Feedback, Stability, and Pole Placement

How state feedback changes system dynamics, why closed-loop stability is read through eigenvalues or poles, and when controllability allows us to place those poles where we want.
Modified

April 26, 2026

Keywords

feedback, stability, pole placement, state feedback, stabilizability

1 Role

This is the third page of the Control and Dynamics module.

Its job is to explain what we do once a system is both modeled and controllable:

we feed the state back into the input in order to reshape the dynamics

That is the first serious design move in control.

2 First-Pass Promise

Read this page after Controllability, Reachability, and Observability.

If you stop here, you should still understand:

  • what full-state feedback means
  • how feedback changes the system matrix
  • how closed-loop stability is read through poles or eigenvalues
  • why controllability is the condition that makes pole placement possible

3 Why It Matters

Open-loop steering can be fragile.

Even if a system is reachable, an open-loop input plan can fail badly when:

  • the model is not exact
  • the initial state is uncertain
  • disturbances are present
  • numerical or measurement errors accumulate

Feedback changes that story.

Instead of choosing the input once and hoping the model behaves, we let the current state influence the next input:

\[ u(t) = -Kx(t) + r(t). \]

This means the system no longer evolves by the original matrix A alone.

It evolves by a new closed-loop matrix.

That is the central idea of this page:

feedback does not only react to the system; it changes the system's own dynamics

4 Prerequisite Recall

  • a state-space model has the form \dot x = Ax + Bu
  • controllability asks whether the input channel can influence every relevant state direction
  • stability for linear systems is read from the long-time behavior of the modes
  • in continuous time, stable linear modes have eigenvalues with negative real part

5 Intuition

5.1 Feedback Rewrites The Dynamics

For a linear system

\[ \dot x = Ax + Bu, \]

if we choose the control law

\[ u = -Kx, \]

then the dynamics become

\[ \dot x = (A-BK)x. \]

So the design question becomes:

can we choose K so that A-BK has better behavior than A?

5.2 Poles Are The Dynamical Modes We Are Trying To Shape

For an LTI system, the poles or eigenvalues tell us whether modes:

  • decay
  • grow
  • oscillate

Moving poles left in the continuous-time complex plane usually means faster decay, but doing that aggressively can demand more control effort and amplify sensitivity.

5.3 Pole Placement Is Possible Only If The Input Reaches The Relevant Modes

If a mode is uncontrollable, no state feedback can move its pole.

So controllability is not just a technical side condition.

It tells us whether feedback design has enough authority to reshape the dynamics at all.

5.4 Stability And Pole Placement Are Connected But Not Identical

Sometimes we only care about making an unstable system stable.

Sometimes we want more:

  • a certain decay rate
  • a certain damping ratio
  • faster transient response
  • less overshoot

Pole placement is the first-pass language for that stronger design problem.

6 Formal Core

Definition 1 (Definition: Full-State Feedback) For the LTI system

\[ \dot x = Ax + Bu, \]

a full-state feedback law has the form

\[ u = -Kx + r, \]

where K is the feedback gain and r is an optional reference or external command.

Definition 2 (Definition: Closed-Loop Matrix) Under the feedback law u=-Kx, the closed-loop dynamics are

\[ \dot x = (A-BK)x. \]

The matrix

\[ A_{\mathrm{cl}} = A-BK \]

is the closed-loop matrix.

Definition 3 (Definition: Continuous-Time Internal Stability For LTI Systems) At a first pass, a continuous-time LTI system is internally asymptotically stable when every eigenvalue of its dynamics matrix has negative real part.

Definition 4 (Definition: Discrete-Time Internal Stability For LTI Systems) At a first pass, a discrete-time LTI system is internally asymptotically stable when every eigenvalue of its update matrix lies strictly inside the unit disk.

Theorem 1 (Theorem Idea: Pole Placement By State Feedback) For a finite-dimensional controllable LTI system, state feedback can assign the closed-loop poles of A-BK arbitrarily, subject to the usual complex-conjugate pairing when K is real.

This is the first-pass pole-placement theorem.

Theorem 2 (Theorem Idea: Uncontrollable Modes Cannot Be Moved) If a mode is uncontrollable, state feedback cannot change its pole.

So uncontrollable unstable modes block stabilization entirely.

Definition 5 (Definition: Stabilizable) At a first pass, a pair (A,B) is stabilizable if every uncontrollable mode is already stable.

This is weaker than full controllability but still enough for stabilization.

7 Worked Example

Consider the unstable system

\[ \dot x = Ax + Bu, \]

with

\[ A= \begin{bmatrix} 1 & 1 \\ 1 & 2 \end{bmatrix}, \qquad B= \begin{bmatrix} 1 \\ 0 \end{bmatrix}. \]

Choose full-state feedback

\[ u=-Kx, \qquad K=\begin{bmatrix}k_1 & k_2\end{bmatrix}. \]

Then

\[ A-BK= \begin{bmatrix} 1-k_1 & 1-k_2 \\ 1 & 2 \end{bmatrix}. \]

The closed-loop characteristic polynomial is

\[ \det(sI-(A-BK)) = s^2 + (k_1-3)s + (1-2k_1+k_2). \]

Suppose we want the closed-loop poles at -5 and -6.

Then the desired polynomial is

\[ (s+5)(s+6)=s^2+11s+30. \]

Matching coefficients gives

\[ k_1-3=11 \quad\Rightarrow\quad k_1=14, \]

and

\[ 1-2k_1+k_2=30 \quad\Rightarrow\quad k_2=57. \]

So one valid feedback gain is

\[ K=\begin{bmatrix}14 & 57\end{bmatrix}. \]

This example shows the main control-design pattern:

  • write the closed-loop matrix A-BK
  • compute its characteristic polynomial
  • choose desired poles
  • solve for K

The whole strategy works because the input channel has enough authority over the modes.

8 Computation Lens

When you see a feedback-design problem, ask:

  1. are we in continuous time or discrete time?
  2. what is the open-loop matrix, and where are its unstable modes?
  3. is the pair (A,B) controllable or at least stabilizable?
  4. what does the chosen feedback law change: only stability, or also transient speed and damping?
  5. are the requested pole locations reasonable, or do they imply excessive control effort or sensitivity?

These questions often matter more than the raw algebra of solving for K.

9 Application Lens

9.1 Stabilizing Unstable Systems

Aircraft, inverted pendulums, and many balancing systems rely on feedback because open-loop plans alone are too fragile.

9.2 Fast Versus Gentle Control

Pole placement lets us talk about tradeoffs between rapid response and aggressive control effort.

9.3 Bridge To Optimal Control

This page sets up the next module step naturally:

LQR can be read as a principled way to pick feedback gains instead of hand-choosing pole locations.

10 Stop Here For First Pass

If you can now explain:

  • what full-state feedback does to the dynamics
  • why the closed-loop matrix is A-BK
  • how stability is read from poles or eigenvalues
  • why controllability enables pole placement
  • why uncontrollable unstable modes block stabilization

then this page has done its job.

11 Go Deeper

The next natural step in this module is:

The strongest adjacent live pages right now are:

12 Optional Deeper Reading After First Pass

The strongest current references connected to this page are:

13 Sources and Further Reading

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