Exercise 4: 2.order Systems (Solutions)

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1 Exercise 4: 2.order Systems (Solutions) A second order transfer function is given on the form: Where is the gain zeta is the relative damping factor [rad/s] is the undamped resonance frequency. The value of is critical for stability of the system: Faculty of Technology, Postboks 203, Kjølnes ring 56, N-3901 Porsgrunn, Norway. Tel: Fax:

2 2 The overshoot factor ( oversvingsfaktoren ) of the step response is defined as: MathScript: We can easily implement and analyze 2.order systems in MathScript using built-in functions. Example: We have the following 2.order system: i.e., We can use the tf function or the sys_order2 function in MathScript: num=[1]; den=[1, 1, 1]; H = tf(num, den) step(h) or: dr = 1 wn = 1 [num, den] = sys_order2(wn, dr) H = tf(num, den) step(h) This should give the same results. [End of Example] Task 1: Basic 2.order properties Given the following transfer function: Task 1.1 Find the following parameters (pen and paper): The gain The relative damping factor The undamped resonance frequency [rad/s]

3 3 Based on the general case: We get: The undamped resonance frequency [rad/s]: The relative damping factor : The gain Task 2: Response Time Given the following transfer function: Task 2.1 Find the total response time for the given system. Note! The response time for a 2.order system is approximately: We do the following: The total response time for the given system is: Where

4 4 We need to find : We have: This means: Then we get: Task 3: Transfer function to Differential equation Given the following transfer function: Task 3.1 Find the differential equation for the system. We do as follows: [ ] [ ] This gives: This gives the following differential equation: Task 4: 2.order transfer functions Task 4.1 Define the transfer function below using the tf and the sys_order2 functions (2 different methods that should give the same results).

5 5 Set Do you get the same results using tf() and sys_order2()? clear clc K = 1; w = 1; z = 1; num = [K]; den = [(1/w)^2, 2*z*(1/w), 1]; H = tf(num, den) step(h) or: clear clc dr = 1 wn = 1 H = sys_order2(wn, dr) step(h) Task 4.2 Plot the step response (use the step function in MathScript) for different values of. Select follows: as Explain the results.

6 6 We see the results are as expected. gives a underdamped system gives a critically damped system gives a overdamped system Task 5: More 2.order transfer functions For the transfer functions given below, find the following parameters: The gain The relative damping factor The undamped resonance frequency [rad/s] You may also try to implement the systems in MathScript and perform a step response. Task 5.1 Based on the general case: We get:

7 7 1. The undamped resonance frequency [rad/s]: ( has no relevance) 2. The relative damping factor : 3. The gain 4. The overshoot factor : Task 5.2 Based on the general case: We transform our transfer function as follows: Then we get: 1. The undamped resonance frequency [rad/s]: ( has no relevance) 2. The relative damping factor :

8 8 3. The gain 4. The overshoot factor : MathScript Code: clear clc % System 1 num1 = [5]; den1 = [1, 4, 1] H1 = tf(num1, den1) figure(1) step(h1) % System 1 num2 = [9]; den2 = [3, 4, 2] H2 = tf(num2, den2) figure(2) step(h2) Task 6: Differential equation to Transfer function Given the following differential equation: Task 6.1 Find the transfer function: We get:

9 9 Further: [ ] This gives the following transfer function: Task 7: Stability Given the following system: Task 7.1 Find poles and zeroes for the system (check your answer using MathScript) and draw them in the complex plane. Tip! In MathScript you can use the built-in functions poles(), zero() and pzgraph(). Solutions: Zeros: Poles:

10 10 MathScript: MathScript code: clear clc % Transfer function num = 5*[10, -1]; den1 = [2, 1]; den2 = [5, 1]; den = conv(den1,den2); H = tf(num, den) p = poles(h) z = zero(h) pzmap(h) We get the same answer in MathScript. Task 7.2 Is the system stable or not? Why/Why not? The system is stable because both the poles are in the left half plane. Task 8: Mass-spring-damper system Given the following system: is the position is the speed/velocity is the acceleration F is the Force (control signal, u) d and k are constants Task 8.1 Draw a block diagram for the system using pen and paper.

11 11 The block diagram becomes: You may also use this notation: Task 8.2 Based on the block diagram, find the transfer function for the system. Where the force may be denoted as the control signal. Set the transfer function on the on the following standard form: Find, and as functions of, and. In order to find the transfer function for the system, we need to use the serial and feedback rules.

12 12 We start by using the serial rule: Next, we use the feedback rule: Next, we use the serial rule: Finally, we use the feedback rule: Or if we want it on the standard 2.order form: We get: This means: Task 8.3 Simulate the system in MathScript (step response). Try with different values for, and.

13 13 MathScript code: % Mass-spring-damper system clear clc % Define variables m = 1; d = 1; k = 1; % Define Transfer function num = 1/m ; den = [1, (d/m), (k/m)]; H = tf(num, den); % Step Response step(h) This gives the following results: Additional Resources Here you will find tutorials, additional exercises, etc.

Exercise 8: Frequency Response

Exercise 8: Frequency Response Introduction We can find the frequency response of a system by exciting the system with a sinusoidal signal of amplitude A and frequency ω [rad/s] (Note: ω = 2πf) and observing