Finite-dimensional Discrete-time Systems Research Articles

Autoencoder-Based Reduced Order Observer Design for a Class of Diffusion-Convection-Reaction Systems

The application of autoencoders in combination with Dynamic Mode Decomposition for control (DMDc) and reduced order observer design as well as Kalman Filter design is discussed for low order state reconstruction of a class of scalar linear diffusion-convection-reaction systems. The general idea and conceptual approaches are developed following recent results on machine-learning based identification of the Koopman operator using autoencoders and DMDc for finite-dimensional discrete-time system identification. The resulting linear reduced order model is combined with a classical Kalman Filter for state reconstruction with minimum error covariance as well as a reduced order observer with very low computational and memory demands. The performance of the two schemes is evaluated and compared in terms of the approximated L2 error norm in a numerical simulation study. It turns out, that for the evaluated case study the reduced-order scheme achieves comparable performance with significantly less computational load.

DiffEq: A MATLAB-Based Discrete-Time System Simulator [Testing Ourselves

A MATLAB-based discrete-time system simulator (DiffEq) is introduced. This tool investigates linear constant-coefficient difference equations, which represent linear time-invariant finite-dimensional discrete-time systems. Closed-form analytical solutions can be expressed as a superposition of homogeneous and particular solutions, and this distinction gives valuable insights into the transient and steady-state behavior of the system. The transfer function, impulse response, characteristic roots, and simulation diagram of the system are obtained for different input types and initial conditions.

Analysis and stabilization for networked linear hyperbolic systems of rationally dependent conservation laws

In this paper, for a networked linear hyperbolic partial differential equations (PDEs) system of conservation laws, the propagation periods of which are rationally dependent, with coupled boundary conditions, we propose a novel approach to analyze its controllability and observability. In addition, we propose a design method of a stabilizing controller, where a boundary-input with boundary-valued feedback is considered. First, we characterize the control properties, such as controllability of such a PDE system, in terms of the corresponding ones of a finite-dimensional discrete-time system defined on the boundaries of the PDE system, which is derived by fully exploiting the method of characteristics. Since the obtained discrete-time system is low-dimensional, its analysis is relatively easier. Next, we propose a design method of a stabilizing controller based on this discrete-time system. Finally, numerical simulations are presented to show that the proposed method is effective.

Non-uniform robust global asymptotic stability for discrete-time systems and applications to numerical analysis

The notion of non-uniform Robust Global Asymptotic Stability (RGAS) presented in this paper generalizes the notion of non-uniform in time RGAS for finiteor infinite-dimensional discrete-time systems. Lyapunov characterizations for this stability notion are provided. The results are applied to finitedimensional discrete-time systems obtained by time discretization of continuous-time systems by the explicit Euler method.

System order and structure indices of linear systems in polynomial form

Linear time-invariant, finite-dimensional discrete time systems are very often specified in a polynomial form representation in either a forward or backward shift operator (sometimes called MFD and ARMA form). In this paper it is shown that there exists a unifying theory for the determination of McMillan degree and Kronecker observability indices of systems represented by polynomial matrices in the two shift operators, considering the MFD and ARMA forms as special cases. Treating dynamical systems in terms of their behaviour, i.e. the set of admissible signal trajectories, the notions of McMillan degree and observability indices can be generalized to non-controllable as well as to non-causal systems.

On the Order and Structural Indices of Linear Systems Represented in Polynomial Form

Linear time-invariant, finite dimensional discrete time systems very often are specified in a polynomial form representation in either a forward or backward shift operator (sometimes called MFD and ARMA form). Considering dynamical systems in terms of their behaviour, i.e. the set of admissible signal trajectories, it is shown that there exists a unifying theory for the determination of McMillan degree and Kronecker indices of systems represented by polynomial matrices in the two shift operators, considering the MFD and ARMA forms as special cases. Moreover the notions of McMillan degree and Kronecker indices can be generalized to noncontrollable and noncausal systems.

On the stability of linear discrete-time systems

In Anderson a connection of two types of stability: Liapunov stability, reflecting the internal stability of a system, and bounded-input bounded-output stability, reflecting the external stability of a system, has been established for time-varying linear finite-dimensional dynamic systems which may be represented in a special canonical form. This correspondence extends the connection to time-varying linear finite-dimensional discrete-time systems.