On the minimal partial realization of 2D discrete linear shift-invariant systems

IDENTIFICATION AND MINIMAL REALIZATION OF MULTIVARIABLE SYSTEMS

Linear discrete controlled systems and linear discrete pursuit games are considered in the problem where the control vector of the pursuer is subject to a geometrical constraint and the control of the pursued object is subject an integral constraint. Necessary and sufficient conditions of solvability of the 0-controllability problem and that of coming together at the zero point are proved by way of constructive selection of the control vector.

The authors develop the theory of minimal realization of linear discrete event systems. Some investigations on the analyses of eigenvalues and periods of matrices in the Z-domain are carried out. >

Mathematical model in the form of discrete linear system of equations with multiple input and multiple output (MIMO) describes different important for practice dynamic processes. Model predictive control (MPC) in recent years is a very intensively developing field of control theory and application. In given chapter, some of the possible realizations of MPC for discrete MIMO linear system are considered. Developed approach is based on the Cauchy formula for discrete linear system. Problem of control design is reduced to solving series of extremal problems including constrained optimization problems. It has succeeded to obtain final results in analytical form which is appropriate for solving different problems including control analysis and design.

Consider a discrete dynamical system given by a square matrix $M \in \mathbb{Q}^{d \times d}$ and a starting point $s \in \mathbb{Q}^d$. The orbit of such a system is the infinite trajectory $\langle s, Ms, M^2s, \ldots\rangle$. Given a collection $T_1, T_2, \ldots, T_m \subseteq \mathbb{R}^d$ of semialgebraic sets, we can associate with each $T_i$ an atomic proposition $P_i$ which evaluates to true at time $n$ if, and only if, $M^ns \in T_i$. This gives rise to the LTL Model-Checking Problem for discrete linear dynamical systems: given such a system $(M,s)$ and an LTL formula over such atomic propositions, determine whether the orbit satisfies the formula. The main contribution of the present paper is to show that the LTL Model-Checking Problem for discrete linear dynamical systems is decidable in dimension 3 or less.

: This report considers a feedback control law for linear time-varying and time invariant discrete systems based on a receding horizon concept applied to a minimum energy problem with fixed terminal constraints. The control law is shown to be asymptotically stable and to result in a new method for stabilizing linear time-varying systems as well as extending some well known methods for stabilizing time invariant systems. In particular, the stabilizing gains of the feedback control law are obtained from the solution to a discrete Riccati equation over an arbitrary finite time interval, which is relatively easy to compute. The gain matrix reduces to a constant matrix for linear time invariant systems. Some stability results will turn out to be special cases of these results. The results parallel those for linear continuous time systems, although the technical details are tedious and more involved.

A procedure based on neural networks for the classification of linear and nonlinear systems is presented, using excitation and response data under swept sine excitation. Special attention is paid to the classification and identification of linear and bilinear systems, the latter being considered since they exhibit typical characteristics of cracked systems. The computer simulations show that: (1) using the procedure presented in this paper the trained classification network can reliably classify a linear system and different nonlinear systems; (2) the output of the trained identification neural network for a linear system and a bilinear system can be used as a quantitative indicator of characteristics of bilinear systems having different stiffness ratios (k(x>0)/k(x<0)) with respect to the bilinear system used in the training stage; (3) for two-degree-of-freedom systems, the trained network can not only determine the existence of a bilinear stiffness and the magnitude of its stiffness ratio, but also specify which stiffness is bilinear, i.e. indicate its position. These results provide a possibility of using the trained neural networks to detect and locate structural cracks which have the characteristics of bilinear systems.

The paper extends the Routh approximation method for one-dimensional (1-D) discrete systems to two-dimensional (2-D) discrete systems for finding stable reduced-order models from a stable high-order 2-D linear discrete separable-denominator system (SDS). The extension is achieved by exploring new properties of the 1-D Routh canonical model and establishing new 2-D bilinear Routh canonical models. Without explicitly performing bilinear transformations, a computationally-efficient procedure is presented for finding the bilinear Routh reduced-order models. The properties of the obtained 2-D bilinear Routh approximants are discussed in detail. In addition, a new 2-D bilinear Routh canonical state-space realisation is presented from which the low-dimensional state-space models corresponding to the bilinear Routh approximants can be obtained by a direct truncation procedure. Furthermore, the relationships among the states of the bilinear Routh reduced-dimension model, the aggregated model, and the original system are explored. Numerical examples are given to demonstrate the effectiveness of the proposed method.

Control of industrial plants is an important engineering field of study, where the optimization of such processes can lead to better product quality and higher profit. The present work deals with the optimization of an industrial gasifier, proposed as a benchmark challenge by ALSTOM Power Technologies in 2002. Researchers from around the globe proposed different methods for optimizing such system, using linear and nonlinear system identification techniques, different controller schemes, such as proportional-integral-derivative and model predictive controllers, and search methods for controller tuning. The authors care for the study of a methodology that can generalize the controller optimization process of real industrial systems, where linear system identification can lead to simpler systems, which are used to design more robust controllers. The proposed work deals with linear system identification and multiobjective optimization for the gasifier's controller design. To do so, the authors divide the problem in two steps, the linear system identification, where step functions are fed to the gasifier in order to detect the output response of the system, and the design of a proportional-integral-derivative controller, where spherical pruned multi-objective differential evolution algorithm is used in order to optimize the controller gains. The proposed method, unfortunately, does not provide an optimal controller, which can be explained by the complexity of the benchmark system. With such results, the authors believe a better multi-objective problem definition is necessary. Also, it is concluded that such benchmark presents a real challenge for future controller design methndologies.

The paper is concerned with the essential spectral points of singular discrete linear Hamiltonian systems. Several sufficient conditions for a real point to be in the essential spectrum are obtained in terms of the number of linearly independent square‐summable solutions of the corresponding homogeneous linear system, and a sufficient and necessary condition for a real point to be in the essential spectrum is given in terms of the number of linearly independent square‐summable solutions of the corresponding nonhomogeneous linear system. As a direct consequence, the corresponding results for singular higher‐order symmetric vector difference expressions are given.

In this work, we consider linear, shift-invariant and complete two-dimensional (2D) discrete systems from a behavioral point of view. In particular, we examine behaviors with two types of variables: the variables that we are interested to control (the to-be-controlled variables) and the variables on which we are allowed to enforce restrictions (the control variables). The main purpose of this contribution is to derive necessary and sufficient conditions for the stabilization of the to-be-controlled variables by ‘attaching’ a controller to the control variables. This problem turns out to be related to the decomposition of a given behavior into the sum of two sub-behaviors. Moreover, we show that under certain conditions, it is possible to obtain a constructive solution and characterize the structure of the to-be-controlled behavior.

Finite-time stability and stabilization of linear discrete time-varying stochastic systems

Orthonormal basis functions models: A transformation analysis

The problem of linear-quadratic optimal terminal control of a linear discrete dynamic system with a vector input is considered. Solutions of the problem based on the apparatus of the optimization form of pseudo-circulation of matrices are given. The solution of the optimal terminal control problem is obtained, which ensures the achievement of the terminal control goal and the passage of the trajectory of the system as close as possible to the specified intermediate points. A complete set of solutions to the problem of optimal terminal control of a bundle of trajectories is obtained.

A new solution to the quadratic optimal tracking problem in linear multivariable discrete-time systems is presented. The approach is based on polynomial input-output models. The optimal control law consists of a plant output feedback and a reference feedforward. It is obtained by performing spectral factorization and then solving two matrix polynomial equations. The design procedure is simple and well suited for systems with inaccessible states.