Controllability and Observability Fundamentals Quiz Quiz

Assess your understanding of controllability and observability in control systems, covering key concepts, mathematical tests, and practical implications in state-space analysis. Strengthen foundational knowledge with clearly explained questions tailored for students and enthusiasts in systems engineering and control theory.

1. Kalman Controllability Criterion

   Which condition must be met for a linear time-invariant system represented by matrices A and B to be completely controllable according to the Kalman criterion?

   1. The observability matrix formed with A and C must have full rank.
   2. The controllability matrix formed with A and B must have full rank.
   3. The system's eigenvalues must all be real and negative.
   4. The matrix A must be invertible.

   Explanation: According to the Kalman controllability criterion, a system is completely controllable if the controllability matrix (constructed from A and B) has full rank, meaning all states can be manipulated. Having all real and negative eigenvalues (option B) implies stability, not controllability. The observability matrix and matrices A and C (option C) relate to observability, not controllability. The invertibility of matrix A (option D) is not a requirement for controllability.

2. Distinguishing Observability

   Given a dynamic system with state matrix A and output matrix C, what does it mean if the system is observable?

   1. The initial state can be uniquely determined from output measurements over a finite time interval.
   2. External inputs have no effect on the system’s output.
   3. The output matrix C has to be a diagonal matrix.
   4. The system’s future outputs can be exactly predicted without knowing the states.

   Explanation: Observability means the initial state can be reconstructed from the outputs over a finite period, allowing for state estimation. Predicting future outputs without knowing the states (option B) is not correct, as knowing states is essential. Option C, about external inputs’ effect, is unrelated to observability. The requirement for C to be a diagonal matrix (option D) is not true; observability does not impose this constraint.

3. Controllable Canonical Form Example

   In state-space representation, which property does the controllable canonical form guarantee for a single-input system?

   1. The system’s transfer function always has real poles only.
   2. The system is always both controllable and observable regardless of parameter choices.
   3. The observability matrix is diagonal by design.
   4. The controllability matrix is always in a convenient form for checking full rank.

   Explanation: The controllable canonical form arranges system matrices so the controllability matrix is simple to construct and check for full rank, aiding controllability analysis. While it often assists with observability, it does not guarantee both controllability and observability by itself (option A). The structure does not restrict system poles to be real (option C). The observability matrix is not required or designed to be diagonal in this form (option D).

4. Physical Implications of Uncontrollable States

   What is a practical consequence of a state being uncontrollable in a real-world system, such as a drone or a robot?

   1. That state cannot be influenced by manipulating the system’s inputs.
   2. That state can be directly measured by any output sensor.
   3. The system will become unstable no matter what.
   4. The system automatically resets that state to zero.

   Explanation: If a state is uncontrollable, changing system inputs will not affect that state, which could impact performance or safety. Uncontrollable states do not necessarily make the system unstable (option B); stability depends on other factors. Option C is incorrect, as systems do not auto-reset states. Direct measurement by sensors (option D) is related to observability, not controllability.

5. Duality between Controllability and Observability

   Which statement best describes the duality between controllability and observability in linear system theory?

   1. Duality means the input vector B and output vector C are always identical.
   2. The controllability of (A, B) is mathematically equivalent to the observability of (Aᵗ, Cᵗ).
   3. If a system is controllable, it must also be observable.
   4. Observability and controllability are unrelated concepts.

   Explanation: The concept of duality in linear systems means controllability analysis for (A, B) corresponds directly to observability analysis for the transposed pair (Aᵗ, Cᵗ). Controllability does not imply observability in general (option B), as systems can be one but not the other. Option C is incorrect because duality does not require B and C to be identical. Saying observability and controllability are unrelated (option D) is false, as duality connects them mathematically.