Discrete-time State-space Model Research Articles

Dynamic Short Circuit Ratio for Stability Assessment of Grid Connected VSC-HVDC Systems Considering the Impact of Time-Varying Phase Shift

As more renewable resources are integrated into the receiving-end AC grid through VSC-HVDC, the grid connected VSC-HVDC system is facing a critical dynamic stability issue involving multiple-frequency oscillatory behaviors, which mainly stems from the low-system-strength and the time-varying phase shift. This paper proposes a general stability analysis method to measure the dynamic stability of grid connected VSC-HVDC system from the point of view of input to state stability (ISS). A nonlinear time-varying discrete-time state space model (NTDS <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> M) of the grid connected VSC-HVDC system is built considering the impact of the time-varying phase shift. Then, the NTDS <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> M is decomposed into several interconnected subsystems. The general asymptotic gain matrix (GAGM), whose elements are derived from the proposed dissipative-form sum-type inequalities (DSIs), is constructed to quantify the interactions between multiple subsystems. Finally, a general dynamic stability analysis index, called dynamic short circuit ratio (DSCR), is proposed to effectively explain the critical dynamic instability point of the grid connected VSC-HVDC system, based on the spectral radius property of the GAGM. Moreover, the real-time simulations and multiple comparative studies are implemented in this paper to verify the effectiveness of the proposed NTDS <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> M and stability analysis method by using RT-LAB platform.

Predictive digital twins for autonomous surface vessels

This paper presents an approach to improve the reliability and efficiency of autonomous surface vessels (ASV) by leveraging predictive digital twin and state estimation techniques. The predictive digital twin is based on a graphical model that shows the relationship between variables, and it is implemented and tested in an ASV based on a nonlinear discrete-time model and an adaptive exogenous Kalman filter to estimate the states and parameters associated with faults in the system. The state estimation method consists of a cascade of a nonlinear adaptive observer and a linearized adaptive Kalman filter, and the predictive algorithm is developed by creating a discrete-time state-space model from measured data and using the validated model to forecast the system response for a future time span. Experimental results in the Otter ASV developed by Maritime Robotics demonstrate the potential of the predictive digital twin as a decision support system for the predictive maintenance of ASVs.

Interpretable Real-Time Modeling of the Diffusion Overpotential in Lithium Batteries

Fractional-order dynamics can form physically-interpretable equivalent-circuit models of the diffusion overpotential in lithium-ion batteries but have complex formulations in the time-domain. Meanwhile, resistor-capacitor circuits have simple implementations but little physical meaning. Thus we propose a discrete-time state-space diffusion model, named ‘receding-horizon diffusion’ (RHD). It combines physical interpretability with computational simplicity. Analogous to theWarburg element in impedance spectroscopy, the RHD constant is explicitly linked with the lithium ion diffusion coefficient. Fivefold validation of the RHD model using simulated and experimental from lithium NiMnCo and NiCoAl cells up to 3 C-rate, temperatures from 0 to 25°C, and wide ranges of states of health and charge. The model has less than 1% modelling error. Ohmic, charge-transfer, and diffusion overpotentials are tracked in real time. The RHD model could be integrated into battery management systems in electric vehicles and used in standard state estimation techniques.

Split-KalmanNet: A Robust Model-Based Deep Learning Approach for State Estimation

Estimation of the state of a discrete-time state-space model from noisy measurements is a crucial aspect of signal processing. The extended Kalman filter (EKF) is widely used as a low-complexity solution based on a state evolution and measurement model of the state-space model. However, obtaining precise information about these models can be difficult in practice, and model mismatch greatly reduces state estimation accuracy. In this paper, we introduce Split-KalmanNet, a robust EKF algorithm that utilizes the power of deep learning for state estimation. Split-KalmanNet is inspired by the recent KalmanNet and calculates the Kalman gain using the Jacobian matrix of a measurement function and two recurrent neural networks (RNNs) with a split structure. The RNNs independently learn the covariance matrices of the prior state estimate and the innovation from data. The proposed split structure in calculating the Kalman gain allows for compensation of both state and measurement model mismatch effects. Numerical simulations show that Split-KalmanNet outperforms traditional EKF and the state-of-the-art KalmanNet algorithm in various scenarios of model mismatch.

A signal compensation-based balance control for the rotary inverted pendulum system

In this paper, we study the balance control for the rotary inverted pendulum system with two joints both subject to uncertainty. To facilitate the experiment, we apply a classical energy-based swing-up control, eschewing compensation for the uncertainty. In balance control, we employ a novel approach in the form of a modified signal compensation-based balance control, which models the uncertainty as its previously sampled term and its variation. The signal compensation-based balance control involves three steps. First, for the linearized discrete-time state-space model around the upright equilibrium point, we design the main control signal by using the pole-placement technique. Second, we estimate the discrete-time model of the system by using the least squares identification method, thereby unveiling key points in the identification procedure. The discrete-time model presents the uncertainty as a previously sampled term and its variation. Third, we design two discrete-time compensation signals for the previously sampled term and the variation. By combining the main control signal and the two compensation signals, we formulate the balance control law. We further optimize the signal compensation-based balance control by adding an extra parameter, successfully circumventing the critical stability problem, and analyze the performance of the closed-loop system. Numerical simulation and comparative experiments demonstrate its effectiveness and strong robustness against the uncertainty in frictions and modeling errors of the system.

Tracking Control Method for Greenhouse Environment Prediction Model Based on Real-Time Optimization Error Constraints

Model predictive control, without strict constraints on the control model, effectively overcomes problems, such as poor system dynamic quality caused by time delay, can improve control accuracy to a certain extent, and can directly process input and output constraints of the system online. It is applied in greenhouse system control. The appropriate objective optimization function and its corresponding constraint conditions have a direct impact on the solution of the optimal control rate of the model predictive control. In response to this issue, this paper proposes a simple and fast optimal predictive tracking control method. Based on the current prediction model of the greenhouse system, which reflects the dynamic relationship between various control equipment actions and greenhouse environmental factors, a multi degree of freedom discrete time state space model with tracking errors is established. Based on this model, in establishing the corresponding objective optimization function, the gradient descent theory and the two-norm definition are applied, and combined with actual constraints, iterative constraint conditions for real-time error tracking updates are established. Compared with traditional constraint ranges, a constraint function with real-time update characteristics is formed, achieving more accurate constraint conditions. By using rolling optimization and iterative methods, the optimal control rate corresponding to the minimum value of the objective optimization function within a finite time is solved. Through simulation examples, it is demonstrated that the model predictive control with optimization constraints can achieve a more accurate prediction and tracking control of indoor environmental parameters. This method has the advantages of simple control, energy-saving optimization, stable control, and accurate tracking, providing a reference for online real-time prediction and tracking control of future greenhouse environmental parameters.

Linear feedback control of spreading dynamics in stochastic nonlinear network models: epileptic seizures.

The development of models and approaches for controlling the spreading dynamics of epileptic seizures is an essential step towards new therapies for people with pharmacologically resistant epilepsy. Beyond resective neurosurgery, in which epileptogenic zones (EZs) are the target of surgery, closed-loop control based on intracranial electrical stimulation, applied at the very early stage of seizure evolution, has been the main alternative, e.g. the RNS system from NeuroPace (Mountain View, CA). In this approach the electrical stimulation is delivered to target brain areas after detection of seizure initiation in the EZ. Here, we examined, on model simulations, some of the closed-loop control aspects of the problem. Seizure dynamics and spread are typically modeled with highly nonlinear dynamics on complex brain networks. Despite the nonlinearity and complexity, currently available optimal feedback control approaches are mostly based on linear approximations. Alternative machine learning control approaches might require amounts of data beyond what is commonly available in the intended application. We thus examined how standard linear feedback control approaches perform when applied to nonlinear models of neural dynamics of seizure generation and spread. In particular, we considered patient-specific epileptor network models for seizure onset and spread. The models incorporate network connectivity derived from (diffusion MRI) white-matter tractography, have been shown to capture the qualitative dynamics of epileptic seizures and can be fit to patient data. For control, we considered simple linear quadratic Gaussian (LQG) regulators. The LQG control was based on a discrete-time state-space model derived from the linearization of the patient-specific epileptor network model around a stable fixed point in the regime of preictal dynamics. We show in simulations that LQG regulators acting on the EZ node during the initial seizure period tend to be unstable. The LQG solution for the control law in this case leads to global feedback to the EZ-node actuator. However, if the LQG solution is constrained to depend on only local feedback originating from the EZ node itself, the controller is stable. In this case, we demonstrate that localized LQG can easily terminate the seizure at the early stage and prevent spread. In the context of optimal feedback control based on linear approximations, our results point to the need for investigating in more detail feedback localization and additional relevant control targets beyond epileptogenic zones.

State-space modeling for degrading systems with stochastic neural networks and dynamic Bayesian layers

To monitor the dynamic behavior of degrading systems over time, a flexible hierarchical discrete-time state-space model (SSM) is introduced that can mathematically characterize the stochastic evolution of the latent states (discrete, continuous, or hybrid) of degrading systems, dynamic measurements collected from condition monitoring sources (e.g., sensors with mixed-type outputs), and the failure process. This flexible SSM is inspired by Bayesian hierarchical modeling and recurrent neural networks without imposing prior knowledge regarding the stochastic structure of the system dynamics and its variables. The temporal behavior of degrading systems and the relationship between variables of the corresponding system dynamics are fully characterized by stochastic neural networks without having to define parametric relationships/distributions between deterministic and stochastic variables. A Bayesian filtering-based learning method is introduced to train the structure of the proposed framework with historical data. Also, the steps to utilize the proposed framework for inference and prediction of the latent states and sensor outputs are discussed. Numerical experiments are provided to demonstrate the application of the proposed framework for degradation system modeling and monitoring.

A Methodology for Extracting the Physical Parameters of Soil-Foundation-Pier Systems from Dynamic Tests

The paper presents a methodology for the identification of the physical parameters of a model describing the transverse dynamics of soil-foundation-pier systems founded on piles, starting from results of dynamic experimental tests. In detail, the procedure exploits the identified state-space models of the systems obtained from results of dynamic tests through consolidated identification techniques available in the literature. The procedure permits to compute the physical parameters of the real system (e.g. masses, pier stiffness matrix, and soil-foundation impedance) by directly comparing the components of the analytical and identified stiffness, mass and damping matrices; the latter extracted from the identified state-space models. The proposed approach allows the direct definition of the numerical model that best fits the experimental data, and permits the identification of the soil-foundation compliance. Firstly, the dynamics of the analytical model, which includes the frequency-dependent behaviour of the soil-foundation system through the introduction of a lumped parameter model, is formulated adopting the continuous-time first-order state-space form. Then, an identification strategy of the physical parameters of a soil-foundation-pier system is proposed, starting from the discrete-time state-space model identified from dynamic tests. The procedure allows the identification of the soil-foundation system compliance through experimental tests.

Model Predictive Phase Control for Single-Phase Electric Springs

In this paper, model predictive control (MPC) is proposed for single-phase electric springs (ESs) with the help of the existing δ control, which is realized by controlling the instantaneous phase angle of the predefined sinusoidal reference of a certain controller. System modeling is analyzed first to get differential forms of state variables. The discrete-time state space model is obtained through first-order approximation. Critical load (CL) voltage can be predicted by the prediction of ES voltage and line current. The operating modes of ESs can be determined and the reference signal for CL voltage can be provided by δ control. As a result, cost function is obtained as the absolute value of the error between predicted CL voltage and its predefined reference. Two typical operating functions such as pure reactive power compensation mode and power factor correction (PFC) mode are selected and simulated to validate the proposed control and analysis. It is revealed that both control objectives can be achieved with the proposed MPC and δ control. Additionally, the total harmonic distortion on the critical load is limited to about 0.5%, which is better than other existing methods.

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

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

Machine prognostics under varying operating conditions based on state-space and neural network modeling

Prognostics based on deep learning models usually assume that training and testing data come from the same equipment with similar working conditions. In practice, some new working conditions of test equipment may not be recorded in the training dataset, and thus the learned remaining useful life (RUL) prediction model may not work well. This work is the first to propose a novel prognostic model based on state-space modeling and reinforcement learning to predict the RUL of equipment operating under extrapolated new conditions without corresponding training data. Instead of directly constructing a supervised model that relates monitoring measurements to their RUL, a discrete-time state-space model is built using possible failure histories. The contribution of this paper is twofold. First, on the basis of the state-space model, an interpretable prognostic model that combines Lyapunov constraint and reinforcement learning is proposed to predict the equipment RUL. Second, an adversarial training method based on the idea of H∞ robustness is integrated to reduce the effect of state-space modeling errors. The proposed model aims to reduce the causality between RUL and operating parameters and increase the causality between RUL and unobserved degradation characteristics. Consequently, the proposed model is interpretable and has the potential to predict RUL for the equipment operating under conditions beyond the record. The proposed approach is demonstrated and validated using a simulation study and an experimental dataset.

Discrete-Time State-Space Construction Method for SSO Analysis of Renewable Power Generation Integrated AC/DC Hybrid System

Construction of the linearized state-space model is the key step of eigenvalue analysis for SSO study of power systems with large-scale renewable power generation (RPG) integrated. This paper proposes a novel method to construct the state matrix of the AC/DC hybrid power system with RPG within the discrete-time domain. The continuous-time state-space model of each AC/DC component as well as RPG units in the power system, is discretized together with their corresponding control. The discretized equation of each component is then represented by an equivalent circuit consisting of parallel historical current and conductance. According to the power system configuration, the discrete-time equivalent circuit of the whole AC/DC hybrid system is established, and the nodal analysis method is applied for the construction of the discrete-time state-space model of the power system. In this way, the state matrix can be acquired much more conveniently than that of continuous-time state-space method, which needs topology analysis to find independent state variables of the network from the proper tree. The proposed method has been applied in the eigenvalue analysis for the SSO study of a case system adapted from the AC/DC hybrid transmission system in northwest China. Eigenvalue results are in high accordance with time-domain simulation results in PSCAD/EMTDC.

On the Combination of Random Matrix Theory With Measurements on a Single Structure

Abstract An approach is proposed for the evaluation of the probability density functions (PDFs) of the modal parameters for an ensemble of nominally identical structures when there is only access to a single structure and the dispersion parameter is known. The approach combines the Eigensystem realization algorithm on sets of dynamic data, with an explicit nonparametric probabilistic method. A single structure, either a mathematical model or a prototype, is used to obtain simulated data or measurements that are employed to build a discrete time state-space model description. The dispersion parameter is used to describe the uncertainty due to different sources such as the variability found in the population and the identification errors found in the noisy measurements from the experiments. With this approach, instead of propagating the uncertainties through the governing equations of the system, the distribution of the modal parameters of the whole ensemble is obtained by randomizing the matrices in the state-space model with an efficient procedure. The applicability of the approach is shown through the analysis of a two degrees-of-freedom mass-spring-damper system and a cantilever system. The results show that if the source of uncertainty is unknown and it is possible to specify an overall level of uncertainty, by having access to a single system's measurements, it is possible to evaluate the resulting PDFs on the modal parameters. It was also found that high values of the dispersion parameter may lead to nonphysical results such as negative damping ratios values.

Cooperative Control of Interconnected Air Suspension Based on Energy Consumption Optimization

In this study, a cooperative control method based on model predictive control and multiagent theory is proposed to control an interconnected air suspension system with three controllable structures of interconnection mode, damping, and vehicle height. The model predictive controller is constructed based on a discrete-time state-space model. The optimal interval for suspension force is obtained through solving cost functions while satisfying a set of constraints on controlled variables and thereby reducing the coupling complexity of a multivariable control system. Deliberative agents are involved in building cost functions of interconnection mode, vehicle height adjustment, and damping force, and the energy consumption control strategy is established to realize suspension force distribution with low energy consumption. Finally, the test results show that the proposed control method can significantly improve vehicle ride comfort and restrain rollover on the premise of ensuring energy efficiency. Compared with traditional control, the peak value of the sprung mass acceleration speed decreases by 70% and the peak value of the unsprung mass acceleration speed decreases by 75% under straight-driving condition. The roll angle decreases by 40% under the steering condition. As for the traditional control, they are skyhook, imitation skyhook, and PID-PWM control strategies.

A methodology for the identification of physical parameters of soil-foundation-bridge pier systems from identified state-space models

A methodology for the identification of the physical parameters of a model describing the transverse dynamics of soil-foundation-pier systems founded on piles is presented in this paper, starting from identified state-space models of the systems obtained from results of dynamic experimental tests and identification techniques available in the literature. By assuming the order of the model to be compliant with that of a numerical one suitably developed to capture the dynamics of the bridge pier, the procedure allows the identification of the stiffness, mass and damping matrices of the analytical model from which the physical parameters of the real system (e.g. masses, pier stiffness matrix and soil-foundation impedance) can be extracted by directly comparing the components of the identified and analytical matrices. The procedure allows the direct definition of the numerical model that best fits the experimental data without the need of any model calibration, and permits the identification of the soil-foundation compliance taking into account its intrinsic frequency-dependent behaviour. Firstly, the dynamics of the analytical model is formulated adopting the continuous-time first-order state-space form. The model includes the frequency-dependent behaviour of the soil-foundation system through the introduction of a lumped parameter model able to reproduce the typical soil-foundation impedances of pile foundations. Then, an identification strategy of the physical parameters of a generic realistic soil-foundation-pier system is proposed starting from the discrete-time state-space model identified from dynamic tests. The procedure is illustrated with some applications, simulating results of forced vibration and ambient vibration tests executed on a known system whose parameters are then identified according to the proposed approach. The procedure revealed to be effective to identify the stiffness, mass and damping matrixes of the known system, and consequently its physical parameters.

Embedding information onto a dynamical system

The celebrated Takens’ embedding theorem concerns embedding an attractor of a dynamical system in a Euclidean space of appropriate dimension through a generic delay-observation map. The embedding also establishes a topological conjugacy. In this paper, we show how an arbitrary sequence can be mapped into another space as an attractive solution of a nonautonomous dynamical system. Such mapping also entails a topological conjugacy and an embedding between the sequence and the attractive solution spaces. This result is not a generalisation of Takens embedding theorem but helps us understand what exactly is required by discrete-time state space models widely used in applications to embed an external stimulus onto its solution space. Our results settle another basic problem concerning the perturbation of an autonomous dynamical system. We describe what exactly happens to the dynamics when exogenous noise perturbs continuously a local irreducible attracting set (such as a stable fixed point) of a discrete-time autonomous dynamical system.

A novel adaptive intelligent MPC scheme for frequency stabilization of a microgrid considering SoC control of EVs

In the recent years, the vehicle-to-grid (V2G) technology has been successfully implemented to stabilize the frequency deviations in a microgrid (MG) whereby charging/discharging of battery of electric vehicles (EVs) is utilized depending upon their state-of-charge (SoC). Compared to the conventional battery energy storage technology, lower degradation tendency and lesser cost of an EV battery are obvious reasons for its utilization as an alternate. This paper proposes a novel adaptive intelligent model predictive control (AIMPC) scheme for frequency stabilization of an MG considering the SoC control of the battery of the EVs. The MPC scheme operates by predicting the future behavior of a plant whereby an explicit discrete-time state-space model of the plant is utilized. Since optimal performance of the MPC depends upon the tuning parameter (τw) present in its cost function, an intelligent optimization algorithm is implemented to dynamically optimize the parameter τw and simultaneously the proposed control scheme is made adaptive. Effect of the SoC control on the frequency deviation response (FDR) of the MG is demonstrated. Further, competence of the proposed control scheme is established over the adaptive fuzzy MPC and PID controller considering diverse loading conditions in the MG. Simulation results clearly establish that the FDRs of the MG are improved with the implementation of the proposed control scheme. Lastly, sensitivity of the proposed scheme is corroborated considering parametric uncertainties in the MG.

Influence of Thermal Characteristics on Optimal Scheduling of Integrated Energy System

The complementarity of heat and electricity is the key to an integrated energy system (IES) to improve performance and accommodate renewables. However, the individual, interactive and total effects of thermal inertia, transition delay, and heat loss, the three typical characteristics of thermal processes, are rarely compared. Therefore, a dynamic scheduling model of an IES is innovatively proposed here, where the thermal inertia of heat exchangers is considered via a discrete-time state-space model. A mixed integer linear problem is then solved and eight operation modes are performed to elaborately compare the effect of thermal characteristics. Based on the results of a case study, we found that heat loss increases system operation cost, whereas thermal inertia and transition delay both reduce the cost but with certain fluctuation. When considering heat loss, the interactive effect of thermal characteristics will deteriorate the economy. The reliability of optimal scheduling results is increased when all the three thermal features are considered, though this will burden solution time and increase operation cost.

Model Predictive Control for Blending Processes in Cement Plants

In this paper, we discuss model predictive control applied to blending processes. Blending processes are ubiquitous in the chemical process industries since reactants usually need to be mixed before entering a reactor. Many times, the blending is trivial as pure streams of reactants are mixed. We consider non-trivial blending problems in which non-pure streams with the reactants are to be mixed. The motivating example is the blending problem that occurs in cement production. The raw mix for the cement kiln must have a specified chemical composition. This composition is obtained by mixing piles with different chemical compositions and economic value such that the raw mix meets specifications in the cheapest possible way. We formulate the blending problem as a nonlinear optimization problem that can be approximated well as a convex quadratic optimization problem. We implement the corresponding nonlinear and linear model predictive controllers (NMPC, LMPC) using a continuous-time transfer function description that is realized as a discrete-time linear state space model. The controller obtains feedback by combining regularly sampled online measurements and irregularly sampled laboratory measurements using a time variant dynamic Kalman filter with memory. Numerical simulations demonstrate that the NMPC and LMPC have similar performance.