Linear Dynamical Systems Research Articles

On the External Positivity of SISO Linear Dynamic Systems under a Class of Nonzero and Possibly Negative Initial Conditions Eventually Subject to Incommensurate Point Internal and External Delays

The property of external positivity of dynamic systems is commonly defined as the non-negativity of the output for all time under zero initial conditions and any given non-negative input for all time. This paper investigates the extension of that property for a structured class of initial conditions of a single-input single-output (SISO) linear dynamic system which can include, in general, certain negative initial conditions. The above class of initial conditions is characterized analytically based on the structure of the transfer function. The basic study is performed in the delay-free case, but extensions are then given for systems subject to a finite number of internal and external, in general incommensurate, point delays and for the closed-loop dynamic systems which incorporate a feedback compensator. The formulation relies on calculating the output based on the impulse responses by considering the relation of the mentioned sets of structured initial conditions with the zero-state response which allows to keep the non-negativity of the zero-input response and that of the total response provided the non-negativity for all time of the zero-state response.

Controllability of Brain Neural Networks in Learning Disorders—A Geometric Approach

The human brain can be interpreted mathematically as a linear dynamical system that shifts through various cognitive regions promoting more or less complicated behaviors. The dynamics of brain neural network play a considerable role in cognitive function and therefore of interest in the bid to understand the learning processes and the evolution of possible disorders. The mathematical theory of systems and control makes available procedures, concepts, and criteria that can be applied to ease the perception of the dynamic processes that administer the evolution of the brain with learning and its control with treatment in case of disorder. In this work, a geometric study through the conception of exact controllability is comprehended to detect the minimum set and the location of the driving nodes of learning. We will describe the different roles of the nodes in the control of the paths of brain networks and show the transition of some driving nodes and the preservation of the rest in the course of learning in patients with some learning disability.

Robustness guarantees for structured model reduction of dynamical systems with applications to biomolecular models

AbstractModel reduction methods usually focus on the error performance analysis; however, in presence of uncertainties, it is important to analyze the robustness properties of the error in model reduction as well. This problem is particularly relevant for engineered biological systems that need to function in a largely unknown and uncertain environment. We give robustness guarantees for structured model reduction of linear and nonlinear dynamical systems under parametric uncertainties. We consider a model reduction problem where the states in the reduced model are a strict subset of the states of the full model, and the dynamics for all of the other states are collapsed to zero (similar to quasi‐steady‐state approximation). We show two approaches to compute a robustness guarantee metric for any such model reduction—a direct linear analysis method for linear dynamics and a sensitivity analysis based approach that also works for nonlinear dynamics. Using the robustness guarantees with an error metric and an input‐output mapping metric, we propose an automated model reduction method to determine the best possible reduced model for a given detailed system model. We apply our method for the (1) design space exploration of a gene expression system that leads to a new mathematical model that accounts for the limited resources in the system and (2) model reduction of a population control circuit in bacterial cells.

Event-Triggered Impulsive Optimal Control for Continuous-Time Dynamic Systems with Input Time-Delay

Time-delay is an inevitable factor in practice, which may affect the performance of optimal control. In this paper, the event-triggered impulsive optimal control for linear continuous-time dynamic systems is studied. The event-triggered impulsive optimal feedback controller with input time-delay is presented, where the impulsive instants are determined by some designed event-triggering function and condition depending on the state of the system. Some sufficient conditions are given for guaranteeing the exponential stability with the optimal controller. Moreover, the Zeno-behavior for the impulsive instants is excluded. Finally, an example with numerical simulation is given to verify the validity of the theoretical results.

Rayleigh’s quotients and eigenvalue bounds for linear dynamical systems

Rayleigh’s quotients and eigenvalue bounds for linear dynamical systems

Дослідження плавності ходу багатоосних технологічних систем

Goal. To improve the smooth running of the modular energy unit by creating an elastic-dissipative connection of the energy module with the technological one. Methods. Mathematical formalization of the studied process was based on the provisions of theoretical mechanics, tractor theory, statistical dynamics, frequency methods of the theory of automatic control of linear dynamical systems for testing statistically random perturbing effects. The experimental studies were based on both generally accepted standards and developed original methods using tensometric, vibrographic equipment with the recording of signals recorded on a PC using an analog-to-digital converter. Processing of experimental results was performed on a PC using probability theory and correlation spectral analysis. Results. According to the results of the study, an adequate dynamic model of a modular power tool in the plow unit as a multi-axis dynamic system is obtained, which allows studying its motion in the longitudinal-vertical plane. The values were obtained of the amplitude characteristics of the vertical displacements of the rear axle of the power module during the reproduction of the oscillations of the path profile and traction resistance of the plow at different values of the coefficient of dissipative resistance of the two modules of the modular power vehicle. Based on these experimental studies, the dependences of the normalized correlation functions and the spectral densities of the accelerations of the vertical oscillations of the rear axle of the energy module at different values of the coefficient of resistance of the dissipative connection of the two modules were obtained. Their analysis made it possible to substantiate the value of the rational rigidity of the elastic-dissipative connection of the energy and technological modules of the modular energy unit of universal use. Conclusions. Studies proved that with increasing the coefficient of resistance of the dissipative connection of the two modules to 1.80 kN·s·m–1, the amplitude of oscillations of the rear axle of the power module decreased 10 times, and resonant peaks of frequency characteristics were shifted to low frequencies from 16.5 s–1 to 6.0 s–1 with simultaneous reduction of their maximum values by 4 times. The peaks of the amplitude-frequency characteristics for the reproduction of the modular energy unit of oscillations of the traction resistance of the plow shifted from 13 to 10 s–1 with a simultaneous decrease in their maximum values by 3 times. It is proved that the rational stiffness of the elastic-dissipative connection of two modules falls on the value of the resistance coefficient 1.65 kN·s·m–1. The dispersion of the accelerations of the vertical oscillations of the modular energy unit was reduced by 3 times. It is established that the oscillations of the traction resistance of agricultural tools have 10 times less effect on the smoothness of the modular power tool in the plowing unit than the oscillations of the longitudinal profile of the track.

O-Minimal Invariants for Discrete-Time Dynamical Systems

Termination analysis of linear loops plays a key rôle in several areas of computer science, including program verification and abstract interpretation. Already for the simplest variants of linear loops the question of termination relates to deep open problems in number theory, such as the decidability of the Skolem and Positivity Problems for linear recurrence sequences, or equivalently reachability questions for discrete-time linear dynamical systems. In this article, we introduce the class of o-minimal invariants , which is broader than any previously considered, and study the decidability of the existence and algorithmic synthesis of such invariants as certificates of non-termination for linear loops equipped with a large class of halting conditions. We establish two main decidability results, one of them conditional on Schanuel’s conjecture is transcendental number theory.

Dynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations.

We analyze the stability of linear dynamical systems defined on sparse, random graphs with predator-prey, competitive, and mutualistic interactions. These systems are aimed at modeling the stability of fixed points in large systems defined on complex networks, such as ecosystems consisting of a large number of species that interact through a food web. We develop an exact theory for the spectral distribution and the leading eigenvalue of the corresponding sparse Jacobian matrices. This theory reveals that the nature of local interactions has a strong influence on a system's stability. We show that, in general, linear dynamical systems defined on random graphs with a prescribed degree distribution of unbounded support are unstable if they are large enough, implying a tradeoff between stability and diversity. Remarkably, in contrast to the generic case, antagonistic systems that contain only interactions of the predator-prey type can be stable in the infinite size limit. This feature for antagonistic systems is accompanied by a peculiar oscillatory behavior of the dynamical response of the system after a perturbation, when the mean degree of the graph is small enough. Moreover, for antagonistic systems we also find that there exist a dynamical phase transition and critical mean degree above which the response becomes nonoscillatory.

What’s decidable about linear loops?

We consider the MSO model-checking problem for simple linear loops, or equivalently discrete-time linear dynamical systems, with semialgebraic predicates (i.e., Boolean combinations of polynomial inequalities on the variables). We place no restrictions on the number of program variables, or equivalently the ambient dimension. We establish decidability of the model-checking problem provided that each semialgebraic predicate either has intrinsic dimension at most 1, or is contained within some three-dimensional subspace. We also note that lifting either of these restrictions and retaining decidability would necessarily require major breakthroughs in number theory.

A simple robust method of fractional time-delay estimation for linear dynamic systems

This paper is concerned with estimating the parameters of single-input, continuous-time systems from sampled input–output data, where any input time delays may be a fraction of the sampling interval. The proposed method estimates the parameters by minimizing a cost function defined by the sum of squared errors between the predicted and measured outputs using the optimal Refined Instrumental Variable method for Continuous-time models (RIVC). Potential difficulties may be encountered when standard gradient-based optimization is used for solving this optimization problem directly. These arise because the cost function is normally multi-modal with respect to the fractional time delay and they can occur for several reasons, such as aliasing errors, the nature of the system dynamics, and the selection of the excitation signal. In order to avoid the problems caused by these local minima, a two-stage procedure is proposed combining RIVC estimation of the model parameters with a grid search for the fractional time delay. Although this algorithm is not quite as computationally efficient as standard gradient-based optimization, it is simple, robust, and more widely applicable, as illustrated by numerical examples.

Computing Different Realizations of Linear Dynamical Systems with Embedding Eigenvalue Assignment

In this paper we investigate realizability of discrete time linear dynamical systems (LDSs) in fixed state space dimension. We examine whether there exist different Θ = (A,B,C,D) state space realizations of a given Markov parameter sequence Y with fixed B, C and D state space realization matrices. Full observation is assumed in terms of the invertibility of output mapping matrix C. We prove that the set of feasible state transition matrices associated to a Markov parameter sequence Y is convex, provided that the state space realization matrices B, C and D are known and fixed. Under the same conditions we also show that the set of feasible Metzler-type state transition matrices forms a convex subset. Regarding the set of Metzler-type state transition matrices we prove the existence of a structurally unique realization having maximal number of non-zero off-diagonal entries. Using an eigenvalue assignment procedure we propose linear programming based algorithms capable of computing different state space realizations. By using the convexity of the feasible set of Metzler-type state transition matrices and results from the theory of non-negative polynomial systems, we provide algorithms to determine structurally different realization. Computational examples are provided to illustrate structural non-uniqueness of network-based LDSs.

Block-Wise Parallel Semisupervised Linear Dynamical System for Massive and Inconsecutive Time-Series Data With Application to Soft Sensing

Linear dynamical system (LDS) has established itself as a powerful tool for soft sensing dynamic industrial processes, which however still faces some practically pivotal issues. Firstly, traditional LDS-based soft sensors require training samples to be labeled and ignore time delays (TDs) between explanatory variables (EVs) and primary variables (PVs), whereas labeled samples are usually scarce and the TDs could have appreciable impacts on the predictive accuracy. Secondly, LDS models are trained in a serial way, rendering significant computational deficiency when dealing with massive data that imply a wealth of information on process dynamics. What’s worse, such learning paradigm may fail to take full advantage of all available data, because the time-series data chain is often broken due to malfunctions of data communication system or measurement sensors. These issues lead the LDS-based soft sensors to degraded performance such as low accuracy, computational deficiency and poor interpretability. To this end, this paper first proposes a semisupervised training framework for the LDS with optimized TDs (TD-SsLDS), so as to make up for the deficiencies of supervised learning and overlooking of TDs. Further, a block-wise parallel TD-SsLDS (BwP-TD-SsLDS) is proposed, where a highly efficient learning algorithm is designed to extract patterns from multiple data blocks in a parallel way such that the LDS is enabled to learn from massive and inconsecutive time-series data. Case studies are conducted on both artificial example and real-life industrial process for evaluating the performance of the TD-SsLDS and BwP-TD-SsLDS. The results demonstrate that the proposed schemes could significantly improve the predictive and computational performance for the LDS-based soft sensors.

Modeling and State Estimation of Linear Destination-Constrained Dynamic Systems

Goal-guided behavior and destination-directed motion appear in a wide range of human activities and physical processes. Taking advantage of such predictive information can produce better system models and state estimates. This paper focuses on modeling and filtering issues of linear dynamic systems with linear destination constraints. We examine deficiencies of the existing destination constrained (DC) estimation methods. Basically, they resort to post-treatment by imposing destination constraints on updated states only. Viewing destination constraints as an attribute implicit of the state evolution, we propose to incorporate the destination constraints accordingly, particularly on the predictive distribution of the whole state sequence. We construct a congruous DC dynamic model by sufficiently refining the relaxed dynamics with the destination constraint using the state augmentation technique, and analyze its characterization and properties. Next, for the proposed DC model, we develop an optimal DC state estimator and describe its properties. Finally, in the context of aerial surveillance, the superiority of the proposed estimator to existing DC estimators is verified by simulation results, and the effectiveness of the proposed DC dynamic model is demonstrated using real data.

Bayesian Algorithms Learn to Stabilize Unknown Continuous-Time Systems

Linear dynamical systems are canonical models for learning-based control of plants with uncertain dynamics. The setting consists of a stochastic differential equation that captures the state evolution of the plant understudy, while the true dynamics matrices are unknown and need to be learned from the observed data of state trajectory. An important problem is to ensure that the system is stabilized and destabilizing control actions due to model uncertainties are precluded as soon as possible. A reliable stabilization procedure for this purpose that can learn from unstable data to stabilize the system in finite time, is not currently available. In this work, we establish sufficient learning-accuracy bounds for guaranteeing stabilization and propose a novel Bayesian algorithm that stabilizes unknown continuous-time stochastic linear systems. The presented algorithm is flexible and exposes effective stabilization performance after a remarkably short time period of interacting with the system.

Spatiotemporal Cardiac Statistical Shape Modeling: A Data-Driven Approach.

Clinical investigations of anatomy's structural changes over time could greatly benefit from population-level quantification of shape, or spatiotemporal statistic shape modeling (SSM). Such a tool enables characterizing patient organ cycles or disease progression in relation to a cohort of interest. Constructing shape models requires establishing a quantitative shape representation (e.g., corresponding landmarks). Particle-based shape modeling (PSM) is a data-driven SSM approach that captures population-level shape variations by optimizing landmark placement. However, it assumes cross-sectional study designs and hence has limited statistical power in representing shape changes over time. Existing methods for modeling spatiotemporal or longitudinal shape changes require predefined shape atlases and pre-built shape models that are typically constructed cross-sectionally. This paper proposes a data-driven approach inspired by the PSM method to learn population-level spatiotemporal shape changes directly from shape data. We introduce a novel SSM optimization scheme that produces landmarks that are in correspondence both across the population (inter-subject) and across time-series (intra-subject). We apply the proposed method to 4D cardiac data from atrial-fibrillation patients and demonstrate its efficacy in representing the dynamic change of the left atrium. Furthermore, we show that our method outperforms an image-based approach for spatiotemporal SSM with respect to a generative time-series model, the Linear Dynamical System (LDS). LDS fit using a spatiotemporal shape model optimized via our approach provides better generalization and specificity, indicating it accurately captures the underlying time-dependency.

Data-Driven Design of Model Predictive Control for Powertrain-Aware Eco-Driving Considering Nonlinearities Using Koopman Analysis

Eco-driving is a highly nonlinear control problem. The nonlinearities include the complex energy conversion/dissipation in the powertrain, environmental influences such as road grade and aerodynamic drag, constraints due to traffic signs, safety issues, and physical limits of the vehicle system. In recent years, researchers have increasingly revisited the Koopman operator to linearize nonlinear dynamics. This paper adopts such an approximation technique to construct the lifted state space in a data-driven procedure that allows us to incorporate nonlinearities and system perturbations in the cost function. In addition, the nonlinear constraints in states can also be handled linearly. The resultant formulation of a linearly constrained quadratic program can be readily applied to design a model predictive control that enjoys a low computation load as with a linear dynamic system. Simulation results demonstrate additional energy saving potential compared to a linear approach.

Features of the phase dynamics of fractional two-dimensional linear control systems for various differentiation operator

Features of the phase dynamics of fractional two-dimensional linear control systems for various differentiation operator

Joint Stabilization and Regret Minimization through Switching in Over-Actuated Systems

Adaptively controlling and minimizing regret in unknown dynamical systems while controlling the growth of the system state is crucial in real-world applications. In this work, we study the problem of stabilization and regret minimization of linear over-actuated dynamical systems. We propose an optimism-based algorithm that leverages possibility of switching between actuating modes in order to alleviate state explosion during initial time steps. We theoretically study the rate at which our algorithm learns a stabilizing controller and prove that it achieves a regret upper bound of O(√T).

Kalman-Like Filter Under Binary Sensors

This article is concerned with the Kalman-like filtering (KLF) problem for linear and nonlinear dynamic systems observed by binary sensors. Binary sensors are a special class of sensors that output only one bit of data and have considerable advantages in terms of energy consumption and economic costs. However, it is difficult to directly use binary sensors to estimate system states since the available information is compressed to the extreme and is difficult to be extracted. To address this problem, this article proposes an uncertainty measurement model to capture the innovations generated from binary sensors by analyzing their characteristics. Based on the proposed model, the KLFs are constructed for linear and nonlinear dynamical systems. Then, to deal with the uncertainties induced by binary sensors, conservative error covariances with adjustable parameters are derived for the KLFs via matrix inequalities and unscented transform. The optimal filter gains and some adjustable parameters are obtained by minimizing the traces and the upper bounds of the conservative covariances, respectively. Finally, arterial O <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> system and damped mass-spring system are employed to show the effectiveness and advantages of the proposed methods.

Efficient Solution and Computation of Models with Occasionally Binding Constraints

Structural macroeconometric analysis and new HANK-type models with extremely high dimensionality require fast and robust methods to efficiently deal with occasionally binding constraints (OBCs), especially since major developed economies have again hit the zero lower bound on nominal interest rates. This paper shows that a linear dynamic rational expectations system with OBCs, depending on the expected duration of the constraint, can be represented in closed form. Combined with a set of simple equilibrium conditions, this can be exploited to avoid matrix inversions and simulations at runtime for significant gains in computational speed. An efficient implementation is provided in Python programming language. Benchmarking results show that for medium-scale models with an OBC, more than 150,000 state vectors can be evaluated per second. This is an improvement of more than three orders of magnitude over existing alternatives. Even state evaluations of large HANK-type models with almost 1000 endogenous variables require only 0.1 ms.