Optimization of sampled-data systems with finite-dimensional linear continuousperiodic process and delay

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The project concerns the development of methods for the solution of H2 and L2 optimization problems for linear sampled-data systems. The systems consist of a linear continuous periodic multi-input multi-output (MIMO) process with arbitrary pure delays, and a MIMO controller, which is realized as a program on a digital processor. The connection between the continuous and digital parts is accomplished by an ideal analogue-to-digital converter and a zero-order hold with free selectable shape of the pulses. The sampling period Ts is synchronized with the period of the continuous process T, such that both commensurate. Let N>1 be a natural number. Then the following cases should be investigated: Ts=T (equally clocked), Ts=T/N (interpolating), and Ts=NT (decimating).The procedures should yield exact solutions and they also have to consider the intersample behavior. Despite of the theoretical and practical importance of the named class of problems, there does not exist systematic literature about it. The reason for that could be the absence of an appropriate mathematical tool.The chosen approach for the investigations is based on the parametric transfer matrix (PTM) concept. In preceding research projects, the applicant has refined this concept in theoretical directions, and in cooperation with his partners he also successfully applied it to various sampled-data control problems. Alternative approaches are not known, which are able to provide exact solutions, when all three difficulties are present:continuous periodic process + delay + sampling.On the way to the solution of the above formulated H2 and L2 optimization problems, the following steps have to be taken:* Floquet-Lyapunov transformation for extracting the dynamics and hosting them in the time-invariant part* Establishing the PTM for the open chain, and then for the closed loop* Construction of the parametrized set of all stabilizing controllers* Construction of the H2 norm and of the cost functional* Extension and application of the Wiener-Hopf method for the analytical solution* Programming of software tools and their integration into the Matlab-Toolbox DirectSD