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Digital Control Systems
z-Plane Analysis of Discrete Time Control
Systems
Digital Control System
A control system which uses a digital computer as a controller or compensator is known as digital control
system. The advantages of using a digital computers for compensation include: accuracy, reliability, economy
and most importantly, flexibility.
Sampler
Converts an anolog signal to a digital signal (train of pulses)
A practical sampler acts like a switch closing every T seconds for a short duration of p seconds. Therefore,
sampled signal can represented as follows:
(u(t) is unit step function)
The output of an ideal sampler is given by
Sampler
Signal r(t) after samling
Signal r(t) after ideal samling
• In practice p is much smaller and can be neglected. This leads to the Ideal Sampler.
• The Ideal Sampling process can be considered as the multiplication of a pulse train with a continuous signal
r* (t) = P(t) r(t)
Sampler
r(t) = 0, for t < 0
Hold Device
The function of hold device is to convert sampled signal into continuous signal. The values of continuous time
signal in between the sampling instants are calculated by extrapolation.
The continuous signal h(t) during the time interval  ≤t≤  + 1  may be approximated by a polynomial in :
0≤  ≤ 
Since the continuous time output signal, ℎ() at sampling instants, must equal input signal, () :
Therefore:
The hold device is called n-th order hold if it uses an n-th order polynomial extrapolator.
n=1 first order hold
n=0 zero order hold
Hold Device
Zero Order Hold (ZOH)
Input-Output signals for a ZOH
The output of ZOH:
Hold Device
Zero Order Hold (ZOH)
The output of ZOH:
We know from Laplace transforms that
ZOH Transfer function
Hold Device
Zero Order Hold (ZOH)
Suppose the transfer function G(s) follows the ZOH. Then the product of ZOH and G(s) becomes:
Obtain the z transform of X(s):
Let
Hold Device
Zero Order Hold (ZOH)
Hold Device
Zero Order Hold (ZOH)
Example:The response of a sampler and a ZOH to a ramp input for two different values of a sampling period.
A sampler and ZOH can accurately follow the input signal if the sampling time T is small compares to the transient
changes in the signal
Example: Ideal sampler followed by a ZOH
Pulse Transfer Function
Transfer function of a continuous time system relates the Laplace transform of the continuous-time output to that of the
continuous time input.
Pulse transfer function relates the z-transform of the output at the sampling instants to that of the sampled input.
Starred Laplace Transform
Assumption: Zero initial conditions.
• While taking the starred Laplace tranform of a product of transforms, where some are ordinary Laplace transforms and
others are starred Laplace transform, the functions already in starred Laplace transforms can be factored out of the
starred Laplace transform.
• The output of a sampled-data system is continuous with respect to time. However the pulse transform (starred LT) of
the output, Y (s) and z-transform of the output Y (z), gives the values of output y(t) only at sampling instants.
Pulse Transfer Function
Starred Laplace Transform
Since the z transform is starred Laplace transform with esT replaced by z,
Pulse Transfer Function
Starred Laplace Transform
The presence or absence of the input sampler is crucial in determining the pulse transfer function of a system
Example:
Pulse Transfer Function
Pulse transfer function of cascaded elements
Taking starred Laplace transforms on both sides of above equations
Pulse Transfer Function
Pulse transfer function of cascaded elements
where
Pulse Transfer Function
Pulse transfer function of closed loop systems
Pulse Transfer Function
Example:
Pulse Transfer Function
Pulse transfer function of a digital controller
Pulse tranfer function of a digital controller can be easily obtained from its input-ouput characteristic which is
specified by means of difference equation:
where m(k) and e(k) are the output and input signals respectively. Taking z-transforms and simplifying we get the
pulse transfer function of the digital controller:
Pulse Transfer Function
Closed-loop Pulse transfer function of a digital control system
A general block diagram of a digital control system
Mathematical block diagram of a digital control system
Pulse Transfer Function
Closed-loop Pulse transfer function of a digital control system
Realization of Digital Controllers and Digital Filters
The general form of the pulse transfer function between the output Y(z) and X(z) is given by
• Direct programming
• Standart programming
In these programmings, coefficients appear as multipliers in the block diagram realization. Those block diagram
schemes where the coefficients appear directly as multipliers are called direct structures.
Realization of Digital Controllers and Digital Filters
Direct programming
Realize the numerator and denominator of the pulse transfer function using separate sets of delay elements.
Total number of delay elementsin direct programming=n+m
Digital filter:
Block diagram realization:
Realization of Digital Controllers and Digital Filters
Standart programming
The number of delay elements required in direct programming can be reduced.
The number of delay elements used in realizing the pulse transfer function can be reduced from n+m to n (where
n≥m) by rearranging the block diagram.
Realization of Digital Controllers and Digital Filters
Standart programming

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