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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.
euraad
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    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 '18 at 14:56
  • So you mean that lyapunov AC and MIT AC are obsolete? – euraad Jan 14 '18 at 15:42
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    "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 '18 at 16:01
  • Can you recommend a modern AC algorithm for me? – euraad Jan 14 '18 at 16:04
  • What are you trying to control? Most industrial AC is autotuned PID (also very old). – JMJ Jan 14 '18 at 16:05
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    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? – euraad Jan 14 '18 at 16:07
  • Autotuned PID is adaptive PID. I'm not an expert in robotics, but I have many friends who are. AC is mostly used when a model is unreliable or unavailable, or if model parameters change in an unpredictable way (e.g. industrial plants which use destructive processes or where actuators degrade quickly). In robotics models are more reliable, so I think most people use model predictive control or some variant. – JMJ Jan 14 '18 at 16:13
  • @SZN does AC works well according to you? – euraad Jan 14 '18 at 18:22
  • For robots? It depends on how reliable of a model you can create. If you have a very reliable model, then AC is not useful, since the controller will not need to adapt to changing model parameters. – JMJ Jan 14 '18 at 20:08
  • @SZN Let's say I have a moving robot in different environments? What do you prefer, MIT rule or Lyapunov design? – euraad Jan 14 '18 at 20:37
  • If your problem is navigation in different environments then you might look up some of the robot path planning literature. There are many specialized methods to solve these sort of problems. – JMJ Jan 14 '18 at 21:17

1 Answers1

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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.

euraad
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    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. – Preston Roy Jan 14 '18 at 15:54
  • So MIT is more preferred that Lyapunov AC because it's hard do find the lyapunov function? – euraad Jan 14 '18 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 '18 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. – Preston Roy Jan 14 '18 at 16:25
  • I have heard that lyapunov/MIT is not the future of adaptive control. – euraad Jan 14 '18 at 16:44
  • @PrestonRoy any AC with a sys id block will "learn" in the sense that it will estimate parameter values and response order in real time. I have a friend who did his PhD thesis applying ML to AC. My guess is he did something with ANNs. I have to admit, I don't know anything about how ANN learning so I can't speak to its strengths compared with more basic AC. – JMJ Jan 14 '18 at 20:11
  • Adaptive control requires the nonlinearities to be linear in parameters, then those parameters are learned over time using update techniques. Many systems aren’t linear in parameters (especially in robotics), so traditional adaptive methods won’t work for them. ANN control doesn’t have this requirement, and it basically applies the same parameter update methods as adaptive control. So it can handle a wider variety of systems. The research mostly focuses on alternate parameter update methods and different ANN architectures. – Preston Roy Jan 15 '18 at 00:02
  • So ANN is much better that AC when it comes to robotics ? – euraad Jan 15 '18 at 06:18
  • In my opinion (and the opinion of the research group I am in), however, I’m sure many people would debate that. Of course you can still obtain good results with PI or PID controls, but it requires much more tuning. – Preston Roy Jan 15 '18 at 20:07