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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So that equation has no solutions mod This process will repeat, step by step, until you reach the original step of the Euclidean algorithm. The values that must be multiplied by the coefficients are the x and y solutions to the equation. This paper has highly influenced 25 other papers. The last divisor that divides evenly is the greatest common factor GCF of the two numbers.

You need to multiply the terms of your last equation by 3 to get a solution: Suppose, for example, that the GCF had worked out to be 5.

Recognize that infinitely many solutions exist. Diophantine-ness refers to the domain of the variable s – it’s those that have to be integers. Check for the impossibility of a solution. Apply the Euclidean algorithm to the coefficients A and B.


Semantic Scholar estimates that this publication has citations based on the available data. Add the y-coefficient B to the x solution. References Publications referenced by this paper.


In some cases, you may be able to tell immediately if there is no solution to your problem. To verify that your new ordered pair is a solution to the equation, insert the values into the equation and see if it works.

Polynomial solution of 2-D Kalman-Bucy filtering problem. How do I find solutions to word kuucera involving linear Diophantine equations? Begin with the last step that has a remainder. Because the Euclidean algorithm for this pair continues all the way down to dividing by 1, the GCF between 87 and 64 is 1.

The values for x will fit a pattern of the original solution, plus any multiple of the B coefficient. Continuing in this manner, the remaining steps are as follows: Jucera find a new solution for x, add the kuvera of the coefficient of y.

For example, if all three terms are even, you can at least divide by 2, as follows: These are linear equations in a ring and result from a fractional representation of the systems involved.

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Rewrite that equation so the remainder stands alone, as equal to the rest of the information in the equation. Cookies make wikiHow better. When you return to the first step of the Euclidean algorithm, you should notice that the resulting equation contains the two coefficients of the original problem. Continue repeating substitution and simplification. Write the equation in standard form. Diolhantine find dilphantine solution of the linear equation, you will use your work on the Euclidean algorithm as the basis for a repeated process of renaming and simplifying values.


A wikiHow Staff Editor reviewed this article to make sure it’s helpful and accurate. Divide the previous divisor 16 by the previous remainder 4.

This is the Step 6 revision.

Diophantine equations in control – A survey – Semantic Scholar

You should notice that your revision of Step 6 contains the number 2, and your revision of Step 5 is equal to 2. You can write this algebraically as follows: Identify the integral solution to the equation.

Finding integral solutions is more difficult than a standard solution and requires an ordered pattern of steps. The pattern of infinite solutions begins with the single solution that you identified.