Linear Time-periodic Systems Research Articles

Linear time-periodic dynamical systems: an H2 analysis and a model reduction framework

ABSTRACTLinear time-periodic (LTP) dynamical systems frequently appear in the modelling of phenomena related to fluid dynamics, electronic circuits and structural mechanics via linearization centred around known periodic orbits of nonlinear models. Such LTP systems can reach orders that make repeated simulation or other necessary analysis prohibitive, motivating the need for model reduction. We develop here an algorithmic framework for constructing reduced models that retains the LTP structure of the original LTP system. Our approach generalizes optimal approaches that have been established previously for linear time-invariant (LTI) model reduction problems. We employ an extension of the usual H2 Hardy space defined for the LTI setting to time-periodic systems and within this broader framework develop an a posteriori error bound expressible in terms of related LTI systems. Optimization of this bound motivates our algorithm. We illustrate the success of our method on three numerical examples.

A Numerical Matrix-Based Method for Stability and Power Quality Studies Based on Harmonic Transfer Functions

Some couplings exist between the positive- and negative-sequence impedances of a voltage-sourced power converter especially in the low frequency range due to the nonlinearities and low-bandwidth control loops such as the phase-locked loop. In this paper, a new numerical method based on the harmonic transfer function for analysis of the linear time periodic systems is presented, which is able to handle these couplings. In a balanced three-phase system, there is only one coupling term, but in an unbalanced (asymmetrical) system, there are more couplings, and therefore, in order to study the interactions between these couplings a matrix-based method should be used. No information about the structure of the converter is needed and elements are modeled as black boxes with known terminal characteristics. The proposed method is applicable for both power quality (harmonic and interharmonic emissions) and stability studies, which is verified by simulations in this paper.

Aeroelastic Stability Estimation of Control Surfaces with Freeplay Nonlinearity

This work discusses the quantitative stability evaluation of aeroelastic problems with freeplay in the control surfaces. Stability estimation of linear time invariant and linear time periodic systems rely on eigenanalysis of state transition matrices and implies simplifications on the problems governed by nonlinear non-autonomous equations. Lyapunov Characteristic Exponents directly provide quantitative information on the stability of nonlinear non-autonomous dynamical systems. Stability estimation using Lyapunov Characteristic Exponents does not require a special reference solution and is consistent with the eigensolution of linear time invariant and Floquet-Lyapunov analysis of linear time periodic systems. Thus, they represent a natural generalization of conventional stability analysis. The Discrete QR method is used to practically estimate the Lyapunov Characteristic Exponents. The method is applied to a three-dimensional aeroelastic problem with freeplay introduced to the control surface.

Linear Time Invariant Approximations of Linear Time Periodic Systems

Several methods for analysis of linear time periodic (LTP) systems have successfully been demonstrated using harmonic decompositions. One method recently examined is to create a linear time invariant (LTI) model approximation by expansion of the LTP system states into various harmonic state representations, and formulating corresponding linear time invariant models. Although this method has shown success, it relies on a second order formulation of the original LTP system. This second order formulation can prove problematic for degrees of freedom not explicitly represented in second order form. Specifically, difficulties arise when performing the harmonic decomposition of body and inflow states as well as interpretation of LTI velocities. Instead this paper will present a more generalized LTI formulation using a first order formulation for harmonic decomposition. The new first order approach is evaluated for a UH-60 rotorcraft model, and is used to show the significance of particular harmonic terms; specifically that the coupling of harmonic components of body and inflow states with the rotor states has a significant contribution to the LTI model fidelity in the prediction of vibratory hub loads.

Realization of Actuator‐Fault Estimation for Distributed Wireless Networked Control Systems

AbstractThis paper addresses a study of fault estimation for distributed wireless networked control systems (WNCSs) with actuator multiple faults during industrial automatic processes. Firstly, the model of WNCSs is formulated as linear time periodic systems (LTPSs) according to the periodicity of subsystems in WNCSs. Afterwards, the distributed fault estimation (FE) method for LTPSs is concerned. The strategy of avoiding the problem of structure‐limitation in the FE gain and the realization of the proposed method in the real process are especially detailed. Finally, the proposed FE method is verified on a wireless network control (WiNC) platform.

Trigonometric spline and spectral bounds for the solution of linear time-periodic systems

Linear time-periodic systems arise whenever a nonlinear system is linearized about a periodic trajectory. Examples include anisotropic rotor-bearing systems and parametrically excited systems. The structure of the solution to linear time-periodic systems is known due to Floquet’s Theorem. We use this information to derive a new norm which yields two-sided bounds on the solution and in this norm vibrations of the solution are suppressed. The obtained results are a generalization for linear time-invariant systems. Since Floquet’s Theorem is non-constructive, the applicability of the aforementioned results suffers in general from an unknown Floquet normal form. Hence, we discuss trigonometric splines and spectral methods that are both equipped with rigorous bounds on the solution. The methodology differs systematically for the two methods. While in the first method the solution is approximated by trigonometric splines and the upper bound depends on the approximation quality, in the second method the linear time-periodic system is approximated and its solution is represented as an infinite series. Depending on the smoothness of the time-periodic system, we formulate two upper bounds which incorporate the approximation error of the linear time-periodic system and the truncation error of the series representation. Rigorous bounds on the solution are necessary whenever reliable results are needed, and hence they can support the analysis and, e.g., stability or robustness of the solution may be proven or falsified. The theoretical results are illustrated and compared to trigonometric spline bounds and spectral bounds by means of three examples that include an anisotropic rotor-bearing system and a parametrically excited Cantilever beam.

Fault‐tolerant control for wireless networked control systems with an integrated scheduler

SummaryThis paper addresses a study of fault‐tolerant control (FTC) for wireless networked control systems (WNCSs) in industrial automatic processes. The WNCSs is composed of many subsystems, which operate with different sampling cycles. In order to meet the real‐time requirements and ensure a deterministic data transmission, the time division multiple access (TDMA) mechanism is adopted in WNCSs. The data in WNCSs are transmitted following a TDMA‐based scheduler. According to the periodicity, WNCSs integrated with the scheduler is first formulated as discrete linear time periodic systems (LTPSs). Afterwards, a fault estimation method for LTPSs is developed under a H∞ performance specification with a regional pole constraint. With the achieved state observer and fault estimator, an FTC strategy for LTPSs is explored. Finally, the proposed methods are verified on a physical experimental WiNC platform. Copyright © 2016 John Wiley & Sons, Ltd.

Solution of Linear Time Periodic Dynamical System with Appliction to Electrical Machines Using Chebyshev Polynomial Approch

The magnetic field within electrical machines causes an interaction between the electrical and mechanical dynamics of the system. A unified rotor dynamic model can be developed combining the electrical dynamics and the mechanical dynamics. The electromagnetic behaviour of an electrical machine is a linear time-dependent model which is then easily coupled with a linear model for the mechanical dynamics. The developed coupled model is a linear time periodic system. This work uses the Chebyshev polynomial for solving LTP system. The solution to Mathieu equation is presented before solving dynamics of electrical machines. The Chebyshev polynomial method has been developed and tested for Mathieu equation. Later the method has been applied for electrical machine. The obtained results also compared with modified Runge-Kutta that is available in MATLABTM software package.

A novel approach for the prediction of the milling stability based on the Simpson method

In this paper a new approach is presented for predicting the milling stability based on the Simpson method. Generally the milling dynamic process is described as a linear time-periodic system with a single discrete time delay. By dividing the tooth passing period equally into a finite set of time intervals, the Simpson method is utilized in each time interval to estimate the state items. Then the state transition matrix over one tooth passing period is constructed, and the milling stability could be predicted by the Floquet theory. The convergence rate of the proposed method is discussed, and two benchmark examples are conducted. The results show that the proposed method achieves satisfactory accuracy and efficiency.

Identification of a vertical hopping robot model via harmonic transfer functions

A common approach to understanding and controlling robotic legged locomotion is the construction and analysis of simplified mathematical models that capture essential features of locomotor behaviours. However, the representational power of such simple mathematical models is inevitably limited due to the non-linear and complex nature of biological locomotor systems. Attempting to identify and explicitly incorporate key non-linearities into the model is challenging, increases complexity, and decreases the analytic utility of the resulting models. In this paper, we adopt a data-driven approach, with the goal of furnishing an input–output representation of a locomotor system. Our method is based on approximating the hybrid dynamics of a legged locomotion model around its limit cycle as a Linear Time Periodic (LTP) system. Perturbing inputs to the locomotor system with small chirp signals yield the input–output data necessary for the application of LTP system identification techniques, allowing us to estimate harmonic transfer functions (HTFs) associated with the local LTP approximation to the system dynamics around the limit cycle. We compare actual system responses with responses predicted by the HTF, providing evidence that data-driven system identification methods can be used to construct models for locomotor behaviours.

Stability analysis and backward whirl investigation of cracked rotors with time-varying stiffness

The dynamic stability of dynamical systems with time-periodic stiffness is addressed here. Cracked rotor systems with time-periodic stiffness are well-known examples of such systems. Time-varying area moments of inertia at the cracked element cross-section of a cracked rotor have been used to formulate the time-periodic finite element stiffness matrix. The semi-infinite coefficient matrix obtained by applying the harmonic balance (HB) solution to the finite element (FE) equations of motion is employed here to study the dynamic stability of the system. Consequently, the sign of the determinant of a scaled version of a sub-matrix of this semi-infinite coefficient matrix at a finite number of harmonics in the HB solution is found to be sufficient for identifying the major unstable zones of the system in the parameter plane. Specifically, it is found that the negative determinant always corresponds to unstable zones in all of the systems considered. This approach is applied to a parametrically excited Mathieu’s equation, a two degree-of-freedom linear time-periodic dynamical system, a cracked Jeffcott rotor and a finite element model of the cracked rotor system. Compared to the corresponding results obtained by Floquet’s theory, the sign of the determinant of the scaled sub-matrix is found to be an efficient tool for identifying the major unstable zones of the linear time-periodic parametrically excited systems, especially large-scale FE systems. Moreover, it is found that the unstable zones for a FE cracked rotor with an open transverse crack model only appear at the backward whirl. The theoretical and experimental results have been found to agree well for verifying that the open crack model excites the backward whirl amplitudes at the critical backward whirling rotational speeds.

On the Robust Stability of the Hill Equation with a Delay Term: A Frequency-Domain Approach

The paper considers the problem of robust stability of the Hill equation with a damping and a time-delay term. Analytical results are presented in terms of the coefficients of the equation. The derivation is based on the frequency-domain approach, specifically, the geometric interpretation of the Yakubovich stability criterion for linear time-periodic systems. There are two approaches to this interpretation: One involves analysis of the so-called Lipatov plots and the other is based directly on the Nyquist hodograph.

Stability analysis of linear systems subject to regenerative switchings

This paper investigates the stability of switched linear systems whose switching signal is modeled as a stochastic process called a regenerative process. We show that the mean stability of such a switched system is characterized by the spectral radius of a matrix. The matrix is obtained by taking the expectation of the transition matrix of the system on one cycle of the underlying regenerative process. The characterization generalizes Floquet’s theorem for the stability analysis of linear time-periodic systems. We illustrate the result with the stability analysis of a linear system with a failure-prone controller under periodic maintenance.

Switching control and time-delay identification

The unknown time delay makes the control design a difficult task. When the lower and upper bounds of an unknown time delay of dynamical systems are specified, one can design a supervisory control that switches among a set of controls designed for the sampled time delays in the given range so that the closed-loop system is stable and the control performance is maintained at a desirable level. In this paper, we propose to design a supervisory control to stabilize the system first. After the supervisory control converges, we start an algorithm to identify the unknown time delay, either on-line or off-line, with the known control being implemented. Examples are shown to demonstrate the stabilization and identification for linear time invariant and periodic systems with a single control time delay.

Accurate frequency domain measurement of the best linear time-invariant approximation of linear time-periodic systems including the quantification of the time-periodic distortions

Time-periodic (TP) phenomena occurring, for instance, in wind turbines, helicopters, anisotropic shaft-bearing systems, and cardiovascular/respiratory systems, are often not addressed when classical frequency response function (FRF) measurements are performed. As the traditional FRF concept is based on the linear time-invariant (LTI) system theory, it is only approximately valid for systems with varying dynamics. Accordingly, the quantification of any deviation from this ideal LTI framework is more than welcome. The “measure of deviation” allows us to define the notion of the best LTI (BLTI) approximation, which yields the best – in mean square sense – LTI description of a linear time-periodic LTP system. By taking into consideration the TP effects, it is shown in this paper that the variability of the BLTI measurement can be reduced significantly compared with that of classical FRF estimators. From a single experiment, the proposed identification methods can handle (non-)linear time-periodic [(N)LTP] systems in open-loop with a quantification of (i) the noise and/or the NL distortions, (ii) the TP distortions and (iii) the transient (leakage) errors. Besides, a geometrical interpretation of the BLTI approximation is provided, leading to a framework called vector FRF analysis. The theory presented is supported by numerical simulations as well as real measurements mimicking the well-known mechanical Mathieu oscillator.

Identifying parameters of multi-degree-of-freedom nonlinear structural dynamic systems using linear time periodic approximations

The authors recently presented a new nonlinear system identification method, here dubbed the NL-LTP method, in which the system of interest is forced harmonically so that it responds in a stable periodic orbit, and then it is perturbed slightly and its response is recorded as it returns to the orbit. Under mild assumptions the response about the periodic orbit can be approximated using a linear time periodic system model, which can be identified from the measurements using techniques that are akin to linear modal analysis. While the prior work focused on simulated measurements from single degree-of-freedom systems, this work presents several tools that are needed in order to use this approach on multi-degree-of-freedom systems and focuses on applying the method to experimental hardware. The proposed system identification methodology is unique in that it identifies both the order of the nonlinear system and a mathematical model for the nonlinear restoring forces without assuming the mathematical form for the nonlinearities a priori. Towards these ends, this work explains how to extract the underlying nonlinear system model, or nonlinear restoring force versus displacement relationships, from the time periodic model that governs deviations of the system from its periodic orbit, and presents various metrics that can be used to determine which terms in the model are meaningful. These new tools are used to apply the identification method to a continuous, multi-degree-of-freedom structure with a discrete geometric nonlinearity, using both simulated and experimental measurements. The experimental hardware consists of a cantilever beam with a nonlinear spring attached to its tip, which is driven in a periodic limit cycle by an electromagnetic shaker.

Frequency domain, parametric estimation of the evolution of the time-varying dynamics of periodically time-varying systems from noisy input–output observations

This paper presents a frequency domain, parametric identification method for continuous- and discrete-time, slow linear time-periodic (LTP) systems from input–output measurements. In this framework, the output as well as the input is allowed to be corrupted by stationary noise (i.e. an errors-in-variables approach is adopted). It is assumed that the system under consideration can be excited by a broad-band periodic signal with a user-defined amplitude spectrum (i.e. a multisine), and that the periodicity of the excitation signal Texc can be synchronized with the periodicity of the time-variation Tsys (i.e. Texc/Tsys∈Q), such that the system reaches a steady state (a periodic solution). Tsys is also known as the pumping period. Once the parametric estimation of the time-evolution of the system parameters has been performed, the system model is evaluated at the level of the instantaneous transfer function (also known as system function, or parametric transfer function), which rigorously characterizes LTP systems. If the dynamics of the LTP system are slowly varying or the system is linear parameter varying (LPV), a frozen transfer function approach is provided to easily visualize and assess the quality of the estimated model. To give the estimated quantities a quality label, uncertainty bounds on the model-related quantities (such as the time-periodic (TP) system parameters, the frozen transfer function, the frozen resonance frequency, etc.) are derived in this paper as well. Besides, a clear distinction between the instantaneous and the frozen transfer function concept is made, and both can be estimated with the proposed identification scheme. The user decides which transfer function definition suits best its purpose in practice. Finally, the identification algorithm is applied to a simulation example and to real measurements on an extendible robot arm.

On the finite element modeling of the asymmetric cracked rotor

The advanced phase of the breathing crack in the heavy duty horizontal rotor system is expected to be dominated by the open crack state rather than the breathing state after a short period of operation. The reason for this scenario is the expected plastic deformation in crack location due to a large compression stress field appears during the continuous shaft rotation. Based on that, the finite element modeling of a cracked rotor system with a transverse open crack is addressed here. The cracked rotor with the open crack model behaves as an asymmetric shaft due to the presence of the transverse edge crack. Hence, the time-varying area moments of inertia of the cracked section are employed in formulating the periodic finite element stiffness matrix which yields a linear time-periodic system. The harmonic balance method (HB) is used for solving the finite element (FE) equations of motion for studying the dynamic behavior of the system. The behavior of the whirl orbits during the passage through the subcritical rotational speeds of the open crack model is compared to that for the breathing crack model. The presence of the open crack with the unbalance force was found only to excite the 1/2 and 1/3 of the backward critical whirling speed. The whirl orbits in the neighborhood of these subcritical speeds were found to have nearly similar behavior for both open and breathing crack models. While unlike the breathing crack model, the subcritical forward whirling speeds have not been observed for the open crack model in the response to the unbalance force. As a result, the behavior of the whirl orbits during the passage through the forward subcritical rotational speeds is found to be enough to distinguish the breathing crack from the open crack model. These whirl orbits with inner loops that appear in the neighborhood of the forward subcritical speeds are then a unique property for the breathing crack model.

State space model identification: from unstructured to structured models with an H∞ approach

Subspace model identification algorithms have become extremely popular in the last few years thanks to the ease with which they can provide consistent estimates for MIMO state space models in a non iterative way, exploiting a full parameterisation of the system matrices. The only, well known, downside of this approach is the impossibility to impose a fixed basis to the state space representation, and therefore the difficulties in recovering physically-motivated models. The problem considered in this paper is the one of recovering the numerical values of the physical parameters of a structured representation of the system starting from a fully parameterised identified model. It will be shown that this can be achieved without explicitly constructing the similarity transformation, for linear time-invariant systems, linear time-periodic systems and linear parameter-varying systems identified from a periodic scheduling sequence. Two numerical examples are presented to illustrate the approach.

Robust Stability Analysis of a Linear Time-PeriodicActive Helicopter Rotor

Robust stability of a helicopter hingeless rotormodel used for air resonance alleviation is investigated in this paper. This aeroelastic phenomenon can be described naturally considering the rotor as a whole through a time-periodic change of coordinates called multiblade coordinates transformation. To assess robust stability of the resulting continuous linear time-periodic system, this paper defines a set of uncertainties specific to the helicopter rotor problem. The uncertain system is written in the form of a linear fractional representation and transformed using a frequency-lifting technique. This leads to a time-invariant representation of the uncertain system, the so-called harmonic transfer function. The extension of the standard -analysis to such lifted systems is proposed in the paper. Compared with a similar approach from literature based on an elementary example, the technique requires a moderate increase of the number of inputs and outputs, thus reducing the numerical effort involved in the computation of the -bounds. Themethodology, applied to the uncertain closed-loop helicopter rotor, shows robust stability for the defined set of uncertainty. The results are compared to an analysis performed on the base of a linear time-invariant system to quantify how strongly robust stability bounds are underestimated when the periodicity is not accounted for.