Linear System Theory Research Articles

Robust Linear Discrete Control for a Hexacopter: Experimental Results

This paper presents a discrete-time robust linear control method for tracking a hexacopter's trajectory in the presence of external disturbances. The control of multi-rotor type unmanned aerial vehicles (UAVs) has gained considerable attention recently due to their various applications, such as crop spraying in precision agriculture. The control of UAVs requires robustness to reject disturbances and accommodate dynamic uncertainties. To achieve this goal, the robust discrete-time control action is designed in two stages. The first stage utilizes the solution of a difference Riccati-equation to guarantee system stability in an optimal sense. The second stage provides system robustness against external disturbances and uncertain dynamics. Furthermore, the Lyapunov stability theory for discrete linear systems is used to derive system asymptotic stability. Finally, experimental results of the hexacopter flight are provided to illustrate the effectiveness of the presented control law.

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.

On embedding the Koopmanization into controlled nonlinear systems, its comparison with the Carleman linearisation and concerning results: beyond the feedback linearisation

The feedback linearisation has found its striking applications in many practical situations for control problems that exploit the ideas from the linear system theory. Towards linearisation techniques for controlled nonlinear systems, operator-theoretic frameworks from functional analysis have received attentions as well. The Koopman operator is a linear, but an infinite dimensional operator. Most notably, the novelties of the paper are the following. (i) Sketching a rigorous and comprehensive proof of the Koopmanization of the controlled nonlinear system using the notion of the operator-invariant subspace and unifying the minimiser inequality formula with the Koopmanized system. The proof unfolds that the Koopmanization meets the bilinearisation in control context with the augmented state space. (ii) A formal construction of an observer encompassing the generalised Riccati equation in the Koopman setting of the original controlled nonlinear system. The Carleman version of the generalised Riccati equation is rephrased as well. (iii) Unfolding the usefulness of the Koopman observer. To achieve extensive illustrations of the theory of the paper, we construct two Koopman observers for two practical control problems, i.e. a controlled Duffing system with a cubic nonlinearity and a controlled system with a square nonlinearity. Then, we compare the Koopman observers with the Carleman observers.

Control approach to well-posedness and asymptotic behavior of a queueing system

In this paper, we investigate well-posedness and asymptotic behavior of an M/G/1 retrial queueing system characterized by reserved idle time and setup time. This system is intricately described by an infinite system of integro-partial differential equations. Our methodology is rooted in the feedback theory of well-posed and regular infinite-dimensional linear systems. Firstly, we convert the system into a control framework, incorporating both input and output variables. Subsequently, we establish its well-posedness by leveraging the feedback theory specific to regular infinite-dimensional linear systems. Secondly, by scrutinizing the spectral distribution of the system operator along the imaginary axis, we demonstrate that the time-evolving solution of the system converges robustly to its steady-state solution. Lastly, we also discuss the essential properties of the semigroup induced by the system operator, encompassing stability and hyperbolicity, thereby shedding light on its regularity.

Elastic flow of curves with partial free boundary

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Decoding of MDP Convolutional Codes over the Erasure Channel under Linear Systems Point of View

This paper attempts to highlight the decoding capabilities of MDP convolutional codes over the erasure channel by defining them as discrete linear dynamical systems, with which the controllability property and the observability characteristics of linear system theory can be applied, in particular those of output observability, easily described using matrix language. Those are viewed against the decoding capabilities of MDS block codes over the same channel. Not only is the time complexity better but the decoding capabilities are also increased with this approach because convolutional codes are more flexible in handling variable-length data streams than block codes, where they are fixed-length and less adaptable to varying data lengths without padding or other adjustments.

Efficient Vibration Measurement and Modal Shape Visualization Based on Dynamic Deviations of Structural Edge Profiles.

As a non-contact method, vision-based measurement for vibration extraction and modal parameter identification has attracted much attention. In most cases, artificial textures are crucial elements for visual tracking, and this feature limits the application of vision-based vibration measurement on textureless targets. As a computation technique for visualizing subtle variations in videos, the video magnification technique can analyze modal responses and visualize modal shapes, but the efficiency is low, and the processing results contain clipping artifacts. This paper proposes a novel method for the application of a modal test. In contrast to the deviation magnification that exaggerates subtle geometric deviations from only a single image, the proposed method extracts vibration signals with sub-pixel accuracy on edge positions by changing the perspective of deviations from space to timeline. Then, modal shapes are visualized by decoupling all spatial vibrations following the vibration theory of continuous linear systems. Without relying on artificial textures and motion magnification, the proposed method achieves high operating efficiency and avoids clipping artifacts. Finally, the effectiveness and practical value of the proposed method are validated by two laboratory experiments on a cantilever beam and an arch dam model.

Nonexistence of integrable nonlinear magnetic fields with invariants quadratic in momenta

Nonlinear, completely integrable Hamiltonian systems that serve as blueprints for novel particle accelerators at the intensity frontier are promising avenues for research, as Fermilab’s Integrable Optics Test Accelerator (IOTA) example clearly illustrates. Here, we show that only very limited generalizations are possible when no approximations in the underlying Hamiltonian or Maxwell equations are allowed, as was the case for IOTA. Specifically, no such systems exist with invariants quadratic in the momenta, precluding straightforward generalization of the Courant–Snyder theory of linear integrable systems in beam physics. We also conjecture that no such systems exist with invariants of higher degree in the momenta. This leaves solenoidal magnetic fields, including their nonlinear fringe fields, as the only completely integrable static magnetic fields, albeit with invariants that are linear in the momenta. The difficulties come from enforcing Maxwell equations; without constraints, we show that there are many solutions. In particular, we discover a previously unknown large family of integrable Hamiltonians.

Momentum and angular momentum reciprocity theorems for acoustic fields.

As a universal theorem applicable to linear physical systems, reciprocity theorems describe the reciprocity between the input and output of linear systems. As for the electromagnetic field, there have been detailed studies on the reciprocity theorems of energy, momentum, and angular momentum, which reveal the correlation between two electromagnetic field sources from different aspects. Based on the fundamentals of linear acoustic theory, the momentum-based and angular momentum-based reciprocity theorems for acoustic fields are derived. They indicate the correlation between two acoustic systems from the perspective of momentum and angular momentum, respectively. On this basis, a systematic summary and analogical study is also performed on various reciprocity theorems for acoustic and electromagnetic fields, which deepens our understanding of the reciprocity theorem as a universal law for linear physical fields. The results obtained in this work can further improve the reciprocity theory of linear acoustic systems, as well as provide theoretical guidance for practical applications, such as audio signal processing and acoustic measurement.

Complete Synchronization, Anti-Synchronization and Hybrid Synchronization of Non-Identical Chaotic Financial Systems

Chaos theory, being a branch of mathematics that deals with disordered or random-seeming mathematical systems, is receiving more attention in market-related circumstances as financial markets become more unstable and the level of unpredictability becomes more prevalent. Understanding the potential of chaos theory, its limitations, and its relationship to traditional economic theories is essential for anyone working in the finance sector. Chaos theory is ideally suited for comprehending the financial market, which is subject to both internal and external influences due to its high degree of instability and growing randomness. This work examines the complete synchronization, anti-synchronization and hybrid synchronization of two non-identical financial systems. The nonlinear active controllers are designed, and the error dynamics stability for each phenomenon is accomplished by two theoretical approaches - linear system theory and Lyapunov second method. Controllers are designed by using the relevant variables of drive and response systems in such a way that the error variables are stable. The controllers, when activated, enable the drive and response state variables to achieve identical dynamics despite starting from different initial conditions. Numerical simulations are performed using the ODE45 algorithm embedded in MATLAB software package to show the feasibility and effectiveness of the designed controllers.

State‐space realization of periodic input/output quaternionic linear systems

The investigation of linear periodic systems is a prominent subject of research within the domain of linear systems theory. In this context, the state‐space realization of such systems is of particular significance. In fact, after successfully treating the realization problem for time‐invariant systems, various researchers have directed their attention toward investigating the case of linear periodic systems. Meanwhile, in the late 1980s, Jan C. Willems proposed an approach that broadened the range of systems studied, now referred to as the behavioral approach. This approach views the behavior of a system as its fundamental element, including all signals that adhere to the system laws (also called system trajectories). More recently, the behavioral framework has also been extended separately to periodic and quaternionic behavioral systems. Our work consists of considering linear periodic input/output quaternionic behavioral systems and using recent developments in order to obtain quasi‐minimal and uniform state‐space realizations.

Operator-valued multiplier theorems for causal translation-invariant operators with applications to control theoretic input-output stability

AbstractWe prove an operator-valued Laplace multiplier theorem for causal translation-invariant linear operators which provides a characterization of continuity from $$H^\alpha ({\mathbb {R}},U)$$ H α ( R , U ) to $$H^\beta ({\mathbb {R}},U)$$ H β ( R , U ) (fractional U-valued Sobolev spaces, U a complex Hilbert space) in terms of a certain boundedness property of the transfer function (or symbol), an operator-valued holomorphic function on the right-half of the complex plane. We identify sufficient conditions under which this boundedness property is equivalent to a similar property of the boundary function of the transfer function. Under the assumption that U is separable, the Laplace multiplier theorem is used to derive a Fourier multiplier theorem. We provide an application to mathematical control theory, by developing a novel input-output stability framework for a large class of causal translation-invariant linear operators which refines existing input-output stability theories. Furthermore, we show how our work is linked to the theory of well-posed linear systems and to results on polynomial stability of operator semigroups. Several examples are discussed in some detail.

Аналіз точності самоналаштування динамічної моделі газотурбінного двигуна

One of the most promising directions in the development of aircraft engines is related to the introduction of adaptive automatic control systems (ACS). The defining element of these systems are dynamic mathematical models of engines capable of self-adjustment based on engine operating parameters measured in flight. A number of leading researchers have developed a concept of using such models called STORM (Self-Tuned On-board Real-time Model). However, in the corresponding works, clearly insufficient attention is paid to solving the problem of checking the sufficiency of the information used to ensure the necessary accuracy of the models. This check must be performed a priori (to predict the composition of engine operating modes, and the volume of registered information), as well as posteriori. The subject of this study is the process of forming dynamic mathematical models (MM) of gas turbine engines using real data for the subsequent use of these models to solve problems related to the control and diagnostics of on-board systems. The goal of this study is to determine the dependence of estimation errors of dynamic parameters of mathematical models on influencing factors. Tasks considered in the work: forming the structure of a mathematical model, dividing the identification process into stages according to the structure of the model (estimating the parameters that determine the static and dynamic properties of the object), forming the least-square functional for the assessment task, determining the errors in estimating dynamic coefficients, analyzing influencing factors, and determining dependencies between factors and errors. For this purpose, the methods of the theory of air-jet engines, the theory of linear dynamic systems, and statistical evaluation are used. The following results were obtained: a mathematical model of a turboshaft engine with a reciprocating gas generator was formed, and ratios were obtained that allow determining the errors in estimating the time constant of a reciprocating engine or gas generator. Scientific and practical innovation: for the first time, a ratio was obtained that determines the errors in estimating the time constant based on the specified values of the measurement errors, the intensity of the jump-like change in fuel consumption, and the frequency and duration of observation. These relations are presented in dimensionless coordinates, which makes them universal and able to be applied to any single-stroke turbojet engine or single-stroke gas generator during a priori or a posteriori analysis of results, as well as planning experiments and debugging on-board self-tuning algorithms of models.

NEW KIND OF IIR DIGITAL FILTERS INTENDED FOR PULSE PERIOD FILTERING

This paper describes a new kind of infinite impulse response (IIR) digital filter designed for pulse period filtering. The IIR digital filter is designed using a third-order IIR frequency locked loop (IIR FLL), which is based on the time measurement and processing of both, the input and output periods. A general form of the difference equation describing this type of IIR FLL of any order is developed and compared with the corresponding difference equation of classical digital filters. The mathematical analyses in the time domain were performed using the Z transform approach and the theory of linear discrete systems. An analysis of the IIR digital filter was performed in the time and frequency domain. The transfer functions and Z transform of the third-order IIR FLL outputs are developed. The main part of the article is devoted to designing the appropriate IIR FLL digital filter using the corresponding IIR FLL. For this purpose, the theory of IIR digital filter and the corresponding MATLAB tools are used, but taking into account the differences between these systems. The filtering abilities of the designed IIR FLL digital filter are demonstrated. Computer simulation of the designed IIR FLL is made in the time domain to enable precise insight into its properties.

A simple and scalable particle swarm optimization structure based on linear system theory

A simple and scalable particle swarm optimization structure based on linear system theory

Model-based linear control of nonlinear pneumatic soft bending actuators

Advanced model-based control techniques hold great promise for the precise control of pneumatic soft bending actuators (PSBAs) with strong nonlinearities. However, most previous controllers were designed in a cumbersome nonlinear form. Considering the simplicity of linear system theory, this paper presents a novel perspective on using model-based linear control to handle nonlinear PSBAs, and for the first time, summarizes two methodologies, global linearization and pseudo-linear construction. Derived from them, Koopman-based and hysteresis-based linear control approaches are proposed, respectively. For the former, a novel fusion prediction equation is developed to build a high-fidelity Koopman model, realizing global linearization, and then the linear model predictive control (MPC) is deployed. For the latter, the inverse of the generalized Prandtl–Ishlinskii (GPI) model cascades with the PSBA to construct a pseudo-linear system, eliminating the asymmetric hysteresis, which activates the linear proportional-integral-derivative (PID) control. It is worth noting that the above two are based on data-driven models adapted to various PSBAs with material and structural customization. Finally, the two model-based linear control approaches are verified and compared through a series of experiments. The results show that the proposed linear controls, with more concise designs, achieve comparable or even superior performance than nonlinear controls.

Chaotic control problem of BEC system based on Hartree–Fock mean field theory

Abstract Due to the difficulty of studying nonlinear quantum systems and the unique composition of Bose–Einstein condensate (BEC) systems, BECs face significant difficulties in solving dynamic analysis and chaotic control problems. Therefore, Hartree–Fock mean field theory is introduced to study the chaotic characteristics, control, and synchronization issues of BEC systems loaded on optical lattices. First, the stability and chaos of BECs in optical lattices were analyzed. Subsequently, constant shift method and activation control were introduced based on the Gross–Pitaevskii equation to achieve control and synchronization of the BEC system. Second, based on the Lyapunov exponent theory, offset parameters are added to BEC chaotic control to achieve control of particle density. Finally, based on the stability theory of linear systems, a control term is introduced to achieve variable analysis of the system’s drive–response system, ensuring that chaotic systems with different initial conditions can still achieve good synchronization and anti-synchronization control. The chaotic problem of BEC system was analyzed using numerical and theoretical methods in the experiment. The effect of adjusting the parameters of the BEC system under the constant shift method is significant. The system exhibits a chaotic state under the Lyapunov exponent, which is mainly concentrated between [3.4, 4.5], demonstrating good system stability. When the offset constant range is [4.21, 5.67], the maximum Lyapunov exponent value is below 0. In the problem of chaotic synchronization, adding activation control causes the system’s time series to exhibit anti-synchronization with spatiotemporal variable variation, while adding control terms leads the system to tend towards synchronization and anti-synchronization with time evolution. The analysis of chaotic control problems in BEC systems can provide reference value and theoretical basis for the dynamic research of quantum physics and related nonlinear systems.

Towards standardised design of wave energy converters: A high-fidelity modelling approach

Within this study, an analysis of the global standard panorama for wave energy converters (WECs) is presented, in order to develop design methodologies as close as possible to the state-of-the-art. In particular, an analysis of such international standards panorama exhibits a lack of information and detail regarding WEC system design, and the specific simulations procedures that shall be followed accordingly. In the light of this, this study proposes a standardised design process that can be followed to define the design loads which characterise a structural analysis, which are obtained by high-fidelity models. The device assessment in extreme states is analysed on intact and damaged conditions, where the use of a computational fluid dynamics (CFD) software is proposed to encompass any non-linear behaviour related to extreme events. Furthermore, this article also describes a fatigue assessment based on linear system theory, with the inclusion of linearised significant terms, such as, e.g. mooring influence on device dynamics.

Duality for Nonlinear Filtering I: Observability

This paper is concerned with the development and use of duality theory for a hidden Markov model (HMM) with white noise observations. The main contribution of this work is to introduce a backward stochastic differential equation (BSDE) as a dual control system. A key outcome is that stochastic observability (resp. detectability) of the HMM is expressed in dual terms: as controllability (resp. stabilizability) of the dual control system. All aspects of controllability, namely, definition of controllable space and controllability gramian, along with their properties and explicit formulae, are discussed. The proposed duality is shown to be an exact extension of the classical duality in linear systems theory. One can then relate and compare the linear and the nonlinear systems. A side-by-side summary of this relationship is given in a tabular form (Table II).

Analytical and numerical modelling of the electrostatic behaviour of highly insulating materials in the time domain

The usual response of an insulator subjected to a voltage step involves time power laws. We present here mathematical tools allowing to calculate this time domain response in open circuit after an initial charge deposit, within the framework of linear systems theory, using linear fractional transfer functions. In the time domain, the inverse Laplace transform of the data taken from the frequency domain leads to Mittag-Leffler functions, generalizing the Debye exponential response to an extended fractional α-order response. The open-circuit boundary conditions are different from the closed-circuit ones. We nevertheless demonstrate that using a transfer function deduced from the Cole-Cole response in closed-circuit, a precise analytical formula of the potential decay after an initial charge deposit may be established, and a numerical computation of this decay may be performed using easily available software. Applying the superposition principle, the voltage return following a brief short circuit may also be deduced. Experimental results are presented and the limits of the superposition principle applied to real materials are discussed.