Applications of Transfer Function Data

For mechatronic systems, accurate frequency response data (FRD) can be obtained at low cost. Controller design techniques exist that can be applied directly to FRD, e.g. classical loop-shaping. However, loop-shaping can not guarantee that the designed controller is optimal in the sense of closed-loop norm speci cations, and the closed-loop pole locations cannot be computed directly. Besides analysis, applying controller synthesis methods on TFD such as pole-placement for example is even more challenging. A di erent approach is to construct a parametric model of the system. This enables the use of advanced controller synthesis methods such as pole-placement, optimal control, robust control, etc. However, in the modeling phase it is often not clear what aspects of the model are relevant for controller design. It would be advantageous to combine the best of both approaches; to use advanced synthesis methods on the commonly used and well accepted measured FRD. In previous research a way to connect these two approaches is found [12]. In [12], a method has been developed to compute a data-based equivalent of the transfer function for lightly damped mechanical systems, by extrapolating FRD. The data-based version of the transfer function gives transfer function data (TFD) which gives information on the transfer function for the whole s-plane, while FRD only gives information on the value of the transfer function on the imaginary axis. Model-based results described in terms of a transfer function can be applied data-based using TFD. The potential of TFD has been demonstrated in previous research, therefore this project is started to explore further applications of TFD. In this thesis, the computation of TFD and three applications of TFD are studied and simulations are performed to demonstrate the proposed theory. Each of these parts will be described shortly. Regarding the computation of TFD, various methods can be used to compute the TFD for the RHP. Three methods that are discussed are Laplace transformation, convolution and Cauchy contour integral methods. The transfer function of lightly damped mechanical systems is symmetric with respect to the imaginary axis. This makes it possible to compute TFD in the LHP from the RHP. The rst application that is studied, is an extension of the Nyquist stability theorem. It is derived based on two known generalizations of the conventional Nyquist stability theorem; the generalization to MIMO systems and the generalization to alternative contours. It is shown that while the conventional MIMO Nyquist method can give no interpretation of stability margins, the proposed method can give such an interpretation; it shows that the poles in the system have a guaranteed amount of relative damping if it is determined from the generalized Nyquist plot that the system is relative stable. The proposed method can be used data-based, by using the TFD. The second application of the TFD is the symmetric root locus (SRL). The SRL gives the optimal pole locations that minimize a quadratic performance criterion. The SRL can be computed data-based by searching for points at which the TFD is negative and real. The gain of the resulting root locus is equal to the parameter in the criterion that determines the relative importance of the output compared to the input. The data-based SRL lies close to the value of the SRL computed with at model of the system and is therefore a good approximation. Finally, it is shown that a generalization of the SRL exists, the optimal return di erence equation, which is the frequency domain version of the Riccati equation. This equation is used in the next section to an optimal controller that can be computed data-based. The data-based computation of an optimal controller is the third application of TFD that is studied in this thesis. An optimal regulator, that can be computed data-based because it is described completely in terms of H(s), is found in literature. This optimal controller can be derived from the optimal return di erence equation that is the frequency domain version of the Riccati equation. The optimal controller can be computed data-based, by computing a data-based spectral decomposition. A data-based method to choose the optimality parameter ρ is proposed too. Closed-loop pole locations can be selected on the SRL and the corresponding value of ρ can be computed. Two examples of application of this theory are