A reachability test for systems over polynomial rings using Gröbner bases

Conditions for the reachability of a system over a polynomial ring are well known in the literature. However, the verification of these conditions remained a difficult problem in general. Application of the Gröbner Basis method from constructive commutative algebra makes it possible to carry out this test explicitly. In this paper it is shown how this can be done in an efficient way. In comparison with a very simple and rather straightforward method, the algorithm proposed in this paper has an enormous advantage: it has a good performance for both reachable and non-reachable systems. Moreover, the method can be used to obtain a right- or left-inverse of a general non-square polynomial matrix. Such inverse matrices are often required for the design of feedback compensators. Finally, a modification of the reachability test is given to speed up the computations in the non-reachable case

Testing reachability and stabilizability of systems over polynomial rings using Gröbner bases

Conditions for the reachability and stabilizability of systems over polynomial rings are well-known in the literature. For a system $\Sigma = (A,B)$ they can be expressed as right-invertibility cconditions on the matrix $(zI - A \mid B)$. Therefore there is quite a strong algebraic relationship between both conditions, but unfortunately they are difficult to check explicitly. In this paper we introduce for each system $\Sigma = (A,B)$ a corresponding polynomial ideal I which characterizes both reachability and stabilizability in a very straightforward way. Moreover, methods are given to compute this ideal and its variety explicitly using Gröbner Bases techniques. With help of the Gröbner Basis of the ideal I, conclusions on the reachability and stabilizability of the system $\Sigma = (A,B)$ are easy to draw

A control problem for affine dynamical systems on a full-dimensional simplex

Given an affine system on a simplex, the problem of reaching a particular facet of the simplex, using affine state feedback is studied. Necessary and sufficient conditions for the existence of a solution are derived in terms of linear inequalities on the input vectors at the vertices of the simplex. If these conditions are met, a constructive procedure yields an affine feedback control law, that solves this reachability problem

On compact models for high-voltage MOS devices

Fast evaluation of integrated circuits(ICs) requires the validity of so-called compact models, i.e. simple-to-evaluate relations between the voltages and the currents in the IC-components. In this paper the compact model for a particular IC part, the LDMOS device, is studied

Discrete-State Abstractions of Nonlinear Systems Using Multi-resolution Quantizer

Abstract. This paper proposes a design method for discrete abstrac-tions of nonlinear systems using multi-resolution quantizer, which is ca-pable of handling state dependent approximation precision requirements. To this aim, we extend the notion of quantizer embedding, which has been proposed by the authors ’ previous works as a transformation from continuous-state systems to discrete-state systems, to a multi-resolution setting. Then, we propose a computational method that analyzes how a locally generated quantization error is propagated through the state space. Based on this method, we present an algorithm that generates a multi-resolution quantizer with a specified error precision by finite refine-ments. Discrete abstractions produced by the proposed method exhibit non-uniform distribution of discrete states and inputs.