Time-invariant State-space Models Research Articles

Linear Time-Invariant Models of a Large Cumulus Ensemble

Abstract Methods in system identification are used to obtain linear time-invariant state-space models that describe how horizontal averages of temperature and humidity of a large cumulus ensemble evolve with time under small forcing. The cumulus ensemble studied here is simulated with cloud-system-resolving models in radiative–convective equilibrium. The identified models extend steady-state linear response functions used in past studies and provide accurate descriptions of the transfer function, the noise model, and the behavior of cumulus convection when coupled with two-dimensional gravity waves. A novel procedure is developed to convert the state-space models into an interpretable form, which is used to elucidate and quantify memory in cumulus convection. The linear problem studied here serves as a useful reference point for more general efforts to obtain data-driven and interpretable parameterizations of cumulus convection.

Proportional-Integral Control and ARX-based Model Predictive Control for Extractive Dividing Wall Column

Extractive dividing wall column (EDWC) is a typical process intensification technology and possesses significant energy-saving prospects, while the poor dynamic controllability hinders its industrialization goal. In this work, taking the benzene and cyclohexene azeotrope separation as an example, a proportional-integral (PI) control structure based on four-point temperature control loops is proposed for EDWC by considering temperature difference as the discriminant criterion. Subsequently, to improve the control performance of EDWC, two model predictive control (MPC) structures based on auto-regressive with external input (ARX) model obtained from closed-loop identification and linear time-invariant state-space (LTI-SS) model, respectively, are studied for EDWC. The performance of three control structures is tested by ±20 % feed flow rate and composition disturbances, and further quantitatively evaluated by integral absolute error (IAE). The dynamic responses demonstrate that the ARX-based MPC structure achieves the best control performance in the presence of feed flow rate and composition disturbances, which can be attributed to the capability of the identified ARX model to track the dynamic features of EDWC accurately.

System Identification of Unmanned Air Systems at Texas A&M University

This paper presents a summary of system identification flight testing and results for a variety of large and small fixed-wing and multirotor unmanned air systems at Texas A&M University from 1999 to 2023. The six different types of vehicles range from a large powered parafoil, to a fixed-wing vehicle with synthetic jet-actuated roll control effectors, to a radially asymmetric multirotor, to large and small fixed-wing vehicles, and to a Steppe eagle. The observer/Kalman filter identification algorithm is used to generate linear time-invariant state-space models, and the results for both near-real-time online model generation and postflight offline model generation are presented. The use and efficacy of a variety of test input types and their sensitivity to exogenous inputs such as turbulence, in addition to identified model evaluation and selection criteria, are discussed. Several generations of low size, weight, power, and cost flight-test instrumentation including the Developmental Flight-Test Instrumentation data acquisition package are also presented. Challenges that arose from the flight-testing campaigns along with solutions are highlighted in the paper.

An Automatable Approach for Small-Signal Stability Analysis of Power Electronic-Based Power Systems Using EMT Companion Circuits

The increasing presence of power electronic (PE) controlled devices in power systems has widened the bandwidth of transients to be studied for grid stability. While these transients are normally studied using time-domain simulation (TDS), wide bandwidth linear time-invariant (LTI) state-space models are a useful complement to TDS for diagnostics, controller design and parametric analysis. A general automatable method to compute an LTI sampled-data model of a PE-based circuit is presented in this paper. The proposed method is based on the techniques employed in electro-magnetic-transient (EMT) programs to perform TDS of PE-based circuits. The method uses the building blocks of EMT nodal formulation such as conductance matrix of companion circuit and its LU factors computed during TDS, interpolation and chatter elimination corrections, and EMT model of the firing angle control system. The method yields a small-signal LTI sampled-data model which is amenable to eigenanalysis and is illustrated using simple PE-based circuits such as a buck converter and a grid-connected STATCOM with closed-loop control.

Design, implementation and performance evaluation of multi-function boost converter

This paper introduces a single-phase multi-function converter (MFC) circuit that can perform the power electronics functions of a DC–DC, a DC–AC, an AC–AC, and an AC–DC boost converter. The power circuit of the proposed converter contains only four bi-directional switches, which is sufficient side by side with a switched logic-based control system to execute converter functions. The proposed converter can implement the boosting converter function, which is a drawback of the conventional single-phase matrix converter, which is limited to performing the buck converter functions. The mathematical analysis of each converter function is presented. A state space averaging technique is employed to obtain the time-invariant state space model for a better understanding of the converter characteristics. The frequency response approach is used to demonstrate the effect of circuit parameters on converter stability using the time-invariant state-space model. Moreover, the performance analysis of the system is checked through the building of a simulation model of the proposed converter via MATLAB/Simulink and MATLAB/Simscape toolbox integration. Additionally, the actual performance of the proposed converter is verified through a laboratory prototype experimental study. The experimental results confirm those obtained by computer simulation, clearly demonstrating the successful operation of the proposed circuit as a multi-function converter with acceptable levels of total harmonic distortion (THD).

A physics-based domain adaptation framework for modeling and forecasting building energy systems

Abstract State-of-the-art machine-learning-based models are a popular choice for modeling and forecasting energy behavior in buildings because given enough data, they are good at finding spatiotemporal patterns and structures even in scenarios where the complexity prohibits analytical descriptions. However, their architecture typically does not hold physical correspondence to mechanistic structures linked with governing physical phenomena. As a result, their ability to successfully generalize for unobserved timesteps depends on the representativeness of the dynamics underlying the observed system in the data, which is difficult to guarantee in real-world engineering problems such as control and energy management in digital twins. In response, we present a framework that combines lumped-parameter models in the form of linear time-invariant (LTI) state-space models (SSMs) with unsupervised reduced-order modeling in a subspace-based domain adaptation (SDA) approach, which is a type of transfer-learning (TL) technique. Traditionally, SDA is adopted for exploiting labeled data from one domain to predict in a different but related target domain for which labeled data is limited. We introduced a novel SDA approach where instead of labeled data, we leverage the geometric structure of the LTI SSM governed by well-known heat transfer ordinary differential equations to forecast for unobserved timesteps beyond available measurement data by geometrically aligning the physics-derived and data-derived embedded subspaces closer together. In this initial exploration, we evaluate the physics-based SDA framework on a demonstrative heat conduction scenario by varying the thermophysical properties of the source and target systems to demonstrate the transferability of mechanistic models from physics to observed measurement data.

Study on Efficient Sensor Node Selection for Observability Gramian Optimization⋆

This study attempts a practical comparison of optimization methods for sensor node selection to efficiently monitor large-scale dynamical systems represented by linear time-invariant state space models. Sensor measurements are evaluated based on an observability measure, the matrix determinant of the observability Gramian. This study confirms the applicability of selection strategies, namely, a convex relaxation method using semidefinite programming, a greedy maximization and its approximation that considers the gradient of the observability measure. Examples based on numerical and real-world experiments illustrate the effectiveness of the selection algorithms in terms of their optimization measures and the run time for the selection.

On the herdability of linear time-invariant systems with special topological structures

In this paper we investigate the herdability property, namely the capability of a system to be driven towards the (interior of the) positive orthant, for linear time-invariant state space models. Herdability of certain matrix pairs (A,B), where A is the adjacency matrix of a multi-agent network, and B a selection matrix that singles out a subset of the agents (the “network leaders”), is explored. The cases when the graph associated with A, G(A), is directed and clustering balanced (in particular, structurally balanced), or it has a tree topology and there is a single leader, are investigated.

Simplified complex-valued modal model for operating wind turbines through aerodynamic decoupling and multi-blade coordinate transformation

An efficient modelling methodology for steady-state operating wind turbines is proposed, combining aerodynamic decoupling, multi-blade coordinate transformation, and modal reduction. This leads to a complex-valued, reduced-order modal model for prediction of dynamic responses of the tower-rotor-blade wind turbine system, considering rotor rotation, blade flexibility, and vibration coupling between rotor and tower. A fully coupled finite element model was first developed, with aerodynamic forces linearised and expressed as a viscous aerodynamic damping matrix. A time-invariant state space model is formed using multi-blade coordinate transformation, allowing standard modal analysis. The complex-valued eigenvalues and mode shapes were obtained, and it is shown that the operating wind turbine modes exhibit a combination of tower and blade vibrations. Various degrees of modal reduction are applied to the state space model to obtain a modal model with fewer degrees of freedom, whose performance was evaluated in the time and frequency domain for operating wind turbines in normal condition. The displacement and stress responses are in close agreement with those of the fully coupled model with the first 21 modes included. The model already performs well with 8 modes considered to capture relevant fundamental frequency peaks. This allows significantly reduced computational effort and can be particularly beneficial for fatigue prediction, reliability analysis, and structural identification.

Modeling and control of a class of urban air mobility tiltrotor aircraft

This paper aims at developing a control-centric analytical formulation for aeromechanics and flight dynamics of urban air mobility (UAM) vehicles. Such vehicles feature a hybrid configuration with tiltrotors and fixed wings, enabling vertical takeoff/landing capability while attaining level flight range and endurance. A comprehensive nonlinear rigid-body dynamic model is developed by incorporating multiple tiltrotor dynamics and their gyroscopic and inertial coupling effects. A quasi-steady aerodynamic formulation is used to derive the aerodynamic loads on all lifting surfaces. In addition to the conventional control surfaces of the fixed-wing aircraft, the tiltrotor angular positions and rotational speeds are also considered additional control inputs in the derivation of nonlinear flight dynamic equations. These nonlinear equations are then linearized with respect to a set of trimmed conditions to generate the corresponding linear time-invariant state-space models, which are best suited for flight control design. In particular, to capture the critical dynamical behaviors during the transition from vertical to forward flight, the linear models are attained at various angular positions of the tiltrotors. These linear models are then utilized to develop a linear parameter-varying (LPV) model in which the tiltrotor angular position is considered the scheduling parameter. Subsequently, the adaptive model predictive control (MPC) methodology is used to design the flight controllers to achieve a stable and smooth transition flight. The numerical simulations demonstrate the efficacy of the proposed modeling and control approach.

Discretization and Model Reduction Error Estimation of Interconnected Dynamical Systems

In this paper a novel approach for estimating both discretization and model order reduction error is presented. Discretization is carried out using finite element method. Finite element models are converted to linear time-invariant state space models. Using the generalized square root balanced truncation method a series of reduced models are obtained. It is shown that for special class of passive and dissipative dynamical systems upper a-posteriori reduction error bound can be modeled less conservatively. Estimation of both discretization and model reduction error is further refined using v-gap metric. Quality of the obtained reduced order model is verified in the frequency domain. The proposed error estimation approach is demonstrated on a numerical example.

Maximum—A Posteriori Estimation of Linear Time-Invariant State-Space Models via Efficient Monte-Carlo Sampling

Abstract This article addresses maximum-a-posteriori (MAP) estimation of linear time-invariant state-space (LTI-SS) models. The joint posterior distribution of the model matrices and the unknown state sequence is approximated by using Rao-Blackwellized Monte-Carlo sampling algorithms. Specifically, the conditional distribution of the state sequence given the model parameters is derived analytically, while only the marginal posterior distribution of the model matrices is approximated using a Metropolis-Hastings Markov Chain Monte-Carlo sampler. From the joint distribution, MAP estimates of the unknown model matrices as well as the state sequence are computed. The performance of the proposed algorithm is demonstrated on a numerical example and on a real laboratory benchmark dataset of a hair dryer process.

Optimal experimental design for linear time invariant state–space models

The linear time invariant state–space model representation is common to systems from several areas ranging from engineering to biochemistry. We address the problem of systematic optimal experimental design for this class of model. We consider two distinct scenarios: (i) steady-state model representations and (ii) dynamic models described by discrete-time representations. We use our approach to construct locally D-optimal designs by incorporating the calculation of the determinant of the Fisher Information Matrix and the parametric sensitivity computation in a Nonlinear Programming formulation. A global optimization solver handles the resulting numerical problem. The Fisher Information Matrix at convergence is used to determine model identifiability. We apply the methodology proposed to find approximate and exact optimal experimental designs for static and dynamic experiments for models representing a biochemical reaction network where the experimental purpose is to estimate kinetic constants.

Scalable design methods for online data‐driven wide‐area control of power systems

A novel online system-identification-based control design for damping inter-area oscillations in power systems using supplementary wide-area excitation control is presented. The identification is based on a partly reduced-order and a partly full-order model of the grid. Assuming the grid to be divided into coherent areas with phasor measurement units located across the grid, the coherent states of the generators are averaged while the non-coherent states, such as the internal states of each power system stabilizer, are not. A hybrid state vector consisting of these averaged and non-averaged states is computed, and used for identifying a hybrid linear time-invariant state-space model. A linear quadratic regulator based on this hybrid model is, thereafter, designed. The reduced-order part of the model is found to save significant amount of online time for both learning and control, while the full-order part serves to enhance closed-loop stability. N4SID with randomized singular value decomposition (rSVD) is used for making the identification loop fast. The reduced-dimensional controller is finally implemented using a broadcast control strategy. The design is extended to non-linear models of power systems using Carleman bilinearization. Results for linear and bilinear control designs are validated using the IEEE 68-bus power system model.

Artificial Neural Network Hysteresis Operators for the Identification of Hammerstein Hysteretic Systems

Abstract This paper introduces explicit neural representations of fundamental hysteresis operators such as the play and stop operators. The hysteresis neurons are represented by recurrent artificial neural networks (ANN) using classical activation functions and are trained with gradient-based learning algorithms. The hysteresis neurons are combined into a single ANN hysteretic layer which, from a mathematical point of view, is equivalent to a classical Prandtl-Ishlinskii model expression, but has all the advantages of ANNs. One such benefit is the flexibility it offers to be combined with other ANN layers to obtain various complex model structures. In this paper, this is illustrated by combining it with a linear recurrent neural network to obtain a Hammerstein model structure, which is often considered for the characterization of systems driven by a class of smart material-based actuators. The input nonlinearity in the Hammerstein model is represented by a Prandtl-Ishlinskii ANN hysteretic layer while the linear dynamics are captured using a linear recurrent ANN representing a single-input multiple-output linear time-invariant state-space model. The identification approach of the Hammerstein model with a hysteretic layer is illustrated both numerically and experimentally.

Linear processes on complex networks

AbstractThis article studies the dynamics of complex networks with a time-invariant underlying topology, composed of nodes with linear internal dynamics and linear dynamic interactions between them. While graph theory defines the underlying topology of a network, a linear time-invariant state-space model analytically describes the internal dynamics of each node in the network. By combining linear systems theory and graph theory, we provide an explicit analytical solution for the network dynamics in discrete-time, continuous-time and the Laplace domain. The proposed theoretical framework is scalable and allows hierarchical structuring of complex networks with linear processes while preserving the information about network, which makes the approach reversible and applicable to large-scale networks.

Robust decentralized voltage control for uncertain DC microgrids

A decentralized voltage control scheme to achieve robust stability and robust performance of islanded direct current (DC) microgrids is presented in this paper. The investigated microgrid consists of multiple distributed generation (DG) units with a general topology, each one comprising a local uncertain ZIP (constant impedance (Z), constant current (I), and constant power (P)) load. The proposed controller confers the following main advantages: 1) the design procedure is scalable, 2) it has a completely decentralized structure, 3) it prepares stability and desirable performance of the nominal closed-loop microgrid, 4) it preserves robust stability as well as robust performance of microgrid system under different sources of uncertainty, including plug-and-play (PnP) functionalities of DGs, microgrid topology changes, uncertain ZIP load, and unmodeled load dynamics, 5) every local controller is the solution of a unique convex optimization problem, resulting in the optimal performance and robustness to several different successive changes. First, a linear time-invariant (LTI) state-space model is developed for each DG subsystem with capturing disturbances, and different uncertainty sources are modeled as a new single polytope. Then, all control objectives are converted into a robust dynamic output-feedback-based controller for an LTI polytopic system with H∞ performance criterion. Finally, the obtained nonconvex problem is reduced to a linear matrix inequality (LMI) based optimization problem. Several simulation case studies are carried out in MATLAB to demonstrate the effectiveness of the proposed controller.

Uncertainties Investigation and µ-Synthesis Control Design for a Full Car with Series Active Variable Geometry Suspension

Linear robust control schemes, for example the H∞ control, are commonly utilized in the control design of an active suspension system, with a linearized and time-invariant state-space model of the system adopted. However, the vehicle parameter uncertainties are mainly ignored and their effect on the control robustness is not investigated. In this paper, a µ-synthesis-based control scheme is synthesized for a full car with the recently introduced Series Active Variable Geometry Suspension (SAVGS), to mainly enhance the ride comfort and road holding performance, with two significant practical uncertainties in the sprung mass and the suspension damping taken into account. Numerical simulations with a high fidelity nonlinear vehicle model are performed, with the cases of the fixed and swept values of the sprung mass tested, to assess the control robustness and performance of the developed scheme against the passive suspension as well as the H∞ -controlled SAVGS. The proposed µ-synthesis control scheme is proved to be more effective for realistic applications as it is capable of maintaining the suspension performance improvement regardless of variations of system parameters associated with the uncertainties, while the H∞ control performance tends to deteriorate when a notable deviation from the nominal values occurs.

Harmonic Interaction Analysis of Delta-Connected Cascaded H-Bridge-Based Shunt Active Power Filter

The harmonic interaction analysis of delta-connected cascaded H-bridge (CHB) multilevel converter as shunt active power filter (APF) is helpful for design and control purposes. This paper presents a harmonic interaction analysis based on the derivation of a current and energy-based time-invariant state-space model with switching functions unknown beforehand. Based on that, we are capable to predict the harmonics propagation through the system and provide a precise calculation of various electrical quantities and switching functions that contain multiple frequency components with possible imbalances, giving an accurate reference for understanding, designing, and controlling of delta-connected CHB. The validity and accuracy of the harmonic interaction analysis are verified by the simulations of delta-connected CHB-based APF equipped with dc-link capacitors of reduced values.

Dynamic analysis model of a class E 2 converter for low power wireless charging links

A dynamic response analysis model of a Class E2 converter for wireless power transfer applications is presented. The converter operates at 200 kHz and consists of an induction link with its primary coil driven by a class E inverter and the secondary coil with a voltage-driven class E synchronous rectifier. A seventh-order linear time invariant state-space model is used to obtain the eigenvalues of the system for the four modes resulting from the operation of the converter switches. A participation factor for the four modes is used to find the actual operating point dominant poles for the system response. A dynamic analysis is carried out to investigate the effect of changing the separation distance between the two coils, based on converter performance and the changes required of some circuit parameters to achieve optimum efficiency and stability. The results show good performance in terms of efficiency (90–98%) and maintenance of constant output voltage with dynamic change of capacitance in the inverter. An experiment with coils of the dimension of 53 × 43 × 6 mm3 operating at a resonance frequency of 200 kHz, was created to verify the proposed mathematical model and both were found to be in excellent agreement.