2  The Damped Mass problem

Author

Iacopo Moles

2.1 Modeling

To better understand the problem let’s take a peek at how the simulated model works.

BlockOnSlope.js
Models.BlockOnSlope.prototype.vars = 
1{
    g: 9.81,
    x: 0,          // distance from objective s
    dx: 0,         // velocity v
    slope: 1,      // slope coefficient alpha = dy/dx in the cartesian plane
    F: 0,          // Requested u
    F_cmd: 0,      // Saturated u
    friction: 0,   // Coulomb friction coefficient mu
    T: 0,          // Simulation Time
};

Models.BlockOnSlope.prototype.simulate = function (dt, controlFunc)
{
    this.F_cmd = controlFunc({x:this.x,dx:this.dx,T:this.T});
    if(typeof this.F_cmd != 'number' || isNaN(this.F_cmd)) throw "Error: The controlFunction must return a number.";
2    this.F_cmd = Math.max(-20,Math.min(20,this.F_cmd));
    integrationStep(this, ['x', 'dx', 'F'], dt);
}

Models.BlockOnSlope.prototype.ode = function (x)
{
    return [
3        x[1],
        (x[2]) - (Math.sin(this.slope) * this.g) - (this.friction * x[1]),
        20.0 * (this.F_cmd - x[2])
    ];
}
1
The model has obviously some default values for the parameters that can be modified for the different scenarios.
2
The control command \(u\) is generated by the controlFunction(block) function provided by us. There are some checks to see if it’s a number. If it’s acceptable then it passes through a saturation between \(\pm20\).
3
The model is a simple ODE with equations:

\[ \begin{cases} \dot s = v \\ \dot v = F - sin(\alpha) \cdot g - \mu \cdot v \\ \dot F = -20 \cdot F + 20 \cdot u_{sat} \\ \end{cases} \]

Converting it in state-space representation:

\[ \begin{bmatrix} \dot s \\ \dot v \\ \dot F \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 \\ 0& -\mu & 1\\ 0 & 0 & -20 \end{bmatrix} \begin{bmatrix} s \\ v \\ F \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ u_{sat} \end{bmatrix} + \begin{bmatrix} 0 \\ - sin(\alpha) \cdot g \\ 0 \end{bmatrix} \]

\[ \begin{bmatrix} s \\ v \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0& 1 & 0 \end{bmatrix} \begin{bmatrix} s \\ v \\ F \end{bmatrix} \]

Obviously the gravitational term acts as a disturbance.

Converting it in transfer function form is not necessary.

\[ \alpha + \]