1Synthesis of Embedded Control Systems with High Sampling FrequenciesJavad Lavaei, Somayeh Sojoudi and Richard M. MurrayAbstract— Motivated by current technological advances in thedesign of real-time embedded systems, this work deals withthe digital control of a continuous-time linear time-invariant(LTI) system whose output can be sampled at a high frequency.Since a typical sampled-data controller operating at a highsampling frequency needs heavy (high-precision) computation toalleviate its sensitivity to measurement and computational errors,the objective is to design a robust hybrid controller for high-frequency applications with limited computational power. To thisend, we exploit our recent results on delay-based controller designand propose a digital-control scheme that can implement everycontinuous-time stabilizing (LTI) controller. This robust hybridcontroller, which consists of an ideal sampler, a digital controller,a number of modified second-order holds and possibly a unityfeedback, can operate at arbitrarily high sampling frequencieswithout requiring expensive, high-precision computation. Lateron, it is discussed how to find a continuous-time LTI controllersatisfying prescribed design specifications so that its correspond-ing digital controller requires the least processing time.I. INTRODUCTIONSince the invention of digital circuits and digital computersin 1937, there has been an every-growing interest in the digitalcontrol of continuous-time systems. Computer controlled sys-tems have been widely used in a broad range of applicationsfrom robotics, autopilot and radar to anti-lock braking systems[1], [2]. A typical digital-control scheme for a continuous-time system is composed of an analog-to-digital converter(sampler), a digital processor and a digital-to-analog converter(holder). This configuration is referred to as sampled-datacontrol system and has been long studied in the literature [3],[4], [5].Among many problems that have been investigated inthe context of sampled-data control systems, one can namestability, robustness, sensitivity and frequency-domain char-acterization. For instance, the paper [6] introduces a liftingtechnique to design H2and H∞sampled-data controllers.The work [7] tackles the H2sampled-data control problemusing a new frequency response operator. The best achievabletracking performance in sampled-data systems are studied in[8]. The works [9] and [10] tackle the stability and trackingcapabilities of sampled-data systems with uncertain and time-varying sampling frequencies. Sparked by the pioneering work[11] on multi-rate systems, a linear-matrix-inequality methodis proposed in [12] for the robust synthesis of multi-ratesampled-data systems.Although the works surveyed above consider the hold deviceto be an ideal zero-order hold, there are other types of holdersstudied in the literature. A generalized sampled-data holdfunction (GSHF) is a common substitute for a zero-order holdThe authors are with the Department of Control and DynamicalSystems, California Institute of Technology, Pasadena, USA (emails:[email protected]; [email protected]; [email protected]).[13], [14]. The early work [15] shows that a GSHF acts as astate feedback controller without requiring a state estimator.Furthermore, it is known that GSHFs are very effective insimultaneous stabilization [16] and decentralized stabilizationof systems with unstable decentralized fixed modes [17].The current silicon technology has enabled the designof embedded systems operating at very high frequencies[18]. However, the conventional methods for the synthesis ofsampled-data control systems require high processing powerto cope with numerical issues if the sampling rate is relativelyfast. More precisely, increasing the sampling frequency makesthe digital controller extremely sensitive to measurement noiseand computational round-off errors. Based on our recent resultin [19], the present work aims to propose a robust digital-control scheme for continuous-time systems that can be used intwo important scenarios: (i) having a high sampling frequencywith limited computational power (ii) having a slow processorwith jitter and irregular sampling times. Note that the secondscenario occurs when the sampling frequency is relativelyfaster than the slow processing rate and, in addition, thesampling times are prone to delays and irregularities [20]. Themain focus of this work will be on the first application (sce-nario), while the second application can be treated similarly.In this paper, the sampled-data control of a continuous-time LTI system is studied, where the output of the system issampled at a high rate. It is shown that every continuous-timestabilizing (LTI) controller can be implemented in a hybridform consisting of a sampler, a digital processor, some so-called “modified second-order holds” and possibly a unityfeedback from the holder to the sampler. This hybrid controllerbenefits from the fact that the increase of the samplingfrequency has a direct influence only on the memory sizeof the controller, as opposed to its parameters. This propertymakes the parameters of the controller robust to the samplingrate. Moreover, it is proven that designing a continuous-timecontroller whose associated digital controller requires the leastprocessing time amounts to a well-studied control problem.The rest of the paper is organized as follows. Somepreliminaries on conventional sampled-data control systemsare provided in Section II and the problem is formulatedaccordingly. The main results are derived in Section III, whichare illustrated with a numerical example in Section IV. Finally,some concluding remarks are given in Section V.Notation: Throughout this paper, the letters t, κ, s, ω andz denote the continuous-time, discrete-time, Laplace-domain,Fourier-domain and Z-domain arguments, respectively. More-over, time-domain signals are denoted by small letters, whiletheir corresponding frequency/Laplace/Z-domain signals arerepresented by capitalized letters. For a continuous-time sig-nal, say h(t), the following notations are used:• h[κ]: A discrete signal obtained from h(t) by sampling.Submitted, 2010 Conference on Decision and Controlhttp://www.cds.caltech.edu/~murray/papers/lsm10-cdc.html2Fig. 1. This figure illustrates a conventional sampled-data control system.• H(s): Laplace transform of h(t).• H(jω): Fourier transform of h(t).• H[z]: Z-transform of h[κ].II. PRELIMINARIES AND