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

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

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.

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.

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.

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

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.

PID Controller Proportional gain Integral gain Derivative gain

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.

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

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

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., http://www.academia.edu/6334047/Demonstrating_PID_Con trol_Principles_using_an_Air_Heater_and_LabVIEW

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

Ziegler-Nichols Tuning Table Kp 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

Linear Time-Invariant Dynamical Systems Guest lecture “Overview of LTI System“ From Stanford Video link https://www.youtube.com/watch?v=bf1264iFr-w Start at 23:00