Keywords: multirate systems, frequency response, discrete-time systems, sampled-data systems, inferential control, multivariable control.

Frequency analysis is an appropriate tool to study the robustness of a controlled system to either disturbances or uncertainties. When a sampled-data system (SD) is analysed in the frequency domain, the dimension of the frequency operators often leads to difficult implementation problems. Among the different techniques to deal with the SD frequency response, two main approaches can be singled out. The approach proposed by Araki and his co-workers [1], is based on the definition of an infinite dimension frequency response (FR) operator. The SD impulse input, by the so-called impulse modulation [2], is decomposed into an infinite sum of sinusoidal components. Clearly, the computation of this operator is not straight forward. The second, and also very popular, approach is the so-called continuous lifting [3], which is based on transforming the sampled-data system to a discrete LTI system, taking into account the input and output expansion into functional spaces. Again the model leads to an infinite dimension operator, with the corresponding difficulty in its computation. Both approaches allow for the steady-state computation of the frequency response and can be used to obtain the continuous time output response, leading to equivalent operators [4]. In this paper, the main objective is to present an approximate analysis by assuming the addition of a fictitious high rate sampler at the output of the system. In this way, a multirate based approach allows the intersampling behaviour of the system to be taken into account by selecting a faster sampling rate at the output. Following this approach, the controlled process robustness to model uncertainties and disturbances can be derived by using the frequency analysis of multirate systems. The proposed methodology by using singular values of the multirate discrete lifted representation [5], greatly simplifies the aforementioned study. The proposed methodology has been used in this paper to compare different control structures [6,7]. As a result of this comparison it is proved, with a counter example, that the claim that a dual rate inferential control could be more robust than a fast single rate one assuming some kind of parametric uncertainty [6], is not always correct. For the sake of comparison, the control performances using a non-conventional dual-rate controller are also computed.

1

Araki, M., Ito, Y., Hagiwara T., "Frequency Response of Sampled-data Systems", Automatica, Vol. 32, No. 4, pp. 483-497, 1996. doi:10.1016/0005-1098(95)00162-X

2

Goodwin, G.C., Salgado, M., "Frequency Domain Sensitivity Functions for Continuous Time Systems under Sampled Data Control", Automatica, Vol. 30, No. 8, pp. 1263-1270, 1994. doi:10.1016/0005-1098(94)90107-4

3

Yamamoto, Y., "On the state space and frequency domain characterization of H_inf-norm of a sampled data system", Systems and Control letters, Vol. 21, pp. 163-173, 1993. doi:10.1016/0167-6911(93)90119-Q

4

Yamamoto, Y., Araki, M., "Frequency response of sampled-data systems. Their equivalence and relationships", Linear Algebra and its Applications, Vol. 205-206, pp. 1319-1339, 1994. doi:10.1016/0024-3795(94)90389-1

5

Meyer, R.A., Burrus, C.S., "A unified analysis of multirate and periodically time-varying digital filters", IEEE Trans. Circuits Syst., CAS-22, pp. 162-168, 1975. doi:10.1109/TCS.1975.1084020

6

Li, D., Shah, S.L., Chen, T., "Analysis of Dual-Rate Inferential Control Systems", Automatica, Vol. 38, No., pp. 1053-1059, 2002. doi:10.1016/S0005-1098(01)00295-3

7

Salt, J., Albertos, P., "Multirate. Model-based multirate controllers design", IEEE Trans. Contr Syst Technol, 13(6), pp. 988-997, 2005. doi:10.1109/TCST.2005.857410

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