Tartalomjegyzék

Computer controlled systems (2020)

Control Tutorials for Matlab & Simulink (University of Michigan)

Lecture notes of the seminar

Seminar 1. - Laplace transformation, non-dimensional ODEs

Important functions of Matlab's Symbolic Math Toolbox (SMT):

• syms a x y z real; syms f(x) g(x,y,z) - Declare (create) multiple scalar symbolic variables or functions
• A = sym('a_%d%d',[3 4]) - Create a single symbolic matrix, vector or scalar variable
• simplify - Simplify symbolic expression
• simplifyFraction - Simplify fraction of polynomials expression to its irreducible form
• f_sym = x^2+y*z; symvars(f_sym) - List symbolic variables appearing in the object passed as an argument
• subs(f_sym, [x z], [x+y x^2]) - Substitute variables or expression by other variables or expression in a symbolic expression
• f_fh = matlabFunction(f_sym, 'vars', {x [y z]}) - Generate a function handle from a symbolic expression
• charpoly - Compute the characteristic polynomial of a matrix
• partfrac - Partial fractional decomposition of a rational function (useful when producing inverse Laplace transformation of a transfer function). FactorMode: real/complex/full/rational. See also its numerical pair: residue.
• laplace
• ilaplace

Matlab examples

1. Compute a polynomial division (see lecture notes nr. 3).
2. Compute the partial fractional decomposition of $H(s) = \frac{5s^2+3s+1}{s^3+6s^2+11s+6}$.

   syms s
   H = (5*s^2 + 3*s + 1)/(s^3 + 6*s^2 + 11*s + 6);
   partfrac(H)

   Output:

   ans = 3/(2*(s + 1)) - 15/(s + 2) + 37/(2*(s + 3))

   $H(s) = \frac{3}{2(s+1)}-\frac{15}{s+2}+\frac{37}{2(s+3)}$

3. Compute the Laplace transformation of $h(t) = \cos(at)+\sin(at)+e^{-ct}\cos(bt)+e^{-ft}\sin(et)$.

   syms t a b c e f
   h = cos(a*t) + sin(a*t) + exp(-c*t)*cos(b*t) + exp(-f*t)*sin(e*t)
   laplace(h,t)

   Output:

   ans = a/(a^2 + t^2) + (c + t)/((c + t)^2 + b^2) + e/((f + t)^2 + e^2)

   $H(s) = \frac{a}{a^2+t^2}+\frac{t}{a^2+t^2}+\frac{c+t}{{(c+t)}^2+b^2}+\frac{e}{{(f+t)}^2+e^2}$

4. Compute the inverse Laplave transformation of $H(s) = \frac{1}{s^2 + s + 1}$.

   syms s
   ilaplace(1/(s^2+s+1))

   Output:

   $h(t) = \frac{2\sqrt{3}{e}^{-\frac{t}{2}}\sin(\frac{\sqrt{3}t}{2})}{3}$