the role of diophantine equations in the synthesis of feedback control systems. 12 20 18 atom c. e-mail [email protected] that evolve in discrete time. This relationship, termed canonical Diophantine equations, can be used to resolve a [11] V. KUCERA, Discrete Linear Control, John Wiley,New York, of linear control systems has revied an interest in linear Diophantine equations for polynomials. Vladimir Kučera; Jan Ježek; Miloš Krupička.

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Isolate the remainder of the previous step. Semantic Scholar estimates that this publication has citations based on the available data. For example, if all three terms are even, you can at least divide by 2, as follows: Introduce a second variable equtions convert the modular equation to an equivalent diophantine equarion. In that case, the equation would have no integral solutions. Moving through the Euclidean algorithm steps in reverse, repeat the process.

## Diophantine equations in control – A survey

Substitute the equality in Step 5 into the place of the 2 in your Step 6 revision: See our FAQ for additional information. So that equation has no solutions mod As you will see below, if an equation has one integral solution, then it also has infinitely many integral solutions.

Continue repeating substitution and simplification. Pay particular attention to the 64 term. From This Paper Figures, tables, and topics from this paper.

Here is a brief algebraic statement of the proof: Rewrite the result in terms of the original coefficients. Write the equation in standard form. In some cases, you may be able to tell immediately if there is no solution to your problem.

kuceta The Euclidean algorithm is a system of repeated divisions, using the remainder each time as the divisor of a new division.

You should notice that your revision of Step 6 contains the number 2, and your revision of Step 5 is equal to 2. Notice that the greatest common divisor for this problem was 1, so the solution that you reached is equal to 1.

The pattern of infinite solutions begins with the single solution that you identified. By clicking accept or continuing to use the site, you agree to the terms outlined in our Privacy PolicyTerms of Serviceand Dataset License. This paper has citations. Linear system Search for additional papers on this topic. Not Helpful 0 Helpful 3.

### LINEAR DIOPHANTINE EQUATIONS

The cornerstone of the exposition is a simple parametrization of all stabilizing controllers for a given plant. If not, then there will be no solution. How do I find solutions to word problems involving linear Diophantine equations?

One can then choose, in principle, the best controllers for various applications. Cookies make wikiHow better. Topics Discussed in This Paper. Skip to search form Skip to diophatine content.

### Diophantine equations in control – A survey – Semantic Scholar

Write a general solution. If a linear equation has one integral solution, then it must have infinitely many integral solutions. Solving a linear Diophantine equation means that you need to find solutions for the variables x and y that are integers only. Not Helpful 0 Helpful 0. In the original problem, that term is subtracted, but the Euclidean algorithm treats it as a positive term.

This is another way of saying that 87 and 64 are relatively prime. Divide the previous divisor 20 by the previous remainder The left side is always a multiple of 14, but 38 is not. Help answer questions Learn more.

The final equation looks like this: Article Info Featured Article Categories: Diophantine-ness refers to the domain of the variable s – it’s those that have to be integers. However, that is not the solution to the problem, since the original problem sets 87xy equal to 3.