and formation of the MM structure of a two-engine helicopter power plant. For this, the methods of the theory of airjet engines and the theory of linear dynamic systems are used. The following results were obtained: the structure of a multimode high-speed MM of a turboshaft engine and a two-engine power plant was formed and tested. The scientific and practical novelty of the obtained results is as follows: the structure of the multimode linearized MM of the turboshaft engine is formed, which consists of static and dynamic submodels implemented in corrected parameters; the modeling technique was worked out on a simplified model, compiled considering expert information about the static and dynamic properties of the engine in the considered operation area. Formulas were obtained that relate the coefficients of the linear dynamic model to the values of the time constants of the rotors and the sensitivities obtained from the static characteristics; transient characteristics of the engine based on changes in fuel consumption and load power are determined, which correspond to physical knowledge about the engine; the modeling methodology and MM structure of a two-engine power plant were formed, which is distinguished by the combination of individual static and linear dynamic models of two engines with a single nonlinear dynamic model of the helicopter rotor; a simplified MM load necessary for testing the MM of the engine installation is proposed, which provides the calculation of the power consumed by the rotor, depending on the angular position of the blades.

Countably Generated Algebras of Analytic Functions on Banach Spaces

In the paper, we study various countably generated algebras of entire analytic functions on complex Banach spaces and their homomorphisms. Countably generated algebras often appear as algebras of symmetric analytic functions on Banach spaces with respect to a group of symmetries, and are interesting for their possible applications. Some conditions of the existence of topological isomorphisms between such algebras are obtained. We construct a class of countably generated algebras, where all normalized algebraic bases are equivalent. On the other hand, we find non-isomorphic classes of such algebras. In addition, we establish the conditions of the hypercyclicity of derivations in countably generated algebras of entire analytic functions of the bounded type. We use methods from the theory of analytic functions of several variables, the theory of commutative Fréchet algebras, and the theory of linear dynamical systems.

A quasi-dynamic air traffic assignment model for mitigating air traffic complexity and congestion for high-density UAM operations

In spite of the significant effects of COVID-19, UAM operations are still expected to grow smoothly and healthily in the near future. If such dense UAM traffic relies on tactical planning to resolve conflicts in a decentralized control scheme, urban airspace could soon be heavily congested and airspace complexity could be overwhelming. In this paper, we propose a Quasi-dynamic Air Traffic Assignment (QATA) model, which aims to allocate traffic flows among air routes in the planning horizon in order to organize UAM traffic flows and reduce air traffic congestion and complexity within a centralized strategic planning scheme while meeting the demand and respecting some criteria. Firstly, UAM traffics are modeled as flows that operate on a 3D two-way UAM route network. Next, the QATA problem is formulated as an optimization problem involving network dynamics to minimize the air traffic complexity evaluated by the linear dynamical system and congestion defined by traffic density and energy consumption. A simulation-based rolling horizon framework is introduced to decompose the QATA problem into several modified static air traffic assignment problems in each time interval. In order to overcome the limitations of conventional dynamic traffic assignment algorithms, a simulated annealing algorithm using parallel computing and a novel neighborhood generation strategy is proposed to efficiently optimize the problem. By applying the model to a pre-designed large-scale UAM route network in Singapore’s urban airspace, Experimental studies demonstrate the performance of the proposed framework and its applicability. Parallel computing can achieve up to three times faster than the original algorithm. The proposed algorithm significantly reduces the value of the objective function by (32.20±0.29)% in 143.47±3.74 seconds at the 95% confidence interval of 100 experiments, far better compared to the representative conventional dynamic traffic assignment algorithms. This study could be useful to assist air traffic control authorities and air navigation service providers in addressing various issues in unmanned traffic management.

Global correlation analysis of strongly nonlinear frequency responses using the arclength-based separation and the Correlation-Map

Global correlation analysis is an important technique to quantify both the shape and amplitude differences between two response vectors. In linear dynamic systems, differences between two Frequency Response Functions (FRFs) are quantified as scalar number curves of the Global Shape Criterion (GSC) and the Global Amplitude Criterion (GAC), to represent FRF similarities at different frequencies. From linear to nonlinear, responses are usually obtained at different frequencies to form the Frequency Response Curve (FRC), replacing the FRF for dynamic analysis. Extending the concept of global correlation analysis from linear FRFs to nonlinear FRCs could quantify shape and amplitude similarities between nonlinear models. However, global correlation analysis for multivalued FRCs with a strong nonlinearity is hard to conduct, as strongly nonlinear correlation functions have complex multivalued phenomena with real/fake characteristics. In this paper, the Global Shape Curve Criterion (GSCC) and Global Amplitude Curve Criterion (GACC) are proposed for the correlation analysis of strongly nonlinear FRCs, which can quantify the similarity between two FRCs with different and complex multivalued phenomena. Through the arclength-based separation, multivalued FRCs are separated to single-valued response branches, in order to compute single-valued correlation functions that form the multivalued correlation function. The computed correlations contain the GSCC and GACC, which separately represent shape and amplitude differences between two FRCs at each frequency. The multivalued correlation function is represented as a Correlation-Map (C-MAP) to extract real correlation characteristics, for accurate correlation analysis. The multivalued correlation analysis is first verified on a 3 DOF model with a strong nonlinearity. Differences between the reference and initial multivalued FRCs are successfully quantified as scalar curves and the GACC may be more sensitive than the GSCC on models with a local nonlinearity. Then, the proposed method is further validated on an experimental 3 DOF model. Very complex 15-valued correlation functions between FRCs with different multivalued phenomena are established. Even so, the real correlations are still successfully extracted by the C-MAP. These show the validity and superiority of the proposed method.

Quantitative Aeroelastic Stability Prediction of Wings Exhibiting Nonlinear Restoring Forces

In engineering practice, eigen-solution is used to assess the stability of linear dynamical systems. However, the linearity assumption in dynamical systems sometimes implies simplifications, particularly when strong nonlinearities exist. In this case, eigen-analysis requires linerisation of the problem and hence fails to provide a direct stability estimation. For this reason, a more reliable tool should be implemented to predict nonlinear phenomena such as chaos or limit cycle oscillations. One method to overcome this difficulty is the Lyapunov Characteristic Exponents (LCEs), which provide quantitative indications of the stability characteristics of dynamical systems governed by nonlinear time-dependent differential equations. Stability prediction using Lyapunov Characteristic Exponents is compatible with the eigen-solution when the problem is linear. Moreover, LCE estimations do not need a steady or equilibrium solution and they can be calculated as the system response evolves in time. Hence, they provide a generalization of traditional stability analysis using eigenvalues. These properties of Lyapunov Exponents are very useful in aeroelastic problems possessing nonlinear characteristics, which may significantly alter the aeroelastic characteristics, and result in chaotic and limit cycle behaviour. A very common nonlinearity in flexible systems is the nonlinear restoring force such as cubic stiffness, which would substantially benefit from using LCEs in stability assessment. This work presents the quantitative evaluation of aeroelastic stability indicators in the presence of nonlinear restoring force. The method is demonstrated on a two-dimensional aeroelastic problem by comparing the system behaviour and estimated Lyapunov Exponents.

Stabilizability of Linear Dynamical Systems Using Sparse Control Inputs

Stabilizability of a linear dynamical system (LDS) refers to the existence of control inputs that drive the system state to zero. In this work, we analyze both theoretical and algorithmic aspects of the stabilizability of an LDS using sparse control inputs with potentially time-varying supports. We show that an LDS is stabilizable using sparse control inputs if and only if it is stabilizable (using unconstrained inputs). For a stabilizable LDS, we present an algorithm to determine the sparse control inputs that steer the system state to zero. We show that all stabilizable LDSs are also sparse mean square stabilizable when the process noise has zero mean and bounded second moment. For such an LDS, we devise a method to sequentially estimate the sparse control inputs to stabilize the LDS in the mean square sense. We prove that a detectable and stabilizable LDS is sparse stabilizable through output feedback and develop an algorithm for finding the corresponding sparse control inputs. Finally, we analyze the stabilizability of an LDS using sparse control inputs with common support. Our results shed light on the conditions under which a given LDS is stabilizable using sparse control inputs and the design of the corresponding control inputs.