Kernel Regression

Chris wrote a very kind article explaining kernel regression. Plus, thank you for referring my blog.

Chris McCormick

Having learned about the application of RBF Networks to classification tasks, I’ve also been digging in to the topics of regression and function approximation using RBFNs. I came across a very helpful blog post by Youngmok Yun on the topic of Gaussian Kernel Regression.

Gaussian Kernel Regression is a regression technique which interestingly does not require any iterative learning (such as gradient descent in linear regression).

I think of regression as simply fitting a line to a scatter plot. In Andrew Ng’s machine learning course on Coursera, he uses the example of predicting a home’s sale value based on its square footage.

Note that the data points don’t really lie on the line. Regression allows for the fact that there are other variables or noise in the data. For example, there are many other factors in the sale price of a home besides just the square footage.

Gaussian Kernel Regression…

View original post 979 more words

HW equation test

 

1. Analysis of the system characteristics regarding observability, controllability, and unforced motion.
   

Motivation: Before building a controller and an observer, we need to analyze the system whether it is observable, controllable, and its unforced motion. Based on the analysis, we can built the best control system.

1. Equilibrium points
   

    

   We can find the equilibrium points satisfying the below equation.
   

    

   
   

    

   The equilibrium points are
   

    

   , where is any real number
   

    

2. Study of observability
   

    

   Observability can be calculated by matrix
   

   
   

   
   

   If
   

   
   

   rank() = 3
   

   if , or rand(4,1)
   

   
   

   rank() = 4
   

   if u is not zero, and
   

   
   

   rank() = 4
   

    

3. Study of controllability
   

   
   

   If
   

   
   

   Rank = 3,
   

    

   if , or rand(4,1)
   

   
   

   Rank = 4,
   

4. Unforced motion
   

Simulation result is attached on the next page.

 

Discussion: First, I found the equilibrium points satisfying the equation, $f=0$, and found the obsevability matrix and controllability matrix at the equilibrium points. The result shows that the system is not observable and not controllable at the equilibrium point. Except the equilibrium the system is observable and controllable. One interesting point is that the system can be observable even at the equilibrium points with control input. This is a unique characteristic which we cannot find in the linear system. Simulation result shows that once the system start from the equilibrium points, the system is completely stable. If the initial state is different with the equilibrium points, x1 (stator current), and x2 (rotor current) converged into the equilibrium points, and once x1 and x2 converged, then x3 (angular velocity) becomes also stable. Because x4 indicates the angular position it becomes larger or smaller continuously.

 

1. Normal form, reduced order manifold (ROM), and zero dynamics.
   

Motivation: Because the system does not have a typical controllability canonical form, I need to transform the system equation into a “normal form” to use a feedback-linearization technique. Feedback linearization is performed when the control input is discovered while differentiating . If the control input is discovered at the -th order differentiation, dimensions are not controllable by the feedback linearization technique. Therefore, the dynamics of the remaining part, zero dynamics, should be validated whether it is stable or not.

,     

,

 

If , , stable.

 

Discussion: The system has the three relative orders, and one remaining order. x2 is not controllable state, and fortunately it converges into zero.

 

1. Discussion about the paper
2. Reasons of rejection
3.                                                     i.     Novelty
4.                                                    ii.     Application
5.                                                  iii.     Technical treatment
6.                                                  iv.     Reviewers
7.                                                    v.     Tendency of JoB
8. What would be the best next step?
9.                                                     i.     Resubmission?
10.                                                    ii.     Other type submission? Such a short communication
11.                                                  iii.     No more JoB
12. Possible applications with the database?
13. Gait pattern prediction
14.                                                     i.     Exploring latent variables in human body (reduced order manifold) for the gait pattern prediction
15. Proof of biomechanics theories,
16.                                                     i.     Jerk minimization theory
17. Others?

Meeting with PhD. Dingwell

1. Discussion about the paper
   

Test it includes equations and figures

 

1. Analysis of the system characteristics regarding observability, controllability, and unforced motion.
   

Motivation: Before building a controller and an observer, we need to analyze the system whether it is observable, controllable, and its unforced motion. Based on the analysis, we can built the best control system.

1. Equilibrium points
   

    

   We can find the equilibrium points satisfying the below equation.
   

    

   
   

    

   The equilibrium points are
   

    

   , where is any real number
   

    

2. Study of observability
   

    

   Observability can be calculated by matrix
   

   
   

   
   

   If
   

   
   

   rank() = 3
   

   if , or rand(4,1)
   

   
   

   rank() = 4
   

   if u is not zero, and
   

   
   

   rank() = 4
   

    

3. Study of controllability
   

   
   

   If
   

   
   

   Rank = 3,
   

    

   if , or rand(4,1)
   

   
   

   Rank = 4,
   

4. Unforced motion
   

Simulation result is attached on the next page.

 

Discussion: First, I found the equilibrium points satisfying the equation, $f=0$, and found the obsevability matrix and controllability matrix at the equilibrium points. The result shows that the system is not observable and not controllable at the equilibrium point. Except the equilibrium the system is observable and controllable. One interesting point is that the system can be observable even at the equilibrium points with control input. This is a unique characteristic which we cannot find in the linear system. Simulation result shows that once the system start from the equilibrium points, the system is completely stable. If the initial state is different with the equilibrium points, x1 (stator current), and x2 (rotor current) converged into the equilibrium points, and once x1 and x2 converged, then x3 (angular velocity) becomes also stable. Because x4 indicates the angular position it becomes larger or smaller continuously.

 

1. Normal form, reduced order manifold (ROM), and zero dynamics.
   

Motivation: Because the system does not have a typical controllability canonical form, I need to transform the system equation into a “normal form” to use a feedback-linearization technique. Feedback linearization is performed when the control input is discovered while differentiating . If the control input is discovered at the -th order differentiation, dimensions are not controllable by the feedback linearization technique. Therefore, the dynamics of the remaining part, zero dynamics, should be validated whether it is stable or not.

,     

,

 

If , , stable.

 

Discussion: The system has the three relative orders, and one remaining order. x2 is not controllable state, and fortunately it converges into zero.

 

1. Full-state feedback controller.
   

    

what is the post

I wnat to know the difference between the post and the page

HW3

1. Analysis of the system characteristics regarding observability, controllability, and unforced motion.
   

Motivation: Before building a controller and an observer, we need to analyze the system whether it is observable, controllable, and its unforced motion. Based on the analysis, we can built the best control system.

1. Equilibrium points
   

   We can find the equilibrium points satisfying the below equation.
   

   
   

   The equilibrium points are
   

   , where is any real number
   

2. Study of observability
   

   Observability can be calculated by matrix
   

   
   

   
   

   If
   

   
   

   rank() = 3
   

   if , or rand(4,1)
   

   
   

   rank() = 4
   

   if u is not zero, and
   

   
   

   rank() = 4
   

3. Study of controllability
   

   
   

   If
   

   
   

   Rank = 3,
   

   if , or rand(4,1)
   

   
   

   Rank = 4,
   

4. Unforced motion
   

Simulation result is attached on the next page.

Discussion: First, I found the equilibrium points satisfying the equation, $f=0$, and found the obsevability matrix and controllability matrix at the equilibrium points. The result shows that the system is not observable and not controllable at the equilibrium point. Except the equilibrium the system is observable and controllable. One interesting point is that the system can be observable even at the equilibrium points with control input. This is a unique characteristic which we cannot find in the linear system. Simulation result shows that once the system start from the equilibrium points, the system is completely stable. If the initial state is different with the equilibrium points, x1 (stator current), and x2 (rotor current) converged into the equilibrium points, and once x1 and x2 converged, then x3 (angular velocity) becomes also stable. Because x4 indicates the angular position it becomes larger or smaller continuously.

1. Normal form, reduced order manifold (ROM), and zero dynamics.
   

Motivation: Because the system does not have a typical controllability canonical form, I need to transform the system equation into a “normal form” to use a feedback-linearization technique. Feedback linearization is performed when the control input is discovered while differentiating . If the control input is discovered at the -th order differentiation, dimensions are not controllable by the feedback linearization technique. Therefore, the dynamics of the remaining part, zero dynamics, should be validated whether it is stable or not.

,     

,

If , , stable.

Discussion: The system has the three relative order, and one remaining order. x2 is not controllable state, and fortunately it converges into zero.