# Control Systems

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REVIEW PROBLEMS

00A-Laplace Transform Analysis and Block Diagram Reduction Techniques

00B-Modeling and Second-order Systems

00C-Routh-Hurwitz Tests and Steady-state Errors

00D-Nyquist Stability Criterion

00E-Bode Plots

00F-Root Locus

LECTURE NOTES & HOMEWORK SOLUTIONS

Laplace Transform Review, Transfer Function, and Stability

Topics include: Unilateral Laplace Transform, Partial Fraction Expansion Method, Laplace Transform Pairs, Properties of Unilateral Laplace Transform, Transfer Function, Characteristic Polynomial, Characteristic Equation, Characteristic Roots, Zero-state Response, Zero-input Response, Forced Response, Natural Response, Strictly Proper Transfer Function, Coprime (Irreducible) Transfer Function, Poles, Zeros, Pole-Zero Cancellation, Asymptotic Stability, Bounded Input Bounded Output Stability, Internal Stability, Examples, Homework Assignment.

01-Laplace Transform Review, Transfer Function, and Stability - Notes

01-Laplace Transform Review, Transfer Function, and Stability - Homework Solutions

Time-domain Response

Topics include: First-order System, Step Response of First-order System, Examples of First-order System Step Response, Second-order System, Examples of Second-order System Step Response, Prototype Second-order System - General Forms of Step Response, Prototype Second-order System - Underdamped Step Response Specifications, Prototype Second-order System Underdamped Step Response - Effects of Varying Complex Conjugate Pole Locations, Examples with Underdamped Step Response Specifications, Effects on Step Response of Adding Left-half Plane Zero to Closed-loop Transfer Function of Prototype Second-order System, Effects on Step Response of Adding Right-half Plane Zero to Closed-loop Transfer Function of Prototype Second-order System, Effects on Step Response of Adding Left-half Plane Zero to Forward-path Transfer Function of Prototype Second-order System, Effects on Step Response of Adding Left-half Plane Pole to Closed-loop Transfer Function of Prototype Second-order System, Effects on Step Response of Adding Left-half Plane Pole to Forward-path Transfer Function of Prototype Second-order System.

02-Time-domain Response - Notes

02-Time-domain Response - Homework Solutions

Steady-state Error

Topics include: System Type, Steady-state Error and Position Error Constant Under Unit Step (Position) Input, Steady-state Error and Velocity Error Constant Under Unit Ramp (Velocity) Input, Steady-state Error and Acceleration Error Constant Under Parabolic (Acceleration) Input, Examples of Steady-state Error Calculations.

03-Steady-state Error - Notes

03-Steady-state Error - Homework Solutions

Design Using Root Locus Plot: Cascade Proportional Compensator

Topics include: Examples of Dominant Pole-pair Approximation of High-order Systems, Cascade Proportional Compensator, Example of Transient Design Via Gain Adjustment Using Root Locus Plot.

04-Design Using Root Locus Plot: Cascade Proportional Compensator - Notes

04-Design Using Root Locus Plot: Cascade Proportional Compensator - Homework Solutions

Design Using Root Locus Plot: Cascade PI Compensator

Topics include: Ideal Integral Compensator, Compensator Pole at Origin of Complex Plane, Steady-state Error Zero, Cascade Proportional-plus-Integral (PI) Compensator, Example of Cascade PI Compensator Design Using Root Locus Plot.

05-Design Using Root Locus Plot: Cascade PI Compensator - Notes

05-Design Using Root Locus Plot: Cascade PI Compensator - Homework Solutions

Design Using Root Locus Plot: Cascade Lag Compensator

Topics include: Non-ideal Integral Compensator, Compensator Pole Near Origin of Complex Plane, Steady-state Error Improved, Cascade Lag Compensator, Example of Cascade Lag Compensator Design Using Root Locus Plot.

06-Design Using Root Locus Plot: Cascade Lag Compensator - Notes

06-Design Using Root Locus Plot: Cascade Lag Compensator - Homework Solutions

Design Using Root Locus Plot: Cascade PD Compensator

Topics include: Ideal Derivative Compensator, Design Procedure to Select Cascade PD Compensator Zero, Improve Transient Response Performance, Cascade Proportional-plus-Derivative (PD) Compensator, Example of Cascade PD Compensator Design Using Root Locus Plot.

07-Design Using Root Locus Plot: Cascade PD Compensator - Notes

07-Design Using Root Locus Plot: Cascade PD Compensator - Homework Solutions

Design Using Root Locus Plot: Cascade Lead Compensator

Topics include: Non-ideal Derivative Compensator, Design Procedure to Select Cascade Lead Compensator Zero and Pole, Improve Transient Response Performance, Cascade Lead Compensator, Example of Cascade Lead Compensator Design Using Root Locus Plot.

08-Design Using Root Locus Plot: Cascade Lead Compensator - Notes

08-Design Using Root Locus Plot: Cascade Lead Compensator - Homework Solutions

Design Using Root Locus Plot: Cascade PID Compensator

Topics include: PID Compensator Composed of Ideal Derivative and Ideal Integral Compensator Sections, Design Procedures to Select Zeros of PD and PI Compensator Sections, Improve Transient Response Performance with PD Compensator Section and Drive Steady-state Error to Zero with PI Compensator Section, Cascade Proportional-plus-Integral-plus-Derivative (PID) Compensator, Example of Cascade PID Compensator Design Using Root Locus Plot.

09A-Design Using Root Locus Plot: Cascade PID Compensator - Notes

09B-Design Using Root Locus Plot: Cascade PID Compensator - Homework Solutions

09C-Design Using Root Locus Plot: Cascade PID Compensator - Example

Design Using Root Locus Plot: Cascade Lag-Lead Compensator

Topics include: Lag-Lead Compensator Composed of Non-ideal Derivative and Non-ideal Integral Compensator Sections, Design Procedures to Select Poles and Zeros of Lead and Lag Compensator Sections, Improve Transient Response Performance with Lead Compensator Section and Improve Steady-state Error with Lag Compensator Section, Cascade Lag-Lead Compensator, Example of Cascade Lag-Lead Compensator Design Using Root Locus Plot.

10-Design Using Root Locus Plot: Cascade Lag-Lead Compensator - Notes

10-Design Using Root Locus Plot: Cascade Lag-Lead Compensator - Homework Solutions

Design Using Bode Plots: Frequency Response Analysis Review

Topics include: Nyquist Path; Nyquist Plot; Procedure to Sketch Complete Nyquist Plot; Applying Nyquist Criterion to Assess Stability with Complete Nyquist Plot; Simplified Nyquist Plot and Nyquist Criterion for Loop Transfer Function with No Pole in Right-half of s-plane; Minimum Phase System; Using Simplified Nyquist Plot to Evaluate Phase Crossover Frequency, Gain Margin, Gain Crossover Frequency, and Phase Margin; Using Bode Plots to Evaluate Phase Crossover Frequency, Gain Margin, Gain Crossover Frequency, Phase Margin, and Steady-state Error.

11-Design Using Bode Plots: Frequency Response Analysis Review - Notes

Design Using Bode Plots: Cascade Proportional Compensator

Topics include: Frequency-domain Specifications, Relationship of Peak Resonance to Damping Ratio, Relationship of Resonant Frequency to (Undamped) Natural Frequency and Damping Ratio, Relationship of Bandwidth to (Undamped) Natural Frequency and Damping Ratio, Relationship of Bandwidth to Settling Time and Damping Ratio, Relationship of Bandwidth to Peak Time and Damping Ratio, Relationship of Bandwidth to Rise Time and Damping Ratio, Relationship of Phase Margin to Damping Ratio, Using Bode Plots to Find Static Error Constants of Unity Feedback Systems, Using Bode Plots to Design Cascade Proportional Compensator, Example of Transient Design Via Gain Adjustment Using Bode Plots.

12-Design Using Bode Plots: Cascade Proportional Compensator - Notes

Design Using Bode Plots: Cascade Lag & PI Compensators

Topics include: Procedure to Design Cascade Lag Compensator Using Bode Plots, Procedure to Design Cascade PI Compensator Using Bode Plots, Example of Cascade Lag Compensator Design Using Bode Plots, Example of Cascade PI Compensator Design Using Bode Plots.

13A-Design Using Bode Plots: Cascade Lag & PI Compensators - Notes

13B-Design Using Bode Plots: Cascade Lag Compensator - Example

13C-Design Using Bode Plots: Cascade PI Compensator - Example

Design Using Bode Plots: Cascade Lead & PD Compensators

Topics include: Procedure to Design Cascade Lead Compensator Using Bode Plots, Procedure to Design Cascade PD Compensator Using Bode Plots, Example of Cascade Lead Compensator Design Using Bode Plots, Example of Cascade PD Compensator Design Using Bode Plots.

14A-Design Using Bode Plots: Cascade Lead & PD Compensators - Notes

14B-Design Using Bode Plots: Cascade Lead Compensator - Example

14C-Design Using Bode Plots: Cascade PD Compensator - Example

Design Using Bode Plots: Cascade Lag-Lead & PID Compensators

Topics include: Procedure to Design Cascade Lag-Lead Compensator Using Bode Plots, Procedure to Design Cascade PID Compensator Using Bode Plots, Example of Cascade Lag-Lead Compensator Design Using Bode Plots, Example of Cascade PID Compensator Design Using Bode Plots.

15A-Design Using Bode Plots: Cascade Lag-Lead & PID Compensators - Notes

15B-Design Using Bode Plots: Cascade Lag-Lead Compensator - Example

15C-Design Using Bode Plots: Cascade PID Compensator - Example