Lecture 5:
Basic Dynamical Systems
CS 344R: Robotics
Benjamin Kuipers
Dynamical Systems
• A dynamical system changes continuously
(almost always) according to
xÝ F(x) where
x
n
• A controller is defined to change the
coupled robot and environment into a
desired dynamical system.
xÝ F(x,u)
y  G(x)
u  H i (y)
xÝ F(x,Hi (G(x)))
In One Dimension
• Simple linear system
xÝ kx
xÝ
• Fixed point
x  0  xÝ 0
• Solution
x(t)  x 0 e
kt
– Stable if k < 0
– Unstable if k > 0
x
In Two Dimensions
• Often, position and velocity:
T
x  (x,v) where v  xÝ
• If actions are forces, causing acceleration:
xÝ xÝ  v 


xÝ 
 
Ý
 
Ý



v  xÝ forces
The Damped Spring
• The spring is defined by Hooke’s Law:
F  ma  mx  k1 x
• Include damping friction
mx  k1 x  k2 x
• Rearrange and redefine constants
x  bx  cx  0
xÝ xÝ  v 


xÝ 
 
Ý
 
Ý



v  xÝ bxÝ cx 
The Linear Spring Model
Ý
c0
xÝ bxÝ cx  0
• Solutions are:
x(t)  Ae  Be

• Where r1, r2 are roots of the characteristic
2
equation   b  c  0
r1 t
r2 t
b  b  4c
r1,r2 
2
2
Qualitative Behaviors
b  b 2  4c
r1,r2 
2
• Re(r1), Re(r2) < 0
means stable.
• Re(r1), Re(r2) > 0
means unstable.
• b2-4c < 0 means
complex roots,
means oscillations.
unstable
c
stable
spiral
nodal
nodal
unstable
b
Generalize to Higher Dimensions
Ý Ax
• The characteristic equation for x
generalizes to det( A  I)  0
– This means that there is a vector v such that
Av  v
• The solutions  are called eigenvalues.
• The related vectors v are eigenvectors.
Qualitative Behavior, Again
• For a dynamical system to be stable:
– The real parts of all eigenvalues must be negative.
– All eigenvalues lie in the left half complex plane.
• Terminology:
– Underdamped = spiral (some complex eigenvalue)
– Overdamped = nodal (all eigenvalues real)
– Critically damped = the boundary between.
Node
Behavior
Focus
Behavior
Saddle
Behavior
Spiral
Behavior
(stable
attractor)
Center
Behavior
(undamped
oscillator)
The Wall Follower
(x,y)

The Wall Follower
• Our robot model:
xÝ
v cos 
 


xÝ yÝ F (x,u)  v sin  
Ý



  
u = (v )T
y=(y )T
  0.
• We set the control law u = (v )T = Hi(y)
Ý yÝ
Ý
e  y  yset so eÝ yÝ and eÝ
The Wall Follower
• Assume constant forward velocity v = v0
– approximately parallel to the wall:   0.
• Desired distance from wall defines error:
e  y  yset
so eÝ yÝ and
Ý
eÝ Ý
yÝ
• We set the control law u = (v )T = Hi(y)
– We want e to act like a “damped spring”
Ý k1 eÝ k2 e  0
eÝ
The Wall Follower
Ý
Ý k1 eÝ k2 e  0
• We want e
• For small values of 
eÝ  yÝ  v sin 
 v
Ý  v
Ý  yÝ
Ý  v cos 
eÝ
• Assume v=v0 is constant. Solve for 
 v 0

v  
u    k   k 2 e H i (e, )
   1 v 

0 
– This makes the wall-follower a PD controller.
Tuning the Wall Follower
eÝ k1 eÝ k2 e  0
• The system is Ý
2
k
• Critically damped is 1  4k 2  0
k1  4k 2
• Slightly underdamped performs better.
– Set k2 by experience.
– Set k1 a bit less than
4k2
An Observer for Distance to Wall
• Short sonar returns are reliable.
– They are likely to be perpendicular reflections.
Experiment with Alternatives
• The wall follower is a PD control law.
• A target seeker should probably be a PI
control law, to adapt to motion.
• Try different tuning values for parameters.
– This is a simple model.
– Unmodeled effects might be significant.
Ziegler-Nichols Tuning
• Open-loop response to a step increase.
d
T
K
Ziegler-Nichols Parameters
• K is the process gain.
• T is the process time constant.
• d is the deadtime.

Tuning the PID Controller
• We have described it as:
u(t)  kP e(t)  kI
t
 e dt  k
D
eÝ(t)
0
• Another standard form is:
t


u(t)  P e(t)  TI  e dt  TD eÝ(t)


0
• Ziegler-Nichols says:
1.5  T
P
TI  2.5  d
TD  0.4  d
Kd
Ziegler-Nichols Closed Loop
1. Disable D and I action (pure P control).
2. Make a step change to the setpoint.
3. Repeat, adjusting controller gain until
achieving a stable oscillation.
•
•
This gain is the “ultimate gain” Ku.
The period is the “ultimate period” Pu.
Closed-Loop Z-N PID Tuning
• A standard form of PID is:

u(t)  P e(t)  TI

• For a PI controller:
P  0.45  K u


 e dt  TDeÝ(t)
0
t
Pu
TI 
1.2
• For a PID controller:
Pu
P  0.6  K u
TI 

2
Pu
TD 
8