Linear Time-varying Research Articles

Improved order-reduction method for cooperative tracking control of time-delayed multi-spacecraft network

This article investigates the order-reduction method for multi-spacecraft cooperative tracking control problems considering non-uniform time delays. The tracking error system is constructed as a linear time-varying (LTV) system since the orbit of the reference point is an ellipse. To facilitate the controller design, a model transformation method is proposed to transform the LTV system into a linear time-invariant (LTI) system with norm-bounded uncertainties. By using the sliding-mode control (SMC) technique, a delay-dependent cooperative tracking controller is designed to guarantee multiple followers to track the leader. Then, an order-reduction method is proposed to reduce the order of sufficient conditions in the form of linear matrix inequalities (LMIs), which make sure that the tracking error system is asymptotically stable. A numerical example is finally provided to illustrate the effectiveness of the designed controller and the improved performance of the order-reduction method.

Regression Factor Analysis With an Application to Continuous HRIR Measurement

Linear time-varying (LTV) regression models play a central role in input–output analysis of many real-world dynamical systems. Most existing models for such systems consider either a switching dynamics with a discrete latent variable driving the system or a linear dynamics directly applied to the regression coefficients. These models usually fall short of capturing structural regularities or the total variability in the dynamics of the system. Addressing these issues in this paper, we propose a method to parametrize joint variations of regression coefficients in LTV systems with continuous latent variables based on factor analysis technique, i.e., regression factor analysis . By using a linear Gaussian dynamical model for the time evolution of factor weights, our model constrains the dynamics of the process. Our inference scheme takes advantage of the expectation–maximization algorithm to estimate the model parameters. We show how our proposed algorithm can be utilized in a key application for fast continuous measurement of personalized head-related impulse responses.

State Estimation for Stochastic Time Varying Systems with Disturbance Rejection

State estimation in the presence of unknown disturbances is useful for the design of robust systems in different engineering fields. Most results available on this topic are restricted to linear time invariant (LTI) systems, whereas linear time varying (LTV) systems have been studied to a lesser extent. Existing results on LTV systems are mainly based on the minimization of the state estimation error covariance, ignoring the important issue of the stability of the state estimation error dynamics, which has been a main focus of the studies in the LTI case. The purpose of this paper is to propose a numerically efficient algorithm for state estimation with disturbance rejection, in the general framework of LTV stochastic systems, including linear parameter varying (LPV) systems, with easily checkable conditions guaranteeing the stability of the algorithm. The design method is conceptually simple: disturbance is first rejected from the state equation by appropriate output injection, then the Kalman filter is applied to the resulting state-space model after the output injection.

On the Identification of Self-Adjoint Linear Time-Varying State Models

A novel approach to the identification of linear time-varying (LTV) systems is illustrated, based on the concept of duality. Generically, if N input-output trajectories (uk, yk), k = 1,...,N of a self-adjoint LTV system are known, then the duality relation can be used to derive state trajectories xk, k = 1, ., N corresponding to such data. Such state trajectories are computed by factorizing a matrix directly constructed from input-output data of the primal and the dual system. From such input-state-output trajectories an “unfalsified” linear time-varying model can be obtained solving a system of functional equations.

New necessary and sufficient conditions for finite‐time stability of impulsive switched linear time‐varying systems

This note considers the problem of finite-time stability (FTS) for impulsive switched linear time-varying (LTV) systems. First, two improved necessary and sufficient conditions are proposed to verify the FTS of LTV by employing an indefinite Lyapunov function. Then, the results obtained are extended to study the FTS of impulsive switched LTV systems, two new necessary and sufficient conditions are given by using common indefinite Lyapunov function and piecewise indefinite Lyapunov functions. Compared with the existing results, the conditions obtained allow the time derivative of Lyapunov functions of subsystems to be indefinite and have a tighter upper bound, moreover, all subsystems are also allowed to be not FTS. Furthermore, the construction method of switching signal is given, moreover, using function-dependent dwell time methods, three sufficient conditions of FTS are also proposed for the systems. Two numerical examples are given to illustrate the theoretical results.

Model Quality Assessment and Model Mismatch Detection: A Temporal Smoothness Regularization Approach

Most industrial controllers are designed based on process models, and hence the closed-loop performance closely depends on the model quality. Since process dynamics variations are inevitable in practical applications, plant-model quality assessment is necessary so that model mismatch can be detected in time. In this article, a novel method based on temporal smoothness regularization is presented for model quality assessment. The linear time variant (LTV) model structure is applied to approximate the process dynamics. To avoid an overfitted model, temporal smoothness regularization is imposed on the model parameter changes so that model generalization ability is guaranteed. On the basis of the LTV model structure, a data-based model quality assessment approach is proposed, and the applicability is demonstrated through representative case studies.

Linear time varying model based model predictive control for lateral path tracking

This paper presents a path following application for vehicles based on a model predictive control (MPC) approach with a linear time varying (LTV) model. The control system is designed to guarantee proper path tracking under certain comfort standards and for a wide range of velocities and traces. Therefore, not only lateral error is considered, but also orientation error to the target trajectory. Additionally, strict constraints are considered in the control signal, in its variation and in the lateral path following error. Stability of the proposed controller is guaranteed using terminal cost and constraint approaches.

Equalization Techniques of Control and Non-Payload Communication Links for Unmanned Aerial Vehicles

In the next years, several new applications involving unmanned aerial vehicles (UAVs) for public and commercial uses are envisaged. In such developments, since UAVs are expected to operate within the public airspace, a key issue is the design of reliable control and non-payload communication (CNPC) links connecting the ground control station to the UAV. At the physical layer, CNPC design must cope with timeand frequency-selectivity (so-called double selectivity) of the wireless channel, due to lowaltitude operation and flight dynamics of the UAV. In this paper, we consider the transmission of continuous phase modulated (CPM) signals for UAV CNPC links operating over doubly-selective channels. Leveraging on the Laurent representation for a CPM signal, we design a two-stage receiver: the first one is a linear time-varying (LTV) equalizer, synthesized under either the zero-forcing (ZF) or minimum mean-square error (MMSE) criterion; the second one recovers the transmitted symbols from the pseudo-symbols of the Laurent representation in a simple recursive manner. In addition to LTV-ZF and LTV-MMSE equalizers, their widely-linear versions are also developed, to take into account the possible noncircular features of the CPM signal. Moreover, relying on a basis expansion model of the doubly-selective channel, we derive frequencyshift versions of the proposed equalizers, by discussing their complexity issues and proposing simplified implementations. Monte Carlo numerical simulations show that the proposed receiving structures are able to satisfactorily equalize the doubly-selective channel in typical UAV scenarios.

FDI in Cyber Physical Systems: A Distributed Zonotopic and Gaussian Kalman Filter with Bit-level Reduction

The problem addressed is that of robust Fault Detection and Isolation (FDI) in Cyber Physical Systems (CPS) modeled by interconnected linear time-varying (LTV) dynamics in a noisy environment with both bounded disturbances and random noises. A Distributed Zonotopic and Gaussian Kalman Filter (DZG-KF) framework is proposed where each network node implements a local estimator using Set-membership (Zonotopic) and Probabilistic (Gaussian) Mergers (SPM/ZGM) to estimate its state. Each node communicates its own state information only to its neighbors in the network. In addition to the usual zonotope reduction, a bit-level reduction is introduced for the first time: its aim is to reduce the required communication channel capacity (bit rate) by approximating sets through quantization by the sender and reconstruction by the receiver. In order to deal with potential packet losses, the negotiation of data reconciliation strategies between agents is discussed. Fault detection and a first-level isolation emerge from agents local tests. Numerical simulations show the efficiency of the proposed scheme.

A recursive propagator-based subspace method for vehicle handling dynamic system model identification

A high-precision vehicle handling dynamic model is of great importance in both the analysis of system performance and development of stability controller. This study introduces a new modeling algorithm based on subspace identification. Different from traditional batchwise subspace identification method, the new modeling method is proposed in the framework of online realization and unbiased estimation, which is ensured by a recursive propagator method (PM) combined with a vector autoregressive with exogenous (VARX) input model. Also, an observable and controllable 4-order linear time-varying (LTV) vehicle handling dynamic model, including rolling motion, is presented. To validate the identification algorithm, real vehicle standard road tests, including the step input test and impulse input test, are conducted, and sampling data are collected to apply into the identification procedure. The feasibility and accuracy of this online identification method are demonstrated. Furthermore, the stability of identification method is validated by an identification process based on road test data with measurement noise interference, and the superiority of the proposed identifiable LTV model is proved compared to the linear time-invariant (LTI) model.

Coordinated path-following and direct yaw-moment control of autonomous electric vehicles with sideslip angle estimation

Abstract This paper presents a novel coordinated path following system (PFS) and direct yaw-moment control (DYC) of autonomous electric vehicles via hierarchical control technique. In the high-level control law design, a new fuzzy factor is introduced based on the magnitude of longitudinal velocity of vehicle, a linear time varying (LTV)-based model predictive controller (MPC) is proposed to acquire the wheel steering angle and external yaw moment. Then, a pseudo inverse (PI) low-level control allocation law is designed to realize the tracking of desired external moment torque and management of the redundant tire actuators. Furthermore, the vehicle sideslip angle is estimated by the data fusion of low-cost GPS and INS, which can be obtained by the integral of modified INS signals with GPS signals as initial value. Finally, the effectiveness of the proposed control system is validated by the simulation and experimental tests.

A homotopy-based moving horizon estimation

ABSTRACTMoving horizon estimation (MHE) solves a constrained dynamic optimisation problem. Including nonlinear dynamics into an optimal estimation problem generally comes at the cost of tackling a non-convex optimisation problem. Here, a particular model formulation is proposed in order to convexify a class of nonlinear MHE problems. It delivers a linear time-varying (LTV) model that is globally equivalent to the nonlinear dynamics in a noise-free environment, hence the optimisation problem becomes convex. On the other hand, in the presence of unknown disturbances, the accuracy of the LTV model degrades and this results in a less accurate solution. For this purpose, some assumptions are imposed and a homotopy-based approach is proposed in order to transform the problem from convex to non-convex, where the sequential implementation of this technique starts with solving the convexified MHE problem. Two simulation studies validate the efficiency and optimality of the proposed approach with unknown disturbances.

On the mobile‐to‐mobile linear time‐variant shallow‐water acoustic channel response

SummaryWe expose some concepts concerning the channel impulse response (CIR) of linear time‐varying (LTV) channels to give a proper characterization of the mobile‐to‐mobile underwater channel. We find different connections between the linear time‐invariant (LTI) CIR of the static channel and 2 definitions of LTV CIRs of the dynamic mobile‐to‐mobile channel. These connections are useful to design a dynamic channel simulator from the static channel models available in the literature. Such feature is particularly interesting for overspread channels, which are hard to characterize by a measuring campaign. Specifically, the shallow water acoustic (SWA) channel is potentially overspread because of the signal low velocity of propagation, which prompts long delay spread responses and great Doppler effect. Furthermore, from these connections between the LTI static CIRs and the LTV dynamic CIRs, we find that the SWA mobile‐to‐mobile CIR does not only depend on the relative speed between transceivers, but also on the absolute speed of each of them referred to the velocity of propagation. Nevertheless, publications about this topic do not consider it and formulate their equations in terms of the relative speed between transceivers. We illustrate our find using 2 couples of examples where, even though the relative speed between the mobiles is the same, their CIRs are not.

A globally exponentially stable filter for bearing-only simultaneous localization and mapping with monocular vision

Abstract This paper proposes a novel filter for sensor-based bearing-only simultaneous localization and mapping in three dimensions with globally exponentially stable (GES) error dynamics. A nonlinear system is designed, its output transformed, and its dynamics augmented so that the proposed formulation can be considered as linear time-varying for the purpose of observability analysis. This allows the establishment of observability results related to the original nonlinear system that naturally lead to the design of a Kalman filter with GES error dynamics. The performance of the proposed algorithm is assessed resorting to real experiments based on the Rawseeds dataset as well as further realistic simulations.

Asymptotic Stability of LTV Systems With Applications to Distributed Dynamic Fusion

We investigate the behavior of Linear Time-Varying (LTV) systems with randomly appearing, sub-stochastic system matrices. Motivated by dynamic fusion over mobile networks, we develop conditions on the system matrices that lead to asymptotic stability of the underlying LTV system. By partitioning the sequence of system matrices into slices, we obtain the stability conditions in terms of slice lengths and introduce the notion of unbounded connectivity , i.e., the time-intervals, over which the multi-agent network is connected, do not have to be bounded as long as they do not grow faster than a certain exponential rate. We further apply the above analysis to derive the asymptotic behavior of a dynamic leader-follower algorithm.

Efficient multidimensional regularization for Volterra series estimation

This paper presents an efficient nonparametric time domain nonlinear system identification method. It is shown how truncated Volterra series models can be efficiently estimated without the need of long, transient-free measurements. The method is a novel extension of the regularization methods that have been developed for impulse response estimates of linear time invariant systems. To avoid the excessive memory needs in case of long measurements or large number of estimated parameters, a practical gradient-based estimation method is also provided, leading to the same numerical results as the proposed Volterra estimation method. Moreover, the transient effects in the simulated output are removed by a special regularization method based on the novel ideas of transient removal for Linear Time-Varying (LTV) systems. Combining the proposed methodologies, the nonparametric Volterra models of the cascaded water tanks benchmark are presented in this paper. The results for different scenarios varying from a simple Finite Impulse Response (FIR) model to a 3rd degree Volterra series with and without transient removal are compared and studied. It is clear that the obtained models capture the system dynamics when tested on a validation dataset, and their performance is comparable with the white-box (physical) models.

Stable inversion of LPTV systems with application in position-dependent and non-equidistantly sampled systems

ABSTRACTMany control applications, including feedforward and learning control, involve the inverse of a dynamical system. For nonminimum-phase systems, the response of the inverse system is unbounded. For linear time-invariant (LTI), nonminimum-phase systems, a bounded, noncausal inverse response can be obtained through an exponential dichotomy. For generic linear time-varying (LTV) systems, such a dichotomy does not exist in general. The aim of this paper is to develop an inversion approach for an important class of LTV systems, namely linear periodically time-varying (LPTV) systems, which occur in, e.g. position-dependent systems with periodic tasks and non-equidistantly sampled systems. The proposed methodology exploits the periodicity to determine a bounded inverse for general LPTV systems. Conditions for existence are provided. The method is successfully demonstrated in several application cases, including position-dependent and non-equidistantly sampled systems.

Coprime Factors Model Reduction of Spatially Distributed LTV Systems Over Arbitrary Graphs

This technical note is on the model reduction of distributed systems formed by discrete-time, linear time-varying, heterogeneous subsystems interconnected over arbitrary directed graphs and subjected to communication latency. We give two procedures to construct a strongly stable coprime factorization for a strongly stabilizable and strongly detectable system. One of the procedures ensures the contractiveness of the resulting factorization. Then, we apply the structure-preserving balanced truncation method for distributed systems. We illustrate the proposed methods through an example.

Balanced truncation model reduction of nonstationary systems interconnected over arbitrary graphs

This paper deals with the balanced truncation of discrete-time, linear time-varying, heterogeneous subsystems interconnected over finite arbitrary directed graphs. The information transfer between the subsystems is subject to a communication latency of one time-step. The method guarantees the preservation of the interconnection structure and further allows for its simplification. In addition to truncating temporal states associated with the subsystems, the method allows for the order reduction of spatial states associated with the interconnections between the subsystems and even the removal of whole interconnections. Upper bounds on the ℓ2-induced norm of the resulting error system are derived. The method is illustrated through an example.

Application of input to state stability to reservoir models

Reservoir models play an important role in representing fluxes of matter and energy in ecological systems and are the basis of most models in biogeochemistry. These models are commonly used to study the effects of environmental change on the cycling of biogeochemical elements and to predict potential transitions of ecosystems to alternative states. To study critical regime changes of time-dependent, externally forced biogeochemical systems, we analyze the behavior of reservoir models typical for element cycling (e.g., terrestrial carbon cycle) with respect to time-varying signals by applying the mathematical concept of input to state stability (ISS). In particular, we discuss ISS as a generalization of preceding stability notions for non-autonomous, non-linear reservoir models represented by systems of ordinary differential equations explicitly dependent on time and a time-varying input signal. We also show how ISS enhances existing stability concepts, previously only available for linear time variant (LTV) systems, to a tool applicable also in the non-linear case.