Modern Control Systems: State Feedback u0026 Pole Placement Quiz Quiz

Explore key concepts of state feedback control and pole placement with this focused quiz, designed to reinforce understanding of canonical forms, controllability, closed-loop system behavior, and practical design considerations. Enhance your knowledge of modern control theory using state-space approaches for dynamic system stability and performance.

1. Controllability in State-Space Systems

   Given a state-space system described by matrices A and B, which property must the system satisfy to be able to place its poles arbitrarily using state feedback?

   1. The observability matrix must be singular
   2. The controllability matrix must be full rank
   3. The determinant of matrix A must be zero
   4. Matrix B must be diagonal

   Explanation: For arbitrary pole placement using state feedback, the controllability matrix must be full rank, meaning the system is controllable. This ensures all states can be influenced by the input through feedback. If the observability matrix is singular, it only affects state estimation, not pole placement. The determinant of matrix A being zero relates to A’s eigenvalues, not controllability. Matrix B does not need to be diagonal; it must simply allow the necessary input channels.

2. Purpose of State Feedback Gain

   What is the primary goal of designing a state feedback gain matrix K in a system with the state equation x' = A x + B u?

   1. To directly modify the input-output transfer function zeros
   2. To maximize the system's open-loop gain
   3. To assign the closed-loop system's poles for desired performance
   4. To minimize the number of inputs in the matrix B

   Explanation: The main purpose of designing the feedback gain K is to assign the closed-loop poles, shaping the transient and steady-state behavior according to design specifications. Modifying transfer function zeros is not directly achieved by state feedback. Maximizing open-loop gain is not a typical design goal for K, and the number of inputs in matrix B is fixed by the system, not by K.

3. Effect of State Feedback on System Poles

   When applying state feedback u = -Kx to a system, how does it affect the system's eigenvalues?

   1. It multiplies all eigenvalues by a fixed constant
   2. It changes the closed-loop eigenvalues to those of (A - BK)
   3. It makes all eigenvalues positive regardless of K
   4. It shifts the zeros of the transfer function to the origin

   Explanation: State feedback modifies the system matrix from A to (A - BK), thus changing the closed-loop eigenvalues accordingly. The eigenvalues are not simply scaled by a constant, nor does state feedback shift all transfer function zeros to the origin. Feedback does not guarantee all eigenvalues become positive; placement is determined by the chosen K and the system's controllability.

4. Pole Placement Limitation Example

   Consider a system where the controllability matrix is not full rank. What is the impact on pole placement using state feedback?

   1. The input matrix B can compensate for this limitation
   2. System becomes unobservable
   3. Not all system poles can be placed arbitrarily
   4. All poles can still be freely assigned

   Explanation: If the controllability matrix is not full rank, the system is not fully controllable, and only some poles can be assigned with feedback; others cannot be moved. All poles cannot be freely assigned unless full controllability is present. Unobservability concerns state estimation, not pole placement. While B is involved in controllability, its structure cannot always compensate for a lack of controllability.

5. Canonical Forms for Pole Placement

   Why are controllable canonical forms helpful when designing state feedback controllers in modern control systems?

   1. They eliminate the need for state measurements
   2. They guarantee the system is stable
   3. They automatically make the system observable
   4. They simplify the computation and implementation of the feedback gain

   Explanation: Controllable canonical forms organize the system equations so that designing and applying state feedback gains for pole placement is more straightforward. These forms do not inherently guarantee system stability, nor do they make a system automatically observable. Also, state measurement is still necessary unless an observer or estimator is used.