# Control engineer’s one-year observation of my baby

As a control engineer, I see different features from my one year old baby, Chloe. When she was first born, she did not have a model for her biomechanical system, and the controller in her brain was immature. Nonetheless, her finger did not vibrate or diverge. The kinematics of her finger is same with that of a three-link pendulum, commonly referred to describe a chaotic dynamic system. She had never tried to make it stable, no, actually she disturbed the system. But, her biomechanical systems were amazingly stable. For the last one year, her control ability has been dramatically improved. Now she can walk and interact with highly uncertain dynamic objects like another baby. It is an extremely difficult control task. Furthermore, in someday, she would be able to drive a car and to manage (control) a business. She has a great system, modeling ability, and controller, three essential elements for control. As a control engineer, I cannot help being surprised and admiring.

Feb 11, 2014

-Mok-

# Sliding Mode Control (SMC), Robust control algorithm for nonlinear system

Sliding Mode Control algorithm is a robust controller for nonlinear systems.

Robust control means that even though the system model has a certain error, if the controller can control the system, we say that the controller is robust.

SMC has two fundamental ideas.

1. To attract the system states to the surface.

2. To make the state slide on the surface toward the origin.

To explain the above two ideas more easily, let’s assume a typical control problem.

$\dot&space;x&space;=&space;f_1&space;(x,\dot&space;x)$   — (1)

$\ddot&space;x&space;=&space;f_2&space;(x,&space;\dot&space;x)&space;+&space;u$

$y=Cx$ — (2) if it is nonlinear, you need to know Lie Derivatives.

To achieve the first idea, we need to define a surface like the below. (Let’s assume that $y$ is a scalar and differentiable.)

$s(x)=a_0&space;e&space;+&space;a_1&space;\dot{e}$  where  $e=y-y_d$ — (3)

Here we need to select a_0 and a_1 to make y->0 as t-> inf,s=0. You will see the reason after more several lines.

Let’s take the derivative of s(x) w.r.t. x. Then we can get the below equation.

$\dot{s}(x)=a_0&space;\dot&space;e&space;+&space;a_1&space;\ddot&space;e$  — (4)

In addition, if we add the additional term -$\eta&space;\text{sign}(s)$ we can obtain the below equation,

$\dot{s}(x)=a_0&space;\dot&space;e&space;+&space;a_1&space;\ddot&space;e=-\eta&space;\text{sign}(s)$ — (5)

Through a simple Lyapunov theorem, we can easily prove s-> 0 at t->inf .

if s=0, e-> 0 and \dot(e)-> 0, because we selected a_0 and a_1 to be like this.

Let’s look at (5),

$a_0&space;\frac{d}{dt}(y-y_d)+a_1&space;\frac{d^2}{dt^2}(y-y_d)=-\eta&space;\text{sign}(s)$

=> $C(a_0&space;\dot&space;x&space;+a_1&space;\ddot&space;x)=-\eta&space;\text{sign}(s)$

=>$C(a_0&space;f_1&space;+a_1&space;(f_2+u)))=-\eta&space;\text{sign}(s)$

=>$u=((-\eta&space;\text{sign}&space;(s))/C&space;-&space;a_0&space;f_1)/a_1&space;-f_2$

With this control input, we can control a nonlinear system with SMC.

it is difficult to explain quickly within short explanations…., if you need more explanation or questions, please leave me a reply.

# Feedback linearization control (FBL) for nonlinear system control proof, practical implementation, and easy example part1

This article explains about Feedback linearization control (FBL) method for control of a nonlinear system. This is one of the easiest strategy to control nonlinear systems, but pretty powerful. The basic idea is that we can cancel control input by manipulating control input. The below is its proof.

This is the most basic concept to explain easily. If the nonlinear system is not fully controllable, we should use another strategy which finds reduced order manifold (ROM). About the more advanced technique and examples will be explained in the next articles.

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I am Youngmok Yun, and writing about robotics theories and my research.

My main site is http://youngmok.com, and Korean ver. is  http://yunyoungmok.tistory.com.

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