This paper discusses a connection between the Wronski matrix and observability of linear time-invariant systems as well as their unobservable subspaces. For a linear time-invariant single-output system, its observability is closely related to linear independence of a set of functions associated with the output. Hence, there is a connection between observability and the Wronski matrix, which is often used for deciding linear independence of a set of functions. In discussing observability and the unobservable subspace in this context, it matters whether the Wronski matrix can describe the necessary and sufficient condition for an arbitrary set of functions to be linearly independent. This paper first reviews the known facts about the relationship between linear independence of a set of functions and the Wronski matrix, as well as an additional condition under which linear independence can be decided in a necessary and sufficient fashion through the Wronski matrix. Then, well-known conditions on observability and characterization of the unobservable subspace of linear time-invariant systems are revisited from the viewpoint of the null space of the Wronski matrix.

In this paper, we deal with initial state estimation problems of the heat equation in metric graphs. Particularly, we aim to reveal proper assignments of observation points on metric graphs to identify the peak positions of initial states. From numerical simulations by singular value decompositions, it is predicted that a kind of symmetricity of metric graphs increases a minimum number of proper assignments of observation points. We also discuss relations between initial state estimation problems in partial differential equations in metric graphs and some problems in discrete graphs.

This paper studies on the problem of consensus state for the event-triggered multi-agent system. Firstly, the method of analyzing consensus state of a simple linear multi-agent system with undirected topology is introduced. Then, such method is applied into a general linear system with directed graph. It is found that the consensus state is a constant vector or a periodic varying vector. Such difference depends on the system matrix, the topology as well as the triggering function. Simulations are performed to verify these relations.

This paper proposes a discrete state predictive control by introducing a virtual system. To determine the state-space model and the time delay, the subspace identification method, called CCA(Canonical correlation analysis) method, is applied. The CCA method can judge orders of the system by checking how many singular values of a covariance matrix among input-output data are near 1. First, the time system is shown to be identified by a higher-dimensional system, and this model can be used to the LQG control for the time delay system. After that, by confirming the number of singular values near 1 according to time-shift operations between input-output data, the time delay can be identified and used it for the discrete state predictive control. The proposed method is applied to the stable vibration cart type pendulum with time delay inserted in artificially.

Since unexpected machine failures are huge losses for users, maintenance activities are essential. If the failures can be predicted in advance using a supervised learning, the machines can be maintained before they break down and some failures can be prevented. However, although a large number of failure data are required to predict failures using a supervised learning, failures rarely occur in the actual field. In this study, we propose to detect the failure of a hydraulic excavator using an autoencoder, which is an unsupervised learning. By using the autoencoder to model normal state data, the failure can be predicted in advance. This paper shows the results of evaluating failure predictions using the LSTM (Long Short-Term Memory) autoencoder model for actual failure of hydraulic excavators.