The Jury stability criterion is a method of determining the stability of a linear discrete time system by analysis of the coefficients of its characteristic polynomial

Characteristic polynomial

In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....

. It is the discrete time analogue of the Routh-Hurwitz stability criterion

Routh-Hurwitz stability criterion

The Routh–Hurwitz stability criterion is a necessary and sufficient method to establish the stability of a single-input, single-output , linear time invariant control system. More generally, given a polynomial, some calculations using only the coefficients of that polynomial can lead to the...

. The Jury stability criterion requires that the system poles are located inside the unit circle centered at the origin, while the Routh-Hurwitz stability criterion requires that the poles are in the left half of the complex plane. The Jury criterion is named after Eliahu Ibraham Jury

Eliahu Ibraham Jury

Eliahu Ibraham Jury is an American engineer, born in Baghdad, Iraq. He received his Doctor of Engineering Science degree from Columbia University of New York in 1953...

.

Method

If the characteristic polynomial of the system is given by

then the table is constructed as follows:

That is, the first row is constructed of the polynomial coefficients in order, and the second row is the first row in reverse order and conjugated.

The third row of the table is calculated by subtracting times the second row from the first row, and the fourth row is the third row with the first n elements reversed (as the final element is zero).

The expansion of the table is continued in this manner until a row containing only one non zero element is reached. If the first element of the first row of every row pair is positive at this point, then the system is stable.

Note the {a_n}/{a_0} is for the 1st 2 rows. Then for 3rd and 4th row the coefficient changes. This can be viewed as the new polynomial which has one less degree and then continuing. This is very easy to implement using dynamic arrays on a computer. It also tells whether all the modulus of the roots (complex and real) lie inside the unit disc. Vector v contains the real coefficients of the original polynomial in the order from highest degree to lowest degree.

/* vvd is the jury array */
vvd.push_back(v); // Store the first row
reverse(v.begin,v.end);
vvd.push_back(v); // Store the second row

for(i=2;;i+=2)
{
v.clear;
double mult=vvd[i-2][vvd[i-2].size-1]/vvd[i-2][0]; // This is an/a0 as mentioned in the article.

for( j=0;j v.push_back(vvd[i-2][j] - vvd[i-1][j]*mult);

vvd.push_back(v);
reverse(v.begin,v.end); // reverse the next row
vvd.push_back(v);
if(v.size

1) break;
}

Check is done using
for(i=0;i {
if(vvd[i][0]<=0) break;
}

if(i

vvd.size)
"All roots lie inside unit disc "
else
"no"

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