# MIT rule VS Lyapunov design - Adaptive Control

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.

1. Assume that we have a open loop state space model:

$$\dot{x} = Ax + Bx$$

1. We want to design our control law $L$ at

$$u = -L\hat{x} + r$$

Here we assume that $r$ is our reference variable.

1. 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.

1. $P$ is the solution to our Lyapunov function

$$PA_m + A_m^TP = -Q$$

Where $$A_m = A-BL^T$$

1. The whole closed loop feedback adaptive system is:

$$\dot{x} = (A-BL^T)x - Bx^T\tilde{L}$$

1. 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:

1. What's the difference between MIT rule and Lyapunov based adaptive control? Benefits for example.
• Both of these are really old versions of AC. Not all AC necessarily performs sys id per se either. Some schemes use instantaneous optimization to compute a control output and never worry about sys id. – SZN Jan 14 '18 at 14:56
• So you mean that lyapunov AC and MIT AC are obsolete? – Daniel Mårtensson Jan 14 '18 at 15:42
• "obsolete" is not really the right way to think about it. PID is one of the oldest control algorithms and is still used in 98% of controllers. it's just that there are newer algorithms which you should also look at. As always there's no one "best" algorithm for all cases. – SZN Jan 14 '18 at 16:01
• Can you recommend a modern AC algorithm for me? – Daniel Mårtensson Jan 14 '18 at 16:04
• I trying to control robots with my Arduino. Self going robot which going to stand against disturbances and can be in different environments. Is autotuned PID an adaptive PID or is it only autotuned once? – Daniel Mårtensson Jan 14 '18 at 16:07

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.