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EE-561: Digital Control Systems


Dr. Abubakr Muhammad, Assistant Professor of Electrical Engineering

Email: abubakr [at]

Office: Room 9-351A, 3rd Floor, SSE Bldg

TA: Usaman Bin Sikandar, Mudassar Ejaz

Course Details

Year: 2013-14

Semester: Fall

Category: Graduate

Credits: 3

Elective course for electrical engineering majors

Course Website:

Course Description

Design and digital implementation of multiple-input, multiple-output (MIMO) linear feedback control systems for specified dynamic response; z-transform and sampling; exposure to embedded control systems; state-space based models; introduction to advanced concepts of multi-variable control. Design and implementation project on real-time digital control.


  • To build on students’ undergraduate exposure to feedback control and teach advanced design techniques
  • To impart knowledge of practical issues related to the implementation of feedback controllers using digital processors.
  • To introduce advanced techniques of linear multivariable control system design.
  • To expose students to aspects of embedded control systems as practiced in robotics, automotive, aerospace, process industries.
  • To prepare students for advanced courses in mathematical control theory and practical control engineering.

Learning Outcomes

  • Identify state, measurement and control in a given problem.
  • Design controllers for linear models of systems using MATLAB and SIMULINK.
  • Select and program real-time digital controllers on platforms similar to PC, microcontroller, DSP.
  • Predict performance for complex multivariable control tasks.


EE-361. Feedback Control Systems

Text book

The course will be taught from the following textbooks.

  • FranklinF. Feedback control of dynamic systems by Franklin, Powell and Emami-Naeni (5th edition), Pearson, 2006.
  • FranklinD. Digital control of dynamic systems by Franklin, Powell and Workman (3rd edition), Addison Wesley, 2000.

Other important references include

  • Astrom. Computer controlled systems by Karl Astrom and Bjorn Witternmark, Prentice Hall, 1997.
  • Oppenheim. Signals and Systems by Oppenheim, Wilsky, Nawab (2nd edition), Pearson, 1997.
  • Chen. Linear Systems Theory and Design by C.T. Chen. Holt, Rinehart & Winston, 1999.

Grading Scheme

Homeworks+Quiz : 15%

Project: 20%

Midterm: 30%

Final : 35%

Policies and Guidelines

  • Quizzes will be announced. There will be no makeup quiz.
  • Homework will be due at the beginning of the class on the due date. Late homework will not be accepted.
  • You are allowed to collaborate on homework. However, copying solutions is absolutely not permitted. Offenders will be reported for disciplinary action as per university rules.
  • Any appeals on grading of homeworks, quiz or midterm scores must be resolved within one week of the return of graded material.
  • Attendance is in lectures and tutorials strongly recommended but not mandatory. However, you are responsible for catching the announcements made in the class.
  • Many of the homeworks will include MATLAB based computer exercise. Some proficiency in programming numerical algorithms is essential for both the homework and project.

Course Delivery Method

Lectures. Mon, Wed: 11:00am-12:15pm. 10-302. SSE Bldg

Recitations. Fri. 9:00am-9:50am. SC-4. Student Center Bldg.


Week 1. Aug 19 Lecture 1. Introduction to concepts of control, feedback, feedforward, uncertainty and robustness;

Recitation. Review of SISO continuous-time signals and systems;

FranklinF Ch1;
Week 2. Aug 26 Lecture 2. Review of SISO feedback control; rational LTI systems; geometry of 2nd order poles; error expression in closed loop and open loop systems; sensitivity function; control design objectives;

Lecture 3. Summary of control design; compensators and PID controllers; introduction to sampled data systems; Naive approaches towards emulation; Euler's forward approximation; a pseudo-algorithm for controller implementation;

FranklinD 2, FranklinF 4.4
Week 3. Sept 2 Lecture 4. Digital control by emulation; Euler's forward and backward approximation; trapezoidal rule; approximation of a continuous time compensator; zero order hold (ZOH) and delays; general difference equations; introduction to the Z-transform;

Lecture 5. Solution of difference equations by Z-transform method; transfer functions; integrator approximation in transform domain; continuous-to-discrete approximations for controller synthesis by emulation; block diagram representations using delays, summers and gain

Recitation / Seminar. Feedback control scheduling of crane control systems. Announcement. Slides

FranklinD Ch 3

Home work #1

Home work #1 solutions

Week 4. Sept 9 Lecture 6. Impulse response and convolution in discrete-time systems; tests for linearity time-invariance, stability, causality; basic block diagrams; canonical forms;

Lecture 7. Frequency response of discrete-time LTI systems; Discrete-time Fourier transform; relation to Z-transform; time and frequency analysis of prototypical first order and second order discrete-time systems;

Lecture 8. Comparison of Z-transform and Laplace transform of sampled signals; mapping between s-plane and z-plane; mappings induced by Euler and trapezoidal approximations; Tustin's approximation of a continuous-time first order system; distortion in frequency response due to trapezoidal approximation;

FranklinD Ch 3; Oppenheim 5.1.1, 6.6.2; FranklinD 6.1;

Home work #2

Home work #2 solutions

Week 5. Sept 16 Lecture 9. Example on frequency response distortion(contd.); introduction to state space analysis; idea of a state; state-space model of Newtonian mechanics; FranklinD 6.1; FranklinF Ch 7;
Week 6. Sept 23 Lecture 10. Examples of state-space modeling; block diagrams and state-space models;

Lecture 11. Control canonical form and modal canonical form for SISO systems; how to find explicit transformations to setup control canonical form; the idea of a controllability matrix;

Recitation. Quiz #1

Quiz #1 solutions

FranklinF Ch 7.2, 7.3;

Home work #3

Home work #3 solutions

Week 7. Sept 30 Lecture 12. Controllability (contd.); invariance of controllability condition under invertible transformations; computing dynamic response from state-equations using Laplace transform; relationship between transfer functions and state-space models for a SISO LTI system;

Lecture 13. Interpretation of transfer function poles in state-space models (contd.); Interpretation of transfer function zeros in state-space models; constructing explicit transformations to obtain Modal canonical form; poles, modes and eigen-decomposition of system matrix; examples;

FranklinF Ch 7.3, 7.4;
Week 8. Oct 7 Lecture 14. Problem solving & review session

Midterm Exam

Midterm Exam solutions

Week 9. Oct 14 Eid/Midterm Break.
Week 10. Oct 21 Lecture 15. Review of canonical forms; the concept of state feedback; pole-placement; Ackermann's formula;

Lecture 16. Review of pole placement; Ackermann's formula and controllability; how to add references for trajectory tracking; state-estimator design; concept of an observer; observer design; observer canonical form; observer design by Ackermann's formula; observability matrix;

FranklinF 7.5.1, 7.5.2;

Home work #4

Home work #4 solutions

Week 11. Oct 28 Lecture 17. Review of combined state feedback and observor design; definitions of controllability and observability; physical interpretation of observability and controllability; examples of uncontrollable and unobservable systems from circuit theory and mechanics;

Lecture 18. Solutions of continuous-time LTI state-space models; a re-look at forced and natural responses; matrix exponential and its properties; Discrete-time state space models; discretized matrix equivalents of continuous-time LTI models;

FranklinF 7.7.1, 7.8; Chen 4.2;
Week 12. Nov 4 Lecture 19. Discrete-time state space models continued; example of double-integrator; a re-look at Zero order hold (ZOH) in state-space models and transfer functions;

Lecture 20. Discrete-time state-space form of Euler and Tustin's approximations; full-state feedback control in discrete-time LTI systems; pole placement; control canonical form; Ackermann's formula re-visited; graphical understanding of z-plane via mapping from s-plane contours of constant damping ratio, natural frequency;

Lecture 21 / Recitation. State-space models from difference equations; FIR and IIR filters; state-space modeling example: Laplacian dynamics in networked control systems

FranklinD 4.3.3; 4.3.1; 4.2.3; 8.1;
Week 13. Nov 11 Lecture 22. Prediction estimators; observability in discrete-time; observability as a dual concept to controllability; derivation of Ackermann's formula for observer design;

Lecture 23. Discrete-time Regulator design; combining control law and estimator; proof of Separation Principle; regulators reinterpreted as classical z-domain compensators;

Quiz #2

Quiz #2 solutions

FranklinD 8.2.1; 8.3;

Home work #5

Home work #5 solutions

Week 14. Nov 18 Lecture 24. Adding reference to standard regulator design for trajectory tracking; Feedforward loop for zero tracking error; determination of pre-filter matrices; state-command structure; output command structure;

Lecture 25. Integral control; state augmentation; disturbance estimation; observability and disturbance estimation;

Recitation. Jordan decomposition method for handling repeated and complex poles.

FranklinD 8.4;

Notes on Jordan decomposition.

Week 15. Nov 25 Lecture 26. Optimal control in discrete-time; Lagrange multiplier method;

Lecture 27. Linear Quadratic Regulator (LQR); steady-state optimal control; algebraic Ricatti equations;

FranklinD 9.2;

Home work #6

Home work #6 solutions

Week 16. Dec 2 Lecture 28. Review Lecture

Quiz #3

Project Presentations.

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