Linear time invariant and z transform in control systems.

1. Analysis of Linear Time Invariable
Systems

2. Linear Time Invariant (LTI) Systems
• Linearity – Linear system is a system that
possesses the property of superposition.
• Time Invariance – A system is time invariant if
the behavior and characteristics of the system
are fixed over time.

3. Discrete – Time LTI systems :
The Convolution Sum

4. System T{.}
System T{.}

7. Continuous– Time LTI systems :
The Convolution Integral
• A similar approach can be drawn for
continuous time LTI systems and following
results can be derived.
y(t) = ∫ x(T)h(t-T)dT
Or,
y(t) = x(t) h(t)

8. Properties of LTI systems :
Commutative
x[n] h[n] = h[n] x[n]
Distributive
x[n] (h1[n]+h2[n]) = x[n] h1[n] + x[n] h2[n]
Associative
x[n] (h1[n] h2[n]) = (x[n] h1[n]) h2[n]

9. Stability for LTI Systems :

10. The output should be bounded for stability

11. Causality for LTI Systems
• A system is casual if the output at any time
depends only on the values of the input at the
present time and in the past.
• Therefore, for LTI systems, y[n] must not
depend upon x[k] for k > n.
• Hence,
h[n] = 0 for n < 0

12. Z - Transform
• Introduction
• Definition
• Region of Convergence and Z Plane
• Pole and Zero
• Example
• Properties

13. Introduction
• Since Fourier Transform has its limitations, a
counterpart of Laplace transform (Continuous
time) was needed for Discrete time systems.
• To perform transform analysis of unstable
systems and to develop additional insight and
tools for LTI systems anlysis.

14. Mathematical Representation

15. Mathematical Representation

16. Z Transform and
Discrete time Fourier Transform
Z = ejω
• Replace Z = rejω
ω
where r = magnitude
ω = angle of Z
The z- transform reduces to the Fourier
transform when the magnitude of the transform
variable z is unity.
• The basic idea is to represent and analyze the
whole system about a unit circle in Z Plane.

17. Region of Convergence
• Z transform of a sequence has associated with
it a range of values of z for which X(z)
converges. This range of values is referred to
as the region of convergence.
• A stable system requires the ROC of z-
transform to include the unit circle.

18. Pole and Zero
• When X(z) is an rational function, then
1.The roots of the numerator polynomial are
referred to as the zeros of X(z).
2.The roots of the denominator polynomial are
referred to as the poles of X(z).
• No poles of X(z) can occur within the region of
convergence since the z-transform does not
converge at a pole.
• The region of convergence is bounded by
poles.

19. Example

20. Properties of Z transform:

21. Properties :

22. Analysis of LTI Systems using
Z- Transform

23. Analysis of LTI Systems using
Z- Transform
• From the convolution property
Y(z) = H(z) X(z)
Where
Y(z)= z-transform of system output.
H(z)= z-transform of impulse response.
X(z) = z-transform of system input .

24. Stability and Causality
• Causality
– A discrete time LTI system is causal if and only if
the ROC of its system function is the exterior of
the circle, including infinity.
• Stability
– The LTI system is stable if and only if the ROC of
the system function H(z) includes the unit circle,
|Z| = 1

25. Stability and Causality for LTI system
with Rational system Function
• Causality
– The ROC is the exterior of the outermost pole.
– With H(z) expressed as a ratio of polynomials in z,
the order of numerator cannot be greater than
the order of denominator.
• Stability
– If it is a causal system, it will be stable if and only if
all the poles of H(z) lie inside the unit circle – i.e.
they must all have magnitude smaller than 1.
– It is possible for a system to be stable but not
casual.

26. THANK YOU
- Pranvendra Champawat
- 08010824
- p.champawat@iitg.ernet.in