General Linear Time-invariant Systems Research Articles

Characterizations of output controllability for LTI systems

The objective of this paper is to make, in a simple and rigorous way, some contributions to the notion of output controllability. We first examine, for general linear time-invariant systems, the notion of state to output controllability, introduced in the 60s by Bertram and Sarachik. More precisely, we extend the Hautus test, well-known in the case of state controllability, to state to output controllability, and propose a controllability Gramian matrix, allowing us to build a continuous control achieving a transfer with minimal energy. We also give two other notions of output controllability, namely output to output and globally output to output controllability. For each of these two other new notions, we give necessary and sufficient conditions, in terms of Kalman rank, Hautus test and Gramian matrices.

Resilient Distributed State Estimation for LTI Systems Under Time-Varying Deception Attacks

This article studies the resilient distributed state estimation over the sensor networks under measurement attacks, which make the measurements of variant subsets of sensors aberrant at different time instants. For this problem, while most of the existing works focus on the static target states that do not change over time, we investigate the estimation for the dynamic ones, which evolve according to general linear time-invariant (LTI) systems. To achieve the resilient distributed state estimation for the general LTI systems under the measurement attacks, we propose a dynamic-target regulative gain estimation (DTRGE) algorithm, in which an attack detector, a regulative gain matrix, and an adaptive gain are designed. The detector helps agents monitor the measurement anomalies, and once the attacks are detected, the adaptive gain can counteract the deviation of the estimates induced by them. The regulative gain matrix restrains the negative effects on the convergence of the estimates caused by the system matrix of the target LTI system, especially the unstable one. We demonstrate that all the sensors can recover the target state by running the DTRGE algorithm, if the topology and the observability of the sensor network satisfy certain conditions. Moreover, we further apply the DTRGE algorithm to the sensor networks with switching topologies, and demonstrate that the estimation task can also be completed by these sensors. Finally, simulation and experiment results are given to illustrate the performance of the DTRGE algorithm.

Multivariate moment-matching for model order reduction of quadratic-bilinear systems using error bounds

We propose an adaptive moment-matching framework for model order reduction of quadratic-bilinear systems. In this framework, an important issue is the selection of those shift frequencies where moment-matching is to be achieved. So far, the choice often has been random or linked to the linear part of the nonlinear system. In this paper, we extend the use of an existing a posteriori error bound for general linear time invariant systems to quadratic-bilinear systems and develop a greedy-type framework to select a good choice of interpolation points for the construction of the projection matrices. The results are compared with standard quadratic-bilinear projection methods and we observe that the approximations obtained by the proposed method yield high accuracy.

On Topological Properties of the Set of Stabilizing Feedback Gains

This article presents a fairly complete account on various topological and metrical aspects of feedback stabilization for single-input single-output (SISO) continuous and discrete time linear time-invariant (LTI) systems. In particular, we prove that the set of stabilizing output feedback gains for a SISO system with n states have at most [n/2] connected components. Furthermore, our analysis yields an algorithm for determining intervals of stabilizing gains for general continuous and discrete LTI systems; the proposed algorithm also computes the number of unstable roots in each unstable interval.

The Hankel-Type Lq/Lp Induced Norms of Positive Systems Across Switching

The Hankel-type Lq/Lp induced norms across a single switching over two linear time-invariant (LTI) positive systems are discussed. The norms are defined as the induced norms from vector-valued Lp-past inputs to vector valued Lq-future outputs across a switching at the time instant zero. The Hankel-type L2/L2 induced norm across a single switching for general LTI systems is studied in details to evaluate the performance deterioration caused by switching. Thanks to the strong positivity property, we successfully characterize the Hankel-type Lq/Lp induced norms for the positive system switching even for p, q being 1, 2, ∞. In particular, we will show that some of them are given in the form of linear program and semidefinite program (SDP). The SDP-based characterizations are useful for the analysis of the Hankel-type Lq/Lp induced norms where the systems of interest are affected by parametric uncertainties.

Weighted Consensus Problem for Multiagent Systems with Edge- and Node-Weighted Directed Graphs

In this paper, the concept of consensus is generalized to weighted consensus, by which the conventional consensus, the bipartite consensus, and the cluster consensus problems can be unified in the proposed weighted consensus frame. The dynamics of agents are modeled by the general linear time-invariant systems. The interaction topology is modeled by edge- and node-weighted directed graphs. Under both state feedback and output feedback control strategies, the weighted consensus problems are transformed into the equivalent conventional consensus problems. Then, some distributed state feedback and output feedback protocols are proposed to solve the weighted consensus problems. For output feedback case, a unified frame to construct the state-observer-based weighted consensus protocols is proposed, and different design approaches are discussed. As special cases, some related results for bipartite consensus and cluster consensus can be obtained directly. Finally, a simple example is given to illustrate the effectiveness of our proposed approaches.

<inline-formula> <tex-math notation="LaTeX">$H_2$</tex-math> </inline-formula> Analysis of LTI Systems via Conversion to Externally Positive Systems

Motivated by recent advances in the study of linear time-invariant (LTI) positive systems, we explore analysis techniques of general, not necessarily positive, LTI systems using positive system theory. Even though a positive system is characterized by its peculiar property that its impulse response is nonnegative, we often deal with nonnegative impulse responses even in general LTI system analysis. A typical example is the computation of the $H_2$ norm, where we focus on squared impulse responses. To deal with such products of impulse responses in a systematic fashion, in this paper, we first establish a construction technique of an LTI system whose impulse response is given by the product of impulse responses of two different LTI systems. Then, as the main result, we reduce the $H_2$ norm computation problem of a general LTI system into the $L_\infty$ -induced norm computation problem (or $L_1$ problem in short) of a positive system, by which we can derive various formulas for the $H_2$ norm computation.

Mixed order robust adaptive control for general linear time invariant systems

We provide a solution to the adaptive control problem of an unknown linear system of a given derivation order, using a reference model or desired poles defined in a possibly different derivation order and employing continuous adjustment of parameters ruled by possibly another different derivation order. To this purpose, we present an extension for the fractional settings of the Bezout’s lemma and gradient steepest descent adjustment. We analyze both the direct and indirect approaches to adaptive control. We discuss some robustness advantages/disadvantages of the fractional adjustment of parameters in comparison with the integer one and, through simulations, the possibility to define optimal derivation order controllers.

<i>H</i><sub>2 </sub>Analysis of LTI Discrete-Time Systems via Conversion to Externally Positive Systems

In this paper, we explore H2 analysis techniques of general, not necessarily positive, discrete-time linear time-invariant (LTI) systems by means of positive system theory. While externally positive systems are a particular class of systems whose impulse responses take only nonnegative values, we often encounter such impulse responses even in the analysis of general LTI systems. A typical example is the computation of the H2 norm where we focus on squared impulse responses. In view of this observation, we first clarify that we can construct an SISO externally positive system whose impulse response is given by the square of the impulse response of a given SISO LTI system. This system conversion allows us to reduce the H2 norm computation problem of a general LTI system to the l∞-induced norm computation problem (or l1 problem in short) of the resulting externally positive system. It also turns out that this problem reduction enables us to derive various alternative formulas for the H2 norm of general discrete-time LTI systems.

Reduced optimal models via cross Gramian for continuous linear time‐invariant systems

In this study, the authors focus on the problem of model order reduction (MOR) suitable for continuous linear time-invariant (LTI) systems. Specifically, for single-input single-output (SISO) LTI systems, a couple of MOR algorithms via the cross Gramian are presented. The authors first give a cross Gramian-based optimal MOR iterative algorithm to reduce the SISO system. Further, the authors explore the optimisation problem on the Stiefel manifold. Based on the geometric notions on this manifold, a conjugate gradient iterative algorithm (CGIAHM) is proposed. The conjugate gradient direction used to search for the minimiser is derived by applying the notion of vector transport. It is worth mentioning that the conjugate gradient used in the CGIAHM algorithm is a decent direction for the cost function due to the ingenious construction of this algorithm. In addition, the authors’ algorithms are extended to solve the optimal MOR problem for general LTI systems. Finally, two numerical examples demonstrate the effectiveness of their algorithms.

Inferring Intent for Novice Human-in-the-Loop Iterative Learning Control

This paper studies human-in-the-loop iterative learning control for output-trajectory tracking of linear time-invariant (LTI) systems. Note that knowledge of the intended output is necessary to ascertain the tracking error and find the corrective input needed to reduce the error at each iteration step. In human-in-the-loop systems, the controller does not have direct access to the intended output, and therefore, the tracking error needs to be inferred from human actions. A challenge, however, in inferring the intent and the tracking error is that the human-response dynamics affects the relation between the tracking error (known to the human) and the resulting human action. The main contribution of this paper is to infer the tracking error by inverting a model of the human response and using it for iteratively controlling general LTI systems. This is enabled by modifying the controlled-system dynamics perceived by the human to be of a form for which the human-response model is known. An advantage of the proposed approach is that output tracking is achieved in spite of (potentially) imperfect corrective actions of the novice human user. Results of human-in-the-loop tracking experiments with nine human subjects show that the proposed approach can estimate the human intent (to within 97% accuracy). Moreover, the iterative approach substantially reduces the tracking error (about 93%) when compared with the tracking error achieved by the human alone.

H2 optimal reduced models of general MIMO LTI systems via the cross Gramian on the Stiefel manifold

In this paper, the optimal H2 model order reduction (MOR) problem for general MIMO linear time-invariant (LTI) systems is studied. The cross Gramian can provide the controllability and observability information of LTI systems at the same time and it thus is used to discuss the MOR problem. We first deal with the general MIMO system, obtaining a set of SISO subsystems. Then, the cost function related to each SISO subsystem is expressed via the cross Gramian. The orthogonality constraint of the cost function makes it is posed on the Stiefel manifold. Then, making full use of the geometry properties of this Stiefel manifold, we propose a feasible and effective iterative algorithm to solve the H2 minimization problem. In addition, we show that our algorithm is rigorously convergent. Finally, a couple of examples related to MIMO LTI systems demonstrate the effectiveness of our algorithm.

A Polynomial Method for Stability Analysis of LTI Systems Independent of Delays

A new method providing necessary and sufficient conditions to test delay-independent stability for general linear time-invariant systems with constant delays is proposed. The method is utilized for single delay and incommensurate multiple delay systems. The proposed method offers an approach to determine the exact boundaries of unknown parameters such as controller gains or system parameters ensuring delay-independent stability, in addition to exhibiting an efficient test for real parameters. The technique is based on nonexistence of unitary complex zeros of an auxiliary characteristic polynomial obtained via extended Kronecker summation. A special feature of the polynomial, i.e., the self-inversive property, is proved and utilized to check its unitary zeros to determine delay-independent stability by an efficient zero location test. The methodology is executed employing simple algebraic operations and inspection of the number of sign variations in the obtained sequence. For the single delay case, the pro...

Gradient and shim pre-emphasis by inversion of a linear time-invariant system model.

The goal of this contribution is to enhance the fidelity and switching speed of gradient and shim fields by advancing pre-emphasis toward broadband and full cross-term correction. The proposed approach is based on viewing gradient and shim chains as linear, time-invariant (LTI) systems. Pre-emphasis is accomplished by inversion of a broadband digital system model. In the multiple-channel case, it amounts to a matrix of broadband filters that perform concerted self- and cross-term correction. This approach is demonstrated with gradients and shims up to the third order in a 7 Tesla whole-body MR system. Pre-emphasis by LTI model inversion is first verified by studying settling speeds and response behavior without and with the correction. It is then demonstrated for rapid shim updating, achieving substantially enhanced fidelity of field dynamics and shim settling within approximately 1 ms. In single-shot echo-planar imaging (EPI) acquisitions in vivo, this benefit is shown to translate into enhanced geometric fidelity. The fidelity of gradient and shim dynamics can be greatly enhanced by pre-emphasis based on inverting a general LTI system model. Permitting shim settling on the millisecond scale, broadband multiple-channel pre-emphasis promises to render higher-order shimming fully versatile at the level of MRI sequence design. Magn Reson Med 78:1607-1622, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

Compact tunable silicon photonic differential-equation solver for general linear time-invariant systems.

We propose and experimentally demonstrate an all-optical temporal differential-equation solver that can be used to solve ordinary differential equations (ODEs) characterizing general linear time-invariant (LTI) systems. The photonic device implemented by an add-drop microring resonator (MRR) with two tunable interferometric couplers is monolithically integrated on a silicon-on-insulator (SOI) wafer with a compact footprint of ~60 μm × 120 μm. By thermally tuning the phase shifts along the bus arms of the two interferometric couplers, the proposed device is capable of solving first-order ODEs with two variable coefficients. The operation principle is theoretically analyzed, and system testing of solving ODE with tunable coefficients is carried out for 10-Gb/s optical Gaussian-like pulses. The experimental results verify the effectiveness of the fabricated device as a tunable photonic ODE solver.

A Linear State-Space Analysis of the Migration Model in an Island Biogeography System

Biogeography deals with the study of the distribution of biodiversity over space and time and has been well studied by naturists and biologists for over the last five decades. Recently, the theory of biogeography has been applied to solve difficult engineering optimization problems in the form of a nature-inspired metaheuristic, known as biogeography-based optimization (BBO) algorithm. In this correspondence paper, we present an in-depth analysis of the linear time-invariant (LTI) system model of immigration and emigration of organisms in an island biogeography system that forms the basis of BBO. We find the bound of the eigenvalues of the general LTI system matrix using the Perron-Frobenius theorem from linear algebra. Based on the bounds of the eigenvalues, we further investigate four important properties of the LTI biogeography system, including the system reasonability with probability distribution vectors, stability, convergence, and nature of the equilibrium state. Our analysis gives a better insight into the dynamics of migration in actual biogeography systems and also helps in the understanding of the search mechanism of BBO on multimodal fitness landscapes.

Control via Interconnection and Damping Assignment of Linear Time–Invariant Systems is Equivalent to Stabilizability

AbstractIn this brief note it is shown that the linear matrix inequalities that result from the application of interconnection and damping assignment passivity-based control to general linear time-invariant systems are feasible if and only if the system is stabilizable. A very simple proof of this fact is given that, in contrast to previous results, does not require the assumption that the system has no uncontrollable pole at s = 0.

Incremental bond graph approach to the derivation of state equations for robustness study

Aiming at an automated bond graph based determination of unnormalized frequency domain sensitivities in symbolic form Borutzky and Granda proposed the systematic construction of a so-called incremental bond graph from an initial bond graph for the increments (Δ e)( t), (Δ f)( t) of power variables e( t) and f( t) associated with each bond. This paper shows that the incremental bond graph can serve also as a starting point for setting up symbolically the canonical form as well as the standard interconnection form of state equations used for robustness study. The approach applicable to general linear time-invariant systems is illustrated by means of a fairly small example.

A result on common quadratic Lyapunov functions

In this note, we define strong and weak common quadratic Lyapunov functions (CQLFs) for sets of linear time-invariant (LTI) systems. We show that the simultaneous existence of a weak CQLF of a special form, and the nonexistence of a strong CQLF, for a pair of LTI systems, is characterized by easily verifiable algebraic conditions. These conditions are found to play an important role in proving the existence of strong CQLFs for general LTI systems.

Generalization of a real modal analysis technique

A synthesis of real eigenvector matrices enabling real modal analysis for systems with complex conjugate eigenvalues, is extended to encompass general linear time-in variant systems An exclusively time-domain treatment introducing block matrix invariant operator faculates the modeling and analyzing of the system response.