Root locus for Discrete Time Systems stability.

The root locus method can also be used for discrete-time systems without many modifications since the characteristic equation of a discrete control system is of the same form as that of a continuous-time control system.

In many LTI discrete time control systems, the characteristics equation may possess any of the below 2 forms.

1 + G(z)H(z) = 0

1 + GH(z) = 0

To combine both, let us define the characteristics equation as:

1 + L(z) = 0 ----------(1)

where,L(z) = G(z)H(z) or L(z) = GH(z).

 L(z) is popularly known as the loop pulse transfer function.

From equation (1), we can write

L(z) = −1

L(z) is a complex quantity that can be split into two equations by equating angles and magnitudes of two sides. This gives us the angle and magnitude criteria as

Angle Criterion: ∠ L(z) = ±180°(2k + 1), k = 0, 1, 2....

Magnitude Criterion :  |L(z)| = 1

The values of z that satisfy both criteria are the roots of the characteristics equation or close

loop poles. Before constructing the root locus, the characteristics equation 1+L(z) = 0 should

be rearranged in the following form

where z’s and p’s are zeros and poles of open-loop transfer function, m is the number of zeros

n is the number of poles.

Construction Rules for Root Locus

1. The root locus is symmetric about the real axis. The number of root locus branches equals the number of open-loop poles.
2. The root locus branches start from the open-loop poles at K = 0 and end at open-loop

   zeros at K = 1. In the absence of open-loop zeros, the locus tends to 1 when K ! 1.

   Number of branches that tend to 1 is equal to the difference between the number of poles

   and the number of zeros.
3. A portion of the real axis will be a part of the root locus if the number of poles plus

   the number of zeros to the right of that portion is odd.
4. If there are n open-loop poles and m open-loop zeros then n−m root locus branches tend
   to 1 along the straight-line asymptotes drawn from a single point s =  which is called
   the centroid of the loci.

   Angle of asymptotes  

   
5. Breakaway (Break in) points or the points of multiple roots are the solution of the following
   equation:

   

    where K is expressed as a function of z from the characteristic equation. This is a necessary
   but not sufficient condition. One has to check if the solutions lie on the root locus.
   
6. The intersection (if any) of the root locus with the unit circle can be determined from the

   Routh array.
7. The angle of departure from a complex open loop pole is given by
   
   where Φ is the net angle contribution of all other open-loop poles and zeros to that pole. 

   

   are the angles contributed by zeros are the angles contributed by the poles
8. The angle of arrival at a complex zero is given by

   

   where Φ is the as in the above rule.
9. The gain at any point z0 on the root locus is given by 

   

    Consider a Discrete control system for determining values of Gain K and Sampling time T.

A discrete-time control system

For example consider T=0.5 sec.

Let us assume a controller as the integral controller for which 

Thus, 

The characteristic equation can be written as 

When T=0.5 Sec, 

L(z) has poles at z = 1 and z = 0.605 and zero at z = 0.

Break away/ break in points are calculated by putting

z^2 = 0.6065 which implies z^1 = 0.7788 and z^2 = −0.7788

Critical value of K can be found out from the magnitude criterion.

Critical gain corresponds to point z = −1. Thus

Fig. 1 shows the root locus of the system for K = 0 to K = 10.

Two root locus branches start from two open loop poles at K = 0. If we further increase K one branch will go towards the zero and the other one will tend to infinity. The blue circle represents the unit circle. Thus the stable range of K is 0 < K < 8.165.

If T = 1 sec,

Break away/ break in points:

z2 = 0.3679 ) z1 = 0.6065 and z2 = −0.6065 Critical gain (Kc) = 4.328 Fig. 2 shows the root locus for K = 0 to K = 10. It can be seen from the figure that the radius of the inside circle reduces and the maximum value of stable K also decreases to K = 4.328.

Similarly if T = 2 sec, 

One can find that the critical gain in this case further reduces to 2.626.