Engineering Math Series 7: Systems of Differential Equations and State Space

Real-world systems often do not have just one variable moving in isolation. Current and voltage change together, position and velocity change together, and the concentrations of substances in multiple tanks influence each other. In such cases, systems of differential equations are natural.

Why Write Systems

For example, if you think of position $x$ and velocity $v$ separately,

$$x' = v$$

$$v' = -\omega^2 x$$

the two equations are interconnected. You could combine them into a single second-order equation, but writing them as two first-order equations makes the structure clearer and easier to organize with matrices.

State-Space Representation

Bundling the variables into a vector gives

$$

\mathbf{y} =

\begin{pmatrix}

x \\

v

\end{pmatrix}

$$

and the equations can be written as

$$\mathbf{y}' = A\mathbf{y}$$

In the example above,

$$

A =

\begin{pmatrix}

0 & 1 \\

-\omega^2 & 0

\end{pmatrix}

$$

This representation connects directly to eigenvalues and diagonalization, which we will learn later.

Worked Example

Consider the following system.

$$x' = y$$

$$y' = -x$$

Differentiating the first equation once more gives

$$x'' = y' = -x$$

so

$$x'' + x = 0$$

Therefore

$$x(t) = C_1 \cos t + C_2 \sin t$$

Since $y = x'$,

$$y(t) = -C_1 \sin t + C_2 \cos t$$

If the conditions are

$$x(0)=1, \quad y(0)=0$$

then

$$C_1 = 1, \quad C_2 = 0$$

and the result is

$$x(t) = \cos t, \quad y(t) = -\sin t$$

This system is the typical form of simple harmonic motion where energy is conserved.

Engineering Applications

Control Engineering

Modern control almost always writes systems in state-space form. The states read by sensors, inputs, outputs, and noise are each represented as vectors, and the overall behavior is expressed as matrices.

Chemical Processes

Concentration transfer between multiple tanks and reaction rates are also represented as systems of equations.

Computer Systems

State variables that influence each other, such as queue length, throughput, backlog, and retry count, can be viewed as coupled dynamical models when simplified.

Phase Plane Intuition

Systems of differential equations have the appeal of visualizing trajectories as pictures, not just numbers. Looking at the flow of solutions in the $(x, y)$ plane reveals whether a point is stable, whether there is circular motion, or whether there is divergence.

This perspective becomes very important when learning about eigenvalues and linear system stability later.

Common Mistakes

Only viewing systems as a single equation

Converting to a single second-order equation is fine, but becoming familiar with the state-space form allows you to handle a much wider range of systems in a unified way.

Frequently changing variable order

The order of the vector and the meaning of the matrix must match. The habit of fixing the state vector definition first is necessary.

Finding the solution but not interpreting the interactions

In systems, the interaction between variables is the key. You must connect the analysis to how each variable pushes and pulls the others.

One-Line Summary

Systems of differential equations are the starting point for viewing systems where multiple states move together through matrices and state space.

Next Post Preview

In the next post, we will introduce the **series solution and ordinary point** perspective as preparation for cases where a closed-form solution is not readily available.

References

- Erwin Kreyszig, _Advanced Engineering Mathematics_, 10th Edition

- Gilbert Strang, _Linear Algebra and Its Applications_

- Hassan K. Khalil, _Nonlinear Systems_