0.5 Computational Preparation and Readiness Project

CCD studies combine mathematical models with numerical simulation and optimization. You do not need advanced software-development experience to begin, but you should be comfortable reading and modifying short computational scripts.

Minimal MATLAB or Python preparation¶

Before continuing, you should be able to:

• define scalars, vectors, and matrices;

• write and call a simple function;

• evaluate a mathematical formula;

• simulate an ODE numerically;

• plot a result; and

• compute a discrete objective function.

A simple computational workflow¶

A basic modeling study follows a repeatable sequence:

1. Define parameters and initial conditions.

2. Write the model equations.

3. Simulate the model and plot the results.

4. Evaluate one or more performance measures.

5. Refine the model, parameters, or numerical settings.

Python example¶

The following script defines a mass–spring–damper model and integrates it numerically. It is a preparation exercise rather than a full CCD implementation.

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

m = 1.0
c = 0.4
k = 4.0


def model(t, x):
    u = 1.0
    x1, x2 = x
    dx1 = x2
    dx2 = -(k / m) * x1 - (c / m) * x2 + (1 / m) * u
    return [dx1, dx2]


sol = solve_ivp(model, [0, 10], [0, 0], max_step=0.02)

plt.plot(sol.t, sol.y[0, :])
plt.xlabel("time")
plt.ylabel("displacement")
plt.show()

The script performs four basic tasks:

• specifies the model parameters $m$, $c$, and $k$;

• defines the first-order state equations;

• integrates the equations from an initial state; and

• plots the displacement trajectory.

MATLAB example¶

An analogous MATLAB script is

m = 1.0;
c = 0.4;
k = 4.0;

f = @(t,x) [x(2);
           -(k/m)*x(1) - (c/m)*x(2) + 1/m];

[t,x] = ode45(f,[0 10],[0;0]);

plot(t,x(:,1))
xlabel('time')
ylabel('displacement')

Both implementations express the same mathematical model. The syntax differs, but the modeling workflow is unchanged.

Computing a discrete performance measure¶

Suppose the continuous control-effort objective is

After sampling $u(t)$ on a time grid, a trapezoidal approximation can be computed in Python with

J = np.trapezoid(u**2, t)

or in MATLAB with

J = trapz(t, u.^2);

This simple pattern—simulate a trajectory, compute a performance measure, and compare alternatives—later becomes part of a numerical optimization loop.

Readiness mini-project¶

Before moving to Chapter 1, model and simulate a mass–spring–damper system and identify how its quantities would appear in a CCD formulation.

Suggested tasks¶

1. Choose values for $m$, $c$, and $k$.

2. Write the second-order ODE and convert it to first-order state-space form.

3. Select initial conditions and a force input $u(t)$.

4. Simulate displacement and velocity.

5. Plot both state trajectories.

6. Compute at least one performance measure, such as peak displacement, RMS displacement, or control effort.

7. Explain which quantities could become decision variables and which conditions could become constraints.

Connection to a CCD formulation¶

Activity 0.6: State and Parameter Sensitivities for a Readiness Mini-Project¶