In this post, the focus shifts from looking at the basic properties of the Laplace Transform to the application of it to dynamical systems of the form<\/p>\n
\\[ \\frac{d}{dt} \\bar S = A \\bar S \\; ,\\]<\/p>\n
where the matrix $$A$$ can be time-varying but does not depend on the state $$\\bar S$$.<\/p>\n
To be sure, some differential equations were already touched upon in passing in earlier posts, but this was done more as a way to motivate the possible transforms that would result and which would require a subsequent inverse rather than being examined in their own right. <\/p>\n
As a prototype for multi-variable dynamical systems and a prelude to the general theory, this column will look at the application of the Laplace Transform in the simplest multi-variable structure – the simple harmonic oscillator (SHO). This tried and true system should always be the first one explored based on one simple fact – if the theory is not understandable or workable for this fundamental system then it won’t be understandable or workable for any other application.<\/p>\n
The familiar state for the SHO is given by:<\/p>\n
\\[ \\bar S = \\left[ \\begin{array}{c} x(t) \\\\ v(t) \\end{array} \\right] \\; \\] <\/p>\n
and its corresponding equation of motion <\/p>\n
\\[ \\frac{d}{dt} \\left[ \\begin{array}{c} x \\\\ v \\end{array} \\right] = \\left[ \\begin{array}{cc} 0 & 1 \\\\ -\\omega_0^2 & 0 \\end{array} \\right] \\left[ \\begin{array}{c} x \\\\ v \\end{array} \\right] \\; ,\\]<\/p>\n
where the matrix $$A$$ is given by<\/p>\n
\\[ A = \\left[ \\begin{array}{cc} 0 & 1 \\\\ -\\omega_0^2 & 0 \\end{array} \\right] \\; .\\]<\/p>\n
Taking the Laplace Transform of the oscillator’s equation of motion (using the properties established in earlier posts) yields<\/p>\n
\\[ \\left[ \\begin{array}{c} s X(s) – x_0 \\\\ s V(s) – v_0 \\end{array} \\right] = \\left[ \\begin{array}{cc} 0 & 1 \\\\ -\\omega_0^2 & 0 \\end{array} \\right] \\left[ \\begin{array}{c} X(s) \\\\ V(s) \\end{array} \\right] \\; .\\]<\/p>\n
Note that, as expected, the differential equation now becomes a (linear) algebraic equation. <\/p>\n
Some basic re-arrangement leads to the following matrix equation <\/p>\n
\\[ (s Id – A) \\left[ \\begin{array}{c} X(s) \\\\ V(s) \\end{array} \\right] = \\left[ \\begin{array}{c} x_0 \\\\ v_0 \\end{array} \\right] \\; ,\\]<\/p>\n
where $$Id$$ is the $$2×2$$ identity matrix. <\/p>\n
The solution to this equation is obtained by finding the inverse of the matrix $$s Id – A$$.<\/p>\n
\\[ \\left[ \\begin{array}{c} X(s) \\\\ V(s) \\end{array} \\right] = (s Id – A)^{-1} \\left[ \\begin{array}{c} x_0 \\\\ v_0 \\end{array} \\right] \\; .\\]<\/p>\n
It is convenient to define<\/p>\n
\\[ M = s Id – A = \\left[ \\begin{array}{cc} s & -1 \\\\ \\omega_0^2 & s \\end{array} \\right] \\; .\\]<\/p>\n
The inverse then follows from the usual formula for a $$2×2$$ matrix<\/p>\n
\\[ M^{-1} = \\frac{1}{s^2 + \\omega_0^2} \\left[ \\begin{array}{cc} s & 1 \\\\ -\\omega_0^2 & s \\end{array} \\right] \\; ,\\]<\/p>\n
where the pre-factor is $$1\/det(M)$$.<\/p>\n
Multiply the right-hand side out leads to the following expressions for $$X(s)$$<\/p>\n
\\[ X(s) = \\frac{s x_0}{s^2 + \\omega_0^2} + \\frac{v_0}{s^2 + \\omega_0^2} \\; \\]<\/p>\n
and $$V(s)$$<\/p>\n
\\[ V(s) = -\\frac{x_0 \\omega_0^2 }{s^2 + \\omega_0^2} + \\frac{s v_0}{s^2 + \\omega_0^2} \\; .\\]<\/p>\n
The last piece is to take the inverse Laplace Transform of the individual components to get <\/p>\n
\\[ x(t) = x_0 \\cos(\\omega_0 t) + \\frac{v_0}{\\omega_0} \\sin(\\omega t) \\; \\]<\/p>\n
and<\/p>\n
\\[ v(t) = -x_0 \\omega_0 \\sin(\\omega_0 t) + v_0 \\cos(\\omega t) \\; ,\\]<\/p>\n
which, happily, are the expected results.<\/p>\n
Next week, I’ll begin my examination of the general theory use in controls engineering.<\/p>\n","protected":false},"excerpt":{"rendered":"