Osami Saito

Output feedback stabilizability and stabilization algorithms for 2D systems

Alternative methods are proposed for test of output feedback stabilizability and construction of a stable closed-loop polynomial for 2D systems. By the proposed methods, the problems can be generally reduced to the 1D case and solved by using 1D algorithms or Gröbner basis approaches. Another feature of the methods is that their extension to certain specialnD (n>2) cases can be easily obtained.Moreover, the “Rabinowitsch trick,” a technique ever used in showing the well-known Hilberts Nullstellensatz, is generalized in some sense to the case of modules over polynomial ring. These results eventually lead to a new solution algorithm for the 2D polynomial matrix equationD(z, w)X(z, w)+N(z, w)Y(z, w)=V(z, w) withV(z, w) stable, which arises in the 2D feedback design problem. This algorithm shows that the equation can be effectively solved by transforming it to an equivalent Bezout equation so that the Gröbner basis approach for polynomial modules can be directly applied.

The design of practically stable n D feedback systems

This paper deals with, by using the matrix fractional description (MFD) approach, the problem of feedback practical stabilization of nD (multidimensional) discrete systems whose input and output signals are unbounded in, at most, one dimension. A constructive algorithm is first presented for solving the Bezout equation over the ring of practically stable rational functions. Then, a necessary and sufficient condition for an nD system to be practically stabilizable is derived and the parametrization of all nD practically stabilizing compensators is given. These results make it clear that the nD practical stabilization problem can be essentially solved by using 1D approaches.

Bilateral polynomial matrix equations in two indeterminates

Two special cases of the bilateral 2-D polynomial matrix equationDU +VN=C whenC=I andC=αI withα being a Ω-stable 2-D polynomial, which are related respectively to deadbeat and asymptotic control problems of 2-D systems, are first considered. By generalizing the concepts of factor coprimeness, zero coprimeness and zero skew primeness in the 2-D polynomial ring to the ring of causal Ω-stable 2-D rational functions, a constructive solution of these two problems mentioned is proposed. Based on these results, we derive a solvability condition for the bilateral equiation whereC is a general 2-D polynomial matrix. The general solutions are investigated as well.

2D Model-Following Servo System

Abstract The paper gives, in the practical sense that only one independent variable of the considered 2D (2-dimensional) systems is unbounded, a formulation for 2D model-following servo problem. That is to determine a control input such that the outputs of a given 2D plant asymptotically track, with tracking error as small as possible, the stop response of a given 2D model system as the unbounded variable approaches infinite. It is shown that this problem can be transformed into an equivalent 1D LQR problem, and thus can be essentially solved by 1D theory. The relationship between the solvability conditions obtained for the equivalent 1D system and the local controllability and observability of the original 2D plant is clarified. A numerical example is also shown.

Symbolic manipulation CAD of control engineering by using REDUCE

Computer aided design (CAD) is an indispensable tool for control engineering. Recently various CAD systems have been developed by using numerical computation languages, but these are not capable of treating symbolic computation. However, control theories provide many symbolic computation algorithms. This study develops a new CAD system which can perform symbolic manipulation, based on REDUCE 3.2. It is shown that this CAD system can be applied to wider problems than the conventional ones, by means of two examples.

A Sufficient Condition for a Graph to Be Weakly k-Linked

Abstract For a pair ( s , t ) of vertices of a graph G , let λ G ( s , t ) denote the maximal number of edge-disjoint paths between s and t . Let ( s 1 , t 1 ), ( s 2 , t 2 ), ( s 3 , t 3 ) be pairs of vertices of G and k > 2. It is shown that if λ G ( s i , t i ) ≥ 2 k + 1 for each i = 1, 2, 3, then there exist 2 k + 1 edge-disjoint paths such that one joins s 1 and t 1 , another joins s 2 and t 2 and the others join s 3 and t 3 . As a corollary, every (2 k + 1)-edge-connected graph is weakly ( k + 2)-linked for k ≥ 2, where a graph is weakly k -linked if for any k vertex pairs ( s i , t i ), 1 ≤ i ≤ k , there exist k edge-disjoint paths P 1 , P 2 ,…, P k such that P i joins s i and t i for i = 1, 2,…, k .

Comments on "Stability tests of N-dimensional discrete time systems using polynomial arrays

For original paper see X. Hu, ibid., vol.42, p.261-8 (1995). In this brief, we wish to point out that Hu overlooked a mistake in the stability test procedure for N-dimensional (N-D, N>2) systems proposed in the above paper, which made the polynomial array approach not general. It is shown that Hus test procedure applies only to a very restricted class of N-D stability test problems, and for a general case, instead of necessary and sufficient conditions, it provides only sufficient conditions. A counterexample is also given.

Practical internal stability of n-D discrete systems

This paper deals with the problem of asymptotic stability for n-D discrete systems in the practical sense that the system input and output signals are unbounded in, at most, one dimension. A definition of practical internal stability is introduced, and necessary and sufficient conditions are derived. The obtained results show that practical internal stability is less restrictive and more relevant for practical applications than the conventional two-dimensional (2-D) internal stability.

Symbolic computation application for the design of linear multivariable control systems

The application of symbolic computation to the algebraic design of linear multivariable control systems is presented. A software package for manipulating polynomial and rational function matrices and for solving the linear matrix equations referred to as unilateral and bilateral equations is implemented on the basis of the symbolic manipulation system known as REDUCE. Several basic functions in the package are explained and their usage is demonstrated by using an example of the design of a discrete-time control system.

Design of practically stable n-dimensional feedback systems : a state-space approach

This paper investigates, by using the state-space approach, some basic properties and state feedback compensation problem of n-dimensional discrete systems in the practical sense that the input and output signals of the systems are unbounded in, at most, one dimension. Notions of practical controllability, practical observability, practical stabilizability by state feedback and practical detectability are introduced and corresponding necessary and sufficient conditions are derived. Moreover, a connection between the state-space representation and doubly coprime matrix fractional description on the ring of practically stable rational functions is shown. The obtained results clarify that all the n-dimensional problems considered in the practical sense are in fact equivalent to the corresponding problems of n one-dimensional systems, hence can be essentially solved by using one-dimensional methods.