I wonder what's the difference between MIT rule and Lyapunov design when it comes to adaptive control law?
As I get it, MIT rule is more like "transfer function"-based control system and Lyapunov design is more like "state space model"-based control system.
Before I asking my complete question, I just want to tell you that Adaptive system combines control law + system identification. You can build a very easy adaptive control system, or a much more difficult if you want. All depends on what you want to control.
I can write out the methods to do a adaptive control system with Lyapunov design.
- Assume that we have a open loop state space model:
$$\dot{x} = Ax + Bx$$
- We want to design our control law $L$ at
$$ u = -L\hat{x} + r$$
Here we assume that $r$ is our reference variable.
- One simply thing we can to is to create our control law by doing
$$\dot{\hat{x}} = S^{-1}xB^TPx, S = S^T > 0 $$ $$ u = -L\hat{x} + r$$
Where $S$ is a matrix. I don't know if the matrix $S$ need to be identical or something. Please correct me if I'm wrong.
- $P$ is the solution to our Lyapunov function
$$PA_m + A_m^TP = -Q$$
Where $$A_m = A-BL^T$$
- The whole closed loop feedback adaptive system is:
$$ \dot{x} = (A-BL^T)x - Bx^T\tilde{L}$$
To prove the stability we using this:
$$V(x, \tilde{L}) = \frac{1}{2}x^TPx + \frac{1}{2}\tilde{L}^TS\tilde{L}$$
I don't know where $\tilde{L}$ comes from, but's is in the professor's lecture notes and books. I assume that $\tilde{L} = L - \hat{L}$. Not sure.
Anyway! My question are:
- What's the difference between MIT rule and Lyapunov based adaptive control? Benefits for example.
The main advantage of Lyapunov design is that it guarantees a closed-loop system.
The main drawback of Lyapunov design is that there is no systematic way of finding a suitable Lyapunov function $V$ leading to a specific adaptive law.
For example, if one wants to add a proportional term to the adaptive law, it is not trivial to find the corresponding Lyapunov function.