Solutions. MathScript. So You Think You Can. Part III: Frequency Response HANS-PETTER HALVORSEN,

Transcription

1 Telemark University College Department of Electrical Engineering, Information Technology and Cybernetics Solutions So You Think You Can HANS-PETTER HALVORSEN, MathScript Part III: Frequency Response Faculty of Technology, Postboks 203, Kjølnes ring 56, N-3901 Porsgrunn, Norway. Tel: Fax:

2 Table of Contents Table of Contents...ii 1 Control Design in MathScript Frequency Response Standard Transfer functions Frequency Response Analysis Stability Analysis of Feedback Systems Additional Tasks ii

3 1 Control Design in MathScript No Tasks. 3

4 2 Frequency Response Task 1: 1.order We have the following transfer function: ( ) Pen & Paper: What is the break frequency (Norwegian: knekkfrekvens )? Find poles and zeroes. From the transfer function we get: Poles: Zeros: None MathScript: Using the poles and zero functions in MathScript gives: % Transfer function num = [4]; den = [2,1]; H = tf(num,den) % Poles and Zeros poles(h) zero(h) ans =

5 5 Frequency Response ans = empty matrix 0 by 1 Set up the mathematical expressions for ( ) and ( ). ( ) ( ) ( ) ( ) Method 1: Bode plot in MathScript Using built-in bode function: Plot the frequency response of the system in a Bode plot using the bode function in MathScript. Discuss the results. MathScript Code: % Transfer function num=[4]; den=[2, 1]; H = tf(num, den) % Bode Plot bode(h) subplot(2,1,1) grid subplot(2,1,2) grid Bode Plot:

6 6 Frequency Response We see that the plot is correct according to our knowledge about a 1.order system. We see that the phase converge to -90 degrees, which is standard for such a 1.order system. Find ( ) and ( ) for the following frequencies using MathScript code (use, e.g., the bode function): ( ) ( )( ) Make sure ( ) is in db. MathScript Code:

7 7 Frequency Response % Margins and Phases wlist=[0.1, 0.16, 0.25, 0.4, 0.625, 2.5]; [mag, phase, w] = bode(h, wlist); magdb=20*log10(mag); %convert to db mag_data = [w, magdb] phase_data = [w, phase] From the code above we get: mag_data = phase_data = Which gives: ( ) ( ) Total Code list: clear clc % Transfer function num=[4]; den=[2, 1]; H = tf(num, den) % Bode Plot bode(h) subplot(2,1,1) grid subplot(2,1,2) grid

8 8 Frequency Response % Margins and Phases wlist=[0.1, 0.16, 0.25, 0.4, 0.625, 2.5]; [mag, phase, w] = bode(h, wlist); magdb=20*log10(mag); %convert to db mag_data = [w, magdb] phase_data = [w, phase] Method2: Bode plot in MathScript Create your own Bode plot from scratch: Find ( ) and ( ) for the same frequencies above using the mathematical expressions for ( ) and ( ). Tip: Use a For Loop and/or define a vector. Plot a Bode diagram using the built-in semilogx function for the frequencies given above. Do you get the same results? MathScript Code: clear clc % Transfer function K = 4; T = 2; num = [K]; den = [T, 1]; H = tf(num, den) % Frequency List wlist=[0.1, 0.16, 0.25, 0.4, 0.625,2.5]; N= length(wlist); for i=1:n end gain(i) = 20*log10(4) - 20*log10(sqrt((2*wlist(i))^2+1)); phase(i) = -atan(2*wlist(i)); phasedeg(i) = phase(i) * 180/pi; %convert to degrees % Print to Screen gain_data = [wlist; gain]' phase_data=[wlist; phasedeg]' % Plot Bode diagram %Gain Plot subplot(2,1,1) semilogx(gain_data(:,1), gain_data(:,2)) grid

9 9 Frequency Response %Phase Plot subplot(2,1,2) semilogx(phase_data(:,1), phase_data(:,2)) grid % % Check with results from the bode function [gain2, phase2,w] = bode(h, wlist); gain2db=20*log10(gain2); %convert to db This gives: We see that this method gives the same answers. Bode plot:

10 10 Frequency Response If we compare the bode plot we created from scratch with the built-in bode plot, and change the axis scaling, we see that this gives the same results: [End of Task] Task 2: 2.order We have the following transfer function: ( ) ( )( ) Pen & Paper: What is the break frequencies (Norwegian: knekkfrekvenser )? Find poles and zeroes.

11 11 Frequency Response Use the poles and zero functions in MathScript. Set up the mathematical expressions for ( ) and ( ). Use Pen & Paper for this Assignment. ( ) ( ) ( ) ( ) ( ) ( ) Method 1: Bode plot in MathScript Using built-in bode function: Plot the frequency response of the system in a bode plot using the bode function in MathScript. Tip! Use the conv and the tf functions in MathScript in order to create the transfer function. Discuss the results. Find ( ) and ( ) for some given frequencies using MathScript code (use the bode function) MathScript Code: clear clc % Transfer function K = 5; T1 = 1; T2 = 10; num = [K]; den1 = [T1, 1]; den2 = [T2, 1]; den = conv(den1,den2); H = tf(num, den) % Bode Plot

12 12 Frequency Response bode(h) subplot(2,1,1) grid subplot(2,1,2) grid % Margins and Phases wlist=[0.01, 0.1, 1, 10, 100]; [mag, phase, w] = bode(h, wlist); magdb=20*log10(mag); %convert to db mag_data = [w, magdb] phase_data = [w, phase] Merk! Vi har brukt conv for å slå sammen de 2 leddene i nevnere: % Transfer function K = 5; T1 = 1; T2 = 10; num = [K]; den1 = [T1, 1]; den2 = [T2, 1]; den = conv(den1,den2); H = tf(num, den) Vi kunne også gjort dette manuelt : ( )( ) Nevneren kan da defineres slik: den=[10, 11, 1] Bode Plot:

13 13 Frequency Response We see that the plot is correct according to our knowledge about a 2.order system. We see that the phase converge to -180 degrees, which is standard for such a 2.order system. Frequencies: ( ) ( )( )

14 14 Frequency Response Method2: Bode plot in MathScript Create your own Bode plot from scratch: Find ( ) and ( ) for the same frequencies above using the mathematical expressions for ( ) and ( ). Tip: Use a For Loop or define a vector. Plot a Bode diagram using the built-in semilogx function for the frequencies given above. Same procedure as previous Task. MathScript Code: clear clc % Transfer function K = 5; T1 = 1; T2 = 10; num = [K]; den1 = [T1, 1]; den2 = [T2, 1]; den = conv(den1,den2); H = tf(num, den) % Frequency List wlist=[0.01, 0.1, 1, 10, 100]; N= length(wlist); for i=1:n gain(i) = 20*log10(5) - 20*log10(sqrt((wlist(i))^2+1)) - 20*log10(sqrt((10*wlist(i))^2+1)); phase(i) = -atan(wlist(i)) - atan(10*wlist(i)); phasedeg(i) = phase(i) * 180/pi; %convert to degrees end % Print to Screen gain_data = [wlist; gain]' phase_data=[wlist; phasedeg]'

15 15 Frequency Response % Plot Bode diagram %Gain Plot subplot(2,1,1) semilogx(gain_data(:,1), gain_data(:,2)) grid %Phase Plot subplot(2,1,2) semilogx(phase_data(:,1), phase_data(:,2)) grid % % Check with results from the bode function [gain2, phase2,w] = bode(h, wlist); gain2db=20*log10(gain2); %convert to db We get the same results as in method1 Bode plot: [End of Task] Task 3: Frequency Response

16 16 Frequency Response We have the following transfer function: ( ) ( ) Pen & Paper: What is the break frequencies (Norwegian: knekkfrekvenser )? Find poles and zeroes. Use the poles and zero functions in MathScript. Set up the mathematical expressions for ( ) and ( ). Use Pen & Paper for this Assignment. ( ) ( ) ( ) ( ) ( ) ( ) Method 1: Bode plot in MathScript Using built-in bode function: Plot the frequency response of the system in a bode plot using the bode function in MathScript. Discuss the results. Find ( ) and ( ) for some given frequencies using MathScript code (use the bode function).

17 17 Frequency Response Bode Plot: Method2: Bode plot in MathScript Create your own Bode plot from scratch:

18 18 Frequency Response Find ( ) and ( ) for the same frequencies above using the mathematical expressions for ( ) and ( ). Tip: Use a For Loop and/or define a vector w=*0.01, 0.1, +. Plot a Bode diagram using the built-in semilogx function for the frequencies given above. Same procedure as previous Tasks. [End of Task] Task 4: Time-delay Given the following system with time-delay: ( ) Plot the Bode diagram. Discuss the results Set up the mathematical expressions for ( ) and ( ). Use Pen & Paper for this Assignment. Find ( ) and ( ) for the following frequencies using MathScript code (use the bode function) [End of Task] MathScript Code: s=tf('s'); K=3.2; T=3; H1=tf(K/(T*s+1)); delay=2; H2=set(H1,'inputdelay',delay); bode(h2); Bode Plot: ( ) ( )( )

19 19 Frequency Response We see that the plot is correct according to our knowledge about a system with delay. We see that the phase curve is very steep, which is standard for such a system. Frequencies: % Margins and Phases wlist=[0.01, 0.1, 1, 10]; [mag, phase,w] = bode(h, wlist); magdb=20*log10(mag); %convert to db mag_data = [w, magdb] phase_data = [w, phase] Here are the Gains and Phases for the specific frequencies:

20 20 Frequency Response Mathematical expressions: ( ) ( ) ( ) ( ) Or in degrees: ( ) ( ( )) Task 5: Calculate Frequency response from sinusoidal input and output signals Given the following system: Set, I this task we will use 4 different methods to find and for a given frequency. Method 1: Calculate Frequency Response from sinusoidal input and output The input signal is given by: ( ) The steady-state output signal will then be: ( ) ( ) The gain is given by: The phase lag is given by: Plot the input signal and the resulting output signal in the same plot. Create a MathScript program where you define the transfer function and define the input signal. Set and.

21 21 Frequency Response Use the lsim function is MathScript to plot the output signal for a given frequency. Make sure you get in decibel and in degrees. For we get: MathScript Code: % Define Transfer function K = 1; T = 1; num = [K]; den = [T, 1]; H = tf(num, den); % Define input signal t = [1: 0.1 : 12]; w = 1; U = 1; u = U*sin(w*t); figure(1) plot(t, u) % Output signal hold on lsim(h, 'r', u, t) grid on hold off legend('input signal', 'output signal') % Values found from plot1 for w=1 Y = 0.68; A = Y/U; AdB = 20*log10(A) dt = 0.8; phi = -w*dt; %[rad] phi_degrees = phi*180/pi %[degrees] This gives the following plot:

22 22 Frequency Response We calculate and from the plot below From the plot we get: Using: and We get: We create a simple script that calculate the values (or you can do it manually):

23 23 Frequency Response MathScript Code: % Values found from plot1 for w=1 Y = 0.68; A = Y/U; AdB = 20*log10(A) dt = 0.8; phi = -w*dt; %[rad] phi_degrees = phi*180/pi %[degrees] The result from the script is: AdB = phi_degrees = Method 2: Find Gain and Phase from the Bode plot Plot the bode plot (use the bode function), and see if you get the same results. bode(h) Read the values for and for from the Bode plot. Make sure you get in decibel and in degrees. For we get: MathScript Code: %Bode plot figure(2) bode(h) subplot(2,1,1) grid on subplot(2,1,2) grid on This gives:

24 24 Frequency Response We see that the results we get is correct Method 3: Calculate the Gain and Phase from the mathematical expressions Set up the mathematical expressions for ( ) and ( ) and calculate the values for. Make sure you get in decibel and in degrees. For we get: Mathematical expressions: ( ) ( ) ( ) ( ) MathScript Code: gain = 20*log10(1) - 20*log10(sqrt(w^2+1)) phase = -atan(w);

25 25 Frequency Response phasedeg = phase * 180/pi %convert to degrees Results: gain = phasedeg = -45 Method 4: Find Gain and Phase using the bode function directly Use also the bode function to calculate the exact values. The bode function can be used like this: [mag, phase, wout] = bode(h, wlist); Compare and see if you get the same results as in the methods above. Make sure you get in decibel and in degrees. For we get: We use the bode function to get the magnitudes and phases: %Calculated magnitude and phase values for some given frequencies wlist = [1]; [mag, phase, wout] = bode(h, wlist); magdb = 20*log10(mag) phase This gives: magdb = phase = The procedure is exactly the same for, etc., and will not be shown here. [End of Task]

26 26 Frequency Response 2.1 Standard Transfer functions Task 6: 1.order system The transfer function for a 1.order system is as follows: ( ) Where is the gain T is the Time constant Tasks: Find the mathematical expressions for ( ) and ( ). Use Pen & Paper for this Assignment. ( ) ( ) ( ) ( ) ( ) Plot the Bode plots

27 27 Frequency Response Find the break frequencies (Norwegian: knekkfrekvenser ) Set, e.g., K=1 and T=1 Discuss the results [End of Task] Task 7: 2.order system The transfer function for a 2.order system is as follows: ( ) ( ) Where is the gain zeta is the relative damping factor [rad/s] is the undamped resonance frequency. Tasks:

28 28 Frequency Response Find the mathematical expressions for ( ) and ( ). Use Pen & Paper for this Assignment. We have: ( ) ( ) Then we get: ( ) ( ) ( ) ( ) ( ) ( ) ( ) Plot the Bode plots Solution: Find the break frequencies (Norwegian: knekkfrekvenser ) Set, e.g., K=1,,

29 29 Frequency Response Discuss the results [End of Task] Task 8: Time delay The transfer function for a Time Delay is as follows: ( ) Where is the gain is the time-delay Tasks: Find the mathematical expressions for ( ) and ( ). Use Pen & Paper for this Assignment. ( ) ( ) ( ) ( ) Note! Plot the Bode plots

30 30 Frequency Response Find the break frequencies (Norwegian: knekkfrekvenser ) Set, e.g., K=1 and Discuss the results [End of Task]

31 3 Frequency Response Analysis Task 9: Frequency Response Analysis Given the following system: Process transfer function: Where, where,, and Measurement (sensor) transfer function: Where Km = 1 %/m. Controller transfer function (PI Controller): Set Kp = 1,5 og Ti = 1000 sec. Define the different transfer functions in MathScript. We define the different transfer functions. There are multiple ways to define the different transfer functions, and we will show some alternative solutions. 31

32 32 Frequency Response Analysis The tricky part is the time delay in the process transfer function. Here we can use different approaches and different functions. We can use the built-in sys_order1, we can use the built-in pade function or create our own Pade approximation (e.g. a 1.order or 2.order approximation). Method1: Here we use combinations of functions tf and sys_order1. clear clc close all % Model parameters: Ks=0.556; %(kg/s)/% A=13.4; %m2 rho=145; %kg/m3 transportdelay=250; %sec %Defining the process transfer function: K=Ks/(rho*A); num1 = [K]; den1 = [1, 0]; H1 = tf(num1, den1); H2 = sys_order1(1, 0, transportdelay); disp('process:') Hp = series(h1, H2) % Defining sensor transfer function: Km=1; %percent per meter disp('sensor:') Hs=tf(Km) % Defining controller transfer function: Kp=1.5; Ti=1000; num = Kp*[Ti, 1]; den = [Ti, 0]; disp('controller:') Hc = tf(num,den) Note! ( ) We can also use the pade function in order to create a Pade approximation: n=5; % Order of Pade approximation H2 = pade(transportdelay, n)

33 33 Frequency Response Analysis Or we can create our own Pade approximation and then use the tf function: % 2.order approx. using tf k1=transportdelay/2; k2=transportdelay^2/12; num=[k2, -k1, 1]; den=[k2, k1, 1]; H2=tf(num, den) For a 2.order Pade approximation ( ) we get the following transfer function: We get: Where: Method 2: This is a new method we haven t used before. We define s as the Laplace operator and then we can use s directly in our equations. clear clc close all s=tf('s'); %Model parameters: Ks=0.556; %(kg/s)/% A=13.4; %m2 rho=145; %kg/m3 transportdelay=250; %sec %Defining the process transfer function: K=Ks/(rho*A); padeorder=5; %Order of Pade-approximation of time-delay. Order 5 is usually ok. Hp1=set(tf(K/s),'inputdelay',transportdelay);%Including transportdelay in process transfer function Hp=pade(Hp1,padeorder);%Deriving process transfer function incl. Pade-approx of time-delay

34 34 Frequency Response Analysis %Defining sensor transfer function: Km=1; Hs=tf(Km); %Defining sensor transfer function (just a gain in this example) %Defining controller transfer function: Kp=1.5; Ti=1000; Hc=Kp+Kp/(Ti*s); %PI controller transfer function Set up the mathematical expression and define the Loop transfer function ( ). Tip! Use the built-in function series in Mathscript. Set up the mathematical expression and define the Sensitivity transfer function ( ) Tip! Use the built-in function feedback in Mathscript. Set up the mathematical expression and define the Tracking transfer function ( ) Tip! Use the code located here as an example: Introduction to MathScript by Finn Haugen. [End of Task] Control system: Defining ( ): ( )

35 35 Frequency Response Analysis We get the following compact system: Mathscriot Code: % Calculating loop tranfer function L=series(Hc,series(Hp,Hs)); %Calculating tracking transfer function T=feedback(L,1); % Calculating sensitivity transfer function S=1-T; Bode plot: Plot the Loop transfer function ( ), the Tracking transfer function ( ) and the Sensitivity transfer function ( ) in a Bode diagram. Use, e.g., the bodemag function in MathScript. Discuss the results. Code: % Bode Diagram figure(1) bodemag(l,t,s) grid Bode diagram:

36 36 Frequency Response Analysis Step Response: Plot the step response for the Tracking transfer function ( ) Discuss the results. Code: % Simulating step response for control system (tracking transfer function) figure(2) step(t) grid Plot:

37 37 Frequency Response Analysis Stability Margins: Find the stability margins (GM, PM) of the system ( ( )). Discuss the results. Code: % Calcutating stability margins and crossover frequencies: [gm, pm, w180, wc] = margin(l) % Plotting L and stability margins and crossover frequencies in Bode diagram figure(3) margin(l) grid This gives:

38 38 Frequency Response Analysis Shown in the Ouput Window of Mathscript: GM=12.764, and PM = deg. The system is asymptotically stable GM is ok, but PM somewhat too small (should have been at least 30 deg). May try to increase Ti to e.g Decreasing Kp does not help (normally it does, but it can be shown that for this system,

39 39 Frequency Response Analysis containing two integrators in series (PI controller and process), reducing gain may actually reduce stability (normally the stability is increased if gain is reduced). Bandwidths: Find the different bandwidths (see the sketch below). crossover-frequency the frequency where the gain of the Loop transfer function ( ) has the value: the frequency where the gain of the Tracking function ( ) has the value: - the frequency where the gain of the Sensitivity transfer function ( ) has the value: Discuss the results. Values for can be found in the plot as shown below:

40 40 Frequency Response Analysis We change the scaling for more details: I get the following values:

41 41 Frequency Response Analysis Below we see the complete code for the Task: clear clc close all % Model parameters: Ks=0.556; %(kg/s)/% A=13.4; %m2 rho=145; %kg/m3 transportdelay=250; %sec %Defining the process transfer function: K=Ks/(rho*A); num1 = [K]; den1 = [1, 0]; H1 = tf(num1, den1); H2 = sys_order1(1, 0, transportdelay); disp('process:') Hp = series(h1, H2) % Defining sensor transfer function: Km=1; %percent per meter disp('sensor:') Hs=tf(Km) % Defining controller transfer function: Kp=1.5; Ti=1000; num = Kp*[Ti, 1]; den = [Ti, 0]; disp('controller:') Hc = tf(num,den) % Calculating loop tranfer function L=series(Hc,series(Hp,Hs)); %Calculating tracking transfer function T=feedback(L,1); % Calculating sensitivity transfer function S=1-T;

42 42 Frequency Response Analysis % Bode Diagram figure(1) bodemag(l,t,s) grid % Simulating step response for control system (tracking transfer function) figure(2) step(t) grid % Calcutating stability margins and crossover frequencies: [gm, pm, w180, wc] = margin(l) % Plotting L and stability margins and crossover frequencies in Bode diagram figure(3) margin(l) grid [End of Task]

43 4 Stability Analysis of Feedback Systems Task 10: Gain and phase margins Given the following system: ( ) ( ) We will find the crossover-frequencies for the system using MathScript. We will also find also the gain margins and phase margins for the system. Method 1: Use the standard bode function and find the crossover-frequencies, the gain margins and the phase margins for the system. GM, PM, and are found as illustrated below: Method 2: Plot a bode diagram where the crossover-frequencies, GM and PM are illustrated. Tip! Use the margin function in MathScript. Use also the margins function to find gmf, gm, pmf, pm. 43

44 44 Stability Analysis of Feedback Systems Compare you results. [End of Task] Code: clear, clc % Transfer function num=[1]; den1=[1,0]; den2=[1,1] den3=[1,1] den = conv(den1,conv(den2,den3)); H = tf(num, den) % Bode Plot bode(h) % Margins and Phases wlist=[0.01, 0.1, 0.2, 0.5, 1, 10, 100]; [mag, phase,w] = bode(h, wlist); magdb=20*log10(mag); %convert to db % [mag, phase,w] = bode(h); mag_data = [w, magdb] phase_data = [w, phase] % Crossover Frequency margin(h) [gm, pm, w180, wc] = margin(h); wc w180 % Convert to db. gm_db = 20*log10(gm) pm Note! Using help margin in MathScript does not give the correct information about the return paramaters return by the margin function!! The correct is: [gm, pm, w180, wc] = margin(h); The Bode diagram:

45 45 Stability Analysis of Feedback Systems We have to find GM, PM, wc and w180 manually from the plot. From the graph above we find the following (an image editor is used to draw on the chart):

46 46 Stability Analysis of Feedback Systems Using the margin function (gm, fm, gmf and pmf are automatically plotted in the Bode chart):

47 47 Stability Analysis of Feedback Systems Results: wc = w180 = gm_db = pm = Note! gm has to be converted to db. gm_db = 20*log10(gm) So the results are as follows:

48 48 Stability Analysis of Feedback Systems Task 11: Time-delay Given the following system with time-delay: ( ) We will find the crossover-frequencies for the system using MathScript. We will also find the gain margins and phase margins for the system. Method 1: Use the standard bode function and find the crossover-frequencies, the gain margins and the phase margins for the system. Method 2: Plot a bode diagram where the crossover-frequencies, GM and PM are illustrated. Tip! Use the margin function in MathScript. Use also the margins function to find gmf, gm, pmf, pm. Compare you results. [End of Task] Code: clear, clc s=tf('s'); K=2.5; T=3; H1=tf(K/(T*s+1)); delay=1; H=set(H1,'inputdelay',delay); bode(h); margin(h) [gm, pm, w180, wc] = margin(h); wc w180 % Convert to db. gm_db = 20*log10(gm)

49 49 Stability Analysis of Feedback Systems pm Ordinary Bode Plot using the bode function (I have manually adjusted the scaling and added grids): We have to find GM, PM, wc and w180 manually from the plot. Using the margin function (gm, fm, gmf and pmf are automatically plotted in the Bode chart):

50 50 Stability Analysis of Feedback Systems Using [gm, pm, w180, wc] = margin(h); gives: wc = w180 = gm_db = pm = Note! gm has to be converted to db. gm_db = 20*log10(gm) So the results are as follows:

51 51 Stability Analysis of Feedback Systems

52 5 Additional Tasks Here are some additional tasks. Task 12: Amplifier The transfer function for an Amplifier is as follows: ( ) Where is the gain Find the mathematical expressions for ( ) and ( ). Use Pen & Paper for this Assignment. ( ) ( ) ( ) ( ) Plot the Bode plots 52

53 53 Additional Tasks [End of Task] Task 13: Integrator The transfer function for an Integrator is as follows: ( ) Where is the gain Tasks: Find the mathematical expressions for ( ) and ( ). Use Pen & Paper for this Assignment. ( ) ( ) ( ) ( ) Plot the Bode plots

54 54 Additional Tasks Find the break frequencies (Norwegian: knekkfrekvenser ) Set, e.g., K=1. Discuss the results [End of Task] Task 14: Derivator The transfer function for a Derivator is as follows: ( ) Where is the gain Tasks: Find the mathematical expressions for ( ) and ( ). Use Pen & Paper for this Assignment.

55 55 Additional Tasks ( ) ( ) ( ) ( ) Plot the Bode plots Solution: Find the break frequencies (Norwegian: knekkfrekvenser ) Set, e.g., K=1 Discuss the results [End of Task] Task 15: Zero part The transfer function for a Zero part (Norwegian: Nullpunktsledd ) system is as follows: ( ) ( ) Where is the gain T is the Time constant

56 56 Additional Tasks Tasks: Find the mathematical expressions for ( ) and ( ). Use Pen & Paper for this Assignment. Gain: ( ) ( ) ( ) or in db: ( ) ( ) Phase: ( ) ( ) ( ) Plot the Bode plots Find the break frequencies (Norwegian: knekkfrekvenser ) Set, e.g., K=1 and T=1

57 57 Additional Tasks Discuss the results [End of Task] Task 16: Frequency Response We have the following transfer function: ( ) ( ) ( )( ) What is the break frequencies (Norwegian: knekkfrekvenser )? Find poles and zeroes. Use the poles and zero functions in MathScript. Set up the mathematical expressions for ( ) and ( ). Use Pen & Paper for this Assignment. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Plot the frequency response of the system in a bode plot using the bode function in MathScript. Discuss the results.

58 58 Additional Tasks Find ( ) and ( ) for some given frequencies using MathScript code (use the bode function). Bode Plot:

59 59 Additional Tasks Find ( ) and ( ) for the same frequencies above using the mathematical expressions for ( ) and ( ). Tip: use a For Loop or define a vector w=*0.01, 0.1, +. Same procedure as previous Task. [End of Task] Task 17: Frequency Response We have the following transfer function: ( ) ( ) ( ) ( ) ( )( ) Plot the frequency response of the system in a bode plot using the bode function in MathScript. Discuss the results. Set Add a title for the plot using the title function in MathScript. Set up the mathematical expressions for ( ) and ( ). Use Pen & Paper for this Assignment. Find the break frequencies? [End of Task]

60 60 Additional Tasks Bode Plot: Task 18: Bode Diagram Given the following system: ( ) Plot the Bode diagram. Is it possible? Discuss the results. (Tip: The system is unstable)

61 61 Additional Tasks [End of Task] The system is unstable and Frequency Response gives meaning only for stable systems. Note! The frequency response of a system is defined as the steady-state response of the system to a sinusoidal input signal. The Bode diagram for unstable systems don t show what happens with the sinusoidal signal of a given frequency when the system input is transferred through the system because it never reach steady state. We see that the system is unstable because some of the coefficients in the denominator polynomial are negative. We confirm this by some simulations and finding the poles for the system: poles(h) pzgraph(h) This gives: We see the poles are complex conjugate and that they lies in the right half-plane. We plot the step response for the transfer function using the step function:

62 62 Additional Tasks num=[1,1]; den=[1,-1,3]; H=tf(num,den); t=[0:0.01:10]; step(h,t); This gives the following plot: We see the system is unstable Task 19: PID Controller Plot the Bode diagram for a P, PI, PD and a PID controller using the pid and bode functions. ( ) Set up the mathematical expressions for these controllers. Use Pen & Paper for this Assignment. Discuss the results. [End of Task]

63 63 Additional Tasks Plots: P controller PI controller

64 64 Additional Tasks PD controller PID controller Task 20: Stability Analysis Given the following transfer function: Set ( ) ( ) ( ) ( ) ( )( ) Find the crossover-frequencies for the system using MathScript. Find also the gain margins and phase margins for the system. Plot a Bode diagram where the crossover-frequencies, GM and PM are illustrated. Tip! Use the margin function in MathScript. [End of Task] Solution:

65 65 Additional Tasks Bode Plot: The phase never crosses 180 degrees. Task 21: Crossover-frequencies Given the following system: ( ) Find the crossover-frequencies for the system using MathScript. Find also the gain margins and phase margins for the system. Is it possible? (Tip: The system is unstable) [End of Task] The system is unstable and Frequency Response gives meaning only for stable systems We see that the system is unstable because some of the coefficients in the denominator polynomial are negative. We plot the step response for the transfer function using the step function: num=[1,1]; den=[1,-1,3];

66 66 Additional Tasks H=tf(num,den); t=[0:0.01:10]; step(h,t); We see the system is unstable We also find the poles for the system: poles(h) pzgraph(h)

67 67 Additional Tasks

68 Telemark University College Faculty of Technology Kjølnes Ring 56 N-3914 Porsgrunn, Norway Hans-Petter Halvorsen, M.Sc. Telemark University College Department of Electrical Engineering, Information Technology and Cybernetics Phone: Blog: Room: B-237a