Problem Of Linear Time-invariant Systems Research Articles

A robust finite–time model reference adaptive controller for arbitrary order disturbed LTI systems

This manuscript deals with the trajectory-tracking problem for linear time-invariant systems with parameter uncertainties and time-dependent external perturbations. A robust finite-time model reference adaptive controller is proposed. In the absence of external perturbations, the proposed controller ensures finite-time convergence to zero of the tracking and parameter identification errors. In presence of time-dependent external perturbations, the tracking and parameter identification errors converge to a region around the origin in a finite time. The convergence proofs are developed based on Lyapunov and input-to-state stability theory. Finally, simulation results in an academic example and a flexible-joint robot manipulator show the feasibility of the proposed approach.

Optimal networked state estimation in the presence of multi-channel correlated packet losses

This paper studies the optimal estimation problem for linear time-invariant systems over lossy communication networks subject to multi-channel packet losses. In its full generality, we assume that the random packet losses are spatially correlated across different channels. The problem is addressed by employing a modified Kalman filter to estimate the system’s states, and the primary issue under study is the convergence of the state estimate under a mean-square criterion. This convergence is shown to be equivalent to the existence of a stabilizing solution to a modified algebraic Riccati equation. We provide a necessary and sufficient condition to characterize the existence of the stabilizing solution, and show that the solution can be computed by solving a set of linear matrix inequalities. Additionally, from a robust convergence perspective, we consider further packet losses whose probabilities are unknown and uncertain. We develop a matrix measure referred to as the critical matrix to characterize the maximal tolerance for the modified Kalman filter to converge. Specifically, we show that the mean-square estimation error is bounded if and only if the critical matrix satisfies a positive semi-definiteness condition.

A data-driven fault isolation and estimation approach for unknown linear systems

This paper considers the data-driven fault isolation and estimation problem for linear time-invariant systems with unknown dynamic matrices and multiple actuator faults. In most of existing fault isolation methods, how to accurately identify the types of faults has not been solved well when the system matrices are unknown. To deal with this problem, a neural network-based fault isolation method is proposed by analyzing and extracting features of different fault models in terms of constructing sparse vectors and function libraries using the available input–output data. Then, a fault estimator is designed to estimate the fault signals within the data-driven framework, where its parameters are computed by the system’s Markov parameters and the identified types of faults. Finally, two examples are used to verify the advantages and effectiveness of the proposed fault isolation and estimation approach.

Complete stability assessment of LTI systems with multiple constant-coefficient distributed delays using the improved frequency sweeping framework

The stability problem of linear time-invariant systems with multiple constant-coefficient distributed delays is investigated from a new perspective. We present an equivalent system with multiple lumped delays and then apply the Dixon resultant to determine the exact upper bound of the projected imaginary spectra of the system. We then adopt the improved frequency sweeping framework over the obtained range to get the kernel and offspring hypersurfaces (KOH). Furthermore, the singularity at the zero characteristic root is scrutinised, leading to what we call the stationary root boundary (SRB) in the domain of the delays. With these, we resort to the Cluster Treatment of Characteristic Roots (CTCR) paradigm to determine the complete stability map of the system with the complete knowledge of the KOH and SRB. Finally, the effectiveness of our proposed procedure is shown by two case studies.

Optimal output regulation for unknown continuous-time linear systems by internal model and adaptive dynamic programming

This paper addresses an optimal output regulation problem for linear time-invariant systems with unknown dynamics. First, a new augmented virtual system is designed to replace the original system and the internal model. Then, by incorporating the data from the augmented virtual system with adaptive dynamic programming (ADP) method, a new iterative learning equation without requiring integral operations or constructing internal model in advance is proposed for establishing the data-driven learning algorithm. Compared with existing ADP-based learning algorithms for linear continuous-time systems, the proposed learning algorithm relaxes both requirements on recording the complete continuous data and setting an initial stabilizing control policy. Finally, the effectiveness of the proposed algorithm is illustrated by an example.

Stabilisation of distributed time-delay systems: a smoothed spectral abscissa optimisation approach

We address the stabilisation problem of linear time-invariant systems with pointwise and distributed delays. We first introduce the concept of the smoothed spectral abscissa, a robust stability measure previously studied for pointwise delay systems. We show that the new stability measure for distributed systems is smooth with respect to the system parameters, and robust in the sense that it provides a trade-off between the decay rate of the solution and the norm of the system. Next, we reformulate the stabilisation problem as an unconstrained minimisation problem considering the smoothed spectral abscissa as the objective function, and show that its properties enable solving the stabilisation problem by using standard gradient-based optimisation techniques. We show the potential of the proposed approach by using it for the stabilisation of a fleet of vehicles, and an oscillator with a class of time-varying delays in the input.

Stability analysis of linear time-invariant systems in the presence of polytopic uncertainty and a time delay state

This paper deals with the robust stability analysis problem of linear time-invariant systems with a state delay and polytopic uncertainty. The delay parameter is constant which does not limited by any upper bound. The system’s matrices linearly depend on uncertain parameters which belong to the unit simplex known as the uncertain space. An extended uncertain system is introduced based on the original delay model that contains the uncertain parameters and two slack parameters. The slack parameters belong to the boundary of the unit circle in two dimensional real space. Then, it has been proven that the delay system is robust stable if and only if the extended uncertain system is robust stable over the uncertain space and the unit circle’s boundary. Since the unit circle’s boundary is not convex, a containing polygonal space is developed that consists of a group of polygonal subspaces. This paper proposes a novel algorithm to check system’s robust stability via linear dependent Lyapunov functions that correspondingly defined for each polygonal subspace. The algorithm is able to establish robust stability of the model for all positive values of the delay parameter. There simulation examples are provided to clarify the notations and facts of this paper. First example presents the notation and results in an instance uncertain delay system. Two extensive examples compare feasibility performances of the proposed algorithm to some existing methods that reveal the superiority of the prosed algorithm.

Remote State Estimation With Stochastic Event-Triggered Sensor Schedule and Packet Drops

This article studies the remote state estimation problem of linear time-invariant systems with stochastic event-triggered sensor schedules in the presence of packet drops between the sensor and the estimator. Due to the existence of packet drops, the Gaussianity at the estimator side no longer holds. It is proved that the system state conditioned on the available information at the estimator side is Gaussian mixture distributed. The minimum-mean-square-error (MMSE) estimator can be obtained from the bank of Kalman filters. Since the optimal estimators require exponentially increasing computation and memory with time, suboptimal estimators to reduce computational complexities by limiting the length and numbers of hypotheses are further provided. In the end, simulations are conducted to illustrate the performance of the optimal and suboptimal estimators.

Eigenvalue Placement by Quantifier Elimination - the Static Output Feedback Problem

This contribution deals with the static output feedback problem of linear time-invariant systems. This is still an area of active research, in contrast to the observer-based state feedback problem, which has been solved decades ago. We consider the formulation and solution of static output feedback design problems using quantifier elimination techniques. Stabilization as well as more specified eigenvalue placement scenarios are the focus of the paper.

Sign-Consensus of Linear Multiagent Systems under a State Observer Protocol

The sign-consensus problem for linear time-invariant systems under signed digraph is considered. The information of the agents’ states is reconstructed, and then, a state observer-type sign-consensus protocol is proposed, whose performance is analyzed using matrix analysis and ordinary differential equation theory. Sufficient conditions for ensuring sign-consensus are given. It is proven that if the adjacency matrix of the signed digraph has strong Perron–Frobenius property or is eventually positive, sign-consensus can be achieved under the proposed protocol. In particular, conventional consensus is a special case of sign-consensus under mild conditions.

Consensus Problem for Linear Time-Invariant Systems with Time-Delay

This work deals with the consensus problem of networks of agents with linear time-invariant dynamics and input time-delay. A predictor-observer scheme that estimates the future value of the system state is considered. The partitioned nature of the predictor allows dealing with larger time-delays than those reported in the literature. The estimated future state of the system is later used in the consensus protocol with the aim of compensating the system input delay. The effectiveness of the solution is shown by means of numerical evaluations.

확률적 불완전 통신 채널에서의 강인제어

In this paper, we consider the robust (HSUB∞/SUB) control problem for linear time-invariant systems over unreliable communication channels that are subject to packet losses from sensors to the controller and from the controller to the actuators. Moreover, the acknowledgments of control packet transmissions are also sent over reverse unreliable communication channels, which is known as the quasi-TCP model. The robust control problem is formulated within a stochastic zero-sum dynamic game framework, from which we obtain an HSUB∞/SUB controller that minimizes the worst-case cost function in the presence of an adversary in the dynamical system. We characterize a set of existence conditions of the HSUB∞/SUB controller in terms of the disturbance attenuation parameter γ and packet loss rates. For the least disturbance attenuation scenario (γ →∝), we show that the HSUB∞/SUB control system converges to the corresponding LQG system. We also show that the HSUB∞/SUB controller for the TCP- and UDP-cases can be arrived at as appropriate limits of the solution for the quasi-TCP problem. Numerical examples are presented to illustrate the theoretical results.

Distributed Kalman Filtering Over Sensor Networks With Unknown Random Link Failures

In this letter we consider the distributed consensus-based filtering problem for linear time-invariant systems over sensor networks subject to random link failures when the failure sequence is not known at the receiving side. We assume that the information exchanged, traveling along the channel, is corrupted by a noise and hence, it is no more possible to discriminate with certainty if a link failure has occurred. Therefore, in order to process the only significant information, we endow each sensor with detectors which decide on the presence of link failures. At each sensor the proposed approach consists of three steps: (1) failure detection; (2) local data aggregation; and (3) Kalman consensus filtering. Numerical examples show the effectiveness of this method.

Observability and stabilisability of networked control systems with limited data rates

ABSTRACTThis paper investigates the observability and stabilisability problem for linear time-invariant systems, where sensors and controllers are geographically separated and connected by a stationary memoryless digital communication channel. The limited information of the plant state is transmitted over such a channel to the controller. We focus on explaining the effect imposed by limited data rates. A new quantisation, coding and control scheme is presented to minimise the required data rate for observability and stabilisability. Different from prior research, our study shows that, the required data rate is determined by the state prediction error. Namely, the smaller prediction error requires the smaller codeword length, which leads to the smaller data rate. It is shown that, there exists a lower bound on the average data rate above which the system is observable and stabilisable. Illustrative examples are given to demonstrate the effectiveness of the proposed quantisation, coding and control scheme.

Approximate causal output tracking for linear perturbed systems via sliding mode control

ABSTRACT A new approach to the solution of what is termed causal output tracking problem for linear time-invariant systems in the presence of both matched and unmatched unmodelled disturbances is presented. The proposed solution is addressed through the design of a dynamic steady state estimator based on the desired system structure and an observer, considering the reference as the system output. The unmatched disturbance is estimated and used in the steady state estimator to achieve invariance, while the matched disturbance robustness is provided via sliding mode control using the super-twisting algorithm. A useful application of the presented techniques is output tracking for non-minimum phase systems; thus, the performed simulation results confirm the effectiveness of the proposed control scheme for this kind of systems.

Minimum information rate for observability of linear systems with stochastic time delay

ABSTRACTThis paper investigates the state estimation problem for linear time-invariant systems where sensors and controllers are geographically separated and connected via stationary memoryless uncertain digital communication channels with the information rate limitation. Such channels introduce data packet dropout, stochastic time delay, and data packet reordering. In the TCP/IP model, the transport layer provides a reliable service on top of an unreliable communication channel. However, the performance of such systems is still affected by packet reordering and stochastic time delay. In particular, we present an optimal encoding and decoding scheme to guarantee observability by employing the minimum information rate, and derive a necessary and sufficient condition on the delay distribution and information rate for observability. Illustrative examples are given to demonstrate the effectiveness of the proposed scheme.

Delay-Dependent Functional Observer Design for Linear Systems With Unknown Time-Varying State Delays.

Partial state estimation has numerous applications in practice. Nevertheless, designing delay-dependent functional observers (FOs) for systems with unknown time delays is rigorous and still an open dilemma. This paper addresses the problem for linear time-invariant systems with state time-varying delays. The delay is assumed to be bounded in an interval with a bounded derivative. A sliding mode FO structure that is robust against the delay uncertainties is established to this aim. The structure employs an auxiliary delay function that can be defined based on the existing knowledge on the actual delay values. Delay-dependent sufficient conditions for the stability of the observer are obtained using the Lyapunov Krasovskii approach, and are expressed in terms of a linear matrix inequality and two rank conditions. The delay-free observer structure is additionally studied and the necessary and sufficient conditions for its stability are obtained. Two descriptive numerical examples and simulation results demonstrate the design procedure and emphasize the effectiveness of the proposed observer design algorithm.

An Augmented Lagrangian Method for the Optimal H∞ Model Order Reduction Problem

This paper treats the computational method of the optimal H∞ model order reduction (MOR) problem of linear time-invariant (LTI) systems. Optimal solution of MOR problem of LTI systems can be obtained by solving the LMIs feasibility coupling with a rank inequality constraint, which makes the solutions much harder to be obtained. In this paper, we show that the rank inequality constraint can be formulated as a linear rank function equality constraint. Properties of the linear rank function are discussed. We present an iterative algorithm based on augmented Lagrangian method by replacing the rank inequality with the linear rank function. Convergence analysis of the algorithm is given, which is distinct to the now available heuristic methods. Numerical experiments for the MOR problems of continuous LTI system illustrate the practicality of our method.

A moment-based approach to ensemble controllability of linear systems

In this paper we introduce a new perspective on the L2-ensemble controllability problem of linear time-invariant systems using an ℓ2-framework. The reformulation results from focussing on the controllability of the ensemble’s moments, which evolve under a linear system defined on the space of square-summable sequences. For a specific class of ensembles, a necessary and sufficient condition can be stated in terms of truncations of infinite Kalman matrices. We illustrate the analysis in this framework on several examples which highlight the possibility of using elementary structural arguments in proving or disproving the controllability of an ensemble’s moments, as well as essential differences to the uniform ensemble controllability property which is mostly considered in the literature.

Output feedback negative imaginary synthesis under structural constraints

The negative imaginary property is a property that many practical systems exhibit. This paper is concerned with the negative imaginary synthesis problem for linear time-invariant systems by output feedback control. Sufficient conditions are developed for the design of static output feedback controllers, dynamic output feedback controllers and observer-based feedback controllers. Based on the design conditions, a numerical algorithm is suggested to find the desired controllers. Structural constraints can be imposed on the controllers to reflect the practical system constraints. Also, the separation principle is shown to be valid for the observer-based design. Finally, three numerical examples are presented to illustrate the efficiency of the developed theory.