Linear Time-invariant Systems Research Articles

Towards efficient control synthesis for nonlinear wave energy conversion systems: impedance-matching meets the spectral-domain

AbstractExisting studies within the literature that focus on designing parametric energy-maximizing controllers for Wave Energy Converter (WEC) systems predominantly rely on the impedance-matching (IM) principle, originally developed for linear time-invariant systems. Alternatively, iterative optimization routines are commonly employed for nonlinear WECs. However, these approaches often face a trade-off between effectiveness in maximizing energy extraction and computational efficiency. To address this limitation, this study proposes a computationally efficient controller tuning method for analogous synthesis in the case of nonlinear WECs. The proposed approach combines a statistical linearization technique known as spectral-domain modeling with the IM principle, to synthesize a Proportional–Integrative (PI) controller for a nonlinear WEC. Furthermore, a comparison is performed with two other synthesis methods: one based on a standard (i.e. linear) frequency-domain representation of the WEC that incorporates the IM principle, and the other employing a gradient-free optimization routine applied to the nonlinear time-domain model of the WEC for PI parameter tuning through exhaustive numerical search. A discussion on the effectiveness of each tuning method in maximizing energy absorption is provided, including an appraisal of their associated computational time requirements. Numerical analyses demonstrate that the proposed method, which integrates spectral-domain modeling and IM, can achieve (almost) optimal PI controller design for a nonlinear WEC. Furthermore, this study addresses the inaccuracies inherent in the frequency-domain approach and significantly reduces the computational time compared to the exhaustive search procedure. The findings of this research represent a significant advancement towards the development of simple, effective, and efficient IM-based techniques for synthesis of controllers in nonlinear WEC systems

Data-driven calibration of decoupling matrix for MIMO precision motion stages

Decoupling control is a commonly employed technique for achieving high precision in multiple-input multiple-output (MIMO) motion control systems. A static decoupling matrix, which can be determined using geometric construction, is widely used due to its practicality and simplicity. However, inaccurate geometric parameters will lead to a coarse decoupling matrix, and result in interactions among the system axes and performance deterioration. To tackle this challenge, various attempts have been undertaken to calibrate the decoupling matrix. Data-driven on-line approaches have gained considerable attention for their ability to calibrate the decoupling matrix without interrupting the normal operation of the system. This paper presents a data-driven approach to calibrate the decoupling matrix for MIMO and linear time invariant (LTI) systems. Through some reasonable assumptions, a calibrated static decoupling matrix can be derived to improve the performance of the system. Moreover, considering the inevitable presence of measurement noise, the consistency of the proposed method has been analyzed. As a result, the instrument variable is introduced in the improved method to eliminate the impact of the measurement noise. Finally, the effectiveness and practicality of the proposed method are demonstrated through both numerical simulations and experiments carried out on an ultraprecision wafer stage.

Advanced order diminution technique for linear time-invariant systems with applications in lag/lead compensators and PID controller design

Advanced order diminution technique for linear time-invariant systems with applications in lag/lead compensators and PID controller design

Existence conditions for functional ODE observer design of descriptor systems revisited

This paper is devoted to the problem of designing functional observers for linear time-invariant (LTI) descriptor systems. The observers are realized by using state–space systems governed by ordinary differential equations (ODEs). Available existence results for functional ODE observers in the literature are extended by introducing new and milder sufficient conditions. These conditions are purely algebraic and provided directly in terms of the system coefficient matrices. The proposed observer has an order less than or equal to the dimension of the functional vector to be estimated. The observer parameter matrices are obtained by using simple matrix theory, and the design algorithm is illustrated by numerical examples.

Linearized Models of the Coupled Rotorcraft Flight Dynamics and Acoustics for Real-Time Noise Prediction

This article demonstrates the linearization of the coupled rotorcraft flight dynamics and aeroacoustics to provide realtime acoustic predictions in generalized maneuvering flight. To demonstrate the methodology, the study makes use of a nonlinear simulation model of a generic utility helicopter (PSUHeloSim) that is coupled with an aeroacoustic solver based on a marching cubes algorithm. A periodic equilibrium of the coupled rotorcraft flight dynamics and acoustics is first found at a desired flight condition using a modified harmonic balance trim solution method. Next, the nonlinear time-periodic dynamics are linearized about that periodic equilibrium and transformed into an equivalent higher order linear time-invariant system in harmonic decomposition form. Composite aeroacoustic measures are included as an output of this system. To speed up runtime and make control design tractable, the order of these harmonic decomposition models is reduced via residualization to an 8-state model where the states are representative of the rigid-body dynamics of the aircraft. This 8-state model is shown to provide accurate acoustic response predictions for small-amplitude pilot inputs and to abate runtime by a factor of approximately 104, thus enabling acoustic predictions in generalized maneuvering flight that are significantly faster than real time. The 8-state model is subsequently used to demonstrate the use of linear system tools for the dynamic analysis of the coupled rotorcraft flight dynamics and acoustics.

Leveraging the Hankel norm approximation and data‐driven algorithms in reduced order modeling

SummaryLarge‐scale linear time‐invariant (LTI) dynamical systems are widely used to characterize complicated physical phenomena. Glover developed the Hankel norm approximation (HNA) algorithm for optimally reducing the system in the Hankel norm, and we study its numerical issues. We provide a remedy for the numerical instabilities of Glover's HNA algorithm caused by clustered singular values. We analyze the effect of our modification on the degree and the Hankel error of the reduced system. Moreover, we propose a two‐stage framework to reduce the order of a large‐scale LTI system given samples of its transfer function for a target degree of the reduced system. It combines the adaptive Antoulas–Anderson (AAA) algorithm, modified to produce an intermediate LTI system in a numerically stable way, and the modified HNA algorithm. A carefully computed rational approximation of an adaptively chosen degree gives us an algorithm for reducing an LTI system, which achieves a balance between speed and accuracy.

Theoretical Development and Numerical Validation of an Asymmetric Linear Bilateral Control Model- Case Study for an Automated Truck Platoon

In this paper, we theoretically develop and numerically validate an asymmetric linear bilateral control model (LBCM) , in which the motion information (e.g., position and speed) from the immediate leading vehicle and the immediate following vehicle of a subject vehicle in an automated vehicle platoon are weighted differently to determine the control inputs to the subject vehicle. The novelty of the asymmetric LBCM is that using this model all the follower vehicles in a platoon can adjust their acceleration and deceleration to closely follow a constant desired time gap to improve platoon operational efficiency while maintaining local and string stability. We theoretically analyze the local stability of the asymmetric LBCM using the condition for asymptotic stability of a linear time-invariant system and prove the string stability of the asymmetric LBCM using a space gap error attenuation approach. Then, we evaluate the efficacy of the asymmetric LBCM by simulating a closely coupled cooperative adaptive cruise control (CACC) platoon of fully automated trucks in various non-linear acceleration and deceleration states. We choose automated truck platooning as a case study since heavy-duty trucks experience higher delays and lags in the powertrain system, and limited acceleration and deceleration capabilities than passenger cars. To evaluate the platoon operational efficiency of the asymmetric LBCM, we compare the performance of the asymmetric LBCM to a baseline model, i.e., the symmetric LBCM, for different powertrain delays and lags. Our analyses showed that the asymmetric LBCM can handle any combined powertrain delays and lags up to 0.6 sec while maintaining a constant desired time gap during a stable platoon operation, whereas the symmetric LBCM fails to ensure stable platoon operation as well as maintain a constant desired time gap for any combined powertrain delays and lags over 0.2 sec. These findings demonstrate the potential of the asymmetric LBCM in improving platoon operational efficiency and stability of an automated truck platoon.

Positive State Controllability of Discrete Linear Time-Invariant Systems

Abstract Positive state controllability is the controllability of systems where the state is positive and the input remains in ℝn. Under some conditions, we established a relation between the reachability map of systems with only the positive state and the reachability map of a related positive system where the state and input are both positive. Using this connection, necessary and sufficient conditions are obtained for the positive state reachability of discrete linear time-invariant (LTI) systems, and then we deduced the positive state controllability. These conditions are evaluated over some numerical examples that support the theoretical results.

The causal α-exponential and the solution of fractional linear time-invariant systems

Closed-form expressions for the impulse and step responses of commensurable linear time-invariant systems are deduced and exemplified. The algorithm is based on obtaining the solution in terms of the α-exponential monomials that generate a vector space containing the solutions of such systems. Several examples are considered, together with the numerical aspects involved in calculating the related series, which show the accuracy and effectiveness of the approach.

Controllability and Observability of Linear Time-Invariant Control System on Superspace

Controllability and Observability of Linear Time-Invariant Control System on Superspace

An anomaly affected discrete LTI systems: a moving horizon approach for estimating position and temperature measurements

A real-time system may be subjected to various anomalies that can affect the quality of the observations. The main motivation of the article arises from the need in addressing challenges posed by the presence of anomalies in discrete linear time-invariant (LTI) systems with a focus on the estimation processes, in the context of position and temperature measurements. The proposed approach leverages the properties of discrete LTI systems and takes advantage of the predictive capabilities of the moving horizon strategy (MHS). It operates recursively updating estimates of new measurement while fairly considering its past estimates that occur within the window of the moving horizon. The estimation framework will be designed to handle disturbances and provide robust estimates, to ensure the effectiveness of the system. In order to validate the proposed approach simulation studies were conducted on different and only scenarios in different order LTI system. Comparative studies with different estimation techniques demonstrate the capability of the proposed approach in terms of performance and efficiency. The proposed approach can be applied to systems with changing system dynamics. Future research may be conducted to utilize this strategy in other domains to mitigate anomalies while enhancing performance.

Optimal Excitation and Readout of Resonators Used as Wireless Passive Sensors.

Resonators are passive time-invariant components that do not produce a frequency shift. However, they respond to an excitation signal close to resonance with an oscillation at their natural frequencies with exponentially decreasing amplitudes. If resonators are connected to antennas, they form purely passive sensors that can be read remotely. In this work, we model the external excitation of a resonator with different excitation signals and its subsequent decay characteristics analytically as well as numerically. The analytical modeling explains the properties of the resonator during transient response and decay behavior. The analytical modeling clarifies how natural oscillations are generated in a linear time-invariant system, even if their spectrum was not included in the stimulation spectrum. In addition, it enables the readout signals to be optimized in terms of duration and bandwidth.

Error-Based Switched Fractional Order Model Reference Adaptive Control for MIMO Linear Time Invariant Systems

This paper presents the design and analysis of Switched Fractional Order Model Reference Adaptive Controllers (SFOMRAC) for Multiple Input Multiple Output (MIMO) linear systems with unknown parameters. The proposed controller uses adaptive laws whose derivation order switches between a fractional order and the integer order, according to a certain level of control error. The switching aims to use fractional orders when the control error is larger to improve transient response and system performance during large disturbed states, and to obtain smoother control signals, leading to a better control energy usage. Then, it switches to the integer order when the control error is smaller to improve steady state. Boundedness of all the signals in the scheme is analytically proved, as well as convergence of the control error to zero. Moreover, these properties are extended to the case when system states are affected by a bounded non-parametric disturbance. Simulation studies are carried out using different representative plants to be controlled, showing that fractional orders and switching error levels can be found in most of the cases, such as when SFOMRAC achieves a better balance among control energy and system performance than the non-switched equivalent strategies.

Underapproximate Reachability Analysis for a Class of Linear Systems With Inputs

Under-approximations of reachable sets and tubes have been receiving growing research attention due to their important roles in control synthesis and verification. Available under-approximation methods applicable to continuous-time linear systems typically assume the ability to compute transition matrices and their integrals exactly, which is not feasible in general, and/or suffer from high computational costs. In this note, we attempt to overcome these drawbacks for a class of linear time-invariant (LTI) systems, where we propose a novel method to under-approximate finite-time forward reachable sets and tubes, utilizing approximations of the matrix exponential and its integral. In particular, we consider the class of continuous-time LTI systems with an identity input matrix and initial and input values belonging to full dimensional sets that are affine transformations of closed unit balls. The proposed method yields computationally efficient under-approximations of reachable sets and tubes, when implemented using zonotopes, with first-order convergence guarantees in the sense of the Hausdorff distance. To illustrate its performance, we implement our approach in three numerical examples, where linear systems of dimensions ranging between 2 and 200 are considered.

Distributed State Estimation Under Jointly Connected Switching Networks: Continuous-Time Linear Systems and Discrete-Time Linear Systems

In this paper, we address the distributed state estimation problem for both continuous-time linear time-invariant (LTI) systems and discrete-time LTI systems under switching networks. The observed system is jointly observable, i.e., each agent can only access a part of the measurement output of the observed system and cannot recover the full state by itself. The full state estimation has to be achieved by network communication of neighboring agents. In contrast to existing works, the salient feature of this work is that the developed approach can deal with the jointly connected switching network and thus is more resilient to unreliable communication. First, we propose a new observability decomposition method for linear systems in modal canonical form. Then, we design distributed observers for both the continuous-time and the discrete-time system. Based on the common Lyapunov function approach, we show that the switched estimation error system is asymptotically stable and thus, the full state estimation can be achieved under jointly connected switching networks.

Guaranteed Robust Performance of $\mathcal {H}_{\infty }$ Filters With Sparse and Low Precision Sensing

The performance of estimation algorithms depends on the available sensors and their precisions. While higher precisions of sensors provide better estimation accuracy, it may lead to unnecessarily expensive designs and higher operational costs due to higher costs of high-precision sensing modalities. Also, higher precision can cause interference for other sensors in the environment, such as RADARS, resulting in degradation in the overall system's performance. This paper presents a framework for co-designing a sparse sensing network with the least precise sensors and the filter that guarantees the prescribed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_{\infty }$</tex-math></inline-formula> estimation accuracy. Convex optimization problems for minimizing sensor precisions are formulated for continuous and discrete-time linear time-invariant systems, with and without model uncertainties. Different heuristics for determining a sparse sensor set with the least precise sensors are discussed, and their performances are compared using numerical simulations. The application of the proposed framework is demonstrated using numerical examples.

Asymptotic Output Tracking Control of a Class of Linear Systems by Finite-and-Quantized Output Feedback

This paper presents a novel finite-and-quantized output feedback asymptotic tracking control method for a general class of continuous-time linear time-invariant systems. First, we construct a finite quantizer with time-varying thresholds and design a pole placement control law that exclusively utilizes the finite-and-quantized output signal and an external reference signal. Then, we establish the boundedness of all closed-loop signals and prove the asymptotic convergence of the output tracking error to zero. The proposed method combines the advantages of classical pole placement control technique and finite quantization feedback technique. It not only reduces the requirement for feedback information compared with existing tracking control methods but also effectively handles unstable poles and zeros in controlled systems, thereby achieving asymptotic output tracking. Finally, we provide a representative example to validate the effectiveness and new features of our proposed method.

Supplemental Stability Criteria with New Formulation for Linear Time-Invariant Fractional-Order Systems

In this paper, new stability criteria for linear time-invariant fractional-order systems (LTIFOSs) based on linear matrix inequalities (LMIs) are derived. The solved variable of the existing LMI formulations is generalized to a complex one. In addition, based on the congruent transformation, a new LMI formulation is obtained, which is different from those in the existing literature. To deal with the above LMIs more conveniently with simulation software, the complex LMIs are converted to equivalent real LMIs. Finally, numerical examples are presented to validate the effectiveness of our theoretical results.

Reset-free data-driven gain estimation: Power iteration using reversed-circulant matrices

A direct data-driven iterative algorithm is developed to accurately estimate the H∞ norm of a linear time-invariant system from continuous operation, i.e., without resetting the system. The main technical step involves a reversed-circulant matrix that can be evaluated in a model-free setting by performing experiments on the real system.

On the Relative Stability of Linear Time Invariant Systems

In this paper, state space representation based approach to relative stability of Linear Time Invariant (LTI) systems is discussed. Specifically, given a bound on the output, relative Bounded Input-Bounded Output (BIBO) stability of LTI systems is discussed. The bound on dominant pole is first determined using input-output description of LTI system. Using state space representation of homogeneous (no external input) LTI system, the components of state vector are upper bound. The results are applied to Continuous Time Markov Chains (i.e. homogeneous stochastic linear systems). Finally, using state space representation of arbitrary (with external input) Linear Time Invariant systems, the location of dominant pole is bounded. Using the upper bound, the location of line (parallel to imaginary-axis jω and to the left of it) is utilized for determining relative stability.