What is the frequency response of a first-order system of ODEs?What is the frequency response of a first-order system of ODEs? Specifically, given the differential equation:y′(t)=Ay(t)+x(t)where y:&nbsp;y:[0,∞]→Rm&nbsp;and&nbsp;A∈Rm×m, what is the solution when&nbsp;x(t)=0⋮cos(ωt)⋮0=cos(ωt)bi?Essentially, if we have a first-order system of ODEs and we activate a single input with a sine wave of a given frequency, what is the solution? Normally, the frequency response is worked out for a linear time invariant system of higher order and the solution is just a phase shifted and amplitude scaled version of the sine wave. I'm interested in what this result looks like for a first-order system rather than a single equation.

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What is the frequency response of a first-order system of ODEs?What is the frequency response of a first-order system of ODEs? Specifically, given the differential equation:y′(t)=Ay(t)+x(t)where y:&amp;nbsp;y:[0,∞]→Rm&amp;nbsp;and&amp;nbsp;A∈Rm×m, what is the solution when&amp;nbsp;x(t)=0⋮cos(ωt)⋮0=cos(ωt)bi?Essentially, if we have a first-order system of ODEs and we activate a single input with a sine wave of a given frequency, what is the solution? Normally, the frequency response is worked out for a linear time invariant system of higher order and the solution is just a phase shifted and amplitude scaled version of the sine wave. I&#039;m interested in what this result looks like for a first-order system rather than a single equation.

PietAppeteoth22m · Accepted Answer

The solution you are probably looking for has the formy(t)=ucos⁡(ωt)+vsin⁡(ωt).Insert into the DE to get the linear system for the vectors u,v. Take care of what happens if iω is an eigenvalue of A.

nida0694ii5 · Accepted Answer

Step 1I use Y(s) and X(s) to denote the Laplace transform L{y(t)}(s) and L{y(t)}(s) respectively. Furthermore, it is worth nothing that for&amp;nbsp;A(s)=L{a(t)}(s)&amp;nbsp;it hold that&amp;nbsp;L{a′(t)}(s)=s,A(s)−a(0), with a&#039;(t) denoting the derivative of a(t) with respect to time (t). Combing this with your original system of first order ODEs yieldss,Y(s)−y(0)=A,Y(s)+X(s).&amp;nbsp;&amp;nbsp;(1)Solving (1) for Y(s) yields&amp;nbsp;Y(s)=(s I-A)-1(X(s)+y(0)).&amp;nbsp;(2)Often for frequency response analysis it is assumed that&amp;nbsp;y(0)=0. And for the steady state frequency response one can substitute in&amp;nbsp;s=j,ωStep 2This is also related to converting state space models into transfer functions. So when starting with the following state space model                                        x            ′                    (          t          )                                    =          A                    x          (          t          )          +          B                    u          (          t          )                                      y          (          t          )                                    =          C                    x          (          t          )          +          D                    u          (          t          )                    then the corresponding transfer function can be obtained using&amp;nbsp;G(s)=C,(s,I−A)−1B+D.

What is the frequency response of a first-order

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