Controllability, Reachability, and Observability

How control asks whether states can be reached by inputs, whether hidden state can be reconstructed from outputs, and why rank tests turn these questions into matrix conditions.
Modified

April 26, 2026

Keywords

controllability, reachability, observability, Kalman matrix, state-space

1 Role

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

Its job is to answer two foundational control questions:

  • can the input move the system where we want it to go?
  • can the measurements reveal the hidden internal state?

Those are the questions behind reachability, controllability, and observability.

2 First-Pass Promise

Read this page after State-Space Models, Inputs, and Outputs.

If you stop here, you should still understand:

  • what reachability asks
  • what observability asks
  • why these are matrix-rank questions for LTI systems
  • why controllability and observability are dual ideas

3 Why It Matters

A state-space model tells us how a system evolves:

\[ \dot x = Ax + Bu, \qquad y = Cx + Du. \]

But writing the model down is only the start.

Control then asks:

  • can a suitable input drive the state to desired places?
  • are some directions in state space impossible to excite?
  • are some directions invisible at the output?
  • can two different initial states produce the same measured behavior?

If the answer to those questions is bad, then later steps like feedback design or state estimation may fail before they begin.

So this page is where control stops being just notation and starts testing what the model can actually do.

4 Prerequisite Recall

  • the state stores the information needed for future prediction
  • the input channel B tells us how external forcing enters the dynamics
  • the output map C tells us which parts of the state are measured
  • linear systems evolve by superposition, so input effects and initial-condition effects can be separated

5 Intuition

5.1 Reachability Is About Steering From Zero

Imagine starting from the zero state and asking:

what states can I create by choosing the input over a time interval?

That is the reachability question.

If some state direction can never be produced, then that direction is unreachable no matter how clever the controller is.

5.2 Controllability Is The Stronger Steering Story

At a first pass, controllability is the idea that we can steer the system where we need it to go in finite time.

For finite-dimensional LTI systems, the reachability and controllability tests line up cleanly, so many first-pass courses treat them together.

The important intuition is:

the input must touch enough dynamical modes of the system

5.3 Observability Is About Inference From Output

Suppose the full state is hidden and we only see the output y(t).

Then the observability question is:

can the output history tell us which initial state the system had?

If two different initial states always produce the same output behavior, then the output is not rich enough to distinguish them.

5.4 Input Geometry And Output Geometry Are Dual

The reachability problem asks whether the input directions, after being propagated through the dynamics, span the state space.

The observability problem asks whether the output map, when composed with the dynamics, can probe every relevant state direction.

That is why the two rank tests look like transposes of each other.

6 Formal Core

Definition 1 (Definition: Reachability) For the continuous-time LTI system

\[ \dot x = Ax + Bu, \qquad x(0)=0, \]

a state x_d is reachable over a time interval if there exists an input u(t) that drives the system from x(0)=0 to x(T)=x_d.

Definition 2 (Definition: Observability) For the system

\[ \dot x = Ax + Bu, \qquad y = Cx + Du, \]

the system is observable if the initial state can be uniquely determined from the output over a time interval when the input is known.

Definition 3 (Definition: Reachability Matrix) For an n-dimensional LTI system, the Kalman reachability matrix is

\[ \mathcal R = \begin{bmatrix} B & AB & A^2B & \cdots & A^{n-1}B \end{bmatrix}. \]

Definition 4 (Definition: Observability Matrix) The Kalman observability matrix is

\[ \mathcal O = \begin{bmatrix} C \\ CA \\ CA^2 \\ \vdots \\ CA^{n-1} \end{bmatrix}. \]

Theorem 1 (Theorem Idea: Kalman Reachability Test) For a finite-dimensional LTI system, the system is reachable at first pass exactly when the reachability matrix \mathcal R has full rank n.

This means the input directions, after repeated action by the dynamics matrix A, span the whole state space.

Theorem 2 (Theorem Idea: Kalman Observability Test) For a finite-dimensional LTI system, the system is observable at first pass exactly when the observability matrix \mathcal O has full rank n.

This means the output map, together with repeated action by the dynamics, can distinguish every state direction.

Theorem 3 (Theorem Idea: Duality) Observability of (A,C) is the controllability or reachability story of the transposed pair (A^T,C^T).

This duality is why many proofs and rank tests come in matching pairs.

7 Worked Example

Consider the two-state system

\[ \dot x = Ax + Bu, \qquad y = Cx, \]

with

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

7.1 Reachability

The reachability matrix is

\[ \mathcal R= \begin{bmatrix} B & AB \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}. \]

Its rank is 2, so the system is reachable.

Interpretation:

  • the input directly pushes the second state through B
  • the dynamics matrix A then transfers that influence into the first state

So the combination of B and the internal dynamics reaches both state directions.

7.2 Observability

The observability matrix is

\[ \mathcal O= \begin{bmatrix} C \\ CA \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}. \]

Its rank is also 2, so the system is observable.

Interpretation:

  • the output directly measures the first state
  • the time evolution couples the second state into what the output reveals over time

So even though y=Cx only shows one coordinate instantaneously, the output history contains enough information to recover both coordinates.

This is the key control lesson:

  • one-time measurement and observability are not the same thing
  • direct forcing and reachability are not the same thing

The dynamics matrix A matters in both stories.

8 Computation Lens

When you see a control model, ask:

  1. does the input matrix B touch every important mode directly or indirectly?
  2. does the output matrix C expose every important mode directly or indirectly?
  3. what do the rank tests say for the reachability and observability matrices?
  4. if the rank is deficient, which directions are hidden or uncontrollable?
  5. is the issue in the actuator geometry, the sensor geometry, or the system dynamics?

These questions usually reveal whether later controller or observer design is even worth attempting.

9 Application Lens

9.1 Actuator Placement

Reachability language helps decide where actuators must be placed so inputs can influence all important modes.

9.2 Sensor Design

Observability language helps decide whether available measurements are enough to reconstruct the internal state.

9.3 Estimation And Feedback

Kalman filtering, observer design, and state feedback all rely on the system not hiding the important state directions.

10 Stop Here For First Pass

If you can now explain:

  • what reachability asks
  • what observability asks
  • why the rank tests depend on A together with B or C
  • why observability is not the same as seeing the whole state instantaneously
  • why reachability and observability are dual

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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