Linear Time-invariant Systems Research Articles

Distinguishability criteria of conformable hybrid linear systems

AbstractWe relate this article to the emerging idea of distinguishability of conformable linear hybrid time-invariant control systems. To obtain the necessary and sufficient conditions ofα\alpha-distinguishability for fractional cases, we develop the Leibnitz rule for conformable derivatives. Furthermore, with the help of a study of Laplace techniques, a more simple criterion ofα\alpha-distinguishability for the fractional linear system is developed.

NSGA-II BASED MULTI-OBJECTIVE OPTIMIZATION OF WEIGHT MATRICES FOR LQR CONTROLLER APPLIED TO DC MOTOR

LQR kontrol yaklaşımıyla, Lineer Zamanla Değişmeyen bir sistemin kararlılığının temin edilmesi yanında, sisteme uygulanacak geri besleme kazancına, karesel bir performans ölçütünün minimizasyonu yoluyla ulaşılıyor olması, bu kontrol yaklaşımını, kapalı çevrim sistemden beklenen performansı şekillendirme kabiliyetine sahip kılmaktadır. Bu noktada, minimize edilecek karesel performans ölçütünün ya da maliyet fonksiyonunun, içerdiği ağırlık matrisleri üzerinden, amaca uygun biçimde oluşturulması kontrolcü performansı bakımından önem arz etmektedir. O yüzden, LQR kontrolcünün tasarımında kullanılan ağırlık matrislerinin optimizasyonu, kontrolcüden beklenen çoklu performans amaçları doğrultusunda gerçekleştirilmelidir. Bu durum çok amaçlı bir optimizasyon problemini doğurmaktadır. Karesel maliyet fonksiyonunun ağırlık matrisleri, optimizasyon yapılmaksızın, deneme yanılma, kutup atama gibi klasik yöntemler yardımıyla ayarlanabilir olsa da bu durum yorucu ve zaman alıcı olabilmektedir. Bu zorluğun üstesinden gelebilmek adına, çeşitli çok amaçlı optimizasyon tekniklerinden istifade edilmesi mümkündür.
 
 Yapılan bu çalışma kapsamında, LQR tabanlı optimal DC motor kontrolü amaçlanmıştır. LQR kontrolcü için söz konusu karesel maliyet fonksiyonuna dair Q ve R ağırlık matrisi parametrelerinin ayarlanması gerekmektedir. Çok amaçlı optimizasyon tekniklerinden biri olan ve Non-Dominated Sorted Genetic Algorithm (NSGA-II) olarak bilinen optimizasyon algoritması yardımıyla ilgili parametreler ayarlanmıştır. Elde edilen optimum parametreler kullanılarak sentezlenen LQR kontrolcünün sistem üzerindeki performans bulguları, simülasyon sonuçları ile sunulmuştur.

A recursive identification algorithm for linear discrete periodic systems

The parameter identification problem of linear discrete period time-varying (LDPV) systems is studied in this paper and a two-step identification algorithm based on the recursive least squares principle is proposed. The recursive principle-based identification algorithm is mostly used to deal with the parameter identification of time-invariant systems. Combine with the cyclic reconstruction method, the LDPV system is converted to a linear time-invariant system. Then, based on this and combined with recursive least squares identification, a discrimination algorithm is proposed for the parameter estimation of the LDPV system. Finally, a numerical simulation example is given to demonstrate the effectiveness of the proposed algorithm.

The Kv10.1 voltage-gated ion channel modeling based on control system theory

Abstract In this paper, we present a control system theory based modeling approach for the voltage-gated ion-channel Kv10.1 at different temperatures. Kv10.1 is expressed in the central nervous system and the brain of healthy humans. The study of the activity of this ion channel is important, because its activity is associated with the occurrence of cancer in different organs or tissues. According to systems and control theory, the voltage-gated channel Kv10.1 was assumed to be a linear time-invariant system and, as such, exhibits dynamic behavior. The experimental results show that Kv10.1 operates as a first-order model based on the input voltage step protocol and the measured macroscopic ion current output.

A quantized output feedback MRAC scheme for discrete-time linear systems

This paper presents a quantized output feedback model reference adaptive control (MRAC) scheme for a class of single-input and single-output discrete-time linear time-invariant systems with unknown parameters. Our method, firstly, integrates the well-known MRAC and quantized control techniques to construct a quantized output feedback adaptive control law with parameter update laws. Then, some vital technical lemmas are developed, fundamentally applicable to finite and infinite quantized output feedback MRAC. Moreover, we prove that in the case of infinite quantization, appropriately choosing the output quantizer’s sensitivity affords the proposed adaptive control law to ensure closed-loop stability and achieve bounded or asymptotic output tracking. The significant advantage of the developed adaptive control scheme is combining the benefits of the classic MRAC and quantized control. Compared with current adaptive tracking control schemes, the developed scheme not only reduces the feedback information requirement, but also has full capability to achieve closed-loop stability and output tracking. The effectiveness of the proposed MRAC scheme is verified through several simulations.

A Lyapunov Approach to Consensus of Linear Multi-Agent Systems with Infinite Distributed Communication Delays

This paper considers the leaderless consensus problem of linear time-invariant multi-agent systems with infinite distributed communication delays. A novel distributed low gain controller is proposed based on the solution to a parametric algebraic Riccati equation. It is shown via the newly developed Lyapunov-like method that not only the consensus of linear time-invariant multi-agent systems can be achieved exponentially under some mild assumptions but also an estimate of the exponential convergence rate of consensus is given in this work. The Lyapunov-like method is also extended to handle a special case of linear time-varying multi-agent systems. In addition, the obtained results include the results on the leaderless consensus of linear multi-agent systems with bounded distributed communication delays as special cases. To the best of our knowledge, this is the first work that develops the Lyapunov-like method for the leaderless consensus problems of both time-invariant and time-varying linear multi-agent systems with infinite distributed communication delays. Finally, a numerical example is presented to illustrate the effectiveness of the proposed controller.

A Calibration Method of Short‐Time Waveform Signals Passed Through Linear Time‐Invariant Systems: 1. Methodology and Simple Examples

AbstractWe propose an accurate calibration method for short‐time waveform signals passed through a linear time‐invariant (LTI) system that has a non‐negligible group delay. Typically, the calibration process of waveform data is expressed by the deconvolution in the time domain, and there is an equivalent operation in the frequency domain with the Fourier transform. For the short‐time data, if the short‐time Fourier transform is applied to the waveform data in the calibration process, multiplying the data by a window function is highly recommended to reduce side‐lobe effects. However, the multiplied window function is also modified in the calibration process. We analyzed the modification mathematically and derived a method to eliminate the modification of the multiplied window function. In the method, calibrated data in the frequency domain are inverse‐transformed into waveform data at each frequency, divided by a modified window function at each frequency, and accumulated over the frequencies. The principle of this method derived quantitatively indicates that the calibration accuracy depends on the transfer function of the system, frequency resolution of the Fourier transform, type of the window function, and typical frequency of the waveform data. Compared with conventional calibration methods, the proposed method provides more accurate results in various cases. This method should be useful for the calibration of general radio wave signals passed through LTI systems as well as for the calibration of plasma waves observed in space.

On the Energy or Power Efficiency of a Linear System

A linear time-invariant (LTI) system is usually characterized by its gains, which are ratios of various output and input signal magnitudes. When the main function of the system is energy conversion, one is more interested in the power or energy gain or loss between input and output terminals. Such a "gain" represents the efficiency of the system as an energy conversion device. In this paper, we show that the power or energy "gain" of an LTI system can be determined as a nonlinear function of three signal gains of the system. This results in an explicit formula for the power (energy) gain in terms of the system parameters. This can then be optimized over the design parameters to maximize the gain and thus the system energy efficiency. A detailed DC circuit example is included for illustration. The energy efficiency of an LTI system operating in a steady state in response to a sinusoidal signal is also determined.

Extension of the Observability Rank Condition to Time-Varying Nonlinear Systems

This article provides an extension of the observability rank condition to time-varying nonlinear systems. Previous conditions to check the state observability only hold for nonlinear systems that do not explicitly depend on time, or, for time-varying systems, they only account for the linear case. In this article, a general analytic condition is provided. The new condition reduces to the well-known observability rank condition in the two simpler cases of time-varying linear systems and time-invariant nonlinear systems. The proposed condition is applied to aerospace robotics to study the observability of a lunar-module-type system equipped with monocular vision and inertial sensors.

Complete Robust Stability Domain of Fractional-Order Linear Time-Invariant Single Parameter-Dependent Systems With the Order 0 < α < 2

The complete robust stability domain of fractional-order linear time-invariant single parameter-dependent systems with the system order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha \in (0,2)$ </tex-math></inline-formula> is investigated in this brief. Firstly, the complete robust stability domain of such parameter-dependent systems is given using the Kronecker sum and the Bialternate sum respectively. The complexity of the algorithm can be reduced with Bialternate sum. This method can be easily extended to the case of multiple parameters. The final numerical examples are used to demonstrate the method proposed in this brief is effective and less conservative than the previous results.

Second-Order NonConvex Optimization for Constrained Fixed-Structure Static Output Feedback Controller Synthesis

For linear time-invariant systems, the design of an optimal controller is a commonly encountered problem in many applications. Among all the optimization approaches available, the linear quadratic regulator (LQR) methodology certainly garners much attention and interest. As is well known, standard numerical tools in linear algebra are readily available to determine the optimal static LQR feedback gain matrix when all the system state variables are measurable. However, in various certain scenarios where some of the system state variables are not measurable, and consequent prescribed structural constraints on the controller structure arise, the optimization problem can become intractable due to the nonconvexity characteristics that can then be present. In such cases, some first-order methods have been proposed to cater to these problems, but all of these methods, if at all successful, are limited to linear convergence. To speed up the convergence, a second-order approach in the matrix space is essential, with appropriate methodology to solve the linear equality constrained static output feedback (SOF) problem with a suitably defined linear quadratic cost function. Thus, in this article, an efficient method is proposed in the matrix space to calculate the Hessian matrix by solving several Lyapunov equations. Then, a new optimization technique is applied to deal with the indefiniteness of the Hessian matrix. Subsequently, through Newton’s method with linear equality constraints, a second-order optimization algorithm is developed to effectively solve the constrained SOF LQR problem. Finally, two numerical examples are described, which demonstrate the applicability and effectiveness of the proposed method.

H-Infinity Loop-Shaped Model Predictive Control With HVAC Application

We formulate a model predictive control (MPC) for linear time-invariant systems based on H-infinity loop-shaping. The design results in a closed-loop system that includes a state estimator and attains an optimized stability margin. Input and output weights are designed in the frequency domain to satisfy steady-state and transient performance requirements, in lieu of standard MPC plant model augmentations. The H-infinity loop-shaping synthesis results in an observer-based state feedback structure. An inverse optimal control problem is solved to construct the MPC cost function, so that the control input computed by MPC is equal to the H-infinity control input when the constraints are inactive. The MPC inherits the closed-loop performance and stability margin of the loop-shaped design when constraints are inactive. We apply the methodology to a multizone heat pump, and validate the results in simulations and laboratory experiments. The design rejects constant unmeasured disturbances, tracks constant references with zero steady-state error, meets transient performance requirements, provides an excellent stability margin, and enforces input and output constraints.

Intelligent Scanning Detection System of Muscle Exercise Fatigue Based on Surface Electromyography.

In order to use the surface EMG signal to automatically detect the muscle fatigue state, a research method of the muscle exercise fatigue intelligent scanning detection system based on surface EMG was proposed, and the sEMG signal features of 10 subjects before and after fatigue were extracted. A time-varying parameter autoregressive model is established. By introducing the Legendre basis function, the parameter identification of the linear nonstationary process is transformed into the parameter identification of the linear time-invariant system. Combined with the correlation index, the optimal Legendre base function dimension of the time-varying system parameter estimation can be obtained, then the best model fitting effect can be obtained, and the time-invariant parameters are solved by the least square method. Using the rate of change of the first time-varying parameter (ARC1) of the autoregressive model before and after fatigue as an index to detect muscle fatigue sensitivity, a two-tailed t test was used to compare the mean power frequency (MPF) and the median frequency (MF) with the rate of change. The results showed that the change rates of ARC1, MPF, and MF before and after fatigue were34.33% ± 2.5%, 68% + 2.03%, and 22.80% + 2.19%, which were 41% and 25%, respectively. The rate of change of ACR1 was significantly higher than that of MPF and MF (P < 0.05). When detecting muscle fatigue by sEMG signal, it has the advantages of short time and high sensitivity. It can be used for online real-time analysis of muscle fatigue, providing a potential analysis tool for limb muscle strain, rehabilitation, and ergonomics assessment.

A generalized dynamic robust observer for uncertain linear time invariant descriptor systems

This paper presents a generalized dynamic robust observer design for uncertain linear time-invariant (LTI) singular systems. In this approach, the state equation of the singular system can consist of parametric uncertainties in three matrices namely the derivative, the system, and the input. The proposed method is according to a new parameterization in the system equations and converting it to a new descriptor model so that in the new structure, the derivative matrix is known. A generalized dynamic robust observer is suggested to estimate the state variables of the system which has more flexibility in contrast with proportional and proportional–integral observers. Also, in this method, in addition to the state variables, whose derivatives are also estimated. A sufficient condition is given in a linear matrix inequality (LMI) form to show the convergence of the observer. Numerical simulation demonstrates the efficacy of the proposed observer.

Characterization and correction of diffusion gradient-induced eddy currents in second-order motion-compensated echo-planar and spiral cardiac DTI.

Very high gradient amplitudes played out over extended time intervals as required for second-order motion-compensated cardiac DTI may violate the assumption of a linear time-invariant gradient system model. The aim of this work was to characterize diffusion gradient-related system nonlinearity and propose a correction approach for echo-planar and spiral spin-echo motion-compensated cardiac DTI. Diffusion gradient-induced eddy currents of 9 diffusion directions were characterized at b values of 150 s/mm2 and 450 s/mm2 for a 1.5Tesla system and used to correct phantom, ex vivo, and in vivo motion-compensated cardiac DTI data acquired with echo-planar and spiral trajectories. Predicted trajectories were calculated using gradient impulse response function and diffusion gradient strength- and direction-dependent zeroth- and first-order eddy current responses. A reconstruction method was implemented using the predicted -space trajectories to additionally include off-resonances and concomitant fields. Resulting images were compared to a reference reconstruction omitting diffusion gradient-induced eddy current correction. Diffusion gradient-induced eddy currents exhibited nonlinear effects when scaling up the gradient amplitude and could not be described by a 3D basis alone. This indicates that a gradient impulse response function does not suffice to describe diffusion gradient-induced eddy currents. Zeroth- and first-order diffusion gradient-induced eddy current effects of up to -1.7rad and -16 to +12 rad/m, respectively, were identified. Zeroth- and first-order diffusion gradient-induced eddy current correction yielded improved image quality upon image reconstruction. The proposed approach offers correction of diffusion gradient-induced zeroth- and first-order eddy currents, reducing image distortions to promote improvements of second-order motion-compensated spin-echo cardiac DTI.

Internally Hankel $k$-Positive Systems

There has been an increased interest in the variation diminishing properties of controlled linear time-invariant (LTI) systems and time-varying linear systems without inputs. In controlled LTI systems, these properties have recently been studied from the external perspective of $k$-positive Hankel operators. Such systems have Hankel operators that diminish the number of sign changes (the variation) from past input to future output if the input variation is at most $k-1$. For $k=1$, this coincides with the classical class of externally positive systems. For linear systems without inputs, the focus has been on the internal perspective of $k$-positive state-transition matrices, which diminish the variation of the initial system state. In the LTI case and for $k=1$, this corresponds to the classical class of (unforced) positive systems. This paper bridges the gap between the internal and external perspectives of $k$-positivity by analyzing internally Hankel $k$-positive systems, which we define as state-space LTI systems where controllability and observability operators as well as the state-transition matrix are $k$-positive. We show that the existing notions of external Hankel and internal $k$-positivity are subsumed under internal Hankel $k$-positivity, and we derive tractable conditions for verifying this property in the form of internal positivity of the first $k$ compound systems. As such, this class provides new means to verify external Hankel $k$-positivity, and lays the foundation for future investigations of variation diminishing controlled linear systems. As an application, we use our framework to derive new bounds for the number of over- and undershoots in the step responses of LTI systems. Since our characterization defines a new positive realization problem, we also discuss geometric conditions for the existence of minimal internally Hankel $k$-positive realizations.

A Note on Order and Index Reduction for Descriptor Systems

We present order reduction results for linear time invariant descriptor systems. Results are given for both forced and unforced systems as well methods for constructing the reduced order systems. Our results establish a precise connection between classical and new results on this topic, and lead to an elementary construction of quasi-Weierstrass forms for a descriptor system. Examples are given to illustrate the usefulness of our results.

Event-Triggered Stabilizing Bit Rate Conditions for an n-Dimensional Linear System With i.i.d. Feedback Dropouts.

This article aims to stabilize an n -dimensional linear time-invariant (LTI) system, whose feedback packets are transmitted through a digital communication network. The digital network suffers from network delay and independent and identically distributed (i.i.d.) feedback dropouts, which may destabilize the system. The coupling among multiple state variables may further harm the stability of the system. In order to deal with these issues and save the occupied bandwidth of the feedback network, we propose a periodic event-triggering strategy. In our strategy, the state is measured periodically, but only quantized and transmitted when a certain condition is triggered. By well balancing the state coupling and making full use of both the information inside transmitted feedback packets and the one carried by sampling time instants, our strategy can maintain the desired mean square stability at a lower bit rate than conventional periodic sampling policies. The obtained stabilizing bit rate conditions are determined by the processing and network delays, the dropout rate, and the unstable eigenvalues of the system matrix, but independent of the process noise. Moreover, the lack of the direct state access does not incur any additional stabilizing bit rate. Simulations are done to confirm the effectiveness of the obtained stabilizing bit rate conditions.

Resilient Control Under Quantization and Denial-of-Service: Codesigning a Deadbeat Controller and Transmission Protocol

This article is concerned with the problem of stabilizing continuous-time linear time-invariant (LTI) systems subject to quantization and denial-of-service (DoS) attacks. In this context, two DoS-induced challenges emerge in the design of resilient encoding schemes, namely, the coupling between encoding strategies of different signals, and the synchronization between the encoder and decoder. These challenges are addressed by a novel proposed structure based on a deadbeat controller as well as a delicate transmission protocol for the input and output channels, and codesigned leveraging the controllability index. When both input and output channels are subject to DoS attacks and quantization, the proposed structure is shown able to decouple the encoding schemes for input, output, and estimated output signals. This property is further corroborated by designing encoding schemes as well as conditions ensuring exponential stability of the closed-loop system. On the other hand, when only the output channel is subject to network attack phenomena, the proposed structure can achieve exponential stabilization without acknowledgment (ACK) signals, in contrast to existing ACK-based results. Finally, a numerical example is given to demonstrate the practical merits of the proposed theoretical and practical approach.

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

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