Jury Stability Criterion
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In
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
and
control theory Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the applic ...
, the Jury stability criterion is a method of determining the stability of a
discrete-time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "poi ...
,
linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstractio ...
by analysis of the
coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
s of its
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...
. It is the discrete time analogue of the
Routh–Hurwitz stability criterion In the control theory, control system theory, the Routh–Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stable polynomial, stability of a linear time-invariant system, linear time-invarian ...
. The Jury
stability criterion In control theory, and especially stability theory, a stability criterion establishes when a system is stable polynomial, stable. A number of stability criteria are in common use: *Circle criterion *Jury stability criterion *Liénard–Chipart cri ...
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 In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
. The Jury criterion is named after
Eliahu Ibraham Jury Eliahu Ibrahim Jury (May 23, 1923 – September 20, 2020) was an Iraqi-born American engineer. He received his the E.E. degree from the Technion – Israel Institute of Technology, Haifa, Mandatory Palestine (now, Israel), in 1947, the M.S. degr ...
.


Method

If the characteristic polynomial of the system is given by f(z) = a_n + a_z^1 + a_z^2 + \dots + a_1z^ + a_0z^n then the table is constructed as follows:Discrete-time control systems (2nd ed.), pg. 185. Prentice-Hall, Inc. Upper Saddle River, NJ, USA ©1995 \begin \underline\text & \ \underline \ & \ \underline \ & \ \underline \ & \ \cdots \ & \ \underline \ & \ \underline \ \\ pt 1 & a_0 & a_1 & a_2 & \cdots & a_ & a_n \\ pt 2 & a_n & a_ & a_ & \cdots & a_1 & a_0 \\ 0pt 3 & b_0 & b_1 & \cdots & b_ & b_ \\ pt 4 & b_ & b_ & \cdots & b_1 & b_0 \\ 0pt 5 & c_0 & c_1 & \cdots & c_ & \\ pt 6 & c_ & c_ & \cdots & c_0 & & \\ 0pt \ \!\vdots & \vdots & \vdots & \vdots & \vdots & & \\ 0pt 2n-5 \quad & p_0 & p_1 & p_2 & p_3 & & \\ pt 2n-4 & p_3 & p_2 & p_1 & p_0 & & \\ 0pt 2n-3 & q_2 & q_1 & q_0 & & & \end 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 \tfrac times the second row from the first row, and the fourth row is the third row with the first elements reversed (as the final element is zero). \begin a_0 & a_1 & \dots & a_ & \quad a_n \quad \\ pt a_n & a_ & \dots & a_1 & a_0 \\ pt a_0 - a_n \frac & a_1 - a_ \frac & \dots & a_ - a_1 \frac & 0 \\ pt a_ - a_1 \frac & \dots & a_1 - a_ \frac & a_0 - a_n \frac & 0 \end The expansion of the table is continued in this manner until a row containing only one non-zero element is reached. Note the \tfrac is a_n for the 1st two rows. Then for 3rd and 4th row the coefficient changes (i.e. \tfrac). This can be viewed as the new polynomial which has one less degree and then continuing.


Stability test

If >0 then for every value of a_0, b_0, c_0... that is negative, the polynomial has one
root In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...
outside of the unit disc. This implies that the method can be stopped after the first negative value is found when checking for stability.


Sample implementation

This method is very easy to implement using dynamic arrays on a computer. It also tells whether all the modulus of the roots (
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
and real) lie inside the unit disc. The vector 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) // Check is done using for (i=0; i

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


See also

* Liénard–Chipart criterion, another stability criterion derived from Routh-Hurwitz (for continuous-time systems)


References

For more details please check these references:
A note on the reduced Schur–Cohn criterion

Wikibooks on Control Systems - Jury's Test
For advanced resources: * * * http://www.laas.fr/~henrion/Papers/lyap.ps.gz For implementations: * http://www.ticalc.org/archives/files/fileinfo/426/42696.html (TI-83+/84+ graphing calculators) {{DEFAULTSORT:Jury Stability Criterion Stability theory