Observability and Controllability in Linear Systems

Introduction

In control and robotics, systems are commonly described using a state-space model:

Here,

• $x$ represents the internal state
• $u$ is the control input
• $y$ is the measured output

Two fundamental questions immediately arise:

1. Can we infer the internal state from what we measure?
2. Can we drive the system to desired states using control inputs?

These questions are formalized by the concepts of observability and controllability.

One-Sentence Definitions

• Observability

  A system is observable if the output $y(t)$, measured over a finite time interval, uniquely determines the initial state $x(0)$.

• Controllability

  A system is controllable if an appropriate choice of input $u(t)$ can drive the state $x(t)$ from any initial state to any desired final state.

Although simple to state, these properties have deep implications for estimation and control.

Observability: Can We See the State?

At first glance, the output equation

might suggest that observability depends only on $C$.
However, this is not sufficient.

Observability is a property of the pair $(A, C)$, not just $C$.

Key intuition

• The output at a single instant is only a snapshot.
• Observability depends on how the state evolves over time through $A$, and whether hidden components eventually influence the output.

Example

Let

• The sensor directly measures only $x_1$.
• The dynamics cause $x_2$ to influence $x_1$ over time.

Although $x_2$ is not directly measured, it leaves a trace in the output trajectory, making the system observable.

If $A$ were zero, $x_2$ would never affect the output, and the system would be unobservable.

Controllability: Can We Drive the State?

Similarly, controllability is not determined by the presence of $B$ alone.

It is a property of the pair $(A, B)$.

Key intuition

• $B$ determines where control input enters the system.
• $A$ determines how that influence propagates through the state.

A system is controllable if control inputs can reach all state directions, possibly indirectly through the dynamics.

Example

Consider

• The input directly affects $x_2$.
• Through the dynamics, changes in $x_2$ propagate to $x_1$.

Even though $x_1$ is not directly actuated, it remains controllable via $x_2$.

If a state neither receives input nor is dynamically coupled to an actuated state, it becomes uncontrollable.

Matrix-Based Tests (Practical Criteria)

To conveniently determine these properties, linear systems use rank-based tests.

Observability Matrix

The system is observable if

This matrix collects all ways the state can influence the output over time.

Controllability Matrix

The system is controllable if

This matrix represents all state directions reachable by propagating the input through the dynamics.

Why These Concepts Matter

• Observability is a prerequisite for state estimation methods such as the Kalman filter.
• Controllability is required to design stabilizing controllers such as LQR.
• In practice, weaker notions like detectability and stabilizability are often sufficient, ensuring that unstable modes are at least observable or controllable.

Summary

• Observability answers: Can we reconstruct the state from outputs over time?
• Controllability answers: Can we drive the state using inputs?
• Both depend on system dynamics, not just sensors or actuators.
• Rank conditions provide a convenient and rigorous test.

Together, observability and controllability form the structural backbone of modern estimation and control theory, especially in robotics and autonomous driving.