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MATLAB BASICS

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

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MATLAB BASICS

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  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.

  3. 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];

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

  5. 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);

  6. 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.

  7. 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;

  8. 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)

  9. PLOTTING • plot • xlabel • ylabel • title • grid • axis • stem • subplot

  10. 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');

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

  12. Transfer Function Representation • Tf2zp • Zp2tf • Cloop • Feedback • Parallel • series

  13. 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)

  14. Cont… • Where B(s) = bMsM+bM-1sM-1+…+b0 and A(s) = aNsN+aN-1sN-1+…+a0 Store the coefficients of B(s) and A(s) in the vectors num = [bM bM-1 … b0] den = [aN aN-1 … a0]

  15. Example 5s+6 H(s) = ---------------- s3+10s2+5 • num = [5 6]; • den = [1 10 0 5]; • all coefficients must be included in the vector, even zero coefficients

  16. 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)

  17. 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

  18. Example Cont… (s-z1)(s-z2)...(s-zn) H(s) = K -------------------------- (s-p1)(s-p2)...(s-pn) 5*(s+1.2) ---------------------------------------------------------- (s+10.0495)(s-{0.0248+0.7049i})(s-{0.0248-0.7049i})

  19. 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);

  20. 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);

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

  22. Time Simulations • residue • Step • Impulse • lsim

  23. 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.

  24. Example • If the ratio of two polynomials is expressed as b(s) 5s3+3s2-2s+7 ------- = ------------------- a(s) -4s3+8s+3 • b = [ 5 3 -2 7] • a = [-4 0 8 3]

  25. 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)

  26. 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)

  27. 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);

  28. Frequency Response Plots • Freqs • Bode • Logspace • Log10 • Semilogx

  29. 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)

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

  31. 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);

  32. 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)

  33. 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)

  34. State Space Representation • Ss • Step • Lsim • Ss2tf • Tf2ss • ss2ss

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