Let us consider to control the position of a cart with mass placed on a slope with angle as illustrated in Figure 1. We can move the cart by force along the -axis parallel to the slope. Assume that can be sufficiently large to pull up the cart. Friction between the cart and the slope and air resistance are negligible. We let and denote the force , the position, and the velocity of the cart at time , respectively. The magnitude of gravity acceleration is denoted by .
Suppose that and at time . We consider a method to move the cart to the position . Answer the following questions.
(1) Find the position and the velocity of the cart when we accelerate it with a constant force () until the time .
(2) We want to deaccelerate the cart with a constant force () from the time in Question (1) until the time () so that and . Find and which realize this motion.
Next, we consider to give a force proportional to the displacement from the target position . Specifically, we give . is a positive constant.
(3) Write down the equations of motion for this case.
(4) Draw a graph of .
Next, we consider to further add a force proportional to the velocity of the cart. Specifically, we give . and are positive constants.
(5) Explain an effect caused by adding and the reason why this effect occurs.
(6) Find the condition regarding and so that does not oscillate. You can use the following facts if necessary.
The general solution of a differential equation
can be represented by the solution of the quadratic equation
as follows:
When Eq.(B) has two different real roots and ,
When Eq.(B) has two different imaginary roots ,
When Eq.(B) has a double root ,
Here, and are constants of integration.
(7) Draw a graph of under the condition obtained in Question (6).
Next, we consider to further add a force proportional to the integral of the displacement from the target position. Specifically, we give . and are positive constants.
(8) Explain an effect caused by adding and the reason why this effect occurs.
We apply Newton's second law to the motion of the cart along the slope.
The forces acting on the cart are the applied force (up the slope) and the component of gravity (down the slope).
The equation of motion is:
The acceleration is:
Since initial velocity and initial position , we integrate with respect to time :
Velocity:
Position:
(2) Bang-Bang Control (Acceleration and Deceleration)
Let the acceleration during the first phase () be .
Let the acceleration during the second phase () be . The force is , so:
At time , the position and velocity are:
For , the velocity is given by . We require :
Substituting :
Let's evaluate the ratio term:
So,
Now consider the position. The total distance traveled is .
This is a harmonic oscillator equation centered at an equilibrium point where the net force is zero:
Given and , the motion is a cosine wave shifted to start at zero and oscillating around .
The maximum peak is . The graph oscillates indefinitely between and . It does not settle at .
Graph:
The vertical axis is , horizontal is . The curve is a sinusoidal wave starting at , peaking at (minus the gravity offset), and centered at a level slightly below .
(Note: As an AI text model, I describe the graph. The key feature is sustained oscillation centered below the target L due to gravity).
Effect: The term acts as a damper (viscous friction). It suppresses the oscillation of the cart, causing the amplitude of the vibration to decay over time so that the position converges to a steady value.
Reason: The force is always opposite to the direction of motion. This performs negative work on the system, dissipating kinetic energy until the cart stops moving ().
Under the condition (overdamped or critically damped), the system approaches the equilibrium without oscillating.
The equilibrium position is found by setting derivatives to zero:
The cart starts at and asymptotically approaches , which is slightly less than the target due to gravity (steady-state error).
Graph:
The curve starts at with zero slope, rises smoothly, and flattens out to approach the horizontal asymptote from below. It never crosses .
Effect: The addition of the integral term eliminates the steady-state error, causing the cart to converge exactly to the target position .
Reason: In the previous cases (P and PD control), the controller relied on the position error to generate force. To counteract gravity (), a non-zero error was required (steady-state error).
With the integral term, if there is any steady error , the integral value grows over time, increasing the applied force . This accumulation continues until the force is sufficient to balance gravity exactly when the error is zero (). In steady state, , and , making the integral term provide the constant force .
(Note: As an AI text model, I describe the graph. The key feature is sustained oscillation centered below the target L due to gravity).