Sampling of Continuous-Time Signals

A common quadratic Lyapunov function (CQLF) guarantees the asymptotic stability of a set of systems. A complete characterization of the set of systems with such a property have been unsuccessful (except for second-order systems). Thus, for both the continuous-time and discrete-time cases, several subsets of linear systems which have a CQLF are known. Some results indicate that there is a parallelism between the continuous-time case and the discrete-time case. In this paper, we show a new subclass for continuous-time systems which have a CQLF by using a property of M-matrices. We also show the discrete-time counterpart of the above new subclass. Next, it is shown that the whole class of continuous-time linear systems having a CQLF is connected directly with its discrete-time counterpart by using a bilinear transformation. For some known subclasses of systems having a CQLF, the transformation gives a one-to-one correspondence between the continuous-time and discrete-time cases. We further show relationships among the obtained results and other, known results.

Analog Signals and Systems

Byrnes et al. (1986) showed that there is no smooth, finite-dimensional, nonlinear time-invariant (NLTI) controller which asymptotically stabilizes every finite-dimensional, stabilizable and detectable, linear time-invariant (LTI) plant (with a fixed number of inputs and outputs). Here we construct a finite-dimensional nonlinear time-varying (NLTV) controller which does exactly that; we treat both the discrete-time and continuous-time cases. With p equal to one in the discrete-time case and the number of plant outputs in the continuous-time case, we first show that for every stabilizable and detectable plant, there exists a p-dimensional linear time-varying (LTV) compensator which provides exponential stabilization; we then construct a (p+1)-dimensional NLTV controller which asymptotically stabilizes every admissible plant by switching between a countable number of such LTV compensators. >

Continuous-time or analog signals are processed using analog devices such as amplifiers, filters, etc. It is impossible to process signals multiplexed from various sources using a single hardware system in the analog domain. On the other hand, digital signals can be processed using both special-purpose hardware and software systems. Worldwide use of Internet, mobile communications, etc. demands all kinds of data such as video, audio, graphics, etc. In order to receive this information on a single device, computer, for instance, it is impossible to use analog signals and techniques. In order to be able to design and implement digitally based systems, it is absolutely necessary to have an understanding of digital signals and systems. Digital signals are discrete in time and amplitude. However, we will assume discrete-time signals to have a continuum of amplitude in order to be able to analyze such signals and systems mathematically. In this chapter we will describe typical discrete-time signals mathematically and then use them to describe and analyze linear time-invariant discrete-time systems. To help the readers understand the mathematical details, we will work out examples followed by MATLAB-based examples. Since digital signals are obtained from analog sources, we will also discuss the conversion of continuous-time signals into digital signals using analog-to-digital converters (ADC).

Chapter 9 - Discrete-Time Signals and Systems

In this paper, the problem of sampling a continuous-time generalized almost-cyclostationary (GACS) signal is addressed. The class of such nonstationary signals includes, as a special case, the almost-cyclostationary (ACS) signals. ACS signals filtered by some linear time-variant channels are further examples. It is shown that the discrete-time signal constituted by the samples of a GACS signal is a discrete-time ACS signal. Thus, the nonstationarity kind of a continuous-time GACS signal cannot be deducted from that of the discrete-time signal of its samples. However, in the paper it is shown how, starting from the sampled signal, the GACS or ACS nature of the continuous-time signal can be conjectured, provided that analysis parameters such as sampling period, padding factor, and data-record length are properly chosen.

Gabor's expansion and the Zak transform for continuous-time and discrete-time signals

– Despite its small size [adult: 60–130 mm, total length (TL)] and limited home range, the round goby (Neogobius melanostomus) has quickly become established throughout the Laurentian Great Lakes. Little information is available, however, on the natural dispersal pattern of this species. This capture–mark–recapture study utilised alphanumeric tags subcutaneously inserted into round gobies (n = 1228) along a 550‐m stretch of the Duluth–Superior Harbor shoreline to observe their movement over a 13‐month period. Recaptured round gobies (n = 415) exhibited highly leptokurtic movement distributions, and movement events were not correlated with fish size, gender or month. Our work indicates that round gobies >50 mm (TL) generally occupy an area less than the minimum sampling interval (25 m); however, occasional movement up to 50 m per day could facilitate range expansion in the Laurentian Great Lakes.

A linear dynamical system resulting from the interconnection of subsystems is con- sidered. Assuming that this interconnection is ''temporal'', i.e. starting at a given initial time in the continuous-time case and ending at a given final time in the discrete-time case, such a system is also said to be ''temporal''. Temporal interconnections generate ''uncontrollable im- pulsive behaviors'' which are not found in the classical theory, though they have been studied for more than 20 years in the case of systems with constant coe‰cients. Determining the struc- ture of the impulsive behavior of a temporal system is a key problem in the theory of linear dynamical systems. It is addressed here, using module theory, for systems with time-varying coe‰cients, in both the continuous- and discrete-time cases. These two cases are merged into a general framework. The impulsive behavior of a temporal system satisfying a suitable regu- larity condition has a structure which is fully elucidated. It turns out that the determination of this structure in practice is an algebraic—not an analytic—problem, which makes the calcu- lations simpler and easier to computerize. The theory is illustrated through several examples.

Global exponential stability of static neural networks with delay and impulses: Discrete-time case

This paper is concerned with the discrete time $H_\infty $ control problem with measurement feedback. It follows that, as in the continuous time case, the existence of an internally stabilizing controller that makes the $H_\infty $ norm strictly less than 1 is related to the existence of stabilizing solutions to two algebraic Riccati equations. However, in the discrete time case, the solutions of these algebraic Riccati equations must satisfy extra conditions.

We offer sufficient and realistic criteria for the existence of positive periodic solutions for a class of non-autonomous scalar equations with deviating arguments. Both the continuous time case and the discrete time case are discussed. Our approach is based on the theory of coincidence degree and the related continuation theorem. To show the usefulness of the results obtained, we include applications to several famous population models.

On the minimum rate of impatience for complicated optimal growth paths

This chapter deals in complete detail with the first two steps involved in converting an analogue signal in to a sequence of binary numbers (called bit stream), specifically the processes of anti-alias filtering and sampling.The sampling theorem that governs distortion-free sampling can only be obeyed in strictly band limited signals that have no frequency components outside a bandwidth B. However, most naturally occurring analogue signals are not strictly band limited in the sense that they do contain stray frequency components outside the bandwidth B. In that case alias distortion is unavoidable and the design goal is to reduce this distortion to a negligible level in order to allow a signal of acceptable quality to be reconstructed from samples even when the original signal is not strictly band limited. This and other issues to do with the sampling of base band and band pass signals as well as measures to mitigate practical constraints and distortions have been discussed in full.

So far, Two-Degree-of-Freedom control system has been used to realize desirable transient response in the tracking system of a Hard Disk Drive (HDD). It is easy to realize a high speed seek, if the actuator of the HDD has a resonance mode at a much higher frequency domain than the desired control domain. However, it is generally difficult to design an actuator that has a resonance mode in a high frequency domain. So, in order to realize a high speed seek, it is important to design a digital control system in consideration of the resonance mode of the actuator. In this case, if the Nyquist frequency of the control system is higher than the resonance mode of the actuator, the control system can be designed by using a method such as a H∞ control theory in consideration of the resonance mode. However, if the Nyquist frequency isn't higher than the resonance mode of the actuator, it is very difficult to realize a high speed seek without mechanical vibration using a conventional method. Then, in this paper the method of applying the N-Delay control theory to the design of the feedforward controller in a Two-Degree-of-Freedom system, and searching for the optimum time allocation of the N-Delay controller based on an evaluation function are proposed. Using these methods, the frequency component of the feedforward control signal can be diminished at the desired frequency point higher than the Nyquist frequency. The validity of the proposed method is confirmed by the numerical and experimental results with a miniature 2.5-inch hard disk drive.