Transfer Function Zeros in Control Systems Theory

Control Systems

• A closed-loop system is a system where the output is defined and has parameters like rise time, peak time, and settling time.

Loop Paths

• A '23 '24 loop is a closed path that originates and terminates on the same node, with no node being met twice along the path.
• Non-touching loops are loops that do not have a common node.

Mason's Rule

• Mason's Rule is used to analyze and design control systems.

Signal-Flow Graph

• A signal-flow graph is a graphical representation of a control system.

Transfer Function

• The transfer function is a mathematical representation of a control system.
• Zeros are the values of s that cause G(s) to become zero.
• Poles are the values of s that cause G(s) to become infinite.
• The transfer function can be written in zero-pole form.

Time Response Analysis

• The output response of a system is a sum of a forced response and a natural response.
• The forced response is the system response to an external stimulus with zero initial conditions.
• The natural response is the system response to initial conditions with all external forces set to zero.

Standard Test Signals

• Standard test signals are used to verify a design and observe the dynamic behavior of a system.
• Examples of standard test signals include impulse, step, ramp, parabolic, and sinusoidal.

First-Order Systems

• A first-order system without zeroes is a unit step response.
• The transient response of a first-order system can be classified into three types: overdamped, underdamped, and critically damped.

Second-Order Systems

• The general second-order transfer function is given as: $$\frac{K}{s^2 + 2\zeta\omega_n s + \omega_n^2}$$
• The output time response of a second-order system is given as: $$c_1e^{(-\zeta\omega_n + \omega_d)t} + c_2e^{(-\zeta\omega_n - \omega_d)t}$$
• The time specifications of a second-order system include rise time, peak time, percent overshoot, and settling time.

Time Specifications

• Rise time is the time required for the waveform to go from 0.1 to 0.9 of the final value.
• Peak time is the time required to reach the first or maximum peak.
• Percent overshoot is the amount that the waveform overshoots the steady-state, or final, value at the peak time.
• Settling time is the time required for the transient's damped oscillations to reach and stay within 2% of the steady-state value.