5.2 Controllability and observability

Controllability and Observability

Controllability and observability answer two fundamental questions about a state-space system: Can you actually drive the system where you want it to go? and Can you figure out what's happening inside the system from its outputs? If either answer is "no," your controller or observer design will hit a wall. These two properties are the gatekeepers for nearly every advanced control technique you'll encounter.

Controllability

Definition of Controllability

A system is controllable if you can steer it from any initial state to any desired final state in a finite amount of time by choosing the right control input. If even one state exists that you can't influence through the input, the system is not fully controllable.

Controllability Matrix

The controllability matrix is the primary tool for testing controllability of a linear time-invariant (LTI) system. Given a state-space model $\dot{x} = Ax + Bu$, the controllability matrix is:

$\mathcal{C} = \begin{bmatrix} B & AB & A^2B & \cdots & A^{n-1}B \end{bmatrix}$

where $n$ is the number of states (the dimension of $x$). Each successive column $A^kB$ captures how the input propagates through the system dynamics over additional time steps.

Kalman's Controllability Criterion

Kalman's criterion gives a clean pass/fail test:

1. Form the controllability matrix $\mathcal{C}$.
2. Compute its rank.
3. If $\text{rank}(\mathcal{C}) = n$ (full row rank), the system is controllable.

Full row rank means the columns of $\mathcal{C}$ span the entire state space, so the input can push the state in every possible direction.

Controllability of Linear Systems

For linear systems, the rank test above is both necessary and sufficient. If $\text{rank}(\mathcal{C}) = n$, you can reach any target state from any starting state with the right input signal. If the rank is less than $n$, some states are unreachable no matter what input you apply.

Controllability of Nonlinear Systems

Nonlinear systems don't have a single matrix test. The typical approach:

• Local controllability: Linearize the system around an operating point to get $A$ and $B$ matrices, then apply the standard rank test. This tells you about controllability near that operating point only.
• Global controllability: Requires more advanced tools from Lie algebra and differential geometry. The idea is to check whether the Lie brackets of the system's vector fields span the full state space.

Controllability Gramian

The controllability Gramian $W_c$ quantifies how much control effort is needed to reach different states. For a continuous-time stable system, it's defined as:

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

• If $W_c$ is positive definite, the system is controllable.
• The eigenvalues of $W_c$ tell you which state directions are easy to control (large eigenvalues) versus hard to control (small eigenvalues).
• This is useful for optimal control and model reduction, where you want to know which states matter most from an energy perspective.

Controllability vs. Reachability

These terms are often used interchangeably, but there's a subtle distinction:

• Reachability: Can you drive the system from the origin to a desired state?
• Controllability: Can you drive the system from any initial state to any final state?

For LTI systems, controllability and reachability are equivalent. The distinction matters more in discrete-time systems or certain nonlinear settings.

Observability

Definition of Observability

A system is observable if you can determine its initial state entirely from knowledge of the output $y(t)$ and input $u(t)$ over a finite time interval. If two different initial states produce identical outputs, you can't tell them apart, and the system is not fully observable.

Observability Matrix

For an LTI system $\dot{x} = Ax + Bu$, $y = Cx$, the observability matrix is:

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

Each successive row $CA^k$ captures how the system dynamics reveal additional state information through the output over time.

Kalman's Observability Criterion

The test mirrors the controllability case:

1. Form the observability matrix $\mathcal{O}$.
2. Compute its rank.
3. If $\text{rank}(\mathcal{O}) = n$ (full column rank), the system is observable.

Full column rank means every initial state produces a distinct output trajectory, so no state information is hidden.

Observability of Linear Systems

For linear systems, the rank test is again necessary and sufficient. If $\text{rank}(\mathcal{O}) = n$, the initial state can be uniquely reconstructed from the output measurements. If the rank falls short, some internal states are invisible to the output.

Observability of Nonlinear Systems

As with controllability, nonlinear observability analysis is harder:

• Local observability: Linearize around an operating point and apply the rank test.
• Global observability: Requires tools from differential geometry, such as the observability rank condition using Lie derivatives of the output function.

Observability Gramian

The observability Gramian $W_o$ quantifies how sensitive the output is to different initial states:

$W_o = \int_0^{\infty} e^{A^T t} C^T C \, e^{At} \, dt$

• If $W_o$ is positive definite, the system is observable.
• Large eigenvalues of $W_o$ correspond to state directions that strongly affect the output (easy to observe), while small eigenvalues correspond to directions that barely show up in the output.

Observability vs. Reconstructability

Similar to the controllability/reachability distinction:

• Observability: Can you determine the initial state from the outputs?
• Reconstructability: Can you determine the current state from past inputs and outputs?

For LTI systems these are equivalent. The distinction becomes relevant in time-varying or discrete-time systems.

Duality of Controllability and Observability

Controllability and observability are dual properties. The duality principle states:

• A system $(A, B, C)$ is controllable if and only if its dual system $(A^T, C^T, B^T)$ is observable, and vice versa.

This means any theorem you prove about controllability has a corresponding observability result (and vice versa) just by transposing matrices and swapping input/output roles. In practice, this saves a lot of work: once you understand one concept deeply, you get the other almost for free.

Controllability and Observability Tests

Popov-Belevitch-Hautus (PBH) Test

The PBH test provides an eigenvalue-by-eigenvalue check instead of computing the full controllability/observability matrix.

• Controllability: The system is controllable if and only if $\text{rank}\begin{bmatrix} A - \lambda I & B \end{bmatrix} = n$ for every eigenvalue $\lambda$ of $A$.
• Observability: The system is observable if and only if $\text{rank}\begin{bmatrix} A - \lambda I \\ C \end{bmatrix} = n$ for every eigenvalue $\lambda$ of $A$.

The PBH test is especially useful because it tells you which modes are uncontrollable or unobservable (whichever eigenvalue causes the rank to drop).

Hautus Lemma

The Hautus lemma generalizes the PBH test by requiring the rank condition to hold for all complex numbers $\lambda$, not just the eigenvalues of $A$. In practice, the rank can only drop at eigenvalues of $A$, so the Hautus lemma and the PBH test give the same result for controllability and observability of LTI systems.

Eigenvector Tests

These offer a geometric interpretation:

• Controllability: The system is controllable if and only if no left eigenvector of $A$ is orthogonal to every column of $B$.
• Observability: The system is observable if and only if no right eigenvector of $A$ is orthogonal to every row of $C$.

Think of it this way: if a mode's eigenvector is orthogonal to $B$, the input can't excite that mode. If it's orthogonal to $C$, the output can't see that mode.

Controllability and Observability in State-Space

Controllable Canonical Form

The controllable canonical form is a specific state-space structure where controllability is immediately obvious. The $A$ matrix takes a companion form and $B$ has a single nonzero entry. You can convert any controllable system to this form via a similarity transformation $\bar{x} = Tx$. This form is particularly useful for pole placement design, since the characteristic polynomial coefficients appear directly in the last row of $A$.

Observable Canonical Form

The observable canonical form is the dual structure where observability is immediately apparent. Here, $C$ has a simple structure and the characteristic polynomial coefficients appear in the last column of $A$. It's obtained through a similarity transformation and is especially convenient for observer design.

Kalman Decomposition

Kalman decomposition separates a system into four subsystems based on controllability and observability:

1. Controllable and observable
2. Controllable but not observable
3. Not controllable but observable
4. Neither controllable nor observable

This decomposition uses a similarity transformation and reveals the internal structure of the system. Only subsystem 1 (controllable and observable) affects the transfer function. The other subsystems represent hidden dynamics that don't appear in the input-output relationship.

Controllability and Observability in Transfer Functions

When you convert a state-space model to a transfer function, pole-zero cancellations signal a loss of controllability or observability. A cancelled pole corresponds to a mode that is either uncontrollable, unobservable, or both.

This is why a transfer function can look well-behaved while the underlying state-space model has hidden unstable modes. A transfer function representation is always a minimal realization only if the state-space model is both controllable and observable.

Minimal Realization and Model Reduction

Minimal Realization Problem

A minimal realization is the smallest state-space model (fewest states) that exactly reproduces a given transfer function. A state-space model is minimal if and only if it is both controllable and observable. Any uncontrollable or unobservable states are redundant from an input-output perspective and can be removed.

Balanced Truncation

Balanced truncation is a systematic model reduction method:

1. Compute the controllability and observability Gramians.
2. Apply a balancing transformation so that both Gramians become equal and diagonal. The diagonal entries (called Hankel singular values) rank each state by how important it is to the input-output behavior.
3. Truncate states with small Hankel singular values.

The method preserves stability and provides a guaranteed bound on the approximation error.

Hankel Norm Approximation

Hankel norm approximation minimizes the Hankel norm of the error between the original and reduced models. The Hankel norm captures the maximum energy transfer from past inputs to future outputs, making it a natural measure tied to both controllability and observability. Like balanced truncation, this method preserves stability and provides error bounds, but it can sometimes achieve a lower-order approximation for the same error tolerance.

Stabilizability and Detectability

Stabilizability vs. Controllability

Stabilizability is a relaxed version of controllability. A system is stabilizable if all its unstable modes are controllable. The stable modes don't need to be controllable because they decay on their own.

• Controllable $\Rightarrow$ Stabilizable (but not the other way around).
• Stabilizability is the actual requirement for designing a stabilizing state feedback controller. Full controllability is needed only if you want to place all closed-loop poles arbitrarily.

Detectability vs. Observability

Detectability is the dual relaxation. A system is detectable if all its unstable modes are observable. Stable unobservable modes are acceptable because their estimation errors decay naturally.

• Observable $\Rightarrow$ Detectable (but not the other way around).
• Detectability is sufficient for designing an asymptotically stable observer. Full observability is needed only if you require arbitrary observer pole placement.

Applications of Controllability and Observability

Controller Design

Controllability is a prerequisite for pole placement: if the system is controllable, you can design a state feedback gain $K$ such that $A - BK$ has any desired eigenvalues. Techniques like LQR and MPC also require controllability (or at minimum stabilizability) to guarantee a solution exists.

Observer Design

Observability is a prerequisite for state estimation. If the system is observable, you can design a Luenberger observer or Kalman filter to reconstruct the full state from output measurements. The observer's convergence rate can be set arbitrarily through pole placement of $A - LC$.

Optimal Control

Both properties come together in LQG (Linear Quadratic Gaussian) control, which combines an LQR controller with a Kalman filter. The LQR requires controllability (or stabilizability) to find the optimal feedback gain. The Kalman filter requires observability (or detectability) to find the optimal estimator gain. This separation principle only works when both conditions are met.

Fault Detection and Isolation

Observability concepts underpin fault detection and isolation (FDI). The core idea: if a fault changes the system's state, an observer should detect the discrepancy between predicted and measured outputs. Designing residual generators that are sensitive to specific faults while rejecting disturbances relies on careful analysis of which fault modes are observable from the available sensors.