Linear Time-varying Systems Research Articles

An approximate minimum mean-square error estimator for linear discrete time-varying systems: Handling Try-Once-Discard protocol

This paper is concerned with the remote state estimation problem for a class of linear discrete time-varying stochastic systems under communication constraints. In order to mitigate the undesired phenomena of data collisions, a Try-Once-Discard (TOD) protocol is utilized to regulate the signal transmissions over the sensor-to-estimator communication channel, where the TOD protocol is based on the scheduling rule described by a specific switching function. Under the introduced TOD protocol, the approximate minimum mean-square error (MMSE) state estimation problem is investigated and a recursive algorithm is developed for the MMSE estimator design whose computational complexity is comparable to that of the conventional Kalman filter. Furthermore, some conditions are established for the boundedness of the estimation error covariance. A numerical example is presented to illustrate the effectiveness of the proposed MMSE estimator.

Application of General Unit Hydrograph Model for Baseflow Separation from Rainfall and Streamflow Data

Baseflow separation from rainfall and streamflow data is a fundamental problem in applied hydrology, unsolved despite extensive investigations. For example, a part of baseflow is interflow, which results from rainfall infiltration, but almost all of the existing baseflow separation methods, such as graphic and filter methods, have nothing to do with it, which is a serious flaw from the viewpoint of science. The objective of this research is thus to present an innovative baseflow separation method based on the recent general unit hydrograph (UH) model and the classic Green-Ampt infiltration equation. Specifically, we divided a rainfall hyetograph into two parts using the Green-Ampt infiltration equation: one for surface flow that generates direct runoff and the other for subsurface flow that recharges groundwater and generates interflow. As with direct runoff from the surface system, we approximated the subsurface system as a linear system and thus applied the general UH model for interflow in the unsaturated soil zone, where excess infiltration is defined by analogy to excess rainfall. We assumed that groundwater flow could be described by the classic recession curve; we then added interflow and the groundwater flow to obtain the baseflow. We validated the proposed method with six real-world case studies representing four interflow patterns. Particularly, we found that, unlike direct runoff, the interflow UH model parameters are not unique, depending on stream stages. This implies that the subsurface system is a time-variant linear system, which can even make a negative interflow if groundwater is recharged from streams. We expect that this research will provide a better baseflow separation method and thus improve our understanding of watershed processes.

Distributed Kalman Filtering for Interconnected Dynamic Systems.

This article is concerned with the distributed Kalman filtering problem for interconnected dynamic systems, where the local estimator of each subsystem is designed only by its own information and neighboring information. A decoupling strategy is developed to minimize the impact of interconnected terms on the estimation performance, and then the recursive and distributed Kalman filter is derived in the minimum mean-squared error sense. Moreover, by using Lyapunov criterion for linear time-varying systems, stability conditions are presented such that the designed estimator is bounded. Finally, a heavy duty vehicle platoon system is employed to show the effectiveness and advantages of the proposed methods.

An Active Perception Framework for Autonomous Underwater Vehicle Navigation Under Sensor Constraints

Inertial navigation for autonomous underwater vehicles (AUVs) is challenging because of the drift error caused by the noise and measurement errors of inertial sensors, typically packaged as an inertial measurement unit (IMU), integrated over time. To mitigate the drift error, recent AUV state estimation approaches incorporate external references or environmental information obtained from exteroceptive sensors, with increased costs and limited operational domains. For improved navigation under sensor constraints, this article proposes an active perception framework that exploits vehicle motion to estimate the flow state together with the vehicle state using IMU and depth sensors only. The proposed framework uses the estimated flow state as external information to improve vehicle state estimation. We construct a linear time-varying system for the flow state, separated from a nonlinear system for the vehicle state. This formulation allows us to analyze uniform complete observability for the flow state, which is found to depend on vehicle motion. Then, along with vehicle and flow state estimators, we design a vehicle controller to enable vehicle motion to maximize an information metric pertaining to estimation performance based on either observability or constructability Gramian for the flow state. The proposed framework is validated through simulations for a case study with a vehicle descending through the water column in a time-varying flow field. The effectiveness of the framework is demonstrated by comparing results obtained from its four implementations with those from baseline approaches without active perception.

Nonconvex, Fully Distributed Optimization Based CAV Platooning Control Under Nonlinear Vehicle Dynamics

CAV platooning technology has received considerable attention, driven by the next generation smart transportation systems. This paper considers nonlinear vehicle dynamics and develops fully distributed optimization based CAV platooning control schemes via the platoon centered MPC approach for a possibly heterogeneous CAV platoon. The nonlinear vehicle dynamics leads to major difficulties in distributed algorithm development and control analysis. Specifically, the underlying MPC optimization problem is nonconvex and densely coupled. Further, the closed loop dynamics becomes a time-varying nonlinear system with non-vanishing external perturbations, making stability analysis rather complicated. To overcome these difficulties, we formulate the underlying MPC optimization problem as a locally coupled, albeit nonconvex, optimization problem and develop a sequential convex programming based fully distributed scheme for a general MPC horizon. Such a scheme can be effectively implemented for real-time computing using operator splitting methods. To analyze the closed loop stability, we apply various tools from global implicit function theorems, stability of linear time-varying systems, and Lyapunov theory for input-to-state stability to show that the closed loop system is locally input-to-state stable uniformly in all small coefficients pertaining to the nonlinear dynamic effects. Numerical tests on a heterogeneous CAV platoon in a real traffic condition illustrate the effectiveness of the proposed method.

Computing valuations of the Dieudonné determinants

This paper addresses the problem of computing valuations of the Dieudonné determinants of matrices over discrete valuation skew fields (DVSFs). Under a reasonable computational model, we propose two algorithms for a class of DVSFs, called split. Our algorithms are extensions of the combinatorial relaxation of Murota (1995) and the matrix expansion by Moriyama and Murota (2013), both of which are based on combinatorial optimization. While our algorithms require an upper bound on the output, we give an estimation of the bound for skew polynomial matrices and show that the estimation is valid only for skew polynomial matrices.We consider two applications of this problem. The first one is the noncommutative weighted Edmonds' problem (nc-WEP), which is to compute the degree of the Dieudonné determinants of matrices having noncommutative symbols. We show that the presented algorithms reduce the nc-WEP to the unweighted problem in polynomial time. In particular, we show that the nc-WEP over the rational field is solvable in time polynomial in the input bit-length. We also present an application to analyses of degrees of freedom of linear time-varying systems by establishing formulas on the solution spaces of linear differential/difference equations.

Bl-homogeneous sliding-mode observers for SISO linear time-varying systems with unknown input

In this article, a family of state observers with bl- homogeneous properties is proposed for a class of linear systems with time-varying coefficients, in presence of an unknown input and with an arbitrary relative degree. The proposed observers converge globally and at least asymptotically to the true states for systems that satisfy the structural property of strong observability. With an adequate selection of gains and parameters the convergence is global and theoretically exact, in finite time or even more, in fixed time.

Optimal control design for linear time-varying systems by interpolated variational iteration method

Optimal control of linear time-varying (LTV) systems is one of the most common problems in engineering, economics, biology, etc. Solving this problem is not always possible, and this motivates us to improve the existing approximate solution methods. In this paper, the optimality conditions are first obtained, and then the well-known variational iteration method, interpolated by B-spline functions, is applied to solve these conditions. The method is also accelerated through the elimination of redundant calculations. Our method provides a feedback controller for LTV optimal control problems and does not need to solve any system of equations or optimization problems at each iteration. The convergence of the proposed method is proved, and the algorithm’s efficiency is illustrated through several examples.

Set-membership state and parameter estimation for discrete time-varying systems based on the constrained zonotope

In this paper, the set-membership state and parameter estimation problems are considered for linear time-varying systems. The noises in the system are unknown but bounded and the variation of parameters is considered as a bounded expansion. The entire estimator is an interactive estimation of parameters and states. In parameter estimation, the exact set of parameter estimates may be non-convex because of the uncertain states. A polytopic linear parameter varying (LPV) enclosure of the system regression expression is constructed as a convex relaxation. The result of parameter estimation is a set to include the true parameter of the model. Meanwhile, in the state estimation, a similar LPV enclosure of the state-space model is also used to bound the state matrix uncertainties. This LPV enclosure in the state estimation is narrowed by the parameter estimation, and then the set of state estimates is computed to include the true state. The constrained zonotope (CZ) is used to describe sets of interest and the necessary set operations are expanded to LPV mapping. Because of the highly adjustable accuracy of CZ, the proposed algorithm can make a better trade-off between computational accuracy and efficiency. Finally, a numerical example is given to illustrate the effectiveness of the proposed algorithm.

Functional interval observer design for linear time-varying systems with additive disturbances

This paper investigates the problem of the functional interval observer design for linear time-varying (LTV) systems with additive disturbances in both input and output channels. First, sufficient conditions for the existence of a functional interval observer for LTV systems are proposed. Based on the solution of a type of Sylvester matrix equations (generalized Sylvester equations, GSEs), completely parameterized expressions of functional interval observer coefficient matrices are established, yielding a simple and effective design approach for solving the interval estimation issue of the LTV system, while the free parameters in the expressions provide the design degrees of freedom that can be utilized to achieve additional system specifications. Furthermore, the developed observer may execute state estimation at the desired convergence rate, considerably improving estimate performance. Finally, a numerical example and a back-to-turn (BTT) aircraft control example are presented, with the results showing that the upper and lower bounds provided by the designed observer provide a better performance of interval estimation of the functional state variables, demonstrating the effectiveness of the proposed method.

Analysis of Rehabilitation Occupational Therapy Techniques Based on Instrumental Music Chinese Tonal Language Spectrogram Analysis.

This paper provides an in-depth analysis of timbre-speech spectrograms in instrumental music, designs a model analysis of rehabilitation occupational therapy techniques based on the analysis of timbre-speech spectrograms in instrumental music, and tests the models for comparison. Starting from the mechanism of human articulation, this paper models the process of human expression as a time-varying linear system consisting of excitation, vocal tract, and radiation models. The system's overall architecture is designed according to the characteristics of Chinese speech and everyday speech rehabilitation theory (HSL theory). The dual judgment of temporal threshold and short-time average energy realized the phonetic length training. Tone and clear tone training were achieved by linear predictive coding technique (LPC) and autocorrelation function. Using the DTW technique, isolated word speech recognition was achieved by extracting Mel-scale Frequency Cepstral Coefficients (MFCC) parameters of speech signals. The system designs corresponding training scenes for each training module according to the extracted speech parameters, combines the multimedia speech spectrogram motion situation with the speech parameters, and finally presents the training content as a speech spectrogram, and evaluates the training results through human-machine interaction to stimulate the interest of rehabilitation therapy and realize the speech rehabilitation training of patients. After analyzing the pre- and post-test data, it was found that the p-values of all three groups were <0.05, which was judged to be significantly different. Also, all subjects changed their behavioral data during the treatment. Therefore, it was concluded that the music therapy technique could improve the patients' active gaze communication ability, verbal command ability, and active question-answering ability after summarizing the data, i.e., the hypothesis of this experiment is valid. Therefore, it is believed that the technique of timbre-speech spectrogram analysis in instrumental music can achieve the effect of rehabilitation therapy to a certain extent.

Distributed Cooperative Learning for Discrete-Time Strict-Feedback Multi Agent Systems Over Directed Graphs

This paper focuses on the distributed cooperative learning (DCL) problem for a class of discrete-time strict-feedback multi-agent systems under directed graphs. Compared with the previous DCL works based on undirected graphs, two main challenges lie in that the Laplacian matrix of directed graphs is nonsymmetric, and the derived weight error systems exist n-step delays. Two novel lemmas are developed in this paper to show the exponential convergence for two kinds of linear time-varying (LTV) systems with different phenomena including the nonsymmetric Laplacian matrix and time delays. Subsequently, an adaptive neural network (NN) control scheme is proposed by establishing a directed communication graph along with n-step delays weight updating law. Then, by using two novel lemmas on the extended exponential convergence of LTV systems, estimated NN weights of all agents are verified to exponentially converge to small neighbourhoods of their common optimal values if directed communication graphs are strongly connected and balanced. The stored NN weights are reused to structure learning controllers for the improved control performance of similar control tasks by the “mod” function and proper time series. A simulation comparison is shown to demonstrate the validity of the proposed DCL method.

Horizonwise Model-Predictive Control With Application to Autonomous Driving Vehicle

In this article, we present an innovative approach, i.e., horizonwise model-predictive control (H-MPC), to solve the model-predictive control (MPC) problem of a linear time-varying (LTV) system. In H-MPC, we regard the time-varying parameters as time invariant within the prediction horizon. To solve the MPC problem of the time-varying system, the decision variable is decomposed into two terms: one for linear time-invariant optimization and the other for compensating LTV uncertainties with an introduction to a uniform compensation condition. The proposed H-MPC solves the time-varying problem by removing the uncertainty due to the future parameter variations within the horizon and by updating the time-invariant MPC at each sampling time. To validate the usefulness of the proposed H-MPC, it is applied to lane tracking control for an autonomous driving vehicle. From a comparative study of the H-MPC and conventional MPCs in lane tracking control, it is confirmed that the proposed H-MPC has a competitive performance compared to LTV-MPC despite its much simpler structure.

An Optimization Framework for Resilient Batch Estimation in Cyber-Physical Systems

This article proposes a class of resilient state estimators for linear time-varying discrete-time systems. The dynamic equation of the system is assumed to be affected by a bounded process noise. As to the available measurements, they are potentially corrupted by a noise of both dense and impulsive natures. The latter, in addition to being arbitrary in its form, need not be strictly bounded. In this setting, we construct the estimator as the set-valued map, which associates with the measurements the minimizing set of some appropriate performance functions. We consider a family of such performance functions, each of which yielding a specific instance of the proposed general estimation framework. It is then shown that the proposed class of estimators enjoys the property of resilience, i.e., it induces an estimation error, which, under certain conditions, is independent of the extreme values of the (impulsive) measurement noise. Hence, the estimation error may be bounded, while the measurement noise is virtually unbounded. Moreover, we provide several error bounds (in different configurations), whose expressions depend explicitly on the degree of observability of the system being observed and on the considered performance function. Finally, a few simulation results are provided to illustrate the resilience property.

L2-L∞ Filter Design With Adjustable Convergence Rate for Linear Stochastic Systems

The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {L}_{2}-\mathcal {L}_{\infty }$ </tex-math></inline-formula> filter design for linear stochastic time-varying delay systems based on the filter velocity constraint is considered in this dissertation. Different from existing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {L}_{2}\,-\,\mathcal {L}_{\infty }$ </tex-math></inline-formula> performance analysis and filter design, a new criterion is proposed to make a more accurate judgment, which can not only guarantee the stability of the filtering error system with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {L}_{2}-\mathcal {L}_{\infty }$ </tex-math></inline-formula> performance level but also can estimate the convergence speed of the filter. The ideal filter should has the same convergence speed as the original system, so as to ensure the rapid convergence of the error system, but also not to reduce the anti-interference ability and increase the bandwidth of the filter system. By the new filter design method, an ideal filter can be designed, which can regulate the convergence speed of the filter system to the desired effect. A simulation instance is provided to exhibit the validity of the new approach.

Finite-Time Stabilization of Linear Systems With Unknown Control Direction via Extremum Seeking

In this article, the finite-time stabilization problem is solved for a linear time-varying system with unknown control direction by exploiting a modified version of the classical extremum-seeking algorithm. We propose to use a suitable oscillatory input to modify the system dynamics, at least in an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">average</i> sense, so as to satisfy a differential linear matrix inequality condition, which in turn guarantees that the system’s state remains inside a prescribed time-varying hyperellipsoid in the state space. The finite-time stability (FTS) of the averaged dynamics implies the FTS of the original system, as the distance between the original and the averaged dynamics can be made arbitrarily small by choosing a sufficiently high value of the dithering frequency used by the extremum-seeking algorithm. The main advantage of the proposed approach resides in its capability of dealing with systems with unknown control direction, and/or with a control direction that changes over time. Being FTS a quantitative approach, this article also gives an estimate of the necessary minimum dithering/mixing frequency provided, and the effectiveness of the proposed finite-time stabilization approach is analyzed by means of numerical examples.

Relaxed design conditions using Takagi-Sugeno fuzzy models and Lyapunov-Floquet transformation for controlling time-varying periodic systems

This manuscript approaches the Takagi-Sugeno fuzzy control technique, the Lyapunov-Floquet transformation, and their applications for controlling linear time-varying systems. Each of the techniques is applied in the control of a linear time-varying periodic system in order to compare their effectiveness. A hybrid control method involving the Takagi-Sugeno fuzzy control and the Lyapunov-Floquet transformation is proposed. In this method, the stability of the closed-loop system is mathematically guaranteed. In addition, this new procedure can reduce the conservatism of time-varying periodic systems with uncertainties in the input matrix. Finally, an application of the hybrid method is presented in a time-varying periodic system considering a constraint on input and uncertainties on the input matrix.

Time-varying stability and vibration properties of a pyrotechnic ejection plate in narrow space

This work simulates the pyrotechnic launch process which is commonly seen in aerospace engineering, and the dynamic responses of the projectile during pyrotechnic ejection are analyzed by studying the aeroelastic stability and nonlinear vibration of a rectangular plate subjected to time-varying axial blast load in a narrow space. During the launch, the plate is accelerated under the explosion effect, which makes the plate and airflow in narrow space a complicated time-varying fluid structure interaction system. The change of the plate velocity is derived from the momentum theorem. Considering the Von Karman nonlinear thin plate theory and the linear potential flow theory, the governing equation of the pyrotechnic ejection plate in narrow space is established. In the process of solving the equation, the Hamilton’s principle and the assumed mode method are employed. By applying the generalized eigenvalue method to linear time-varying system, the variations of the natural frequencies of the plate during pyrotechnic ejection are studied, from which the stability mechanism of the system is obtained. For nonlinear vibration of the system, the displacement time responses of the plate during pyrotechnic ejection are discussed by employing the Runge-Kutta method. From the results, the plate loses stability rapidly when the explosion occurs, and then with the increase of the velocity of the plate, the plate exhibits the flutter instability. In the process of pyrotechnic ejection, the amplitude jumping phenomenon of the plate is observed. Besides, the explosion strengthen, explosion position, slit height and aspect ratio of the plate have significant effects on the stability and displacement jumping amplitude of the pyrotechnic ejection plate.

Predictive control of uncertain systems over networks with redundant link

Networked systems using redundant channels to transmit data can effectively reduce the probability of data loss and improve system reliability and control margin. However, the structural complexity and economic cost of the system are also increased. To balance the redundancy and feasibility, the ratio of attraction domain to packet loss rate is defined as a balanced feasibility index. In this paper, single-channel packet loss is considered as Bernoulli distribution and a bounded packet loss network system control model is constructed as the arbitrary bounded packet loss control problem for redundant channel transmission network system. Therefore, the robust conditions of the closed-loop system and the constraints of the input and state are established under the framework of robust predictive control to construct the linear matrix inequality (LMI) optimization problem. Finally, to verify the effectiveness of the design method proposed in this paper, the discrete time-varying linear system and the main steam control system with redundant channels are used as study cases.

A recursive identification algorithm for linear discrete periodic systems

The parameter identification problem of linear discrete period time-varying (LDPV) systems is studied in this paper and a two-step identification algorithm based on the recursive least squares principle is proposed. The recursive principle-based identification algorithm is mostly used to deal with the parameter identification of time-invariant systems. Combine with the cyclic reconstruction method, the LDPV system is converted to a linear time-invariant system. Then, based on this and combined with recursive least squares identification, a discrimination algorithm is proposed for the parameter estimation of the LDPV system. Finally, a numerical simulation example is given to demonstrate the effectiveness of the proposed algorithm.