Definition

Suppose a fixed, linear, analog system,

𝐱˙(t)=A𝐱(t)+B𝐮(t)𝐲(t)=C𝐱(t)+D𝐮(t) \begin{aligned} \dot{\mathbf{x}}(t) & = A \mathbf{x}(t) + B \mathbf{u}(t) \\ \mathbf{y}(t) & = C \mathbf{x}(t) + D \mathbf{u}(t) \end{aligned}

take Laplace transform (ℒ\mathscr{L}-transform) of both sides

ℒ{𝐱˙(t)}=Aℒ{𝐱(t)}+Bℒ{𝐮(t)}ℒ{𝐲(t)}=Cℒ{𝐱(t)}+Dℒ{𝐮(t)}\begin{aligned} \mathscr{L}\{\dot{\mathbf{x}}(t)\} & = A \mathscr{L}\{\mathbf{x}(t)\} + B \mathscr{L}\{\mathbf{u}(t)\} \\ \mathscr{L}\{\mathbf{y}(t)\} & = C \mathscr{L}\{\mathbf{x}(t)\} + D \mathscr{L}\{\mathbf{u}(t)\} \end{aligned}

then, let ℒ{𝐮(t)}=𝐔(s)\mathscr{L}\{\mathbf{u}(t)\} = \mathbf{U}(s), ℒ{𝐲(t)}=𝐘(s)\mathscr{L}\{\mathbf{y}(t)\} = \mathbf{Y}(s), ℒ{𝐱(t)}=𝐗(s)\mathscr{L}\{\mathbf{x}(t)\} = \mathbf{X}(s)

using Laplace differentiation theorem, where ℒ{𝐱˙(t)}=s𝐗(s)−𝐱(0)\mathscr{L}\{\dot{\mathbf{x}}(t)\} = s \mathbf{X}(s) - \mathbf{x}(0),

s𝐗(s)−𝐱(0)=A𝐗(s)+B𝐔(s)𝐘(s)=C𝐗(s)+D𝐔(s)\begin{aligned} s \mathbf{X}(s) - \mathbf{x}(0) & = A \mathbf{X}(s) + B \mathbf{U}(s) \\ \mathbf{Y}(s) & = C \mathbf{X}(s) + D \mathbf{U}(s) \end{aligned}

#incomplete

Then

[C(sI−A)−1B+D]=H(s)[C (sI - A)^{-1} B + D] = H(s)

ℒ\mathscr{L}-transform of convolution integral gives

𝐘(s)=ℒ{H(t)}𝐔(s)\mathbf{Y}(s) = \mathscr{L}\{H(t)\}\mathbf{U}(s)

where ℒ{H(t)}=H(s)\mathscr{L}\{H(t)\} = H(s).

See also

References

  1. P. E. Sarachik, Principles of Linear Systems, Cambridge Press, 1996, pp. 101-102, 125.