Linear Time-invariant Dynamical Systems Research Articles

Joint learning of linear time-invariant dynamical systems

Linear time-invariant systems are very popular models in system theory and applications. A fundamental problem in system identification that remains rather unaddressed in extant literature is to leverage commonalities amongst related systems to estimate their transition matrices more accurately. To address this problem, we investigate methods for jointly estimating the transition matrices of multiple systems. It is assumed that the transition matrices are unknown linear functions of some unknown shared basis matrices. We establish finite-time estimation error rates that fully reflect the roles of trajectory lengths, dimension, and number of systems under consideration. The presented results are fairly general and show the significant gains that can be achieved by pooling data across systems, in comparison to learning each system individually. Further, they are shown to be robust against moderate model misspecifications. To obtain the results, we develop novel techniques that are of independent interest and are applicable to similar problems. They include tightly bounding estimation errors in terms of the eigen-structures of transition matrices, establishing sharp high probability bounds for singular values of dependent random matrices, and capturing effects of misspecified transition matrices as the systems evolve over time.

\(\boldsymbol{\mathcal{L}_2}\)-Optimal Reduced-Order Modeling Using Parameter-Separable Forms

We provide a unifying framework for -optimal reduced-order modeling for linear time-invariant dynamical systems and stationary parametric problems. Using parameter-separable forms of the reduced-model quantities, we derive the gradients of the cost function with respect to the reduced matrices, which then allows a nonintrusive, data-driven, gradient-based descent algorithm to construct the optimal approximant using only output samples. By choosing an appropriate measure, the framework covers both continuous (Lebesgue) and discrete cost functions. We show the efficacy of the proposed algorithm via various numerical examples. Furthermore, we analyze under what conditions the data-driven approximant can be obtained via projection.

Reinforcement Learning of Structured Stabilizing Control for Linear Systems With Unknown State Matrix

This article delves into designing stabilizing feedback control gains for continuous-time linear systems with unknown state matrix, in which the control gain is subjected to a structural constraint. We bring forth the ideas from reinforcement learning (RL) in conjunction with sufficient stability and performance guarantees in order to design these structured gains using the trajectory measurements of states and controls. We first formulate a model-based linear quadratic regulator (LQR) framework to compute the structured control gain. Subsequently, we transform this model-based LQR formulation into a data-driven RL algorithm to remove the need for knowing the system state matrix. Theoretical guarantees are provided for the stability of the closed-loop system and the convergence of the structured RL (SRL) algorithm. A remarkable application of the proposed SRL framework is in designing distributed static feedback control, which is necessary for automatic control of many large-scale cyber–physical systems. As such, we validate our theoretical results with numerical simulations on a multiagent networked linear time-invariant dynamical system.

Order Reduction of Large-Scale Linear Dynamic Systems Using Balanced Truncation with Modified Cauer Continued Fraction

This article discusses a new technique to reduce the order of high-dimensional, continuous, linear time-invariant dynamic systems. The proposed approach utilizes the merits of the balanced truncation method, and modified Cauer continued fraction expansion. The reduced model denominator was retrieved via the concept of Lyapunov balancing, and state truncation and numerator coefficients were obtained by using a modified Cauer continued fraction. The proposed technique circumvents the steady-state gain discrepancy and steady-state error mismatch of the conventional balanced truncation method, and the instability issue of continued fraction expansion. The proposed method assures steady-state gain approximation, few moments, Markov parameters, stability, and other essential control features of a high-order system. The efficacy and applicability of the proposed technique are demonstrated using some typical test systems adopted from the literature. This method gives improved performance indices like minimum H∞ norm, H2 norm, a minimum ISE (integral squared error), IAE (integral absolute error), and retains approximately the full IRE (impulse response energy) of the existing system in the reduced model. The simulation findings of the proposed technique are compared with other current model reduction methods in recent literature using MATLAB.

Balancing a Stick With Eyes Shut: Inverted Pendulum on a Cart Without Angle Measurement

Abstract We consider linear time-invariant dynamic systems in the single-input, single-output (SISO) framework. In particular, we consider stabilization of an inverted pendulum on a cart using a force on the cart. This system is easy to stabilize with pendulum angle feedback. However, with cart position feedback it cannot be stabilized with stable and proper compensators. Here, we demonstrate that with an additional compensator in a parallel feedforward loop, stabilization is possible with such compensators. Sensitivity to noise seems to be about three times worse than for the situation with angle feedback. For completeness, discussion is presented of compensator parameter choices, robustness, fragility, and comparison with another control approach.

Tight computationally efficient approximation of matrix norms with applications

We address the problems of computing operator norms of matrices induced by given norms on the argument and the image space. It is known that aside of a fistful of ``solvable cases,'' most notably, the case when both given norms are Euclidean, computing operator norm of a matrix is NP-hard. We specify rather general families of norms on the argument and the images space (``ellitopic'' and ``co-ellitopic,'' respectively) allowing for reasonably tight computationally efficient upper-bounding of the associated operator norms. We extend these results to bounding ``robust operator norm of uncertain matrix with box uncertainty,'' that is, the maximum of operator norms of matrices representable as a linear combination, with coefficients of magnitude $\leq1$, of a collection of given matrices. Finally, we consider some applications of norm bounding, in particular, (1) computationally efficient synthesis of affine non-anticipative finite-horizon control of discrete time linear dynamical systems under bounds on the peak-to-peak gains, (2) signal recovery with uncertainties in sensing matrix, and (3) identification of parameters of time invariant discrete time linear dynamical systems via noisy observations of states and inputs on a given time horizon, in the case of ``uncertain-but-bounded'' noise varying in a box.

Semi-Explicit Model Predictive Control of Quasi Linear Parameter Varying Systems

Explicit linear model predictive control (MPC) is a proven method for constrained optimal control of fast, linear time invariant (LTI) dynamic systems. However, an extension to quasi-linear parameter varying (Q-LPV) systems requires overcoming significant challenges, since neither the explicit MPC law is piecewise linear in state nor are the critical regions polytopic. In this work, we propose a semi-explicit solution to a sub-optimal MPC solution for a class of Q-LPV systems, which are obtained as multiple linear model approximations to nonlinear systems. The sub-optimal MPC for Q-LPV systems, which has been shown to be stabilizing in an earlier work, requires an online solution of a quadratic program (QP). Casting the QP into a multiparametric form leads to a new class of multiparametric QP (mpQP), termed herein as generalized mpQP (GmpQP). Information about the critical regions and the MPC law are obtained offline using a semi-explicit solution of the GmpQP based on an implicit enumeration of the active sets. Three algorithms are compared for the online evaluation of the control law. Offline and online complexity of the proposed method are presented for an example, including comparison of computational speeds with well-known implicit QP solvers.

External drivers of BOLD signal's non-stationarity.

A fundamental challenge in neuroscience is to uncover the principles governing how the brain interacts with the external environment. However, assumptions about external stimuli fundamentally constrain current computational models. We show in silico that unknown external stimulation can produce error in the estimated linear time-invariant dynamical system. To address these limitations, we propose an approach to retrieve the external (unknown) input parameters and demonstrate that the estimated system parameters during external input quiescence uncover spatiotemporal profiles of external inputs over external stimulation periods more accurately. Finally, we unveil the expected (and unexpected) sensory and task-related extra-cortical input profiles using functional magnetic resonance imaging data acquired from 96 subjects (Human Connectome Project) during the resting-state and task scans. This dynamical systems model of the brain offers information on the structure and dimensionality of the BOLD signal’s external drivers and shines a light on the likely external sources contributing to the BOLD signal’s non-stationarity. Our findings show the role of exogenous inputs in the BOLD dynamics and highlight the importance of accounting for external inputs to unravel the brain’s time-varying functional dynamics.

Stability analysis of linear time-invariant dynamic systems using the matrix sign function and the Adomian decomposition method

Stability analysis of linear time-invariant dynamic systems using the matrix sign function and the Adomian decomposition method

Study of the amplitude modulation method for kurtosis control purposes

The amplitude modulation method is used to create non-stationary and non-Gaussian signals with desired kurtosis values, which act as non-stationary excitations for shaker-table fatigue tests. However, the non-stationary signals produced by the method have been proved to possess a relatively narrow band of kurtosis values, thus limiting its potential in accelerated life test. To solve the problem, the kurtosis model of a non-stationary signals produced by the amplitude modulation method is investigated. It is found that the feature of the beta distribution imposes restrictions on the kurtosis range. Based on the finding, two types of updates are put forward in this article. The first one is to utilize the gamma-distributed random numbers to scale the amplitudes of the modulating waves. The second update is to employ stationary non-Gaussian signals, instead of the stationary Gaussian signals, to synthesize the non-stationary and non-Gaussian signals. The two alternatives are all verified with numerical and experimental examples. The results demonstrate that both methods can widen the band of kurtosis. But the non-stationary excitations produced by the second method turn out to have different kurtosis transferring rates in a linear time-invariant dynamical system. Especially, the excitations created with sub-Gaussian signals have larger kurtosis transferring rates than the excitations made by the original amplitude modulation method do, making them suitable to be applied in an accelerated fatigue test.

Teaching transfer functions without the Laplace transform

ABSTRACT Transfer functions are convenient representations to analyse cause-and-effect relationships of linear time-invariant dynamic systems. Traditionally, transfer functions are introduced using the Laplace transform. In this paper, we argue that the Laplace transform method is not a necessary prerequisite to understanding the topic ‘transfer functions’. We offer an inquiry-based learning method to represent transfer functions without formally using the full machinery of Laplace transforms. The method is used in an introductory engineering course on system modelling and simulation at Aalborg University, Denmark. The paper also presents an initial assessment of the students' understanding of transfer functions without knowledge of Laplace transforms and the experiences gained from implementing inquiry-based learning in the course. Finally, we conclude with a discussion of the impact of CAS tools on mathematics and engineering science teaching and learning.

An Ellipsoidal Predictor–Corrector State Estimation Scheme for Linear Continuous-Time Systems With Bounded Parameters and Bounded Measurement Errors

For linear time-invariant dynamic systems with exactly known coefficients of their system matrices for which measurements with bounded errors are available at discrete time instants, an optimal polygonal state estimation scheme was recently published. This scheme allows for tightly enclosing all possible state trajectories in presence of uncertain, but bounded, system inputs which may be varying arbitrarily within in their bounds. Moreover, this approach is also capable of accounting for uncertainty related to the measurement time instants. However, the drawback of this polygonal technique is its rapidly increasing complexity for larger system dimensions. For that reason, the polygonal state enclosures are replaced by a computationally less expensive, but nearly optimal, ellipsoidal enclosure technique in this paper. Numerical simulations for representative benchmark examples focusing both on applications with precisely known and uncertain parameters conclude this contribution.

A linear algorithm for the minimal realization problem in physical coordinates with a non-invertible output matrix

In this paper we present a linear algorithm that estimates some physical parameters of a continuous-time system, described by an analytical mathematical model, when not all the state variables can be measured. The algorithm starts from the well-known subspace methods and applies some linear transformations to recover, at least partially, the estimated model in physical coordinates. Some analytical investigations and numerical experiments are shown for this method, which has general application within linear time-invariant (LTI) dynamical systems.

Observer Design in LTI Time-Delay Systems using Partial Pole Placement with Applications

In order to estimate the state of a dynamical system, we address the problem of designing an observer for linear time-invariant (LTI) dynamical systems including a time-delay. Sufficient conditions for the existence of the proposed observer are given using partial placement of the poles for the error. Namely, we exploit the multiplicity-induced-dominance property of the characteristic function corresponding to the system's error. The effectiveness of the proposed observer design is shown in both state lag and input lag respectively through the problems of Mach number control in a wind tunnel and stabilization of the inverted pendulum on a cart by using the delay.

Resource-aware time-optimal control with multiple sparsity measures

The paper proposes a novel time-optimal control that is also sparse in both time and space domain to take account of resource constraints in networked control systems. The control problem is described as minimization of a weighted sum of the terminal time and the L0 norm of multi-input control for a linear time-invariant dynamical system, with state and control constraints. In particular, we treat the constraint on the number of actuations at each time, which is described as an ℓ0 norm constraint. Since the L0/ℓ0 optimization problem is highly non-convex, we propose to solve a convex relaxation using L1 and ℓ1 norms. We give sufficient conditions for the equivalence between the original L0/ℓ0 problem and the relaxed L1/ℓ1 problem. Based on the relaxation, we also propose a numerical computation method for the L1/ℓ1 problem by sequential linear programming. We show numerical examples to illustrate the effectiveness of the proposed control method.

Dynamical system parameter identification using deep recurrent cell networks

In this paper, we investigate the parameter identification problem in dynamical systems through a deep learning approach. Focusing mainly on second-order, linear time-invariant dynamical systems, the topic of damping factor identification is studied. By utilizing a six-layer deep neural network with different recurrent cells, namely GRUs, LSTMs or BiLSTMs; and by feeding input-output sequence pairs captured from a dynamical system simulator, we search for an effective deep recurrent architecture in order to resolve damping factor identification problem. Our study results show that, although previously not utilized for this task in the literature, bidirectional gated recurrent cells (BiLSTMs) provide better parameter identification results when compared to unidirectional gated recurrent memory cells such as GRUs and LSTM. Thus, indicating that an input-output sequence pair of finite length, collected from a dynamical system and when observed anachronistically, may carry information in both time directions for prediction of a dynamical systems parameter.

A compute-bound formulation of Galerkin model reduction for linear time-invariant dynamical systems

This work aims to advance computational methods for projection-based reduced-order models (ROMs) of linear time-invariant (LTI) dynamical systems. For such systems, current practice relies on ROM formulations expressing the state as a rank-1 tensor (i.e., a vector), leading to computational kernels that are memory bandwidth bound and, therefore, ill-suited for scalable performance on modern architectures. This weakness can be particularly limiting when tackling many-query studies, where one needs to run a large number of simulations. This work introduces a reformulation, called rank-2 Galerkin, of the Galerkin ROM for LTI dynamical systems which converts the nature of the ROM problem from memory bandwidth to compute bound. We present the details of the formulation and its implementation, and demonstrate its utility through numerical experiments using, as a test case, the simulation of elastic seismic shear waves in an axisymmetric domain. We quantify and analyze performance and scaling results for varying numbers of threads and problem sizes. Finally, we present an end-to-end demonstration of using the rank-2 Galerkin ROM for a Monte Carlo sampling study. We show that the rank-2 Galerkin ROM is one order of magnitude more efficient than the rank-1 Galerkin ROM (the current practice) and about 970 times more efficient than the full-order model, while maintaining accuracy in both the mean and statistics of the field.

Equivalent circuits for electromigration

Electromigration continues to be a major concern for integrated circuit design. The susceptibility to electromigration is assessed by tracking the stress in metal lines under the influence of applied currents, which can be computationally expensive for large chips. Over the last few years, an efficient approach for tracking stress in large interconnect networks has been developed, and well-received, in part because it provides a model for stress dynamics in the form of a standard linear time-invariant dynamic system. In the context of this model, we will show that the dynamic behavior of the stresses and fluxes in metal lines is exactly identical to the dynamic behavior of voltages and currents in certain RC circuits that can be easily constructed for the metal lines. Thus, electromigration assessment for any metal interconnect structure can be done by simply simulating the corresponding equivalent RC circuit. This opens the door for benefiting from well-known techniques for fast circuit simulation, as well as methods for macro-modelling and analysis of RC circuits, in order to improve the performance and capacity of practical methods for electromigration assessment in large circuits.

Revisiting Ho–Kalman-Based System Identification: Robustness and Finite-Sample Analysis

Weconsider the problem of learning a realization for a linear time-invariant (LTI) dynamical system from input/output data. Given a single input/output trajectory, we provide finite time analysis for learning the system’s Markov parameters, from which a balanced realization is estimated using the classical Ho–Kalman algorithm. By proving a robustness result for the Ho–Kalman algorithm and combining it with the sample complexity results for Markov parameters, we show how much data are needed to approximate the balanced realization of the system up to a desired accuracy with high probability.

Statistical and Nonstatistical Methods for Estimating the Impulse Characteristic of a Dynamical System

We consider the problem of estimating the impulse characteristic of a linear time-invariant discrete-time dynamical system, using data on its input and output signals. We propose versions of estimation methods for asymptotically stable dynamical systems in the context of various approaches to the presentation of parametric uncertainty, including approaches based on the interval analysis and the fuzzy set theory. A comparative analysis and numerical illustration of the methods are presented.