Controller Design Steps

Below are steps to design a controller for a system. An example is also provided.

Steps

Create a model. Express the real system on paper. The model should:

• Capture the characteristics of the control target.
• Be easy to analyze and design a controller for. Do not directly apply dynamics equations if it complicates analysis.

Study the characteristics of the control target using the model. Then, design a control method.

• Applying different inputs and observe how the system behaves.
• Find input, u(t), that result in the desired behavior.
• Choose parameters that satisfy criteria specified in the design specification.

Write code according to the previously determined control method.

Example

A car moves along a line. Design a controller that moves the car to y = 0

From Newton's second law:

Control the force by using the car's position and velocity as feedback (as an example):

Substitute into the model and solve for the position, y(t) because we want y = 0:

Where C₁ and C₂ depends on y(0) and ẏ(0) so we leave them.

λ₁ and λ₂ are:

The objective is y = 0 so we want,

So we want λ₁, λ₂ < 0

So choose k₁, k₂ > 0

C code:

loop()
{
  y = get_position();
  vy = get_velocity();
  u = -k1 * y - k2 * vy;
  set_force(u);
}

Simulation:

from scipy.integrate import solve_ivp
import matplotlib.pyplot as plt


# Constants
m  =  1 # Mass (kg)
k1 = 10
k2 =  3


# dx/dt = f(t, x)
# 
# t     : Current time (seconds), scalar
# x     : Current state, [y, vy]
# return: First derivative of state, [vy, ay]
def xdot(t, x):
    # Controller (input, u)
    u = -k1 * x[0] - k2 * x[1]
    
    # Dynamics (dx/dt = xdot = [vz, az])
    return [x[1], u/m]


x0     = [5, 0] # Initial state, [y0, vy0]
t_span = [0, 5] # Simulation time (seconds), [from, to]


# Solve for the states, x(t) = [y(t), vy(t)]
sol = solve_ivp(xdot, t_span, x0)


# Plot z vs t
plt.plot(sol.t, sol.y[0], 'k-o')
plt.show()