Linear Time-varying Dynamic Systems Research Articles

Controllability of time-varying fractional dynamical systems with distributed delays in control

Abstract The objective of this study is to assess controllability results in the realm of Caputo fractional derivatives, specifically focusing on time-varying linear and nonlinear fractional dynamical systems with distributed control delays. In the setting of a linear system, using the Grammian matrix's positive definiteness makes it possible to ascertain both the necessary and sufficient conditions. It is possible to determine, for nonlinear systems, the sufficient conditions for the presence of a solution by making use of the Schauder fixed point theorem. In addition to that, the paper discusses the controllability of a nonlinear integrodifferential system for a particular case. A few examples and the graphs that correspond to them are shown here to make it easier to conceptualize the conclusions of the theoretical discussion.

Identification of linear time-varying dynamic systems based on the WKB method

This work proposes a new parameter identification method based on the Wentzel-Kramers-Brillouin (WKB) approximation for slow linear time-varying (LTV) dynamic systems. The considered time period is divided into a series of short time windows. In each time window, the assumption of “short time linearly varying” parameters is employed, and a nonlinear optimization problem is solved using the WKB results for the slow LTV dynamic system. A search algorithm is developed to find the optimal solution. In the identification process, only one type of response signal (displacement, velocity or acceleration) is required. Thus, numerical differentiation or integration of the measured signal, which leads to truncation or cumulative errors in noise environment, is avoided. The accuracy and robustness of the new identification method are validated by applying it to a particular LTV system with time-varying stiffness.

Optimal Sensor Precision for Multirate Sensing for Bounded Estimation Error

We address the problem of determining optimal sensor precisions for estimating the states of linear time-varying discrete-time stochastic dynamical systems, with guaranteed bounds on the estimation errors. This is performed in the Kalman filtering framework, where the sensor precisions are treated as variables. They are determined by solving a constrained convex optimization problem, which guarantees the specified upper bound on the posterior error variance. Optimal sensor precisions are determined by minimizing the l1 norm, which promotes sparseness in the solution and indirectly addresses the sensor selection problem. The theory is applied to realistic flight mechanics and astrodynamics problems to highlight its engineering value. These examples demonstrate the application of the presented theory to a) determine redundant sensing architectures for linear time invariant systems, b) accurately estimate states with low-cost sensors, and c) optimally schedule sensors for linear time-varying systems.

Optimal trajectory design accounting for the stabilization of linear time-varying error dynamics

This study is dedicated to the development of a direct optimal control-based algorithm for trajectory optimization problems that accounts for the closed-loop stability of the trajectory tracking error dynamics already during the optimization. Consequently, the trajectory is designed such that the Linear Time-Varying (LTV) dynamic system, describing the controller’s error dynamics, is stable, while additionally the desired optimality criterion is optimized and all enforced constraints on the trajectory are fulfilled. This is achieved by means of a Lyapunov stability analysis of the LTV dynamics within the optimization problem using a time-dependent, quadratic Lyapunov function candidate. Special care is taken with regard to ensuring the correct definiteness of the ensuing matrices within the Lyapunov stability analysis, specifically considering a numerically stable formulation of these in the numerical optimization. The developed algorithm is applied to a trajectory design problem for which the LTV system is part of the path-following error dynamics, which is required to be stable. The main benefit of the proposed scheme in this context is that the designed trajectory trades-off the required stability and robustness properties of the LTV dynamics with the optimality of the trajectory already at the design phase and thus, does not produce unstable optimal trajectories the system must follow in the real application.

Laplace transform collocation and Daftar-Gejii-Jafaris method for fractional order time varying linear dynamical systems

In this article, the fractional order time-varying linear dynamical system defined by Caputo derivative is investigated. Laplace transform collocation method (LTCM) and Daftar-Gejii-Jafaris method (DGJM) are used to find the approximation solution of this equation. Using the Laplace transform collocation method, a new form of trial function from the original equation is presented. The unknown coefficients in the trial functions are calculated by using collocation method. LTCM gives a good result for the numerical solution of this equation. Providing DGJM converges, it is shown that obtained approximate solution is effective which is close to the exact solution. Then, the exact solution is compared with these approximate solutions. The results showed that the methods are effective and useful. These methods produced better approximations than the ones produced with the standard weighted residual methods.

Recursive formulation of the WKB solution for linear time-varying dynamic systems

This work proposes a new calculation algorithm of the Wentzel–Kramers–Brillouin (WKB) solution for slow linear time-varying (LTV) systems based on the recursive formulation. For a linear dynamic system with time-varying (TV) mass, damping or stiffness, the recursive relations in real function form among the components of the WKB solution are obtained. The integral terms within the sampling intervals that cannot be solved analytically are approximated by simple algebraic calculations with high precision. Thus, the conventional expression of the WKB solution involving complex numerical integral terms is reduced to an analytical recursive formulation. An explicit function relationship between the TV system parameters and the dynamic response can be obtained when the external excitation is given. In the premise of similar accuracy, the recursive formulation has much higher efficiency than the conventional calculation process, which is verified by applying the proposed method to a particular LTV dynamic system.

Is My System of ODEs k-Cooperative?

A linear dynamical system is called positive if its flow maps the non-negative orthant to itself. More precisely, it maps the set of vectors with zero sign variations to itself. A linear dynamical system is called $k$-positive if its flow maps the set of vectors with up to $k-1$ sign variations to itself. A nonlinear dynamical system is called $k$-cooperative if its variational system, which is a time-varying linear dynamical system, is $k$-positive. These systems have special asymptotic properties. For example, it was recently shown that strongly $2$-cooperative systems satisfy a strong Poincar\'{e}-Bendixson property. Positivity and~$k$-positivity are easy to verify in terms of the sign-pattern of the matrix in the dynamics. However, these sign conditions are not invariant under a coordinate transformation. A natural question is to determine if a given~$n$-dimensional system is $k$-positive up to a coordinate transformation. We study this problem for two special kinds of transformations: permutations and scaling by a signature matrix. For any $n\geq 4$ and~$k\in\{2,\dots, n-2\}$, we provide a graph-theoretical necessary and sufficient condition for $k$-positivity up to such coordinate transformations. We describe an application of our results to a specific class of Lotka-Volterra systems.

Model-free data reconstruction of structural response and excitation via sequential broad learning

In this study, a novel sequential broad learning (SBL) approach is proposed to reconstruct the missing signal of damaged sensors in structural health monitoring (SHM) sensory networks. It is capable to reconstruct the structural response and external excitation of linear/nonlinear time-varying dynamical systems under stationary/nonstationary excitation for sensory networks. The proposed approach is a model-free data-driven machine learning methodology and the data reconstruction is executed sequentially with moving time windows. The learning algorithm is developed by adopting the recently developed broad learning system (BLS) (Chen and Liu, 2018). In contrast to deep learning that suffers from excessive computational cost for training the stacks of hierarchical layers, BLS is established with a broadly expandable network and can be modified incrementally based on the inherited results from the previous trained architecture. Therefore, BLS provides a computationally very efficient alternative to deep learning. Taking the benefit of BLS, the proposed SBL approach can efficiently handle the massive data stream generated in long-term monitoring. To demonstrate the efficacy and applicability of the proposed approach, simulated examples that cover linear and nonlinear time-varying dynamical systems subjected to stationary/nonstationary wind-load/base excitation with different types of sensing devices are discussed. Moreover, the SHM database of the field measurement monitored from the MIT Green Building is utilized to examine the performance of the proposed approach in realistic application. It is demonstrated that the proposed approach offers a powerful data reconstruction tool for challenging data missing situations encountered in SHM.

Moment Analysis of Linear Time-Varying Dynamical Systems with Renewal Transitions

Stochastic dynamics of several systems can be modeled via piecewise-deterministic time evolution of the state, interspersed by random discrete events. Within this general class of systems, we consi...

A Bayesian estimator of operational modal parameters for linear time-varying mechanical systems based on functional series vector TAR model

Operational modal analysis of time-varying mechanical dynamic systems is a useful but challenging task. This paper presents a Bayesian estimator for modal parameters of linear time-varying mechanical dynamic systems in the framework of the functional-series vector time-dependent autoregressive (FS-VTAR) model with output-only measurements. The proposed Bayesian estimator cannot only give the modal parameters with the estimation of mean value, but also supplies the uncertainty with the creditable interval estimation. A series of numerical examples have illustrated the advantages of the proposed Bayesian estimator against the overestimate, the better performance for the short data and the capability of supplying uncertainty of estimates. An experimental example validates the proposed estimator further.

Fast Kalman-like optimal FIR filter for time-variant systems with improved robustness

In this paper, a fast Kalman-like iterative OFIR algorithm is proposed for discrete-time filtering of linear time-varying dynamic systems. The batch OFIR filter is re-derived in an alternative way to show that this filter is unique for such systems. A computationally efficient fast iterative form is found for the OFIR filter using recursions. It is shown that each recursion has the Kalman filter (KF) predictor/corrector format with initial conditions specified via measurements on a horizon of N nearest past points. In this regard, the KF is considered as a special case of the iterative OFIR filtering algorithm when N goes to infinity. Applications are given for the 3-state target tracking and three-degree-of-freedom (DOF) hover system. It has been shown experimentally that the proposed iterative OFIR algorithm operates much faster than the batch OFIR filter and has the computational complexity acceptable for real-time applications. It has also been demonstrated by simulations that an increase in the number of the states results in better robustness of the OFIR filter against temporary model uncertainties and in higher immunity against errors in the noise statistics.

Heart rate from face videos under realistic conditions for advanced driver monitoring

Abstract The role of physiological signals has a large impact on driver monitoring systems, since it tells something about the human state. This work addresses the recursive probabilistic inference problem in time-varying linear dynamic systems to incorporate invariance into the task of heart rate estimation from face videos under realistic conditions. The invariance encapsulates motion as well as varying illumination conditions in order to accurately estimate vitality parameters from human faces using conventional camera technology. The solution is based on the canonical state space representation of an Itô process and a Wiener velocity model. Empirical results yield to excellent real-time and estimation performance of heart rates in presence of disturbing factors, like rigid head motion, talking, facial expressions and natural illumination conditions making the process of human state estimation from face videos applicable in a much broader sense, pushing the technology towards advanced driver monitoring systems.

Gap metric computation for time-varying linear systems on finite horizons

It is shown how to compute the aperture, or gap, between the finite-horizon graphs of linear time-varying dynamical systems, given state-space models of the continuous-time input-output maps. The approach involves four quadratic matrix first-order differential equations, three with the symplectic structure of Riccati equations. All four are subject to single-ended boundary conditions. So standard numerical methods can be used to compute the solutions as required. Three of the differential equations, two Riccati and the other not, need to be solved once to construct normalized graph representations, and to test an invertibility condition, respectively. When this invertibility condition is verified, the directed gaps are equal, and the remaining Riccati differential equation is solved repeatedly in a bisection search to determine one of these. Computation of both directed gaps to find the maximum would, by contrast, involve repeatedly solving two such Riccati differential equations, each in a bisection search.

Adaptive Observers with Uncertainty in Loop Tuning for Linear Time-Varying Dynamical Systems

Adaptive Observers with Uncertainty in Loop Tuning for Linear Time-Varying Dynamical Systems

Канонические формы уравнений состояния линейных нестационарных динамичексих систем управления

Канонические формы уравнений состояния линейных нестационарных динамичексих систем управления

An Iterative Technique for Solving a Class of Nonlinear Quadratic Optimal Control Problems Using Chebyshev Polynomials

In this paper, a method for solving a class of nonlinear optimal control problems is presented. The method is based on replacing the dynamic nonlinear optimal control problem by a sequence of quadratic programming problems. To this end, the iterative technique developed by Banks is used to replace the original nonlinear dynamic system by a sequence of linear time-varying dynamic systems, then each of the new problems is converted to quadratic programming problem by parameterizing the state variables by a finite length Chebyshev series with unknown parameters. To show the effectiveness of the proposed method, simulation results of a nonlinear optimal control problem are presented.

On Nonnegative Solutions of Fractionalq-Linear Time-Varying Dynamic Systems with Delayed Dynamics

This paper is devoted to the investigation of nonnegative solutions and the stability and asymptotic properties of the solutions of fractional differential dynamic linear time-varying systems involving delayed dynamics with delays. The dynamic systems are described based onq-calculus and Caputo fractional derivatives on any order.

Frequency-domain least-squares support vector machines to deal with correlated errors when identifying linear time-varying systems

A Least-Squares Support Vector Machine (LS-SVM) estimator, formulated in the frequency domain is proposed to identify linear time-varying dynamic systems. The LS-SVM aims at learning the structure of the time variation in a data driven way. The frequency domain is chosen for its superior robustness w.r.t. correlated errors for the calibration of the hyper parameters of the model.The time-domain and the frequency-domain implementations are compared on a simulation example to show the effectiveness of the proposed approach. It is demonstrated that the time-domain formulation is mislead during the calibration due to the fact that the noise on the estimation and calibration data sets are correlated. This is not the case for the frequency-domain implementation.

Comments on “Decentralized iterative learning control for a class of large scale interconnected dynamical systems” by Hansheng Wu [J. Math. Anal. Appl. 327 (2007) 233–245

In the reference paper Wu (2007) [1], a decentralized iterative learning control (ILC) algorithm is proposed aiming at achieving the asymptotic convergence of output errors for a class of linear time-varying large scale interconnected dynamic systems. The key points in the analysis of this decentralized ILC are the adoption of a specific time-weighted norm, and the use of a property of nondecreasing real functions that leads to the cancellation of interactions between subsystems. Here, on the one hand, we show that there exists a derivation problem in the proof, thus the asymptotic convergence property cannot be obtained. On the other hand, we provide an alternative analysis method, the classical contraction-mapping based ILC analysis with the lambda-norm, to demonstrate that the decentralized ILC algorithm is still valid and able to achieve the asymptotic convergence.

Robust adaptive learning of feedforward neural networks via LMI optimizations

Feedforward neural networks (FNNs) have been extensively applied to various areas such as control, system identification, function approximation, pattern recognition etc. A novel robust control approach to the learning problems of FNNs is further investigated in this study in order to develop efficient learning algorithms which can be implemented with optimal parameter settings and considering noise effect in the data. To this aim, the learning problem of a FNN is cast into a robust output feedback control problem of a discrete time-varying linear dynamic system. New robust learning algorithms with adaptive learning rate are therefore developed, using linear matrix inequality (LMI) techniques to find the appropriate learning rates and to guarantee the fast and robust convergence. Theoretical analysis and examples are given to illustrate the theoretical results.