Homework 8

1 / 27

# Homework 8 - PowerPoint PPT Presentation

Chapter 5. Discrete-Time Process Models. Homework 8. (a) Find the discrete-time transfer functions of the following continuous-time transfer function, for T s = 0.25 s and T s = 1 s. Use the Forward Difference Approximation.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Homework 8' - devika

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

Chapter 5

Discrete-Time Process Models

Homework 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 Models

Solution to Homework 8

(a)

Chapter 5

Discrete-Time Process Models

Solution to Homework 8

Chapter 5

Discrete-Time Process Models

Solution to Homework 8

(b) The step response of both transfer functions for 0 ≤ t ≤ 5 s.

Using the following command in Matlab workspace:

Y1 = dlsim([0.625],[1 –1.5 1.125],ones(1,21))

Y1 = [0 0 0.6250 1.5625 2.2656 2.2656 1.4746 0.2881 –0.6018 –0.6018 0.3993 1.9010 3.0273 3.0273 1.7602 –0.1403 –1.5658 –1.5658 0.0378 2.4433 4.2473]

Using the following command in Matlab workspace:

Y2 = dlsim([10],[1 0 9],ones(1,6))

Y2 = [ 0 0 10 10 –80 –80 ]

Chapter 5

Discrete-Time Process Models

Solution to Homework 8

(c) Comparing the step responses

Ts = 0.25 s

Ts = 1 s

• Possible solutions can be the use of smaller sampling time Ts or the use of ZOH or TA

Chapter 5

Discrete-Time Process Models

Solution to Homework 8
• FDA with smaller sampling time Ts

Chapter 5

Discrete-Time Process Models

Solution to Homework 8
• Using TA or ZOH, with reasonably large sampling time Ts

System Modeling and Identification

Chapter 6

Process Identification

Chapter 6

Process Identification

Process Identification
• Industry processes can be modeled in various ways, such as in state-space description or in transfer functions.
• The models mostly used for control purposes are in form of linear differential or difference equations, with parameters assumed as known and constant.
• In real conditions, it is often necessary to measure or estimate these parameters from input and output signals of the process.
• This case is referred to as parameter estimation or process identification.

Chapter 6

Process Identification

Process Identification
• The objective of process identification is to find a model that can describe the process.
• The information provided to do that is the inputs and the outputs of the process.

independent,

arbitrary,

measurable,

known

dependent,

measurable,

known

• The ideal result of a process identification will be:

Chapter 6

Process Identification

Identification Procedure
• A general procedure in process identification includes:
• Determination of model structure

→ Based on mathematical origin or artificial intelligence

• Estimation of model parameter
• → Based on the chosen model structure
• Model verification

→ A model must be able to produce accurate output if “unseen” input data is given to it

Chapter 6

Process Identification

Classification of Identification Methods
• Based on input signals
• Natural, generated during the process and measured
• Artificial, generated especially for the identification purpose
• Based on mathematics point of view
• Deterministic, assuming exact knowledge about process outputs, inputs, disturbance, etc, and do not consider random sources and influences
• Stochastic, assuming some properties and some knowledge of random disturbances, statistical approach
• Based on data processing
• Batch method, one calculation using the whole data at once, off-line
• Recursive method, gradual use of data, estimated parameters are improved from each experiment, can be on-line or off-line

Chapter 6

Process Identification

Identification from Step Response
• The methods in this category aim to provide first estimate of the process and provide approximate information about the process gain, dominant time constant, and time delay.
• The input signal used to excite the process is a step change of the process input.
• It is necessary that the process is in a steady-state before the step change occurs.
• The measured step response needs to be normalized for unit step change and zero initial conditions.

Chapter 6

Identification from Step Response

“First Order + Time Delay” Approximation
• The approximation model for the identified process is given in s-Domain as:

where K is the process gain, τ denotes time constant, and Td is the time delay.

• The step response of the transfer function G(s) given above in time domain is:

Chapter 6

Identification from Step Response

“First Order + Time Delay” Approximation

Unit step response

Approximation of unit step responseFirst order + time delay

• If the step response is a normalized one, the process gain K is equal to the new steady-state output, K = y(∞).
• The actual unit step response and its approximation will always have two crossing points.
• Time constant τ and time delay Td can be calculated if the two crossing points are already chosen.
• The two crossing points should be chosen thoughtfully, to avoid large difference between the two step responses.

Chapter 6

Identification from Step Response

“First Order + Time Delay” Approximation

Unit step response

Approximation of unit step responseFirst order + time delay

• From two freely-chosen points (t1,y1) and (t2,y2), after some manipulations, we can also obtain τ and Td through calculations as follows:

Chapter 6

Identification from Step Response

“First Order + Time Delay” Approximation
• Easy calculation, straightforward after two points are chosen
• Low accuracy, the higher the process order, the lower the accuracy of the model
• Time delay will always present in the model

Chapter 6

Identification from Step Response

Time-Percent Value Method
• The approximation model for the identified process is given in s-Domain as:
• From the unit step response, empirical values h∞, t10, t30, t50, t70, and t90 are obtained.

Step response

Chapter 6

Identification from Step Response

Time-Percent Value Method
• The values of parameters K, τ, and n are determined as follows:
• K is obtained from the steady-state value of the step response of the process divided by the magnitude of the input step.
• Using the “t/t Table”, up to 6 points of ti/tj can be located → the model order n can be determined.
• Using the “t/τ Table”, up to 5 points of ti/τ for the previously determined model order n can be located → the time constant τ can be determined.

Chapter 6

Identification from Step Response

Time-Percent Value Method

t/t Table

t/τ Table

Chapter 6

Identification from Step Response

Example: Time-Percent Value Method

A step function u(t) = 3(t) is fed in a process. As the step response, the following graph is obtained.

Determine the approximate transfer function of the process by using the Time-Percent Value Method.

Chapter 6

Identification from Step Response

Example: Time-Percent Value Method

Chapter 6

Identification from Step Response

Example: Time-Percent Value Method

From 6 ti/tj points, the most representative order for the model is 5

t/t Table

Chapter 6

Identification from Step Response

Example: Time-Percent Value Method

5 values of ti/τ can be located for n = 5

Result:

t/τ Table

Chapter 6

Identification from Step Response

Homework 9
• Time Percent Value MethodDetermine the approximation of the model in the last example, if after examining the t/t table, the model order is chosen to be 4 instead of 5.

Chapter 6

Identification from Step Response

Homework 9A
• “First Order + Time Delay” ApproximationDetermine the approximation of the model in the last example, using the data from t1= 2*(last 2 digits of Student ID), t2= arbitrary.
• Perform calculations to get your model.
• Print the graph (Slide 9/22) and draw the response of your model on it.