Continuous-time Linear Dynamical Systems Research Articles

Strong versions of impulsive controllability and sampled observability

We give a simple proof of the (perhaps not so) well known fact that exponential polynomials of order k with real exponents have at most k−1 real zeros. We deduce several results that relate to impulsive controllability and sampled observability of finite-dimensional linear time-invariant dynamical systems. We prove that the initial state of a continuous-time linear time-invariant dynamical system of dimension n can be uniquely reconstructed from the sampled output regardless of its sampling time sequence of length n if and only if the system is observable and all the eigenvalues of the system matrix are real. This result thus characterizes sampled observability with arbitrary sampling times. Likewise, we prove that the system is controllable by means of n impulses regardless of when they occur if and only if the system is controllable and all the eigenvalues of the system matrix are real. We also show that, if the system is observable and all the eigenvalues of the system matrix are real, then there exists an initial state such that the times where the output crosses a prescribed threshold are prescribed m times with m<n.

Convergence Analysis of Ensemble Filters for Linear Stochastic Systems with Poisson-Sampled Observations

For continuous-time linear stochastic dynamical systems driven by Wiener processes, we consider the problem of designing ensemble filters when the observation process is randomly time-sampled. We propose a continuous-discrete McKean–Vlasov type diffusion process with additive Gaussian noise in observation model, which is used to describe the evolution of the individual particles in the ensemble. These particles are coupled through the empirical covariance and require less computations for implementation than the optimal ones based on solving Riccati differential equations. Using appropriate analysis tools, we show that the empirical mean and the sample covariance of the ensemble filter converges to the mean and covariance of the optimal filter if the mean sampling rate of the observation process satisfies certain bounds and as the number of particles tends to infinity.

Analysis of average consensus of multi-agent systems with time delay using packet selection and synchronization protocol

A stabilization approach for the average consensus of homogeneous multi-agent systems (MASs) entailing non-uniform, asymmetric, and time-varying delays in communication networks is proposed. Agents are modeled using a continuous-time linear dynamical system with a local discrete-time controller. To address communication delays, we propose two logic functions, namely, a packet selection algorithm that constantly selects and updates the latest received control signal packets and a synchronization algorithm that asymmetrically synchronizes these packets to preserve the average value of state variables. Based on the Lyapunov theory, we establish a linear matrix inequality (LMI) condition that sufficiently guarantees the stability and consensus of the overall system. To establish this condition, only the time delay upper bound is required, without any special assumptions on the delay pattern. We thoroughly discuss the proposed algorithm and evaluate its effectiveness via numerical simulations.

Kernel-based identification of asymptotically stable continuous-time linear dynamical systems

ABSTRACT In many engineering applications, continuous-time models are preferred to discrete-time ones, in that they provide good physical insight and can be derived also from non-uniformly sampled data. However, for such models, model selection is a hard task if no prior physical knowledge is given. In this paper, we propose a non-parametric approach to infer a continuous-time linear model from data, by automatically selecting a proper structure of the transfer function and guaranteeing to preserve the system stability properties. By means of benchmark simulation examples, the proposed approach is shown to outperform state-of-the-art continuous-time methods, also in the critical case when short sequences of canonical input signals, like impulses or steps, are used for model learning.

State-Space Network Topology Identification From Partial Observations

In this article, we explore the state-space formulation of a network process to recover from partial observations the network topology that drives its dynamics. To do so, we employ subspace techniques borrowed from system identification literature and extend them to the network topology identification problem. This approach provides a unified view of network control and signal processing on graphs. In addition, we provide theoretical guarantees for the recovery of the topological structure of a deterministic continuous-time linear dynamical system from input-output observations even when the input and state interaction networks are different. Our mathematical analysis is accompanied by an algorithm for identifying from data,a network topology consistent with the system dynamics and conforms to the prior information about the underlying structure. The proposed algorithm relies on alternating projections and is provably convergent. Numerical results corroborate the theoretical findings and the applicability of the proposed algorithm.

Parametrical Non-Complex Tests to Evaluate Partial Decentralized Linear-Output Feedback Control Stabilization Conditions from Their Centralized Stabilization Counterparts

This paper formulates sufficiency-type linear-output feedback decentralized closed-loop stabilization conditions if the continuous-time linear dynamic system can be stabilized under linear output-feedback centralized stabilization. The provided tests are simple to evaluate, while they are based on the quantification of the sufficiently smallness of the parametrical error norms between the control, output, interconnection and open-loop system dynamics matrices and the corresponding control gains in the decentralized case related to the centralized counterpart. The tolerance amounts of the various parametrical matrix errors are described by the maximum allowed tolerance upper-bound of a small positive real parameter that upper-bounds the various parametrical error norms. Such a tolerance is quantified by considering the first or second powers of such a small parameter. The results are seen to be directly extendable to quantify the allowed parametrical errors that guarantee the closed-loop linear output-feedback stabilization of a current system related to its nominal counterpart. Furthermore, several simulated examples are also discussed.

Stackelberg Game Approach to Mixed H2/H∞ Problem for Continuous-Time System

This paper studies the mixed H2/H∞ control for continuous-time linear dynamic systems. By applying Stackelberg game approach, the control input is treated as the leader and the disturbance is treated as the follower, respectively. Under standard assumptions and maximum principle, a necessary and sufficient existence condition which is based on three decoupled Riccati equations is obtained. Explicit expression of controllers and solutions to forward backward differential equations (FBDES) are obtained by homogeneous analysis of variables. A numerical example is finally given to verify the efficiency of the proposed approach.

A Min–Max Approach to Event- and Self-Triggered Sampling and Regulation of Linear Systems

This paper presents both an event- and a self-triggered sampling and regulation scheme for continuous time linear dynamic systems by using zero-sum game formulation. A novel performance index is defined wherein the control policy is treated as the first player and the threshold for control input error due to aperiodic dynamic feedback is treated as the second player. The optimal control policy and sampling intervals are generated using the saddle point or Nash equilibrium solution, which is obtained from the corresponding game algebraic Riccati equation. To determine the optimal event-based sampling scheme, an event-triggering condition is derived by utilizing the worst case control input error as the threshold. To avoid the additional hardware for the event-triggering mechanism, a near optimal self-triggering condition is derived to determine the future sampling instants given the current state vector. To guarantee Zeno-free behavior in both the event- and self-triggered closed-loop systems, the minimum intersample times are shown to be lower bounded by a nonzero positive number. Asymptotic stability of the closed-loop system is ensured using Lyapunov stability analysis. Finally, simulation examples are provided to substantiate the analytical claims.

Calibrating linear continuous-time dynamical systems via perturbation analysis

This work considered the continuous-time linear dynamical systems described by the matrix differential equations, and aimed at studying the perturbation analysis via solving perturbed linear dynamical systems. In specific, we solved Riccati differential equations and continuous-time algebraic Riccati equations with finite and infinite times respectively. Moreover, we stated some assumptions on the existence and uniqueness of the solutions of the perturbed Riccati equations. Similar techniques were applied to the discrete-time linear dynamical systems. Two numerical examples illustrated the efficiency and accuracy.

Robust Multi-Parametric Control of Continuous-Time Linear Dynamic Systems

We extend a recent methodology called multi-parametric NCO-tracking for the design of parametric controllers for continuous-time linear dynamic systems in the presence of uncertainty The approach involves backing-off the path and terminal state constraints based on a worst-case uncertainty propagation determined using either interval analysis or ellipsoidal calculus. We address the case of additive uncertainty and we discuss approaches to handling multiplicative uncertainty that retain tractability of the mp-NCO-tracking design problem, subject to extra conservatism. These developments are illustrated with the case study of a fluidized catalytic cracking (FCC) unit operated in partial combustion mode.

An Observer/Predictor-Based Model of the User for Attaining Situation Awareness

Situation awareness (SA) is essential for the safe operation of systems involving human-automation interaction. In this paper, using the theory of functional observers, we model SA for the user interacting with a continuous-time linear time-invariant dynamical system. For systems under human control or shared control, we use the proposed model to determine the required information to be displayed in the user interface for achieving SA. The user interface provides the user with the ability to observe the continuous-time outputs of the system, as well as the ability to enter continuous-time control inputs. In some systems, due to inadequacy of the displayed information, the user may not be able to accomplish the desired task. To determine the required information to be displayed and the necessary states to be tracked, we propose a model of attaining SA for the users by modeling the user as a specific type of estimator (i.e., the extended delayed functional observer/predictor). We then evaluate what information is needed for such an estimator and how the desired functional of the states have to be expanded so that the user can precisely reconstruct and accurately predict the desired task. As an application example, we investigate the problem of controlling the depth of anesthesia during surgery and determine whether there exists a feasible combination of the expanded task and the displayed information that allows the anesthetist to precisely predict the depth of anesthesia of the patient.

Consensus Control for Heterogeneous Multiagent Systems

We study distributed output feedback control for a heterogeneous multi-agent system (MAS), consisting of N different continuous-time linear dynamical systems. For achieving output consensus, a virtual reference model is assumed to generate the desired trajectory that the MAS is required to track and synchronize. A distinct feature of our results lies in the local optimality and robustness achieved by our proposed consensus control algorithm. In addition our study is focused on the case when the available output measurements contain only relative information from the neighboring agents and reference signal. Indeed by exploiting properties of strictly positive real (SPR) transfer matrices, conditions are derived for the existence of distributed output feedback control protocols, and solutions are proposed to synthesize the stabilizing and consensus control protocol over a given connected digraph. It is shown that design techniques based on the LQG, LQG/LTR and H-infinity loop shaping can all be directly applied to synthesize the consensus output feedback control protocol, thereby ensuring the local optimality and stability robustness. Finally the reference trajectory is required to be transmitted to only one or a few agents and no local reference models are employed in the feedback controllers thereby eliminating synchronization of the local reference models. Our results complement the existing ones, and are illustrated by a numerical example.

From physical linear systems to discrete-time series. A guide for analysis of the sampled experimental data.

Modeling physical data with linear discrete-time series, namely, the autoregressive fractionally integrated moving average (ARFIMA) model, is a technique that has attracted attention in recent years. However, this model is used mainly as a statistical tool only, with weak emphasis on the physical background of the model. The main reason for this lack of attention is that the ARFIMA model describes discrete-time measurements, whereas physical models are formulated using continuous-time parameters. In order to eliminate this discrepancy, we show that time series of this type can be regarded as sampled trajectories of the coordinates governed by a system of linear stochastic differential equations with constant coefficients. The observed correspondence provides formulas linking ARFIMA parameters and the coefficients of the underlying physical stochastic system, thus providing a bridge between continuous-time linear dynamical systems and ARFIMA models.

Non-Asymptotic Kernel-Based Parametric Estimation of Continuous-time

In this paper, a novel framework to address the problem of parametric estimation for continuous-time linear time-invariant dynamic systems is dealt with. The proposed methodology entails the design of suitable kernels of non-anticipative linear integral operators thus obtaining estimators showing, in the ideal case, “non-asymptotic” (i.e., “finite-time”) convergence. The analysis of the properties of the kernels guaranteeing such a convergence behaviour is addressed and a novel class of admissible kernel functions is introduced. The operators induced by the proposed kernels admit implementable (i.e., finite-dimensional and internally stable) state-space realizations. Extensive numerical results are reported to show the effectiveness of the proposed methodology. Comparisons with some existing continuous-time estimators are addressed as well and insights on the possible bias affecting the estimates are provided.

Stable Interval Observers in ${\BBC}$ for Linear Systems With Time-Varying Input Bounds

This technical note deals with the design of stable interval observers and estimators for continuous-time linear dynamic systems under uncertain initial states and uncertain inputs enclosed within time-varying zonotopic bounds. No monotony assumption such as cooperativity is required in the vector field: the interval observer stability directly derives from the stability of the observer state matrix, where any poles (real or complex, single or multiple) are handled in the same way.

Harltley 함수를 이용한 선형시스템의 상태해석에 관한 연구

In this paper Hartley functions are used to approximate the solutions of continuous time linear dynamical system. The Hartley function and its integral operational matrix are first presented, an efficient algorithm to solve the Stein equation is proposed. The algorithm is based on the compound matrix and the inverse of sum of matrices. Using the structure of the Hartley's integral operational matrix, the full order Stein equation should be solved in terms of the solutions of pure algebraic matrix equations, which reduces the computation time remarkably. Finally a numerical example is illustrated to demonstrate the validity of the proposed algorithm.

ON THE CALCULATION OF LYAPUNOV CHARACTERISTIC EXPONENTS FOR CONTINUOUS-TIME LTV DYNAMICAL SYSTEMS USING DYNAMIC EIGENVALUES

The concept of the dynamic eigenvalues may be used, in principle, to formulate in a general way analytic solutions of continuous-time linear time-varying dynamical (LTV) systems. It has also been suggested that the mean value of these quantities may be used to calculate Lyapunov characteristic exponents for the aforementioned systems. In this article, it will be demonstrated that this conjecture is not necessarily valid.

Adaptive Control for Uncertain Continuous-Time Systems Using Implicit Inversion of Prandtl-Ishlinskii Hysteresis Representation

In this note, an implicit inversion approach is introduced to avoid difficulties associated with stability analysis in the direct application of inversion for operator-based hysteresis models. Based on this implicit inversion, an adaptive control algorithm is formulated for continuous-time linear dynamical systems preceded with hysteresis nonlinearities described by the Prandtl-Ishlinskii model. A stability analysis of the controlled system is performed to show that zero-output tracking error can be achieved. Simulation results show the effectiveness of the proposed algorithm.

Memory in linear recurrent neural networks in continuous time

Reservoir Computing is a novel technique which employs recurrent neural networks while circumventing difficult training algorithms. A very recent trend in Reservoir Computing is the use of real physical dynamical systems as implementation platforms, rather than the customary digital emulations. Physical systems operate in continuous time, creating a fundamental difference with the classic discrete time definitions of Reservoir Computing. The specific goal of this paper is to study the memory properties of such systems, where we will limit ourselves to linear dynamics. We develop an analytical model which allows the calculation of the memory function for continuous time linear dynamical systems, which can be considered as networks of linear leaky integrator neurons. We then use this model to research memory properties for different types of reservoir. We start with random connection matrices with a shifted eigenvalue spectrum, which perform very poorly. Next, we transform two specific reservoir types, which are known to give good performance in discrete time, to the continuous time domain. Reservoirs based on uniform spreading of connection matrix eigenvalues on the unit disk in discrete time give much better memory properties than reservoirs with random connection matrices, where reservoirs based on orthogonal connection matrices in discrete time are very robust against noise and their memory properties can be tuned. The overall results found in this work yield important insights into how to design networks for continuous time.

An improved phase method for time-delay estimation

A promising method for estimation of the time-delay in continuous-time linear dynamical systems uses the phase of the all-pass part of a discrete-time model of the system. We have discovered that this method can sometimes fail totally and we suggest a method for avoiding such failures.