Now let us calculate the transient response of a combined discrete-time and continuous-time - PowerPoint Presentation

Slide1

Now let us calculate the transient response of a combined discrete-time and continuous-time system, as shown below.

Chapter 5

Discrete-Time Process Models

Discrete-Time Transfer Functions

The input to the continuous-time system

G(s) is the signal:

The system response is given by the convolution integral:

With

Chapter 5

Discrete-Time Process Models

Discrete-Time Transfer Functions

For 0 ≤

τ

≤

t,

We assume that the output sampler is ideally synchronized with the input sampler.

The output sampler gives the signal

y

*

(

t

) whose values are the same as

y

(

t

) in every sampling instant

t

= jTs.

Applying the

Z

-transform yields:

Chapter 5

Discrete-Time Process ModelsDiscrete-Time Transfer Functions

Taking

i

= j – k

, then:

For zero initial conditions,

g(iT

s) = 0, i < 0, thus:

where

Discrete-time Transfer Function

The

Z

-transform

of Continuous-time Transfer Function

g

(

t

)

The

Z

-transform

of

Input Signal

u

(

t

)

Chapter 5

Discrete-Time Process ModelsDiscrete-Time Transfer Functions

Y

(

z

) only indicates information about y(t) in sampling times, since G(z

) does not relate input and output signals at times between sampling times.

When the sample-and-hold device is assumed to be a zero-order hold, then the relation between G(s) and G(z) is

Chapter 5

Discrete-Time Process ModelsExample

Find the discrete-time transfer function of a continuous system given by:

where

Chapter 5

Discrete-Time Process ModelsInput-Output Discrete-Time Models

A general discrete-time linear model can be written in time domain as:

where

m

and

n

are the order of numerator and denominator, k denotes the time instant, and d is the time delay.

Defining a shift operator

q–1, where:

Then, the first equation can be rewritten as:

or

Chapter 5

Discrete-Time Process ModelsInput-Output Discrete-Time Models

The polynomials

A

(q–1) and

B(q–1) are in descending order of q–1, completely written as follows:

The last equation on the previous page can also be written as:

Hence, we can define a function:

Identical, with the difference only in the use of notation for shift operator,

q

-1

or

z

–1

Chapter 5

Discrete-Time Process ModelsApproximation of

Z-Transform

Previous example shows how the

Z-transform of a function written in s

-Domain can be so complicated and tedious.Now, several methods that can be used to approximate the Z-transform will be presented.

Consider the integrator block as shown below:

The integration result for one sampling period of

T

s

is:

Chapter 5

Discrete-Time Process Models

Forward Difference Approximation

(Euler Approximation)

The exact integration operation presented before will now be approximated using Forward Difference Approximation.

This method follows the equation given as:Approximation of Z-Transform

Taking the

Z

-transform

of the above equation:

while

Thus, the Forward Difference Approximation is done by taking

or

Chapter 5

Discrete-Time Process Models

Backward Difference Approximation

The exact integration operation will now be approximated using Backward Difference Approximation.

This method follows the equation given as:

Approximation of Z-Transform

Taking the

Z

-transform

of the above equation:

while

Thus, the

Backward Difference

Approximation is done by taking

or

Chapter 5

Discrete-Time Process Models

Trapezoidal Approximation

(Tustin Approximation, Bilinear Approximation)

The exact integration operation will now be approximated using Backward Difference Approximation.

This method follows the equation given as:

Taking the

Z

-transform

,

while

Thus, the

Trapezoidal Approximation

is done by taking

or

Approximation of

Z

-Transform

Chapter 5

Discrete-Time Process ModelsExample

Find the discrete-time transfer function of

for the sampling time of

T

s = 0.5 s, by using (a) ZOH, (b) FDA, (c) BDA, (d) TA.

(a) ZOH

(b) FDA

Chapter 5

Discrete-Time Process ModelsExample

(c) BDA

(d) TA

Chapter 5

Discrete-Time Process ModelsExample

ZOH

FDA

TA

BDA

Chapter 5

Discrete-Time Process ModelsExample: Discretization of Single-Tank System

Retrieve the

linearized model

of the single-tank system. Discretize the model using trapezoidal approximation, with

Ts = 10 s.

Laplace Domain

Z

-Domain

Chapter 5

Discrete-Time Process ModelsExample: Discretization of Single-Tank System

Chapter 5

Discrete-Time Process ModelsExample: Discretization of Single-Tank System

Chapter 5

Discrete-Time Process ModelsExample: Discretization of Single-Tank System

:

Linearized

model

:

Discretized l

inearized model

Chapter 5

Discrete-Time Process ModelsExample: Discretization of Single-Tank System

:

Linearized

model

:

Discretized l

inearized model

Chapter 5

Discrete-Time Process ModelsHomework 8

(a) Find

the discrete-time transfer functions of the following continuous-time transfer function, for

Ts = 0.25 s and

Ts = 1 s. Use the Forward Difference Approximation

(b) Calculate

the step response of both

discrete transfer functions for

0 ≤ t ≤ 5 s.(c) Compare the step response of both transfer functions with the step response of the continuous-time transfer function G(s

) in one plot/scope for 0 ≤

t

≤ 0.5 s.

Chapter 5

Discrete-Time Process ModelsHomework 8A

Find

the discrete-time transfer functions of the following continuous-time transfer function, for

Ts = 0.1 s and

Ts = 0.05 s. Use the following approximation:Forward Difference

(Andre, Burawi, Deom, Indah, Jagat)

Backward Difference (Arief, Arwin, Keanu, Yeza, Wilbert)

(b) Calculate

the step response of both

discrete transfer

functions for

0

≤

t

≤ 0.5

s. The calculation for

t

=

kT

s, k = 0 until k

= 5 in each case must be done manually. The rest may be done by the help of Matlab Simulink.(c) Compare the step response of both discrete transfer functions with the step response of the continuous-time transfer function

G(s) in one plot/scope for 0 ≤ t ≤ 0.5 s.

Deadline: Thursday,

21 March 2019

.