# Direction-dependent processes - Theory and application

[Doktorsavhandling]

In this thesis, direction-dependent processes are described as processes whose responses differ in shape, strength and speed, depending on whether the input signal or output signal increases or decreases. The way in which these dynamics arise is discussed for three different applications; these applications are a thermomechanical pulp refiner, in which the motor load response to the hydraulic pressure is direction dependent, a paper machine, in which the calliper control in the calender section is direction dependent, and an electronic nose, in which the response to substrate concentration is direction dependent. In addition, it has been shown that direction-dependent dynamics originate not only from the process itself but sometimes also from sensors or actuators.

Direction-dependent processes have been put into the framework of piecewise-linear dynamic systems, which consist of two linear time-invariant submodels. The sign of the rate of change of the input signal schedules the switching between the submodels. Parameter estimation methods have been derived, where the existing algorithms for parameter estimation in piecewise-linear processes have been extended in order to cope with direction-dependent dynamics. The dynamics in the above-mentioned industrial processes have been justified using direction-dependent parameter estimation methods. In these processes, the direction-dependent modelling approach has been compared with the standard-procedure linear time-invariant approach.

Theoretical aspects, such as the realisation of the input/output form into the state-space form and the convergence of the time-averaged norm, have been considered. Even though a general piecewise-linear state-space model cannot be converted into input/output form, it has been shown that it is possible to find state-space models in two special cases. The convergence of the averaged mean-square error to its expected value, when the number of data points approaches infinity, has been proven for the direction-dependent parameter estimation method. Further, the control of direction-dependent processes is discussed in terms of a new concept called trend controllability. This concept deals with the ability to govern the output in any direction, even though the input signal is bounded. Finally, it can be concluded that direction-dependent models in general and the piecewise-linear ones presented here are suitable when it comes to identifying processes that exhibit direction dependence.

**Nyckelord: **direction-dependent dynamic processes, piecewise-linear modelling, realisation, parameter estimation, pulp and paper applications, electronic noses

Denna post skapades 2006-10-23. Senast ändrad 2013-09-25.

CPL Pubid: 1902