Time-invariant Case Research Articles

Structure at infinity and defect of transfer matrices with time-varying coefficients, with application to exact model-matching

We study the structure at infinity of transfer matrices with time-varying coefficients. Such transfer matrices have their entries in a skew field F of rational fractions, i.e. of quotients of skew polynomials. Any skew rational fraction is the quotient of two proper ones, the latter forming a ring Fpr (a subring of F) on which a “valuation at infinity” is defined. A transfer matrix G has both a “generalized degree” and a valuation at infinity, the sum of which is the opposite of the “defect” of G. The latter was first defined by Forney in the time-invariant case to be the difference between the total number of poles and the total number of zeros of G (poles and zeros at infinity included and multiplicities accounted for). In our framework, which covers both continuous- and discrete-time systems, the classic relation between the defect and Forney's left- and right-minimal indices is extended to the time-varying case. The exact model-matching problem is also completely solved. These results are illustrated through an example belonging to the area of power systems.

Discrete-time systems properties defined on state-space regions

In this paper various properties of discrete-time dynamic control systems trajectories with respect to state-space corner regions are considered. The known notion of state-space invariance serves as a basis for derivation of the whole family of dynamics behaviors for which both necessary and sufficient conditions are derived in a general nonlinear case as well as in the linear time-invariant case, shortly LTI-case. Specific examples are given for every case considered, and the proposed notions are analyzed with both theoretical and practical usefulness in mind. For the general nonlinear case a geometric approach is used, which provides a direct insight into the nature of trajectories’ behavior. In the LTI-case a geometric approach is used as well, but it is also translated into the purely algebraic set of conditions allowing for a direct analysis of system matrices. The presented family of control systems expands on the classical state-space invariance (and positive systems) analysis, thus potentially opening new research venues in this branch of control theory and system dynamics in general.

On the Feasibility of Self-Powered Linear Feedback Control

A control system is called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">self-powered</i> if the only energy it requires for operation is that which it absorbs from the plant. For a linear feedback law to be feasible for a self-powered control system, its feedback signal must be colocated with the control inputs, and its input-output mapping must satisfy an associated passivity constraint. The imposition of such a feedback law can be viewed equivalently as the imposition of a linear passive shunt admittance at the actuation ports of the plant. In this paper we consider the use of actively-controlled electronics to impose a self-powered linear feedback law. To be feasible, it is insufficient that the imposed admittance be passive, because parasitic losses must additionally be overcome. We derive sufficient feasibility conditions which explicitly account for these losses. In the finite-dimensional, time-invariant case, the feasibility condition distills to a more conservative version of the Positive Real Lemma, which is parametrized by various loss parameters. Three examples are given, in which this condition is used to determine the least-efficient loss parameters necessary to realize a desired feedback law.

Multiagent Consensus Over Time-Invariant and Time-Varying Signed Digraphs via Eventual Positivity

Laplacian dynamics on signed digraphs have a richer behavior than those on nonnegative digraphs. In particular, for the so-called "repelling" signed Laplacians, the marginal stability property (needed to achieve consensus) is not guaranteed a priori and, even when it holds, it does not automatically lead to consensus, as these signed Laplacians may loose rank even in strongly connected digraphs. Furthermore, in the time-varying case, instability can occur even when switching in a family of systems each of which corresponds to a marginally stable signed Laplacian with the correct corank. In this paper we present conditions guaranteeing consensus of these signed Laplacians based on the property of eventual positivity, a Perron-Frobenius type of property for signed matrices. The conditions cover both time-invariant and time-varying cases. A particularly simple sufficient condition valid in both cases is that the Laplacians are normal matrices. Such condition can be relaxed in several ways. For instance in the time-invariant case it is enough that the Laplacian has this Perron-Frobenius property on the right but not on the left side (i.e., on the transpose). For the time-varying case, convergence to consensus can be guaranteed by the existence of a common Lyapunov function for all the signed Laplacians. All conditions can be easily extended to bipartite consensus.

Fault-Tolerant Coherent $H^\infty$ Control for Linear Quantum Systems

Robustness and reliability are two key requirements for developing practical quantum control systems. The purpose of this article is to design a coherent feedback controller for a class of linear quantum systems suffering from Markovian jumping faults so that the closed-loop quantum system has both fault tolerance and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H^\infty$</tex-math></inline-formula> disturbance attenuation performance. This article first extends the physical realization conditions from the time-invariant case to the time-varying case for linear stochastic quantum systems. By relating the fault-tolerant <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H^\infty$</tex-math></inline-formula> control problem to the dissipation properties and the solutions of Riccati differential equations, an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H^\infty$</tex-math></inline-formula> controller for the quantum system is then designed by solving a set of linear matrix inequalities. In particular, an algorithm is employed to introduce additional quantum inputs and to construct the corresponding input matrices to ensure the physical realizability of the quantum controller. Also, we propose a real application of the developed fault-tolerant control strategy to quantum optical systems. A linear quantum system example from quantum optics, where the amplitude of the pumping field randomly jumps among different values due to the fault processes, can be modeled as a linear Markovian jumping system. It is demonstrated that a quantum <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H^\infty$</tex-math></inline-formula> controller can be designed and implemented using some basic optical components to achieve the desired control goal for this class of systems.

Networked robust stability for linear time-varying systems

A networked control system (NCS) with the communication channels described by cascaded two-port networks for linear time-varying systems is studied. In this framework, a system is considered as a (possibly unbounded) linear operator that maps an input space to an output space. We consider the operator norm bounded uncertainties in the transmission matrices of two-port networks as well as the gap-type uncertainties in the plant and controller. A necessary and sufficient condition for robust stability of the NCS is generalized from the time-invariant case.

DNN-kWTA With Bounded Random Offset Voltage Drifts in Threshold Logic Units.

The dual neural network-based k -winner-take-all (DNN- k WTA) is an analog neural model that is used to identify the k largest numbers from n inputs. Since threshold logic units (TLUs) are key elements in the model, offset voltage drifts in TLUs may affect the operational correctness of a DNN- k WTA network. Previous studies assume that drifts in TLUs follow some particular distributions. This brief considers that only the drift range, given by [-∆, ∆] , is available. We consider two drift cases: time-invariant and time-varying. For the time-invariant case, we show that the state of a DNN- k WTA network converges. The sufficient condition to make a network with the correct operation is given. Furthermore, for uniformly distributed inputs, we prove that the probability that a DNN- k WTA network operates properly is greater than (1-2∆)n . The aforementioned results are generalized for the time-varying case. In addition, for the time-invariant case, we derive a method to compute the exact convergence time for a given data set. For uniformly distributed inputs, we further derive the mean and variance of the convergence time. The convergence time results give us an idea about the operational speed of the DNN- k WTA model. Finally, simulation experiments have been conducted to validate those theoretical results.

Tracking and Regret Bounds for Online Zeroth-Order Euclidean and Riemannian Optimization

We study numerical optimization algorithms that use zeroth-order information to minimize time-varying geodesically convex cost functions on Riemannian manifolds. In the Euclidean setting, zeroth-order algorithms have received a lot of attention in both the time-varying and time-invariant cases. However, the extension to Riemannian manifolds is much less developed. We focus on Hadamard manifolds, which are a special class of Riemannian manifolds with global nonpositive curvature that offer convenient grounds for the generalization of convexity notions. Specifically, we derive bounds on the expected instantaneous tracking error, and we provide algorithm parameter values that minimize the algorithm's performance. Our results illustrate how the manifold geometry in terms of the sectional curvature affects these bounds. Additionally, we provide dynamic regret bounds for this online optimization setting. To the best of our knowledge, these are the first regret bounds even for the Euclidean version of the problem. Lastly, via numerical simulations, we demonstrate the applicability of our algorithm on an online Karcher mean problem.

Suppression of several different modal responses in split-path transmissions by mesh phasing

This paper studies the effectiveness of adopting mesh phasing for suppressing several particular modal responses in a typical split-path transmission. The dynamic model admits three planar degrees of freedom for each gear and considers time-varying mesh stiffnesses, which captures the effect of mesh phasing. The vibration modes of the system are analyzed for the time-invariant case with average mesh stiffnesses, and the characteristics of each mode shape are summarized. Then, according to the characteristic of rigid-body modes, a new method for eliminating the rigid-body displacement of a time-variant system is proposed. This method has a clear physical interpretation and is convenient to use. The modal equations are solved numerically after removal of the equation of the rigid-body mode. It is found that appropriate mesh phasing can effectively suppress several different modal responses, and a physical interpretation of this phenomenon is provided by using the modal equations and the modal analysis results.

Controllability on a class of switched time-varying systems with impulses and multiple time delays

The problem of complete controllability is addressed for a class of switched time-varying impulsive systems under multiple input delays. Several controllability criteria are obtained with the aid of algebraic and geometric tools. Firstly, the explicit expression of solution on every impulsive interval for such systems is derived by mathematical induction and variation of parameters. Then, complete controllability is investigated for the underlying system based on the solution established. Specifically, the new necessary or/and sufficient algebraic controllability conditions are obtained without assuming that all impulse-dependent matrices are non-singular. In addition, the geometric criteria with regard to the constant matrices are further discussed when the considered system reduces to a time-invariant case. Finally, numerical examples are worked out to demonstrate the controllability tests derived in this paper. It is shown that the existence of the delayed term will potentially affect the controllability of the studied system. In contrast to the existing literatures, the criteria derived in this paper removing some constrained assumptions have less conservatism and can be used to verify the controllability of a more general linear hybrid system.

Simultaneous input & state estimation, singular filtering and stability

Input estimation is a signal processing technique associated with deconvolution of measured signals after filtering through a known dynamic system. Kitanidis and others extended this to the simultaneous estimation of the input signal and the state of the intervening system. This is normally posed as a special least-squares estimation problem with unbiasedness. The approach has application in signal analysis and in control. Despite the connection to optimal estimation, the standard algorithms are not necessarily stable, leading to a number of recent papers which present sufficient conditions for stability. In this paper we complete these stability results in two ways in the time-invariant case: for the square case, where the number of measurements equals the number of unknown inputs, we establish exactly the location of the algorithm poles; for the non-square case, we show that the best sufficient conditions are also necessary. We then draw on our previous results interpreting these algorithms, when stable, as singular Kalman filters to advocate a direct, guaranteed stable implementation via Kalman filtering. This has the advantage of clarity and flexibility in addition to stability. En route, we decipher the existing algorithms in terms of system inversion and successive singular filtering. The stability results are extended to the time-varying case directly to recover the earlier sufficient conditions for stability via the Riccati difference equation.

Properties and computation of continuous-time solutions to linear systems

We investigate solutions to the system of linear equations (SoLE) in both the time-varying and time-invariant cases, using both gradient neural network (GNN) and Zhang neural network (ZNN) designs. Two major limitations should be overcome. The first limitation is the inapplicability of GNN models in time-varying environment, while the second constraint is the possibility of using the ZNN design only under the presence of invertible coefficient matrix. In this paper, by overcoming the possible limitations, we suggest, in all possible cases, a suitable solution for a consistent or inconsistent linear system. Convergence properties are investigated as well as exact solutions.

Vector Extensions of Halanay’s Inequality

We provide two extensions of Halanay’s inequality, where the scalar function in the usual Halanay’s inequality is replaced by a vector valued function, under a Metzler condition. We provide an easily checked necessary and sufficient condition for asymptotic convergence of the function to the zero vector in the time-invariant case. For the time-varying cases, we provide a sufficient condition for this convergence, which can be easily checked when the systems are periodic. We illustrate our results in cases that are beyond the scope of prior asymptotic stability results.

Controllability Gramian and Kalman rank condition for mean-field control systems

This paper is concerned with the exact controllability of linear mean-field stochastic systems with deterministic coefficients. With the help of the theory of mean-field backward stochastic differential equations (MF-BSDEs, for short) and some delicate analysis, we obtain a mean-field version of the Gramian matrix criterion for the general time-variant case, and a mean-field version of the Kalman rank condition for the special time-invariant case.

GMM Estimation of Continuous-Time Bilinear Processes

This paper examines the moments properties in frequency domain of the class of first order continuous-timebilinear processes (COBL(1,1) for short) with time-varying (resp. time-invariant) coefficients. So, we used theassociated evolutionary (or time-varying) transfer functions to study the structure of second-order of the process and its powers. In particular, for time-invariant case, an expression of the moments of any order are showed and the continuous-time AR (CAR) representation of COBL(1,1) is given as well as some moments properties of special cases. Based on these results we are able to estimate the unknown parameters involved in model via the so-called generalized method of moments (GMM) illustrated by a Monte Carlo study and applied to modelling two foreign exchange rates of Algerian Dinar against U.S-Dollar (USD/DZD) and against the single European currency Euro (EUR/DZD).

Parallel computation of time-varying convolution

This paper introduces a method for computing the time-varying convolution in parallel. It discusses the motivations for this approach, detailing the limitations with the current serial implementation. A detailed review of the signal processing involved is presented, describing the time-varying filter as a modification of the time-invariant case. This is followed by description of the parallel method, which is then implemented in the Open Computing Language. An analysis of tests result is provided, detailing the improvements on the existing approach and noting the cases where it is not the most suitable option.

New Passivity Conditions for Linear Time Varying Output Feedback Systems

The paper presents new passivity conditions for square linear time varying (LTV) output feedback systems. The new conditions enable the formulation of a new simple test for almost strict passivity, which is necessary for the closed loop to be output strictly passive. The new test requires the solution of an algebraic Riccati equation in the linear time invariant (LTI) case and the solution of a forward differential Riccati equation in the LTV case. The proposed test simplifies the synthesis and design of output strictly passive systems. The examples discussed in the paper demonstrate the efficiency of the test.

Controllability analysis of complex-valued impulsive systems with time-varying delays

This paper studies the controllability of complex-valued impulsive systems with time-varying delays in control input. The impulsive systems considered here have a complex-valued state space, for some typical examples such as biological neural networks and quantum systems can be described by such systems. Using the method of variation of parameters, an explicit expression of the solution for the given system is established. Based on the solution so obtained, several controllability criteria are presented for such systems. Necessary and sufficient conditions for controllability are further derived for the time-invariant case. It is shown that controllability of such systems is dependent on the impulsive function in discrete time, the system matrices in continuous time, and the time delay in control. Three numerical examples are presented to demonstrate the effectiveness of the developed controllability results.

Backstepping Control of Coupled Linear Parabolic PDEs With Space and Time Dependent Coefficients

In this article, the backstepping design of stabilizing state feedback controllers for coupled linear parabolic PDEs with spatially varying distinct diffusion coefficients as well as space and time dependent reaction is presented. The selected target system is a cascade of exponentially stable, time-invariant systems, with time dependent couplings, that is uniformly exponentially stable with a prescribed rate of convergence. To determine the state feedback controller, the kernel equations are derived, which results in a set of coupled PDEs for a time dependent and spatially varying kernel. For this, the method of successive approximations is extended from the time-invariant case to the present problem. The applicability of the method is demonstrated by the stabilization of two coupled unstable parabolic PDEs with space and time dependent coefficients.

Higher-Order ZNN Dynamics

Several improvements of the Zhang neural network (ZNN) dynamics for solving the time-varying matrix inversion problem are presented. Introduced ZNN dynamical design is termed as ZNN models of the order p, $$p\ge 2$$, and it is based on the analogy between the proposed continuous-time dynamical systems and underlying discrete-time pth order hyperpower iterative methods for computing the constant matrix inverse. Such ZNN design is denoted by $$\hbox {ZNN}_H^p$$. Particularly, the $$\hbox {ZNN}_H^2$$ design coincides with the standard ZNN design. Moreover, $$\hbox {ZNN}_H^3$$ design represents a time-varying generalization of the previously defined ZNNCM model. In addition, an integration-enhanced noise-handling $$\hbox {ZNN}_H^p$$ model, termed as $$\hbox {IENHZNN}_H^p$$, is introduced. In the time-invariant case, we present a hybrid enhancement of the $$\hbox {ZNN}_H^p$$ model, shortly termed as $$\hbox {HZNN}_H^p$$, and investigate it theoretically and numerically. Theoretical and numerical comparisons between the improved and standard ZNN dynamics are considered.