This is MRAC – Model Reference Adaptive Control for SISO systems.
$G_m(s)$ is our reference model. It's is a first order system because they don't have overshoot. $G_m(s)$ is a desired wish how then output $y$ should behave.
$\gamma > 0$ is the tuning parameter and $G(s)$ is our unknown plant.
In other words. This is a PI-regulator, where the $P$ is also an integral. But still adaptive. MRAC is the most used adaptive regulators becuase it it's the simpliest of them all. It works for almost all systems. But must used in slow systems that slowly changes over time.
- https://www.youtube.com/watch?v=BIpnjvQ1uhI
- https://www.youtube.com/watch?v=y9VLEOqBmpM
- https://www.youtube.com/watch?v=2GkDGpCSDbU
The formula for MRAC is:
$$u = u_c(t)\int (-\gamma u_c(t) e(t)) – y(t)\int (\gamma y(t) e(t))$$
Question:
If I want to apply this to MIMO systems. How should I change MRAC then?
Best Answer
Although there exist papers on mimo MRAC, the theory hasn't in my opinion progressed very much. That is because MRAC depends on cancelling zeroes, an approach that is questionable already for siso systems, and messier in the mimo case.
I'd say that, until more readily applicable directo adaptive control theory is developed, one would be better off with indirect adaptive control for multivariable systems - combine some type of parameter estimation with certainty-equivalence feedback.
This is just my opinion, if someone knows of successful uses of multivariable MRAC I'd be interested.