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Frequency Responses for Sampled-Data Systems -- Their Equivalence And Relationships
Abstract
There are two ways to introduce the notion of frequency response for sampled-data systems. One is based on the so-called lifting, and the other based on an interpretation of steady-state response in terms of impulse modulation. This paper proves the equivalence of these two notions; in particular, it establishes a more direct link of the second approach to the H∞ norm, and also provides the first approach with a natural interpretation of steady-state response as an infinite sum of sinusoidal signals. This study also leads to a comprehensive account of impulse modulation from the lifting viewpoint.
... In contrast to shift invariant systems, it is hard to relate this output u.t/ to its harmonic input v.t/ D e j!t . There are generalization of the notion of the frequency response to sampled-data (and other periodic) systems, see [23] and the references therein. To account for possible folding effects, these generalizations determine the frequency response at each frequency as an infinite-dimensional operator, which, again, is hard to visualize. ...
... This is fine for the analysis of system norms, but less appropriate for a harmonic analysis. In fact, if applied to the system from v to e in Fig. 5, the methods presented in [23] would result in the frequency response gain larger than one at each frequency, which makes no sense. ...
... The frequency power responses FPR ev .!/ of the reconstruction for l D 1; 2 are presented in Fig. 7. The plots are calculated by (23) and the dashed lines present the non-causal case, l D 1. We are mostly interested in the frequency range ! ...
This paper studies the problem of reconstructing an analog signal from its sampled measurements, in which the sampler (acquisition device) is given and the reconstructor (interpolator/hold) is the design parameter. We formulate this problem as an {{\mbi L}^{\bf 2}} (Wiener/Kalman filtering like) optimization problem and place the main emphasis on a systematic incorporation of causality constraints into the design procedure. Specifically, the optimization problem is solved under the constraint that the interpolation kernel is {\mbi l}-causal for a given {{\mbi l} \in {\BBN}}, i.e., that its impulse response is zero in the time interval {(-{\mbi \infty},-{\mbi l}{\mbi h})}, where {\mbi h} is the sampling period. We present a closed-form state-space solution of the problem, whose computational complexity does not depend on {\mbi l} and which can be efficiently calculated and implemented.
... A recently introduced concept, which is closely related to L 2 -induced norms, is that of the frequency-gain of a SD operator. This concept extends the LTI notion of frequency response to SD systems, in the sense that the maximum magnitude of the frequency-gain of a SD operator is its L 2 -induced norm (Yamamoto & Khargonekar 1993, Hagiwara, Ito & Araki 1995, Yamamoto & Araki 1994, Chen & Francis 1995). Yamamoto & Khargonekar (1993) used lifting techniques to compute the frequency-gain of a general SD system, while obtained similar results for the class of SD compact operators based on the frequency-domain framework developed in Araki,. ...
... Yamamoto & Khargonekar (1993) used lifting techniques to compute the frequency-gain of a general SD system, while obtained similar results for the class of SD compact operators based on the frequency-domain framework developed in Araki,. Relations between both approaches have been discussed in Yamamoto & Araki (1994). Although more general, the procedures proposed in Yamamoto & Khargonekar (1993) do not seem to have an easy numerical implementation. ...
... In other words, if M is a bounded operator in L 2 , and FMF -1 is the corresponding operator in L 2 ( N ; ` 2 ), then its action can be described as (FMF -1 y)(!) = M(!)y(!). An important consequence of these facts is that the L 2 -induced norm of the operator can then be computed as (Bamieh et al. 1991, Yamamoto & Araki 1994) kMk = sup ...
This paper develops exact, computable formulas for the frequency
gain and L2-induced norm of the sensitivity operator in a
sampled-data control system. With sampled data, we refer to a system
that combines both continuous-time and discrete time signals, which is
studied in continuous time. The expressions are obtained using lifting
techniques in the frequency domain and have application in performance
and stability robustness analysis taking into account full intersample
information
... This seems to suggest that digital control below Nyquist frequency is potentially capable to attenuate disturbances above π/T s . On the other hand, based on sampled-data control [4][5][6][7][8], the inherent periodic sampling partitions the continuous-time frequency into infinite regions of [2kπ/T s , 2(k + 1)π/T s ), k = 0, ±1, ±2,...; and a continuous-time disturbance yields a fundamental mode plus an infinite amount of aliases in the partitioned regions. Literature has analyzed the system characteristics (i) by treating all regions as a set [4][5][6][7] and (ii) in the particular region of [0, 2π/T s ) [8,9]. ...
... On the other hand, based on sampled-data control [4][5][6][7][8], the inherent periodic sampling partitions the continuous-time frequency into infinite regions of [2kπ/T s , 2(k + 1)π/T s ), k = 0, ±1, ±2,...; and a continuous-time disturbance yields a fundamental mode plus an infinite amount of aliases in the partitioned regions. Literature has analyzed the system characteristics (i) by treating all regions as a set [4][5][6][7] and (ii) in the particular region of [0, 2π/T s ) [8,9]. Specifically for case (ii), different from pure continuous-time feedback design, high-gain control is no longer capable of fully rejecting the fundamental disturbance if it occurs in [0, 2π/T s ) [9]. ...
A fundamental challenge in digital and sampled-data control arises when the continuous-time plant is subject to fast disturbances that possess significant frequency components beyond Nyquist frequency. Such intrinsic difficulties are more and more encountered in modern manufacturing applications, where the measurement speed of the sensor is physically limited compared to the plant dynamics. The paper analyzes the spectral properties of the closed-loop signals under such scenarios, and uncovers several fundamental limitations in the process.
... Among them, one of the most important results is the introduction of the notion of the frequency response to sampled-data systems. Two types of definitions are introduced in (Yamamoto and Khargonekar, 1996) and (Araki et al., 1996), and their equivalence is studied in (Yamamoto and Araki, 1994). ...
... Remark 1. Another definition of frequency response gain in (Araki et al., 1996) is equivalent to Definition 1 (Yamamoto and Araki, 1994). ...
This paper provides a condition to determine the frequency response gain of sampled-data systems is less than a given positive number for its bisection computation. In contrast to existing conditions, there is no assumption on the norm of the related state space compression operators. The derived condition also unifies the computation of the frequency response gain of sampled-data systems and the induced norm of state-space compression operators.
... This system is periodically time varying, so its frequency-domain analysis is more complicated than that of LTI systems. There are generalization of the notion of the frequency response to sampled-data (and other periodic) systems, see [12] and the references therein. ...
... This is fine for the analysis of system norms, but less appropriate for a harmonic analysis. In fact, if applied to G ev defined above, the methods from [12] would result in the frequency response gain larger than one at each frequency, which makes no sense. ...
The problem of recovering an analog signal from discrete measurements, known as signal reconstruction, can be formulated as a sampled-data L2 fixed-lag smoothing problem. In this paper the problem is solved under the constraint that the reconstructor is an FIR (finite impulse response) system. An analytic and numerically stable solution, whose computational complexity is linear in the length of the reconstructor impulse response, is derived. The formulation can accommodate steady-state constraints via the use of weights with imaginary axis eigenvalues and imposing the stability requirement on the error system.
... We will omit the dependence on K hereafter. This operator is also equivalent [13] to the notion of the so-called FR operator [2]. We denote this operator by T (ω). ...
... Outline of Proof Follow the proof in Yamamoto and Araki [13] given for the unity feedback case. ...
Despite the existence of methods for a direct optimal design of sampled-data control systems, it is often desired to approximate sampled-data systems with discrete-time ones. This occurs frequently in the context of digital redesign, in which one retains intuition in the continuous-time context. This paper investigates and gives conditions under which such a method works. Under some conditions, we guarantee that fast-sampling approximation works well for H∞ sampled-data design, and give an estimate for such an up sampling factor. As an application, we propose a new method for obtaining an FIR controller (possibly with first order approximation). A comparison is made with an existing method.
... The relationship above shows that the response of a sampled-data system is expressible as an infinite sum of sinusoids consisting of aliasing components. This is the viewpoint adopted in [2,14] to discuss the frequency response of sampled-data systems; see also [39]. It is also closer to the classical treatment based on the impulse modulation [28,34]. ...
... The treatment of frequency response given here is based on [40] and [39]. Some computational aspects are discussed in [19]. ...
this article, however, we confine ourselves to the 0-order hold above
... If this time dependence is expressed in an harmonic basis of the type {e jk2π t/T } k∈Z , we essentially obtain the HTF. Relations of this sort are treated in [Yamamoto and Araki, 1994;. In this paper, we investigate what happens when higher harmonics in this basis are truncated. ...
In this paper the balanced truncation procedure is applied to a time-varying linear system, both in continuous and in discrete time. It is discussed how to obtain the reduced-order systems by using certain projections instead of direct balancing. An approximative zero-order-hold discretization of continuous-time systems is described, and a new a priori approximation error bound for balanced truncation in the discrete-time case is obtained. The case study shows that there are several advantages to work in discrete time, including simpler implementation and less computations.
... Unlike the more conventional notion of frequency response where only the sampled behavior is considered, this new notion takes into account the intersampling bahavior, which fails to be captured by the conventional concept. See [14] for the relationship with the concept of aliasing, and [8] for discussions with various numerical examples. ...
This paper proves that the frequency response gains of fast-sample/fasthold approximations of a sampled-data system converge to that of the original system as the sampling rate gets faster. While this may appear to hold trivially, there is a serious technical di#culty, and the proof is indeed nontrivial. It is also guaranteed that this convergence is uniform on the total frequency range. The latter property is necessary to guarantee that a single approximant can be used for frequency response computation for the overall frequency range. A numerical example is given to illustrate the result
... where Φ P (t, τ ) denotes the state-transition matrix associated with P(t). This formula is obtained by a lifting technique (Khargonekar, Poolla, & Tannenbaum, 1985;Yamamoto, 1994;Yamamoto & Araki, 1994) in continuous time, thanks to the fact that the input of (3) has the same period as the filter dynamics P(t). ...
In this paper, the problem of input reconstruction for the general case of periodic linear systems driven by periodic inputs ẋ=A(t)x+A0(t)w(t),y=C(t)x is addressed where x(t)∈Rnx(t)∈Rn and A(t)A(t), A0(t)A0(t), and C(t)C(t) are T0T0-periodic matrices and ww is a periodic signal containing an infinite number of harmonics. The contribution of this paper is the design of a real-time observer of the periodic excitation w(t)w(t) using only partial measurement. The employed technique estimates the (infinite) Fourier decomposition of the signal. Although the overall system is infinite dimensional, convergence of the observer is proven using a standard Lyapunov approach along with classic mathematical tools such as Cauchy series, Parseval equality, and compact embeddings of Hilbert spaces. This observer design relies on a simple asymptotic formula that is useful for tuning finite-dimensional filters. The presented result extends recent works where full-state measurement was assumed. Here, only partial measurement, through the matrix C(t)C(t), is considered.
... , will not vanish as k → ∞. Let the reference signal r(t) be a sinusoidal function e jω0t , ω 0 := 2π/T , then the steady state response of e(t) is given as follows (Yamamoto and Araki, 1994): ...
Repetitive control has been implemented in various industrial applica-tions to allow systems to track or reject unknown periodic signals with a fixed period. Repetitive controllers are usually implemented by digital computers, in which undesired ripples may occur between sampling instants. In this work, we approach this problem via sampled-data H ∞ control. We show that our approach significantly attenuates the ripples without sacrificing the tracking performance.
... The gain of the linear subsystem R T W V is more complicated than in the proof of Theorem 2, as it consists of both continuous-time and sampled systems. This gain can be calculated in a number of ways, but the most useful result for our purposes is presented in Yamamoto and Araki (1994). The ...
This paper considers the problem of checking stability of linear feedback systems with time-varying but bounded delays. Simple but powerful criteria of stability are presented for both continuous-time and discrete-time systems. Using these criteria, stability can be checked in a closed loop Bode plot. This makes it easy to design the system for robustness.
... Besides the continuous time-variable, there is no time-variable which g i v es a fair judgment f o r every signal and appreciate the eeectiveness of asynchronous multirate control correctly. This standpoint of using continuous-time performance measures is similar to recent research on synchronous sampled-data control, to name a few, 4, 19, 31, 1, 10, 32, 29, 5]. To the author's knowledge, the study of multirate control began in 1950s (see 20] and references therein). ...
This paper focuses on multi-rate sample-data control with nonsynchronous decentralized controllers whose sampler-and-hold elements in di erent stations update their state independently of each other. Phases of discrete-time events in decentralized control stations are di erent because of digital control with multiple processors although each controller in a station is a synchronous single-rate sampled-data controller. The paper analyzes the e ect of the nonsynchronous phase distribution on Lp worst-case performance and stability. New representations of the nonsynchronous sample-data system are presented for analyzing and computing performance degradation caused by the nonsynchronism. The method of analysis is further modi ed to propose an approach to design of Lp-induced norm suboptimal nonsynchronous multi-rate controllers. Some results on stability and performance robustness for the nonsynchronous closed-loop systems are developed. Furthermore, it is shown that the worst-case performance measure is a continuous function of the phase shift provided that anti-aliasing lters are located appropriately. Slight perturbation of the phase only results in a slight degradation or improvement of the closed-loop performance. The analysis of the continuity property enables us to estimate how robust the performance is against the phase perturbation.
... What distinguishes this notion from the classical frequency response of discrete-time systems is that the new notion is defined between the continuous-time input and the continuous-time output of the internally stable sampled-data system as shown in Fig. 1, consisting of the continuous-time plant , the discrete-time controller , the hold device and the sampler with sampling period . Such a notion has been introduced with the lifting approach [32], and independently with the FR-operator approach [1], and it is known that the definitions in these two approaches are essentially equivalent [30]. Manuscript received February 11, 1999; revised November 30, 1999 and August 1, 2000. ...
This paper derives a bisection algorithm for computing the
frequency response gain of sampled-data systems with their intersample
behavior taken into account. The properties of the infinite-dimensional
congruent transformation (i.e., the Schur complement arguments and the
Sylvester law of inertia) play a key role in the derivation.
Specifically, it is highlighted that counting up the numbers of the
negative eigenvalues of self-adjoint operators is quite important for
the computation of the frequency response gain. This contrasts with the
well-known arguments on the related issue of the sampled-data
H∞ problem, where the key role is played by the
positivity of operators and the loop-shifting technique. The
effectiveness of the derived algorithm is demonstrated through a
numerical example
... This formula displays the fundamental fact that the frequency response of a sampled signal is built upon the superposition of innnitely many copies of its continuous-time frequency response. The formula has been known for some time in the literature of digital control systems (e.g., Jury 1958, Ragazzini & Franklin 1958), and it has recently been at the basis of a considerable number of works on sampled-data systems (Leung, Perry & Francis 1991, Araki, Ito & Hagiwara 1993, Goodwin & Salgado 1994, Yamamoto & Araki 1994, Freudenberg, Middleton & Braslavsky 1994, Rosenvasser 1995, Hagiwara, Ito & Araki 1995, Braslavsky, Middleton & Freudenberg 1995a, Braslavsky, Middleton & Freudenberg 1995b). Unfortunately, despite the fact that the result appears in many textbooks (e.g., Astrr om & Wittenmark 1990, Franklin, Powell & Workman 1990, Ogata 1987, Kuo 1992, Chen & Francis 1995), it is diicult to nd in the literature a proof that is rigorous and self-contained, and which clearly delineates the classes of signals to which it is applicable. ...
This note provides a new, rigorous derivation of a key sampling formula for discretizing an analogue system. The required conditions are formulated in time-domain, and give a clear characterization of the classes of signals and systems to which the formula applies. Keywords: Z transform, sampled-data systems, frequency response. 1 Introduction A formula that is crucial to the understanding of the frequency-domain properties of a sampled-data system is the following, G d (e sT ) = 1 T 1 X k=-1 G(s + jk! s ); (1) where G is the Laplace transform of a continuous-time signal g, G d is the Z transform of the sequence of its samples, fg(kT)g 1 k=0 , and T and ! s = 2=T denote the sampling period and sampling frequency, respectively. This formula displays the fundamental fact that the frequency response of a sampled signal is built upon the superposition of infinitely many copies of its continuous-time frequency response. The formula has been known for some time in the literature o...
This article gives a brief account on digital control, particularly sampled‐data control where signals are sampled with a uniform rate. We first review basic notions such as sampling and hold, digital‐to‐analog (DA) converter, z‐transform, and pulse transfer functions. The relationship of sampling theorem with aliasing and controllability is also reviewed. We then discuss standard classical design methodologies and problems inherent to them. The problem lies in the fact that sampled‐data systems operate on two time sets ‐ continuous and discrete. This is usually the cause of difficulties encountered in the classical design methods, in particular the oscillatory intersample behavior called ripples. This problem is resolved by introducing a new technique called lifting. It is seen that lifting makes linear continuous‐time time‐invariant plant an infinite‐dimensional discrete‐time system, thereby making the total system a time‐invariant discrete‐time system. Time‐invariant notions, such as steady‐state response, frequency response, transfer functions, can then be fully recovered. This framework makes it possible to formulate and solve robust control such as H2 and H‐infinity control and improve upon the continuous‐time performance dramatically. The same design philosophy is then applied to digital signal processing, where one can use this robust control method to optimize the intersample signals optimally. Robust sampled‐data control is suitably applied to the signal processing context, and is seen to optimally reproduce intersample high‐frequency signals. Applications to digital signal processing are discussed. The article concludes with reviews on the existing literature and recent advances in sampled‐data control as well as discussing some future directions.
This paper presents a family of simplified stability conditions for the filtered Smith predictor control of uncertain systems with time-varying delay. The delay can vary arbitrarily, but its maximum value is assumed to be bounded. Simple design rules are proposed to ensure robust stability in the presence of time-varying delay and model uncertainty. Frequency analysis and design methods can be directly used. Moreover, if (i) the process is open-loop stable and (ii) the closed-loop system without time-varying delay is stable for a given filtered Smith predictor control, then the robustness filter can always be re-designed to recover robust stability in the presence of any bounded time-varying delays. It is shown that the main results can be extended to multivariable systems. Simulation examples and an experimental case study are presented to illustrate the benefits of the proposed stability conditions and design rules.
A fundamental challenge in sampled-data control arises when a continuous-time plant is subject to disturbances that possess significant frequency components beyond the Nyquist frequency of the feedback sensor. Such intrinsic difficulties create formidable barriers for fast high-performance controls in modern and emerging technologies such as additive manufacturing and vision servo, where the update speed of sensors is low compared to the dynamics of the plant. This paper analyzes spectral properties of closed-loop signals under such scenarios, with a focus on mechatronic systems. We propose a spectral analysis method that provides new understanding of the time- and frequency-domain sampled-data performance. Along the course of uncovering spectral details in such beyond-Nyquist controls, we also report a fundamental understanding on the infeasibility of single-rate high-gain feedback to reject disturbances not only beyond but also below the Nyquist frequency. New metrics and tools are then proposed to systematically quantify the limit of performance. Validation and practical implications of the limitations are provided with experimental case studies performed on a precision mirror galvanometer platform for laser scanning.
Frequency analysis is an appropriated tool to study the robustness of a controlled system to either disturbances or uncertainties. In this paper, a multirate based approach allows to consider the intersampling behaviour of the system by selecting a faster sampling rate at the output signal. Using the frequency analysis of multirate systems, the controlled process robustness to model uncertainties and disturbances can be derived. Although some authors claim that a dual rate inferential control could be more robust than a fast single rate one assuming some kind of parametric uncertainty in the process, a counterexample will show the opposite conclusion. Our approach is used to computationally show that this is not always true, and some assumptions about the model uncertainties should be taken. For the sake of comparison, the control performances using a non conventional dual-rate controller are also computed.
When a feedback controller is implemented in a networked embedded system, the computations and communications induce delays and jitter, which may destabilize the control loop. The majority of previous work on analysis of control loops with time-varying delays has focused on output (actuation) jitter. In many embedded systems, input (sampling) jitter is also an issue. In this paper, we analyze the combined effect of input and output jitter on the stability and performance of linear sampled-data control systems. The analysis is performed via a loop transformation involving two time-varying uncertainties. We show how the input-output gains of the linear part of the transformed system can be computed using a fast-sampling/fast-hold approximation. At the same time, we reduce the conservativeness of a previous stability theorem for pure output jitter.
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.
Stability in presence of (possibly varying) time-delays in a control loop consisting of a continuous-time plant and a discrete-time controller is studied. A simple stability criterion is developed, consisting of a graphical test in a Bode plot of the closed loop system. The graphical test makes it very easy to design the controller for time-delay robustness.
This paper is concerned with new discretization methods of continuous-time controllers taking account of the intersample behavior of the resulting sampled-data systems. We first state how to evaluate the controller discretization error based on the Hilbert-Schmidt norm of the sampled-data frequency response, and introduce what we call the desirable frequency response of the discretized controller. We then show that the desirable frequency response can be determined in a closed form in terms of the state-space representations of the original continuous-time controller and the frequency-domain weight. We further discuss some important properties of the desirable frequency response, whereby we propose a heuristic method for controller discretization. We further propose a more sophisticated method, and the effectiveness of these proposed methods is demonstrated by numerical examples. The relationship of the proposed methods with (as well as difference and advances from) similar existing studies are clarified from the viewpoint of the modern theory on the frequency response of sampled-data systems, and the roles and choice of the frequency-domain weight are discussed for better recovery of control performance
Robust performance of sampled-data control for uncertain
finite-dimensional LTI systems with structured and parametric
uncertainties is considered. It is shown that the small gain condition
with dynamic scaling implies robust performance and robust internal
stability in spite of their time-varying characteristics. It is
demonstrated that to prove the sufficiency, the argument of equivalence
between stability and performance problems is unnecessary. This paper
also shows that dynamic scaling gives a less conservative condition even
when considering stability against unstructured uncertainty. A frequency
response approach to analysis and synthesis problems is proposed
This paper applies the frequency response (FR) operator technique
to the robust stability problem of sampled-data systems against
additive/multiplicative perturbations, where a reasonable class of
perturbations consists of unstable as well as stable ones. Under certain
mild assumptions, we show that a small-gain condition in terms of the
FR-operator representation (which is actually equivalent to a small-gain
condition in terms of the L2-induced norm) is still necessary
and sufficient for the sampled-data system to be robustly internally
stable in the sense of exponential stability, in spite of possible
instability of perturbations. The result is derived by a
Nyquist-stability-criterion type of arguments. For the case where a
stronger assumption is made that the perturbations are linear
time-invariant, a necessary and sufficient condition for robust
stability is given. Furthermore, we show that if the plant is either
single-input or single-output, the condition can be reduced to a readily
testable form. Finally, we clarify when the small-gain condition becomes
a particularly poor measure for robust stability
In this paper, we first study strong positive-realness of sampled-data systems and introduce a measure called positive-realness gap index. We show that this index can be computed efficiently with a bisection method, and provide state space formulas for its computation. The importance of this index lies in that it is useful for robust stability analysis of sampled-data systems. An iterative procedure for computing an exact robust stability margin is given and illustrated through a numerical example.
This paper presents a method for computing the frequency response gains and H∞-norm (L2-induced norm) of a sampled-data system composed of continuous-time and discrete-time systems together with a sampler and a hold. The method is based on the notion of FR-operator, and uses only such elementary frequency-domain notions as (conventional) z-transformation, pulse transfer function and impulse modulation formula, while the numerical computations can be carried out using state-space equations. This method turns out to have a natural parallelism to the usual methods for continuous-time and pure discrete-time systems, and the frequency response gain at each frequency can be computed to any degree of accuracy in a numerically reliable fashion.
A key sampling formula for discretising a continuos-time system is proved when the signals space is a subclass of the space of Distributions. The result is applied to the analysis of an open-loop hybrid system.
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 l 2 , which is different from the usual understanding. Fortunately, however, it turns out that this frequency response operator can have an extension onto l 2 so that the equivalence between the time-domain H 2 norm (respectively, the L 2 -induced norm) and the frequency-domain H 2 norm (respectively, the H ∞ 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 l 1 to l 1 , which can also be established via the steady-state analysis.
In this work a new methodology for tracing the frequency response of a multirate system is presented, a multirate system being that in which different blocks appear operating with different sampling periods. Normally, lifting techniques are used to model this kind of system leading to multivariable systems and, in this way, the procedures from this environment are used to obtain the frequency response. It was possible to demonstrate that this methodology is extremely conservative and is also incapable of explaining specific cases of dual-rate controls. The proposed method is applied to the study of the intersample ripple of a multirate control system.
A novel method for the approximation of the frequency response of a sampled data system is presented. Usually the frequency response calculation of sampled-data systems leads to use of the infinite dimension matrices of cumbersome mathematical treatment. The technique presented is based on the inclusion of a fictitious sampler in the continuous process output and on the signal flow analysis in the consequent Kranc operator multirate model obtained by switch decomposition. The main goal of this sampler is to get as much information as it can about the output continuous signal, but finite. The application of this tool will be demonstrated with the calculation of the frequency response of a dual-rate sampled data system, and its validity will be verified by looking correspondence of the results obtained with its time-domain response to a single frequency sine signal input.
In this paper we study how a system with a time-periodic impulse response may be expanded into a sum of modulated time-invariant systems. This allows us to define a linear frequency-response operator for periodic systems, called the harmonic transfer function (HTF). Similar frequency-response operators have been derived before for sampled-data systems and periodic finite-dimensional state-space systems. The HTF is an infinite-dimensional operator that captures the frequency coupling of a time-periodic system. The paper includes analysis of convergence of truncated HTFs. For this reason the concepts of input/output roll-off are developed and related to time-varying Markov parameters.
This paper presents a method for computing the frequency response
gains and H∞-norm of a sampled-data system with its
intersample behavior taken into account. It turns out to have a natural
parallelism to the usual methods for continuous-time and pure
discrete-time systems, and the frequency response gain at each frequency
can be computed to any degree of accuracy in a numerically reliable
fashion
In this paper, we study convergence of truncated representations of the frequency-response operator of a linear time-periodic system. The frequency-response operator is frequently called the harmonic transfer function. We introduce the concepts of input, output, and skew roll-off. These concepts are related to the decay rates of elements in the harmonic transfer function. A system with high input and output roll-off may be well approximated by a low-dimensional matrix function. A system with high skew roll-off may be represented by an operator with only few diagonals. Furthermore, the roll-off rates are shown to be determined by certain properties of Taylor and Fourier expansions of the periodic systems. Finally, we clarify the connections between the different methods for computing the harmonic transfer function that are suggested in the literature.
The fundamental issues of capability of robust and adaptive
control in dealing with uncertainty are investigated in stochastic
systems. It is revealed that to capture the intrinsic limitations of
adaptive control, it is necessary to use supt types of
transient and persistent performance, rather than lim supt
types which reflect only asymptotic behavior of a system. For clarity
and technical tractability, a simple first-order linear time-varying
system is employed as a vehicle to explore performance lower bounds of
robust and adaptive control. Optimal performances of nominal, robust and
adaptive control are explicitly derived and their implications are
discussed in an information framework. An adaptive strategy is
scrutinized for its achievable performance bounds. The results indicate
that intimate interaction and inherent conflict between identification
and control result in a certain performance lower bound which does not
approach the nominal performance even when the system varies very
slowly. Explicit lower bounds are obtained when disturbances are either
normally or uniformly distributed
Properties of the best achievable performance of sampled-data
control systems are investigated. Contrary to our intuition, the best
sampled-data control performance does not always converge to the best
continuous-time control performance, even as the sampling period
approaches zero. A counterexample is presented and a lifting-based
analysis is performed. It is shown that the performance convergence is
guaranteed if, and only if, a certain Hankel norm and an aliasing effect
computed from a specified antialiasing filter tend to vanish with a
decreasing sampling period. It is also shown that this condition is
reduced to a simpler form if the specified filter has special forms.
These results can be generalized to a system involving a generalized
hold
Applies the FR-operator technique to the robust stability problem
of sampled-data systems against additive/multiplicative perturbations,
where a reasonable class of perturbations consists of unstable as well
as stable ones. Assuming that the number of unstable modes of the plant
does not change, we show that a small-gain condition in terms of the
FR-operator representation (which is actually equivalent to a small-gain
condition in terms of the L2-induced norm) is still necessary
and sufficient for the sampled-data system to be robustly stable against
h-periodic perturbations, in spite of their possible instability. The
result is derived by a Nyquist-type of arguments. Next, a necessary and
sufficient condition for robust stability against linear time-invariant
(LTI) perturbations is also given. Furthermore, we show that if the
plant is either single-input or single-output, the condition can be
reduced to a readily testable form. Finally, we clarify when the
small-gain condition becomes a particularly poor measure for robust
stability
5.33> z-transform to specify its operation in discrete-time. To interface these two kinds of systems, sampling/hold operations are introduced. The former is 1 Supported in part by the Murata Science Foundation. - r + f e Phi Phi - C(z) - H - P (s) q y - 6 Gamma Figure 1. Unity Feedback Sampled-Data System denoted by the slanted line segment while the latter is designated by the box H. The objective here is to analyze and design this control system. The lecture started out with the introduction of z-transform, description of how we can compute the z-transform from a given Laplace transform, and then proceeded to the z-domain representation etc. I can still recall with a rather vivid image that I
The first part of the theory which provides a frequency-domain paradigm for the design of sampled-data control systems is developed. The key idea is to cousider the signal space where -ws/2 < φ ≤ ws/2, ws = 2π/Γ is the sampling augular frequency, and τ is the sampling period. In this paper stable open-loop sampled-data systems equipped with strictly-proper pre-filters before samplers are considered. It is shown that such a sampled-data system maps and that this mapping, denoted by , is bounded. It is also shown that the, norm of the sampled-data system as all operator from L2 to L2 is given by .
This paper studies the question of quasi-(approximate) reachability of the standard observable realizations of pseudorational impulse responses introduced by the author. The framework places the current theory of retarded and neutral delay-differential systems into a unified input/output framework. Several necessary and sufficient conditions for quasi-reachability are derived. In particular, new criteria for quasi-reachability and eigenfunction completeness are obtained for general delay-differential systems with no restriction on the type of delays. Furthermore, as a byproduct, the theory leads to necessary and sufficient conditions for approximate left coprimeness of matrices with distribution entries. Examples are discussed to illustrate the theory.
This paper considers a continuous-time linear system with finite jumps at discrete instants of time. An iterative method to compute the -induced norm of a linear system with jumps is presented. Each iteration requires solving an algebraic Riccati equation. It is also shown that a linear feedback interconnection of a continuous-time finite-dimensional linear time-invariant (FDLTI) plant and a discrete-time finite-dimensional linear shift-invariant (FDLSI) controller can be represented as a linear system with jumps. This leads to an iterative method to compute the -induced norm of a sampled-data system.
By sampled data systems we mean systems with continuous time inputs and outputs, but where some state variables evolve in continuous time, and others evolve in discrete time. This paper presents a method for computing the induced norm of sampled data systems where these are considered as operators relating square integrable signals.
This paper studies the problem of characterization and computation of the H # - norm of sampled-data systems using the time-invariant function space model via lifting. With the advantage of time-invariance, the treatment gives an eigenvalue type characterization, first in the operator form in the frequency domain, and then in the Hamiltonian-type finite-dimensional form. The obtained form can be adopted to the bisection algorithm for actual computation. Keywords: sampled-data systems, H # -norm, singular value equation, generalized eigenvalue problem 1
The synthesis of sampled-data controllers for continuous-time systems is studied in the context of H∞ control theory. The sampled-data H∞ optimal control problem consists of finding a sampled-data controller which minimizes the L2-induced norm from the continuous-time disturbance to the continuous-time output. The problem is here solved via a discrete system representation. This is achieved by describing the closed-loop system as a discrete system with finite state dimension, and with a representation of the disturbance and the output signals of the discrete system as sequences of elements of certain infinite-dimensional spaces. Via finite-rank approximations of a compact integral operator associated with the system, the sampled-data H∞ control problem can be solved to any degree of accuracy. From the structure of the problem it follows that irrespective of the order of the approximation, a feasible sampled-data controller can be found which has the same order as the system.
This brief paper considers multivariable sampled-data systems with bandlimited exogenous inputs. A procedure is given for the computation of the 2 induced input to output norm of such a system. In addition, sufficient conditions are given for input-output stability.The digital implementation of an analog controller will generally degrade the performance of the closed-loop system. In light of this, two examples are given to illustrate how this degradation is quantified as a function of the sampling period. In general, this allows the determination of the minimum sampling rate required to recover the performance designed for using the original continuous-time setup.
This paper presents results on the frequency response of sampled data systems. In particular, two new frequency domain sensitivity functions are defined which give qualitative and quantitative information regarding the continuous-time response of these systems. The functions allow a quantification of intersample behaviour.
Translation and rev. ed. of Analyse harmonique des opérateurs de l'espace de Hilbert
Sampled-data systems with continuous-time inputs and outputs,
where some state variables evolve in continuous time and others evolve
in discrete time, are considered. Methods are presented for computing
and optimizing the L 2-induced norm of such systems,
considered as operators relating square integral signals. These
worst-case analysis and design results are counterparts of the
H ∞-norm analysis and synthesis for purely
continuous- or discrete-time systems. The analysis shows that the L
2-induced norm of a sampled-data system is smaller than
γ if certain discrete-time descriptor systems have no eigenvalue
of magnitude one, and if a certain discrete-time linear time-invariant
plant depending on γ has H∞ norm smaller than
γ. The synthesis shows that the optimal attenuation problem for a
four-block continuous-time plant with a digital controller can be solved
with a γ-iteration on a certain discrete-time four-block plant
which depends on γ
H ∞ control and filtering problems for
sampled-data systems are studied. Necessary and sufficient conditions
are obtained for the existence of controllers and filters that satisfy a
specified H ∞ performance bound. When these
conditions hold, explicit formulas for a controller and a filter
satisfying the H ∞ performance bound are also
given
The authors present a framework for dealing with continuous-time periodic systems. The main tool is a lifting technique which provides a strong correspondence between continuous-time periodic systems and certain types of discrete-time time-invariant systems with infinite-dimensional input and output spaces. Despite the infinite dimensionality of the input and output spaces, a lifted system has a finite-dimensional state space if the original system does. This fact permits rather constructive methods for analyzing these systems. As a demonstration of the utility of this framework, the authors use it to describe the continuous-time (i.e., intersample) behavior of sampled-data systems, and to obtain a complete solution to the problem of parameterizing all controllers that constrain the L/sup 2/-induced norm of a sampled-data system to within a certain bound.< >
The authors study the L p input-output
stability of a continuous-time controller, using the usual arrangement
of periodic sampling and zero-order hold. It is noted that even if the
hybrid system is exponentially stable, this arrangement does not yield
L p (1⩽ p <∞) stability in
general. It is shown that this problem can be alleviated if a strictly
causal stable continuous-time filter (e.g. antialiasing filter) is
introduced prior to the sampler. For various configurations, L
p stability is examined in connection with exponential
stability