Applied estimation of piecewise constant parameters

Abstract: A time-varying linear system is a realistic description of many industrial processes, and nonlinear behavior can then also be accounted for. Then, we can consider a linear system with time-varying parameters as the model uncertainty, \emph{e.g.} an AR(X) model or an affine input-output approximation. In this thesis, we seek to estimate these parameters of the linear time-varying system for two purposes: 1) As uncertainty bounds for use in robust control. 2) Fault detection and isolation.Robustness is a necessary property of a control system in an industrial environment, due to changes of the process such as changes of material quality, aging of equipment, replacing of instrument, manual operation (\emph{e.g.} a valve that is opened or closed) etc. The uncertainties associated with the nominal process model is a concern in most approaches to robust control. One purpose of this research is to achieve a bound of the uncertainty by using a set of measurement data.Change detection is a quite active field, both in research and applications. Faults occur in almost all systems, and change detection often has the aim to locate the fault occurrence in time and to raise an alarm. Examples of faults in an industry are leakage of a valve, clogging of a valve or faults in measurement instruments. The second purpose of this research is thus to detect faults under the assumption that these are manifested as abrupt parameter changes.Many time-varying changes or faults of industrial processes can be described as abrupt changes in parameters. The approach is to model them as piecewise constant parameters which results a sparse structure of their derivative. This quality serves as a cost and regularization on flexibility for parameter estimation.Sparsity can be approximated in different ways, \emph{e.g.} with $l_q$-norm for $q\leq1$. We present several online methods to estimate piecewise constant parameters, based on these approximators. As an application, the parameters of a pump in the flue gas desulphurization process at the Luossavaara-Kiirunavaara AB (LKAB) facility in Malmberget, Sweden, are estimated for the purpose of detecting if the pump is coated or worn.We also present an exact solution of maximization of sparsity by using MILP (Mixed Integer Linear Programming) to minimize the number of non-zero elements in a matrix or vector. The metod is used as a regularization to estimate the time-varying parameters of an AR(X) model. We specifically apply it to detect faults in a blender's hinged-outflow valve which is included in the pelletization of LKAB. Proper function of this valve is essential for the mixing of bentonite and slurry and thus for the quality of the iron ore pellets. Simulation with measurement data from the LKAB facility at Malmberget, Sweden, shows the viability of the algorithm.We consider a time-varying time-delay first-order process model. The gain, time-constant and time-delay are considered as uncertainties in this example. An estimate of the perturbations is produced based on the MILP method. The Pad\'{e}-approximation and orthogonal collocation method are used to approximate the delay. An overhead crane with uncertain parameters is also used as an illustrative example.