EE561Spring2020
From CYPHYNETS
EE561: Digital Control Systems  

Spring 2020 
Instructors
Talha Manzoor, Assistant Professor, Center for Water Informatics & Technology (WIT)
Email: talha.manzoor@lums.edu.pk
Office: 9252, Tesla Wing, 2nd Floor, SSE Bldg
TA: Muhammad Mateen Shahid, MS Electrical Engineering
Email: 18060020@lums.edu.pk
Office: Control Systems Lab, Tesla Wing, 2nd Floor, SSE Bldg
Course Details
Year: 201617
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: http://cyphynets.lums.edu.pk/index.php/EE561Spring2017
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 ztransform, discretization of continuoustime systems, determining the dynamic response of discretetime 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, discretetime rootlocus, frequency based design and statespace methods. Since the transform methods in discretetime follow fairly easily from the transform design methods in continuoustime, a focus will be placed on statespace methods which includes both basic and advanced design techniques.
Objectives
 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 realtime control.
Prerequisites
 EE361. 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, WellesleyCambridge 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: 11:00am12:15pm. 10201. SSE Bldg
Schedule
WEEK  TOPICS  REFERENCES 

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 
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 Ztransform; Physical interpretation of z (delays); Solving difference equations through the Ztransform method; The role of bilateral and unilateral Ztransforms in dealing with initial conditions; Introduction to the Transfer function and zdomain 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 discretetime convolution formula; Inverting the ztransform through longdivision; Internal (asymptotic) and external (BIBO) stability for discrete systems
Lecture 6 The final value theorem for discretetime systems, Deriving a relationship between the s and z variables via the impulse sampler, Corresponding timedomain signals for the s and z planes, mapping regions from the s plane to the z plane. 
Franklin Ch 4 
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 zplane.
Lecture 8 Zeropole matching equivalents; Realizing discrete controllers: direct and standard programming; Block diagrams and pseudo code; Minimizing memory elements: the control canonical form; Uncovering the underlying statespace representation. 
Franklin Ch4, Franklin Ch 6 
Week 5 Feb 20  Lecture 9 Block diagrams; pseudo code and memory elements cntd; the observer canonical form
Lecture 10 Quiz 1 Review: A firstorder 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: freefalling particle, cruise control, mass spring system, inverted pendulum  FranklinF Ch 7 
Week 7 Mar 06  Lecture 13 Transforming the statespace variables; Invariance of the transfer function to nonsingular 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
Midterm 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 polezero cancellations, Modal canonical form  FranklinF Ch 7 
Week 9 Mar 20  Midsemester 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 realvalued block diagonal forms for systems with complex eigenvalues
Lecture 19 Control design in statespace; 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 nonhomogeneous 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 
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 Discretetime equivalents; Discrete equivalents via numerical integration; The forward, backward and Tustin's substitution formulas in statespace form; Solving state space systems in discrete time through recursion, the discretetime statetransition matrix; Solving state space systems using the Ztransform  Franklin Ch 4
Franklin Ch 6 
Week 14 Apr 24 
Lecture 24 Discrete time statespace analysis; canonical forms; relationship between statespace representations and pulse transfer function; internal and BIBO stability; Controllability and Observability in discretetime; Proof of the controllability rank criterion via The CayleyHamilton Theorem Lecture 25 State Space design in discretetime; pole placement: matching coefficients, control canonical form and Ackermann's formula; Discretetime observer design; Prediction estimators; Current estimators; Reducedorder estimators; The separation principle; Compensation: combined control law and observer design  Franklin Ch 8 
Week 15 May 01 
Lecture 26 Discrete time state space design continued; introduction of the reference input; feedforward control loop for handling steady state errors; tracking with estimation: the statecommand and outputerror 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  Franklin Ch 8
Franklin Ch 9 
Week 16 May 08 
Lecture 28 Timevarying optimal control as a constrained minimization problem; conversion to a twopoint boundary problem via Lagrange multipliers; solving the twopoint boundary problem via the sweep method; Calculating the timevarying gains; Steadystate optimal control and solution to the LQR problem  Franklin Ch 9 
Week 17 May 15  Finalexam Week 