DISCRETE-TIME SYSTEMS AND DIGITAL CONTROLLERS
|
|
- Dorthy Nicholson
- 6 years ago
- Views:
Transcription
1 DISCRETE-TIME SYSTEMS AND DIGITAL CONTROLLERS
2 In practice controllers are nowadays almost exclusively implemented digitally. This means that the controller operates in discrete time, although the controlled systems usually operate in continuous time. Therefore, the digital controller has to be connected to the system by interfaces which: transform the continuous-time system output y(t) to a discrete sequence {y k } which the digital controller can process, and transform the digital control sequence {u k } to a continuoustime control signal u(t) sent to the system. 2
3 A typical digital control system is shown in the figure. r k e k u k u H (t) G d D/A + H u(t) G p y(t) A/D y k Here: G p is the system to be controlled. G d is the digital controller. The block A/D transforms the continuous-time (analog) system output y(t) to a discrete-time (digital) sequence y k, k = 0, 1, 2,.... The block D/A transforms the (digital) control sequence u k, k = 0, 1, 2,... to a continuous-time (analog) control signal u H (t). H is a filter to smooth out discontinuities due from the digital-to -analog transformation. 3
4 The system output y(t) has first to be sampled at discrete time instants kh in an A/D (analog-to-digital) transformer to generate a discrete sequence, y k = y(kh), k = 0, 1, 2,... Here h is the sampling time. Sampling of a continuous-time signal implies loss of information, since several analog signals can give rise to the same discrete-time signal Example: Sampling of three different sinusoidal signal give the same discrete sequence ( o ). 4
5 Therefore, an analog low-pass prefilter is used to filter out high frequency components before sampling: y(t) y f (t) F A/D y k The digital controller G d then takes the sequence e k = r k y k, k = 0, 1, 2,... as its input to generate a discretetime control sequence u k, k = 0, 1, 2,.... The discrete-time control sequence u k is then transformed to a continuous-time control signal u H (t) using a D/A (digital-to-analog) transformer. Typically the D/A transformer generates a piecewise constant control signal given by u H (t) = u k, kh t < kh + h Finally, the discontinuities in u H (t) are smoothed out by a continuous-time filter H. See figure: 5
6 u k u H (t) D/A H u(t) H t 6
7 Digital controllers The continuous-time standard P-, I-, PI- and PID-controllers can be generalized in a straightforward way to obtain their discrete-time counterparts: Discrete-time (digital) P-controller: Digital I-controller: Digital PID-controller: u k = K p e k + K i u k = K p e k u k = K i k n=1 k e n n=1 e n + K d [e k e k 1 ] The digital PID-controller is usually implemented using the so-called velocity form, u k = u k 1 +K p [e k e k 1 ]+K i e k +K d [e k 2e k 1 + e k 2 ] which is obtained by subtracting u k 1 from u k. An advantage of the velocity form is that there is no need to keep track of the sum. 7
8 Discrete-time systems In order to study systems under digital control we should describe how the sampled, discrete-time, system output y k depends on the discrete-time input u k. The discrete-time counterpart of continuous-time systems described by differential equations are systems described by difference equations. A first-order discrete-time system is described by the difference equation y k+1 + ay k = bu k Similarly, a second-order discrete-time system is described by the difference equation y k+2 + a 1 y k+1 + a 2 y k = b 1 u k+1 + b 2 u k Remark: Observe that the systems above are discrete-time systems, but not digital, as the signals real numbers. A digital system has the additional property that the signals and parameters are represented digitally. 8
9 Discrete-time transfer functions The theory for continuous-time dynamical systems has a counterpart for discrete-time systems. In particular, instead of the differential operator, we can for discrete-time systems define the forward shift operator q such that qy k = y k+1 Then the first-order system takes the form y k+1 + ay k = bu k qy k + ay k = bu k or y k = b q + a u k Here G[q] = b q + a is the discrete-time transfer function. 9
10 Similarly, the second-order system y k+2 + a 1 y k+1 + a 2 y k = b 1 u k+1 + b 2 u k can be written as q 2 y k + a 1 qy k + a 2 y k = b 1 qu k + b 2 u k or where y k = G[q]u k b 1 q + b 2 G[q] = q 2 + a 1 q + a 2 10
11 Response of discrete-time systems In analogy with continuous-time systems, the response of a discrete-time systems is obtained by solving the difference equation. Example - First-order system As a simple example, we can determine the output of the first-order system y k+1 + ay k = bu k By applying the system equation recursively to compute y 1, y 2,..., y k we obtain y k = ( a) k y 0 + k 1 ( a) k n 1 bu n n=0 In particular, the step response for the input u k = u step with initial condition y 0 = 0 is y k = k 1 n=0 ( a) k n 1 b u step = ( 1 a + ( a) 2 + ( a) ( a) k 1) b u step = 1 + ( a)k 1 + a b u step The response of second-order and higher-order discretetime systems can be determined in a way similar to the continuous-time case. 11
12 Stability The above example shows that the output of the firstorder system remains bounded for bounded inputs if and only if a < 1. Equivalently, the pole q p = a of the transfer function G[q] = b q + a should be less than 1 in magnitude. This feature generalizes to higher-order discrete-time systems, and we have: A discrete-time system is stable if and only if all the poles q p,i of its transfer function G[p] (i.e., the zeros of its denominator) satisfy q p,i < 1. Observe that for second and higher order systems, the poles may be complex numbers, and q p,i is then the absolute value of this complex number. 12
13 Backward shift operator An alternative to the forward shift operator q is to use the backward shift operator q 1, such that Then the first-order system or, equivalently, takes the form q 1 y k = y k 1 y k+1 + ay k = bu k y k + ay k 1 = bu k 1 y k + aq 1 y k = bq 1 u k 1 or y k = bq aq 1u k = G[q]u k where the discrete transfer function G[q] is given by G[q] = bq aq 1 Expressing G[q] in terms of q (instead of q 1 ), we have G[q] = b q + a which is the same expression as before. 13
14 Construction of discrete-time models for continuous-time systems When dealing with digital controller, an important question is to construct a discrete-time model which relates the inputs, u(t) and d(t), to the sampled outputs y k = y(kh) of a system described by a differential equation. Example consider the first-order system dy(t) dt + ay(t) = bu(t) + cd(t) A simple way to construct an approximate discrete-time model is to introduce the finite-difference approximation dy(t) dt 1 [y(t + h) y(t)] h Using this approximation at time t = kh would give the discrete-time approximation 1 [y(kh + h) y(kh)] + ay(kh) = bu(kh) + cd(kh) h or y(kh + h) + (ha 1)y(kh) = hbu(kh) + hcd(kh) 14
15 Second derivative: similarly, a finite-difference approximation of a second derivative is given by dy 2 (t) dt 2 1 h y(t + h) y(t) h y(t) y(t h) h = 1 [y(t + h) 2y(t) + y(t h)] h2 Problems with finite-difference approximations of the derivatives: the resulting discrete-time model is often not very accurate. the discrete-time model may be unstable even though the original system is stable, or vice versa. For example, in the above first-order example, the continuoustime system is stable if a > 0, but the discrete-time model will be unstable if the sampling time is too long, so that ha 1 > 1 (or h > 2/a) holds. 15
16 Tustin s method (bilinear transform) A more accurate way to discretize a continuous-time system is to use the trapezoidal rule. Again, we illustrate the method on the first-order system dy(t) dt + ay(t) = bu(t) (omitting d(t) for simplicity). Given y(kh) at time t = kh, we obtain y(kh + h) by integration: y(kh + h) = y(kh) + kh+h kh [ ay(τ) + bu(τ)] dτ Here we can approximate the integral using the trapezoidal rule: t 2 t 1 f(τ)dτ t 2 t 1 2 [f(t 2 ) + f(t 1 )] 16
17 This gives: I = kh+h kh h 2 [ ay(τ) + bu(τ)] dτ [ ay(kh) + b(kh) ay(kh + h) + b(kh + h)] Introducing the approximation into the expression for y(kh): y(kh + h) = y(kh) + h [ ay(kh) + bu(kh) 2 ay(kh + h) + bu(kh + h)] Solving for y(kh + h): y(kh+h) = 1 ah/2 1 + ah/2 or where bh/2 y(kh)+ [u(kh + h) + u(kh)] 1 + ah/2 y k+1 + a d y k = b d u k+1 + b d u k a d = 1 ah/2 1 + ah/2, b d = bh/2 1 + ah/2 17
18 There is an interesting and important relation between the transfer function of the continuous-time system and the discrete-time transfer function of the discretized system obtain by the trapezoidal approximation. Recalling the transfer function of a discrete-time system, the discrete-time system can be expressed as y(kh) = G[q] u(kh) where the transfer function is given by G[q] = b dq + b d q + a d bh/2 = 1 + ah/2 q + 1 q 1 ah/2 1+ah/2 This can be simplified to: G[q] = b 2 q 1 h q+1 + a Recalling that the original continuous-time system has the transfer function G(p) = b p + a we see that G[q] and G(p) are related by according to: G[q] = G(p) p= 2 h q 1 q+1 18
19 This property can be generalized, so that approximating the solution of a linear differential equation by the trapezoidal rule is equivalent to substituting the differential operator p in the continuous-time transfer function by p = 2 q 1 h q + 1 to obtain the corresponding discrete-time transfer function. This transformation is known as the Tustin transform or the bilinear transform. An important and valuable property of the transformation is that it preserves stability properties. We have: The discrete-time system G[q] = G(p) p= 2 h q 1 q+1 is stable if and only if the continuous-time system G(p) is stable 19
Exam in Automatic Control II Reglerteknik II 5hp (1RT495)
Exam in Automatic Control II Reglerteknik II 5hp (1RT495) Date: August 4, 018 Venue: Bergsbrunnagatan 15 sal Responsible teacher: Hans Rosth. Aiding material: Calculator, mathematical handbooks, textbooks
More informationEXAMPLES EXAMPLE - Temperature in building
DYNAMICAL SYSTEMS EXAMPLES EXAMPLE - Temperature in building Energy balance: Rate of change = [Inflow of energy] [Outflow of energy] of stored energy where Rate of change of stored energy = cρv dt (c =
More informationExam in Systems Engineering/Process Control
Department of AUTOMATIC CONTROL Exam in Systems Engineering/Process Control 27-6-2 Points and grading All answers must include a clear motivation. Answers may be given in English or Swedish. The total
More informationAN EXTENSION OF GENERALIZED BILINEAR TRANSFORMATION FOR DIGITAL REDESIGN. Received October 2010; revised March 2011
International Journal of Innovative Computing, Information and Control ICIC International c 2012 ISSN 1349-4198 Volume 8, Number 6, June 2012 pp. 4071 4081 AN EXTENSION OF GENERALIZED BILINEAR TRANSFORMATION
More informationDistributed Real-Time Control Systems
Distributed Real-Time Control Systems Chapter 9 Discrete PID Control 1 Computer Control 2 Approximation of Continuous Time Controllers Design Strategy: Design a continuous time controller C c (s) and then
More information9 Controller Discretization
9 Controller Discretization In most applications, a control system is implemented in a digital fashion on a computer. This implies that the measurements that are supplied to the control system must be
More informationAutomatique. A. Hably 1. Commande d un robot mobile. Automatique. A.Hably. Digital implementation
A. Hably 1 1 Gipsa-lab, Grenoble-INP ahmad.hably@grenoble-inp.fr Commande d un robot mobile (Gipsa-lab (DA)) ASI 1 / 25 Outline 1 2 (Gipsa-lab (DA)) ASI 2 / 25 of controllers Signals must be sampled and
More informationLecture: Sampling. Automatic Control 2. Sampling. Prof. Alberto Bemporad. University of Trento. Academic year
Automatic Control 2 Sampling Prof. Alberto Bemporad University of rento Academic year 2010-2011 Prof. Alberto Bemporad (University of rento) Automatic Control 2 Academic year 2010-2011 1 / 31 ime-discretization
More informationControl Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich
Control Systems I Lecture 6: Poles and Zeros Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 27, 2017 E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/2017
More informationNorm invariant discretization for sampled-data fault detection
Automatica 41 (25 1633 1637 www.elsevier.com/locate/automatica Technical communique Norm invariant discretization for sampled-data fault detection Iman Izadi, Tongwen Chen, Qing Zhao Department of Electrical
More informationLecture 4 Continuous time linear quadratic regulator
EE363 Winter 2008-09 Lecture 4 Continuous time linear quadratic regulator continuous-time LQR problem dynamic programming solution Hamiltonian system and two point boundary value problem infinite horizon
More informationOrdinary Differential Equations I
Ordinary Differential Equations I CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Ordinary Differential Equations I 1 / 27 Theme of Last Three
More informationSAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015
FACULTY OF ENGINEERING AND SCIENCE SAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015 Lecturer: Michael Ruderman Problem 1: Frequency-domain analysis and control design (15 pt) Given is a
More informationDESIGN OF CMOS ANALOG INTEGRATED CIRCUITS
DESIGN OF CMOS ANALOG INEGRAED CIRCUIS Franco Maloberti Integrated Microsistems Laboratory University of Pavia Discrete ime Signal Processing F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete
More informationExam in Systems Engineering/Process Control
Department of AUTOMATIC CONTROL Exam in Systems Engineering/Process Control 7-6- Points and grading All answers must include a clear motivation. Answers may be given in English or Swedish. The total number
More informationOptimal Polynomial Control for Discrete-Time Systems
1 Optimal Polynomial Control for Discrete-Time Systems Prof Guy Beale Electrical and Computer Engineering Department George Mason University Fairfax, Virginia Correspondence concerning this paper should
More informationNotice the minus sign on the adder: it indicates that the lower input is subtracted rather than added.
6.003 Homework Due at the beginning of recitation on Wednesday, February 17, 010. Problems 1. Black s Equation Consider the general form of a feedback problem: + F G Notice the minus sign on the adder:
More informationLecture Discrete dynamic systems
Chapter 3 Low-level io Lecture 3.4 Discrete dynamic systems Lecture 3.4 Discrete dynamic systems Suppose that we wish to implement an embedded computer system that behaves analogously to a continuous linear
More informationControl Systems I. Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback. Readings: Emilio Frazzoli
Control Systems I Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 13, 2017 E. Frazzoli (ETH)
More informationChapter 4 Optimal Control Problems in Infinite Dimensional Function Space
Chapter 4 Optimal Control Problems in Infinite Dimensional Function Space 4.1 Introduction In this chapter, we will consider optimal control problems in function space where we will restrict ourselves
More informationSampling of Linear Systems
Sampling of Linear Systems Real-Time Systems, Lecture 6 Karl-Erik Årzén January 26, 217 Lund University, Department of Automatic Control Lecture 6: Sampling of Linear Systems [IFAC PB Ch. 1, Ch. 2, and
More informationOrdinary Differential Equations I
Ordinary Differential Equations I CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Ordinary Differential Equations I 1 / 32 Theme of Last Few
More informationELEG 305: Digital Signal Processing
ELEG 305: Digital Signal Processing Lecture : Design of Digital IIR Filters (Part I) Kenneth E. Barner Department of Electrical and Computer Engineering University of Delaware Fall 008 K. E. Barner (Univ.
More informationECE 301 Fall 2010 Division 2 Homework 10 Solutions. { 1, if 2n t < 2n + 1, for any integer n, x(t) = 0, if 2n 1 t < 2n, for any integer n.
ECE 3 Fall Division Homework Solutions Problem. Reconstruction of a continuous-time signal from its samples. Consider the following periodic signal, depicted below: {, if n t < n +, for any integer n,
More informationLecture 8 - IIR Filters (II)
Lecture 8 - IIR Filters (II) James Barnes (James.Barnes@colostate.edu) Spring 2009 Colorado State University Dept of Electrical and Computer Engineering ECE423 1 / 27 Lecture 8 Outline Introduction Digital
More informationRational Implementation of Distributed Delay Using Extended Bilinear Transformations
Rational Implementation of Distributed Delay Using Extended Bilinear Transformations Qing-Chang Zhong zhongqc@ieee.org, http://come.to/zhongqc School of Electronics University of Glamorgan United Kingdom
More informationELEC2400 Signals & Systems
ELEC2400 Signals & Systems Chapter 7. Z-Transforms Brett Ninnes brett@newcastle.edu.au. School of Electrical Engineering and Computer Science The University of Newcastle Slides by Juan I. Yu (jiyue@ee.newcastle.edu.au
More informationMethods for analysis and control of. Lecture 6: Introduction to digital control
Methods for analysis and of Lecture 6: to digital O. Sename 1 1 Gipsa-lab, CNRS-INPG, FRANCE Olivier.Sename@gipsa-lab.inpg.fr www.lag.ensieg.inpg.fr/sename 6th May 2009 Outline Some interesting books:
More information10.4 Controller synthesis using discrete-time model Example: comparison of various controllers
10. Digital 10.1 Basic principle of digital control 10.2 Digital PID controllers 10.2.1 A 2DOF continuous-time PID controller 10.2.2 Discretisation of PID controllers 10.2.3 Implementation and tuning 10.3
More informationEE263: Introduction to Linear Dynamical Systems Review Session 6
EE263: Introduction to Linear Dynamical Systems Review Session 6 Outline diagonalizability eigen decomposition theorem applications (modal forms, asymptotic growth rate) EE263 RS6 1 Diagonalizability consider
More information24 Butterworth Filters
24 Butterworth Filters Recommended Problems P24.1 Do the following for a fifth-order Butterworth filter with cutoff frequency of 1 khz and transfer function B(s). (a) Write the expression for the magnitude
More informationLinear dynamical systems with inputs & outputs
EE263 Autumn 215 S. Boyd and S. Lall Linear dynamical systems with inputs & outputs inputs & outputs: interpretations transfer function impulse and step responses examples 1 Inputs & outputs recall continuous-time
More informationNear Optimal LQR Performance for Uncertain First Order Systems
Near Optimal LQR Performance for Uncertain First Order Systems Li Luo Daniel Miller e-commerce Development Dept. of Elect. and Comp. Eng. IBM Canada Toronto Laboratory University of Waterloo Markham, Ontario
More informationarxiv: v1 [math.oc] 5 Jul 2017
Variational discretization of a control-constrained parabolic bang-bang optimal control problem Nikolaus von Daniels Michael Hinze arxiv:1707.01454v1 [math.oc] 5 Jul 2017 December 8, 2017 Abstract: We
More informationOn the Stability of Linear Systems
On the Stability of Linear Systems by Daniele Sasso * Abstract The criteria of stability defined in the standard theory of linear systems aren t exhaustive and show some inconsistencies. In this article
More informationV. IIR Digital Filters
Digital Signal Processing 5 March 5, V. IIR Digital Filters (Deleted in 7 Syllabus). (dded in 7 Syllabus). 7 Syllabus: nalog filter approximations Butterworth and Chebyshev, Design of IIR digital filters
More informationComputer Control: An Overview
Computer Control: An Overview Wittenmark, Björn; Årzén, Karl-Erik; Åström, Karl Johan Published: 22-1-1 Link to publication Citation for published version (APA): Wittenmark, B., Årzén, K-E., & Åström,
More informationTime Response of Systems
Chapter 0 Time Response of Systems 0. Some Standard Time Responses Let us try to get some impulse time responses just by inspection: Poles F (s) f(t) s-plane Time response p =0 s p =0,p 2 =0 s 2 t p =
More informationLecture 6: Deterministic Self-Tuning Regulators
Lecture 6: Deterministic Self-Tuning Regulators Feedback Control Design for Nominal Plant Model via Pole Placement Indirect Self-Tuning Regulators Direct Self-Tuning Regulators c Anton Shiriaev. May, 2007.
More informationCONTINUOUS TIME D=0 ZOH D 0 D=0 FOH D 0
IDENTIFICATION ASPECTS OF INTER- SAMPLE INPUT BEHAVIOR T. ANDERSSON, P. PUCAR and L. LJUNG University of Linkoping, Department of Electrical Engineering, S-581 83 Linkoping, Sweden Abstract. In this contribution
More informationRaktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Dynamic Response
.. AERO 422: Active Controls for Aerospace Vehicles Dynamic Response Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. . Previous Class...........
More informationLTI Systems (Continuous & Discrete) - Basics
LTI Systems (Continuous & Discrete) - Basics 1. A system with an input x(t) and output y(t) is described by the relation: y(t) = t. x(t). This system is (a) linear and time-invariant (b) linear and time-varying
More informationDiscrete-time linear systems
Automatic Control Discrete-time linear systems Prof. Alberto Bemporad University of Trento Academic year 2-2 Prof. Alberto Bemporad (University of Trento) Automatic Control Academic year 2-2 / 34 Introduction
More informationControl System Design
ELEC ENG 4CL4: Control System Design Notes for Lecture #27 Wednesday, March 17, 2004 Dr. Ian C. Bruce Room: CRL-229 Phone ext.: 26984 Email: ibruce@mail.ece.mcmaster.ca Discrete Delta Domain Models The
More informationAn Iir-Filter Example: A Butterworth Filter
An Iir-Filter Example: A Butterworth Filter Josef Goette Bern University of Applied Sciences, Biel Institute of Human Centered Engineering - microlab JosefGoette@bfhch February 7, 2017 Contents 1 Introduction
More information/ / MET Day 000 NC1^ INRTL MNVR I E E PRE SLEEP K PRE SLEEP R E
05//0 5:26:04 09/6/0 (259) 6 7 8 9 20 2 22 2 09/7 0 02 0 000/00 0 02 0 04 05 06 07 08 09 0 2 ay 000 ^ 0 X Y / / / / ( %/ ) 2 /0 2 ( ) ^ 4 / Y/ 2 4 5 6 7 8 9 2 X ^ X % 2 // 09/7/0 (260) ay 000 02 05//0
More informationUNIT - III PART A. 2. Mention any two techniques for digitizing the transfer function of an analog filter?
UNIT - III PART A. Mention the important features of the IIR filters? i) The physically realizable IIR filters does not have linear phase. ii) The IIR filter specification includes the desired characteristics
More informationProblem Set 3: Solution Due on Mon. 7 th Oct. in class. Fall 2013
EE 56: Digital Control Systems Problem Set 3: Solution Due on Mon 7 th Oct in class Fall 23 Problem For the causal LTI system described by the difference equation y k + 2 y k = x k, () (a) By first finding
More informationOrdinary Differential Equations II
Ordinary Differential Equations II CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Ordinary Differential Equations II 1 / 29 Almost Done! No
More informationSOLVING DIFFERENTIAL EQUATIONS. Amir Asif. Department of Computer Science and Engineering York University, Toronto, ON M3J1P3
SOLVING DIFFERENTIAL EQUATIONS Amir Asif Department of Computer Science and Engineering York University, Toronto, ON M3J1P3 ABSTRACT This article reviews a direct method for solving linear, constant-coefficient
More informationChapter 13 Digital Control
Chapter 13 Digital Control Chapter 12 was concerned with building models for systems acting under digital control. We next turn to the question of control itself. Topics to be covered include: why one
More informationOptimal Discretization of Analog Filters via Sampled-Data H Control Theory
Optimal Discretization of Analog Filters via Sampled-Data H Control Theory Masaaki Nagahara 1 and Yutaka Yamamoto 1 Abstract In this article, we propose optimal discretization of analog filters or controllers
More informationControl Systems I. Lecture 5: Transfer Functions. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich
Control Systems I Lecture 5: Transfer Functions Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 20, 2017 E. Frazzoli (ETH) Lecture 5: Control Systems I 20/10/2017
More informationUsing fractional delay to control the magnitudes and phases of integrators and differentiators
Using fractional delay to control the magnitudes and phases of integrators and differentiators M.A. Al-Alaoui Abstract: The use of fractional delay to control the magnitudes and phases of integrators and
More informationDiscrete-time first-order systems
Discrete-time first-order systems 1 Start with the continuous-time system ẏ(t) =ay(t)+bu(t), y(0) Zero-order hold input u(t) =u(nt ), nt apple t
More informationSUCCESSIVE POLE SHIFTING USING SAMPLED-DATA LQ REGULATORS. Sigeru Omatu
SUCCESSIVE POLE SHIFING USING SAMPLED-DAA LQ REGULAORS oru Fujinaka Sigeru Omatu Graduate School of Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, 599-8531 Japan Abstract: Design of sampled-data
More informationDiscrete-time Controllers
Schweizerische Gesellschaft für Automatik Association Suisse pour l Automatique Associazione Svizzera di Controllo Automatico Swiss Society for Automatic Control Advanced Control Discrete-time Controllers
More informationData-driven methods in application to flood defence systems monitoring and analysis Pyayt, A.
UvA-DARE (Digital Academic Repository) Data-driven methods in application to flood defence systems monitoring and analysis Pyayt, A. Link to publication Citation for published version (APA): Pyayt, A.
More informationIntermediate Process Control CHE576 Lecture Notes # 2
Intermediate Process Control CHE576 Lecture Notes # 2 B. Huang Department of Chemical & Materials Engineering University of Alberta, Edmonton, Alberta, Canada February 4, 2008 2 Chapter 2 Introduction
More information2.161 Signal Processing: Continuous and Discrete Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. I Reading:
More informationLecture 8 - IIR Filters (II)
Lecture 8 - IIR Filters (II) James Barnes (James.Barnes@colostate.edu) Spring 24 Colorado State University Dept of Electrical and Computer Engineering ECE423 1 / 29 Lecture 8 Outline Introduction Digital
More informationLecture 9 Infinite Impulse Response Filters
Lecture 9 Infinite Impulse Response Filters Outline 9 Infinite Impulse Response Filters 9 First-Order Low-Pass Filter 93 IIR Filter Design 5 93 CT Butterworth filter design 5 93 Bilinear transform 7 9
More informationECEEN 5448 Fall 2011 Homework #4 Solutions
ECEEN 5448 Fall 2 Homework #4 Solutions Professor David G. Meyer Novemeber 29, 2. The state-space realization is A = [ [ ; b = ; c = [ which describes, of course, a free mass (in normalized units) with
More informationDifferential and Difference LTI systems
Signals and Systems Lecture: 6 Differential and Difference LTI systems Differential and difference linear time-invariant (LTI) systems constitute an extremely important class of systems in engineering.
More informationGATE EE Topic wise Questions SIGNALS & SYSTEMS
www.gatehelp.com GATE EE Topic wise Questions YEAR 010 ONE MARK Question. 1 For the system /( s + 1), the approximate time taken for a step response to reach 98% of the final value is (A) 1 s (B) s (C)
More informationLecture 7 - IIR Filters
Lecture 7 - IIR Filters James Barnes (James.Barnes@colostate.edu) Spring 204 Colorado State University Dept of Electrical and Computer Engineering ECE423 / 2 Outline. IIR Filter Representations Difference
More informationECEN 420 LINEAR CONTROL SYSTEMS. Lecture 2 Laplace Transform I 1/52
1/52 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 2 Laplace Transform I Linear Time Invariant Systems A general LTI system may be described by the linear constant coefficient differential equation: a n d n
More informationDigital Control & Digital Filters. Lectures 13 & 14
Digital Controls & Digital Filters Lectures 13 & 14, Professor Department of Electrical and Computer Engineering Colorado State University Spring 2017 Systems with Actual Time Delays-Application 2 Case
More informationControl Systems Lab - SC4070 Control techniques
Control Systems Lab - SC4070 Control techniques Dr. Manuel Mazo Jr. Delft Center for Systems and Control (TU Delft) m.mazo@tudelft.nl Tel.:015-2788131 TU Delft, February 16, 2015 (slides modified from
More informationOptimized Compact-support Interpolation Kernels
Optimized Compact-support Interpolation Kernels Ramtin Madani, Student Member, IEEE, Ali Ayremlou, Student Member, IEEE, Arash Amini, Farrokh Marvasti, Senior Member, IEEE, Abstract In this paper, we investigate
More informationSection 6.5 Impulse Functions
Section 6.5 Impulse Functions Key terms/ideas: Unit impulse function (technically a generalized function or distribution ) Dirac delta function Laplace transform of the Dirac delta function IVPs with forcing
More informationInfinite-dimensional methods for path-dependent equations
Infinite-dimensional methods for path-dependent equations (Università di Pisa) 7th General AMaMeF and Swissquote Conference EPFL, Lausanne, 8 September 215 Based on Flandoli F., Zanco G. - An infinite-dimensional
More informationModels for Sampled Data Systems
Chapter 12 Models for Sampled Data Systems The Shift Operator Forward shift operator q(f[k]) f[k +1] In terms of this operator, the model given earlier becomes: q n y[k]+a n 1 q n 1 y[k]+ + a 0 y[k] =b
More informationSTABILITY ANALYSIS. Asystemmaybe stable, neutrallyormarginallystable, or unstable. This can be illustrated using cones: Stable Neutral Unstable
ECE4510/5510: Feedback Control Systems. 5 1 STABILITY ANALYSIS 5.1: Bounded-input bounded-output (BIBO) stability Asystemmaybe stable, neutrallyormarginallystable, or unstable. This can be illustrated
More informationELEC-E8101 Digital and Optimal Control (5 cr), autumn 2015
ELEC-E80 Digital and Optimal Control (5 cr), autumn 205 Lectures Fridays at 2.5-4.00, room AS2 Lecturer: Kai Zenger, TuAS-house, room 3567, kai.zenger(at)aalto.fi Exercise hours Wednesdays at 4.5-6.00
More information12/20/2017. Lectures on Signals & systems Engineering. Designed and Presented by Dr. Ayman Elshenawy Elsefy
//7 ectures on Signals & systems Engineering Designed and Presented by Dr. Ayman Elshenawy Elsefy Dept. of Systems & Computer Eng. Al-Azhar University Email : eaymanelshenawy@yahoo.com aplace Transform
More informationSection 6.4 DEs with Discontinuous Forcing Functions
Section 6.4 DEs with Discontinuous Forcing Functions Key terms/ideas: Discontinuous forcing function in nd order linear IVPs Application of Laplace transforms Comparison to viewing the problem s solution
More informationControl Systems Design
ELEC4410 Control Systems Design Lecture 18: State Feedback Tracking and State Estimation Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 18:
More informationChapter 8. Feedback Controllers. Figure 8.1 Schematic diagram for a stirred-tank blending system.
Feedback Controllers Figure 8.1 Schematic diagram for a stirred-tank blending system. 1 Basic Control Modes Next we consider the three basic control modes starting with the simplest mode, proportional
More informationExercise 5: Digital Control
Gioele Zardini Control Systems II FS 017 Exercise 5: Digital Control 5.1 Digital Control 5.1.1 Signals and Systems A whole course is dedicated to this topic (see Signals and Systems of professor D Andrea).
More informationLecture A1 : Systems and system models
Lecture A1 : Systems and system models Jan Swevers July 2006 Aim of this lecture : Understand the process of system modelling (different steps). Define the class of systems that will be considered in this
More informationEE 380. Linear Control Systems. Lecture 10
EE 380 Linear Control Systems Lecture 10 Professor Jeffrey Schiano Department of Electrical Engineering Lecture 10. 1 Lecture 10 Topics Stability Definitions Methods for Determining Stability Lecture 10.
More informationLecture 8: Discrete-Time Signals and Systems Dr.-Ing. Sudchai Boonto
Dr-Ing Sudchai Boonto Department of Control System and Instrumentation Engineering King Mongut s Unniversity of Technology Thonburi Thailand Outline Introduction Some Useful Discrete-Time Signal Models
More informationOperator based robust right coprime factorization and control of nonlinear systems
Operator based robust right coprime factorization and control of nonlinear systems September, 2011 Ni Bu The Graduate School of Engineering (Doctor s Course) TOKYO UNIVERSITY OF AGRICULTURE AND TECHNOLOGY
More informationBasic Procedures for Common Problems
Basic Procedures for Common Problems ECHE 550, Fall 2002 Steady State Multivariable Modeling and Control 1 Determine what variables are available to manipulate (inputs, u) and what variables are available
More informationSingle-Input-Single-Output Systems
TF 502 Single-Input-Single-Output Systems SIST, ShanghaiTech Introduction Open-Loop Control-Response Proportional Control General PID Control Boris Houska 1-1 Contents Introduction Open-Loop Control-Response
More informationOrdinary Differential Equations II
Ordinary Differential Equations II CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Ordinary Differential Equations II 1 / 33 Almost Done! Last
More informationMath 461 Homework 8. Paul Hacking. November 27, 2018
Math 461 Homework 8 Paul Hacking November 27, 2018 (1) Let S 2 = {(x, y, z) x 2 + y 2 + z 2 = 1} R 3 be the sphere with center the origin and radius 1. Let N = (0, 0, 1) S 2 be the north pole. Let F :
More informationOrdinary Differential Equations I
Ordinary Differential Equations I CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Doug James (and Justin Solomon) CS 205A: Mathematical Methods Ordinary Differential Equations I 1 / 35
More informationDigital Control & Digital Filters. Lectures 21 & 22
Digital Controls & Digital Filters Lectures 2 & 22, Professor Department of Electrical and Computer Engineering Colorado State University Spring 205 Review of Analog Filters-Cont. Types of Analog Filters:
More informationMath 461 Homework 8 Paul Hacking November 27, 2018
(1) Let Math 461 Homework 8 Paul Hacking November 27, 2018 S 2 = {(x, y, z) x 2 +y 2 +z 2 = 1} R 3 be the sphere with center the origin and radius 1. Let N = (0, 0, 1) S 2 be the north pole. Let F : S
More informationRank Tests for the Observability of Discrete-Time Jump Linear Systems with Inputs
Rank Tests for the Observability of Discrete-Time Jump Linear Systems with Inputs Ehsan Elhamifar Mihály Petreczky René Vidal Center for Imaging Science, Johns Hopkins University, Baltimore MD 21218, USA
More informationEL1820 Modeling of Dynamical Systems
EL1820 Modeling of Dynamical Systems Lecture 9 - Parameter estimation in linear models Model structures Parameter estimation via prediction error minimization Properties of the estimate: bias and variance
More informationSmooth Path Generation Based on Bézier Curves for Autonomous Vehicles
Smooth Path Generation Based on Bézier Curves for Autonomous Vehicles Ji-wung Choi, Renwick E. Curry, Gabriel Hugh Elkaim Abstract In this paper we present two path planning algorithms based on Bézier
More informationLinear Algebra Review (Course Notes for Math 308H - Spring 2016)
Linear Algebra Review (Course Notes for Math 308H - Spring 2016) Dr. Michael S. Pilant February 12, 2016 1 Background: We begin with one of the most fundamental notions in R 2, distance. Letting (x 1,
More information1 Continuous-time Systems
Observability Completely controllable systems can be restructured by means of state feedback to have many desirable properties. But what if the state is not available for feedback? What if only the output
More informationReview: Continuous Fourier Transform
Review: Continuous Fourier Transform Review: convolution x t h t = x τ h(t τ)dτ Convolution in time domain Derivation Convolution Property Interchange the order of integrals Let Convolution Property By
More informationEECE Adaptive Control
EECE 574 - Adaptive Control Recursive Identification in Closed-Loop and Adaptive Control Guy Dumont Department of Electrical and Computer Engineering University of British Columbia January 2010 Guy Dumont
More informationDiscrete-time signals and systems
Discrete-time signals and systems 1 DISCRETE-TIME DYNAMICAL SYSTEMS x(t) G y(t) Linear system: Output y(n) is a linear function of the inputs sequence: y(n) = k= h(k)x(n k) h(k): impulse response of the
More informationComputing inverse Laplace Transforms.
Review Exam 3. Sections 4.-4.5 in Lecture Notes. 60 minutes. 7 problems. 70 grade attempts. (0 attempts per problem. No partial grading. (Exceptions allowed, ask you TA. Integration table included. Complete
More information