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EE-561: Digital Control Systems
Spring 2017


Talha Manzoor, Teaching Fellow & PhD Candidate in Electrical Engineering


Office: CyPhyNetS Lab, Right Wing, 2nd Floor, SSE Bldg

TA: Syed Muhammad Ahsan Razvi, MS Electrical Engineering


Office: RA-Room 9-239, Power Systems Lab, Right Wing, 2nd Floor, SSE Bldg

Office Hours: 3:00-5:00 PM, Monday and Wednesday

Course Details

Year: 2016-17

Semester: Spring

Category: Graduate

Credits: 3

Elective course for electrical engineering majors. Core course for electrical engineering students pursuing an MS in the "Systems and Controls" stream.

Course Website:

Course Description

This course involves the design and analysis of control to be implemented by digital computers for systems that operate on continuous signals. Much of the material is an extension of the concepts learned in a undergraduate course on linear feedback control. The first part of the course focuses on the analysis of discrete time systems and the tools employed to study them. These include the language of difference equations, the z-transform, discretization of continuous-time systems, determining the dynamic response of discrete-time systems and the effects of sampling and quantization. The second part of the course covers the design of feedback control in discrete time domain which includes emulation of controllers designed in continuous time domain, discrete-time root-locus, frequency based design and state-space methods. Since the transform methods in discrete-time follow fairly easily from the transform design methods in continuous-time, a focus will be placed on state-space methods which includes both basic and advanced design techniques.


  • To expose undergraduate students finished with linear feedback control to advanced methods in control design.
  • To introduce graduate students to the perspective of digitization in control.
  • To impart the students with an appreciation of the adverse effects of sampling and quantization.
  • 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.
  • Understand the advantages and pitfalls of following a particular design methodology for digital controllers.
  • Be able to develop a satisfactory design with minimal compensation.
  • Intelligent selection of sample rates and quantization levels for real-time control.


  • EE-361. Feedback Control Systems (for undergrads)
  • A working knowledge of ordinary differential equations and linear algebra will be assumed while delivering the lectures.
  • Experience in programming with MATLAB will be required to solve some components of the assignments.

Text book

The course will be taught from the following textbook.

(Franklin) Digital control of dynamic systems by Franklin, Powell and Workman (3rd edition), Addison Wesley, 2000.

Other references

(Strang) Computational Science and Engineering, Wellesley-Cambridge Press, 2007

(FranklinF) Feedback Control of Dynamics Systems, Pearson Prentice Hall, 2013

(Ogata) Modern control engineering, Pearson Prentice Hall, 2010

Grading Scheme

Homeworks+Quiz : 30%

Midterm: 35%

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 in lectures is strongly recommended but not mandatory. However, you are responsible for catching the announcements made in the class.

Course Delivery Method

Lectures. Tue, Thu: 11:00am-12:15pm. 10-201. SSE Bldg


Week 1 Jan 23 Lecture 1 Digital Control: Motivation; Feedback control with DAC and ADC; The effect of discretization on stability and other response features; Introduction to difference equations; Example: Discretizing the Logistic Equation - solution by the recursive method;

Lecture 2 Introduction to the finite difference approximations (forward, backward and centered); Finite difference matrices of the first and second derivatives; Testing the accuracy of approximations through application to common signals (steps, ramps and parabolas); How computers solve difference equations: conversion to a linear algebra problem

Franklin Ch 1

discretizing the logistic equation

Week 2 Jan 30 Lecture 3 Example: solving a discrete IVP through finite difference matrices; The method of undetermined coefficients; Example: obtaining a closed form expression for the voltages in a resistor network; The link between finite difference approximations and rectangular rules

Lecture 4 Introduction to the Z-transform; Physical interpretation of z (delays); Solving difference equations through the Z-transform method; The role of bilateral and unilateral Z-transforms in dealing with initial conditions; Introduction to the Transfer function and z-domain representations of the integrator;

Strang Sec 1.2 example

Franklin Ch 4

Week 3 Feb 06 Lecture 5 The transfer function as the response to the discrete impulse; recovering the discrete-time convolution formula; Inverting the z-transform through long-division; Internal (asymptotic) and external (BIBO) stability for discrete systems

Lecture 6 The final value theorem for discrete-time systems, Deriving a relationship between the s and z variables via the impulse sampler, Corresponding time-domain signals for the s and z planes, mapping regions from the s plane to the z plane.

Franklin Ch 4

Homework 1

Week 4 Feb 13 Lecture 7 Discrete control design by emulation; Z.O.H equivalents; obtaining discrete equivalents through numerical integration; forward, backward and Tustin's substitution rules and the induced mappings from s to z-plane.

Lecture 8 Zero-pole matching equivalents; Realizing discrete controllers: direct and standard programming; Block diagrams and pseudo code; Minimizing memory elements: the control canonical form; Uncovering the underlying state-space representation.

Franklin Ch4, Franklin Ch 6

Week 5 Feb 20 Lecture 9 Block diagrams; pseudo code and memory elements cntd; the observer canonical form

Quiz 1

Lecture 10 Quiz 1 Review: A first-order hold equivalent; Discretizing the PID controller through numerical integration; Introduction to the concept of a state

Franklin Ch4
Week 6 Feb 27 Lecture 11 Introduction to state space representations; Concept and technical definitions; Deriving state space representations from physical models; The use of Newton's second law to determine system dynamics; Free body diagrams; Deriving state space representations from transfer functions; Dealing with derivatives of the input

Lecture 12 State space representations ctd; Using the control objective to select state variables; Obtaining State Space models through linearization; Deriving the Transfer Function from the State Space Matrices; Examples: free-falling particle, cruise control, mass spring system, inverted pendulum

FranklinF Ch 7

Homework 2

Week 7 Mar 06 Lecture 13 Transforming the state-space variables; Invariance of the transfer function to non-singular transformations of the state; State space representation of a series RLC circuit; Invariance of the characteristic equation under different choices of input and output; Eigenvalues of the system matrix and poles of the transfer function; Rules for internal stability (stable, marginally stable and unstable systems); Invariance of the eigenvalues under similarity transformations

Mid-term Exam

FranklinF Ch 7
Week 8 Mar 13 Lecture 14 Stability analysis from state space representations, Introduction to canonical forms: the control canonical form, The controllability matrix and transformation to the control canonical form

Lecture 15 Invariance of controllability under state transformations, The observer canonical form, Alternate definitions of controllability, observability and physical implications, Loss of controllability/observability and pole-zero cancellations, Modal canonical form

FranklinF Ch 7
Week 9 Mar 20 Mid-semester Break.
Week 10 Mar 27 Lecture 16 Controllabillity and Observability: review, Modal canonical and diagonal forms, Matrix diagonalization and transformation to modal form, Identifying controllable/observable modes of a system

Lecture 17 Commons reasons for loss of controllability/observability, diagonalizing systems with repeated eigenvalues, Algebraic and geometric multiplicity of eigenvalues, Introduction to the Jordan form

FranklinF Ch 7
Week 11 Apr 03 Lecture 18 Jordan forms ctd, Generalized eigenvectors and corresponding Jordan blocks, Obtaining real-valued block diagonal forms for systems with complex eigenvalues

Lecture 19 Control design in state-space; pole placement for regulator design; pole placement in control canonical form; Ackermann's formula for control design; Introduction of the reference input for tracking systems

Reading Material

FranklinF Ch 7

Week 12 Apr 10 Lecture 20 Observer design in state space; Observer pole placement, the observer canonical form and Ackermann's formula; Duality in control and estimation problems; The separation principle

Lecture 21 Solution to homogeneous state equations; The matrix exponential; Calculating the matrix exponential through power series, diagonalization and the Laplace transform; Solution to non-homogeneous state equations; Obtaining discrete state space models from continuous counterparts; The zero order hold equivalent in state space form

FranklinF Ch 7

Ogata Ch 9

Franklin Ch 4

Homework 3

Week 13 Apr 17

Lecture 22 Alternate formulas for state space discretization, Discretizing the constant acceleration model for a single particle, review: a generic algorithm for transformation to the Jordan form

Lecture 23 Revisiting Discrete-time equivalents; Discrete equivalents via numerical integration; The forward, backward and Tustin's substitution formulas in state-space form; Solving state space systems in discrete time through recursion, the discrete-time state-transition matrix; Solving state space systems using the Z-transform

Franklin Ch 4

Franklin Ch 6

Jordan form Algo

Week 14 Apr 24

Lecture 24 Discrete time state-space analysis; canonical forms; relationship between state-space representations and pulse transfer function; internal and BIBO stability; Controllability and Observability in discrete-time; Proof of the controllability rank criterion via The Cayley-Hamilton Theorem

Quiz 2

Lecture 25 State Space design in discrete-time; pole placement: matching coefficients, control canonical form and Ackermann's formula; Discrete-time observer design; Prediction estimators; Current estimators; Reduced-order estimators; The separation principle; Compensation: combined control law and observer design

Franklin Ch 8

Homework 4

Week 15 May 01

Lecture 26 Discrete time state space design continued; introduction of the reference input; feed-forward control loop for handling steady state errors; tracking with estimation: the state-command and output-error command structures; The history and evolution of mathematical programming: from calculus to optimal control

Lecture 27 Constrained minimization: the method of Lagrange multipliers; Setting up the Lagrangian; Conversion of constrained optimization problems to a set of simultaneous equations; Introduction to the LQR problem; Cost functionals and weighing matrices

Quiz 3

Franklin Ch 8

Franklin Ch 9

Week 16 May 08

Lecture 28 Time-varying optimal control as a constrained minimization problem; conversion to a two-point boundary problem via Lagrange multipliers; solving the two-point boundary problem via the sweep method; Calculating the time-varying gains; Steady-state optimal control and solution to the LQR problem

Franklin Ch 9
Week 17 May 15 Final-exam Week
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