Introduction:
In a first-order dynamic linear system with a proportional controller, an offset is observed when a unit step input is applied. This offset refers to a steady-state error, where the output of the system does not reach the desired value. To reduce this offset, different control strategies can be employed. Among these strategies, adding an integral mode is the most effective method.

Explanation:
To understand why adding an integral mode can reduce the offset in a first-order dynamic linear system, let's first discuss the characteristics of the proportional controller.

1. Proportional Controller:
A proportional controller adjusts the output based on the error between the desired setpoint and the measured value. The control effort is proportional to this error, which means that the controller output is directly proportional to the error. However, a proportional controller alone cannot eliminate the steady-state error.

2. Offset in a First-Order System:
In a first-order system, the output response to a unit step input is characterized by a time constant and a gain. The time constant determines the speed at which the output approaches the final value, while the gain determines the magnitude of the output. However, due to the presence of a proportional controller, the output does not exactly reach the desired value, resulting in an offset or steady-state error.

3. Integral Mode:
Adding an integral mode to the control system allows the controller to integrate the error over time. This means that the control effort is not only based on the instantaneous error but also on the accumulated error over time. As a result, the integral mode can eliminate the steady-state error and bring the output closer to the desired setpoint.

4. Effect of Integral Mode:
When the integral mode is added to the proportional controller, the integrated error drives the controller output. As the error persists, the integral term continuously increases, leading to a larger control effort. This increased control effort compensates for the offset in the system and brings the output closer to the desired value.

Conclusion:
In a first-order dynamic linear system with a proportional controller, the offset or steady-state error can be reduced by adding an integral mode. The integral mode allows the controller to accumulate and integrate the error over time, resulting in an increased control effort that compensates for the offset. This control strategy is effective in minimizing the steady-state error and improving the system's response to a unit step input.