Controllability & Observability: Matrix vs Gramian

In control theory, controllability and observability are two fundamental concepts that describe how inputs influence the state of a system, and how outputs reveal the system’s internal state.

There are two main mathematical tools to analyze them:

• Controllability / Observability Matrices
• Controllability / Observability Gramians

Although they serve the same purpose (checking controllability and observability), they provide different perspectives.

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1. Controllability Matrix

For a linear system

$$\dot{x}(t) = A x(t) + B u(t),$$

the controllability matrix is

$$\mathcal{C} = [B, AB, A^2B, \dots, A^{n-1}B].$$

• The system is controllable if

  $$\text{rank}(\mathcal{C}) = n.$$

👉 Interpretation: This matrix checks whether inputs can span the entire state space.

👉 Limitation: It only gives a binary answer (controllable or not), without quantifying how easy it is to control.

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2. Observability Matrix

For the system output

$$y(t) = Cx(t),$$

the observability matrix is

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

• The system is observable if

  $$\text{rank}(\mathcal{O}) = n.$$

👉 Interpretation: It checks whether outputs provide enough information to reconstruct the entire state.

👉 Limitation: Like controllability matrix, it does not measure the degree of observability.

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3. Controllability Gramian

If the system is stable, the controllability Gramian is defined as

$$W_c = \int_0^\infty e^{A\tau} B B^T e^{A^T \tau} d\tau.$$

It satisfies the Lyapunov equation:

$$A W_c + W_c A^T + B B^T = 0.$$

• The system is controllable if $W_c$ is positive definite.

• The minimum control energy to reach a state $x_T$ is

  $$E_{\min}(x_T) = x_T^T W_c^{-1} x_T.$$

👉 Interpretation: The Gramian describes how much input energy is needed to move the system in each state direction.

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4. Observability Gramian

The observability Gramian is defined as

$$W_o = \int_0^\infty e^{A^T\tau} C^T C e^{A\tau} d\tau,$$

and satisfies

$$A^T W_o + W_o A + C^T C = 0.$$

• The system is observable if $W_o$ is positive definite.
• $x^T W_o x$ measures how much output energy is generated from the state $x$.

👉 Interpretation: The Gramian quantifies how “visible” different state directions are from the outputs.

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5. Matrix vs Gramian

Tool	Purpose	Criterion	Meaning	Limitation
Controllability Matrix $\mathcal{C}$	Check controllability	$\text{rank}(\mathcal{C})=n$	Can input affect all states?	Only Yes/No
Observability Matrix $\mathcal{O}$	Check observability	$\text{rank}(\mathcal{O})=n$	Can output reveal all states?	Only Yes/No
Controllability Gramian $W_c$	Quantify controllability	$W_c > 0$	Input energy required to reach states	Requires stability
Observability Gramian $W_o$	Quantify observability	$W_o > 0$	Output energy generated by states	Requires stability

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6. Conclusion

• Matrices (controllability/observability) → algebraic tools for a binary test.

• Gramians (controllability/observability) → energy-based tools for a quantitative measure.

• Both are equivalent in deciding controllability and observability, but Gramians are essential in advanced applications such as:

  • Optimal control (minimum-energy control problems)
  • Model reduction (balanced truncation)
  • System sensitivity analysis

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👉 In short:

• Matrices tell you if the system is controllable/observable.
• Gramians tell you how much control or observation effort is required.