Linear Time-invariant Dynamical Systems Research Articles

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.

Generalized algorithms for the approximate matrix polynomial GCD of reducing data uncertainties with application to MIMO system and control

Computation of (approximate) polynomials common factors is an important problem in several fields of science, like control theory and signal processing. While the problem has been widely studied for scalar polynomials, the scientific literature in the framework of matrix polynomials seems to be limited to the problem of exact greatest common divisor computation. In this paper, we generalize two algorithms from scalar to matrix polynomials. The first one is fast and simple. The second one is more accurate but computationally more expensive. We test the performances of the two algorithms and observe similar behavior to the one in the scalar case. Finally we describe an application to multi-input multi-output linear time-invariant dynamical systems.

Characteristics That Make Linear Time-Invariant Dynamic Systems Difficult for Humans to Control

We present results from an experiment in which 55 human subjects interact with a dynamic system 40 times over a one-week period. The subjects are divided into five groups. For each interaction, a subject performs a command-following task, where the reference command is the same for all subjects and all trials; however, each group interacts with a different linear time-invariant dynamic system. We use a subsystem identification algorithm to estimate the control strategy that each subject uses on each trial. The experimental and identification results are used to examine the impact of the system characteristics (e.g., poles, zeros, relative degree, system order, phase lag) on the subjects' command-following performance and the control strategies that the subjects learn. Results demonstrate that phase lag (which arises from higher relative degree and nonminimum-phase zeros) tends to make dynamic systems more difficult for humans to control, whereas higher system order does not necessarily make a system more difficult to control. The identification results demonstrate that improvement in performance is attributed to: 1) using a comparatively accurate approximation of the inverse dynamics in feedforward; and 2) using a feedback controller with comparatively high gain. Results also demonstrate that system phase lag is an important impediment to a subject's ability to approximate the inverse dynamics in feedforward, and that a key aspect of approximating the inverse dynamics in feedforward is learning to use the correct amount of phase lead in feedforward.

The impact of static output nonlinearities on the control strategies that humans use in command-following tasks

The results of a human-in-the-loop experiment are used to investigate the control strategies that humans use to interact with nonlinear dynamic systems. Two groups of human subjects interact with a dynamic system and perform a command-following task. The first group interacts with a linear time-invariant (LTI) dynamic system. The second group interacts with a Wiener system, which consists of the same LTI dynamics cascaded with a static output nonlinearity. Both groups exhibit improved performance over the trials, but the average of the linear group’s performance is better on more than three-fourths of the trials. A new nonlinear subsystem identification algorithm is presented and used to identify the feedback and feedforward control strategies used by the subjects in both groups. The identification results for the linear group agree with prior studies suggesting that adaptive feedforward inversion is a primary control strategy used by humans for command-following tasks. The main results of this paper address an open question of whether a similar control strategy is used for nonlinear systems. The identification results for the nonlinear group suggest that those subjects also use adaptive feedforward inversion. However, the static output nonlinearity inhibits the human’s ability to approximate the inverse.

Parameter-disturbance-robust model predictive control of input-saturated MIMO fractional systems

This paper primarily discusses the solution to multi-degree of freedom linear time invariant fractional order dynamic systems with fractional actuators in detail. This solution is of significant importance due its ease of usage and compatibility with digital systems. Subsequently, a modified model predictive control (MMPC) scheme is introduced that is an infusion of a novel discrete-time robust sliding mode control and model predictive control. The proposed modified model predictive control algorithm is proven to illustrate robustness against uncertainty in system’s parameters and external disturbance while abstaining from the undesirable characteristics of sliding mode control such as chattering. By employing the MMPC scheme, it is possible to constrain the control inputs to be less than pre-determined bounds and by this mean, take a step toward making this algorithm more suitable for practical applications. A general and illustrious plant, represented by the famous Torvik’s three parameter model, is selected and the proposed control scheme is applied to further elucidate the merits, qualities and capabilities of MMPC for fractional systems. The corresponding MATLAB code for the simulation section is also presented in the appendix of the paper. A discussion is also included that examines the effectiveness of MMPC and provides insights for future works.

Modeling protein concentrations in cycling cells using stochastic hybrid systems

We analyze a class of time-triggered stochastic hybrid systems where the state-space evolves as per a linear time-invariant dynamical system. This continuous-time evolution is interspersed with two kinds of stochastic resets. The first reset occurs based on an internal timer that measures the time elapsed since it last occurred. Whenever the first reset occurs, the states-space undergoes a random jump, and the timer is reset to zero. The second reset occurs based on an arbitrary timer-depended rate, and whenever this reset fires, the state-space is changed based on a given random map. We provide exact conditions for this class of systems that lead to finite statistical moments and the corresponding exact analytical expressions for the first two moments. This framework is applied to study random fluctuations in the concentration of a protein in a growing cell. In the context of this example, the timer denotes the time elapsed since the cell was born, and the cell division event (first reset) is triggered based on a timer-dependent rate. The second reset corresponds to the protein synthesis in stochastic bursts, and finally, during cell growth, protein concentration continuously decreases due to dilution. Our analysis provides closed-form formulas for the noise in the protein concentration and leads to a striking result - for a constant (timer-independent) protein synthesis rate, the noise in the protein concentration is invariant of the noise in the cell-cycle time. Finally, we provide a rigorous framework for investigating protein noise levels for different forms of timer-dependent synthesis rates, as is the case for cell-cycle regulated genes inside the cell.

A novel two-polynomial criteria for higher-order systems stability boundaries detection and control

There are many methods for identifying the stability of complex dynamic systems. Routh and Hurwitz’s criterion is one of the earliest and commonly used analytical tools analysing the stability of dynamic systems. However, it requires tedious and lengthy derivations of all components of the Routh array to solve the stability problem. Therefore, it is not a simple method to define analytically, stability boundaries for the coefficients of the system characteristic equation.The proposed brand-new criterion is an effective alternaztive technique in identifying stabilityhigher-order linear time-invariant dynamic system that binds the coefficients of the system characteristic polynomial at the stability boundaries by means of an additional single constantk. It defines the necessary and sufficient conditions for the absolute stability of higher-order dynamic systems. It also allows the analysing of the system’s precise marginal stabilityor marginal instability condition when the roots are relocated on imaginary jω-axis of s-plane. The criterion proposed bythe authors, in contrast to Routh criteria, simplifies the identification of maximum and minimum stability limits for any coefficient of the higher-order characteristic equation significantly. The derived in the paperstability boundary formulas for the polynomial coefficients are successfully used for the proportional integral derivative (PID) controller with single or multiple gains selections in closed-loop control system.

Analyzing control properties of multi-agent linear systems

The networked multi-agent systems that they are interconnected via communication channels have great applicability in multiple areas, such as power grids, bioinformatics, sensor networks, vehicles, robotics and neuroscience, for example. Consequently, they have been widely studied by scientists in different fields in particular in the field of control theory. Recently an interest has grown to analyze the control properties as consensus controllability and observability of multi-agent dynamical systems motivated by the fact that the architecture of communication network in engineering multi-agent systems is usually adjustable. In this paper, we analyze how to improve the control properties in the case of multiagent linear time-invariant dynamical systems.

Revisiting IRKA: Connections with Pole Placement and Backward Stability

The iterative rational Krylov algorithm (IRKA) is a popular approach for producing locally optimal reduced-order ${\mathscr{H}}_{2}$ -approximations to linear time-invariant (LTI) dynamical systems. Overall, IRKA has seen significant practical success in computing high fidelity (locally) optimal reduced models and has been successfully applied in a variety of large-scale settings. Moreover, IRKA has provided a foundation for recent extensions to the systematic model reduction of bilinear and nonlinear dynamical systems. Convergence of the basic IRKA iteration is generally observed to be rapid—but not always; and despite the simplicity of the iteration, its convergence behavior is remarkably complex and not well understood aside from a few special cases. The overall effectiveness and computational robustness of the basic IRKA iteration is surprising since its algorithmic goals are very similar to a pole assignment problem, which can be notoriously ill-conditioned. We investigate this connection here and discuss a variety of nice properties of the IRKA iteration that are revealed when the iteration is framed with respect to a primitive basis. We find that the connection with pole assignment suggests refinements to the basic algorithm that can improve convergence behavior, leading also to new choices for termination criteria that assure backward stability.

Controllability of Deep-Coupling Dynamical Networks

This paper explores the state controllability of deep-coupling networks with inter-layer couplings, where the nodes of each layer are higher-dimensional linear time-invariant dynamical systems. The collective effects on the network controllability from intra-layer network topologies, node dynamics, external control inputs and inner interactions are extensively explored under two representative inter-layer coupling modes, where the integrated network can be controllable even if the intra-layer network topology is uncontrollable by external inputs. Some necessary and/or sufficient conditions for the network controllability are obtained for the representative two-layer intra-layer couplings of interdependent and drive-response modes. Simulated examples reveal the important role of inter-layer couplings of a controllable deep-coupling dynamical network assembled from uncontrollable intra-layer networks.

Controllability of Directed Networked MIMO Systems With Heterogeneous Dynamics

This paper studies the controllability of networked multi-input-multi-output (MIMO) systems, in which the network topology is weighted and directed, and the nodes are heterogeneous higher-dimensional linear time-invariant (LTI) dynamical systems. The primary objective is to search for controllability criteria beyond those already known for homogeneous networks. The focus is on the effects of the network topology, node dynamics, external control inputs, as well as the inner interactions on the network controllability. It is found that a network of heterogeneous systems can be controllable even if the corresponding homogeneous network topology is uncontrollable. The finding thus unravels another fundamental property that affects the network controllability---the heterogeneity of the node dynamics. A necessary and sufficient condition is derived for the controllability of heterogeneous networked MIMO LTI systems. For some typical cases, necessary and/or sufficient controllability conditions are specified and presented on the node dynamics, inner interactions, as well as the network topology.

Computational Algorithms for Reducing Rational Transfer Functions’ Order

A problem of reducing a linear time-invariant dynamic system is considered as a problem of approximating its initial rational transfer function with a similar function of a lower order. The initial transfer function is also assumed to be rational. The approximation error is defined as the standard integral deviation of the transient characteristics of the initial and reduced transfer function in the time domain. The formulations of two main types of approximation problems are considered: a) the traditional problem of minimizing the approximation error at a given order of the reduced model; b) the proposed problem of minimizing the order of the model at a given tolerance on the approximation error.
 Algorithms for solving approximation problems based on the Gauss-Newton iterative process are developed. At the iteration step, the current deviation of the transient characteristics is linearized with respect to the coefficients of the denominator of the reduced transfer function. Linearized deviations are used to obtain new values of the transfer function coefficients using the least-squares method in a functional space based on Gram-Schmidt orthogonalization. The general form of expressions representing linearized deviations of transient characteristics is obtained.
 To solve the problem of minimizing the order of the transfer function in the framework of the least squares algorithm, the Gram-Schmidt process is also used. The completion criterion of the process is to achieve a given error tolerance. It is shown that the sequence of process steps corresponding to the alternation of coefficients of polynomials of the numerator and denominator of the transfer function provides the minimum order of transfer function.
 The paper presents an extension of the developed algorithms to the case of a vector transfer function with a common denominator. An algorithm is presented with the approximation error defined in the form of a geometric sum of scalar errors. The use of the minimax form for error estimation and the possibility of extending the proposed approach to the problem of reducing the irrational initial transfer function are discussed.
 Experimental code implementing the proposed algorithms is developed, and the results of numerical evaluations of test examples of various types are obtained.

Real-Time Remote Estimation With Hybrid ARQ in Wireless Networked Control

Real-time remote estimation is critical for mission-critical applications including industrial automation, smart grid and tactile Internet. In this paper, we propose a hybrid automatic repeat request (HARQ)-based real-time remote estimation framework for linear time-invariant (LTI) dynamic systems. Considering the estimation quality of such a system, there is a fundamental tradeoff between the reliability and freshness of the sensor's measurement transmission. We formulate a new problem to optimize the sensor's online transmission control policy for static and Markov fading channels, which depends on both the current estimation quality of the remote estimator and the current number of retransmissions of the sensor, so as to minimize the long-term remote estimation mean squared error (MSE). This problem is non-trivial. In particular, it is challenging to derive the condition in terms of the communication channel quality and the LTI system parameters, to ensure a bounded long-term estimation MSE. We derive an elegant sufficient condition of the existence of a stationary and deterministic optimal policy that stabilizes the remote estimation system and minimizes the MSE. Also, we prove that the optimal policy has a switching structure, and accordingly derive a low-complexity suboptimal policy. Numerical results show that the proposed optimal policy significantly improves the performance of the remote estimation system compared to the conventional non-HARQ policy.

High-performance modeling and discrete-time sliding mode control of uncertain non-commensurate linear time invariant MIMO fractional order dynamic systems

This paper primarily proposes an analytical solution to the set of coupled non-commensurate linear time invariant fractional order differential equations (representing Multi input Multi output (MIMO) fractional order dynamic systems). The proposed solution is reviewed and accurately truncated to have a short finite-memory and to demonstrate further compatibility for practical applications. It is also stated that this discrete-time approach is explicitly superior to any approach in continuous-time in terms of the possibility of implementation. Standards are provided for adequate truncation of the response with respect to the application. A novel discrete-time enhanced sliding mode control (ESMC) scheme is proposed for MIMO fractional order dynamic systems that can cope with parameter uncertainty while illustrating robustness against disturbance. Due to its structure, ESMC shows no sign of chattering in its control signal and employs no fractional operator in its structure; hence, computational cost of ESMC is minuscule. A novel mathematical method is also proposed for acquiring the stability bounds and solving linear matrix inequalities that arise in this problem. A famous MIMO fractional order dynamic system is presented and controlled under considerable disturbance and parameter uncertainty to render the merits of ESMC.

Towards an MID-based Delayed Design for Arbitrary-order Dynamical Systems with a Mechanical Application

In this study we consider the stabilization of nth-order linear time-invariant (LTI) dynamical systems using Multiplicity-Induced-Dominancy (MID)-based controller design in the presence of delays in the input/output channels. A sufficient condition is given for the dominancy of a real root with multiplicity n + 1 using an integral representation of the corresponding characteristic function. Furthermore, this sufficient condition is analyzed in the case when the characteristic function of the open-loop system is real-rooted, and delay intervals are derived for the set of parameters satisfying stabilizability and dominancy conditions. The efficiency of the proposed controller design is shown in the case of a multi-link inverted pendulum.

Online Optimization as a Feedback Controller: Stability and Tracking

This paper develops and analyzes feedback-based online optimization methods to regulate the output of a linear time invariant (LTI) dynamical system to the optimal solution of a time-varying convex optimization problem. The design of the algorithm is based on continuous-time primal-dual dynamics, properly modified to incorporate feedback from the LTI dynamical system, applied to a proximal augmented Lagrangian function. The resultant closed-loop algorithm tracks the solution of the time-varying optimization problem without requiring knowledge of (time varying) disturbances in the dynamical system. The analysis leverages integral quadratic constraints to provide linear matrix inequality (LMI) conditions that guarantee global exponential stability and bounded tracking error. Analytical results show that under a sufficient time-scale separation between the dynamics of the LTI dynamical system and the algorithm, the LMI conditions can be always satisfied. This paper further proposes a modified algorithm that can track an approximate solution trajectory of the constrained optimization problem under less restrictive assumptions. As an illustrative example, the proposed algorithms are showcased for power transmission systems, to compress the time scales between secondary and tertiary control, and allow to simultaneously power rebalancing and tracking of the DC optimal power flow points.

Security Analysis of a Distributed Networked System Under Eavesdropping Attacks

In this brief, we consider the security problem for a network of linear time-invariant dynamical systems under eavesdropping attacks, where an eavesdropper can measure the states of some nodes in the network. We provide a necessary and sufficient condition for the eavesdropper to reveal the states of the networked system. We also present some necessary conditions for the case when the network has tail nodes or at least one node not monitored by the eavesdropper. In addition, we investigate several typical network structures and show that the networked system can be completely observed provided that the eavesdropper can monitor one node in each cycle. Further, we characterize the minimal set of necessarily monitored nodes with the upper bound set of such nodes. Finally, we verify the theoretical results by numerical simulations.

Byzantine-resilient distributed observers for LTI systems

Consider a linear time-invariant (LTI) dynamical system monitored by a network of sensors, modeled as nodes of an underlying directed communication graph. We study the problem of collaboratively estimating the state of the system when certain nodes are compromised by adversaries. Specifically, we consider a Byzantine adversary model, where a compromised node possesses complete knowledge of the system dynamics and the network, and can deviate arbitrarily from the rules of any prescribed algorithm. We first characterize certain fundamental limitations of any distributed state estimation algorithm in terms of the measurement and communication structure of the nodes. We then develop an attack-resilient, provably correct state estimation algorithm that admits a fully distributed implementation. To characterize feasible network topologies that guarantee success of our proposed technique, we introduce a notion of ‘strong-robustness’ that captures both measurement and communication redundancy. Finally, by drawing connections to bootstrap percolation theory, we argue that given an LTI system and an associated sensor network, the ‘strong-robustness’ property can be checked in polynomial time.

Sequential Bayesian estimation of state and input in dynamical systems using output-only measurements

• A sequential Bayesian method is proposed for estimating the state and input forces. • Covariance matrices of the process and measurement noise are updated sequentially. • Estimations of the state and input have decent accuracy and stability. • Low-frequency drift appearing in estimations of the input and state is eliminated. The problem of joint estimation of the state and input in linear time-invariant dynamical systems is revisited proposing novel sequential Bayesian formulations. An appealing feature of the proposed method is the promise it delivers for updating the covariance matrices of the process and measurement noise in a real-time fashion using asymptotic approximations. The proposed method avoids the direct transmission of the input into predictions of the state using a zero-mean Gaussian distribution for the input. This prior distribution aims to eliminate low-frequency drifts from estimations of the state and input. Moreover, the method is outlined in a computational algorithm offering real-time estimations of the state and input forces. Numerical and experimental examples are used to examine and demonstrate the efficacy of the method. It is observed that the proposed method achieves decent accuracy for estimating the state, input forces, and noise covariance matrices when compared to the actual values. Contrary to the present methods that produce significant low-frequency drifts while using noisy acceleration response-only measurements, the proposed method offers drift-free perfect predictions. This Bayesian filtering-technique proposed for the reconstruction of the state and input forces can next be employed in the emerging fatigue prognosis frameworks.