Exercise 3: Transfer functions (Solutions)

Transcription

1 Exercise 3: Transfer functions (Solutions) Transfer functions are a model form based on the Laplace transform. Transfer functions are very useful in analysis and design of linear dynamic systems. A general Transfer function is on the form: Where is the output and is the input. MathScript has several functions for creating transfer functions: Function Description Example tf Sys_order1 Sys_order2 step Example: Creates system model in transfer function form. You also can use this function to state-space models to transfer function form. Constructs the components of a first-order system model based on a gain, time constant, and delay that you specify. You can use this function to create either a state-space model or a transfer function model, depending on the output parameters you specify. Constructs the components of a second-order system model based on a damping ratio and natural frequency you specify. You can use this function to create either a state-space model or a transfer function model, depending on the output parameters you specify. Creates a step response plot of the system model. You also can use this function to return the step response of the model outputs. If the model is in state-space form, you also can use this function to return the step response of the model states. This function assumes the initial model states are zero. If you do not specify an output, this function creates a plot. Given the following transfer function: >num=[1]; >den=[1, 1, 1]; >H = tf(num, den) >K = 1; >tau = 1; >H = sys_order1(k, tau) >dr = 0.5 >wn = 20 >[num, den] = sys_order2(wn, dr) >SysTF = tf(num, den) >num=[1,1]; >den=[1,-1,3]; >H=tf(num,den); >t=[0:0.01:10]; >step(h,t); In MathScript we will use the following code: % Define Transfer function num = [1]; den = [1, 1]; H = tf(num, den) Faculty of Technology, Postboks 203, Kjølnes ring 56, N-3901 Porsgrunn, Norway. Tel: Fax:

2 2 % Step Response step(h) This gives the following step response: A general transfer function can be written on the following general form: The Numerators of transfer function models describe the locations of the zeros of the system, while the Denominators of transfer function models describe the locations of the poles of the system. In MathScript we can define such a transfer function using the built-in tf function as follows: num = [bm, bm_1, bm_2,, b1, b0]; den = [an, an_1, an_2,, a1, a0]; H = tf(num, den) Task 1: 1.order transfer functions Given the following system: Task 1.1 What are the values for the gain and the time constant for this system? Sketch the step response for the system using pen and paper.

3 3 Find the step response using MathScript and compare the result with your sketch. Gain and the time-constant : Step response for a 1.order system: MathScript: clear clc K=2; T=4; num=[k]; den=[t, 1]; H = tf(num, den); step(h) This gives the following plot:

4 4 Task 1.2 Find the differential equation from the transfer function above and draw a block diagram of the system ( pen and paper ). For a 1.order system in general we have: or: Which gives: In the time domain we get the following differential equation (using Inverse Laplace): We can draw the following block diagram of the system:

5 5 Where and for our system: Note! Even when the system is in the time plane we normally use the symbol. Other symbols that are commonly used for the integrator are: or. Task 1.3 From the block diagram in Task 1.2, find the transfer function (The answer shall of course be ) From the block diagram in the previous task we get the following transfer function: As expected, the result is the same as the transfer function given in Task 1.1. Note! We have used both the serial and feedback rules that yield for block diagram reduction.

6 6 Task 1.4 Find the solution for the differential equation and plot it ( pen and paper ). We will use a step for the control signal ( ). Note! The Laplace Transformation pair for a step is as follows: Tip! You also need to use the following Laplace transform pair: Compare to the results from Task 1.1. For a 1.order system in general we have: Here we will find the mathematical expression for the step response ( ): Where We use inverse Laplace and find the corresponding transformation pair in order to find ). We use the following Laplace transform pair:

7 7 This gives: Setting, and gives: ( ) We can plot this in MathScript: clear clc K=2; T=4; U=1; t=0:0.1:20; % Method 1 - Transfer Function num=[k]; den=[t, 1]; H = tf(num, den); figure(1) step(h, t) % Method 2 - Plot the solution of the differential equation y = K*U*(1-exp(-t/T)); figure(2) plot(t,y) We get the same results (of course). Task 2: Transfer functions in MathScript Define the following transfer functions in MathScript. Task 2.1 Given the following transfer function: MathScript Code:

8 8 num = [2, 3, 4]; den = [5, 9]; H = tf(num, den) Task 2.2 Given the following transfer function: MathScript Code: num = [4, 0, 0, 3, 4]; den = [5, 0, 9]; H = tf(num, den) Note! If some of the orders are missing, we just put in zeros. The transfer function above can be rewritten as: Task 2.3 Given the following transfer function: We need to rewrite the transfer function to get it in correct orders: MathScript Code: num = [2, 3, 7]; den = [6, 5, 0]; H = tf(num, den) Task 3: Differential equations to Transfer functions Task 3.1 Given the following differential equation:

9 9 Find the following transfer function: Solution: Laplace gives: Further: Further: Further: This gives: Task 4: PI Controller A PI controller is defined as: Where u is the controller output and is the control error: Task 4.1 Find the transfer function for the PI Controller: Using Laplace gives:

10 10 Then we get: This gives the following transfer function for the PI controller: Additional Resources Here you will find tutorials, additional exercises, etc.