Linear State Space Representation Research Articles

Integrated Control Design for a Partially Turboelectric Aircraft Propulsion System

Abstract Electrified Aircraft Propulsion (EAP) holds great potential for reducing aviation emissions and fuel burn. A variety of EAP architectures have been proposed including partially turbo-electric configurations that combine turbofan engines with motor-driven propulsors. Such architectures exhibit coupling between subsystems and thus require an integrated control solution. To address this need, this paper presents an integrated control design strategy for a commercial single-aisle partially turbo-electric aircraft concept consisting of two wing-mounted turbofan engines and an electric motor driven tailfan propulsor. Within this architecture the turbofans serve the dual purpose of generating thrust and supplying mechanical offtake power used to generate electricity for the tailfan motor. The propulsion control system is tasked with coordinating turbofan and tailfan operation under both steady-state and transient scenarios. The paper introduces a linear state-space representation of the architecture reflecting the coupling between the turbofan and tailfan subsystems along with loop transfer functions reflecting open- and closed-loop system dynamics. Also discussed is an applied strategy for scheduling the tailfan setpoint command based on the average sensed fan speed of the two turbofans. This approach ensures synchronized operation of the turbofan and tailfan subsystems while also allowing the turbofan fuel control design to be simplified. Performance of the integrated control design is assessed through a real-time hardware-in-the-loop test conducted at the NASA Electric Aircraft Testbed. During this test a scaled version of the electrical system and turbomachinery shaft dynamics were implemented in electrical machine hardware and evaluated under closed-loop control. Results from this facility test are presented to illustrate the efficacy of the applied integrated control design approach under steady-state and transient scenarios including a full-flight mission profile.

Modeling the out-of-sample predictive relationship between equity premium, returns on the price of crude oil and economic policy uncertainty using multivariate time-varying dimension models

Researchers increasingly rely on the newspaper-based uncertainty (volatility) measures pioneered by Baker et al. (2016) to forecast economic, financial variables and commodity prices out-of-sample. Among them, equity premium and returns on the price of crude oil have received a great deal of attention given their importance. By combining different linear state-space representations of the multivariate dynamic linear model with the discount factor-based model averaging (selection) technique outlined in Raftery et al. (2010), and using monthly data from 1997m1 through 2022m10, we suggest three multivariate time-varying dimension models, and forecast these variables out-of-sample in a contemporaneous fashion. The time-varying dimension feature allows the number of predictors in each regression of the multivariate system to change over time. From a technical viewpoint, the suggested models are intuitive (flexible), and do not require much subjective input from the researcher. They also produce very accurate one-month ahead out-of-sample density (point) forecasts on average. From an empirical viewpoint, our analysis provides new and interesting insights into the out-of-sample predictive relationship between equity premium, returns on the price of crude oil and economic policy uncertainty.

Learning strange attractors with reservoir systems

This paper shows that the celebrated embedding theorem of Takens is a particular case of a much more general statement according to which, randomly generated linear state-space representations of generic observations of an invertible dynamical system carry in their wake an embedding of the phase space dynamics into the chosen Euclidean state space. This embedding coincides with a natural generalized synchronization that arises in this setup and that yields a topological conjugacy between the state-space dynamics driven by the generic observations of the dynamical system and the dynamical system itself. This result provides additional tools for the representation, learning, and analysis of chaotic attractors and sheds additional light on the reservoir computing phenomenon that appears in the context of recurrent neural networks.

Control-oriented modeling of a LiBr/H[formula omitted]O absorption heat pumping device and experimental validation

Absorption heat pumping devices (AHPDs, comprising absorption heat pumps and chillers) are devices that use thermal energy instead of electricity to generate heating and cooling, thereby facilitating the use of waste heat and renewable energy sources such as solar or geothermal energy. Despite this benefit, widespread use of AHPDs is still limited. One reason for this is partly unsatisfactory control performance under varying operating conditions, which can result in poor modulation and part load capability. A promising approach to tackle this issue is using dynamic, model-based control strategies, whose effectiveness, however, strongly depend on the model being used. This paper therefore focuses on the derivation of a viable dynamic model to be used for such model-based control strategies for AHPDs such as state feedback or model-predictive control. The derived model is experimentally validated, showing good modeling accuracy. Its modeling accuracy is also compared to alternative model versions, that contain other heat transfer correlations, as a benchmark. Although the derived model is mathematically simple, it does have the structure of a nonlinear differential–algebraic system of equations. To obtain an even simpler model structure, linearization at an operating point is discussed to derive a model in linear state space representation. The experimental validation shows that the linear model does have slightly worse steady-state accuracy, but that the dynamic accuracy seems to be almost unaffected by the linearization. The presented new modeling approach is considered suitable to be used as a basis for the design of advanced, model-based control strategies, ultimately aiming to improve the modulation and part load capability of AHPDs.

A wavelet-based dynamic mode decomposition for modeling mechanical systems from partial observations

Dynamic mode decomposition (DMD) has emerged as a popular data-driven modeling approach to identifying spatio-temporal coherent structures in dynamical systems, owing to its strong relation with the Koopman operator. For dynamical systems with external forcing, the identified model should not only be suitable for a specific forcing function but should generally approximate the input–output behavior of the underlying dynamics. A novel methodology for modeling those classes of dynamical systems is proposed in the present work, using wavelets in conjunction with the input–output dynamic mode decomposition (ioDMD). The wavelet-based dynamic mode decomposition (WDMD) builds on the ioDMD framework without the restrictive assumption of full state measurements. Our non-intrusive approach constructs numerical models directly from trajectories of the full model’s inputs and outputs, without requiring the full-model operators. These trajectories are generated by running a simulation of the full model or observing the original dynamical systems’ response to inputs in an experimental framework. Hence, the present methodology is applicable for dynamical systems whose internal state vector measurements are not available. Instead, data from only a few output locations are only accessible, as often the case in practice. The present methodology’s applicability is explained by modeling the input–output response of an Euler–Bernoulli finite element beam model. The WDMD provides a linear state-space representation of the dynamical system using the response measurements and the corresponding input forcing functions. The developed state-space model can then be used to simulate the beam’s response towards different types of forcing functions. The method is further validated on a real (experimental) data set using modal analysis on a simple free–free beam, demonstrating the efficacy of the proposed methodology as an appropriate candidate for modeling practical dynamical systems despite having no access to internal state measurements and treating the full model as a black-box.

Wayset-based guidance of multirotor aerial vehicles using robust tube-based model predictive control

The present paper is concerned with the wayset-based guidance of underactuated multirotor aerial vehicles (MAVs). A hierarchical guidance and control structure is first established, in which the guidance is realized as a supervisory loop. The lower-level stabilizing attitude and position control laws are assumed to be available. On the other hand, the outer-loop guidance is designed based on a fixed-horizon tube-based robust model predictive control (MPC), which conducts the MAV to visit a given sequence of waysets, without violating their state and control bounds, and allowing the vehicle to rest in each wayset for a specified period. The MPC is designed using a reduced-order closed-loop dynamic model describing the vehicle’s translation, which is derived considering the stabilizing position and attitude control laws and the assumption of a time-scale separation between the closed-loop translational and rotational dynamics. This model is put into a discrete-time linear state–space representation subject to additive bounded random disturbance and measurement noise. The properties of the proposed method, which includes the MPC recursive feasibility and robust stability as well as the overall guidance feasibility, are analytically studied. The method is also numerically evaluated using a realistic quadrotor dynamic model, showing its effectiveness and confirming its properties.

Research on Kalman Filter for One-dimensional Discrete Data

Kalman filter processes the input and observation signals with noise on the basis of linear state space representation to obtain the system state or real signal. In the one-dimensional model, due to the lack of multi-dimensional description of the target, the prior estimate of the state is often the measured value of the previous moment. When the target state changes, the divergence phenomenon will occur. Aiming at the problem that the one-dimensional traditional Kalman filter lacks the target observation dimension, which leads to the divergence or imprecision of the filter, this paper focuses on improving the estimation method of the target state, and proposes a real-time prediction model based on the cascade structure. This model can improve the response of Kalman filter to the change of target state and dynamically adjust the Kalman iterative domain to improve the measurement accuracy. The digital signal filtering simulation is carried out and the performance of the filter is verified based on LabVIEW. Experimental results show that the algorithm can maintain the accuracy and real-time performance of filtering when only one dimension observation results are obtained.

LPVcore: MATLAB Toolbox for LPV Modelling, Identification and Control

This paper describes the LPVcore software package for MATLAB developed to model, simulate, estimate and control systems via linear parameter-varying (LPV) input-output (IO), state-space (SS) and linear fractional (LFR) representations. In the LPVcore toolbox, basis affine parameter-varying matrix functions are implemented to enable users to represent LPV systems in a global setting, i.e., for time-varying scheduling trajectories. This is a key difference compared to other software suites that use a grid or only LFR-based representations. The paper contains an overview of functions in the toolbox to simulate and identify IO, SS and LFR representations. Based on various prediction-error minimization methods, a comprehensive example is given on the identification of a DC motor with an unbalanced disc, demonstrating the capabilities of the toolbox. The software and examples are available on www.lpvcore.net.

Performance and Stability Margins Comparison of Pole Placement and Optimal Controllers using Cart-Inverted Pendulum System

In this paper, pole placement and two optimal control techniques which are the linear quadratic regulator and linear quadratic Gaussian are compared. A cart and inverted pendulum which is an inherently unstable dynamical system is used as a case study to analyze their performance and stability margins. Lagrangian equations defining the system dynamics are converted to linear state-space representation. The objective is to keep the pendulum in an upright position as the cart on which it is mounted moves from one position to another. MATLAB is used to solve the optimization problem and simulate the step response of the system. The robustness of both controllers is measured by giving uncertain model parameters to the system and observing the level of uncertainty these controllers can handle. The simulation results justify the relative advantages of these control schemes.

Causality and Network Graph in General Bilinear State-Space Representations

This article proposes an extension of the well-known concept of Granger causality, called GB–Granger causality. GB–Granger causality is designed to relate the internal structure of bilinear state-space systems and statistical properties of their output processes. That is, if such a system generates two processes, where one does not GB–Granger cause the other, then it can be interpreted as the interconnection of two subsystems: one that sends information to the other, and one which does not send information back.This result is an extension of earlier obtained results on the relationship between Granger causality and the internal structure of linear time-invariant state-space representations.

Research on semi-active vibration isolation system based on electromagnetic spring

This paper proposes a semi-active variable stiffness vibration isolation system based on electromagnetic spring for the low-frequency vibration isolation of mass-varying objects. It is achieved by four straight leaf springs in parallel to an electromagnetic spring system composed of a single electromagnet and a permanent magnet. The equivalent magnetic circuit method is used to compute electromagnetic force of the electromagnetic spring system, and mathematical model of the semi-active vibration isolation system is established according to Maxwell's equations. The nonlinear mathematical model is linearized at the equilibrium point by using the Taylor series expansion theorem to establish linear state-space representation of the system, and then using the traditional PID control method, a double closed-loop feedback control system of the inner current loop and outer location loop is designed. By controlling the current in the coil, the equivalent stiffness and electromagnetic force of the system are variable to achieve semi-active control. Furthermore, the control block diagram of the semi-active vibration isolation system is built based on Simulink software, then make a simulation analysis to the vibration isolation performance of the system and compare the effects of vibration isolation with inner current loop control and without inner current loop control, respectively. Finally, the experiments prove the correctness of the theory. It concludes that this semi-active vibration isolation system is a vibration isolation system with broad application prospects, which has fast current response, high vibration isolation efficiency, and an excellent vibration isolation effect for the low-frequency disturbance of mass-varying objects.

Combining Linear Algebra and Numerical Optimization for Gray-Box Affine State-Space Model Identification

Accurately estimating the parameters of a gray-box linear time-invariant state-space representation is still a challenging problem especially if the number of unknowns exceeds ten. The standard nonlinear optimization-based procedures often fail because the initial guesses are not in the domain of attraction of the user-defined cost function global minimum. Herein, a new solution is developed to give access to estimated parameters which are accurate estimates of the unknown parameters directly or, at worst, good initial guesses for any standard nonlinear optimization-based method. By assuming that 1) the gray-box model to identify is minimal and structurally identifiable, 2) the parameter dependency is affine, and 3) the available datasets have been transformed into a reliable and accurate black-box state-space model, our procedure consists in determining the unique similarity transformation between the black-box and gray-box representations of the system to identify. More specifically, we introduce a technique which first extracts a system of equations linear in terms of the similarity transformation; second, it solves this system by using standard linear algebra tools; finally, completes this two-step procedure by a numerical optimization solution (dedicated now to a very small number of unknowns) if the second step does not give access to all the similarity transformation parameters effectively. In this contribution, we first describe our new procedure, prove its capabilities of giving accurate estimates under specific practical constraints, and finally demonstrates its effectiveness with simulation examples.

A model‐based and data‐driven joint method for state‐of‐health estimation of lithium‐ion battery in electric vehicles

Lithium-ion battery state-of-health estimation is one of the vital issues for electric vehicle safety. In this work, a joint model-based and data-driven estimator is developed to achieve accurate and reliable state-of-health estimation. In the estimator, an increase in ohmic resistance extracted from the Thevenin model is defined as the health indicator to quantify the capacity degradation. Then, a linear state-space representation is constructed based on the data-driven linear regression. Furthermore, the Kalman filter is introduced to trace capacity degradation based on the novel state space representation. A series of battery aging datasets with different dynamic loading profiles and temperatures are obtained to demonstrate the accuracy and robustness of the proposed method. Results show that the maximum error of the Kalman filter is 2.12% at different temperatures, which proves the effectiveness of the proposed method.

EPKF: Energy Efficient Communication Schemes Based on Kalman Filter for IoT

The Internet of Things (IoT) has been recognized as the next technological revolution. It faces two challenges: 1) how to achieve energy efficient communication for the battery constrained devices and 2) how to connect a very large number of devices to the Internet with low latency, high efficiency, and reliability. To address these problems, this paper proposes two methods based on Kalman filter (KF), termed as extensions of predicable KF (EPKF). They locally reduce the unnecessary transmission (access) of end devices to the network (Internet) utilizing the spatial and temporal correlations with low algorithmic overhead. Each transmitting device (TD) independently controls its transmission using the temporal correlation; and the receiving device (RD) exploits the spatial correlation among the TDs to further improve the reconstruction quality. The reconstruction problem in the RD is nonlinear. To reduce the computation complexity, an in-depth analysis of the local estimate error is conducted and the approximated linear solutions are thereupon obtained. They are fundamental methods applicable to any IoT monitored/controlled physical system that can be modeled as a linear state space representation. The pedestrian-position application is used as a case study to demonstrate the efficiency in the simulation. Remarkably, the EPKF methods using the linear combinations of the local estimates from multiple TDs reduce the transmission rate to 10%, while achieving the same reconstruction quality as using KF in the traditional manner.

Relationship Between Granger Noncausality and Network Graph of State-Space Representations

The goal of this paper is to explore the relationship between the network graph of a state-space representation of an observed process and the causal relations among the components of that process. We will show that the existence of a linear time-invariant state-space representation, with its network graph being the star graph, is equivalent to (conditional) Granger noncausal relations among the components of the output process. Granger noncausality is a statistical concept, which applies to arbitrary processes and does not depend on the representation of the process. That is, we relate intrinsic properties of a process with the network graph of its state-space representations.

State estimation of lithium-ion cells using a physicochemical model based extended Kalman filter

Two time-varying linear state-space representations of the generally accepted physicochemical model (PCM) of a lithium-ion cell are used to estimate local and global states during different charging scenarios. In terms of computational speed and suitability towards recursive state observer models, the solid-phase diffusion in the PCM of an exemplaric MCMB/LiCoO2 lithium-ion cell is derived with the aid of two different numerical reduction methods in the form of a Polynomial Profile and an Eigenfunction Method. As a benchmark, the PCM using the original Duhamel Superposition Integral approximation serves for the comparison of accuracy and computational speed. A modified spatial discretization via the finite volume method improves handling of boundary conditions and guarantees accurate simulation results of the PCM even at a low level of spatial discretization. The Polynomial Profile allows for a significant speed-up in computational time whilst showing a poor prediction accuracy during dynamic load profiles. The Eigenfunction Method shows a comparable accuracy as the benchmark for all load profiles whilst resulting in an even higher computational effort. The two derived observer models incorporate the state-space representation of the reduced PCM applying both the Polynomial and Eigenfunction approach combined with an Extended Kalman Filter algorithm based on a novel initialization algorithm and conservation of lithium mass. The estimation results of both models show robust and quick reduction of the residual errors for both local and global states when considering the applied current and the resulting cell voltage of the benchmark model, as the underlying measurement signal. The carried out state estimation for a 4C constant charge current showed a regression of the cell voltage error to 1 mV within 30 s with an initial SOC error of 42.4% under a standard deviation of 10 mV and including process noise.

Conditions for Hurwitz Stability/Instability of a Real Matrix via its Sign Pattern with a Necessary and Sufficient Condition for Magnitude Independent Stability

In this paper, we address the issue of discerning the Hurwitz stability/instability of real matrix directly from its sign pattern.In that connection, in this paper, we first classify all sign patterns of matrices into three categories, namely i) Qualitative Sign Unstable (QLSU) matrices (i.e. matrices with a sign pattern that it is unstable for any magnitudes in the entries, that is, magnitude independent instability), then ii) Qualitative Sign Stable (QLSS) matrices (i.e. matrices with a sign pattern that it is stable for any magnitudes in the entries, that is, magnitude independent stability) and finally iii) MDSU matrices (which require magnitudes to determine its stability/instability). We then propose a necessary and sufficient condition for a sign matrix to be QLSS. The proposed necessary and sufficient condition is derived solely based on the nature (signs) of interactions and interconnections (i.e. based only on the signs of the entries of the matrix), borrowed from ecological principles. The proposed condition in this paper, serves as a better (being much simpler) alternative to the necessary and sufficient condition for QLSS matrices that is available in the ecology literature which uses a complicated test labeled ‘the Color test’. Identifying QLSS/MDSU/QLSU sign structures in a necessary and sufficient way has significant implications in many engineering systems whose dynamics are described by linear state space representation.

Active vibration robust control for FGM beams with piezoelectric layers

The dynamic output-feedback robust control method based on linear matrix inequality (LMI) method is presented for suppressing vibration response of a functionally graded material (FGM) beam with piezoelectric actuator/sensor layers in this paper. Based on the reduced model obtained by using direct mode truncation, the linear fractional state space representation of a piezoelectric FGM beam with material properties varying through the thickness is developed by considering both the inherent uncertainties in constitution material properties as well as material distribution and the model error due to mode truncation. The dynamic output-feedback robust H-infinity control law is implemented to suppress the vibration response of the piezoelectric FGM beam and the LMI method is utilized to convert control problem into convex optimization problem for efficient computation. In numerical studies, the flexural vibration control of a cantilever piezoelectric FGM beam is considered to investigate the accuracy and efficiency of the proposed control method. Compared with the efficient linear quadratic regulator (LQR) widely employed in literatures, the proposed robust control method requires less control voltage applied to the piezoelectric actuator in the case of same control performance for the controlled closed-loop system.

Tube-based LPV constant output reference tracking MPC with error bound * *This research was supported by the Impulse 1 program of the Eindhoven University of Technology and ASML.

Abstract We address the constrained output tracking of constant references for linear parameter-varying systems. In model predictive control based on linear parameter-varying state-space representations, offset-free output tracking is generally not possible due to the variations in the scheduling signal. Therefore, we aim instead to guarantee a pre-specified tracking error bound which is achievable for all admissible variations of the scheduling variable. The construction of an invariant set in which such a bound can be satisfied is described. Subsequently a tube-based model predictive controller is designed which brings the state of the system inside this set in finite time. The properties of the approach are demonstrated on numerical examples.

Five degrees of freedom linear state-space representation of electrodynamic thrust bearings

Electrodynamic bearings can provide stable and contactless levitation of rotors while operating at room temperatures. Depending solely on passive phenomena, specific models have to be developed to study the forces they exert and the resulting rotordynamics. In recent years, models allowing us to describe the axial dynamics of a large range of electrodynamic thrust bearings have been derived. However, these bearings being devised to be integrated into fully magnetic suspensions, the existing models still suffer from restrictions. Indeed, assuming the spin speed as varying slowly, a rigid rotor is characterised by five independent degrees of freedom whereas early models only considered the axial degree. This paper presents a model free of the previous limitations. It consists in a linear state-space representation describing the rotor's complete dynamics by considering the impact of the rotor axial, radial and angular displacements as well as the gyroscopic effects. This set of ten equations depends on twenty parameters whose identification can be easily performed through static finite element simulations or quasi-static experimental measurements. The model stresses the intrinsic decoupling between the axial dynamics and the other degrees of freedom as well as the existence of electrodynamic angular torques restoring the rotor to its nominal position. Finally, a stability analysis performed on the model highlights the presence of two conical whirling modes related to the angular dynamics, namely the nutation and precession motions. The former, whose intrinsic stability depends on the ratio between polar and transverse moments of inertia, can be easily stabilised through external damping whereas the latter, which is stable up to an instability threshold linked to the angular electrodynamic cross-coupling stiffness, is less impacted by that damping.