Sampled Data System and Z-Transform

Pulse Transfer Function and Manipulation of Block Diagram Definition • Pulse Transfer Function is • A ratio of the z-transform of the sampled output and input at the sample instants • A

Sampled Data System and the z-Transform

Thanh Vo-Duy Department of Industrial Automation

thanh.voduy@hust.edu.vn

Prior to Lecture

• Conventional Control System

Sampled data control system

The Sampling Process

• A Sampler is a switch that closes every T seconds

The Sampling Process

• The closure time is much smaller than T

The Sampling Process

• In ideal sampling, closure time can be neglected

The Sampling Process

The Sampling Process

The Sampling Process

• From sampled 𝑟∗(𝑡) to continuous output 𝑦(𝑡)

→ Use Zero-Order Hold circuit.

The Sampling Process

The Sampling Process

• Example

• z-Transformation in sampled data system

• Laplace Transformation in continuous-time systems

The z-Transform

Commonly used functions

• Unit step function

The z-Transform

Commonly used functions

• Unit ramp function

The z-Transform

Commonly used functions

http://lpsa.swarthmore.edu/LaplaceZTable/LaplaceZFuncTable.html

of equation 𝐷 𝑠 = 0

The z-Transform

Inverse z-Transform

• Method to inverse Y(z) to y(t)

• Power Series (long division)

𝑌 𝑧 = 𝑦0 + 𝑦1𝑧−1 + 𝑦2𝑧−2 + ⋯

→ 𝑦 𝑡 = 𝑦0𝛿 𝑡 − 𝑇 + 𝑦1𝛿 𝑡 − 2𝑇 + 𝑦2𝛿 𝑡 − 3𝑇 + ⋯

• Expanding Y(z) into partial fraction

• Inversion formula method: using inversion integral

𝑦 𝑛𝑇 = 1

2𝜋𝑗

𝑟

𝑌 𝑧 𝑧−1𝑑𝑧

Pulse Transfer Function and

Manipulation of Block Diagram

Definition

• Pulse Transfer Function is

• A ratio of the z-transform of the sampled output and input at the sample instants

• A ratio of the z-transform of the output and input when both input and output are trains of pulse

• ….

• Pulse Transfer Function = Discrete Transfer Function

Pulse Transfer Function and

Manipulation of Block Diagram

𝑦 𝑧 = 𝑒 𝑧 𝐺 𝑧

→ 𝐺(𝑧) is call Pulse Transfer Function

Pulse Transfer Function and

Manipulation of Block Diagram

𝑎

𝑠 + 𝑎 𝑒(𝑠) 𝑒∗(𝑠)

𝐺1(𝑠) 𝐺2(𝑠)

𝑦(𝑠) 𝑦∗(𝑠)

Pulse Transfer Function and

Manipulation of Block Diagram

Pulse Transfer Function and

Manipulation of Block Diagram

𝑎

𝑠 + 𝑎 𝑒(𝑠) 𝑒∗(𝑠)

𝐺2(𝑠)

𝑦(𝑠) 𝑦∗(𝑠) 𝑥(𝑠) 𝑥∗(𝑠)

Pulse Transfer Function and

Manipulation of Block Diagram

Pulse Transfer Function and

Manipulation of Block Diagram

Open-Loop Time Response

• Consider the system with ZOH as below What will the output response if a unit step input is applied?

Assume that T = 1s

𝐺1(𝑠)

𝑠 + 1 𝑢(𝑠) 𝑢∗(𝑠)

𝐺2(𝑠)

𝑦(𝑠)

Pulse Transfer Function and

Manipulation of Block Diagram

Open-Loop Time Response

• Solution

Transfer function of ZOH:

𝐺1 𝑠 = 1 − 𝑒

−𝑇𝑠 𝑠

Equivalent transfer function of system (with T = 1s):

Pulse Transfer Function and

Manipulation of Block Diagram

Open-Loop Time Response

𝑦 𝑛𝑇

= 0.63𝛿 𝑡 − 1 + 0.86𝛿 𝑡 − 2 + 0.95𝛿 𝑡 − 3 + 0.98𝛿 𝑡 − 4 + ⋯

Pulse Transfer Function and

Manipulation of Block Diagram

Open-Loop Time Response

• Solution (cont.)

The time response in this case:

Pulse Transfer Function and

Manipulation of Block Diagram

Open-Loop Time Response

• Solution (cont.)

Pulse Transfer Function and

Manipulation of Block Diagram

Pulse Transfer Function and

Manipulation of Block Diagram

Closed-Loop System

• Example 2:

Derive the z-transform of closed-loop system below

Pulse Transfer Function and

Manipulation of Block Diagram

Closed-Loop System

• Example 3:

Consider the system as below

Derive the transfer function of the system

Pulse Transfer Function and

Manipulation of Block Diagram

Closed-Loop System

• Solution for Example 3:

The equivalent system:

Thus, the Transfer function of system is:

𝑦 𝑧

𝑟 𝑧 =

𝐷 𝑧 𝐺 𝑧

1 + 𝐷 𝑧 𝐺𝐻 𝑧

Pulse Transfer Function and

Manipulation of Block Diagram

Closed-Loop Time Response

• Example:

What will the output be if Unit Step input is applied to the sampled closed-loop system

Pulse Transfer Function and

Manipulation of Block Diagram

Closed-Loop Time Response

𝑟 𝑧 = 𝑧

𝑧 − 1 ; 𝐺 𝑧 =

𝑧 1 − 𝑒−𝑇

𝑧 − 1 𝑧 − 𝑒−𝑇 ; 𝐻 𝑧 = 1So:

2 1 − 𝑒−𝑇

𝑧2 − 2𝑧𝑒−𝑇 + 𝑒−𝑇 𝑧 − 1

Pulse Transfer Function and

Manipulation of Block Diagram

Closed-Loop Time Response

Pulse Transfer Function and

Manipulation of Block Diagram

Closed-Loop Time Response

• Solution (cont.)

1 A function 𝑦 𝑡 = 2 sin(4𝑡) is sampled every 𝑇 =0.1𝑠.Find the z-transform of the resultant number sequence

2 Find the z-transform of the function 𝑦 𝑡 = 3𝑡

3 Determine the output response of the system

described below when the Unit Step input is applied

𝑦 𝑧 = 𝑧

𝑧 + 0.5 𝑧 − 0.2

8 Derive the transfer function of the closed-loop system whose block diagrams are shown as: