Linear Time-varying Research Articles

Quantitative analysis of incipient fault detectability for time-varying stochastic systems based on weighted moving average approach

In this paper, the problem of incipient fault detection is investigated for linear time-varying (LTV) systems with stochastic noises. The fault detectability in a probabilistic sense is defined for LTV stochastic systems by considering false alarm rate (FAR) and missed detection rate (MDR) simultaneously. Necessary and sufficient conditions are derived to reveal the relationship among the fault amplitude, FAR and MDR, and the reason why incipient faults are difficult to detect is quantitatively analyzed in the model-based framework. To improve the sensibility of the residual to incipient faults, the weighted moving average approach is introduced and its parameters, the optimal weight and the smallest window length, are accurately analyzed in theory. Moreover, the concept of average fault detectability is introduced, which is conducive to providing a feasible scheme for incipient fault detection. Finally, a numerical example and an experiment are given to show the effectiveness of the derived results.

Dynamic learning from neural network‐based control for sampled‐data strict‐feedback nonlinear systems

AbstractThis article focuses on the dynamic learning and control problem for a class of nonlinear sampled‐data systems in strict‐feedback form. To achieve learning of unknown system dynamics in closed‐loop control, two obstacles need to be overcome: the inherent noncausal problem in the backstepping design of discrete‐time nonlinear systems, and the exponential stability of the sampling closed‐loop error system. First, a novel command filtered adaptive neural control is developed to achieve closed‐loop stability and the convergence of output tracking error based on Lyapunov theory, the noncausal problem is overcome by utilizing the command filter technique, and the ‐step delay is avoided in the recursive process. Second, by combining a state transformation method, the closed‐loop error system is divided into perturbed linear time‐varying (LTV) subsystems. Based on the convergence of the tracking errors, the partial persistent excitation (PE) condition of the radial basis function network is verified and established recursively. By extending the exponential stability result of a class of discrete‐time LTV systems, the convergence of the estimated weights to their optimal values is guaranteed with the PE condition. Subsequently, locally accurate approximation/learning of the unknown closed‐loop sampled dynamics can be obtained and stored in the form of constant neural networks. Finally, by reutilizing the acquired knowledge, a learning‐based control scheme for the sampled‐data system is further proposed to pursue improved control performance. Simulation studies are performed to illustrate the validity of the proposed scheme.

Real-Time Optimization-Based Reference Calculation Integrated Control for MMCs Considering Converter Limitations

The paper addresses a real-time optimization-based reference calculation integrated with a control structure for Modular Multilevel Converters (MMC) operating under normal and constrained situations (where it has reached current and/or voltage limitations, as it may occur during system faults). Firstly, a nonlinear optimization problem has been developed in which it prioritizes to satisfy the AC grid current set-points imposed by the transmission System Operator (TSO). The constrained nonlinear optimization problem is formulated based on the steady-state model of the MMC, whereby the prioritization is achieved through distinct weights defined in the Objective Function’s (OF) terms. The resultant optimization problem, however, is highly nonlinear requiring high computation burden to be solved in real-time. To cope with this issue, this paper applies a Linear Time-Varying (LTV) approximation, which permits to represent the nonlinear dynamics of the system as constant parameters, while a Linear Time-Invariant (LTI) system is used to formulate the optimization constraints. The converter’s current references are determined in real-time by solving a constrained linearized optimization problem at each control time step, which considers the TSO’s demands, the current MMC operating point and its physical limitations. Theoretical analyses comparing the responses of the linear and nonlinear optimization problems are performed to validate the accuracy of the LTV approximation. Finally, the linearized-optimization problem is integrated with the MMC controllers, evaluated under different AC and DC network conditions and compared with conventional control strategies, where it is shown that the presented method can be potentially employed to obtain the MMC current references for distinct network scenarios.

Detectability Conditions and State Estimation for Linear Time-Varying and Nonlinear Systems

This work proposes a detectability condition for linear time-varying systems based on the exponential dichotomy spectrum. The condition guarantees the existence of an observer, whose gain is determined only by the unstable modes of the system. This allows for an observer design with low computational complexity compared to classical estimation approaches. An extension of this observer design to a class of nonlinear systems is proposed and local convergence of the corresponding estimation error dynamics is proven. Numerical results show the efficacy of the proposed observer design technique.

Exploring Inductive Linearization for simulation and estimation with an application to the Michaelis–Menten model

Nonlinear ordinary differential equations (ODEs) are common in pharmacokinetic–pharmacodynamic systems. Although their exact solutions cannot generally be determined via algebraic methods, their rapid and accurate solutions are desirable. Thus, numerical methods have a critical role. Inductive Linearization was proposed as a method to solve systems of nonlinear ODEs. It is an iterative approach that converts a nonlinear ODE into a linear time-varying (LTV) ODE, for which a range of standard integration techniques can then be used to solve (e.g., eigenvalue decomposition [EVD]). This study explores the properties of Inductive Linearization when coupled with EVD for integration of the LTV ODE and illustrates how the efficiency of the method can be improved. Improvements were based on three approaches, (1) incorporation of a convergence criterion for the iterative linearization process (for simulation and estimation), (2) creating more efficient step sizes for EVD (for simulation and estimation), and (3) updating the initial conditions of the Inductive Linearization (for estimation). The performance of these improvements were evaluated using single subject stochastic simulation-estimation with an application to a simple pharmacokinetic model with Michaelis–Menten elimination. The reference comparison was a standard non-stiff Runge–Kutta method with variable step size (ode45, MATLAB). Each of the approaches improved the speed of the Inductive Linearization technique without diminishing accuracy which, in this simple case, was faster than ode45 with comparable accuracy in the parameter estimates. The methods described here can easily be implemented in standard software programme such as R or MATLAB. Further work is needed to explore this technique for estimation in a population approach setting.

Improved state augmentation method for buffeting analysis of structures subjected to non-stationary wind

Extreme winds such as hurricanes and thunderstorms often present non-stationary characteristics, having time-varying mean wind speeds and non-stationary wind fluctuations. When concerning the wind-induced vibrations under non-stationary wind, the excitation will be a non-stationary process, and the wind-structure coupled system can be represented by a linear time-varying (LTV) system. The aim of this study is to present a state augmentation method to investigate the non-stationary buffeting of a model bridge tower subjected to non-stationary wind with consideration of the aeroelastic damping. Based on the theory of stochastic differential equations and Itô’s lemma, the statistical moments of the non-stationary buffeting response are derived through solving a first-order ordinary differential equation system. The proposed method is validated by comparisons with the Monte Carlo method and the pseudo excitation method. The result shows that the state augmentation method has higher accuracy and efficiency than the well-accepted time–frequency techniques.

Linear time-varying state transition matrix for spacecraft relative dynamics on highly elliptical orbits

Space exploration is a topic of great interest in the space community and worldwide. For instance, the race to the Lunar surface now includes not only government agencies, but also commercial entities and organizations. Such missions will require robotic spacecraft to perform formation flying, rendezvous and docking to reduce costs of the overall mission and establish space stations within Lunar orbits. Therefore, an accurate and computationally efficient dynamics model, to be used within a spacecraft’s guidance, navigation and control system, to calculate and control the desired relative motion will be required. This paper presents a new, linear time-varying, solution to spacecraft relative motion on Lunar orbits that includes solar radiation pressure, third-body, and J2 perturbations. The solution is obtained by using Hamiltonian dynamics in combination with the Peano–Baker solution for linear time-varying systems, which allows for the inclusion of short and long periodic variations of each perturbation within the dynamical model. The solution resulted in a 75% decrease in computational time when compared to a numerical simulator with the same perturbing effects.

Unique non-negative definite solution of the time-varying algebraic Riccati equations with applications to stabilization of LTV systems

In the context of infinite-horizon optimal control problems, the algebraic Riccati equations (ARE) arise when the stability of linear time-varying (LTV) systems is investigated. Using the zeroing neural network (ZNN) approach to solve the time-varying eigendecomposition-based ARE (TVE-ARE) problem, the ZNN model (ZNNTVE-ARE) for solving the TVE-ARE problem is introduced as a result of this research. Since the eigendecomposition approach is employed, the ZNNTVE-ARE model is designed to produce only the unique nonnegative definite solution of the time-varying ARE (TV-ARE) problem. It is worth mentioning that this model follows the principles of the ZNN method, which converges exponentially with time to a theoretical time-varying solution. The ZNNTVE-ARE model can also produce the eigenvector solution of the continuous-time Lyapunov equation (CLE) since the Lyapunov equation is a particular case of ARE. Moreover, this paper introduces a hybrid ZNN model for stabilizing LTV systems in which the ZNNTVE-ARE model is employed to solve the continuous-time ARE (CARE) related to the optimal control law. Experiments show that the ZNNTVE-ARE and HFTZNN-LTVSS models are both effective, and that the HFTZNN-LTVSS model always provides slightly better asymptotic stability than the models from which it is derived.

Adaptive State Observer for Linear Time-Varying System with Partially Unknown State Matrix and Input Matrix Parameters

In this paper the problem of adaptive state observer synthesis for linear time-varying SISO (single-input-single-output) dynamical system with partially unknown parameters was considered. It is assumed that the input signal and output variable of the system are measurable. It is also assumed that the state matrix of the plant contains known variables and unknown constants when the input matrix (vector) is unknown. Observer synthesis is based on GPEBO (generalized parameter estimation based observer) method proposed in [1]. Observer synthesis provides preliminary parametrization of the initial system and its conversion to a linear regression model with further unknown parameters identification. For identification of the unknown constant parameters classical estimation algorithm — least squares method with forgetting factor — was used. This approach works well in cases, when the known regressor is " frequency poor" (i.e. the regressor spectrum contains r/2 harmonics, where r is a value of the unknown parameters) or does not meet PE (persistent excitation) condition. To illustrate performance of the proposed method, an example is provided in this paper. A time-varying second-order plant with four unknown parameters was considered. Parametrization of the initial dynamical model was made. A linear static regression with six unknown parameters (including unknown state initial conditions vector) was obtained. An adaptive observer was synthesized and the simulation results were provided to illustrate the purpose reached. The main difference with the results, that were published earlier in [2], is the new assumption that not only does the state matrix of the linear time-varying system contain unknown parameters, but input matrix (vector) contains unknown constant coefficients.

Leakage Suppression in Pulse-Shaped OTFS Delay-Doppler-Pilot Channel Estimation

In this letter, we propose a novel channel estimation scheme with leakage suppression for orthogonal time frequency and space (OTFS) modulation. In OTFS, data and pilot symbols are placed in the delay-Doppler (DD) domain and spread over the time-frequency (TF) domain. This increases the achievable accuracy in estimating linear time-varying (LTV) channels. Unfortunately, leakage effects reduce the measurement precision significantly. Therefore, we propose a scheme that exploits smoothness regularization in TF to suppress the leakage observed in the DD domain. It allows us to improve the channel estimation accuracy while reducing signaling overhead.

Practical Prescribed-Time Sampled-Data Control of Linear Systems With Applications to the Air-Bearing Testbed

This article studies the practical prescribed time sampled-data control problem for input-constrained linear time invariant continuous time systems. By using the solution to a parametric Lyapunov equation, a bounded linear time-varying (LTV) sampled-data controller is designed to solve such a problem. In addition, stability analysis and performance analysis of the closed-loop sampled-data control system are presented by using the methodology of scalarization. Finally, the developed LTV sampled-data state controller is utilized to the whole physical simulation system of air-bearing testbed and experimental results verify the effectiveness of the proposed approach.

A novel strategy for identification and control of a class of linear systems

Control of time-varying systems has become increasingly popular in recent years, as various domains such as aviation, automobile and manufacturing industries require a quick and precise response in the presence of uncertainties. Adaptive control, a popular strategy is used to control systems with unknown or time-varying properties. Classic adaptive control is insufficient to cope with time-varying parameters to produce desired results with minimal parametric error. Furthermore, for these strategies to work, identification model must be initialised closer to the plant. Else the transient response of the plant is likely to grow unbounded. This paper mainly concentrates on employing adaptive control for a type of linear time-invariant (LTI) and linear time-varying (LTV) systems. The time variations in LTV plant parameters are either slow, rapid or fast. This research in turn proposes a two-stage adaptive scheme which swiftly determines the compact region in which plant parameters reside. This method in turn ensures atleast one model to be initialised closer to the plant which significantly reduces parametric estimation error. The uniform boundedness of identification error, tracking error as well as the parametric error of LTI and LTV systems, are deduced. Sufficient examples are also considered to demonstrate the efficacy of the proposed two-stage scheme.

Post-resonance backward whirl analysis in cracked overhung rotors

Overhung rotors usually exhibit recurrent transitions through critical whirl rotational speeds during startup and coast down operations, which significantly differ from their steady-state whirl responses. The presence of angular acceleration results in a linear-time-varying (LTV) system, which, although technically linear, still presents complexities often evinced by a nonlinear system. In general, backward whirl zones can either precede the critical forward whirl speed (termed as pre-resonance backward whirl, Pr-BW), or immediately follow the critical forward whirl speed (termed as post-resonance backward whirl, Po-BW). The Po-BW in the whirl response of a cracked overhung rotor with a breathing crack is studied here as distinct from that of geometrically symmetric configurations of other rotor systems. The equations of motion from the finite element (FE) model of an overhung rotor system with a breathing crack are numerically integrated to obtain the whirl response. The transient whirl responses with different bearing conditions are thoroughly investigated for excitation of Po-BW. The Po-BW zones of rotational speeds are determined via the wavelet transform method and full spectrum analysis (FSA) and applied to signals with added noise. The results of this work confirm the excitation of the Po-BW in cracked overhung rotors and confirm the robustness of the employed methods.

Finite-time convergent zeroing neural network for solving time-varying algebraic Riccati equations

Various forms of the algebraic Riccati equation (ARE) have been widely used to investigate the stability of nonlinear systems in the control field. In this paper, the time-varying ARE (TV-ARE) and linear time-varying (LTV) systems stabilization problems are investigated by employing the zeroing neural networks (ZNNs). In order to solve the TV-ARE problem, two models are developed, the ZNNTV-ARE model which follows the principles of the original ZNN method, and the FTZNNTV-ARE model which follows the finite-time ZNN (FTZNN) dynamical evolution. In addition, two hybrid ZNN models are proposed for the LTV systems stabilization, which combines the ZNNTV-ARE and FTZNNTV-ARE design rules. Note that instead of the infinite exponential convergence specific to the ZNNTV-ARE design, the structure of the proposed FTZNNTV-ARE dynamic is based on a new evolution formula which is able to converge to a theoretical solution in finite time. Furthermore, we are only interested in real symmetric solutions of TV-ARE, so the ZNNTV-ARE and FTZNNTV-ARE models are designed to produce such solutions. Numerical findings, one of which includes an application to LTV systems stabilization, confirm the effectiveness of the introduced dynamical evolutions.

An efficient solver of predictive control methods for motion control of autonomous vehicles with experimental verification

Real-time performance is one of the major challenges for optimal control and MPC control. In this paper, a novel general open-source C++ MPC solver based on state-of-the-art sparse QP solver OSQP is designed to solve the motion control problem of autonomous vehicles efficiently. A new linear dynamic model is proposed by replacing heading angle error with planning lateral projection speed, whose measurement is independent of the measurement of heading angle. A general local linearization method for the nonlinear tire model is proposed. Through the theory of offset-free MPC control, the steady-state value problem is solved caused by a disturbance in the time domain systematically. The algorithm in this paper is implemented in ROS, which means that the algorithm can be directly used in the prototype development of autonomous vehicles. The proposed MPC solver can solve a linear time-varying or time-invariant system, with or without disturbances. It is very efficient and can handle different kinds of hard and soft constraints. A real-time simulation platform based on Unreal Engine 4 and CarSim is designed to verify the algorithm’s effectiveness. Real-time simulations and real vehicle experiments verify the efficiency of the algorithm.

Estimation of sideslip angle and cornering stiffness of an articulated vehicle using a constrained lateral dynamics model

In this paper, a constrained lateral dynamics model of articulated vehicles and an algorithm for estimating sideslip angle and cornering stiffness are proposed. The articulated vehicle was modeled using the bicycle model, linear tire model, and modified Dug-off model. The normal force of each axle included in the model was estimated based on the longitudinal load transfer model. Physical constraints were applied to reduce model states. Accurate sideslip angle and cornering stiffness are essential for vehicle control safety and autonomous driving performance. The sideslip angle and cornering stiffness were simultaneously estimated using a dual linear time-varying (LTV) Kalman filter. The observability matrix guaranteed the convergence of the proposed estimation algorithm. The estimation performance was verified by simulation with TruckSim and an experiment using an articulated bus.

On the Stabilization through Linear Output Feedback of a Class of Linear Hybrid Time-Varying Systems with Coupled Continuous/Discrete and Delayed Dynamics with Eventually Unbounded Delay

This research studies a class of linear, hybrid, time-varying, continuous time-systems with time-varying delayed dynamics and non-necessarily bounded, time-varying, time-differentiable delay. The considered class of systems also involves a contribution to the whole delayed dynamics with respect to the last preceding sampled values of the solution according to a prefixed constant sampling period. Such systems are also subject to linear output-feedback time-varying control, which picks-up combined information on the output at the current time instant, the delayed one, and its discretized value at the preceding sampling instant. Closed-loop asymptotic stabilization is addressed through the analysis of two “ad hoc” Krasovskii–Lyapunov-type functional candidates, which involve quadratic forms of the state solution at the current time instant together with an integral-type contribution of the state solution along a time-varying previous time interval associated with the time-varying delay. An analytic method is proposed to synthesize the stabilizing output-feedback time-varying controller from the solution of an associated algebraic system, which has the objective of tracking prescribed suited reference closed-loop dynamics. If this is not possible—in the event that the mentioned algebraic system is not compatible—then a best approximation of such targeted closed-loop dynamics is made in an error-norm sense minimization. Sufficiency-type conditions for asymptotic stability of the closed-loop system are also derived based on the two mentioned Krasovskii–Lyapunov functional candidates, which involve evaluations of the contributions of the delay-free and delayed dynamics.

Data-driven model predictive control design for offset-free tracking of nonlinear systems

We propose a design of data-driven Model Predictive Control (MPC) using a suboptimal trajectory and the linear time-varying (LTV) models from data-driven trajectory optimisation that achieves offset-free tracking. Data-driven constrained differential dynamic programming (CDDP) is exploited to improve the trajectory iteratively without the knowledge of the nonlinear model. A trajectory is divided to the transient and steady state regions, controlled by the Linear time-varying MPC (LTVMPC) and the offset-free linear MPC (LMPC), respectively. We prove the feasibility of the proposed LTVMPC in the transient region, and the offset-free tracking property of LMPC. The proposed scheme is validated to a continuous stirred tank reactor (CSTR) process. Simulation studies show that the suboptimal trajectory and LTV models are generated by CDDP, and the proposed MPC achieves offset-free tracking and disturbance rejection for a set of initial conditions and set points in the operating region.

Estimation and balancing of multi-state differences between lithium-ion cells within a battery pack

This article develops a combined estimation and control strategy for the balancing of cell-to-cell differences within lithium-ion battery packs. Heterogeneity in state of charge, state of health, and temperature reduces both pack lifespan and real-time performance capabilities. Balancing algorithms, typically designed for specific battery models, pack sizes, and heterogeneity types, mitigate these issues by removing differences through manipulation of cell currents. This article provides a generalized approach to balancing by introducing a framework that models state heterogeneity through a linear time-varying (LTV) model. The framework facilitates the development of a state estimation and balancing algorithm based on the Kalman filter (KF) and linear quadratic regulator (LQR). This balancing strategy contains three main benefits derived from the LTV model: (1) The modeling strategy explicitly expresses heterogeneity. Using this, (2) the combined estimator and controller is applicable for a large subset of heterogeneity types and cell models. (3) The form of the LTV heterogeneity model allows for reduction in computing costs of both the estimation and control algorithms, nearly decoupling runtime from pack size. The capabilities of the novel estimation-LQR strategy are evaluated using a realistic electro-thermal pack model with charge, temperature, and electrochemical state heterogeneity. Three case studies are performed to evaluate the accuracy of the heterogeneity estimator, effectiveness of the balancing strategy, and reduction in computation runtime as compared to a baseline strategy.

Bounds of linear time‐varying systems with applications to quantify model gap of synchronous generators

Linear time-varying (LTV) systems play a key role in many engineering problems, for example, trajectory sensitivity analysis of power systems. Because LTV systems do not generally have closed-form solutions, estimates of their bounds are crucial to their analysis. In this paper, bounds for LTV systems with known inputs and unknown but bounded inputs are obtained via Grönwall's inequality. The estimated bounds are then used to quantify the gap between synchronous generators and their models in different scenarios through simulation.