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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.

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 ]

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

Discrete-Time Process Models

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

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 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:

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

- Determination of model structure

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

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.

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:

Identification from Step Response

“First Order + Time Delay” ApproximationUnit 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.

Identification from Step Response

“First Order + Time Delay” ApproximationUnit 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:

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

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

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.

Identification from Step Response

Example: Time-Percent Value MethodA 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.

Identification from Step Response

Example: Time-Percent Value MethodFrom 6 ti/tj points, the most representative order for the model is 5

t/t Table

Identification from Step Response

Example: Time-Percent Value Method5 values of ti/τ can be located for n = 5

Result:

t/τ Table

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.

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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