Structures of commutative linear representation systems and an efficient coding for two-dimensional arrays

Full probabilistic solution of a finite dimensional linear control system with random initial and final conditions

The purpose of this paper is to give four new applications of the Quillen-Suslin theorem to mathematical systems theory. Using a constructive version of the Quillen-Suslin theorem, also known as Serre's conjecture, we show how to effectively compute flat outputs and injective parametrizations of flat multidimensional linear systems. We prove that a flat multidimensional linear system is algebraically equivalent to the controllable 1-D dimensional linear systems obtained by setting all but one functional operator to zero in the polynomial matrix defining the system. In particular, we show that a flat ordinary differential time-delay linear system is algebraically equivalent to the corresponding ordinary differential system without delay, i.e., the controllable ordinary differential linear system obtained by setting all the delay amplitudes to zero. We also give a constructive proof of a generalization of Serre's conjecture known as Lin-Bose's conjecture. Moreover, we show how to constructively compute (weakly) left-/right-/doubly coprime factorizations of rational transfer matrices over a commutative polynomial ring. The Quillen-Suslin theorem also plays a central part in the so-called decomposition problem of linear functional systems studied in the literature of symbolic computation. In particular, we show how the basis computation of certain free modules, coming from projectors of the endomorphism ring of the module associated with the system, allows us to obtain unimodular matrices which transform the system matrix into an equivalent block-triangular or a block-diagonal form. Finally, we demonstrate the package QuillenSuslin which, to our knowledge, contains the first implementation of the Quillen-Suslin theorem in a computer algebra system as well as the different algorithms developed in the paper.

In this paper we attempt to describe a circle of ideas which makes possible a unified exposition of a large part of linear algezbra, operator theory and both finite and infinite dimensional multivariable linear systems. As the title suggests the unifying conzcepts will be those of functional models, module theory and various factorizations of polynomial and analytic matrix valued functions. The full exposition of the ideas presented here will be the theme of a forthcoming monograph [20].

From time series to linear system—Part I. Finite dimensional linear time invariant systems

A new canonical form for the minimal realizations of finite dimensional multi-input multi-output linear systems possessing cyclic state spaces has been developed. This minimal realization is called the ‘observable-pair realization’ and may be used as a direct summand of minimal realizations of general linear systems. An algorithm, suitable for computer implementation, is presented for calculating the observable-pair realizations of transfer function matrices of cyclic systems. It is shown that this algorithm is readily used as part of a second algorithm for calculating the minimal realizations of general transfer functions. Finally, it is proved that in a special sense the observable-pair realizations are unique. Notes † Communicated by Professor J. H. Westcott

In this paper we consider a linear autonomous system with delays and we prove that we can associate a mantainability subspace to each zero of the system matrix. This space is the generalized eigenspace of the closed loop system obtained by using a special feedback. This observation provides a representation of the system which generalizes a known result on finite dimensional linear systems.

Consistent estimation of real NMP zeros in stable LTI systems of arbitrary complexity

Statistical Analysis and H2 - Norm of Finite Dimensional Linear Time-Periodic Systems

The main result of this paper establishes that if a controller $C$ (comprising of a linear feedback of the output and its derivatives) globally stabilizes a (nonlinear) plant $P$, then global stabilization of $P$ can also be achieved by an output feedback controller $C[h]$, where the output derivatives in $C$ are replaced by an Euler approximation with sufficiently small delay $h>0$. This is proved within the conceptual framework of the nonlinear gap metric approach to robust stability. The main result is then applied to finite dimensional linear minimum phase systems with unknown coefficients but known relative degree and known sign of the high frequency gain. Results are also given for systems with nonzero initial conditions.

The minimal time function of a class of semilinear control systems is considered in Banach spaces, with the target set being a closed ball. It is shown that the minimal time functions of the Yosida approximation equations converge to the minimal time function of the semilinear control system. Complete characterization is established for the subdifferential of the minimal time function satisfying the Hamilton---Jacobi---Bellman equation. These results extend the theory of finite dimensional linear control systems to infinite dimensional semilinear control systems.

The theory of balanced realizations is essentially based on co-ordinating controllability and observability properties of a finite dimensional linear asymptotically stable system. By analyzing more general quadratic measures on the external behaviour of a dynamical system, we may obtain state space representations in which other quantities are internally balanced. For the class of L 2-trajectories of finite dimensional linear time invariant systems, we formalize the notion of a balanced state space with respect to these type of quadratic measures. A particular choice of quadratic measures is studied in more detail and it is shown that ‘classical’ balanced realizations are obtained as a special case.KeywordsBalanced realizationsfinite dimensional linear time invariant systemsstate space methodsmodel approximation

Many methods of image generation being mainly intended for the realism of images, this pattern generation is a very different graphic generation from them. It is a trial for a sort of graphic art by using a machine. The equipments with machines are mainly composed of a mathematical model which may be constructed by computer programs or switching and other electric circuits. The model, called commutative linear representation systems (CLRS), is suitable for computer programs and electric circuits for linear operations. It has been already shown that there are relations between the index of CLRS and pattern with periodicity. In this paper, basic systems are introduced with the fact that any finite-sized pattern can be represented by them. And any pattern with periodicity can be represented as direct sum of minimum CLRS, which are called quasi-reachable basic system.

New properties of finite-dimensional linear discrete-time systems under conical control constraints that are similar to the “hidden convexity” of continuous-time systems are studied.

This paper presents a model predictive control (MPC) formulation for the Czochralski (CZ) crystal growth which is given by the discrete finite-dimensional system representation of a class of parabolic partial differential equations (PDEs) with time-varying features. The motivation behind this work is to address the problem of stabilizing, infinite horizon, linear quadratic regulator synthesis for PDEs with time-dependent parameters which affect the dynamics of the underlying system and appear in the state-space representation. The modal decomposition of the PDE facilitates its approximation by a finite-dimensional linear state-space system. A receding horizon regulator is proposed which incorporates the time-dependence of the PDE parameters and numerical simulations are carried out which demonstrate the optimal stabilization of the process under state and input constraints.

SummaryThis paper presents a new observer‐based controller design method for Lipschitz nonlinear systems with uncertain parameters and ‐bounded disturbance inputs. In the presence of uncertain parameters, the separation principle is not applicable even in the case of linear time invariant systems. A state of the art review for uncertain linear systems is first presented to describe the shortcomings and conservatism of existing results for this problem. Then a new LMI‐based design technique is developed to solve the problem for both linear and Lipschitz nonlinear systems. The features of the new technique are the use of a new matrix decomposition, the allowance of additional degrees of freedom in design of the observer and controller feedback gains, the elimination of any need to use equality constraints, the allowance of uncertainty in the input matrix and the encompassing of all previous results under one framework. An extensive portfolio of numerical case studies is presented to illustrate the superiority of the developed design technique to existing results for linear systems from literature and to illustrate application to Lipschitz nonlinear systems. Copyright © 2016 John Wiley & Sons, Ltd.