1. MATLAB BASICS ECEN 605 Linear Control Systems Instructor: S.P. Bhattacharyya

2. STARTING MATLAB You can start MATLAB by double-clicking on the MATLAB icon or invoking the application from the Start menu of Windows. The main MATLAB window will then pop-up.

4. BASIC OPERATION MATLAB is based on matrix and vector algebra; even scalars are treated as 1x1 matrices. e.g. v = [1 3 5 7]; t = 0:.1:10; k = 0:10; M = [1 2 4; 3 6 8];

5. APPROACH TRY EXPLORING ALL THE COMMANDS INITIALLY TO GET FAMILIARIZED. TAKE UP EXAMPLES AND SOLVE THEM TRY THE ASSIGNMENT QUESTIONS

6. SPECIAL MATRICES There are a number of special matrices that can be defined: null matrix: M = [ ]; nxm matrix of zeros: M = zeros(n,m); nxm matrix of ones: M = ones(n,m); nxn identity matrix: M = eye(n);

7. GENERAL INFORMATION Matlab is case sensitive so "a" and "A" are two different names. Comment statements are preceded by a "%". On-line help for MATLAB can be reached by typing help for the full menu or typing help followed by a particular function name. The commands who and whos give the names of the variables that have been defined in the workspace. The command length(x) returns the length of a vector x and size(x) returns the dimension of the matrix x.

8. CREATING FUNCTIONS As example of an M-file that defines a function, create a file in your working directory named yplusx.m that contains the following commands: function z = yplusx(y,x) z = y + x;

9. USING THE FUNCTIONS The following commands typed from within MATLAB demonstrate how this M-file is used: x = 2; y = 3; z = yplusx(y,x)

10. PLOTTING plot xlabel ylabel title grid axis stem subplot

11. EXAMPLE OF A PLOT Z = [0 : pi/50 : 10*pi]; X = exp(-.2.*Z).*cos(Z); Y = exp(-.2.*Z).*sin(Z); plot3(X,Y,Z); grid on; xlabel('x-axis'); ylabel('y-axis'); zlabel('z- axis');

12. Continuous Time System Analysis Transfer Function Representation Time Simulations Frequency Response Plots Control Design State Space Representation

13. Transfer Function Representation Tf2zp Zp2tf Cloop Feedback Parallel series

14. Cont… Transfer functions are defined in MATLAB by storing the coefficients of the numerator and the denominator in vectors. Given a continuous-time transfer function B(s) H(s) = --------- A(s)

15. Cont… Where B(s) = b M s M +b M-1 s M-1 +…+b 0 and A(s) = a N s N +a N-1 s N-1 +…+a 0 Store the coefficients of B(s) and A(s) in the vectors num = [b M b M-1 … b 0 ] den = [a N a N-1 … a 0 ]

16. Example 5s+6 H(s) = ---------------- s 3 +10s 2 +5 num = [5 6]; den = [1 10 0 5]; all coefficients must be included in the vector, even zero coefficients

17. Cont… To find the zeros, poles and gain of a transfer function from the vectors num and den which contain the coefficients of the numerator and denominator polynomials: [z,p,k] = tf2zp (num,den)

18. Example num = [5 6]; den = [1 10 0 5]; [z,p,k] = tf2zp (num,den) z = -1.2000 p = -10.0495 0.0248+0.7049i 0.0248-0.7049i k = 5

19. Example Cont… (s-z 1 )(s-z 2 )...(s-z n ) H(s) = K -------------------------- (s-p 1 )(s-p 2 )...(s-p n ) 5*(s+1.2) ---------------------------------------------------------- (s+10.0495)(s-{0.0248+0.7049i})(s-{0.0248-0.7049i})

20. Cont… To find the numerator and denominator polynomials from z, p, and k: [num,den] = zp2tf (z,p,k) To reduce the general feedback system to a single transfer function: T(s) = G(s)/(1+G(s)H(s)) [numT,denT]=feedback(numG,denG,numH,denH);

21. Cont… For a unity feedback system: [numT,denT] = cloop(numG,denG,-1); To reduce the series system to a single transfer function, T(s) = G(s)H(s) [numT,denT] = series(numG,denG,numH,denH); To reduce the parallel system to a single transfer function, T(s) = G(s) + H(s) [numT,denT] = parallel(numG,denG,numH,denH);

22. Example [num,den] = zp2tf (z,p,k) num = 0 0 5 6 den = 1.0000 10.0000 0.0000 5.0000

23. Time Simulations residue Step Impulse lsim

24. Cont… [R,P,K] = residue (B,A) finds the residues, poles and direct term of a partial fraction expansion of the ratio of two polynomials B(s)/A(s) The residues are stored in r, the corresponding poles are stored in p, and the gain is stored in k.

25. Example If the ratio of two polynomials is expressed as b(s) 5s 3 +3s 2 -2s+7 ------- = ------------------- a(s) -4s 3 +8s+3 b = [ 5 3 -2 7] a = [-4 0 8 3]

26. Example Cont… r = -1.4167 -0.6653 1.3320 p = 1.5737 -1.1644 -0.4093 k = -1.2500 B(s) R(1) R(2) R(n) ------ = -------- + --------- +... + --------- + K(s) A(s) s - P(1) s - P(2) s - P(n)

27. Find the response of a system to a particular input First store the numerator and denominator of the transfer function in num and den, respectively. To plot the step response: step(num,den) To plot the impulse response: impulse(num,den)

28. Cont… For the response to an arbitrary input, use the command lsim (linear simulation) Create a vector t which contains the time values in seconds t = a:b:c; Define the input x as a function of time, for example, a ramp is defined as x = t lsim(num,den,x,t);

29. Frequency Response Plots Freqs Bode Logspace Log10 Semilogx

30. Cont… To compute the frequency response H of a transfer function, store the numerator and denominator of the transfer function in the vectors num and den. Define a vector w that contains the frequencies for which H) is to be computed, for example w = a:b:c where a is the lowest frequency, c is the highest frequency and b is the increment in frequency. H = freqs(num,den,w)

31. Cont… To draw a Bode plot of a transfer function which has been stored in the vectors num and den: bode(num,den)

32. Cont… To customize the plot, first define the vector w which contains the frequencies at which the Bode plot will be calculated. Since w should be defined on a log scale, the command logspace is used. For example, to make a Bode plot ranging in frequencies from 0.1 to 100, define w by w = logspace(-1,2); The magnitude and phase information for the Bode plot can then be found by: [mag,phase] = bode(num,den,w);

33. Cont… To plot the magnitude in decibels, convert mag using the following command: magdb = 20*log10(mag); To plot the results on a semilog scale where the y- axis is linear and the x-axis is logarithmic: semilogx(w,magdb) For the log-magnitude plot : semilogx(w,phase)

34. Control Design Rlocus Consider a feedback loop where G(s)H(s) = KP(s) and K is a gain and P(s) contains the poles and zeros of the controller and of the plant. Suppose that the numerator and denominator coefficients of P(s) are stored in the vectors num and den. rlocus(num,den)

35. State Space Representation Ss Step Lsim Ss2tf Tf2ss ss2ss