Surprise Quiz – 2 (CONTROL SYSTEMS)

Aashi is working on a project to design a control system for a new automated manufacturing process. She is considering different methods for analyzing the system. The primary advantage of the state-space formulation over transfer function methods for analyzing control systems is that it:

• Is only applicable to single-input, single-output (SISO) systems.

• Provides a unified framework for both time-domain and frequency-domain analysis.

• Is a simpler method for designing lead-lag compensators.

• Can elegantly handle multi-input, multi-output (MIMO) systems and is the basis for modern control theory.

In the state equation X΄=AX+BU, the 'A' matrix is also known as:

• The input matrix.

• The output matrix.

• The system matrix.

• The feedforward matrix.

Asher is an engineer working on a new control system for a robotic arm. He recalls that according to Kalman's test for controllability, a system is completely state controllable if and only if:

• The observability matrix has full rank.

• The determinant of the matrix A is non-zero.

• The system matrix A is diagonal.

• The controllability matrix [B AB A2B ... An–1B] has full rank.

In a control system for a smart home, the state transition matrix, Φ(t), fundamentally represents:

• The forced response of the system to a specific input.

• The solution to the homogeneous state equation x'=Ax, describing the free response.

• The output of the system at time t.

• The transfer function of the system in the frequency domain.

A control engineer is analyzing a dynamic system in a manufacturing plant. The observability matrix of the system is found to have a rank of 2. If the system is of order 3, what can be concluded?

• The system is completely observable.

• Only 2 of the 3 states are observable.

• The system is not observable.

• The system is controllable.

Ananya is designing a control system for a robotic arm. She prefers using the "canonical variable form" in state-space representation because it:

• Makes the system matrices (A, B, C) diagonal or in a standard controllable/observable form, simplifying analysis and design

• Directly represents the physical variables of the system, like velocity and position.

• Is the only form that can be used for MIMO systems

• Guarantees that the system is always stable.

In the context of controllability, what does Kalman's test assess?

• Kalman's test evaluates the stability of a system.

• Kalman's test assesses the controllability of a linear time-invariant system.

• Kalman's test measures the performance of a control algorithm.

• Kalman's test determines the optimal control inputs for a system.

Aarush is trying to understand a complex machine that he cannot see inside. What is the implication of this machine being unobservable?

• The machine can be easily controlled without any feedback.

• The machine's internal state cannot be inferred from its outputs.

• The machine's internal state is always visible to Aarush.

• The machine's outputs can be fully predicted from its inputs.

In a manufacturing plant, what does the controllability matrix indicate about the production system?

• It shows how the system's states can be influenced by the inputs.

• It provides the output response of the system.

• It determines the stability of the system.

• It is used to calculate the system's transfer function.

In state-space representation, what is the role of the 'B' matrix?

• It indicates the system's observability.

• It is used to calculate the state transition matrix.

• It defines how the input affects the state of the system.

• It represents the system's output dynamics.