Jury stability criterion

(Control Theory : Digital Control : Jury stability criterion)

Note: you should probably have a look at the Routh stability criterion before reading this writeup.

The Jury stability criterion is essentially the discrete time analogue of the continuous time Routh stability criterion. It is a technique for verifying the stability of a linear discrete time system described in the z-domain (see z-transform).

Instead of testing whether the poles (roots) of a given system are all in the left half of the complex plane (LHP), it tests whether all the poles are contained within the unit circle centered on the origin of the complex plane. The reason for this, without getting into too much detail, is that frequency range in discrete time is finite and once you pass either the positive or negative boundary, the previous frequencies are repeated. Therefore you can think of the imaginary axis of the complex plane in continuous time being "wrapped" around the unit circle of the complex plane in discrete time. So while in continuous time you could increase the frequency to infinity, in discrete time you just keep going around the unit circle infinitely, repeating all the frequencies. Well, if all your poles are in the LHP (giving you a stable system) when you wrap the axis around the unit circle, they are then on the inside when you're done.

Method

Ok, well hopefully that's all clear and we can start with the actual method. Given a closed-loop transfer function in the z-domain for a linear discrete time feedback system, we obtain the characteristic equation as follows:

          B(z-1)    b0 + b1z-1 + ... + bnz-n
G(z-1) = ------- = --------------------------
          A(z-1)    a0 + a1z-1 + ... + anz-n

(Z-domain transfer functions are usually given in the z-1 form since
it tends to be more useful for design purposes. For example, z-1 can
be thought of as a unit delay operator.)

p(z) = zn*A(z-1) = a0zn + a1zn-1 + .. + an

Characteristic equation:  p(z)

Using the coefficients of p(z), we begin to form the Jury table:

1. a0  a1 ... an
2. an an-1 ... a0

(Row 2 is the simply the reverse of Row 1)

Now here's where it gets a little strange. Row 3 is calculated from rows 1 and 2 using the formula:

Row1Element - α_n*Row2Element, where α_n = a_n/a₀

So:

1. a0  a1 ... an
2. an an-1 ... a0
3. a0' a1' ... an'

αn = an/a0

a0' = a0 - αn*an = a0 - (an/a0)*an = a0 - an2/a0
a1' = a1 - αn*an-1 = a1 - (an/a0)*an-1 = a1 - (an*an-1)/a0
...
an' = an - αn*a0 = an - (an/a0)*a0 = 0

Then of course, all you do is reverse Row 3 to get Row 4 and repeat the process until your last row contains only one expression in the first column and the rest are zero. Now for the criterion: assuming a₀ > 0, the first element of the first row of each row pair must also be > 0 for all the poles to be on the unit disc and therefore for the overall system to be stable.