Linear Periodic Time-varying System Research Articles

Full Time-Varying Phase Noise Analysis for MOS Oscillators Based on Floquet and Sylvester Theorems

In this paper a set of full time-varying analyzing methods of phase noise for oscillators based on Floquet and Sylvester theorems are established, it provides a good idea for designing oscillators with perfect phase noise performance. The periodic state solution space of a linear periodic time-varying system is constructed with Floquet and Sylvester theorems, and the phase noise perturbation vectors of an oscillator autonomous system are characterized on this space. The analytical expressions of the phase noise spectrums, both 1/(Δf)2 and Lorentzian forms, are obtained, and the contributions to the phase noise of each noise sources are determined. With a generator approach and some modification, the method could be extended to the flicker noise. For RF front-end oscillators composed of MOS active devices, planar inductors and MOS varactors, the time-varying model parameters of the small signal equivalent circuits are constructed according to the periodic varying working-points. By the means of automatic small-signal equivalent-circuit construction, state-variable selection and periodic time-varying state-matrix generation, the system perturbation vectors and phase noise power spectrums are efficiently calculated. For a 10 GHz MOS oscillator, the 1/(Δf)2 and Lorentzian spectrums are calculated. Comparing with the results of SpectreRF, it indicates the proposed methods are accurate and reliable, especially the Lorentzian spectrum close to the carrier is more reasonable than previous methods. Every noise source contributions to the phase noise are listed and the results are analyzed. At last the applications of the methods to designing low phase noise oscillators and to analyzing the phase noise of composite systems, as well as the difficulty of flicker noise analysis, are addressed.

Efficient model reduction of linear periodically time-varying systems via compressed transient system function

This paper presents a new algorithm to obtain reduced-order model of linear periodically time-varying (LPTV) systems. The proposed algorithm introduces the approach of projection-based reduction algorithms, which have been used for reduction of linear time-invariant (LTI) systems, to the domain of LPTV systems. The key idea in the proposed approach is the utilization of integrated congruence transform to project the original LPTV system matrices onto the Hilbert subspace spanned by the time-dependent derivatives (or moments) of the transfer function. We prove that the transfer function of the resulting reduced-order model has the same derivatives as that of the original system in the Laplace-domain. The new approach presents a computationally efficient method to generate the orthogonal transformation operator (used in the integrated congruence transform) through expanding the time-varying transfer function in the right-half plane of the Laplace-domain. This enables using numerical time-domain integration for a very short transient period to generate the orthogonal transformation operator.

Mixed p-z Approach for Time-Domain Analysis of Voltage Source Inverters With Periodic Pulsewidth Modulation

A new method for time-domain analysis of power converters with periodic pulsewidth modulation (PWM) is developed. The method is based on a mixed p-z description of linear periodic time-varying system. The basic of the model is explained in DC-DC converters with periodical PWM. Next, two applications of the model are shown. First, the model was used for the analysis of three-phase voltage source inverters with a space-vector PWM feeding a three-phase R-L load. Second, the same analysis has been used for the single-phase voltage source inverters with periodic PWM. But it is shown that the model is applicable for all types of converters with an explicitly determined output voltage (converters with forced commutation). The mathematical model uses the Laplace and modified Z-transforms. The solution is not dependent on the number of the pulses of the PWM pattern. Instead of solution of algebraic equations, the change of switching instants is reflected in the solution only by a change in two values m/sub k/ and n/sub k/. All results were visualized from the derived equations by the program MathCAD. The derived equations are validated using a 3-kW three-phase inverter.

Modeling and identification of LPTV systems by wavelets

We propose a novel model for discrete linear periodic time varying (LPTV) systems using wavelets. The new model is compared with the ‘raised model’, which is commonly used for modeling LPTV systems. In fact, it turns out that the new model can be viewed as a generalization of the raised model. The wavelets model will be shown to be particularly suitable for adaptive identification of LPTV systems. It offers a compromise between time- and frequency-based algorithms. Time resolution is needed for modeling reasons and minimizing processing delay. Frequency resolution enables faster convergence of adaptive algorithms in general and the least mean square algorithm used here, in particular. Simulations show that for a colored input using the new model results not only in faster convergence compared to the raised model based algorithm, but also produces a lower steady-state error. This, at no significant additional cost in numerical complexity.

Relationship between an LPTV system and the equivalent LTI MIMO structure

It is understood how to transform a linear periodically time-varying (LPTV) filter/ difference equation into an equivalent multiple-input/multiple-output (MIMO) structure, or transfer matrix, with linear time-invariant (LTI) elements, but no published method exists for the reverse operation. The paper presents a technique to transform from the LTI MIMO structure to the original single-input/single-output LPTV difference equation and discusses its implications. In addition, it is shown how this new result is then used to represent parallel and cascade connections of LPTV systems as single LPTV filters, implement order reduction of an LPTV difference equation, and finally obtain an LPTV difference equation representation that is equivalent to an LPTV state-space structure.

Optimal Estimation of the Parameters of Linear Periodically Time‐varying Systems

This paper discusses the linear periodically time-varying (LPTV) system parameter estimation using a block approach. An block algorithm is proposed for optimal estimation of the parameters of LPTV system from the input sequence and the output sequence corrupted by additive Gaussianly distributed noise. In the proposed method, the least squares error criterion has been used.The algorithm provides a useful computational tool based on an appropriate theoretical foundation for parameter estimation of linear time-invariant (LTI) systems from input and output data. Simulation results are presented that demonstrate the performance of the approach.

A Gradient Flow Approach to Optimal Model Reduction of Discrete-Time Periodic Systems

This paper is concerned with the optimal model reduction for linear discrete periodic time-varying systems and digital filters. Specifically, for a given stable periodic time-varying model, we shall seek a lower order periodic time-varying model to approximate the original model in an optimal H2 norm sense. By orthogonal projections of the original model, we convert the optimal periodic model reduction problem into an unconstrained optimization problem. Two effective algorithms are then developed to solve the optimization problem. The algorithms ensure that the H2 cost decreases monotonically and converges to an optimal (local) solution. Numerical examples are given to demonstrate the computational efficiency of the proposed method. The present paper extends the optimal model reduction for linear time invariant systems to linear periodic discrete time-varying systems.

Robustness of discrete periodically time-varying control under LTI unstructured perturbations

Investigates robustness of linear periodically time varying (LPTV) control of discrete linear time invariant (LTI) plants subject to LTI unstructured perturbations. The note first derives a necessary and sufficient condition for robust stability of an LPTV system subject to LTI perturbations, which is less conservative than the well known small gain condition. It then presents a quantitative analysis on the robustness of LPTV control under LTI unstructured perturbations in comparison with that of LTI control. It is shown that under the normal value of the controller period suggested in the previous literature, the stability margin is deteriorated by LPTV control if LTI unstructured perturbations are considered. Hence LTI control is superior to LPTV control in this respect.

Properties of linear switching time-varying discrete-time systems with applications

We study two discrete-time, linear switching time-varying (LSTV) structures, each of which consists of a periodic switch connected to several linear time-invariant (LTI) systems. Such structures can be used to represent any linear periodically time-varying (LPTV) systems. We give basic properties associated with the LSTV structures in terms of their LTI building blocks, and then apply the results to solve a general approximation problem: How to optimally approximate an LPTV system with period p by an LPTV system with period p̂? The optimality is measured using H 2 norms. The study is extended to general multirate periodic systems.

A new approach to stability analysis of variable speed machining systems

This paper presents a new method for the stability analysis of variable speed machining systems. By using spindle angular position as the independent variable, the system dynamics are modeled as a linear periodic time-varying system with fixed delay. This representation is proven much easier to analyze and to numerically simulate than the time-varying delay representation, which traditionally uses the real-time as the independent variable. With a finite difference scheme, the infinite dimensional periodic time-varying system is approximated by a finite dimensional periodic time-varying discrete system, which in turn is converted to a time-invariant system by multiplying the time-varying state transition matrix over one period of speed variation. System-relative stability becomes tractable by spectral radius analysis. This approach makes possible the quantitative characterization of system stability as a function of variable speed profiles as well as other system parameters such as stiffness and damping of the cutting process and the tool/workpiece structure. Verifications for the face milling process by numerical simulation and experiment for both constant and variable speed are given.

Decentralized Control for Linear Time-varying Systems with m-control Stations

Time-varying decentralized control for linear tine-varying systems with m-control stations is studied. It is obtained that under some conditions a time-varying decentralized output feedback can be found such that the closed-loop system is completely controllable and completely observable for a certain channel. With this result, we get a sufficient condition for completely controllability under decentralized information structure for linear time-varying systems with m-control stations. Furthermore, under similar conditions for a linear periodic time-varying system with m-control stations there exists a periodic time-varying decentralized output feedback such that the closed-loop system is uniformly completely controllable and uniformly completely observable for a certain channel. Therefore the periodic time-varying decentralized controller is obtained.

A Constructive Algorithm for Sensitivity Optimization of Periodic Systems

In a recent paper (A. Feintuch, P. P. Khargonekar and A. Tannenbaum, On the sensitivity minimization problem for linear time-varying periodic systems, this Journal, 24 (1986), pp. 1076–1085), the problem of weighted sensitivity optimization was considered for linear, discrete-time, periodic time-varying systems. Here we present a constructive algorithm for solving this problem.

On the Sensitivity Minimization Problem for Linear Time-Varying Periodic Systems

An optimal control problem is formulated in the context of linear, discrete-time, periodic systems. The cost is the supremum over all exogenous inputs in a weighted ball of plant inputs. The controller is required to be causal, periodic of the fixed order of the system and to achieve internal stability. Existence of an optimal controller is proved and a formula for the minimum cost is derived.