To increase the performance of digitally implemented control laws, it is desirable to directly synthesize the digital control law. Hence, this paper considers the direct synthesis of discrete-time, H{sub}∞ control laws. Previously, two approaches have been proposed to this problem. The most common method uses the bilinear transformation to convert the discrete-time problem into an equivalent continuous-time problem, allowing the controller to be synthesized with software developed forcontinuous-time systems. This method is often effective but requires unnecessary steps and is not applicable in all circumstances. The second approach is to synthesize the discrete-time, H{sub}∞control law by solving discrete-time, H{sub}∞Riccatiequations. Due to numerical ill-conditioning, this approach can fail at sufficiently high sample rates. This numerical ill-conditioning can be eliminated by representing the discrete-time system and H{sub}∞ controller using the delta operator. Hence, the H{sub}∞ design equations for direct synthesis using the delta operator are developed. These equations reduce to standard continuous-time H{sub}∞design equations when the sample period is set to zero. The results are illustrated with a numerical examplewhich also shows that the central continuous-time H{sub}∞controller of the continuous-time system obtained by a bilinear transformation of the original discrete-time system does not necessarily correspond to the central discrete-time controller.
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