Lecture 2
Stability in Z plane
Dept. Of Electrical Engineering/ College
of Engineering/ Misan University
Control Engineering
Fourth Year
OUTLINE
• Overview
• Mapping from the s-plane into the z –plane
• Stability of Real-Time Systems:
• Jury’s stability
• Examples
Overview
• The stability of a control system is defined as the ability of any system
to provide a bounded output when a bounded input is applied to it.
• Bounded value of a signal represents a finite value
• Types of Stable System: stable, unstable and critically stable.
Stable unstable critically stable
The roots of a control system
• Stability is the most important issue in control system design.
• The stability of the following closed loop system:
• It can be determined from the location of closed loop poles in z-plane
which are the roots of the characteristic equation:
• For the system to be stable, the closed loop poles or the roots of the
characteristic equation must lie within the unit circle in z-plane.
Otherwise the system would be unstable.
• The location of such roots or poles on the s-plane will indicate the condition
of stability as shown in Fig. below.
Mapping from the s-plane into the z -plane
• It is possible to map from the s to the z-plane using the relationship:
• The S is complex variable:
• Now :
• It can be written
(7 63)
• where ws is
the sampling frequency.
Stability of Real-Time Systems:
• Suppose we have a closed-loop system transfer function:
• The characteristic equation of the
system:
• We can say that a system in the z-
plane will be stable if all the roots
of the characteristic equation, lie
inside the unit circle.
Example 1: Figure below represents the digital control system. Check the
stability of the system when the T=1 seconds?
Solution:
For T=1 second
• The characteristic equation of the system:
• Now,
• The solution of which is z = -1.594 which is outside the unit circle, i.e.
the system is unstable.
Example 2: For the system given in Example 1, find the value of T for which
the system is stable.
Solution:
• Thus, the system will be stable as long as the sampling time T < 0.549.
The Jury stability test
• In the same way that the Routh-Hurwitz criterion offers a simple method of
determining the stability of continuous systems, the Jury’s stability test is
employed in a similar manner to assess the stability of discrete systems.
• Consider the characteristic equation of a sampled-data system.
• We now form the array shown in the following table.
• The elements of this array are defined as follows:
• The elements of each of the even-numbered rows are the elements
of the preceding row, in reverse order.
Jury’s Table rows = (2n-3) rows
Jury’s test is then applied as follows:
• Check the first three conditions and construct the array given in the Table
and check the fourth conditions given above if any condition is not satisfied,
the system is unstable.
Example 1: Check stability of a system has an open loop transfer function:
Solution:
• The characteristic equation:
• Now apply Jury’s condition test:
Condition 1 Q(1)=1-1+0.7=0.7˃ 0
Condition 2 = 1(1+1+0.7) = 2.7 ˃ 0
Condition 3 / a0 / ˂ an 0.7 ˂ 1 The system is stable
Example 2: the characteristic equation of digital control system is given
by:
• Find the stability of the system?
• Solution:
Now,
Condition 1: Q(1)= 1-2+1.4- 0.1= 0.3 ˃ 0.
Condition 2: = (-1)3 (-1-2 -1.4- 0.1) = 4.5 > 0
Condition 3: / a0 / ˂ an 0.1 ˂ 1 .
• We are Appling condition 4:
• Since / -0.99/ ˂ /-1.2/ the system is unstable
Example 2: Determine the closed loop stability of the system shown in
Figure below when K = 1?
Solution:
Since H(S)= 1
We know that the characteristics equation is 1 + G(z) = 0
Since |Z1| = |Z2| < 1, the system is stable.
• Example 1: Figure below shows the digital control system. Find the value of
the digital compensator gain K to make the system just unstable.
• Solution: K?
G(Z) =
• The characteristic equation of the system:
• Simplify:
• Now, the first row of jury's array:
• Condition 1 Q(1) ˃ 0
• The condition 1 is satisfied if K ˃ 0 (stable)
• Condition 2
• Hance, the system is marginally stable when the K =105.23 and 9.58
