Ülo Nurges

New stability conditions via reflection coefficients of polynomials

The geometry of stable discrete polynomials using their coefficients and reflection coefficients is investigated. Starting from so-called barycentric simplex some necessary stability conditions in terms of unions of polytopes are obtained by splitting the unit hypercube of reflection coefficients. Sufficient stability conditions in terms of linear covers of reflection vectors of a family of stable polynomials improve the Cohn stability criterion.

Reflection Coefficients of Polynomials and Stable Polytopes

The geometry of stable discrete polynomials using their coefficients and reflection coefficients is investigated. Two linear Schur invariant transformations with a free parameter in the polynomial coefficient space are introduced. The first transformation Rfrn times RfrrarrRfrn maps an arbitrary stable polytope into another stable polytope. The second transformation Rfrn times RfrrarrRfrn maps a stable tilted n-dimensional hyperrectangle defined by the discrete Kharitonov theorem into a stable (n+1)- dimensional polytope.

The distance from stability boundary and reflection vectors

A new stability margin for discrete-time systems is proposed in the system characteristic polynomial coefficient space making use of, so-called, reflection vectors of monic Schur polynomials. Reflection vector margins give the distance (not minimal) to the stability boundary in directions of 2n reflection vectors of an n-th degree polynomial. The stability radius can be found by minimizing the reflection vector margin.

Fixed-order stabilising controller design by a mixed randomised/deterministic method

A novel mixed randomised/deterministic method for fixed-order controller design is proposed for discrete-time single-input single-output plants. It is based on the random generation of Schur stable polynomials using reflection coefficients and reflection segments of polynomials. Stable reflection segments are then projected onto the affine set of closed-loop characteristic polynomials and the stable line segments in the controller parameter space are determined. A novel approach is proposed for global and local optimisation over reflection segment bunches on the basis of the weighted sum of absolute values of reflection coefficients.

On the robust stability of discrete-time systems

A sufficient stability condition for monic Schur polynomials is obtained via, so-called, reflection coefficients of polynomials and the discrete version of Kharitonovs weak theorem. The discrete Kharitonov theorem defines only (n-1)-dimensional stable hyperrectangle for n-degree monic polynomials. By the use of a linear Schur invariant transformation we put stable line segments through vertices of this hyperrectangle and find an n-dimensional stable polytope with all vertices on the stability boundary.

Generation of stable polytopes of Hurwitz polynomials via Routh parameters

The paper addresses an important issue in the field of continuous-time linear control systems - convex stability domain approximation problem. The constructive procedure of generating a stable polytope is proposed. The main idea is based on constructing so-called Routh stable line segments (half-lines) starting from a given stable polynomial. It is summarized in the form of a step-by-step algorithm that results in a stable polytope around a given point. Several numerical examples are presented to demonstrate the covered concepts and the effectiveness of the proposed approach. Calculations are performed in a MATLAB environment.

Stability domain approximation for the robust PID controller parameters by reflection segments approach

An innovative method for robust stabilizing PI, PD controller design and its stabilizing area approximation is presented for discrete-time SISO plants. It starts from the random generation of Schur stable polynomials using reflection coefficients and reflection segments of polynomials. Stable reflection segments are projected onto affine set of closed-loop characteristic polynomials which is defined by the controller parameters and the stable line segments in the controller parameter space are then determined. The segments of two systems with different parameters are used to generate a rectangular and polygonal approximations of the stabilizing area of robust controller parameters that stabilize both systems.

ON THE ROBUST STABILITY OF DISCRETE-TIME SYSTEMS

A sufficient stability condition for monic Schur polynomials is obtained via the so-called reflection coefficients of polynomials and the discrete version of Kharitonovs weak theorem. The discrete Kharitonov theorem defines only (n - 1)-dimensional stable hyperrectangle for n-degree monic polynomials. By the use of a linear Schur invariant transformation we put stable line segments through vertices of this hyperrectangle and find an n-dimensional stable polytope with all vertices on the stability boundary.

Reflection segments based robust PID controller solution for benchmark problem

A number of methods has been published in the recent years in order to demonstrate a stable and satisfactory solution for the benchmark test problem of a flexible transmission system. This paper presents a new approach for PID stabilization for the benchmark problem via generation of stable segments in the controller coefficient space and their subsequent optimization methods in order to find the best suited robust solution.

Weighted reflection vector polytopes in robust control

A constructive procedure for robust output controller design is given starting from the unit hypercube of reflection coefficients of monic polynomials. This procedure consists of two main steps: convex approximation of the stability domain by a reflection vector polytope and robust controller design by quadratic programming approach. The roots placement of reflection vectors is studied and the weights for reflection vectors are calculated.