1. A system is said to be completely controllable if it is possible to transfer the system state from any initial state X(to) to any desired state X(t) in specified finite time by a control vector u(t).

2. A system is said to be completely observable, if every state X(to) can be completely identified by measurements of the outputs y(t) over the time interval (0 ≤ t ≤ ∞).

Read the above two statements and select the correct option.

Explanation:

System Controllability and Observability

In control system theory, the concepts of controllability and observability are fundamental for understanding the behavior and capabilities of a system. These properties determine whether a system can be controlled to achieve desired states and whether the internal states of the system can be inferred from external outputs, respectively.

Statement 1 Analysis:

Statement 1: "A system is said to be completely controllable if it is possible to transfer the system state from any initial state X(to) to any desired state X(t) in specified finite time by a control vector u(t)."

This statement defines the concept of controllability. A system is considered completely controllable if, given any initial state, it is possible to drive the system to any final state using an appropriate control input within a finite time. This property ensures that the system can be maneuvered to follow desired trajectories or reach specific states, which is crucial for the effective control and operation of the system.

The given statement accurately captures the essence of controllability. Therefore, Statement 1 is true.

Statement 2 Analysis:

Statement 2: "A system is said to be completely observable if every state X(to) can be completely identified by measurements of the outputs y(t) over the time interval (0 ≤ t ≤ ∞)."

This statement defines the concept of observability. A system is considered completely observable if the current state of the system can be determined from its outputs over a given time interval. This property is essential for system monitoring, state estimation, and feedback control, as it ensures that the internal states of the system can be inferred from external measurements.

The given statement accurately describes the concept of observability. Therefore, Statement 2 is true.

Correct Option Analysis:

The correct option is:

Option 2: Both Statement 1 and Statement 2 are true.

This option correctly identifies that both statements accurately describe the fundamental properties of controllability and observability in control system theory.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: Statement 1 is true and Statement 2 is false.

This option is incorrect because both statements accurately describe the concepts of controllability and observability, so Statement 2 is not false.

Option 3: Both Statement 1 and Statement 2 are false.

This option is incorrect because both statements are true and provide accurate definitions of controllability and observability.

Option 4: Statement 2 is true and Statement 1 is false.

This option is incorrect because Statement 1 is true, as it correctly defines the concept of controllability.

Conclusion:

Understanding the concepts of controllability and observability is essential for the design and analysis of control systems. These properties ensure that the system can be effectively controlled and monitored. Both statements provided in the question accurately describe these concepts, making Option 2 the correct choice.