Optimal Sampled-Data Control Systems

“The fact that the controller is based on output feedback instead of state feedback does not allow for straightforward extensions of existing event-triggering mechanisms if a minimum time between two subsequent events has to be guaranteed.”

“Time-triggered control paves an easy way to design suitable controllers using a fruitful sampled-data system theory [1], [2].”

“The first one is based on lifting [3,4], in which the problem is transformed into an equivalent finite-dimensional discrete-time problem while maintaining the inter-sampling information of the system.”

“Sampled-data systems have been extensively studied in the literature [1, 3, 4, 21, 22] and the references therein.”

“For details on the SDD method for linear systems see [2].”

“Sampled-data systems have been studied extensively over the past decades (see e.g. Chen & Francis, 1995, Fridman, 2010, Fujioka, 2009, Mirkin, 2007 and Naghshtabrizi, Hespanha, & Teel, 2008 and the references therein).”

“We will use a technique similar to lifting [ 2 ] to obtain a discrete linear time-invariant representation of the system.”

“While continuous communication is an ideal assumption, it is more realistic to interact intermittently at discrete sampling instants (Chen & Francis, 1995).”

“Secondly, the sampled-data approach considers discretetime controllers but for a continuous-time model that describes the sampled-data NCS dynamics without exploiting any form of discretization [52].”

“Control design methodologies based on convex embeddings have been presented in Cloosterman et al. (2010), Fujioka and Nakai (2010), Hetel et al. (2006, 2008), and Mustafa and Chen (2013). See also Robert, Sename, and Simon (2010) for an LPV design of controllers that are adapted in real time to the value of the sampli…”

“Sampled-data control has become more and more important due to rapid advancement of the digital hardware technologies (see Chen and Francis, 1995; and reference therein).”

“Digital Object Identifier 10.1109/TAC.2008.919850 a discrete-time controller with sample and hold devices.”

“Lyapunov and Sylvester matrix equations play important roles in system theory [5,6,33–35].”

“As is well known, computers are usually used as digital controllers to control continuous-time systems in modern control systems [4].”

“In stability analysis of control systems and robust control [6], we often need to solve (coupled) matrix equations of the following forms: • Continuous-time (CT) Sylvester equation: AX +XB = C.”

“Sampled-data control of finite-dimensional systems have been studied extensively over the past decades (see e.g. Chen and Francis (1995), Naghshtabrizi, Hespanha, and Teel (2008), Fujioka (2009), Fridman (2010) and the references therein).”

“their wide applications in digital systems and networked control systems in the last two decades [1]–[4].”

“The lifting [12] is a lossless discretization operation converting continuous-time signals into functional space valued sequences.”

“uk decades and numerous results have been reported in the literature [2, 5, 22].”

“Recently, modern sampled-data control theories have been developed, which can optimize the intersample response (e.g., [ 10 ]‐[12]).”

“INTRODUCTION As communication and digital technology have grown, sampled-data systems have been becoming important topic in various research fields (see [2] and reference therein).”

“It is often assumed for the traditional discrete-time sampled-data systems that the control input updating period equals the output sampling period, and such systems are called single-rate systems [1].”

“This special scenario is the one well-studied in the so-called sampled-data control theory [1].”

“1 Introduction There is a strong motivation for the investigation of sampled-data systems due to the prevalence of computer controlled systems (see [4, 5, 6]).”

“On the other hand, in practical and modern control systems, computers are usually used as digital controllers to control continuous-time systems [11].”

“Although using periodic sampling may be adequate for some cases [33], [34], it is usually required to use the nonuniform sampling pattern in resource constrained scenarios to further reduce the energy, computation, and communication costs.”

“Systems with regular missing data can also be viewed as multirate systems which have uniform but various input/output sampling rates (Chen and Francis 1995).”

“By employing the sampled-data system theory (Chen & Francis, 1995), the unmeasurable state can be reconstructed in terms of input/output data, whereby we can iteratively solve the ARE.”

“In conventional sampleddata systems, the sampling periods are always assumed to be fixed [37].”

“Fact 1: If the input is standard (unit variance) white noise, then the root-mean-square (RMS) value of the output equals the H2-norm of the system [2].”

“gif"/> -norm of the system (Chen and Francis, 1995 ).”

“…we discretize via the zero-order hold method with the sampling period to get , and assume that has a state-space model of the form (see, e.g., [50]) (3) where the two matrices and are as follows: For tractability, the lifting technique is used to transform a periodic time-varying system into…”

“In conventional sampled-data control systems, the periodic sampling is popular since it helps simplify the analysis and design of control systems [17]–[20].”

“1 Introduction Emulation is a well-established method to design digital controllers for continuous-time plants (see, for instance [2, 6, 9]).”

“Such control systems are referred to as sampled-data systems [1] and involve both continuous-time and discrete-time signals in the continuous-time framework [28].”

“Among these methods, the most typical ones may be control [ 1 ], [5], [12], variable structure system (VSS) sliding-mode control [6], [16], [18], adaptive control [7], [8], [10], [11], -synthesis method [2], [14], dynamic versus parametric uncertainty control [4], [8], and robust process control [3], [13].”

“…the ZOH function transforms the discrete-time control input u k to a continuous-time control input ( The resulting closed-loop system is (1), (2): In the literature, three main approaches have been used to the sampled-data control (see e.g., [11,14,22,35,50,93,146, 200]) and later to the NCSs.”

“In view of the technological appeal of digital implementations, sampled-data control systems have been the subject of many researches in recent years, see for example, [1]–[9] and the references therein.”

“, [7, 3] and the references therein).”

“rs A+BK Hurwitz and minimizes the H2-norm of the transfer function T (K) : d → z of the closed-loop system " x(k +1) z(k) # = A +BK I " I K # 0 " x(k) d(k) # (5) where [Chen and Francis, 1995, Section 4.4] kT (K)k2 := 1 2π Z 2π 0 trace h ejθ ⊤ h ejθ dθ 1 2 In particular [Chen and Francis, 1995, Section…”

“…progress of computer and digital technologies and non-approximate treatment of the intersample behavior from sampled-data control’s own features, which results in no degradation of the closed-loop performance (Chen & Francis, 1991, 1995; Hara, Yamamoto, & Fujioka, 1996; Hu, Lam, Cao, & Shao, 2003).”

“Hence, sampled-data based control laws become more practical in such cases [23].”

“5κ, κ ∈ N, so that the discretization of the plant remains controllable [46].”

“There are essentially two approaches to deal with this problem; one is loss function sampling [1] and the other is fast lifting [2].”

“Sampled-data systems with periodic sampling have been extensively studied in the literature (Chen and Francis, 1995; Zhang et al., 2001) and this field is now very mature.”

“Sampled-data H∞ control of systems has been studied in a number of papers (see e.g., Bamieh & Pearson, 1992; Basar & Bernard, 1995; Chen & Francis, 1995; Lall & Dullerod, 2001; Sagfors & Toivonen, 1997; Sivashankar & Khargonekar, 1994 and the references therein).”

“[2–4,11,15,16], and the references therein).”

“A nice account of these optimization based approaches for linear systems has been given in the Bode Lecture by Anderson in [1] and later in the book [4].”

“[ 4 ] I. Kanellakopoulos, P. V. Kokotovic, and A. S. Morse, “Systematic de-”

“Sampled-data control has been widely developed and applied in many areas, such as radar tracking systems, time-sharing systems and temperature control, and it has many favourable properties, such as flexibility, robustness and low cost; see Chen and Francis (1995) for further details.”

“Lemma 3 [46]: For the given system (4) in Lemma 2, if the pair (Q1, Q2) is controllable, then there exists a small enough sampling period T such that the pair (eQ1T, ∫ T 0 e 1dsQ2) is Fig.”

“The main issue in sampled-data control of nonlinear systems is that for a continuous-time nonlinear system it is in general not possible to derive an exact discretetime model (Chen & Francis, 1995; Findeisen, 2004; Nesic & Teel, 2001).”

“For example, a multirate sampled-data system can be naturally represented as a periodic system ([7]), many physical dynamics having a cyclic behavior (for example, pendulum ([5])) can be modeled by periodic systems, and in some cases, periodic feedback is used to upgrade the performances of a time-invariant plant, res…”

“The sampled-data approaches used in Araki and Ito (1993), Bamieh, Pearson, Francis, and Tannenbaum (1991), Bamieh and Pearson (1992), Chen and Francis (1995, 1996) , Chen and Qiu (1994), Dullerud (1996), Hara and Kabamba (1990), Kabamba and Hara (1993), Sagfors, Toivonen, and Lennartson (1997c), Toivonen (1992), and Y…”

“The fact that the controller is based on output feedback instead of state feedback does not allow for straightforward extensions of existing event-triggering mechanisms if a minimum time between two subsequent events, the so-called ‘minimum inter-event time’, has to be guaranteed.”

“function as the generalized H2 norm (often abbreviated as the H2 norm) due to their equivalence with proper transformation [ 14 ].”