Key Concepts of Controllability

Why This Matters

Controllability is one of the most fundamental questions you can ask about any dynamic system: can you actually steer it where you want it to go? Before you design a controller, before you pick gains, before you simulate anything, you need to know whether the system is even capable of being controlled. This concept connects directly to state-space analysis, eigenvalue placement, and feedback design. If a system isn't controllable, no amount of clever control design will save you.

You're being tested on your ability to verify controllability using multiple methods (the controllability matrix, Kalman's rank condition, the PBH test, and the Gramian) and to interpret what controllability means for system design. Don't just memorize the formulas. Understand what each test reveals about the system's structure and when you'd choose one method over another.

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Foundational Definitions and Representations

Before testing for controllability, you need to understand what it means and how systems are mathematically described. State-space representation provides the framework; controllability tells you what's possible within that framework.

Definition of Controllability

A system is controllable if you can drive it from any initial state $x_0$ to any desired final state $x_f$ in finite time using admissible inputs. The key word is any: the control input must exist for every possible state transition, not just convenient ones.

If a system is uncontrollable, some portion of the state space is effectively "locked." No input signal, no matter how large or cleverly shaped, can influence those states. Think of it this way: if one state variable has no path connecting it to the input (even indirectly through coupling with other states), you simply can't move it.

State-Space Representation

Four matrices define a linear system:

• $A$ (system/state matrix): governs the internal dynamics
• $B$ (input matrix): maps control inputs into the state space
• $C$ (output matrix): maps states to measured outputs
• $D$ (feedthrough matrix): direct input-to-output path

The state equation $\dot{x} = Ax + Bu$ captures how states evolve under input influence. This is your starting point for all controllability analysis. Without state-space form, systematic controllability testing would be impractical for anything beyond trivial systems.

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Matrix-Based Controllability Tests

The most common exam questions involve constructing matrices and checking their properties. These tests convert the abstract question "is this controllable?" into concrete linear algebra.

Controllability Matrix

The controllability matrix is built by repeatedly multiplying $A$ into $B$:

$\mathcal{C} = [B \quad AB \quad A^2B \quad \cdots \quad A^{n-1}B]$

where $n$ is the number of state variables (the dimension of $A$).

Each column block represents how the input's influence propagates through the system over successive time steps. $B$ captures the immediate effect of the input on the states. $AB$ captures where that effect migrates after one time step through the dynamics. $A^2B$ is two steps out, and so on. By the Cayley-Hamilton theorem, $A^n$ and higher powers can be expressed as linear combinations of $I, A, \ldots, A^{n-1}$, so you never need more than $n$ blocks.

Full rank means controllable. If $\text{rank}(\mathcal{C}) = n$, every state direction is reachable. If the rank falls short, some states are inaccessible from the input.

Kalman's Rank Condition

This is the formal name for the test you just saw: a system $(A, B)$ is controllable if and only if $\text{rank}(\mathcal{C}) = n$. When a problem says "use Kalman's condition," it's asking you to build the controllability matrix and compute its rank.

This condition works identically for both continuous-time and discrete-time systems. The matrices $A$ and $B$ come from whichever domain you're working in, but the rank check is the same.

Controllable Canonical Form

A system in controllable canonical form has a specific structure: $A$ takes companion matrix form (characteristic polynomial coefficients along the last row, ones on the superdiagonal, zeros elsewhere) and $B$ is a unit vector (typically $[0 \; 0 \; \cdots \; 1]^T$).

This structure is useful for two reasons:

• Controllability is guaranteed by construction. If you can transform a system into this form via a similarity transformation $\bar{x} = Tx$, the system must be controllable (and the transformation matrix $T$ is built from $\mathcal{C}$).
• Pole placement becomes straightforward. State feedback gains can be read directly by comparing the desired characteristic polynomial coefficients with the existing ones.

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Eigenvalue-Based and Energy-Based Tests

When the controllability matrix is large or when you need physical insight into which modes are problematic, alternative tests offer computational or conceptual advantages. These methods connect controllability to the system's modal structure and energy requirements.

PBH Test for Controllability

The Popov-Belevitch-Hautus (PBH) test checks controllability one eigenvalue at a time. For every eigenvalue $\lambda_i$ of $A$, verify that:

$\text{rank}([A - \lambda_i I \quad B]) = n$

If this rank condition holds for all eigenvalues, the system is controllable. If it fails for a specific $\lambda_i$, that eigenvalue's associated mode cannot be influenced by the input.

This test is particularly useful when you already know the eigenvalues of $A$ (or when $A$ is diagonal/block-diagonal), because you can pinpoint exactly which mode is uncontrollable rather than just getting a single rank number from $\mathcal{C}$. For repeated eigenvalues, you need to check each one carefully since repeated modes are a common source of lost controllability.

Controllability Gramian

The controllability Gramian provides an energy-based perspective. For continuous-time LTI systems:

$W_c = \int_0^{T} e^{At} B B^T e^{A^T t} \, dt$

For stable systems, you can let $T \to \infty$, and $W_c$ converges to the unique solution of the Lyapunov equation:

$AW_c + W_c A^T + BB^T = 0$

Positive definite $W_c$ (all eigenvalues strictly positive) means the system is controllable. A singular $W_c$ indicates uncontrollable modes.

The Gramian goes beyond a binary yes/no. The eigenvalues of $W_c$ quantify how much energy is needed to reach different state directions. Small eigenvalues correspond to directions that require enormous input energy to reach. So a system can be technically controllable but practically very difficult to control along certain directions.

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System Types and Related Concepts

Controllability analysis adapts to different system classes, and understanding its relationship to stabilizability helps you handle imperfect situations. Real-world systems aren't always fully controllable, so knowing what's still possible matters.

Controllability of Linear Time-Invariant Systems

For LTI systems, $A$ and $B$ are constant matrices. This means a single controllability matrix characterizes the system for all time. All standard tests (Kalman's condition, PBH, Gramian) apply directly without modification.

Most control design techniques assume LTI structure. When you linearize a nonlinear system around an operating point, you get an LTI approximation, and controllability of that linearization tells you whether local control is feasible near that equilibrium.

Controllability of Discrete-Time Systems

For discrete-time systems $x[k+1] = Ax[k] + Bu[k]$, controllability means reaching any state in at most $n$ steps. The controllability matrix has the identical form:

$\mathcal{C} = [B \quad AB \quad A^2B \quad \cdots \quad A^{n-1}B]$

using the discrete $A$ and $B$ matrices. Kalman's rank condition and the PBH test apply unchanged.

The Gramian adapts by replacing the integral with a sum:

$W_c = \sum_{k=0}^{n-1} A^k B B^T (A^T)^k$

The interpretation is the same: positive definite means controllable, and eigenvalue magnitudes reflect control effort requirements.

Relationship Between Controllability and Stabilizability

Controllability implies stabilizability, but not the other way around. Stabilizability is a weaker condition: a system is stabilizable if only its unstable modes are controllable. Stable uncontrollable modes are acceptable because they decay on their own.

Formally, a system is stabilizable if for every eigenvalue $\lambda_i$ of $A$ with $\text{Re}(\lambda_i) \geq 0$ (continuous-time) or $|\lambda_i| \geq 1$ (discrete-time), the PBH rank condition holds.

This distinction matters in practice. Many real systems have stable modes that inputs can't reach, but that's fine for keeping the system bounded. If an exam problem presents uncontrollable but stable modes, argue that stabilizability is what matters for practical control design.

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Quick Reference Table

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Self-Check Questions

1. Given a system with $A$ as a $3 \times 3$ matrix and $B$ as $3 \times 1$, what dimensions does the controllability matrix have, and what rank indicates full controllability?

2. Compare the PBH test and the controllability matrix method. When would you prefer one over the other, and what additional information does each provide?

3. A system has three eigenvalues: one unstable and two stable. The controllability matrix has rank 1. Is the system controllable? Is it stabilizable? Explain your reasoning.

4. How does the controllability Gramian differ from the controllability matrix in terms of what it tells you about a system, and why might a system be "technically controllable" but practically difficult to control?

5. If you're asked to design a state feedback controller using pole placement, why must you first verify controllability, and which representation would make the design process easiest?