Time-invariant Systems Research Articles

Experimental Data-Driven Estimation of Impulse Response in Audio Systems Using Parametric and Non-Parametric Methods

The impulse response is a fundamental tool for characterizing linear time-invariant (LTI) systems, enabling the derivation of a mathematical model that accurately describes system dynamics under arbitrary input conditions. This study used experimental data to estimate the impulse response of an audio system—comprising an amplifier, a speaker, a room, and a microphone. Four methods were employed: two parametric and two non-parametric approaches, applied in both the time and frequency domains. The methods were evaluated quantitatively using the Root Mean Square Error (RMSE) metric and qualitatively through a perceptual analysis with six participants. The parametric frequency-domain method achieved the best perceptual results, with 75% of participants rating the output as good. While this method exhibited slightly higher RMSE compared to other techniques, its low filter order (8) resulted in superior computational efficiency. The findings highlight that perceptual alignment often diverges from purely mathematical error minimization. Real-time implementation of the selected impulse response further demonstrated its practical application in audio processing systems. This research bridges quantitative metrics and human auditory perception, emphasizing the need for balanced decision-making in audio system modeling. The results contribute to advancing data-driven methodologies in acoustics, offering insights into both experimental design and computational efficiency

On Linear Operators in Hilbert Spaces and Their Applications in OFDM Wireless Networks

This paper explores the application of Hilbert topological spaces and linear operator algebra in the modelling and analysis of OFDM signals and wireless channels, where the channel is considered as a linear time-invariant (LTI) system. The wireless channel, when subjected to an input OFDM signal, can be described as a mapping from an input Hilbert space to an output Hilbert space, with the system response governed by linear operator theory. By employing the mathematical framework of Hilbert spaces, we formalise the representation of OFDM signals, which are interpreted as elements of an infinite-dimensional vector space endowed with an inner product. The LTI wireless channel is characterised by using bounded linear operators on these spaces, allowing for the decomposition of complex channel behaviour into a series of linear transformations. The channel’s impulse response is treated as a kernel operator, facilitating a functional analysis approach to understanding the signal transmission process. This representation enables a more profound understanding of channel effects, such as fading and interference, through the eigenfunction expansion of the operator, leading to a spectral characterization of the channel. The algebraic properties of linear operators are leveraged to develop optimal solutions for mitigating channel distortion effects.

Element-wise and recursive solutions for the power spectral density of biological stochastic dynamical systems at fixed points

Stochasticity plays a central role in nearly every biological process, and the noise power spectral density (PSD) is a critical tool for understanding variability and information processing in living systems. In steady state, many such processes can be described by stochastic linear time-invariant (LTI) systems driven by Gaussian white noise, whose PSD is a complex rational function of the frequency that can be concisely expressed in terms of their Jacobian, dispersion, and diffusion matrices, fully defining the statistical properties of the system's dynamics at steady state. Here, we arrive at compact, element-wise solutions of the rational function coefficients for the auto- and cross-spectrum that enable the explicit analytical computation of the PSD in dimensions n=2,3,4. We further present a recursive Leverrier-Faddeev-type algorithm for the exact computation of the rational function coefficients. Crucially, both solutions are free of matrix inverses. We illustrate our element-wise and recursive solutions by considering the stochastic dynamics of neural systems models, namely, FitzHugh-Nagumo (n=2), Hindmarsh-Rose (n=3), Wilson-Cowan (n=4), and the stabilized supralinear network (n=22), as well as an evolutionary game-theoretic model with mutations (n=5,31). We extend our approach to derive a recursive method for calculating the coefficients in the power-series expansion of the integrated covariance matrix for interacting spiking neurons modeled as Hawkes processes on arbitrary directed graphs. Published by the American Physical Society 2024

Finite-time distributed state and disturbance estimation for LTI systems with disturbance: a PEBO-based approach

For a class of linear time-invariant (LTI) systems with disturbance, the finite-time distributed estimation problem of both system state and disturbance was investigated. First, observability decomposition is performed to transform the original system into a cascaded one that is composed of multiple subsystems, and the system disturbance is modelled as the output of an external system. Second, the first substate and external system state are augmented into a new state, and the explicit relationships between state and unknown parameters are established by applying global filtering transformation. Subsequently, the DREM-based parameter estimator and parameter estimation-based observer are developed to simultaneously estimate the first substate and disturbance in finite time. Then, under a unidirectional communication mechanism between adjacent nodes, the subsequent nodes can obtain the estimates of disturbance and substates of all precursor subsystems and only need to estimate the state of current subsystem. Finally, the effectiveness of the proposed methods is illustrated by a numerical example and double-pendulum overhead crane system.

State-Space Solution to Spectral Entropy Analysis and Optimal State-Feedback Control for Continuous-Time Linear Systems

In this paper, a problem of random disturbance attenuation capabilities for linear time-invariant continuous systems, affected by random disturbances with bounded σ-entropy, is studied. The σ-entropy norm defines a performance index of the system on the set of the aforementioned input signals. Two problems are considered. The first is a state-space σ-entropy analysis of linear systems, and the second is an optimal control design using the σ-entropy norm as an optimization objective. The state-space solution to the σ-entropy analysis problem is derived from the representation of the σ-entropy norm in the frequency domain. The formulae of the σ-entropy norm computation in the state space are presented in the form of coupled matrix equations: one algebraic Riccati equation, one nonlinear equation over log determinant function, and two Lyapunov equations. Optimal control law is obtained using game theory and a saddle-point condition of optimality. The optimal state-feedback control, which minimizes the σ-entropy norm of the closed-loop system, is found from the solution of two algebraic Riccati equations, one Lyapunov equation, and the log determinant equation.

Modal Parameter Identification of Jacket-Type Offshore Wind Turbines Under Operating Conditions

Operational modal analysis (OMA) is essential for long-term health monitoring of offshore wind turbines (OWTs), helping identifying changes in structural dynamic characteristics. OMA has been applied under parked or idle states for OWTs, assuming a linear and time-invariant dynamic system subjected to white noise excitations. The impact of complex operating environmental conditions on structural modal identification therefore requires systematic investigation. This paper studies the applicability of OMA based on covariance-driven stochastic subspace identification (SSI-COV) under various non-white noise excitations, using a DTU 10 MW jacket OWT model as a basis for a case study. Then, a scaled (1:75) 10 MW jacket OWT model test is used for the verification. For pure wave conditions, it is found that accurate identification for the first and second FA/SS modes can be achieved with significant wave energy. Under pure wind excitations, the unsteady servo control behavior leads to significant identification errors. The combined wind and wave actions further complicate the picture, leading to more scattered identification errors. The SSI-COV based modal identification method is suggested to be reliably applied for wind speeds larger than the rated speed and with sufficient wave energy. In addition, this method is found to perform better with larger misalignment of wind and wave directions. This study provides valuable insights in relation to the engineering applications of in situ modal identification techniques under operating conditions in real OWT projects.

Data-driven distributed output consensus control for multi-agent systems with unknown internal state

This paper discusses the topic of optimal output consensus control of linear time-invariant discrete-time multi-agent systems (MASs). The attainment of optimal output consensus control for MASs hinges upon the resolution of the interconnected Hamilton–Jacobi-Bellman equation, a task typically precluded by the inherent intractability of analytical solutions. Furthermore, most real-world systems are too complex to obtain the internal states of systems. To address these issues, a modified deep Q-learning network is constructed using current and historical system data rather than a precise model of the system. First, reconstructing the internal state of each agent using an adaptive distributed observer based on output feedback prevents the system instability brought on by the augmented systems. Then the local error system of the agent can be redefined. Based on the redefined error system, a data-driven adaptive dynamic programming (ADP) method is introduced, realized by using the actor–critic neural network structure. In addition, an experience replay strategy is proposed to reduce the propagation of estimation bias and improve the learning speed. Finally, the comparative numerical simulations substantiate the efficacy of the proposed algorithm in a quantifiable manner.

Minimum-Time Control of a Linear System With Input Saturation: A Practical Approach

Abstract A simple algorithm is presented to find near-minimum-time control inputs for a controllable continuous-time linear time-invariant system with any number of states and inputs. The well-known digital deadbeat control algorithm is modified to satisfy input constraints, and the design issue is the selection of sampling period to minimize the time required to reach the desired final state. During each sampling period, the input required to place all the poles at the origin of z-plane (deadbeat control) via state feedback is computed. If the infinity norm of the deadbeat input exceeds the input constraint, the input is found by placing the poles as close to the origin as possible. A sufficient condition for the system stability is developed. Numerical examples illustrate that this algorithm can lead to true time-optimal control or near time-optimal control.

An Enhanced State-Space Modeling for Detecting Abnormal Aging in VRLA Batteries

The knowledge of battery aging is an indicator that allows controlling the performance of large battery banks. State of Health (SOH) is typically the metric used, encompassing all possible mechanisms in a percentage indicator, with the Coulomb Counting as the most common method. Hence, an in-depth study of aging based on known models provides proper information for correctly managing batteries. This article proposes an aging-sensitive 3-RC-array-equivalent electrical circuit model to characterize the behavior of batteries throughout their useful life, identifying parametric changes as complementary information to the state of health. This model was validated based on experimental tests with 2 V and 6 Ah VRLA batteries aged according to the manufacturer’s recommended use. The results reveal a proportionality through capacity degradation. Then, a control group of batteries was subjected to overcharge and over-discharge conditions. The information given by Coulomb Counting SOH and the proposed method were evaluated. The proposed method provides additional information to the SOH, enhancing the distinguishing capability between typical aging performance and misused aging performance, resulting in a useful tool capable of identifying the aging associated with parametric changes in a time-invariant system where aging is treated as an imminent multiplicative fault.

Periodic event-triggered data-driven control for networked control systems with time-varying delays

This article focuses on data-driven analysis and controller design for networked control systems (NCSs) with network-induced delays. The study considers a linear time-invariant (LTI) system controlled through a periodic event-triggering mechanism. First, by leveraging data-based representations, we establish data-based stability conditions for NCSs with time-varying delays. Furthermore, we propose the data-based method for co-designing the controller and the periodic event-triggering scheme. In addition, we present novel data-based conditions for verifying dissipativity properties of NCSs. The effectiveness of our proposed methods is validated through a simulation and a turbofan engine hardware-in-the-loop (HIL) experiment.

Stabilization of LTI systems with both uncertainties and external disturbances via DE-based control method

This paper is concerned with the stabilization of linear time-invariant (LTI) systems with unknown parameters and external disturbances. Firstly, four types of suitable filters are presented and applied to achieve the accuracy estimate of the disturbances. Based on these filters, four types of disturbance estimators are designed and used to asymptotically cancel the corresponding disturbances. Secondly, an adaptive disturbance estimator (DE)-based control method is obtained by combining the adaptive control method with the DE-based control method. It is noted that the algebraic representation of the adaptive feedback control law is given in a more concise form by the tool of the semi-tensor product of matrix. Further, the stabilization of LTI systems is achieved by the obtained adaptive DE-based control method. Finally, three numerical examples with computer simulation are provided to verify the correctness and validity of the obtained theoretical methods.

Improving extensions for the global uniform asymptotic stability of continuous-time nonlinear systems

This paper considers the classical problems of global asymptotic stability (GAS) of nonlinear autonomous (time-invariant) systems and global uniform asymptotic stability (GUAS) of nonlinear non-autonomous (time-varying) systems. By imposing new conditions on the Lyapunov function and its derivative, the state convergence to the origin from any initial condition with bounded norm in the classical setting is extended to the initial conditions with unbounded norms in the present work. The proposed new notions of stability are called extended global asymptotic stability (EGAS) and extended global uniform asymptotic stability (EGUAS) in this paper. The proposed Lyapunov-based criteria yield the global asymptotic stability of the origin and fixed-time/predefined-time attractiveness of any selected neighbourhood of the origin. The result is that the dynamical behaviour of the state trajectories from the convergence time point of view is improved considerably. The relationship between the smoothness and input-to-state stability (ISS) properties is also studied from a quantitative point of view. Under some new sufficient conditions, it is shown that dynamical systems with smaller Lipschitz constants represent smaller ultimate bounds, and as a result, are more robust. The desirable dynamical behaviour and robustness property of the proposed stability notion makes it comparable with the finite/fixed/predefined/prescribed (exact) stable systems that are naturally non-Lipschitz at the origin. Numerical results illustrate the effectiveness of the proposed theoretical results.

On separability of semigroups: application to models of physics and discrete optimization

Nilpotent semigroups are used in various fields of physics to model processes like system decay in quantum mechanics, state transitions in statistical mechanics, and simplifying dynamical systems. They also assist in optimizing control processes. The set of nilpotent matrices forms a semigroup and plays a key role in Jordan decomposition, which has many physical applications. In quantum mechanics, the Jordan form helps analyze linear operators on Hilbert spaces, especially with systems involving eigenvalue multiplicities. In vibration analysis, it aids in studying normal modes of mechanical systems with degenerate frequencies. Additionally, Jordan decomposition simplifies control theory for linear time-invariant systems, stability analysis in dynamical systems, and the behavior of coupled oscillators, as well as solving systems of linear differential equations. Certainly, the semigroup theory has even more applications in theoretical computer science: suffice it to say, for example, that some authors identify semigroups and finite automata. In this paper, we address the problem of P-separability of semigroups with respect to three key predicates: equality-separability, divisibility-separability, and subsemigroup-separability, achieved through homomorphisms into a nilpotent semigroup. We present some significant results that advance the understanding of these concepts.

Some Combined Results from Eneström–Kakeya and Rouché Theorems on the Generalized Schur Stability of Polynomials and the Stability of Quasi-Polynomials-Application to Time-Delay Systems

This paper derives some generalized Schur-type stability results of polynomials based on several forms and generalizations of the Eneström–Kakeya theorem combined with the Rouché theorem. It is first investigated, under sufficiency-type conditions, the derivation of the eventually generalized Schur stability sufficient conditions which are not necessarily related to the zeros of the polynomial lying in the unit open circle. In a second step, further sufficient conditions were introduced to guarantee that the above generalized Schur stability property persists within either the same above complex nominal stability region or in some larger one. The classical weak and, respectively, strong Schur stability in the closed and, respectively, open complex unit circle centred at zero are particular cases of their corresponding generalized versions. Some of the obtained and proved results are further generalized “ad hoc” for the case of quasi-polynomials whose zeros might be interpreted, in some typical cases, as characteristic zeros of linear continuous-time delayed time-invariant dynamic systems with commensurate constant point delays.

A Covariance-Free Strictly Complex-Valued Relevance Vector Machine for Reducing the Order of Linear Time-Invariant Systems

A Covariance-Free Strictly Complex-Valued Relevance Vector Machine for Reducing the Order of Linear Time-Invariant Systems

Model order reduction for the input–output behavior of a geothermal energy storage

In this article, we consider a geothermal energy storage system in which the spatio-temporal temperature distribution is modeled by a heat equation with a time-dependent convection term. Such storage systems are often embedded in residential heating systems. The control and management of such systems requires knowledge of aggregated characteristics of the temperature distribution in the storage. These describe the input–output behavior of the storage, the associated energy flows, and their response to charging and discharging processes. Our aim is to derive an efficient, approximate description of these characteristics by using low-dimensional systems of ordinary differential equations (ODEs). This leads to a model order reduction problem for a large-scale linear system of ODEs resulting from the semidiscretization of the heat equation combined with a linear algebraic output equation. In a first step, we approximated the nonautonomous system of ODEs by a linear time-invariant system. Then, we applied Lyapunov balanced truncation model order reduction to approximate the output by a reduced-order system that has only a few state equations but almost the same input–output behavior. The results of our extensive numerical experiments show the efficiency of the applied model order reduction method. We found that only a few suitably chosen ODEs are sufficient to achieve good approximations of the input–output behavior of the storage.

Reduction in Optimal Time in Systems with Input Redundancy

This paper discusses a reduction in the optimal time due to the presence of input redundancy in time-optimal control problems. By introducing a non-idle channel to represent an active input channel, we establish the necessary and sufficient conditions that ensure a strict reduction in the optimal time for affine nonlinear systems. In cases of identical input redundancy, its impact varies according to the type of input constraint, and certain types may not lead to a reduction in the optimal time. Ultimately, in linear time-invariant (LTI) systems, the extent of the optimal time reduction depends on the system’s controllability.

Robust data-driven predictive control for unknown linear time-invariant systems

This paper presents a new robust data-driven predictive control scheme for unknown linear time-invariant systems by using input–state–output or input–output data based on whether the state is measurable. To remove the need for the persistently exciting (PE) condition of a sufficiently high order on pre-collected data, a set containing all systems capable of generating such data is constructed. Then, at each time step, an upper bound on a given objective function is derived for all systems in the set, and a feedback controller is designed to minimize this bound. The optimal control gain at each time step is determined by solving a set of linear matrix inequalities. We prove that if the synthesis problem is feasible at the initial time step, it remains feasible for all future time steps. Unlike current data-driven predictive control schemes based on behavioral system theory, our approach requires less stringent conditions for the pre-collected data, facilitating easier implementation. The effectiveness of our proposed methods is demonstrated through application to an unknown and unstable batch reactor.

State-space estimation of spatially dynamic room impulse responses using a room acoustic model-based prior

Room impulse responses (RIRs) between static loudspeaker and microphone locations can be estimated using a number of well-established measurement and inference procedures. While these procedures assume a time-invariant acoustic system, time variations need to be considered for the case of spatially dynamic scenarios where loudspeakers and microphones are subject to movement. If the RIR is modeled using image sources, then movement implies that the distance to each image source varies over time, making the estimation of the spatially dynamic RIR particularly challenging. In this paper, we propose a procedure to estimate the early part of the spatially dynamic RIR between a stationary source and a microphone moving on a linear trajectory at constant velocity. The procedure is built upon a state-space model, where the state to be estimated represents the early RIR, the observation corresponds to a microphone recording in a spatially dynamic scenario, and time-varying distances to the image sources are incorporated into the state transition matrix obtained from static RIRs at the start and end points of the trajectory. The performance of the proposed approach is evaluated against state-of-the-art RIR interpolation and state-space estimation methods using simulations, demonstrating the potential of the proposed state-space model.

An online reconstruction method of dynamic loading based on adaptive tracking dual nested Kalman filter

Accurately reconstructing the unknown dynamic loading is crucial for controlling the vibration of mechanical systems. Many methods have been proposed for load identification of time-invariant systems, but in some mechanical systems, their structural parameters exhibit time-varying dynamic characteristics during their operation. Therefore, tracking the time-varying structural parameters to update the transfer function is necessary, which is the most important prerequisite for load reconstruction. However, some existing methods are unstable because of the nonlinear conditions. To overcome this issue, an improved dual-nested Kalman filter framework has been proposed. This method allocates unknown variables to two Kalman filters, allowing them to iterate and loop with each other. As a result, it achieves linear reconstruction of the dynamic loading for a time-varying system. The proposed method has been validated in two numerical examples: one is a spring-mass system with abruptly changing mass parameters, and the other is a milling system with slowly changing mass parameters. The results show that the proposed method can reconstruct loading stably and is not susceptible to noise interference.