1. Dynamical Systems Basics
Notes by
Professor Benjamin Kuipers
(Additional slides from Internet)

4. Phase Portrait: (x,v) space
Shows the trajectory (x(t),v(t)) of the system
– Stable attractor here

12. 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.

18. y x
WALL
Controller can directly change speed v and rate of turn ω in response to sensor input which measures distance to the wall.

19. State variables are x, y, θ
State variables are x, y, θ. Controller can directly change scalar speed v and rate of turn ω. Input is distance from wall y.

21. ω is computed from the spring model (eq. 1)
Notice that the ω component of u is the first derivative of state variable θ, and so the feedback correction has a D component in the controller.
ω is computed from the spring model (eq. 1)
(eq. 1)  v ω + v k1θ + v/v0 k2 e ≈ 0

24. 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.

25. PID Controller
Proportional gain
Integral gain
Derivative gain

26. In the standard form, u(t) is given by:
u(t) = Kp (e(t) + 1/Ti 0 𝑡 𝑒 𝜏 𝑑𝜏 + Td 𝑑𝑒 𝑡 𝑑𝑡 )
where Kp= P , Ki= Kp /Ti , Kd= KpTd
Proportional gain
Integral gain
Derivative gain
The proportional term corrects current error at time t.
The integral term amortizes errors accumulated in the past over the next Ti time units. For example, a wind gust from that lasts for some time may push a vessel off course; the longer the wind gust, the stronger is the integral term to correct the accumulated error.
The derivative term anticipates error in the next Td time units.

27. Effect of Increasing Gain Parameter
Rise Time
Overshoot
Settling
Time
Steady State Error
Stability Kp
Decrease
Increase
Small change
Degrade Ki
Eliminate Kd
Minor
No Effect in theory
Improve if Kd is small

29. kP = P kI = PTI kD = PTD
Definition: Ti ≜ 1/ TI Td ≜ TD
where Ti and Td are called integral time and derivative time respectively.

32. Summary of Concepts Dynamical systems and phase portraits
Qualitative types of behavior
Stable vs unstable; nodal vs saddle vs spiral
Boundary values of parameters
Designing the wall-following control law
Tuning the PI, PD, or PID controller
Ziegler-Nichols tuning rules
Controller gain is the corrective effort per unit error (deviation from set point)
For more, Google controller tuning, e.g.,
trol_Principles_using_an_Air_Heater_and_LabVIEW

33. Manual PID Controller Tuning by Ziegler-Nichols Method
In open-loop manual mode, bring the system to the normal or specified set point by manually adjusting the control input signal.
Determine the direction (positive or negative) of the required Proportional control gain by trying open-loop control. Increase the input u a little bit, under manual control, to see if the resulting steady state value of the process output has increased. If it does, then the steady-state process gain is positive and so is the Proportional control gain, Kp.
To determine Kp, close the feedback loop and set the controller to P-only mode, i.e., both the Integral and Derivative modes are turned off.
Increase the controller gain Kc in small increments (more positive/negative if Kp was decided to be positive/negative in step 1), and observe the corresponding output response when it reaches steady state for each increment. When a value of Kc results in a sustained periodic oscillation in the output, record this critical value of the controller gain Kc and call it Ku, the ultimate gain. Also, record the period of the output oscillation, Pu which we call the ultimate period.
Using the values of the ultimate gain, Ku and the ultimate period, Pu, Ziegler and Nichols prescribes the following values for Kp, Ki and Kd, depending on which type of controller is desired: Kp Ki Kd
P control
0.5 Ku
PI control
0.45 Ku
1.2 Kp / Pu
PID control
0.6 Ku
2.0 Kp / Pu
Kp Pu / 8
Note: The Ziegler-Nochols tuning method is applicable only to systems of order higher than 2 or have control loop delays 33

34. Ziegler-Nichols Tuning TableKp Ti Td
P Controller
0.5 ∞
PI Controller
0.45
Pu/1.2
PID Controller
0.6
Pu/2
Pu/8
Ki = Kp /Ti
Kd= KpTd

35. Linear Time-Invariant Dynamical Systems
Guest lecture “Overview of LTI System“
From Stanford
Video link
Start at 23:00