For starters, can you tell the dimensions of each matrix ? It will make it easier to think further.

Assuming that u(t) and c are both of same dimensions. If they are, you can just make it :
der(x) = F*x(t)+G*[u(t) + c] , because u(t) is an input ( usually the one we control ) and c is also an input, only it's an external one. u(t) suggests it's time varying input, but that's just a general case, it doesn't have to be.

Edit after your comment:

Now that you have explained that u(t) is a 1x1 scalar, I see a solution, that may be too simple to work. You said you are using lsim command in Matlab, so I suppose you pass state space (ss) system model to it. In that case, since according to your comment u(t) is a scalar, G*u(t) is a constant 6x1 vector. Therefore, you can make a substition and say [G*u + c] is your new G matrix. For now I assume you are familiar with this part of using Matlab, but just in case you need a kickstart:

Matlab documentation on this website:
LINK
Says you can pass systems to lsim in this fashion :

[y,t] = lsim(sys,u,t)

[y,t,x] = lsim(sys,u,t) % for state-space models only

[y,t,x] = lsim(sys,u,t,x0) % with initial state

Although you are in frequency domain you still should be able to get all parameters as you were in time domain. They are different domains but they both should represent the same thing. Time domain represent things in terms of amplitude in respect to time. Frequency domain represent things in terms of amplitude AND PHASE in respect to frequency values. Note that you should have both amplitude and phase in frequency domain, since in the time domain the phase can be represented in the same plot by a shift.

One way to represent these things in frequency domain is by dealing with complex numbers. Complex numbers can be viewed as vectors in a 2D space which have a length (as you said) and an angle. The length represents the output/input ratio and the angle represent the phase shift in comparison also to the input.

So, answering your question, you should calculate the H length to find your output/input ratio. To help you, imagine that:

\$e^{jw}=cos(w)+jsin(w)\$

In other words, its a complex number with always length of 1 and angle \$w\$

You can solve this by two methods:

-Vector method:

imagine that number 1 is \$Z=1+0.i\$ which is a vector to the right, with length 1 and angle \$0\$.

Imagine that \$e^{jw}\$ is a vector that I showed right above

Now add them. Then divive vectors 1 by the vector that you've found.

-Cartesian Coordinates:

represent all in terms of \$Z=a+jb\$ and also \$e^{jw}=cos(w)+jsin(w)\$

and imagine that you have:

\$\large Z = \frac{Z_1}{Z_2+Z_3}\$

and then find length of Z by:

\$|Z| = \sqrt{a^2+b^2}\$

## Best Answer

The reason why transfer functions work so well for linear time-invariant (LTI) systems (and don't for non-linear systems) is that they are the Laplace transform (or, in discrete time) the Z-transform of the system's

impulse response, which completely characterizes the behavior of such systems. I.e., the impulse response, or, equivalently, the transfer function is all you need to know to compute the response of the system toanyinput signal. The reason for this is that any input signal can be written as an integral (or a sum in discrete time) of scaled and shifted delta impulses, and due to linearity and time invariance, the response to this integral/sum equals the integral/sum of scaled and shifted impulse responses:$$x(t)=\int_{-\infty}^{\infty}x(\tau)\delta(t-\tau)d\tau\quad\Longrightarrow\quad y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau\tag{1}$$

where \$x(t)\$ is the input signal, \$y(t)\$ is the system's response, and \$h(t)\$ is the impulse response. Equation (1) is the convolution integral, which completely describes the system's behavior. In the transform domain (i.e., Laplace transform, Fourier transform, or Z-transform), convolution becomes multiplication.

For non-linear systems you can of course compute or measure the system's response to an impulse, but this function does not tell you anything about the system's response to other input signals, or even to a scaled impulse. This is why the impulse response and its transform do not have any significance for non-linear systems.