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Password Changed Successfully Your password has been changed. Returning user. Request Username Can't sign in? Forgot your username? In each of these regions, the non-linear function is approximated using a linear affine representation.
These functions allow the development of a systematic and accurate treatment of the approximation. Figure 3 depicts the real curve for the nonlinear gain, as well as the PWL approximation. The parameters identification of the PWL model in Eq.
Wigren presented a structure for the control of Wiener systems. In this scheme see Figure 4 , two static nonlinearities are included in the loop. Under the hypothesis that N. Note that this is the case of the pH neutralizer. Now, the controller design involves two steps: a the inversion of the nonlinear gain and, b the computation of a LTI controller in order to compensate for the linear block model of the process.
To compute f.
Rapid and robust control of single quantum dots
As mentioned above, the linear controller K should be designed to compensate for the behavior of the linear dynamic block of the process model. This controller could be designed using any of the classical techniques found in the literature. Let us consider the reduced loop i. The design transfer function is then. This minimization is performed using the function hinfsyn Gahinet et al.
It is important to note the significance of considering the measurement noise because its existence in the complete scheme which includes the static nonlinearity could produce a mismatch between the PWL sector of N and the PWL sector of f. The robustness of this controller is tested against all the uncertainty sources: the linear model parameters variation Fig.
The system used for robustness analysis is described in Figure 7. Figure 8 shows the simulation results for setpoint changes. Note that the system follows the setpoint with smooth changes in the manipulated variable, even under the wide excursion of the reference signal.
An interesting point is that, when we try to follow the same setpoint using only the linear controller i. Two additional tests are performed; first we consider the effect of the measurement noise. A Gaussian noise of variance 0. The simulation results are shown in Figure 9.
It is important to mention that, when noise variance increases, the performance of the controller deteriorates and the control system could become unstable. In what follows, the effect of perturbations is evaluated. When load changes are applied to the process, a fixed Wiener model no longer represents adequately the process. For example, the titration curve changes drastically Kalafatis et al. To perform a robustness analysis in this case, we consider that q A varies between 0.
The effects of these changes on the LTI model parameters and the nonlinear gain are shown in Figure 10 and Figure 11 , respectively. From these plots, it is clear that the model uncertainties increase. A simulation to evaluate the influence of lower perturbations is shown in Figure In this case, q A is reduced to 0.
In this article, controller design as well as robustness analysis of Wiener systems were considered. Wiener systems modeling and control were treated in the context of the more realistic case in which different sources of uncertainty are present. For this purpose, PWL approximating functions were introduced. As regards the modeling and control approaches proposed in this work, PWL functions proved to be an appropriate and simple tool for uncertainty inclusion, nonlinearity modeling and nonlinearity inversion.
A design technique for the controller synthesis was also proposed. The different topics developed were tackled together in an application example of significant complexity. Biagiola, S. International Journal of Control, 77, Bloemen, H. Journal of Process Control, 11 6 , a.
International Journal of Control, 74, b. Bruls, J. European Journal of Control, 5, Chua, L. CAS, Cheong, M.
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Doyle, J. Gahinet, P. The MathWorks Inc.
Journal of Process Control, 14, International Journal of Systems Science, 31, Journal of Process Control, 14 6 , Gustafsson, T. Chemical Engineering Science, 38, Henson, M. IEEE Trans. Control Syst. Hunter, I.
Biological Cybernetics, 55 , Kalafatis, A. Chemical Engineering Science, 50, Journal of Process Control, 15, a. Automatica, 41, b. Kang, H. Kim, K. Journal of Process Control, 22, Korenberg, M. Journal of Process Control, 13, Mahmoodi, S.
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