State-space Form Research Articles

Data fusion based on short-term memory Kalman filtering using intermittent-displacement and acceleration signal with a time-varying bias

Accurate displacement information plays a vital role in various applications from structural health monitoring to damage detection in physical systems. Due to practical reasons, displacement sensing is often performed using sensors that provide less accuracy or a low sampling rate. This study proposes an algorithm for displacement signal estimation that fuses high sampling rate acceleration data (which consists of unknown time-varying bias) with low sampling rate displacement data. In contrast to the present literature, the state vector in the proposed state-space formulation consists of short-term memory of the past realizations of displacement and velocity, and bias terms which allow (i) online estimation of the displacements by avoiding multi-rate Kalman filtering, and (ii) estimate time-varying bias in the measured acceleration signal. The state equations comprise the finite difference approximations of the time derivative of velocity and displacement which act as a moving average filter to provide smooth estimates, thus preventing the need for offline smoothing procedures. By inclusion of bias terms in the state vector, the drifts in displacement and velocity caused due to time-varying bias in measured acceleration are easily eliminated. The proposed method was found to provide accurate estimates of displacement, velocity, and time-varying bias in measured acceleration signals when tested on different types of bias, including but not limited to constant, linear, and parabolic variations. Additional tests on three oscillators subjected to three earthquake ground motions showed significantly better state estimates for stiff oscillators and improved estimates for flexible oscillators. Experimental validation on a base-excited two-story frame further affirms the capability of the method to estimate the hidden bias in the measured acceleration and provide an accurate estimate of displacement.

A novel approach to model nance drying considering its shrinkage

Air drying is a viable preservation method to boost the accessibility of nance, a seasonal drupe with very short shelf life; however, information about nance drying is scarce, especially from a mass transfer (MT) standpoint. Besides, the lack of simple models considering the stone fraction (radius of inner stone/radius of nance) and the mesocarp shrinkage in drupes has caused most investigations rely on sphere and cube geometries, contributing to the unreliable estimation of MT properties. This study developed a discrete state-space (SS) formulation with analytical solution to describe nance drying which, unlike a variable separation (VS) solution, can consider a variable stone fraction, caused by outer boundary moving while flesh-pit interface remains still. Nance fruits were air dried (60, 70 and 80 °C) to obtain their moisture kinetics and shrinking behavior. Drying data were used to estimate water diffusivity (D) under rigid (VS and SS solutions) and shrinkable (SS and numerical solution) mesocarp assumptions. Nances lost about 60% of their initial volume at the end of drying, while stone fraction increased from 0.44 to 0.62. D values ranged from 5.3 to 6.7 (×10–10) m2/s with SS approach and variable stone fraction. The major source of error in D was not the formulation (SS or VS) when a rigid mesocarp was considered, but the values assigned to the nance radius (length for diffusion) and stone fraction, an initial or average value along process. SS diffusivities did not differ to those estimated with the full numerical solution under variable stone fraction (p>0.05), while VS parameters were overestimated by 44–46 % as a minimum. SS formulation can be applied to hollow cylinders and other geometries with 2D or 3D MT, while able to include non-isotropic shrinkage and evolving external conditions, a problem normally reserved to pure numerical implementations, offering substantial advantages over VS equations.

Networked Evolutionary Game-Based Demand Response via Feedback Controls

In this paper, the demand response considering interactive decision making between residential users and utility companies is modelled and controlled using networked evolutionary game (NEG) theory. The NEG is achieved via a widely used mathematical tool, namely the semi-tensor product (STP) of matrices, through which, feedback controls can be implemented to dynamically regulate the game-based networks. Firstly, the dynamic interactions between utility companies who provide various energy consumption packs and residential users who can intelligently choose preferable packs are modelled in the state-space form, in which both sides are consistently pursuing their maximum payoffs. Then, a quantitive analysis of such system’s equilibria is conducted using rigorous mathematical derivations, followed by the design of a profile feedback controller when the existing equilibria are not satisfactory towards the required energy consumption. Finally, an illustrative example is demonstrated, where the dynamic gaming between utility companies and users is illustrated and the effectiveness of the proposed feedback control is validated. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper addresses a very practical engineering problem, which is the demand response in the modern smart grid. Different from general optimization approaches in the existing works, the method proposed by this paper improves the demand response via feedback control of networked systems, which is based on the rigorous matrix approaches via state space equations. In addition, the work in this paper considers very practical factors in actual engineering (i.e., users with different action choices and decision logics being all together), which also provides much value to practitioners than the relevant works with similar theoretical approaches. The method proposed in this paper provides an effective regulation of the demand response in practice and can be flexibly adjusted according to actual situations.

Stability analysis and vibration suppression in an overhung rotor system using active magnetic damper

The present work demonstrates the effectiveness of an active magnetic damper (AMD) in improving the vibration characteristics of an overhung rotor system (ORS) considering the effects of synchronous mass-unbalance, rotor bow, and viscous internal damping. A finite element-based rotordynamic analysis has been conducted to evaluate the system response using state-space formulation. MATLAB codes are developed for the numerical implementation of the rotordynamic model pertaining to the flexible ORS integrated with an AMD consisting of three parallel feedback loops. The internal damping is found to reduce the vibration amplitudes at the bearing and disk locations. However, it leads to system instability in the absence of AMD beyond the first critical speed (670 RPM) which is indicated by the change in the real part of eigenvalues. Under these circumstances, the use of AMD causes a remarkable improvement and the system stabilizes for almost the entire speed range considered here, 320–10,000 RPM. In addition, the AMD causes a maximum reduction of 95.8%, 96.5%, and 83.3% in the vibration amplitudes at the locations of Bearing 1, Bearing 2, and the overhung disk, respectively. Using an appropriate set of AMD parameters, the system characteristics can be altered effectively as per the requirement.

State-Space Formulation for Buckling and Free Vibration of Axially Functionally Graded Graphene Reinforced Nanocomposite Microbeam under Axially Varying Loads.

This paper focuses on the size-dependent free vibration and buckling behaviors of the axially functionally graded (AFG) graphene platelets (GPLs) reinforced nanocomposite microbeams subjected to axially varying loads (AVLs). With various axial grading patterns, the GPL nano-reinforcements are distributed throughout the polymer matrix against microbeam length, and the improved Halpin-Tsai micromechanics model and the rule of mixture are adopted to evaluate the effective material properties. Eigenvalue equations of the microbeams governing the static stability and vibration are derived based on the modified couple stress Euler-Bernoulli beam theory via the state-space method, and are analytically solved with the discrete equilong segment model. The effects of axial distribution patterns, weight fraction, and geometric parameters of GPLs, as well as different types of AVLs, on the size-dependent buckling load and natural frequency are scrutinized in detail. The results show that the synchronized axial distributions of GPLs and AVLs could improve the buckling resistance and natural frequency more powerfully.

Automated tight Lyapunov analysis for first-order methods

AbstractWe present a methodology for establishing the existence of quadratic Lyapunov inequalities for a wide range of first-order methods used to solve convex optimization problems. In particular, we consider (i) classes of optimization problems of finite-sum form with (possibly strongly) convex and possibly smooth functional components, (ii) first-order methods that can be written as a linear system on state-space form in feedback interconnection with the subdifferentials of the functional components of the objective function, and (iii) quadratic Lyapunov inequalities that can be used to draw convergence conclusions. We present a necessary and sufficient condition for the existence of a quadratic Lyapunov inequality within a predefined class of Lyapunov inequalities, which amounts to solving a small-sized semidefinite program. We showcase our methodology on several first-order methods that fit the framework. Most notably, our methodology allows us to significantly extend the region of parameter choices that allow for duality gap convergence in the Chambolle–Pock method when the linear operator is the identity mapping.

Genetically Tuned Linear Quadratic Regulator for Trajectory Tracking of a Quadrotor

In this paper, a linear quadratic regulator (LQR) controller operating according to the genetically tuned inner-outer loop structure is proposed for trajectory tracking of a quadrotor. Setting the parameters of a linear controller operating according to the inner-outer loop structure is a matter that requires profound expertise. Optimization algorithms are used to cope with the solution of this problem. First, the dynamic equations of motion of the quadrotor are obtained and modelled in state-space form. The LQR controller, which will operate according to the inner-outer loop structure in the MATLAB/Simulink environment, has been developed separately for 6 degrees of freedom (DOF) of the quadrotor. Since adjusting these parameters will take a long time, a genetic algorithm has been used at this point. The LQR controller with optimized coefficients and a proposed LQR controller-based study in the literature are evaluated according to their success in following the reference trajectory and their responses to specific control inputs. According to the results obtained, it was observed that the genetically adjusted LQR controller produced more successful outcomes.

Robust state and output feedback stabilization for Lipschitz nonlinear systems in reciprocal state space: Design and real-time validation

In many engineering problems, modelling the state and output derivative variables in reciprocal state space form can be generated more easily than in standard state space one. The formulation of a robust [Formula: see text] control problem using feedback principle for continuous Lipschitz nonlinear systems with uncertainties in reciprocal state space is presented in this article. In contrast to the existing approaches, the considered model is affected by unknown disturbances, parameter uncertainties and derivative Lipschitz nonlinearities. The asymptotic stability based on the proper Lyapunov functions of the closed-loop system is guaranteed. The [Formula: see text] control design resolution problem is ensured through the linear matrix inequality technique, the lemmas and the use of new variables with S-procedure. The performance of the proposed approach is shown through the experimental results using real-time implementation with a digital signal processing device (DSpace DS 1104).

A Lean Path for Learning Control in Undergraduate Electronics Programmes

In recent years, the development and spread of machine learning, robotics and other novel application areas, made indispensable to include such topics in undergraduate programmes for students in the area of electrical and computer engineering. However, this entails that traditional subjects must be covered in less time while maintaining their relevant topics for getting an engineering diploma. In this paper, one presents an approach to streamline the learning and teaching of automatic control. As surveyed in the paper, many universities divide the study of control theory in an initial course with system theoretic concepts followed by two other dedicated to linear systems. Various approaches appear on how to divide the topics over these two units. The study of single-input single-output systems defined by differential equations and the study of transfer functions can be argued to be mainly rooted on the need to introduce the Proportional Integral Derivative (PID) controller. In this paper, motivated by the concept of differential observation, we propose a new organization of the control curricula where a first course on systems theory gathers all relevant general tools and a second course, specific to automatic control, presents both continuous and discrete-time linear control systems in the state-space formulation together with ways to get working control systems in face of actuator saturation, measurement noise and parametric uncertainty. In this approach, PID controllers are presented as a special case of state feedback. The structure can also be useful for Universities developing their curricula in countries where control theory teaching is still being developed.

Space-Filling Input Design for Nonlinear State-Space Identification

The quality of a model resulting from (black-box) system identification is highly dependent on the quality of the data that is used during the identification procedure. Designing experiments for linear time-invariant systems is well understood and mainly focuses on the power spectrum of the input signal. Performing experiment design for nonlinear system identification on the other hand remains an open challenge as informativity of the data depends both on the frequency-domain content and on the time-domain evolution of the input signal. Furthermore, as nonlinear system identification is much more sensitive to modeling and extrapolation errors, having experiments that explore the considered operation range of interest is of high importance. Hence, this paper focuses on designing space-filling experiments i.e., experiments that cover the full operation range of interest, for nonlinear dynamical systems that can be represented in a state-space form using a broad set of input signals. The presented experiment design approach can straightforwardly be extended to a wider range of system classes (e.g., NARMAX). The effectiveness of the proposed approach is illustrated on the experiment design for a nonlinear mass-spring-damper system, using a multisine input signal.

Identification of Linear Dynamic Systems in State Sate

System identification has been growing in an engineering community over the last 60 years, and an intensive research has been published in this field. This study presents an identification method of a linear time-invariant continuous dynamic system directly from time series data. The considered system is represented in a state space form. All states are assumed to be measured. The proposed identification algorithm is demonstrated and investigated, with noise-free data and noisy data, on a MATLAB simulation environment. The results confirm that the simple procedures of the proposed algorithm give an effective and successful estimation of the system parameter matrices.

State-Space Approaches to Complex Structures in Aerospace

The state-space approach (SSA), traditionally utilized in modern control theory, has been successfully adopted over the last three decades to investigate the mechanical behaviors of complex structures composed of composite or smart materials. This is largely due to their increasing application across various fields, including aerospace, civil and marine engineering, and transportation vehicles. This paper provides a comprehensive review of the establishment of state-space formulations for structures of typical configurations, such as beams, plates, shells, and trusses, with a particular focus on their applications in the mechanical analyses of various complex aerospace or smart structures using the transfer matrix method. The paper first summarizes the three-dimensional SSAs applied to laminated structures without any assumptions on physical fields. By employing structural theories such as various beam, plate, and shell theories, simplified one-dimensional and two-dimensional SSAs for laminated structures are developed. The paper then outlines the advances in generating analytical solutions for the mechanical behaviors of laminated structures. For the sake of completeness, the paper also provides an account of SSAs applied to complex periodic structures, particularly in beam and truss forms. To overcome the limitations of conventional SSAs related to structures with specialized geometric configuration or under arbitrary boundary conditions, state-space based numerical methods have been proposed, for example, the state-space based differential quadrature method and state-space based finite-element method. The applications of these methods in the analyses of static and dynamic responses of complex structures are extensively reviewed. It is observed that there are still intriguing and potential research topics for the development of advanced SSAs with enhanced versatility and the studies on practical complex structures used in modern engineering, particularly in aerospace industry. Therefore, this review is expected to be beneficial for researchers in the fields of analytical and numerical methods, composite structures, aerospace, structural engineering, and more.

Identification of modal parameters and relaxation parameter for exponential non-viscous damping system using stochastic subspace method

Non-viscous damping parameters can be identified approximately based on the frequency response function of structures. However, it is difficult or expensive to perform vibration test and layout enough amount measured points for structures. The stochastic subspace method can identify modal damping parameters only using responses data from partial measured points. This paper extends the stochastic subspace method from the viscous damping system to non-viscous damping system. In addition, the eigenvalues of the second-order form equation of motion are proved equally to the state space formulation. The identified procedure of the proposed method is given in detail. Finally, the method is validated using three numerical examples for assessing its accuracy.

Staying positive: challenges and solutions in using pure multiplicative ETS models

Abstract Accepted by: Zied Babai Exponential smoothing in state space form (ETS) is a popular forecasting technique, widely used in research and practice. While the additive error ETS models have been well studied, the multiplicative error ones have received much less attention in forecasting literature. Still, these models can be useful in cases, when one deals with positive data, because they are supposed to work in such situations. Unfortunately, the classical assumption of normality for the error term might break this property and lead to non-positive forecasts on positive data. In order to address this issue, we propose using Log-Normal, Gamma and Inverse Gaussian distributions, which are defined for positive values only. We demonstrate what happens with ETS(M,*,*) models in this case, discuss conditional moments of ETS with these distribution and show that they are more natural for the models than the Normal one. We conduct the simulation experiments in order to study the bias introduced by point forecasts in these models and then compare the models with different distributions. We finish the paper with an example of application, showing how pure multiplicative ETS with a positive distribution works.

Hybrid Fuzzy PID Sound Radiation Control of a Functionally Graded Porous GPL-Reinforced Plate with Piezoelectric Sensor and Actuator Layers

This paper is the first attempt to investigate the sound radiation control of a functionally graded porous graphene platelet-reinforced composite (FGP-GPLRC) plate integrated with piezoelectric face sheets. A spatial state-space formulation based on the linear 3D piezoelasticity theory is utilized to derive a semi-analytic solution for the coupled vibroacoustic response of the simply-supported arbitrarily thick composite plate. A comprehensive parametric study examines the effect of GPLs’ distribution, weight fraction, and porosity parameters of the core layer on the radiated sound power. Then, the best material configuration is selected to be actively controlled. A hybrid Fuzzy Proportional-Integral-Derivative (FPID) controller, which utilizes an FPID controller integrated with a classical PID controller, is employed to mitigate sound radiation from the smart sandwich plate via the volume velocity approach. The gain parameters of the proposed controller are tuned using the Bat optimization algorithm. Finally, the performance of the active control strategy in attenuating radiated sound power is investigated through several numerical simulations. Results show the proposed control scheme's robustness and rapid disturbance rejection performance.

A new time integration method based on state formulations for dynamic analysis of nonviscously damped systems

The mode superposition method (MSM), which is used for dynamic response calculations of nonviscously damped systems, requires state–space eigensolutions, which incur a high computational cost. This study proposed an alternative symmetric formulation-based time integration method for analyzing structural dynamic systems. An anelastic displacement field (ADF) model was introduced to capture damping mechanisms in nonviscous systems. The differential equation describing the ADF model was transformed into an augmented coupled system via the introduction of dissipative variables. Two symmetric state formulations based on coupled augmented systems were derived for nonviscous systems. Consequently, alternative asymmetric state formulations were derived for nonviscous systems involving the ADF model. Thereafter, a new method was developed based on these state–space formulations by assuming linear approximations. A numerical implementation of the new method for dynamic systems was demonstrated. Three benchmark examples of discrete and continuous systems were considered to evaluate the asymmetric formulation and the computational efficiency, accuracy, stability, and time-step-size sensitivity of the new method. Further, the response of the new method was validated using MSM and precise time integration (PTI) methods. An asymmetric state formulation was also validated against MSM and direct frequency response methods. The results revealed that the new method accurately matched the MSM and PTI methods at various time steps. The results also showed that the new method was more efficient than the MSM and PTI methods, rendering it suitable for solving large-scale structural dynamics problems.

An Investigation of Stability, Controllability and Observability of a Three Degree of Freedom Translational Mechanical System using State Space Approach

State-space modelling approach which is essentially time-domain developed in the late 1960s is a new approach to the analysis and design of complex control systems. Several researches have carried out modelling and analysis of mechanical systems with a number of degrees of freedom of movement using state space approach among which is the work of Sivak and Darina (2012) who modelled a system with two degrees of freedom. In this paper, therefore, we provide an extension of the work of Sivak and Darina to model and analyse a three degree of freedom translational mechanical system using state-space approach. The system was first presented in equivalent free body diagrams, then Newton's second law of motion was used to derive its equations of motion. The State-space formulation in the controllable canonical form obtained from the time-domain differential equations is adopted and the Laplace transform method was used in the analysis to determine the poles (natural frequencies) of the system using two numerical examples in MATLAB software. The software was also used to determine the controllability and observability matrices of the system. The stability, controllability and observability of the system were then discussed from the poles, controllability matrix and observability matrix respectively. Tables of numerical values and line graphs were used to present the results obtained from the analysis. Finally, in these results, the system was found to be stable, controllable and observable suggesting that a state feedback ( pole placement ) control design for the system is possible.

Identification of polynomial nonlinear systems based on center manifold

A center-manifold (CM) based system identification (SID) method is developed to identify multi-input-multi-output continuous-time polynomial nonlinear systems in the state–space form. On one hand, the CM method is a natural extension of the frequency-domain subspace method to nonlinear SID problems. It decomposes the nonlinear SID problem into a series of linear least-square problems (in a precise sense), which can be solved analytically. This is in contrast to many existing methods that rely on iterative search of the global minimum. On the other hand, it is also a generalization of the invariant-subspace (ISP) based SID method. It inherits the nice property of the ISP method that the advantages of time-domain and frequency-domain SID methods are combined. In the paper, a persistence of excitation (PE) condition for the CM method is derived, which is described by the rank of certain Khatri–Rao product. Under the PE condition, the identification error is shown to be bounded within an arbitrarily small neighborhood of the origin when measurement noise is presented.

System modeling and temperature control for a fuel cell system based on local model networks

ABSTRACT Fuel cells have been studied for use in stationary power generation and vehicle propulsion systems. The fuel cell thermal management subsystem is coupled and nonlinear, posing challenges for modeling and temperature control. This paper aims to integrate the physical models of the fuel cell stack, pump, thermostat, and other components combined with intelligent algorithms into an efficient system-level thermal management model framework and develop a model predictive controller to solve the temperature control problem. First, a physics-based nonlinear model of the fuel cell system is developed and used as a basis to identify the linearized model for different operating points. Then, the global model is obtained by fusing the local models with Gaussian validity functions using the local linear model tree method. Third, a multi-step prediction model is derived based on the local model networks, and a parameterized linear state space form is obtained and used for controller design. Furthermore, an online correction method is developed to reduce the model discrepancy. Finally, the accuracy of the system model and the performance of the proposed controller are verified by open-loop experimental data and a series of closed-loop simulation cases.

Generalized, Complete and Accurate Modeling of Non-Ideal Push–Pull Converters for Power System Analysis and Control

Power converters are a basic element for the control and design of any power electronic system. Among the many available topologies, the push–pull converter is widely used due to its versatility, safety and efficiency. For its correct analysis, sizing, simulation and control, models that meet the characteristics of generality, accuracy and simplicity are required, especially if its control is to be optimized by means of some analytical technique. This requires models that consider the practical non-idealities intrinsic to the converter, as well as being intuitive and easy to handle analytically in a control loop. In general, the models reviewed in the scientific literature adopt simplifications in their definition that are detrimental to their accuracy. In response to the posed problem, this work presents a generalized, complete, accurate and versatile model of real (non-ideal) push–pull converters, ideal for the analysis, simulation, and control of power systems. Following the premise of general and complete converters, the proposed model includes all the practical non-idealities of the converter elements, and it is accurate because it faithfully reflects its dynamics. Furthermore, the model is versatile, as its state space formulation allows for its easy adaptability to the converter operating conditions (voltage, current and temperature) for each sampling time. Also, the model is excellent for use in model-based control techniques, as well as for making very accurate simulators. The behavior of the developed model has been contrasted with a real push–pull converter, as well as with reference models present in the scientific literature for both dynamic and steady-state response tests. The results show excellent performance in all the studied cases, with behavior faithful to the real converter and with relative errors that are much lower than those obtained for the reference models. It follows that the model behaves like a digital twin of a real push–pull converter.