Linear 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.

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.

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.

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.

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.

Frequency-domain stability conditions for asynchronously sampled decentralized LTI systems

This paper deals with the exponential stability analysis of decentralized, sampled-data, Linear Time Invariant (LTI) control systems with asynchronous sensors and actuators. We consider the case where each controller in the decentralized setting has its own sampling and actuation frequency, which translates to asynchrony between sensors and actuators. Additionally, asynchrony may be induced by delays between the sampling instants and actuation update instants as relevant in a networked context. The decentralized, asynchronous LTI system is represented as the feedback interconnection of a continuous-time LTI system operator and an operator that captures the effects of asynchrony induced by sampling and delay. By characterizing the properties of the operators using small-gain type Integral Quadratic Constraints (IQC), we provide criteria for exponential stability of the asynchronous, decentralized LTI state-space models. The approach provided in this paper considers two scenarios, namely the ‘large-delay’ case and the ‘small-delay’ case where the delays are larger and smaller than the sampling interval, respectively. The effectiveness of the proposed results is corroborated by a numerical example.

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.

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.

Iterative Residual Generator for Fault Detection With Linear Time-Invariant State–Space Models

In this paper, an iterative residual generator (IRG) is proposed for discrete time-invariant state-space model with the aim of detecting faulty signals. By minimizing the mean square errors subject to unbiasedness constraint, a new filter with finite impulse response structure is derived. The resulting IRG is then obtained by extracting residual signal from the batch filter through several predictor/corrector iterations. It shows that IRG can provide a zero-mean Gaussian process regardless of previous estimation errors. More importantly, it includes the residual generation process in the Kalman filter as its special case. With the chi-square test, a numerical example is simulated to demonstrate that IRG can reduce the false alarm significantly compared with the traditional recursive strategy in the presence of actuator or sensor faults, and the estimation horizon length in IRG serves as a tuning parameter providing a tradeoff between the missed alarm and false alarm.

An input-output linear time invariant model captures neuronal firing responses to external and behavioral events.

Investigating how neurons in different motor regions respond to external stimuli and behavioral events provides insight into motor control. A recent approach to studying neuronal activity is to construct a zero-input linear time invariant (ZI-LTI) state-space model, wherein the state vector consists of firing rate signals for different populations of neurons across motor regions. This approach allows for the populations to influence each other in a dynamical manner given an initial firing rate condition, and the model can accurately reconstruct firing rates within a limited epoch in the motor task during which no event occurs. Here, we generalize this LTI modeling approach to characterize firing responses of neurons to two events (a go cue and movement onset) in a movement task with a non-zero input LTI state-space model, herein referred to as input-output LTI (IO-LTI). Specifically, responses from 196 neurons in the primary motor (M1), ventral premotor (PMv), and dorsal premotor cortex (PMd) were recorded and modeled in two nonhuman primates executing a reach-to-grasp task. We found that a single IO-LTI model can reconstruct neuronal firing rate patterns of six populations of these neurons across the three areas in the presence of multiple events (go cue, movement onset). This is the first step towards constructing generative models of neuronal firing rates in the presence of multiple events, which then can be used to construct better decoders for brain machine interactive control.

Gain Scheduling Dual Mode MPC for a Solar Thermal Power Plant

A nonlinear gain scheduling control strategy is proposed for a concentrated solar thermal power plant. The strategy involves the identification of local linear time-invariant state space models around a family of operating points, the design of corresponding local linear dual mode model-based predictive controllers and the selection of an appropriate scheduling variable to govern the switching. The local models are estimated directly from input-output data using a subspace identification method while taking into account the frequency response of the plant. Input-output data are obtained from a nonlinear simulation model of the plant rather than the plant itself. The effectiveness of the proposed control strategy in terms of tracking and disturbance rejection is evaluated through two different scenarios created in a nonlinear simulation environment.