Input-output representation to state space

Question

How do you create a state space representation of the following input/output equation:

$$ \dot{y} = -ky + u + a\sin(ωt) $$

where the parameters $k$, $a$, and $ω$ are unknown.

The end goal is to create a model reference adaptive controller (MRAC) using only the measurements of the input and output of the system. I believe the first step of this process is to create a state space model for the system from the equation given above. Without the addition of $a\sin(ωt)$ this would be a simple problem but that additional sine term is throwing off my calculation.

Answer 1

You can switch to the frequency domain to look at the transfer function of the system. For you particular case, taking the Laplace transform of both sides gives

$$ sY(s) = -kY(s) + U(s) + \frac{a\omega^2}{s^2+\omega^2}, $$ or

$$ Y(s) = G(s) U(s) + \Delta(s) = \frac{1}{s+k}U(s) + \frac{a\omega^2}{(s^2+\omega^2)(s+k)}. $$

To me, this looks like a linear system with some output disturbance. To get the state space from $G(s)$, note that the transfer function of a linear SISO system is given as

$$ G(s) = c(s - a)^{-1}b + d, $$ so one possible realization is

$$ \dot{x} = -kx + u, \qquad \bar{y} = x, $$ and it follows that the true output $y = \bar{y} + a\sin(\omega t)$. A block diagram of the system is shown below. From there, you can implement any particular controller you're trying to design (either open loop or closed loop).