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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.

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homework 8

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.

solution to homework 8

Chapter 5

Discrete-Time Process Models

Solution to Homework 8

(a)

solution to homework 81

Chapter 5

Discrete-Time Process Models

Solution to Homework 8
solution to homework 82

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 ]

solution to homework 83

Chapter 5

Discrete-Time Process Models

Solution to Homework 8

(c) Comparing the step responses

Ts = 0.25 s

Ts = 1 s

  • FDA delivers bad results
  • Possible solutions can be the use of smaller sampling time Ts or the use of ZOH or TA
solution to homework 84

Chapter 5

Discrete-Time Process Models

Solution to Homework 8
  • FDA with smaller sampling time Ts
solution to homework 85

Chapter 5

Discrete-Time Process Models

Solution to Homework 8
  • Using TA or ZOH, with reasonably large sampling time Ts
slide9

System Modeling and Identification

Chapter 6

Process Identification

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.
process identification1

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:
identification procedure

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

classification of identification methods

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
identification from step response

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.
first order time delay approximation

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:
first order time delay approximation1

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.
first order time delay approximation2

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:
first order time delay approximation3

Chapter 6

Identification from Step Response

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

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

time percent value method1

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.
time percent value method2

Chapter 6

Identification from Step Response

Time-Percent Value Method

t/t Table

t/τ Table

example time percent value method

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.

example time percent value method1

Chapter 6

Identification from Step Response

Example: Time-Percent Value Method
example time percent value method2

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

example time percent value method3

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

homework 9

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.
homework 9a

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.
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