Foundations — Pole placement — Ackermann's formula

This page assumes you have seen nothing. Every symbol in the parent note is built here from the ground up, in an order where each idea leans only on the ones before it.

1. The state vector $x$ — "everything the system is right now"

Picture a cart on a track. To know what happens next you need two facts: where it is and how fast it's moving. Bundle those numbers into a single stacked list:

$x=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{c}\text{position}\\ \text{velocity}\end{array}\right].$

The picture: $x$ is a single point floating in a space whose axes are the state variables. As the system evolves, that point traces a path. Everything else in the topic is about steering this point.

Why the topic needs it: pole placement designs how the point moves, so we first need a name for the point.

2. The dot $\dot{x}$ — "the rate of change" (a first taste of the derivative)

The little dot means "how fast this is changing per second."

Look at the amber arrows in the figure: each one is $\dot{x}$ at that spot. The blue curve just obeys the arrows. The topic needs this because the entire system is written as a rule for $\dot{x}$.

3. Matrices $A$ and $B$ — "the machine that turns state into motion"

A matrix is a rectangular grid of numbers that acts as a transformer: feed it a vector, it hands back another vector. Multiplying $A$ by $x$ means "combine the state variables in fixed proportions to build each component of the answer."

$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{c}a{x}_1+b{x}_2\\ c{x}_1+d{x}_2\end{array}\right].$

Row $i$ of the grid dotted into $x$ gives entry $i$ of the output.

Putting it together, the parent's headline equation:

Why single-column $B$: this page (like Ackermann's classic form) is single-input — exactly one knob $u$. See State feedback control.

4. The input $u$ and feedback gain $K$ — "the push and how you decide it"

$u$ is your one control knob — a single number you get to choose each instant (motor voltage, thrust, steering torque).

The clever move is state feedback: don't pick $u$ by hand — compute it from what you measure.

Substitute $u=-Kx$ into $\dot{x}=Ax+Bu$:

$\dot{x}=Ax-BKx=(A-BK)x.$

5. Eigenvalues $\lambda$ — "the secret numbers = the poles"

Some special vectors, when hit by $A$, come out pointing the same way, only stretched:

$Av=\lambda v.$

Here $v$ is the eigenvector (the special direction) and $\lambda$ is its eigenvalue (the stretch factor). See Eigenvalues and system stability.

Read the complex plane in the figure — this is the full case list:

Why the topic exists: if the poles are in a bad column of this plane, we want to drag them left into the stable, well-damped region. That's pole placement.

6. $e^{\lambda t}$ — "the shape of decay, growth, and wobble"

The exponential $e^{\lambda t}$ is the one function that is its own rate of change ($\frac{d}{dt}{e}^{\lambda t}=\lambda e^{\lambda t}$) — exactly why it solves $\dot{y}=\lambda y$.

7. The characteristic polynomial $\chi (s)$ — "the equation the poles solve"

To find the eigenvalues without guessing, ask: for which $\lambda$ does $Av=\lambda v$ have a nonzero solution? That happens exactly when the matrix $\lambda I-A$ squashes some direction to zero, i.e. its determinant vanishes.

Here $I$ is the identity matrix (1's on the diagonal, 0's elsewhere — the "do-nothing" transformer), and $\det$ is the determinant, a single number measuring how much a matrix stretches area/volume; zero determinant means it collapses a direction.

The parent uses two of these:

• $\chi (s)$ — the system's actual char. poly (from $A$).
• $\alpha (s)=(s-{\mu }_1)\cdots (s-{\mu }_n)=s^n+a_{n-1}{s}^{n-1}+\cdots +a_0$ — the desired one, built from the target poles ${\mu }_i$ you choose.

8. Cayley–Hamilton — "a matrix obeys its own polynomial"

What does "plug a matrix into a polynomial" mean? Replace $s^k$ by the matrix power $A^k$ and the constant $c_0$ by $c_{0}I$:

$\alpha (A)=A^n+a_{n-1}{A}^{n-1}+\cdots +a_{1}A+a_{0}I.$

9. Controllability matrix $\mathcal{C}$ — "can your one push reach every wobble?"

Your push $u$ enters only through direction $B$. But repeated pushes ripple through the dynamics: $B$, then $AB$, then $A^{2}B$, ... reach further into the state space each step. Stack those reach-vectors side by side:

The figure: $B$ and $AB$ as two arrows. When they point different ways (span the plane, area $\ne 0$) you can reach anywhere → controllable → ${\mathcal{C}}^{-1}$ exists. When they're parallel (collapsed, zero area) a whole direction is unreachable → not controllable → no ${\mathcal{C}}^{-1}$ → Ackermann fails.

10. The selector row $[0\cdots 01]$

A single row that, when it multiplies a matrix, plucks out that matrix's bottom row. In Controllable canonical form only the last coordinate is actuated, so the feedback gains land in the bottom row — and this row reads them off. Nothing mystical: it's "look at the last line."

Putting the pieces together

Now every symbol in the boxed formula has a meaning:

$K=\underset{\text{read bottom row}}{\underbrace{[0\cdots 01]}}\underset{\text{return ticket}}{\underbrace{{\mathcal{C}}^{-1}}}\underset{\text{desired dynamics, fed to }A}{\underbrace{\alpha (A)}}.$

Equipment checklist

Cover the right-hand side and see if you can answer each before revealing.