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, all of which are heavily tested topics. 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

• Controllability means reachability—a system is controllable if you can drive it from any initial state to any final state in finite time using admissible inputs
• The control input must exist for every possible state transition, not just some convenient ones
• Uncontrollable systems have "locked" states—portions of the state space that inputs simply cannot influence, no matter what you do

State-Space Representation

• Four matrices define the system: $A$ (system dynamics), $B$ (input mapping), $C$ (output mapping), and $D$ (feedthrough)
• State equation $\dot{x} = Ax + Bu$ captures how states evolve under input influence—this is your starting point for controllability analysis
• Compact matrix form enables systematic testing—without state-space representation, controllability analysis would be impractical for complex 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

• Construction formula: $\mathcal{C} = [B \quad AB \quad A^2B \quad \cdots \quad A^{n-1}B]$ where $n$ is the number of state variables
• Full rank means controllable—if $\text{rank}(\mathcal{C}) = n$, every state is reachable; if rank is less than $n$, some states are inaccessible
• Each column represents input influence at successive time steps—$B$ is immediate effect, $AB$ is one step later, and so on

Kalman's Rank Condition

• The definitive algebraic test: system is controllable if and only if $\text{rank}(\mathcal{C}) = n$
• Named criterion for exam recall—when a problem says "use Kalman's condition," it's asking you to build the controllability matrix and compute its rank
• Works for both continuous and discrete systems—the same rank check applies regardless of system type

Controllable Canonical Form

• Special structure reveals controllability directly—the $A$ matrix takes companion form and $B$ becomes a unit vector
• Transformation always exists for controllable systems—if you can convert to this form, the system is controllable by construction
• Simplifies controller design—state feedback gains can be read directly from desired characteristic polynomial coefficients

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Eigenvalue-Based and Energy-Based Tests

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

PBH Test for Controllability

• Eigenvalue-by-eigenvalue check: for every eigenvalue $\lambda$ of $A$, verify that $\text{rank}([A - \lambda I \quad B]) = n$
• Reveals which modes are uncontrollable—if the test fails for a specific $\lambda$, that eigenvalue's mode cannot be influenced by input
• Particularly useful for systems with known eigenvalues—avoids constructing the full controllability matrix when modal information is available

Controllability Gramian

• Energy interpretation: $W_c = \int_0^{T} e^{At} B B^T e^{A^T t} \, dt$ measures how much control energy reaches each state direction
• Positive definite Gramian means controllable—if $W_c > 0$ (all eigenvalues positive), every state is reachable; singular $W_c$ indicates uncontrollable modes
• Quantifies "ease" of control—small eigenvalues of $W_c$ indicate states that require enormous input energy to reach

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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—knowing what's still possible matters.

Controllability of Linear Time-Invariant Systems

• Constant matrices simplify everything—$A$ and $B$ don't change with time, so a single controllability matrix characterizes the system forever
• All standard tests apply directly—Kalman's condition, PBH test, and Gramian all work without modification for LTI systems
• Foundation for most control design—LTI controllability results extend to linearized nonlinear systems around operating points

Controllability of Discrete-Time Systems

• Same conceptual framework: for $x[k+1] = Ax[k] + Bu[k]$, controllability means reaching any state in finite steps
• Identical matrix construction—$\mathcal{C} = [B \quad AB \quad A^2B \quad \cdots \quad A^{n-1}B]$ uses the discrete $A$ and $B$ matrices
• Gramian becomes a sum: $W_c = \sum_{k=0}^{n-1} A^k B B^T (A^T)^k$ replaces the integral for discrete systems

Relationship Between Controllability and Stabilizability

• Controllability implies stabilizability—if you can reach any state, you can certainly reach stable ones
• Stabilizability is weaker—a system is stabilizable if only its unstable modes are controllable; stable uncontrollable modes are acceptable
• Practical design criterion—many real systems aren't fully controllable, but stabilizability is sufficient for keeping them bounded

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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?