Teaching · Control Systems

Modern Control Systems

From Physical Modeling to Nyquist Stability and Frequency-Domain Design

This nine-chapter course builds a coherent path from feedback fundamentals and mathematical modeling to state-variable analysis, performance, stability, root locus, frequency response, and Nyquist-based design.

System foundationsFeedback · physical modeling

State representationState variables · transition matrix

Feedback analysisPerformance · sensitivity · stability

Design methodsRoot locus · Bode · Nyquist

Course content

Chapters 1–9

The course follows the current nine lecture chapters, moving from control-system fundamentals and mathematical models through state variables, feedback characteristics, performance and stability to root-locus and frequency-domain design.

Chapter 1

Introduction to Feedback Control Theory

Introduces the purpose and architecture of control systems, the meaning of negative feedback, the roles of sensors and actuators, and the effects of disturbances and measurement noise through practical examples.

PDFLecture slides (PDF)

1.1

Purpose and Architecture of a Control System

Reference, output, error, and control signals
Regulation and command following in physical systems

1.2

Negative Feedback and Loop Signals

Why the feedback junction uses a negative sign
Open-loop and closed-loop behavior

1.3

Sensors, Actuators, Disturbances, and Noise

How measured outputs and physical inputs are created
Disturbance paths, measurement noise, and multiloop systems

1.4

Worked Control-System Examples

Thermostat regulation and cruise control
Mapping reference, output, error, and input signals

Chapter 2

Mathematical Models of Systems

Develops mathematical models from physical laws, moving from mechanical and electrical examples to state-space representations, linear approximations around operating points, and transfer-function descriptions.

PDFLecture slides (PDF)

2.1

Modeling Methodology

Define the system boundary, variables, and assumptions
Apply conservation laws and constitutive relations step by step

2.2

Mechanical-System Models

Mass–spring–damper free-body diagrams
Newtonian equations and state-space form

2.3

Electrical-System Models

Series RLC element relations and Kirchhoff laws
Second-order differential equations and state variables

2.4

Equivalent Mathematical Representations

Differential equations, state space, and transfer functions
Mechanical–electrical system analogies

2.5

Linear Approximation of Physical Systems

Taylor expansion and Jacobian matrices
Single-variable and multivariable linearization

2.6

Operating-Point Models and Transfer Functions

Nonlinear spring and pendulum examples
Validity of local models and transfer-function construction

Chapter 3

State Variable Models

Builds time-domain state-space models, develops their solutions through the matrix exponential and state-transition matrix, and connects physical state models, canonical realizations, signal-flow graphs, and transfer functions.

PDFLecture slides (PDF)

3.1

State, State Variables, and State Selection

State as the internal information required to predict motion
Energy variables, model order, and non-unique state choices

3.2

State and Output Equations

From higher-order differential equations to first-order vectors
Constructing the A, B, C, and D matrices

3.3

State Transition and Time Response

Matrix exponential and the general state solution
Zero-input and zero-state response decomposition

3.4

State Space and Transfer Functions

Phase-variable realization from a transfer function
Computing transfer functions from state-space models

3.5

Signal-Flow Graphs and Canonical Forms

Mason's gain formula and integrator chains
Phase-variable and input-feedforward realizations

3.6

Physical and Numerical Examples

Mass–spring–damper, RLC, and DC-motor models
Physical states versus canonical coordinates

Chapter 4

Feedback Control System Characteristics

Explains how feedback changes system behavior through closed-loop signal relations, sensitivity reduction, disturbance and noise paths, transient shaping, and steady-state tracking accuracy.

PDFLecture slides (PDF)

4.1

Open-Loop and Closed-Loop Control

Standard feedback architecture and closed-loop advantages
Reference tracking and internal signal paths

4.2

Error-Signal Analysis

Tracking error in terms of sensitivity functions
Unity and nonunity feedback configurations

4.3

Sensitivity to Parameter Variations

Normalized sensitivity and model uncertainty
How loop gain reduces parameter sensitivity

4.4

Disturbance and Measurement-Noise Responses

Plant-input and plant-output disturbance paths
Frequency-dependent noise attenuation

4.5

Control of the Transient Response

Feedback shaping of time constants and damping
Relating transient specifications to pole locations

4.6

Steady-State Error

Final-value theorem and tracking accuracy
Effect of loop type and low-frequency gain

Chapter 5

The Performance of Feedback Control Systems

Organizes feedback performance around standard test inputs and time-domain specifications, then relates second-order response, pole–zero location, steady-state error, and integral performance indices.

PDFLecture slides (PDF)

5.1

Performance Vocabulary and Specifications

Transient and steady-state measures
Rise time, peak time, overshoot, and settling time

5.2

Standard Test Input Signals

Impulse, step, ramp, and parabolic commands
Selecting an input that exposes the required behavior

5.3

Performance of Second-Order Systems

Natural frequency and damping ratio
Analytical and graphical response specifications

5.4

Effects of Additional Poles and Zeros

How an added zero changes transient shape
Dominant-pole approximation and third-pole effects

5.5

s-Plane Root Location and Transient Response

Pole geometry and response speed
Damping-ratio and natural-frequency design regions

5.6

Steady-State Error of Feedback Systems

System type and static error constants
Step, ramp, and parabolic tracking error

5.7

Performance Indices

Integral error criteria for comparing responses
Performance trade-offs and design interpretation

Chapter 6

Stability of Linear Feedback Systems

Develops stability concepts and algebraic tests for feedback systems, including Routh–Hurwitz analysis, relative stability, state-variable stability, and design examples with parameter constraints.

PDFLecture slides (PDF)

6.1

Stability Definitions

Absolute, relative, internal, and input–output stability
Pole locations and bounded system responses

6.2

Routh–Hurwitz Stability Criterion

Constructing and interpreting the Routh array
First-column sign changes and special cases

6.3

Relative Stability

Shifted-axis Routh tests
Decay-rate and damping requirements in the s-plane

6.4

Stability of State-Variable Systems

Eigenvalues of the state matrix
Hidden modes, minimality, and BIBO stability

6.5

Stability Design Examples

Parameter regions that preserve stability
Tracked-vehicle and motorcycle-balance examples

Chapter 7

The Root-Locus Method

Presents root locus as a graphical description of how closed-loop poles move with gain, develops a complete sketching procedure, and applies the method to gain, lead, lag, PI, and PD design.

PDFLecture slides (PDF)

7.1

Root-Locus Concept and Conditions

Closed-loop characteristic equation with variable gain
Angle and magnitude criteria

7.2

Basic Root-Locus Rules

Branch count, symmetry, and real-axis segments
Asymptote angles and centroid

7.3

Completing the Root-Locus Sketch

Breakaway and break-in points
Imaginary-axis crossings and gain reading

7.4

Angles of Departure and Arrival

Complex-pole and complex-zero geometry
Seven-step sketching workflow and common pitfalls

7.5

Design with Root Locus

Mapping time-domain specifications to the s-plane
Gain, lead/PD, and lag/PI compensation

Chapter 8

Frequency Response Methods

Introduces sinusoidal steady-state response and Bode, polar, and Nichols representations, then develops measured frequency-response estimation, stability margins, bandwidth, and compensator design.

PDFLecture slides (PDF)

8.1

Frequency-Response Foundations

Sinusoidal steady-state magnitude and phase
Frequency response from transfer functions and state-space models

8.2

Bode, Polar, and Nichols Representations

Factor-by-factor asymptotic Bode construction
Equivalent polar and log-magnitude–phase views

8.3

Frequency-Response Measurement

Excitation signals, leakage, and windowing
Spectral estimators, coherence, and closed-loop extraction

8.4

Frequency-Domain Specifications

Gain crossover, phase crossover, and stability margins
Bandwidth, resonance, sensitivity, and time-domain expectations

8.5

Nichols Charts

Mapping open-loop response to closed-loop contours
Reading margins, peaking, and bandwidth

8.6

Frequency-Domain Design Examples

Lead and lag compensation
Lead–lag, notch filtering, and integral-action design

homework

Homework Solutions

This is the homework solutions.

PDFLecture slides (PDF)

Chapter 9

Feedback Control: MATLAB Practice

Use MATLAB and the Control System Toolbox to model, analyze, and design feedback systems through 15 practice problems and two DC motor projects, connecting theoretical predictions with simulation results.

PDFLecture 01 Tutorials (PDF) PDFHomework 01 (PDF) PDFLecture 02 Tutorials (PDF) PDFHomework 02 (PDF)

9.1

Physical Modeling and Simulation

Build transfer-function and state-space models of thermal, mechanical, and electrical systems
Linearize nonlinear dynamics and compare nonlinear and small-signal responses

9.2

State-Space Analysis and Discretization

Check controllability, observability, canonical forms, and equivalent state-space realizations
Discretize continuous-time models and examine the effects of sampling time

9.3

Feedback Performance and Stability

Construct series, parallel, and feedback interconnections and evaluate tracking error
Analyze rise time, overshoot, settling time, and stability using simulated responses and closed-loop poles

9.4

Sensitivity, Disturbances, and Measurement Noise

Evaluate sensitivity to gain variations and compare disturbance rejection at the plant input and output
Use complementary sensitivity and Bode plots to assess measurement-noise amplification

9.5

Controller Design and Performance Trade-offs

Tune PI controllers and compare tracking performance across different closed-loop bandwidths
Design pole-placement and LQR controllers and compare response speed, tracking, and control effort

9.6

Mini Project: DC Motor Speed Control

Derive the electromechanical model and design a PI controller to meet speed-tracking specifications
Verify rise time, overshoot, steady-state error, and load-torque disturbance rejection, with an optional LQR comparison

9.7

Mini Project: DC Motor Position Control

Tune a PI or PID controller to meet settling-time, overshoot, and steady-state position-error requirements
Assess closed-loop stability, disturbance rejection, and noise sensitivity, and document the design with reproducible MATLAB code