A nonlinear optimization approach to the construction of general linear methods of high order

We search for general linear methods with s internal stages and r = s + 1 external stages of order p = s + 1 and stage order q = s. We require that stability function of these methods has only two non-zero roots. This is achieved by imposing the so-called inherent quadratic stability conditions. Examples of such general linear methods which are A- and L-stable up to the order p = 8 and stage order q = p - 1 are derived.

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OPTIMIZATION-BASED SEARCH FOR NORDSIECK METHODS OF HIGH ORDER WITH QUADRATIC STABILITY POLYNOMIALS

Mathematical Modelling and Analysis ◽

10.3846/13926292.2012.685497 ◽

2012 ◽

Vol 17 (3) ◽

pp. 293-308 ◽

Cited By ~ 12

Author(s):

Angelamaria Cardone ◽

Zdzislaw Jackiewicz ◽

Hans Mittelmann

Keyword(s):

State Of The Art ◽

High Order ◽

General Linear Methods ◽

General Linear ◽

Stability Function ◽

Optimization Software ◽

Quadratic Stability ◽

Nordsieck Methods ◽

The Stability

We describe the search for explicit general linear methods in Nordsieck form for which the stability function has only two nonzero roots. This search is based on state-of-the-art optimization software. Examples of methods found in this way are given for order p = 5, p = 6, and p = 7.

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Construction of high-order quadratically stable second-derivative general linear methods for the numerical integration of stiff ODEs

Journal of Computational and Applied Mathematics ◽

10.1016/j.cam.2016.02.054 ◽

2016 ◽

Vol 303 ◽

pp. 218-228 ◽

Cited By ~ 14

Author(s):

A. Abdi

Keyword(s):

Numerical Integration ◽

High Order ◽

General Linear Methods ◽

General Linear ◽

Second Derivative ◽

Stiff Odes

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μ-Synthesis for Fractional-Order Robust Controllers

Mathematics ◽

10.3390/math9080911 ◽

2021 ◽

Vol 9 (8) ◽

pp. 911

Author(s):

Vlad Mihaly ◽

Mircea Şuşcă ◽

Dora Morar ◽

Mihai Stănese ◽

Petru Dobra

Keyword(s):

Control Problem ◽

Fractional Order ◽

Design Procedure ◽

High Order ◽

Numerical Control ◽

Performance Criteria ◽

Iteration Algorithm ◽

Optimization Approach ◽

Cnc Machine ◽

Bee Colony

The current article presents a design procedure for obtaining robust multiple-input and multiple-output (MIMO) fractional-order controllers using a μ-synthesis design procedure with D–K iteration. μ-synthesis uses the generalized Robust Control framework in order to find a controller which meets the stability and performance criteria for a family of plants. Because this control problem is NP-hard, it is usually solved using an approximation, the most common being the D–K iteration algorithm, but, this approximation leads to high-order controllers, which are not practically feasible. If a desired structure is imposed to the controller, the corresponding K step is a non-convex problem. The novelty of the paper consists in an artificial bee colony swarm optimization approach to compute the nearly optimal controller parameters. Further, a mixed-sensitivity μ-synthesis control problem is solved with the proposed approach for a two-axis Computer Numerical Control (CNC) machine benchmark problem. The resulting controller using the described algorithm manages to ensure, with mathematical guarantee, both robust stability and robust performance, while the high-order controller obtained with the classical μ-synthesis approach in MATLAB does not offer this.

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