Spatiotemporal systems are systems whose dynamics depend on time and space and are commonly found in real life. These systems are mathematically modeled using partial differential equations and are also known as distributed-parameter systems. Due to their structure and the high number of variables involved, control and estimation for this class of systems are very challenging. This thesis addresses two problems related to spatiotemporal systems: state estimation and system identification.

Monitoring the states of a control system is important to ensure the behavior of the system is achieving the control objectives. This can be achieved, among others, by using state observers that estimate the states of the systems regularly. First, we present a literature review of observer design methods for distributed parameter systems. In general, the design requires a dimension-reduction approach to implement the observer. From the dimension reduction, the design approaches can be classified into late and early lumping. In the late lumping perspective, model reduction is performed at the end of the observer design. In the early lumping perspective, dimension reduction is applied to the model of the system. We incorporate both approaches in our literature review.

State observer design requires the model of the systems. This thesis also presents a system identification method for distributed-parameter systems. The identification of such systems typically requires spatially dense and regular measurements, followed by selecting sensors that provide significant measurements to the model to reduce the model complexity. However, these requirements may be challenging to fulfill. In case the sensor locations are irregular and sparse in space, we propose the use of lumped-parameter system identification.

For models with a large number of regressors, we propose a method for reducing the number of regressors using a tree representation. The tree is a way to list models with different numbers of regressors. From all possible regressors for the model, the proposed method builds the tree from the simplest models, i.e., models with one regressor. The number of regressors in the models is incrementally increased to one or more models with the best performance. The addition is repeated until the tree contains models with the desired maximum number of regressors.

System identification is typically performed using a complete data set, i.e., for each input sample, there is an associated output sample available. However, there are cases in which some output samples are not recorded in the data set, making the identification data incomplete. This thesis also considers the problem of incomplete data for Takagi-Sugeno (TS) fuzzy system identification using the product space clustering method. This method comprises two steps: fuzzy clustering and rules construction. The first proposed method enables the use of incomplete system identification data to fuzzy c-means clustering algorithm developed for incomplete classification, which yields different estimates for a missing sample. This can be achieved by fusing those different values into a single value. The second proposed method treats missing samples as optimization variables during the identification process. The optimization is repeated until the change of all optimization variables is small.
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