Event-based state reconstruction and control of linear time-invariant continuous system

This thesis deals with event-based state reconstruction and control strategy, motivated by certain bottlenecks in the control loop, especially by the rotational speed sensors with binary-valued outputs and BLDC motor with Hall-Effect binary sensors to measure the angle of the rotor. For event-based state reconstruction of linear time-invariant continuous system, it is first shown that the traditional passive-type state reconstruction that does not require input design will fail in general, even if the system has a full-rank observability matrix. It is proved that under controllability conditions and bounded uncertainty on the initial state, it is always possible to construct a causal input that will cause the output to cross the sensor threshold in a designated time interval and the input design is explored. Then after introducing the main theoretical foundation of this thesis on the number of zeros of a special class of functions in a designated time interval and by using some recent mathematics results on zeros of harmonic functions, it is proved that if the system is observable, such a reconstruction is always possible if the eigenvalues of the system are all real valued and if the eigenvalues of the system contain only purely imaginary and non-repeating values, it is sufficient that threshold crossing occurs within a relatively small time interval. At present, general conclusions for the numbers of zeros of functions that are sums of weighted exponential and sinusoidal functions remain unavailable, but in general without constraints on system eigenvalues, an input can always be randomized to ensure that the state can be reconstructed with probability one. Constructive algorithms are developed for initial state reconstruction under these conditions. Illustrative examples with only one binary sensor are provided and applications of a simplified DC electric machine with the known load or the unknown load and with multiple binary sensors are proved. After a general event-based control strategy is proposed, a hybrid two-stage finite-time controller consisted of two open loop procedures performed sequentially is explained as: 1) Finite-time initial state estimation based on enough recorded pairs of system output values and time instant values generated by the events. Once the state is reconstructed, 2) Finite-time control to drive the system from the current state to the desired state based on the optimal control is calculated and applied. The proposed method is illustrated by the example of a position control of an AGV in the seaport container terminal environment