Approximation of frequency response for sampled-data control systems

doi:10.1016/S0005-1098(98)00206-4

Automatica

Volume 35, Issue 4, April 1999, Pages 729-734

https://doi.org/10.1016/S0005-1098(98)00206-4 Get rights and content

Abstract

This paper proves that the frequency response gains of fast-sample/fast-hold 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 difficulty, 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.

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Cited by (56)

H<inf>∞</inf> and H<inf>2</inf> optimal sampled-data controller synthesis: A hybrid systems approach with mixed discrete/continuous specifications
2021, Automatica
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High performance is warranted by taking a sufficiently high sampling frequency, see, e.g., Chen and Francis (1995) and Skogestad and Postlethwaite (2007), while it is of interest to select low sampling/update frequencies, when computation/communication resources are limited. Extensive research has been done on the topic of sampled-data controller synthesis for linear systems in the last few decades, see, e.g., Bamieh and Pearson (1992), Chen and Francis (1995), Dai et al. (2010), Fridman (2014), Fridman et al. (2005), Geromel et al. (2019), Ichikawa and Katayama (2001), Khargonekar and Sivashankar (1991), Lall and Dullerud (2001), Mirkin and Palmor (1997, 2003), Sivashankar and Khargonekar (1994), Toivonen (1992) and Yamamoto et al. (1999), when focussing on linear systems. The aforecited papers can be categorised into “lifting” approaches, fast-sampling/fast-hold approaches, time-delay approaches and jump/hybrid systems approaches.
In this paper, a tractable sampled-data controller synthesis method is proposed for linear time-invariant plants that gives guarantees for stability and performance of the closed-loop system in terms of the $H_{\infty}$ - and $H_{2}$ -norm. This is done by taking a so-called jump/hybrid systems approach, which allows formulating linear matrix inequalities using an explicit solution to the Riccati differential equation. Furthermore, the sampled-data problem is formulated such that continuous-time design techniques like $H_{\infty}$ loop-shaping can be used in a sampled-data context. To do so, it is essential to consider mixed discrete/continuous specifications, in which both discrete and continuous signals are weighted using weighting filters. This leads to a generalised plant that has both continuous-time and discrete-time dynamics. The controller design method is demonstrated on a benchmark example, in which we compare to existing results in the literature, and on a design example of reference tracking of a two-mass–spring–damper system.
Periodic disturbance rejection to the Nyquist frequency and beyond
2018, Control Engineering Practice
Citation Excerpt :
When the sampling frequency is not sufficiently high, unobservable oscillations or ripples of the controlled variable occurring between samples can degrade the performance or even destabilize the closed-loop control system (Atsumi & Messner, 2012). Since 1990s, researchers started to study the built-in intersample behavior for sampled-data systems (Araki, Ito, & Hagiwara, 1996; Bamieh & Pearson, 1992; Chen & Francis, 1995; Yamamoto, 1994), and many techniques have been proposed (Hagiwara & Okada, 2009), e.g. the lifting technique see Bamieh and Pearson (1992), Bittanti and Colaneri (2009), Hara, Fujioka, Khargonekar, and Yamamoto (1996), Mirkin and Palmor (2002), Toivonen (1992), Yamamoto (1994) and Yamamoto and Khargonekar (1996), the FR-operator technique see Araki et al. (1996) and Hagiwara and Araki (1995), the fast-sample/fast-hold approximation see Hagiwara and Okada (2009) and Yamamoto, Madievski, and Anderson (1999), as well as the methods in Chen and Qiu (1997), Goodwin and Salgado (1994), Hagiwara, Suyama, and Araki (2001), Ito, Hagiwara, Maeda, and Araki (2001), Jia (2005) and Kim and Hagiwara (2015). Various approaches for periodic disturbance rejection have been proposed during the past few decades, such as the phase-locked loop technique-based adaptive methods (Bodson & Douglas, 1997; Bodson, Sacks, & Khosla, 1994), the internal model principle (Brown & Zhang, 2004; Kim, Kim, Chung, & Tomizuka, 2011), and the repetitive control (Hara, Yamamoto, Omata, & Nakano, 1988).
In this paper, we address periodic disturbance rejection for sampled-data control systems, where the frequency of the disturbance can be beyond the Nyquist frequency of the limited measured plant output sampling rate. The disturbance effect on steady-state response of the fictitious fast-rate plant output including intersample information is obtained using discrete-time Fourier series. A sufficient condition for disturbance rejection together with a sufficient and necessary condition for perfect disturbance elimination are provided by employing the steady-state response. A simple but effective controller design procedure combining $H_{\infty}$ loop shaping, Youla–Kucera parameterization, and gradient methods is provided to achieve the two conditions. The proposed approach is applied on vibration control beyond the Nyquist frequency in a commercial hard disk drive. The effectiveness of the proposed approach is verified by our simulation results showing disturbance rejection of $62 %$ and perfect disturbance elimination, as well as by our experimental results showing disturbance rejection of $54 %$ .
Digital repetitive controller design via sampled-data delayed signal reconstruction
2016, Automatica
In this paper, we propose a design method of digital repetitive controllers to track continuous-time periodic references based on sampled-data $H^{\infty}$ delayed signal reconstruction. It is well known that perfect tracking is not possible for arbitrary periodic signals with a fixed period if the plant to be controlled is strictly proper.
For this reason, we adopt the internal model control (IMC), with which the repetitive control is cast into the framework of sampled-data $H^{\infty}$ delayed signal reconstruction problem. We also extend this formulation to multi-rate and robust control. Design examples show the effectiveness of our method.
Fast-lifting approach to the computation of the spectrum of retarded time-delay systemsg
2011, European Journal of Control
This paper discusses a new method for computing the spectrum of retarded time-delay systems based on the discrete-time viewpoint provided by the application of the lifting technique. Stimulated by a recent study that applied (conventional) fast-sample/fast-hold (FSFH) approximation, we consider applying a somewhat alike but completely different method with modified FSFH approximation developed in the fast-lifting approach to sampled-data systems. We show that this leads to a finite-dimensional method that gives an asymptotically exact result. Compared with other existing methods that also lead to discretization approaches, the arguments in this paper are based on the bounded operator treatment of dynamical systems, and more system theoretic. The associated convergence analysis is also carried out in an operator-theoretic framework in a straightforward fashion. An advantage of the method is discussed in dealing with discrete-time controllers, and the effectiveness of the method is demonstrated with numerical examples.
Computing the L<inf>2</inf> gain for linear periodic continuous-time systems
2009, Automatica
A method to compute the $L_{2}$ gain is developed for the class of linear periodic continuous-time systems that admit a finite-dimensional state-space realisation. A bisection search for the smallest upper bound on the gain is employed, where at each step an equivalent discrete-time problem is considered via the well-known technique of time-domain lifting. The equivalent problem involves testing a bound on the gain of a linear shift-invariant discrete-time system, with the same state dimension as the periodic continuous-time system. It is shown that a state-space realisation of the discrete-time system can be constructed from point solutions to a linear differential equation and two differential Riccati equations, all subject to only single-point boundary conditions. These are well behaved over the corresponding one period intervals of integration, and as such, the required point solutions can be computed via standard methods for ordinary differential equations. A numerical example is presented and comparisons made with alternative techniques.
Modified fast-sample/fast-hold approximation for sampled-data system analysis
2008, European Journal of Control
This paper deals with theH∞ norm and frequency response gain analysis of sampled-data systems and provides a new approach, which we call modified fast-sample/fast-hold approximation. The new approximation approach discretizes the continuous-time generalized plant in a “γ-independent fashion” and leads to a discrete-time generalized plant with a similar structure to what is obtained by the conventional fast-sample/fast-hold approximation approach. Unlike the conventional approach, however, the modified fast-sample/fast-hold approximation approach can give both the upper and lower bounds of the H_∞-norm and/or the frequency response gain for sampled-data systems. Furthermore, the gap between the upper and lower bounds can be bounded from above directly from the fast-sample/fast-hold parameter N and is independent of the controller. These features are quite useful when it is applied to control system design, and this study indeed has very close relation with the control system design via noncausal periodically time-varying scaling, a novel notion introduced recently.

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^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor Peter Dorato. This author was supported in part by the Sound Technology Promotion Foundation.

¹: The second and third authors wish to acknowledge the funding of the activities of the Cooperative Research Center for Robust and Adaptive Systems by the Australian Commonwealth Government under the Cooperative Research Centres Program. The second author was previously at the address of the third author.

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Technical CommuniqueApproximation of frequency response for sampled-data control systems☆

Abstract

Technical Communique
Approximation of frequency response for sampled-data control systems☆