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Basic Control Theory

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1. About the Course

what’s in this course

  • control theory

  • mathematical play a important role

  • robot models

  • mobility controllers

  • How to do, not why to do (Not a AI course)

great! I have learned planning in the school so I can focus on the control theory that boost the overall performance

  • application

2. What is control theory

system: something that changes over time

control: influence that change

Building blocks

  • State: $x$ representation of what the system is currently doing
  • Dynamics
  • Reference $r$ what we want the system to do
  • output: $y$ measurement of system
  • input: $u$ control signal
  • Feedback: mapping from output to input

3. Model

model is the approximation the real system.

Why model is important? because we need to design control theory

How we design control system, there are some objectives

  • stability BIBO bounded in bounded out
  • tracking
  • robustness: model are not perfect, so we need to tolerant parameter
  • disturbance rejection
  • optimality: how to do the thing best

In implementation, everything is discrete/sampled (Taylor series )
$$
x_{k+1}=x_k+\delta t\dot x_k
$$

4. Cruise-Controllers

Make a car drive a desired speed $r$

Physics: Newton’s Second Law
$$
F=ma
$$

State: velocity $x$, and we should measure velocity as well

input: gas/brake $u$
$$
F=cu
$$
where $c$ is a electro-mechanical transmission coefficient
$$
cu=m\dot x
$$
simple model has big power

The expected properties of the control signal:

  • small $e$ error gives small $u$ control signal
  • $u$ should NOT be jerky, so the performance would be smooth
  • $u$ should NOT depend on exact $c$ and $m$, so this will be a robust control model

4.1. Control Design

we got car model $\dot x=\frac{c}{m}u$

we want $x\to r \space$ as $t\to \infin$

  • Attempt 1: Bang-Bang Control
  • Nope, turns out bumpy ride and burns out actuators
  • Problem: over-reacts to small errors
  • Attempt 2: P control $u=ke$
  • Nope, has steady error
  • Problem: not tracking
  • Attempt 3: $u=ke+\gamma\frac{m}{c}x$
  • Nope, yet cancel the steady error
  • Problem: not robust
  • Attempt 4: PI controller, even, PID controller

5. PID Controller

Widely used controller
$$
u(t)=k_{P} e(t)+k_{I} \int_{0}^{t} e(\tau) d \tau+k_{D} \frac{d e(t)}{d t}
$$
3 knobs we can tune $k_P, k_I,k_D$

  • P: contributes to stability,medium-rate responsiveness
  • I: tracking and disturbance rejection, slow-rate responsiveness
  • may cause oscillation
  • D: Fast-rate responsiveness. Sensitive to noise

6. Implementation

First, translate into something implementable

  • find the sample time $\Delta t$
  • integral： $E_{new}=E_{old}+\Delta te$
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read e;
e_dot=e-old_e;
E=E+e;
u=kP*e+kD*e_dot+kI*E;
old_e=e;

7. Example: Quadrotor Altitude Control

$$
\ddot x=cu-g
\u=PID
$$

8. Programming Assignments: Week 1

Why assignments worth my time

  • opportunity to apply the equation
  • learn Matlab