Finite-dimensional Linear Continuous-time Periodic Research Articles

Controllability discrepancy and irreducibility/reducibility of Floquet factorisations in linear continuous-time periodic systems

The paper reports interesting but unnoticed facts about irreducibility (resp., reducibility) of Flouqet factorisations and their harmonic implication in term of controllability in finite-dimensional linear continuous-time periodic (FDLCP) systems. Reducibility and irreducibility are attributed to matrix logarithm algorithms during computing Floquet factorisations in FDLCP systems, which are a pair of essential features but remain unnoticed in the Floquet theory so far. The study reveals that reducible Floquet factorisations may bring in harmonic waves variance into the Fourier analysis of FDLCP systems that in turn may alter our interpretation of controllability when the Floquet factors are used separately during controllability testing; namely, controllability interpretation discrepancy (or simply, controllability discrepancy) may occur and must be examined whenever reducible Floquet factorisations are involved. On the contrary, when irreducible Floquet factorisations are employed, controllability interpretation discrepancy can be avoided. Examples are included to illustrate such observations.

Average Floquet factorisations in linear continuous-time periodic systems

The paper examines existence and properties about average Floquet factorisations, determined via average integration instead of matrix logarithms, for the transition matrices of finite-dimensional linear continuous-time periodic (FDLCP) systems. We talk about the following aspects about average Floquet factorisations: existence conditions, properties and relationships with the conventional Floquet factorisations. We reveal that average scaling and modelling may be incomplete or even incorrect for reflecting dynamics in general FDCLP systems, compared to the Floquet factorisations defined through matrix logarithms.

Spectral properties and their applications of frequency response operators in linear continuous-time periodic systems

Spectral properties related to the frequency response operators of finite-dimensional linear continuous-time periodic (FDLCP) systems are examined rigorously and thoroughly in the paper. As applications of the spectral features, positive realness of FDLCP systems is scrutinized and then a harmonic Hamiltonian criterion is derived for the H ∞ norm of FDLCP systems. In particular, positive realness of FDLCP systems is interpreted in term of Toeplitz operators for the first time in this study, together with a testing algorithm. Deriving the harmonic Hamiltonian criterion with a frequency-domain approach bridges the spectra of the frequency response operators in FDLCP systems with their time-domain behaviors.

Zeros and Poles of Linear Continuous-Time Periodic Systems: Definitions and Properties

The paper deals with definitions of zeros and poles and their features in finite-dimensional linear continuous-time periodic (FDLCP) systems under a harmonic framework. More precisely, system and transfer zeros and poles in the harmonic wave-to-wave sense are defined on what we call the regularized harmonic system operators and the harmonic transfer operators of FDLCP systems by means of regularized determinants; then their composition and properties related to system structures are examined via the Floquet theory and controllability/observability decompositions of FDLCP systems. The study shows that under mild assumptions, the harmonic transfer operators of FDLCP systems are analytic and meromorphic, on which zeros and poles are well-defined. Basic zero/pole relationships are established, which are similar to their linear time-invariant counterparts and in particular explicate some interesting harmonic wave-to-wave behaviors of FDLCP systems. The results are significant in analysis and synthesis of FDLCP systems when the harmonic approach is adopted.

Harmonic Lyapunov equations in continuous-time periodic systems: solutions and properties

First, the harmonic Lyapunov equations claimed on the Hilbert space I 2 are restricted to the Banach space I 1 for asymptotic stability of finite-dimensional linear continuous-time periodic (FDLCP) systems. Second, solutions to the harmonic Lyapunov equations are scrutinised. Third, the harmonic Lyapunov equations are connected with periodic matrix differential (PMD) Lyapunov equations. The connection sheds new light on periodic solutions to the PMD Lyapunov equations. Fourth, the trace formula of the H 2 norm for FDLCP systems is shaped with periodic solutions to the PMD Lyapunov equations. Finally, an algorithm for computing periodic solutions to the PMD Lyapunov equations is proposed, which involves only solutions to algebraic Lyapunov equations. There are numerical examples to illustrate the results.

Transfer Functions and Statistical Analysis of FDLCP Systems Described by Higher-order Differential Equations

The paper considers the investigation of finitedimensional linear continuous-time periodic (FDLCP) systems, where the process is described by differential equations of higher than first order, and is influenced by a differentiated input signal. The concept of quasistationary motion mode for such systems is introduced, and closed formulae are derived for calculating statistical characteristics of the quasi-stationary output, when the process is excited by a stationary stochastic signal. Moreover, closed formulae for determining the H2- norm and associated H∞-norms are derived. The methods use a generalization of the parametric transfer matrix (PTM) concept for FDLCP systems of the considered class, and then apply the corresponding classical frequency-domain methods for statistical analysis of continuous-time LTI systems.

Implementing the Hamiltonian test for the [formula omitted] norm in linear continuous-time periodic systems

By the Floquet similarity transformations of finite-dimensional linear continuous-time periodic (FDLCP) systems, the time-domain Hamiltonian test for the H ∞ norm is first interpreted in terms of Toeplitz operators of the system matrices. Based on this novel frequency-domain interpretation, implementing the time-domain Hamiltonian test can be accomplished by working on its frequency-domain counterpart via truncations. This gives us a bisection algorithm for evaluating the H ∞ norm of a class of FDLCP systems through finite-dimensional linear time-invariant continuous-time models. The finite-dimensional Hamiltonian test is necessary and sufficient in the asymptotic sense, and claimed only via Fourier coefficients of the system matrices without the transition matrix of the FDLCP system concerned. There are numerical examples to illustrate the suggested algorithm.

FINITE-DIMENSIONAL MODELS IN EVALUATING THE H2 NORM OF CONTINUOUS-TIME PERIODIC SYSTEMS

Finite-dimensional harmonic models for the H2 norm evaluation of finite-dimensional linear continuous-time periodic (FDLCP) systems are derived, which are expressed explicitly through finitely many Fourier coefficients of the system matrices, and thus dispense with the transition matrix knowledge of any FDLCP models, as opposed to most existing methods in the literature. This paper also shows that the skew- and square-truncated counterparts to the harmonic state operator are invertible in a class of stable FDLCP systems. This invertibility fact, together with the 2-regularized determinant technique about Hilbert-Schmidt operators, plays a key role in justifying the multiple-step truncation on the unbounded harmonic state operators of FDLCP systems and establishing rigorous convergence arguments for the proposed H2 norm formulae and the associated finite-dimensional harmonic models.

STATISTICAL ANALYSIS OF STABLE FDLCP SYSTEMS DESCRIBED BY HIGHER ORDER DIFFERENTIAL EQUATIONS

The paper investigates the response of stable finite dimensional linear continuous-time periodic (FDLCP) systems to white noise input. The FDLCP system is described by differential equations of higher order. Closed formulae for calculating the matrix variance of the output, as well as the mean variance and the H2-norm of the system are derived on basis of the parametric transfer matrix. These formulae only employ matrices of finite dimensions. An example was computed with Matlab.

2-Regularized Nyquist Criterion in Linear Continuous-Time Periodic Systems and Its Implementation

First, by using the 2-regularized determinant technique for Hilbert--Schmidt operators, the computation formula, infinite-product convergence, and analyticity of the 2-regularized determinant of the modified harmonic state operator in finite-dimensional linear continuous-time periodic (FDLCP) systems are derived in this paper. Second, based on these results, a 2-regularized Nyquist criterion is established for asymptotic stability analysis of a class of FDLCP systems for the first time. Third, a numeric implementation algorithm for the 2-regularized Nyquist criterion is also proposed via the staircase truncation on the harmonic transfer operator of the FDLCP system concerned. Finally, to illustrate the results of this paper, asymptotic stability of the lossy Mathieu differential equation is investigated.

Closed formulae for the L2-norm of linear continuous-time periodic systems

The paper presents accurate closed formulae for calculating the L2-norm for solutions of finite dimensional linear continuous-time periodic (FDLCP) systems. The solution is given in form of finite dimensional matrices. The formulae are proven on the basis of the Laplace transformation and the concept of the parametric transfer matrix (PTM).

Existence of harmonic Riccati equations and their solutions in continuous-time periodic systems

By means of the harmonic Lyapunov equation and the contraction mapping theorem, the existence of Toeplitz-operator-valued Riccati equations in finite-dimensional linear continuous-time periodic (FDLCP) systems is verified for the first time in this paper. Properties of solutions of Toeplitz-operator-valued Riccati equations are also derived. Through this novel method different from the frequently adopted Hamiltonian matrix analysis approach, it is revealed that periodic solutions of periodic matrix Riccati equations exist under appropriate assumptions on FDLCP systems matrices and cost function matrices, and analytic properties of the periodic solutions are also investigated. Based on the contraction mapping theorem, an iterative algorithm is suggested for asymptotic computation of the periodic solution of periodic matrix Riccati equations. The iterative algorithm only involves solutions of algebraic Lyapunov equations and Fourier coefficients calculations of periodically time-varying matrices.

Spectral characteristics and eigenvalues computation of the harmonic state operators in continuous-time periodic systems

Spectral characteristics of what we call the harmonic state operators in finite-dimensional linear continuous-time periodic (FDLCP) systems are examined by means of the Fredholm theory for the first time. It is shown that the harmonic state operator is a closed, densely defined Fredholm operator on the Hilbert space l 2, and its spectrum contains only eigenvalues of finite type in any bounded neighborhood of the origin of the complex plane. These spectral characteristics lead us to an asymptotic computation algorithm for the eigenvalues of the harmonic state operator via truncations on its Fredholm regularization. Furthermore, the truncation approach also gives us a necessary and sufficient stability criterion in the FDLCP setting, which only involves the Fourier coefficients of the state matrix. Asymptotic stability of the lossy Mathieu differential equation is investigated to illustrate the results.

Trace formula of linear continuous-time periodic systems via the harmonic Lyapunov equation

In the former part of this paper, a trace formula is established for the H 2 norms of a class of finite-dimensional linear continuous-time periodic (FDLCP) systems based on the solution of the harmonic Lyapunov equations. This trace formula is quite similar in form to what we have for the H 2 norm of an LTI continuous-time system apart from the fact that infinite-dimensional matrices are involved in the FDLCP setting. Based on this formula, in the latter part of this paper some trace formulas are developed via the approximate modelling and truncating approach, which are numerically implementable in most practical FDLCP systems. There are numerical examples to illustrate the efficacy of the trace formulas suggested.

Operational Description and Statistical Analysis of Linear Periodic Systems on the Unboundede Interval −∞

The paper considers finite dimensional linear continuous-time periodic (FDLCP) systems that have no multipliers on the periphery of the unit circle (e-dichotomy). A constructive method for the Green operator of an e-dichotomic FDLCP system is derived that is defined over the set of inputs given on the unbounded interval — ∞ H 2 norm and associated H ∞ norms are given.

H2 and H∞ norm computations of linear continuous-time periodic systems via the skew analysis of frequency response operators

The H 2 and H ∞ norm computations of finite-dimensional linear continuous-time periodic (FDLCP) systems through the frequency response operators defined by steady-state analysis are discussed. By the skew truncation, the H 2 norm can be reached to any degree of accuracy by that of an asymptotically equivalent linear time-invariant (LTI) continuous-time system. The H ∞ norm can be approximated by the maximum singular value of the frequency response of an asymptotically equivalent LTI continuous-time system over a certain frequency range via the modified skew truncation. By the latter result, a Hamiltonian test is proved for FDLCP systems in an LTI fashion, based on which a modified bisection algorithm is developed.

Existence Conditions and Properties of the Frequency Response Operators of Continuous-Time Periodic Systems

The definition of the frequency response operator via the steady-state analysis in finite-dimensional linear continuous-time periodic (FDLCP) systems is revisited. It is shown that the frequency response operator is guaranteed to be well defined only densely on the linear space l2 , which is different from the usual understanding. Fortunately, however, it turns out that this frequency response operator can have an extension onto l2 so that the equivalence between the time-domain H2 norm (respectively, the L2 -induced norm) and the frequency-domain H2 norm (respectively, the $H_{\infty}$ norm of the frequency response operator) is recovered. Under some stronger assumptions, it is also shown that the frequency response operator can be viewed as a bounded operator from l1 to l1 , which can also be established via the steady-state analysis.

Stability analysis of continuous-time periodic systems via the harmonic analysis

Asymptotic stability of finite-dimensional linear continuous-time periodic (FDLCP) systems is studied by harmonic analysis. It is first shown that stability can be examined with what we call the harmonic Lyapunov equation. Another necessary and sufficient stability criterion is developed via this generalized Lyapunov equation, which reduces the stability test into that of an approximate FDLCP model whose transition matrix can be determined explicitly. By extending the Gerschgorin theorem to linear operators on the linear space l/sub 2/, yet another disc-group criterion is derived, which is only sufficient. Stability of the lossy Mathieu equation is analyzed as a numerical example to illustrate the results.

Trace Formulas for the H2 Norm of Linear Continuous-Time Periodic Systems

A trace formula is established for the H2 norms of a class of finite-dimensional linear continuous-time periodic (FDLCP) systems based on the solution of the so-called harmonic Lyapunov equations. The trace formula is similar to what we have for the H2 norm of an LTI continuous-time system apart from the fact that infinite-dimensional matrices are involved in the FDLCP setting. Based on this formula, trace formulas are developed via approximate modeling approach, which are numerically implementable in most practical systems.

H2 and H∞ Norms of Continuous-Time Periodic Systems.: The Skew Truncation Approach: Convergence and Computations

The computations of the H2 and H∞ norms of finite-dimensional linear continuous-time periodic (FDLCP) systems are discussed with the frequency response operators. By the skew truncation approach, it is shown that the H2 norm can be reached to any degree of accuracy by the H2 norm of an asymptotically equivalent LTI continuous-time system. The H∞ norm computation can also be approximated by the maximum singular value of the frequency transfer function of an asymptotically equivalent LTI continuous-time system over a specified frequency range via the modified skew truncation. Hence, a Hamiltonian test is established for FDLCP systems in the LTI continuous-time fashion.