Linear Dynamical Systems Research Articles

Reduced-order modeling via proper generalized decomposition for uncertainty quantification of frequency response functions

This paper presents a new reduced-order modeling methodology for frequency response analysis of linear dynamical systems with parametric uncertainty. The proposed framework consists of offline and online stages in the computation process. The offline stage introduces a variant of proper generalization decomposition (PGD), which approximates the solution of the full-order model (FOM) as a low-rank separated representation. A key feature of the proposed PGD is the collocation representation equipped with the Dirac-delta function, which handles non-smooth characteristics of frequency responses point-wisely. This strategy replaces direct computations of FOM on a large number of samples and frequencies with iterative solving of the subproblems formulated by the progressive Galerkin approach. In the online stage, the PGD modes acquired from the offline stage are used to generate a reduced-order model (ROM). Solutions for unsampled points are evaluated through ROM, which is further utilized to estimate the statistics of the frequency response function. Numerical examples demonstrate that the proposed methodology accelerates computational efficiency while maintaining accuracy over the frequency range including resonance.

Saddle-point equilibrium for Hurwicz model considering zero-sum differential game of uncertain dynamical systems with jump

As an effective vehicle, uncertainty theory is applicable for handling subjective indeterminacy. Based on uncertainty theory, the Hurwicz model of the zero-sum uncertain differential game with jump is formulated, in which the dynamic system is portrayed by an uncertain differential equation satisfying both the canonical Liu process and V-jump uncertain process. An equilibrium equation for solving the saddle-point of the above game is proposed. In addition, the game with a linear dynamic system and the quadratic objective function is further analysed. At last, a resource extraction problem using our theoretical results is described.

Passivity-based Analysis of the ADMM Algorithm for Constraint-Coupled Optimization

We propose a novel, system theoretic analysis of the Alternating Direction Method of Multipliers (ADMM) applied to a convex constraint-coupled optimization problem. The resulting algorithm can be interpreted as a linear, discrete-time dynamical system (modeling the multiplier ascent update) in closed loop with a static nonlinearity (representing the minimization of the augmented Lagrangian). When expressed in suitable coordinates, we prove that the discrete-time linear dynamical system has a discrete positive-real transfer function and is interconnected in closed loop with a static, passive nonlinearity. This readily shows that the origin is a stable equilibrium for the feedback interconnection. Finally, we also show global asymptotic stability of the origin for the closed-loop system and, thus, global asymptotic convergence of ADMM to the optimal solution of the optimization problem.

External drivers of BOLD signal's non-stationarity.

A fundamental challenge in neuroscience is to uncover the principles governing how the brain interacts with the external environment. However, assumptions about external stimuli fundamentally constrain current computational models. We show in silico that unknown external stimulation can produce error in the estimated linear time-invariant dynamical system. To address these limitations, we propose an approach to retrieve the external (unknown) input parameters and demonstrate that the estimated system parameters during external input quiescence uncover spatiotemporal profiles of external inputs over external stimulation periods more accurately. Finally, we unveil the expected (and unexpected) sensory and task-related extra-cortical input profiles using functional magnetic resonance imaging data acquired from 96 subjects (Human Connectome Project) during the resting-state and task scans. This dynamical systems model of the brain offers information on the structure and dimensionality of the BOLD signal’s external drivers and shines a light on the likely external sources contributing to the BOLD signal’s non-stationarity. Our findings show the role of exogenous inputs in the BOLD dynamics and highlight the importance of accounting for external inputs to unravel the brain’s time-varying functional dynamics.

Controllability and observability for fractal linear dynamical systems

In the present paper, we investigate the controllability, and observability of a non-homogeneous continuous-time fractal dynamical system (NC-TFDS). We show that the controllability is equivalent to a controllability matrix that has a full rank. We also show a relationship between controllability and fractal differential Lyapunov equation (F-DLE). We give some theorems for the observability of a NC-TFDS. Finally, we offer two examples are provided to illustrate the effectiveness of our results and compare it with the two papers Al-Zhour (2022) and (Sadek et al., 2022).

Online optimization of switched LTI systems using continuous-time and hybrid accelerated gradient flows

This paper studies the design of feedback controllers to steer a switching linear dynamical system to the solution trajectory of a time-varying convex optimization problem. We propose two types of controllers: (i) a continuous controller inspired by the online gradient descent method, and (ii) a hybrid controller that can be interpreted as an online version of Nesterov’s accelerated gradient method with restarts of the state variables. By design, the controllers continuously steer the system toward a time-varying optimal equilibrium point without requiring knowledge of exogenous disturbances affecting the system. For cost functions that are smooth and satisfy the Polyak–Łojasiewicz inequality, we demonstrate that the online gradient-flow controller ensures uniform global exponential stability when the time scales of the system and controller are sufficiently separated and the switching signal of the system varies slowly on average. For cost functions that are strongly convex, we show that the hybrid accelerated controller can outperform the continuous gradient descent method. When the cost function is not strongly convex, we show that the hybrid accelerated method guarantees global practical asymptotic stability.

On the use of Directional Importance Sampling for reliability-based design and optimum design sensitivity of linear stochastic structures

This contribution focuses on reliability-based design and optimum design sensitivity of linear dynamical structural systems subject to Gaussian excitation. Directional Importance Sampling (DIS) is implemented for reliability assessment, which allows to obtain first-order derivatives of the failure probabilities as a byproduct of the sampling process. Thus, gradient-based solution schemes can be adopted by virtue of this feature. In particular, a class of feasible-direction interior point algorithms are implemented to obtain optimum designs, while a direction-finding approach is considered to obtain optimum design sensitivity measures as a post-processing step of the optimization results. To show the usefulness of the approach, an example involving a building structure is studied. Overall, the reliability sensitivity analysis framework enabled by DIS provides a potentially useful tool to address a practical class of design optimization problems.

Measuring dynamical systems on directed hypergraphs.

Networks and graphs provide a simple but effective model to a vast set of systems in which building blocks interact throughout pairwise interactions. Unfortunately, such models fail to describe all those systems in which building blocks interact at a higher order. Higher-order graphs provide us the right tools for the task, but introduce a higher computing complexity due to the interaction order. In this paper we analyze the interplay between the structure of a directed hypergraph and a linear dynamical system, a random walk, defined on it. How can one extend network measures, such as centrality or modularity, to this framework? Instead of redefining network measures through the hypergraph framework, with the consequent complexity boost, we will measure the dynamical system associated to it. This approach let us apply known measures to pairwise structures, such as the transition matrix, and determine a family of measures that are amenable to such a procedure.

ОЦЕНКА РЕЗОНАНСНЫХ РЕЖИМОВ ПРИВОДА ГЛАВНОГО ДВИЖЕНИЯ ТОКАРНОГО СТАНКА С БЕССТУПЕНЧАТЫМ РЕГУЛИРОВАНИЕМ

The study objective is to define and evaluate the resonant drive modes of the lathe primary motion continuously rated. The problem to which the paper is devoted is to draw up a design scheme of a dynamic torsional drive system for the lathe primary motion with the spindle speed continuously rated and to make a calculation that allows finding out the lowest natural torsional vibration frequencies of the drive. Research methods: analysis of a multi-mass analytical model of the lathe drive primary motion, followed by its reduction to models with a smaller number of masses (inertial elements). The novelty of the work is in the development of software that allows simplifying linear dynamic systems and finding out natural frequencies. Study results: the dynamic quality of the drive is analyzed, which gives the opportunity to find non-resonant zones of the drive by the rotational and toothed frequencies of forced vibrations with the help of the frequency ranges of shafts. Conclusions: the developed software makes it possible at the design stage to simplify the dynamic system, find the lowest natural frequencies of torsional vibrations and take measures to exclude resonant phenomena.

State estimation for dynamic systems with higher-order autoregressive moving average non-Gaussian noise

The classical Kalman filter is a very important state estimation approach, which has been widely used in many engineering applications. The Kalman filter is optimal for linear dynamic systems with independent Gaussian noises. However, the independence and Gaussian assumptions may not be satisfied in practice. On the one hand, modeling physical systems usually results in discrete-time state-space models with correlated process and measurement noises. On the other hand, the noise is non-Gaussian when the system is disturbed by heavy-tailed noise. In this case, the performance of the Kalman filter will deteriorate, or even diverge. This paper is devoted to addressing the state estimation problem of linear dynamic systems with high-order autoregressive moving average (ARMA) non-Gaussian noise. First, a triplet Markov model is introduced to model the system with high-order ARMA noise, since this model relaxes the independence assumption of the hidden Markov model. Then, a new filter is derived based on correntropy, instead of the commonly used minimum mean square error (MMSE), to deal with non-Gaussian noise. Unlike the MMSE, which uses only second-order statistics of error, correntropy can capture second-order and higher-order statistics. Finally, simulation results verify the effectiveness of the proposed algorithm.

Measurement-outlier robust Kalman filter for discrete-time dynamic systems

This paper proposes a recursive filter for discrete-time linear dynamic systems subject to output outliers or heavy-tailed noises. First, we introduce a weight matrix in the conventional MAP estimation. It is shown that this matrix plays an influential role in the innovation whitening and asymptotic variance of the modified MAP estimation and, consequently, can be used in outlier detection. Then, we propose two different constrained optimization problems to obtain this weight. These constraints, stemming from environmental noise characteristics, help to obtain the weight matrix more precisely, which increases the filtering performance significantly. In the first approach, we introduce a convex optimization problem to minimize the estimation upper bound of the error covariance matrix. The second approach converts the modified MAP estimation to a min–min optimization problem with a concave cost function. Consequently, to reduce the effect of outliers in estimation, a semidefinite program (SDP) is proposed for outlier detection. At last, simulation results show the effectiveness and verify the performance of the proposed filter for dynamic systems in the presence of measurement outliers.

Assessment of DeepONet for time dependent reliability analysis of dynamical systems subjected to stochastic loading

Time dependent reliability analysis and uncertainty quantification of structural system subjected to stochastic forcing function is a challenging endeavour as it necessitates considerable computational time. We investigate the efficacy of recently proposed DeepONet in solving time dependent reliability analysis and uncertainty quantification of systems subjected to stochastic loading. Unlike conventional machine learning and deep learning algorithms, DeepONet is an operator network and learns a function to function mapping and hence, is ideally suited to propagate the uncertainty from the stochastic forcing function to the output responses. We use DeepONet to build a surrogate model for the dynamical system under consideration. Multiple case studies, involving both toy and benchmark problems, have been conducted to examine the efficacy of DeepONet in time dependent reliability analysis and uncertainty quantification of linear and nonlinear dynamical systems. Comparisons have also been drawn with Recurrent Neural Network results and with results obtained from Proper Orthogonal Decomposition based Gaussian process. The results obtained indicate that the DeepONet architecture is accurate as well as efficient. Moreover, DeepONet posses zero shot learning capabilities and hence, a trained model easily generalizes to unseen and new environment with no further training.

The first International Conference on Mechanical System Dynamics grandly convened in Nanjing, China

The first International Conference on Mechanical System Dynamics grandly convened in Nanjing, China

Finding Minimum Volume Circumscribing Ellipsoids Using Generalized Copositive Programming

We study the problem of finding the Löwner–John ellipsoid (i.e., an ellipsoid with minimum volume that contains a given convex set). We reformulate the problem as a generalized copositive program and use that reformulation to derive tractable semidefinite programming approximations for instances where the set is defined by affine and quadratic inequalities. We prove that, when the underlying set is a polytope, our method never provides an ellipsoid of higher volume than the one obtained by scaling the maximum volume-inscribed ellipsoid. We empirically demonstrate that our proposed method generates high-quality solutions and can be solved much faster than solving the problem to optimality. Furthermore, we outperform the existing approximation schemes in terms of solution time and quality. We present applications of our method to obtain piecewise linear decision rule approximations for dynamic distributionally robust problems with random recourse and to generate ellipsoidal approximations for the set of reachable states in a linear dynamical system when the set of allowed controls is a polytope.

Multiple Faults Estimation in Dynamical Systems: Tractable Design and Performance Bounds

In this article, we propose a tractable nonlinear fault estimation filter along with explicit performance bounds for a class of linear dynamical systems in the presence of both additive and nonlinear multiplicative faults. We consider the case, where both faults may occur simultaneously and through an identical dynamical relationship, a setting that is relevant to several application domains, including automotive driving, aviation, and chemical plants. The proposed filter architecture combines tools from model-based approaches in the control literature and regression techniques from machine learning. To this end, we view the regression operator through a system-theoretic perspective to develop operator bounds that are then utilized to derive performance bounds for the proposed estimation filter. In the case of constant, simultaneously, and identically acting additive and multiplicative faults, it can be shown that the estimation error converges to zero with an exponential rate. The performance of the proposed estimation filter in the presence of incipient faults is validated through an application on the lateral safety systems of SAE level 4 automated vehicles. The numerical results show that the theoretical bounds of this study are indeed close to the actual estimation error.

Performance evaluation of MDKF based on state estimations for path tracking applications

One of the most difficult tasks for tracking in control systems in the presence of uncertainty is estimating the accuracy and efficiency of hidden variables. For tracking applications, the Kalman filter is the most extensively used estimation algorithm. However, tracking several objects remains a difficult effort in order to improve prediction and corrective results. For tracking several objects, a multi-dimensional Kalman filter (MDKF) based on state estimations is offered as a solution. This research also includes a performance analysis of the MDKF for tracking paths by modifying co-efficients of the steady-state and covariance equations. For linear dynamic systems, MDKF is evaluated. Path tracking using the Kalman filter and MDKF is also investigated. The true and filtered responses of MDKF filtering algorithm for path tracking are observed. After four trials, the output covariance yields steady state values. The simulation results reveal that our proposed filtering algorithm performs 2x better than a typical Kalman filter for objects moving in linear motion, indicating that it is suitable for real-time implementation.

Application of the Tomtit Flock Metaheuristic Optimization Algorithm to the Optimal Discrete Time Deterministic Dynamical Control Problem

A new bio-inspired method for optimizing the objective function on a parallelepiped set of admissible solutions is proposed. It uses a model of the behavior of tomtits during the search for food. This algorithm combines some techniques for finding the extremum of the objective function, such as the memory matrix and the Levy flight from the cuckoo algorithm. The trajectories of tomtits are described by the jump-diffusion processes. The algorithm is applied to the classic and nonseparable optimal control problems for deterministic discrete dynamical systems. This type of control problem can often be solved using the discrete maximum principle or more general necessary optimality conditions, and the Bellman’s equation, but sometimes it is extremely difficult or even impossible. For this reason, there is a need to create new methods to solve these problems. The new metaheuristic algorithm makes it possible to obtain solutions of acceptable quality in an acceptable time. The efficiency and analysis of this method are demonstrated by solving a number of optimal deterministic discrete open-loop control problems: nonlinear nonseparable problems (Luus–Tassone and Li–Haimes) and separable problems for linear control dynamical systems.

Model‐free distributed optimal control for continuous‐time linear systems

This paper studies the distributed optimal consensus problem for a group of continuous-time linear systems in the scenario that the dynamics of linear systems are completely unknown. Both an optimal state feedback control law and an optimal output feedback control law are constructed by solving the Algebraic Riccati Equation (ARE) that contains no global information, which guarantees the consensus of linear systems and the global optimality. Adaptive Dynamic Programming (ADP) method is presented to solve the ARE, thus the control algorithms are not relying on the dynamics of linear systems any more. Finally, numerical simulation results are presented to demonstrate the efficiency of proposed approach.

Probing the Relationship Between Latent Linear Dynamical Systems and Low-Rank Recurrent Neural Network Models

A large body of work has suggested that neural populations exhibit low-dimensional dynamics during behavior. However, there are a variety of different approaches for modeling low-dimensional neural population activity. One approach involves latent linear dynamical system (LDS) models, in which population activity is described by a projection of low-dimensional latent variables with linear dynamics. A second approach involves low-rank recurrent neural networks (RNNs), in which population activity arises directly from a low-dimensional projection of past activity. Although these two modeling approaches have strong similarities, they arise in different contexts and tend to have different domains of application. Here we examine the precise relationship between latent LDS models and linear low-rank RNNs. When can one model class be converted to the other, and vice versa? We show that latent LDS models can only be converted to RNNs in specific limit cases, due to the non-Markovian property of latent LDS models. Conversely, we show that linear RNNs can be mapped onto LDS models, with latent dimensionality at most twice the rank of the RNN. A surprising consequence of our results is that a partially observed RNN is better represented by an LDS model than by an RNN consisting of only observed units.

Stability and Resilience—A Systematic Approach

Stability and resilience are two crucial concepts to the proper functioning and understanding of the behavior of both natural and man-made systems exposed to perturbations and change. However, although the two have covered a similar territory within dynamic systems, the terminology and applications differ significantly. This paper presents a critical analysis of the two concepts by first collating the wealth of modern stability concept literature within dynamics systems and then linking it to resilience thinking, defined as adaptation where the system has the ability to respond perturbations and change through passive and active feedback structures. A lumped mass and simple pendulum, two simple linear and nonlinear dynamic systems following a state-space approach from modern control systems theory, are used to support the analysis and application. The research findings reveal that the two overarching categories of engineering resilience and socio-ecological resilience (extended ecological resilience) are in fact a reinvention of a closed-loop system dynamic stability with different types of active feedback mechanisms. Additionally, structural stability describes some vital aspects of social–ecological resilience such as critical thresholds where, under change, a system loses the ability to return to the starting form or move to another suitable form through active feedback mechanisms or direct management actions.