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Inherent design limitations for linear sampled-data feedback systems
Abstract
There is a well-developed theory describing inherent design limitations for linear time invariant feedback systems consisting of an analogue plant and analogue controller. This theory describes limitations on achievable performance present when the plant has non-minimum phase zeros, unstable poles, and/or time delays. The parallel theory for linear time invariant discrete time systems is less interesting because it describes system behaviour only at sampling instants. This paper develops a theory of design limitations for sampled-data feedback systems wherein the response of the analogue system output is considered. This is done using the fact that the steady-state response of a hybrid feedback system to a sinusoidal input consists of a fundamental component at the frequency of the input together with infinitely many harmonics at frequencies spaced integer multiples of the sampling frequency away from the fundamental. This fact allows fundamental sensitivity and complementary sensitivity functions that relate the fundamental component of the response to the input signal to be defined. These sensitivity and complementary sensitivity functions must satisfy integral relations analogous to the Bode and Poisson integrals for purely analogue systems. The relations show, for example, that design limitations due to non-minimum phase zeros of the analogue plant constrain the response of the sampled-data feedback system regardless of whether the discretized system is minimum phase and independently of the choice of hold function.
... , and a continuous-time disturbance yields a fundamental mode plus an infinite number of shifted replicas in the partitioned regions. Due to the sampled-data architecture, the conventional concept of frequency responses does not apply to evaluate the full system performance here [7]- [12]. Three variations are introduced: (i) the fundamental transfer function (FTF) [11], (ii) the performance frequency gain (PFG) [13], [14], and (iii) the robust frequency gain (RFG) [7]. ...
... . , which equals 0 from (12). For the case where the disturbance is beyond Nyquist frequency in Fig. 6b, because there is little control over 1 − Γ 0 (Ω o ), and Γ ±k (Ω o ) (k = 0) is amplified, the aliasing effect cancels the fundamental component after sampling. ...
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.
... 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]. ...
... 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.
... The progress of the researches has also led to the development of Bode and Poisson type integrals [7,9,10,32], making available a series of integral constraints, in either equality or inequality, on the sensitivity and complementary sensitivity functions applicable to multi-variable systems. Other relevant extensions have been pursued in [26] and [28], toward problems pertaining to sampled-data systems and filter design. ...
... Problems concerning on the fundamental design limitations in sampleddata control systems have been widely investigated since last decade [26,29,47] and one pertains to the tracking performance limitations of stable plant is recently studied in [15], which gives the analytical closed-form expression of the optimal tracking performance. In this paper, the problem of tracking a step reference signal using sampled-data control systems is studied by adopting a frequency domain lifting technique. ...
This thesis is devoted to a research area that studies the fundamental performance limitation and trade-off of feedback control, a subject intensively developed in the linear time-invariant feedback systems, beginning with the classical work of Bode in the 1940s on logarithmic sensitivity integrals. In modern control design, the studies on performance limitation serve as an appendage tool since they help a control system designer specifies reasonable control objectives, and understands the intrinsic limits and the trade-off between conflicting design considerations.
In this thesis we quantify and characterize the fundamental performance limitations arise in H2 optimal tracking and regulation control problems of single-input multiple-output (SIMO) linear time-invariant (LTI) feedback control systems. In tracking problem, the control performance is measured by the tracking error response, possibly under control input constraint, with respect to a step reference input. While in regulation problem, the performance is measured by the energy of the measurement output simultaneously with that of the control input and sensitivity constraints, against an impulsive disturbance input.
Our primary interest is not on how to find the optimal or robust controller. Rather, we are interesting in relating the optimal performance with some simple characteristics of the plant to be controlled. In other words, we provide the analytical closed-form expressions of the optimal performance in terms of dynamics and structure of the plant. The analytical expressions, however, constitute guidelines for designing an easily controllable plant in practical situations, from which the control system designer may rely in determining the optimal design parameters and reasonable control strategies.
We mostly focus our attention on tracking and regulation problems of discrete-time systems. Toward the existing results of continuous-time systems, we make small corrections and perform a few extensions. We then reformulate and resolve both problems in delta domain. An analysis on the continuity property shows that we can completely recover the continuous time expressions from the delta domain expressions stand point as sampling time approaches zero. Frankly speaking, we provide comprehensive and unified expressions on the characterization of the control performance limitations in the H2 tracking and regulation problems. Furthermore, our analytical expressions show that the optimal tracking and regulation performances are explicitly characterized by the plant’s non-minimum phase zeros and unstable poles as well as the plant gain. We confirm the effectiveness of the derived expressions by several illustrative examples. We also show how to apply the analytical expressions to practical applications including the control of three-disk torsional system, the determination of the optimal parameters in inverted pendulum system, and the sensor selection in magnetic bearing system. In addition, by exploiting the delta domain expressions we derive the analytical closed-form expressions of the optimal tracking and regulation performances for delay-time systems and by invoking our discrete-time LTI results we provide the similar expressions to approximate the optimal tracking performance for sampled-data systems.
... Nevertheless, in contrast with FOH and FROH, rather a poor intersample behavior results (Feuer & Goodwin, 1994. Although this can be alleviated (Juan & Kabamba, 1991;Arvanitis, 1998;Liang & Ishitobi, 2004), the fact is that for a sampled-data system with a discrete integrator to be able to reject step disturbances in continuous time, the impulse response of the hold in question must have s-zeros where ZOHs have theirs, which is not met by a generic GSHF (Freudenberg, Middleton & Braslavsky, 1995). ...
... if the hold in question is to reject step disturbances in continuous time (along with a discrete integrator, Freudenberg et al., 1995). With a ZOH we have ...
We present a generalized sampled-data hold function that combines arbitrary z-domain zero-placement ability with zero-order hold behavior under constant input, thus exhibiting minimal intersample ripple by design. Our hold can be regarded as a generalization of the fractional-order hold, with a polynomial instead of a simple linear pattern, and therefore with as many tuning parameters as desired. Moreover, the polynomial approach turns out to provide a simple mechanism of control energy minimization. Among
other benefits, all these features help to achieve qualitatively better perfect model referencing, because
problematic sampling z-zeros or the intersample issues of the conventional generalized holds need no
longer be endured.
... The key tools used in this paper are the fundamental sensitivity and complementary sensitivity functions (denoted S fun s and T fun s) introduced by Freudenberg et al. (1995) in the study of fundamental design limitations for sampled-data feedback control systems; see also (Braslavsky, 1995;Freudenberg et al., 1997). While these functions are not transfer functions in the usual sense, they do play a key role in governing the tracking and disturbance rejection response of sampled-data systems, and are more readily calculated than the complete sampled-data frequency response (Yamamoto and Khargonekar, 1996). ...
... The paper is organized as follows. In x2, we review the notion of the frequency response of sampleddata systems as presented in the work of Freudenberg et al. (1995), Braslavsky (1995), and Braslavsky et al. (1998). In x3 we use this frequency-domain based framework to investigate the implications of cancellation (or near-cancellation) of sampling zeros on the fundamental complementary sensitivity operator T fun s. ...
In this paper, we investigate the implications for robust
sampled-data feedback design of minimum phase sampling zeros appearing
in the transfer function of discrete-time plants. Such zeros may be
obtained by zero-order hold (ZOH) sampling of continuous-time models
having relative degree two or greater. In particular, we address the
robustness of sampled-data control systems to multiplicative uncertainty
in the model of the continuous-time plant. We argue that lightly damped
controller poles, which may arise from attempting to cancel, or almost
cancel, sampling zeros of the discretized plant are likely to introduce
peaks into the fundamental complementary sensitivity function near the
Nyquist frequency. This in turn makes the satisfaction of necessary
conditions for robust stability difficult for all but the most modest
amounts of modeling uncertainty in the continuous-time plant. Some
H2- and H∞-optimal discrete-time and sampled
data designs may lead to (near-) cancellation, and we therefore argue
that their suitability is restricted
... Al contrario que los controladores lineales, los controladores reseteados no están afectados por las llamadas limitaciones lineales fundamentales. Por ejemplo, cuando en bucle abierto, nuestro sistema presenta un integrador, ha quedado probado que independientemente del regulador escogido, la señal de error está restringida por la igualdad ∞ 0 e(τ )dτ = 1/K v [22], donde K v es la ganancia de velocidad. La planta (6) para control lateral contiene un doble integrador y otras raíces, con lo que K v = ∞. ...
... optimal performance indices, concerning, e.g., cheap regulation (K wakernaak and Sivan, 1972), servomechanism (Qiu and Davison, 1993), and optimal tracking (Morari and Zafiriou, 1989;Chen et al., 1996). These developments have been substantial and are continuing to branch to different problems and for different system categories; recent extensions are found for, e.g., filtering problems (Goodwin et al., 1995), and sampled data systems (Freudenberg et al., 1994). The understanding in the limitations of feedback control has also been compelling. ...
Nevanlinna-Pick interpolation techniques are employed in this paper to derive exact expressions and bounds for the best attainable H∞ norms of the sensitivity and complementary sensitivity functions of linear time-invariant multivariable systems. These results improve the previously known performance bounds and provide intrinsic limits on the performance of feedback systems irreducible via compensator design, leading to new insights into the understanding of fundamental limitations in feedback control. It becomes clear that in a multivariable system the best achievable performance is constrained by plant nonminimum phase zeros and unstable poles, and additionally, is determined by the mutual orientation of the zero and pole directions.
... Sampled-data systems are a special type of time-periodic systems. Fundamental limitations for sampled-data systems are studied in [Freudenberg et al., 1995;Ortiz et al., 2000] using transfer function techniques. We study general time-periodic systems in this paper and we use the harmonic transfer function (HTF), see Zhou and Hagiwara, 2002], which formally is a MIMO transfer functionĜ(s) with an infinite amount of inputs and outputs. ...
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.
... A synchronization controller is used to track the measured wafer stage error signals with the aim to reduce the synchronization error between the two stage systems: the reticle stage and the wafer stage; see ) for a data-based approach and Lambregts et al. (2015) for a model-based approach. Disturbance feedforward control suffers from a causality problem, i.e., a waterbed effect related to the Bode sensitivity integral (Freudenberg et al., 1994). The input signals need to be measured before any compensation can take place; see also Butler (2012) for predictive feedforward control. ...
This paper gives an exposition of examples of data-based control and optimization that found their way to successful application in the motion systems of industrial wafer scanners. The examples represent selective works brought together in an overview that highlights the possibilities, challenges, and open issues in data-based motion control of wafer scanners.
... Furthermore, weighted sensitivity (or modulated sensitivity) integrals are considered in [9], [13], [24] for linear sampleddata periodic systems. As opposed to these treatments, our approach is able to consider open-loop unstable processes that are being driven by disturbances that need not be Gaussian. ...
We present a generalization of the Bode integral formula for discrete-time linear periodic systems. It is shown that similar to the classical Bode integral formula, the sensitivity integral for a discrete-time linear periodic system depends only on the open-loop dynamics of the system; in particular, on the open-loop characteristic multipliers outside the unit circle in the complex plane. The integral is derived by using an asymptotic eigenvalue distribution theorem for block Toeplitz matrices, which does not require the open loop system to be stable or the disturbance to be Gaussian. The result is demonstrated through application to a class of multi-rate sampled data systems with commensurate rates.
... decentralized control (Aghdam, Davison & Becerril-Arreola, 2006;Lavaei & Aghdam, 2007), the fact is that they have not yet made their way into the industrial world. This uncomplete success may be due to their unsatisfactory intersample behavior (Feuer & Goodwin, 1994, 1996 as well as because of a certain degree of bad reputation they still bear in the control community since they were proposed to disguise nonminimum-phase continuous models as minimum phase discretized models (Yan, Anderson & Bitmead, 1994;Rossi & Miller, 1999), which collides with essential limitations of control systems, as Freudenberg, Middleton & Braslavsky (1995, 1997) so rightly pointed out. ...
This paper presents an exhaustive experimental study on the performance of a new design of generalized sampled-data hold function (GSHF) introduced in Ugalde, Bárcena, and Basterretxea (2012). First a simple tuning procedure is developed for the GSHF to improve the intersample response while not damaging the plant output response. Next the reconstruction algorithm is enhanced so that the GSHF can remove the steady-state errors that dry friction causes in positioning systems. The experiments conducted on a small-scale lightly damped resonant system show that significant energy savings can be achieved in regulation with no extra hardware, and that no additional computational load results, except having to call the D/A converter more than once per sampling period.
... A frequency domain analysis of the system offigure 3 is performed. The development is to a large extent analogous to the SD frequency response functions; see Goodwin and Salgado (1994), Freudenberg et al. (1995), Araki et al. (1996), Yamamoto and Khargonekar (1996), Lindga¨rdeLindga¨rde and Lennartson (1997), and Cantoni and Glover (1997). Lemma 3: Let ! ...
Control design for high-performance sampled-data systems with continuous time performance specifications is investigated. Direct optimal sampled-data control design explicitly addresses both the digital controller implementation and the intersample behaviour. The model that is required for direct optimal sampled-data control should evolve in continuous time. Accurate models for control design, however, generally evolve in discrete time since they are obtained by means of system identification techniques. The purpose of this paper is the development of a control design framework that enables the usage of models delivered by system identification techniques, while explicitly addressing both the digital controller implementation and the intersample behaviour aspects. Thereto, the incompatibility of the models delivered by system identification techniques and the models used in sampled-data control is analysed. To use models delivered by system identification techniques in conjunction with optimal sampled-data control, tools are employed that stem from multirate system theory. For the actual control design, key theoretical issues in sampled-data control, which include the linear periodically time-varying nature of sampled-data systems, are addressed. The control design approach is applied to the H 1 -optimal feedback control design of an industrial high-performance wafer scanner. Experimental results illustrate the necessity of addressing the intersample behaviour in high-performance control design.
... Freudenberg and Looze also provided similar constraints on the integral of the complementary sensitivity function T (s) = I − S(s). Discrete-time and sampled-data versions of these integrals are found in [3][4][5][6]. While the result in (1.2) applies to multivariable systems, there has also been considerable effort in deriving sensitivity integrals on the individual singular values of S(iω), rather than the weighted integral one obtains from the determinant. ...
A new time-domain interpretation of Bode's integral is presented. This allows for a generalization to the class of time-varying systems which possess an exponential dichotomy. It is shown that the sensitivity function is constrained, on average, by the spectral values in the dichotomy spectrum of the antistable component of the open-loop dynamics.
... Más recientemente se han publicado extensiones de los resultados anteriores a sistemas con ceros en el eje imaginario (Goodwin et al., 1999 ), sistemas multivariables (Chen, 1995; Chen y Nett, 1995; G ´ omez y Goodwin, 1996; Woodyatt et al., 2001; Havre y Skogestad, 2001) sistemas de estimación (Goodwin et al., 1995; ), sistemas muestreados (Goodwin y Salgado, 1994; Freudenberg et al., 1994; Braslavsky et al., 1995 ) y sistemas inestacionarios y no lineales (Iglesias, 2001a; Iglesias, 2001b). Otra línea de trabajo explora límites de desempeño desde el punto de vista del controí optimo. ...
En este trabajo se describen herramientas que permiten determinar límites fundamentales de desempeño en sistemas de control en realimentación. Estos límites dependen de los polos inestables y los ceros de fase no mínima de la planta y pueden usarse para determinar especificaciones alcanzables o no alcanzables y para identificar compromisos de diseño en dichos sistemas.
... In other words, we take the inter-sample behavior into account to evaluate the tracking performance. The study on performance limitations has been done for years, for example, see [20,21]. ...
This paper is concerned with the inherent H2 tracking performance limitation of single-input and multiple-output (SIMO) linear time-invariant (LTI) feedback control systems. The performance is measured by the tracking error between a step reference input and the plant output with additional penalty on control input. We employ the plant augmentation strategy, which enables us to derive analytical closed-form expressions of the best achievable performance not only for discrete-time system, but also for continuous-time system by exploiting the delta domain version of the expressions.
... In theory these difficulties are ameliorated by sub-Nyquist sampling as ω T = π. In practice as the underlying restriction is on the analog channels, poor intersample behavior will persist in the closed loop response, [9], unless the plant bandwidth is significantly below the channel bandwidth. ...
This paper concerns the sampled data control of continuous time plants from a remote location when the feedback and actuation signals are transmitted over finite bandwidth communication channels. We show two results. First that under the theoretical restriction of perfect bandlimitation an unstable plant cannot be stabilized, if the actuation and feedback data are transmitted at rates exceeding the Nyquist frequencies of the communications channels. Under more practical bandlimitations, we show that super-Nyquist rates result in closed loop H ∞ norms that grow with the severity with which band confinement is imposed. In theory these problems do not appear if sub-Nyquist rates are used. We explain in the paper that this loss of Stabilizability is a consequence of a fundamental interplay between communications and control.
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 studies Bode/Poisson type sensitivity and complementary sensitivity integral relations for linear, time-invariant feedback control systems. We call for attention to a seemingly unnoticed link between the well-known argument principle and Bode/Poisson integral relations. In this respect, a generalized version of the argument principle is developed, under which it is shown that various Bode/Poisson type integral relations can be unified, and new integral relations can be derived. The new and classical integral relations together provide useful insights for analyzing control design limitations and tradeoffs.
This paper presents a review on the fundamental performance limitations and design tradeoffs of feedback control systems, ranging from classical performance tradeoff issues to the more recent information-theoretic analysis, and from conventional feedback systems to networked control systems, with an attempt to document some of the key achievements in more than seventy years of intellectual inquiries into control performance limitation studies, as so embodied by the timeless contributions of Bode known as the Bode integral relations.
Performance analysis in the frequency domain is presented for single and multirate sampled-data control systems. The main idea is to compute the input to output power gain for a single sinusoidal input with a specific frequency. This method is therefore closely related to the maximum principal gain for continuous-time MIMO systems. However, in contrast to the continuous-time case, we observe that the maximum gain of a single sinusoidal input does not equal the H∞-norm for sampled-data systems. This implies that robust stability and performance can not be determined by the same kind of frequency analysis. The suggested method for performance analysis, is illustrated on a multirate sampled-data controlled packed-bed distillation column.
This paper is concerned with inherent H 2 tracking performance limitations in feedback control systems. We deal first with the single-input and multiple-output (SIMO) linear time-invariant (LTI) discrete-time system and provide the analytical closed-form expression of the best achievable performance. Then we reformulate and resolve the problem in delta domain by means the delta operator, from which we can completely recover the counterpart expression for the continuous-time case by approaching the sampling time to zero. In addition, we provide a similar result in sampled-data feedback control systems by using the fast sampling.
Sampled-data control systems occasionally exhibit aliased resonance phenomena within the control bandwidth. The aim of this paper is to investigate the aspect of these aliased dynamics with application to a high performance industrial nano-positioning machine. This necessitates a full sampled-data control design approach, since these aliased dynamics endanger both the at-sample performance and the intersample behaviour. The proposed framework comprises both system identification and sampled-data control. In particular, the sampled-data control objective necessitates models that encompass the intersample behaviour, i.e., ideally continuous time models. Application of the proposed approach on an industrial wafer stage system provides a thorough insight and new control design guidelines for controlling aliased dynamics.
This paper is concerned with inherent H2 tracking performance limitations in feedback control systems. We deal flrst with the single- input and multiple-output (SIMO) linear time-invariant (LTI) discrete- time system and provide the analytical closed-form expression of the best achievable performance. Then we reformulate and resolve the problem in delta domain by means the delta operator, from which we can completely recover the counterpart expression for the continuous- time case by approaching the sampling time to zero. In addition, we provide a similar result in sampled-data feedback control systems by using the fast sampling.
This report compiles, refines, and extends a general theory for developing digital servo-tracking controllers which will achieve a high-level of servo-tracking performance that is unmatched by currently available digital servo-design methods. This general theory applies to multiple-input/multiple-output linear time-invariant systems subjected to plant parameter-perturbations and complex, multi-variable, time-varying servo-commands and disturbances. Obstacles to achieve high-performance servo-tracking are identified and discussed, along with key shortcomings inherent in conventional design methods. In addition, a collection of example problems are worked in detail to illustrate the design techniques described and developed in this study. Simulation results are used to demonstrate the performance of the resulting controllers.
We present a set of feedback limitations for linear time-invariant systems controlled by periodic digital controllers based upon an analysis of the inter-sample response of the closed-loop system to sinusoidal inputs. Fundamental sensitivity and complementary sensitivity functions govern the fundamental and harmonic components of the continuous closed-loop response. The continuous and discrete response of the system is sensitive to variations in the analog plant at frequencies integer multiples of ωs/N away from the excitation frequency, where ωs is the sampling frequency and N is the period of the controller. These functions satisfy interpolation and integral constraints due to open-loop non-minimum phase zeros and unstable poles. In addition, the use of periodic digital control may result in a reduction in closed-loop bandwidth. Copyright © 2000 John Wiley & Sons, Ltd.
Nevanlinna-Pick interpolation techniques are employed in this paper to derive for multivariable systems exact expressions
and bounds for the best attainable H
∞ norms of the sensitivity and complementary sensitivity functions. These results improve the previously known performance
bounds and provide intrinsic limits on the performance of feedback systems irreducible via compensator design, leading to
new insights toward the understanding of fundamental limitations in feedback control. It becomes clear that in a multivariable
system the best achievable performance is constrained by plant nonminimum phase zeros and unstable poles, and additionally,
is dependent on the mutual orientation of the zero and pole directions.
It is well known that linear time invariant control of plants with unstable poles results in time domain integral constraints
on the closed loop system. This leads to the question of whether nonlinear or time varying control can be used to ameliorate
these constraints. In this paper a particular class of time varying controllers, namely sampled data feedback controllers,
is studied. An integral constraint, analogous to one for continuous time systems, is derived for sampled data systems with
an unstable plant pole and an output disturbance. Analysis of this constraint in several examples shows that the performance
limitation is often worse than in the continuous time case.
Presents a bisection algorithm for computing the frequency
response gain of sampled-data systems composed of a continuous-time and
a discrete-time system together with a sampler and a hold. Although a
bisection algorithm is known in the literature, it is applicable only to
those frequencies at which the frequency response gain is greater than
||D||, where D, denotes the direct feed-through term in the lifted
transfer function of the sampled-data system. The paper gives a new
bisection algorithm that can be applied to any frequencies. The
derivation given is based on the lifting technique, and the properties
of the congruent transformation (the Schur complement and the Sylvester
law of inertia) play a key role. To facilitate the algorithm, a
bisection algorithm for computing the singular values of D is also
presented with the combined use of the lifting and FR-operator
techniques
In this paper frequency analysis methods for sampled-data control
are compared. The relation between the analysis tools that take the
inter-sample behavior into consideration and the traditional
discrete-time analysis is investigated. It is shown how the plant, the
controller design, and the anti-alias filter affect the inter-sample
behavior. Both analytical results and numerical examples are included
In this paper, we develop a theory of design limitations for sampled-data feedback systems wherein we consider the response of the analog system output. To do this, we use the fact that the steady state response of a hybrid feedback system to a sinusoidal input consists of a fundamental component at the frequency of the input together with infinitely many harmonics at frequencies spaced integer multiples of the sampling frequency away from the fundamental. This fact allows us to define fundamental sensitivity and complementary sensitivity functions that relate the fundamental component of the response to the input signal. These sensitivity and complementary sensitivity functions must satisfy integral relations analogous to the Bode and Poisson integrals for purely analog systems. The relations show, for example, that design limitations due to nonminimum phase zeros of the analog plant constrain the response of the sampled-data feedback system regardless of whether the discretized system is minimum phase and independently of the choice of hold function.
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
Optimal damping of harmonic disturbances of known frequencies is studied for sampled-data systems. A sampled-data output feedback controller is designed to minimize the intersample variations of the controlled variable. The set of all stabilizing optimal controllers is obtained in terms of the Youla parameterization and a set of interpolation conditions at the disturbance frequencies, which ensure that the stationary cost is minimized.
This paper is concerned with the computations of the frequency response gain of sampled-data systems. The exact computations is intrinsically quite burdensome even at a fixed frequency, due to the infinite-dimensional nature involved in this problem. To facilitate an approximate computation instead, this paper gives three independent methods to compute upper and lower bounds of the frequency response gain. The first one is such that there is no parameter through which we can balance accuracy and the computational load, while the second and third are with such a parameter. In spite of this, the first methods will be found, in our numerical studies, to give quite a nice approximations in a reasonably short computations time, compared with other methods given in this paper and those available in the literature. It is also shown that our first methods has very close relationship to the sampled-data H2 and H∞ problems.
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.
This paper is motivated by the problem of computing the frequency response gain of general sampled-data systems with noncompact frequency response operators. We first show that, with the J-unitary transformation, the computation in the noncompact operator case can be reduced, in principle, to that in the compact operator case, to which an existing efficient and reliable bisection method can be applied. At the same time, however, we point out that there arise some critical problems in this reduction to the compact case which could be serious enough to invalidate the apparent success in the reduction. Through some spectral analysis of operators involving or related to the frequency response operators, we eventually prove that these critical problems can be circumvented after all, and we give an explicit result that shows how to compute the frequency response gain with a bisection method dealing only with finite-dimensional matrices. Extending the arguments, we also give a bisection method to compute the singular values of the frequency response operators and the associated compression operators.
This paper studies input/output stability of nonlinear sampled-data systems with a sector nonlinearity. A stability condition of circle-criterion type was derived recently, for the case where the sector nonlinearity is possibly time varying and dynamical. In contrast, this paper deals with the case where it is time invariant and memoryless, and gives a less conservative stability criterion for such a case. It is derived by applying the multiplier technique, and corresponds to the Popov criterion in the continuous-time setting. The arguments make use of the frequency-domain theory of sampled-data systems, and a sort of convexity in the frequency domain plays an important role. A method with the cutting-plane algorithm is provided for finding a multiplier that proves stability. An illustrative example is also given.
For linear time-invariant systems Bode's sensitivity integral is a well-known formula that quantifies some of the limitations in feedback control. In this paper we show that a very similar formula holds for linear time-periodic systems. We use the infinite-dimensional frequency-response operator called the harmonic transfer function to prove the result. It is shown that the harmonic transfer function is an analytic operator and a trace class operator under the assumption that the periodic system has roll-off 2. A periodic system has roll-off 2 if the first time-varying Markov parameter is equal to zero.
Most networked sampled data control systems involve transmission of feedback and actuation signals over finite bandwidth communication channels. In an earlier paper, [S. Dasgupta, 2003] we had demonstrated that this restriction on the transmission bandwidth severely impairs stabilizability should the sampling rate exceed the Nyquist frequency of the communication channels conveying the feedback and actuation signals. In this paper we examine the effect of such bandwidth limitation to the intersample behavior of the closed loop system. While it is well known that slower sampling degrades intersample performance [J.S. Freudenberg et al., 1995] in the presence of delays, the results here quantify the severe performance degradation when the sampling is too fast.
In this paper we study the problem of tracking a step reference signal using sampled-data control systems. We investigate the best achievable tracking performance, where the performance is deemed best for it is the minimal attainable by all possible sampled-data stabilizing controllers. Our primary objective is to investigate the fundamental tracking performance limit in sampled-data systems, and to understand whether and how sampling and hold in a sampled-data system may impose intrinsic barriers to performance. For this optimal tracking performance. The result shows that a performance loss is generally incurred in a sampled-data system, in comparison to the tracking performance achievable by continuous-time controllers. This loss of performance, as so demonstrated by the expression, can be attributed to the non-minimum behaviors and the aliasing effects generated by samplers and hold devices.
In this paper we investigate the best achievable H 2 tracking performance for a step reference signal under physical constraints such as control effort limitation and sampled-data control scheme. We here extend our previous results slightly and summarize the practical control performance limits based on the derived analytical closed form expressions, which consist of a term affected by the non-minimum phase zeros including the time-delay of the continuous-time plant and extra terms which are affected by physical constraints imposed such as control effort constraints and sampling.
This paper studies the tracking performance of sampled-data control systems. We consider the problem of tracking a step reference input by a sampled-data controller, and the goal is to minimize the integral square of the error between the output and input signals. Under this criterion, of particular interest is to investigate whether and when a degradation in the tracking performance may result, due to the use of sampled-data controllers, and whether such a degradation, where it does exist, can be remedied by a sufficiently fast sampler. The degradation thus constitutes a gap between the optimal performance achievable by continuous-time controllers and that by optimal sampled-data controllers. It is shown that for plants with relative degree greater than one, a performance loss does take place and it can never be recovered despite that the sampler is allowed to operate arbitrarily fast. This performance loss is seen to be fundamental of the sampling and hold mechanism, rather than from the plant itself. It is also shown that for plants with relative degree one, the loss can indeed. be retrieved with a sufficiently fast sampler.
A new time-domain interpretation of H.W. Bode's (1945) integral is
presented. This allows for a generalization to the class of time-varying
systems which possess an exponential dichotomy. It is shown that the
sensitivity function is constrained, on average, by the spectral values
in the dichotomy spectrum of the unstable component of the open-loop
dynamics
We study Bode and Poisson type integral relations. We focus on a link between the well-known argument principle and Bode and Poisson integrals, which have been unnoticed previously. We show how various integral constraints may be unified under an extended version of the argument principle. This enables us to derive the classical Bode and Poisson integral relations in a simple manner, and further to discover new integral formulas of significance for an analysis of control design limitation and tradeoff.
In the design of linear sampled data control systems, tradeoffs
between achievable performance and disturbance rejection (sensitivity
reduction) always manifest themselves. However, when disturbance
rejection is considered to be the first priority, then question would be
raised as how to design digital controllers to obtain asymptotic
disturbance rejection in the first place. In this paper, a relation
between loop shaping and internal model control for linear
continuous/sampled-data systems is presented. The design criteria for
achieving asymptotic disturbance rejection in linear sampled data
control systems are also discussed
We extend the classical Bode integral relations to multi-input
multi-output systems. Our main contributions include Bode and Poisson
type integral inequalities for the sensitivity and complementary
sensitivity functions, together with bounds on the sensitivity and
complementary sensitivity magnitudes, which are obtained for both
continuous-time and discrete-time systems. Part I presents Bode type
integral inequalities. These inequalities may be viewed as some variants
to the previously available results, but are also significantly
different. The main difference is two fold. First, while the previous
integral equalities are valid only under a restrictive assumption, the
results herein hold in general. Secondly, in spite of the fact that the
present results are based upon techniques and assumptions similar to
those used in deriving Bode type integral inequalities, the in-depth
study carried out here leads to more explicable expressions particularly
indicative of the adverse effects of open loop unstable poles. As a
direct outcome, the new results confirm that in a multivariable system
design limitation and tradeoff depends on both the locations and
directions of open loop unstable poles, and in particular, in how such
directions may be aligned. The latter is characterized by angles
measuring the mutual orientation between the pole directions
Multirate, sampled-data control systems are analyzed in the
frequency domain, taking the inter-sample behavior into account. The
focus of the analysis is both on the rejection of disturbances within a
certain frequency region, and on the robust stability properties. Due to
the periodic behavior of such systems, a distinction is made between the
performance frequency response (PFR) and the robust frequency response
(RFR). The RFR is related to current frequency response approaches for
sampled-data systems
A paper by Bode (1945) has shown the limitations of using a feedback structure in terms of an integral constraint on the sensitivity function for open-loop stable continuous-time systems. The present paper examines and derives equivalent results for discrete-time feedback systems. These integral constraints also provide some guidelines regarding the philosophy of feedback design specifically for sampled-data systems. For example, it is shown that, for all sampled-data control systems, there is a maximum sampling frequency, beyond which little improvement in performance is gained.
This paper considers the use and design of linear periodic time-varying controllers for the feedback control of linear time-invariant discrete-time plants. We will show that for a large class of robustness problems, periodic compensators are superior to time-invariant ones. We will give explicit design techniques which can be easily implemented. In the context of periodic controllers, we also consider the strong and simultaneous stabilization problems. Finally, we show that for the problem of weighted sensitivity minimization for linear time-invariant plants, time-varying controllers offer no advantage over the time-invariant ones.
Over the past 5 years, there has been substantial interest in the
use of generalized sample hold functions for control. In this
correspondence, the authors use a tool, which is novel in this context,
namely, amplitude modulation theory. The authors employ this tool to
analyze the quantitative and qualitative features of the intersample
behavior in a frequency domain setting. This offers new theoretical and
practical insights into the method. The authors' conclusion is that the
perceived benefits come at substantial cost which makes its practical
use questionable
New conic sectors are presented which can be used to analyze sampled-data feedback systems. First, a cone is constructed which contains just the sampler. Next, by using conic sector results for interconnected systems, the cone is propagated to the entire sampled-data operator. The new cone result is less conservative (has a smaller radius) and removes the restriction of an open-loop stable sampled-data compensator.
Given a generalized sampled-data hold function (GSHF), we define a frequency dependent response function having many properties analogous to those of a transfer function. In particular, the zeros of the response function have transmission blocking properties. We then study the problem of non-pathological sampling with a GSHF. Sampling is said to be pathological if the discretized version of a stabilizable/detectable continuous time plant is not itself stabilizable and detectable. Sufficient conditions for nonpathological sampling with a zero order hold have long been known. We extend these to the case of a GSHF, and describe the role in non-pathological sampling played by right half plane zeros of the response function. The results are presented for square multivariable linear systems and include a generalization to allow for a time delay.
Considering linear time-invariant distributed feedback systems, a necessary and sufficient condition for robust stability is derived with respect to plant perturbations belonging to a specified ball. The conclusion is shown to exhibit design limitations imposed by plant uncertainties.
Sensitivity and complementary sensitivity functions play a key role in feedback control system design. This paper is concerned with the relation between the sampling period and the properties of these functions in digital control systems. Some integral constraints and the lower bounds of the H ∞-norm are derived, which show that the feedback performance for the unstable plant with the stable digital compensator can be improved as the sampling period goes to zero.
A frequency response for linear time periodic (LTP) systems that
exploits the one-to-one map induced by geometrically periodic signals is
developed. This map is described by an integral operator, based on the
GP (geometrically periodic) test input, and a generalized harmonic
balance approach, based on an EMP (exponentially modulated periodic)
input. The singular values or principal gains of the LTP operator are
discussed, and the LTP principal gain diagram is described. Directional
properties of the LTP operator are discussed, and notions of the domain
and range spaces are presented. The framework of linear operators
described has lead to the development of a comprehensive open-loop
analysis theory for LTP systems, including a characterization of poles,
transmission zeros, and their directional properties
For some time now, many practitioners and researchers in the control area have been aware that unstable open loop poles, non-minimum phase zeros, and/or time delays make control systems design difficult. In this paper we examine the nature of these difficulties by discussing the results of Freudenberg and Looze (1987, 1988) and Sung and Hara (1988) on integral constraints on sensitivity functions. One of the key conclusions here is a set of rules of thumb, giving limitations on the closed loop bandwidth which are imposed by unstable open loop poles, non-minimum phase zeros and/or time delays.
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.
The conic-sector analysis of the closed-loop stability and robustness of a multivariable-analog-system controller based on sampled-data feedback compensation is investigated. Conic sectors and sampled-data feedback systems are defined, and the existence of a conic sector containing a sampled-data operator is established mathematically. An example is presented to prove that the conic sector is computable and gives sufficient conditions of closed-loop stability. A procedure for determining sampled-data-operator gain is also derived.
This paper investigates the use of generalized sampled-data hold functions (GSHF) in the control of linear time-invariant systems. The idea of GSHF is to periodically sample the output of the system, and generate the control by means of a hold function applied to the resulting sequence. The hold function is chosen based on the dynamics of the system to be controlled. This method appears to have several advantages over dynamic controllers: it has the efficacy of state feedback without the requirement of state estimation; it provides the control system designer with substantially more freedom; and it requires few on-line computations. This paper focuses on four questions: pole assignment, specific behavior, noise sensitivity, and robustness. Among the problems solved are: simultaneous arbitrary pole assignment for a finite number of systems by a single GSHF controller, exact model matching, decoupling, and optimal noise rejection. Examples are given.
In this paper the ripple-free deadbeat control problem for sampled-data systems is considered. This control objective is to settle the error to zero for all time after some finite settling time, in other words to eliminate the ripples between the sampling instants in deadbeat control of sampled-data systems. The necessary and sufficient conditions to solve this problem are derived and related to the system type, and a method of constructing the ripple-free deadbeat control system is presented.
The application of standard robust servomechanism theory to sampled data systems guarantees asymptotic tracking at sampling instances and between sampling instants, the output will normally contain ripple. In this note, the robust servomechanism theory is extended to sampled-data systems and a technique is proposed for ripple-free tracking of sinusoids and polynomials. It is shown that a continuous internal model is necessary and sufficient to provide ripple-free response.
This paper expresses limitations imposed by right half plane poles and zeros of the open-loop system directly in terms of the sensitivity and complementary sensitivity functions of the closed-loop system. The limitations are determined by integral relationships which must be satisfied by these functions. The integral relationships are interpreted in the context of feedback design.
In this paper the problem of sensitivity, reduction by feedback is studied and related to a problem of decentralized control. A plant will be represented by an N times N matrix of frequency responses, which may be unstable or irrational. The object will be to find conditions on P(s) under which a diagonal feedback F(s) can make the sensitivity parallel{I + P(s)F(s)}^{-1}parallel arbitrarily small over some specified frequency interval [ -jomega_{0}, jomega_{0} ] without violating a global sensitivity, bound parallel{I+ P(s)F(s)}^{-1}parallel leq M , ( Mgeq some const. >1) for Re(s) geq 0 . It will be shown that such a diagonal feedback of the "high gain" type can be constructed whenever P^{-1}(s) is analytic in Re(s)geq 0, P(s) satisfies an attenuation condition near s = infty , and P(s) approaches diagonal dominance at high frequencies. It will also be shown that these conditions on the plant can be interpreted as conditions for the existence of a decentralized wide-band control scheme.
This paper presents a practical design perspective on multivariable feedback control problems. It reviews the basic issue-feedback design in the face of uncertainties-and generalizes known single-input, single-output (SISO) statements and constraints of the design problem to multiinput, multioutput (MIMO) cases. Two major MIMO design approaches are then evaluated in the context of these results.
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First Page of the Article
This paper shows that the use of dynamic compensation based on
generalized sampled-data hold functions (GSHF) can arbitrarily improve
the gain margin for continuous-time nonminimum phase linear systems. The
GSHF compensator is a particular periodic digital controller. The effect
of sampling period on the gain margin is analyzed. Furthermore, it is
proved that under a mild condition, the gain margin improvement can be
achieved without forcing the sampling period small. An important
advantage of periodic compensation over LTI compensation is found to be
the capability of reducing conflict between gain margin and sensitivity
Sampled-data systems with stable, structured, LTI (linear
time-invariant) perturbations in the half-plane algebra applied to the
continuous-time plant are considered. Necessary and sufficient
conditions are derived for robust L 2 stability of
such systems. An example is provided to illustrate the results, which
shows that the small gain theorem can be an extremely conservative
robustness test in this sampled-data context
The authors consider the robust stability and performance analysis
of a general linear interconnection of a continuous-time plant and a
discrete-time controller via sample and hold devices. They obtain
necessary and sufficient conditions for the robust stability of the
feedback system consisting of an uncertain plant represented as a known
linear time-invariant (nominal) model with time-varying norm bounded
uncertainty and a shift-invariant, discrete-time controller. These
results show that the condition arising from the small gain theorem is
necessary and sufficient when the uncertainties belong to a class of
linear systems. Also similar results are derived for a case of
structured uncertainty. Both L 2 and
L ∞ signal norms are considered. Robust
performance analysis problem is solved by converting it to a robust
stability analysis problem
A simple method (based on Floquet theory) for obtaining the
characteristic equation (and hence stability) of periodic discrete-time
systems is presented. Using this method it is shown that 2-periodic
controllers can be used to relocate the zeros of SISO plants. Some
examples are considered to illustrate the use of 2-periodic controllers
for robust control of finite gain margin problems
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
The control of a linear time-invariant plant by a digital
controller composed of a sampler, a periodic discrete-time component,
and a zero-order hold is considered. The stability of such a
configuration is analyzed. It is shown how closed-loop zeros can be
placed using such a scheme. As a consequence, it is proved that the gain
margin can be arbitrarily assigned by suitable choice of sampling time
and digital controller. The design procedure is constructive, using
state-space methods
A new type of controller is proposed which detects the
i th plant output N i times during a period
of T 0 and changes the plant inputs once during T
0. It is shown that an arbitrary state feedback can be
realized by such controllers if the plant is observable. This implies,
for instance, that arbitrary symmetric pole assignment is possible if
the plant is controllable. It is also shown that, if the plant has no
zeros at the origin, the state transition matrix of the controller
itself can be set arbitrarily without changing the state feedback to be
realized. That is to say, inversely expressed, any state feedback can be
equivalently realized by a controller with any prescribed degree of
stability