In control systems, understanding the system response is essential to designing stable and high-performance systems. A system's response can be divided into two primary phases:
1. Transient Response: The behavior of the system immediately after a change in input (such as a step input) before it reaches steady-state. It reflects how the system reacts to initial disturbances and how quickly it settles to its final value.
2. Steady-State Response: The behavior of the system after it has had enough time to settle and reach equilibrium. It reflects how the system behaves once transient effects have subsided.
Both responses are important for assessing system stability, speed, and accuracy.

The transient response of a system describes how the output behaves immediately after a change in the input, such as a sudden step, ramp, or impulse input. It is characterized by the following features:
1. Rise Time (trt_r): The time it takes for the output to go from 10% to 90% of its final value. It indicates how quickly the system responds to changes.
2. Settling Time (tst_s): The time required for the output to stay within a certain percentage (e.g., 2% or 5%) of its final value. It provides an indication of how long the system takes to stabilize after a disturbance.
3. Overshoot (MpM_p): The maximum peak value of the output response, expressed as a percentage of the steady-state value. Overshoot reflects how much the system exceeds the desired output before settling down.
4. Peak Time (tpt_p): The time it takes for the system to reach the first peak of its response.
5. Damping Ratio (ζ): A dimensionless measure that describes the amount of damping in the system. It influences the speed and overshoot of the transient response. Systems with higher damping have less oscillation and faster settling times.
6. Natural Frequency (ωn): The frequency at which the system oscillates in the absence of damping. It is related to the speed of the transient response.

For a second-order system, the transfer function is given by:
G(s)=ωn² / (s² + 2ζωns + ωn²)
The system's time-domain response to a step input (using the inverse Laplace transform) is given by:
y(t)=1−1/√(1−ζ²)e^(−ζωnt)sin(ω_d t + ϕ)
where:
● ωd=ωn√(1−ζ²) is the damped natural frequency.
● ϕ=arccos(ζ) is the phase angle.

● Underdamped (0<ζ<1): The system exhibits oscillations that decay over time.
● Critically damped (ζ=1): The system returns to steady-state without oscillating, but as fast as possible.
● Overdamped (ζ>1): The system returns to steady-state slowly without oscillating.

Once the system has settled and transient effects have subsided, the system reaches a steady-state. The steady-state response describes how the output behaves in response to a constant input after the transient effects die out. Key factors in analyzing steady-state response include:
1. Steady-State Error: The difference between the desired output and the actual output as time approaches infinity.
2. Error Constants: These are used to determine the steady-state error for different types of inputs: Step input, ramp input, or parabolic input.

Error Constants:

1. Position Error Constant Kp: Determines the steady-state error for a step input.
   Kp=lim s→0G(s)H(s)
2. Velocity Error Constant Kv: Determines the steady-state error for a ramp input.
   Kv=lim s→0 s⋅G(s)H(s)
3. Acceleration Error Constant Ka: Determines the steady-state error for a parabolic input.
   Ka=lim s→0 s²⋅G(s)H(s)

Steady-State Error Formulae:

• Step Input: e_ss = 1/(1 + Kp)
• Ramp Input: e_ss = 1/Kv
• Parabolic Input: e_ss = 1/Ka

In this chapter, we examined both the transient and steady-state responses of control systems. The transient response provides insight into the system's behavior during the transition from one state to another, while the steady-state response describes the system's behavior once it has settled. Engineers use various time-domain and frequency-domain methods to analyze these responses, allowing them to design systems with optimal performance in terms of speed, accuracy, and stability.