Control Systems: Static, Dynamic Characteristics and Response
System Characteristics
In measurement and instrumentation, system characteristics are divided into static and dynamic depending on whether the input changes with time.
1. Static Characteristics
These describe the behavior of a system when the input is constant (not changing with time).
Definition: Static characteristics are the properties of a system measured under steady-state conditions.
Examples:
- Accuracy: How close the measurement is to the true value.
- Precision: Repeatability of measurements.
- Sensitivity: Ratio of output change to input change.
- Linearity: How well output follows a straight line.
- Resolution: Smallest detectable change.
- Drift: Change in output over time without input change.
In short: They define how correct and reliable the system is at steady state.
2. Dynamic Characteristics
These describe the behavior of a system when the input changes with time.
Definition: Dynamic characteristics are the properties that show how a system responds to time-varying inputs.
Examples:
- Speed of response: How fast the system reacts.
- Lag: Delay in response.
- Fidelity: Ability to reproduce input accurately over time.
- Dynamic error: Difference during changing conditions.
In short: They define how fast and accurately the system responds to changing inputs.
Time Response of a Control System
Definition: The time response is the output behavior of a system as a function of time when subjected to an input (e.g., step, ramp, or impulse).
Components:
- Transient Response: The temporary behavior occurring immediately after input application until the system stabilizes.
- Steady-State Response: The behavior that remains after transient effects have died out (t → ∞).
Formula: Time Response = Transient Response + Steady-State Response.
First-Order System Response to Ramp Input
A first-order system is represented by the transfer function: G(s) = 1 / (τs + 1), where τ is the time constant.
Output Response: For a ramp input r(t) = At, the time response is c(t) = A(t – τ + τe⁻ᵗ/τ).
Key Findings:
- The output follows the input with a lag.
- The steady-state error is Aτ (constant).
- The system cannot perfectly track a ramp input.
Time Domain Specifications
These describe how a system responds to a standard input:
- Delay Time (td): Time to reach 50% of the final value.
- Rise Time (tr): Time to rise from 0% to 100% of the final value.
- Peak Time (tp): Time to reach the first maximum peak.
- Maximum Overshoot (Mp): The peak value above the final value.
- Settling Time (ts): Time to stay within a specified error band (±2% or ±5%).
- Steady-State Error (ess): The difference between input and output as t → ∞.
Integral Controller
An integral controller produces an output proportional to the integral of the error signal.
Equations:
- Time Domain: u(t) = Ki ∫₀ᵗ e(τ) dτ
- Transfer Function: G(s) = Ki / s
Key Point: It eliminates steady-state error but may slow down the response and cause oscillations.