# 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.
– JMJ
Jan 14, 2018 at 14:56
• So you mean that lyapunov AC and MIT AC are obsolete? Jan 14, 2018 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.
– JMJ
Jan 14, 2018 at 16:01
• Can you recommend a modern AC algorithm for me? Jan 14, 2018 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? Jan 14, 2018 at 16:07

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.

• This is a good answer. To adress a comment above, Lyapunov is the most commonly used for nonlinear control and is not obsolete. I would also say that Lyapunov design is rigorous. In my field of neural network control we most commonly use Lyapunov methods. Jan 14, 2018 at 15:54
• So MIT is more preferred that Lyapunov AC because it's hard do find the lyapunov function? Jan 14, 2018 at 15:58
• @PrestonRoy of the pittance I know about ANNs it seems to me that Lyapunov methods would be particularly useful for such systems, so I don't doubt you at all on that. Many control theorists worry about mathematical rigor in closed-loop stability or nonlinear analysis. I've rarely encountered such concerns in industry however.
– JMJ
Jan 14, 2018 at 16:13
• ANN can be applied to any system, the idea is that the controller learns over time. Industry often isn’t concerned with this, they want something simple to implement. Which is why most controllers are just PI. Jan 14, 2018 at 16:25
• I have heard that lyapunov/MIT is not the future of adaptive control. Jan 14, 2018 at 16:44