# Analysis of Discrete-Time Systems

Size: px
Start display at page:

Transcription

1 TU Berlin Discrete-Time Control Systems TU Berlin Discrete-Time Control Systems 2 Stability Definitions We define stability first with respect to changes in the initial conditions Analysis of Discrete-Time Systems Overview Stability Sensitivity and Robustness Controllability, Reachability, Observability, and Detectabiliy Consider x[k + ] = fx[k], k Let x [k] and x[k] be solutions when the initial conditions are x [k ] and x[k ], respectively Definition Stability: The solution x [k] is stable if for a given ε, there exists a δε, k > such that all solutions with x[k ] x [k ] < δ are such that x[k] x [k] < ε for all k > k Definition Asymptotic Stability: The solution x [k] is asymptotically stable if it is stable and if δ can be chosen such that x[k ] x [k ] < δ implies that x[k] x [k] when k Stability, in general, is a local concept System is asymptotically stable if the trajectories do not change much if the initial condition is changed by a small amount TU Berlin Discrete-Time Control Systems 3 TU Berlin Discrete-Time Control Systems 4 Stability of Linear Discrete Time Systems System x [k + ] = Φx [k] x [] = a System with perturbed initial value x[k + ] = Φx[k] x[] = a The difference x = x x satisfies the equation x[k + ] = Φ x[k] x[] = a a If the solution x [k] is stable every other solution is also stable For linear, time-invariant systems, stability is a property of the system and not of a special trajectory! Solution for the last system: x[k] = Φ k x[] If it is possible to diagonalize Φ then the solution is a combination of λ k i terms, where λ k i, i =,, n are the eigenvalues of Φ If it is not possible to diagonalize Φ then the solution is a linear combination of the terms p ikλ k i where pik are polynomials in k of the order one less the multiplicity of the corresponding eigenvalue To get asymptotic stability, all solution must go to zero as k increases to infinity Eigenvalues must have to property λ i < i =,, n Theorem Asymptotic Stability of Linear Systems: A discrete-time linear time-invariant system is asymptotically stable if and only if all eigenvalues of Φ are strictly inside the unit disk TU Berlin Discrete-Time Control Systems 5 TU Berlin Discrete-Time Control Systems 6 Stability Tests Input-Output Stability Definition Bounded-Input Bounded-Output Stability: A linear time-invariant system is defined bounded-input bounded-output BIBO stable if a bounded input gives a bounded output for every initial value Theorem Relation between Stability Concepts: Asymptotic stability implies stability and BIBO stability Stability does not imply BIBO stability, and vice versa! Numerical computation of the eigenvalues of Φ or the roots of the characteristic equation detzi Φ = a z n + a z n + + a n = Scilab commands spec and roots Algebraic or graphical methods help to understand how parameters of the system or controller will influence the stability: Direct algebraic computation of the eigenvalues Methods based on the properties of characteristic polynomials Root locus method used to determine closed-loop stability for known open-loop system The Nyquist criterion used to determine closed-loop stability for known open-loop system Lyapunov s method

2 TU Berlin Discrete-Time Control Systems 7 Jury s Stability Criterion Characteristic polynomial Az = a z n + a z n + + a n = See blackboard: TU Berlin Discrete-Time Control Systems 8 Remark: If all a k are positive for k >, then the condition a can be shown to be equivalent to the conditions A > n A > These are necessary conditions for stability and hence can be used before forming the table above Example for Jury s Stability Criterion: See blackboard Theorem Jury s Stability Test: If a >, then Eq has all roots inside the unit disk if and only if all a k, k =,,, n are positive If no a k is zero, then the number of negative ak is equal to the number of roots outside the unit disk TU Berlin Discrete-Time Control Systems 9 TU Berlin Discrete-Time Control Systems Nyquist and Bode Diagrams for Discrete-Time Systems Continuous-time system Gs: The Nyquist curve or frequency response of the system is the Nyquist plot: Gjω dashed, He jω solid map Gjω for ω [, This curve is drawn in polar coordinates Nyquist diagram or as amplitude and phase curves as a function of frequency Bode diagram Gjω interpretation: Stationary amplitude and phase of system output when a sinusoidal input signal with frequency ω is applied Discrete-time system with pulse-transfer function Hz: The Nyquist curve or frequency curves is given by the map He jω for ω [, π] up to the Nyquist frequency He jω is periodic with period 2π/ Higher harmonics are created Example: Gs = 66z + 55 s 2 ZOH & = 4s Hz = + 4s + z 2 45z + 57 TU Berlin Discrete-Time Control Systems Bode plot: Gjω dashed, He jω solid TU Berlin Discrete-Time Control Systems 2 Nyquist Criterion Well-known stability test for continuous-time systems To determine the stability of the closed-loop system when the open-loop system is given Can be reformulated to handle discrete-time systems Consider the discrete-time system: H clz = Y z U cz = Hz + Hz with the characteristic polynomial + Hz =

3 TU Berlin Discrete-Time Control Systems 3 TU Berlin Discrete-Time Control Systems 4 Nyquist contour Γ c that encircles the area outside the unit disc instability area: Stability is determined by analysing how Γ c is mapped by Hz The number of encirclements N in the positive direction around -, by the map Γ c is equal to N = Z P where Z and P are the numbers of zeros and poles, respectively, of + Hz outside the unit disk Zeros Z of + H are unstable closed-loop poles Poles P of + H are unstable open-loop poles Indentation at z = : to exclude the integrators in the open-loop system infinitesimal semicircle with decresing arguments from π/2 to π/2 is mapped into the Hz-plane as an infinitely large circle from nπ/2 to nπ/2 n - number of integrators in the open-loop system If the open-loop system is stable P = then N = Z The stability of the system is then ensured if the map of Γ c does not encircle the point -, If Hz when z, the parallel lines III and V do not influence the stability test, and it is sufficient to find the map of the unit circle and the small semicircle at z = The map of the unit circle is He jω for ω, 2π TU Berlin Discrete-Time Control Systems 5 TU Berlin Discrete-Time Control Systems 6 Simplified Nyquist Criterium: If the open-loop system and its inverse are stable then the stability of the closed-loop system is ensured if the point -, in the Hz-plane is to the left of the map of He jω for ω = to π that is, to the left of the Nyquist curve Example: Consider the system 25K Hz = z z 5 with = The black line on the right graphic shows He jω for ω = to π, the Nyquist curve Relative Stability Amplitude and phase margins can be defined for discrete-time systems analogously to continuous-time systems Definition Amplitude Margin: Let the open-loop system have the pulse-transfer function Hz and let ω the smallest frequency such that arg He jω = π The amplitude or gain margin is then defined as A marg = He jω The amplitude margin is how much the gain can be increased before the closed-loop system becomes unstable TU Berlin Discrete-Time Control Systems 7 TU Berlin Discrete-Time Control Systems 8 Sensitivity and Complementary Sensitivity Function Consider the closed-loop system with the feedforward filter H ff and the feedback controller H fb: Definition Phase Margin: Let the open-loop system have the pulse-transfer function Hz and further let the crossover frequency ω c be the smallest frequency such that uc Hff v H x e y He jωc = The phase margin is then defined as Hfb n φ marg = π + arg He jωc The phase margin is how much extra phase lag is allowed before the closed-loop system becomes unstable Pulse-transfer operator from the inputs to y: Open-loop transfer function: y = Hff H + L uc + H + L v + + L e L + L n L = H fbh

4 TU Berlin Discrete-Time Control Systems 9 TU Berlin Discrete-Time Control Systems 2 Closed-loop transfer function: Sensitivity function S: H cl = Hff H + L Sensitivity of H cl with respect to variations in H is given by dh cl dh = Hff + L 2 The relative sensitivity of H cl with respect to H thus can be written as dh cl = dh H cl + L H = S dh H S = + L Pulse-transfer function from e to y, should be small at low frequencies Complementary sensitivity function T : T = S = L + L negative Pulse-transfer function from n to y or e to x, should be small at higher frequencies Remark: Due to T + S =, we can not make T and S for all frequencies and must make a trade-off as outlined above! TU Berlin Discrete-Time Control Systems 2 Robustness The distance r from the Nyquist curve Le jω to the critical point -, is a good measure for the robustness and is given by r = + L Note that the complex number + Le jω can be represented as the vector from the point to the point Le jω on the Nyquist curve Reciprocal of the smallest distance r min from the Nyquist curve Le jω to the critical point: = max r min + Le jω = max Se jω 2 Guaranteed bounds on the margins: A marg / r min, 3 φ marg 2 arcsinr min/2 4 By requiring max Se jω < 2, the system will have at least the robustness margins A marg 2 and φ marg 29 TU Berlin Discrete-Time Control Systems 22 Controllability, Reachability Question: Is it possible to steer a system from a given initial state to any other state? Consider the system x[k + ] = Φx[k] + Γ u[k] 5 y[k] = Cx[k] 6 Assume that the initial state x[] is given At state the sample instant n, where n is the order of the system, is given by where x[n] = Φ n x[] + Φ n Γ u[] + + Γ u[n ] = Φ n x[] + W cu W c = Γ ΦΓ Φ n Γ U = u T [n ] u T [] TU Berlin Discrete-Time Control Systems 23 TU Berlin Discrete-Time Control Systems 24 The matrix W c is referred to as the controllability matrix If W c has rank, then it is possible to find n equations from which the control signal can be found such that the initial state is transferred to the desired state x[n] The solution is not unique if there is more than one input! Definition Controllability: The system 5 is controllable if it is possible to find a control sequence such that the origin can be reached from any initial state in finite time Definition Reachability: The system 5 is reachable if it is possible to find a control sequence such that an arbitrary state can be reached from any initial state in finite time Controllability does not imply reachability: If Φ n x[] =, then the origin will be reached with zero input, but the system is not necessarily reachable See example Reachability is independent of the coordinates: W c = Γ Φ Γ Φn Γ = T Γ T ΦT T Γ T Φ n T T Γ = T W c If W c has rank n, than W c has also rank n for a non-singular transformation matrix T Theorem Reachability: The system 5 is reachable if and only if the matrix W c has rand n The reachable states belong to the linear subspace spanned by the columns of W c Example

5 TU Berlin Discrete-Time Control Systems 25 TU Berlin Discrete-Time Control Systems 26 Assume a SISO system, a non-singular W c and detλi Φ = λ n + a λ n + + a n = 7 Then there exists a transformation on controllable canonical form a a 2 a n a n z[k + ] = z[k] + u[k] y[k] = b b n z[k] State Trajectory Following Question: Does reachability also imply that is possible to follow a given trajectory in the state space? In order to drive a system from x[k] to a desired x[k + ] the matrix Γ must have rank n For a reachable SISO system it is, in general possible, to reach desired states only at each n-th sample instant, provided that the desired points are known n steps ahead Good for computation of I/O model and for the design of a state-feedback-control law TU Berlin Discrete-Time Control Systems 27 TU Berlin Discrete-Time Control Systems 28 Observability and Detectability Output Trajectory Following Assume that a desired trajectory u c[k] is given, the control signal should satisfy or y[k] = Bq u[k] = uc[k] Aq u[k] = Aq Bq uc[k] For a time-delay of d steps the generation of u[k] is only causal if the trajectory is known d steps ahead The signal u[k] is bounded if u c[k] is bounded and if the system has a stable inverse To solve the problem of finding the state of a system from observations of the output, the concept of unobservable states is introduced at first: Definition - Unobservable States: The state x is unobservable if there exists a finite k n such that y[k] = for k k when x = x o and u[k] = for k k The system 5 is observable if there is a finite k such that knowledge of the inputs u[],, u[k ] and the outputs y[],, y[k ] is sufficient to determine the initial state of the system Effect of u[k] always can be determined Without loss of generality we assume u[k] = y[] = Cx[] y[] = Cx[] = CΦx y[n ] = CΦ n x[] TU Berlin Discrete-Time Control Systems 29 Using vector notations: C y[] CΦ y[] x[] = CΦ n y[n ] The initial state x[] can be obtained if and only if the observability matrix has rank n W o = C CΦ CΦ n Theorem - Observability The system 5 is observable if and only if W has rank n TU Berlin Discrete-Time Control Systems 3 Theorem - Detectability A system is detectable if the only unobservable states are such that they decay to the origin That is, the corresponding eigenvalues are stable Observable Canonical Form Assume a SISO system, a non-singular matrix W and detλi Φ = λ n + a λ n + + a n = 8 then there exists a transformation such that the transformed system is a b a 2 b 2 z[k + ] = z[k] + u[k] a n b n a n b n y[k] = z[k] which is called observable canonical form

6 TU Berlin Discrete-Time Control Systems 3 TU Berlin Discrete-Time Control Systems 32 Kalman s Decomposition The reachable and unobservable parts of a system are two linear subspaces of the state space This form has the advantage that it is easy to find the I/O model and to determine a suitable observer The transformation is given by T = W o W o where W o is the observable matrix of the system in observable canonical form Kalman showed that it is possible to introduce coordinates such that a system can be partitioned in the following way: Φ Φ 2 Φ 22 x[k + ] = Φ 3 Φ 32 Φ 33 Φ 34 Φ 42 Φ 44 y[k] = C C 2 x[k] Γ x[k] + u[k] Γ 3 TU Berlin Discrete-Time Control Systems 33 TU Berlin Discrete-Time Control Systems 34 The state space is partitioned into four parts yielding for subsystems: S or Observable and reachable S or Observable but not reachable S or Not observable but reachable S or Neither observable nor reachable The pulse-transfer operator for the observable and reachable subsystem is given by: Hq = C qi Φ Γ See picture Loss of Reachability and Observability Trough Sampling To get a reachable discrete-time system, it is necessary that the continuous-time system is also reachable However it may happen that reachability is lost for some sampling periods Conditions for unobservability are more restricted in the continuous-time case: output has to be zero over a time interval for discrete-time system only at sampling instants Continuous-time system may oscillate between sampling instants and remain zero at sampling instants hidden oscillations The sampled-data system thus can be unobservable even if the corresponding continuous-time system is observable

### Analysis of Discrete-Time Systems

TU Berlin Discrete-Time Control Systems 1 Analysis of Discrete-Time Systems Overview Stability Sensitivity and Robustness Controllability, Reachability, Observability, and Detectabiliy TU Berlin Discrete-Time

### Frequency methods for the analysis of feedback systems. Lecture 6. Loop analysis of feedback systems. Nyquist approach to study stability

Lecture 6. Loop analysis of feedback systems 1. Motivation 2. Graphical representation of frequency response: Bode and Nyquist curves 3. Nyquist stability theorem 4. Stability margins Frequency methods

### 10/8/2015. Control Design. Pole-placement by state-space methods. Process to be controlled. State controller

Pole-placement by state-space methods Control Design To be considered in controller design * Compensate the effect of load disturbances * Reduce the effect of measurement noise * Setpoint following (target

### Control 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

### Design Methods for Control Systems

Design Methods for Control Systems Maarten Steinbuch TU/e Gjerrit Meinsma UT Dutch Institute of Systems and Control Winter term 2002-2003 Schedule November 25 MSt December 2 MSt Homework # 1 December 9

### Digital Control Systems

Digital Control Systems Lecture Summary #4 This summary discussed some graphical methods their use to determine the stability the stability margins of closed loop systems. A. Nyquist criterion Nyquist

### SAMPLE 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

### Control Design. Lecture 9: State Feedback and Observers. Two Classes of Control Problems. State Feedback: Problem Formulation

Lecture 9: State Feedback and s [IFAC PB Ch 9] State Feedback s Disturbance Estimation & Integral Action Control Design Many factors to consider, for example: Attenuation of load disturbances Reduction

### Sampling 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

### Modelling and Control of Dynamic Systems. Stability of Linear Systems. Sven Laur University of Tartu

Modelling and Control of Dynamic Systems Stability of Linear Systems Sven Laur University of Tartu Motivating Example Naive open-loop control r[k] Controller Ĉ[z] u[k] ε 1 [k] System Ĝ[z] y[k] ε 2 [k]

### Lecture 11. Frequency Response in Discrete Time Control Systems

EE42 - Discrete Time Systems Spring 28 Lecturer: Asst. Prof. M. Mert Ankarali Lecture.. Frequency Response in Discrete Time Control Systems Let s assume u[k], y[k], and G(z) represents the input, output,

### Table of Laplacetransform

Appendix Table of Laplacetransform pairs 1(t) f(s) oct), unit impulse at t = 0 a, a constant or step of magnitude a at t = 0 a s t, a ramp function e- at, an exponential function s + a sin wt, a sine fun

### Intro to Frequency Domain Design

Intro to Frequency Domain Design MEM 355 Performance Enhancement of Dynamical Systems Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline Closed Loop Transfer Functions

### CBE507 LECTURE III Controller Design Using State-space Methods. Professor Dae Ryook Yang

CBE507 LECTURE III Controller Design Using State-space Methods Professor Dae Ryook Yang Fall 2013 Dept. of Chemical and Biological Engineering Korea University Korea University III -1 Overview States What

### Linear Control Systems Lecture #3 - Frequency Domain Analysis. Guillaume Drion Academic year

Linear Control Systems Lecture #3 - Frequency Domain Analysis Guillaume Drion Academic year 2018-2019 1 Goal and Outline Goal: To be able to analyze the stability and robustness of a closed-loop system

### SYSTEMTEORI - ÖVNING Stability of linear systems Exercise 3.1 (LTI system). Consider the following matrix:

SYSTEMTEORI - ÖVNING 3 1. Stability of linear systems Exercise 3.1 (LTI system. Consider the following matrix: ( A = 2 1 Use the Lyapunov method to determine if A is a stability matrix: a: in continuous

### Lecture Note #6 (Chap.10)

System Modeling and Identification Lecture Note #6 (Chap.) CBE 7 Korea University Prof. Dae Ryook Yang Chap. Model Approximation Model approximation Simplification, approximation and order reduction of

### Return Difference Function and Closed-Loop Roots Single-Input/Single-Output Control Systems

Spectral Properties of Linear- Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 2018! Stability margins of single-input/singleoutput (SISO) systems! Characterizations

### Control Systems I. Lecture 9: The Nyquist condition

Control Systems I Lecture 9: The Nyquist condition adings: Guzzella, Chapter 9.4 6 Åstrom and Murray, Chapter 9.1 4 www.cds.caltech.edu/~murray/amwiki/index.php/first_edition Emilio Frazzoli Institute

### MEM 355 Performance Enhancement of Dynamical Systems

MEM 355 Performance Enhancement of Dynamical Systems Frequency Domain Design Intro Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University /5/27 Outline Closed Loop Transfer

### Introduction. Performance and Robustness (Chapter 1) Advanced Control Systems Spring / 31

Introduction Classical Control Robust Control u(t) y(t) G u(t) G + y(t) G : nominal model G = G + : plant uncertainty Uncertainty sources : Structured : parametric uncertainty, multimodel uncertainty Unstructured

### DESIGN USING TRANSFORMATION TECHNIQUE CLASSICAL METHOD

206 Spring Semester ELEC733 Digital Control System LECTURE 7: DESIGN USING TRANSFORMATION TECHNIQUE CLASSICAL METHOD For a unit ramp input Tz Ez ( ) 2 ( z ) D( z) G( z) Tz e( ) lim( z) z 2 ( z ) D( z)

### Control Systems. Frequency Method Nyquist Analysis.

Frequency Method Nyquist Analysis chibum@seoultech.ac.kr Outline Polar plots Nyquist plots Factors of polar plots PolarNyquist Plots Polar plot: he locus of the magnitude of ω vs. the phase of ω on polar

### Control Systems I. Lecture 9: The Nyquist condition

Control Systems I Lecture 9: The Nyquist condition Readings: Åstrom and Murray, Chapter 9.1 4 www.cds.caltech.edu/~murray/amwiki/index.php/first_edition Jacopo Tani Institute for Dynamic Systems and Control

### 1 (20 pts) Nyquist Exercise

EE C128 / ME134 Problem Set 6 Solution Fall 2011 1 (20 pts) Nyquist Exercise Consider a close loop system with unity feedback. For each G(s), hand sketch the Nyquist diagram, determine Z = P N, algebraically

### Contents. PART I METHODS AND CONCEPTS 2. Transfer Function Approach Frequency Domain Representations... 42

Contents Preface.............................................. xiii 1. Introduction......................................... 1 1.1 Continuous and Discrete Control Systems................. 4 1.2 Open-Loop

### Lifted approach to ILC/Repetitive Control

Lifted approach to ILC/Repetitive Control Okko H. Bosgra Maarten Steinbuch TUD Delft Centre for Systems and Control TU/e Control System Technology Dutch Institute of Systems and Control DISC winter semester

### 1 (s + 3)(s + 2)(s + a) G(s) = C(s) = K P + K I

MAE 43B Linear Control Prof. M. Krstic FINAL June 9, Problem. ( points) Consider a plant in feedback with the PI controller G(s) = (s + 3)(s + )(s + a) C(s) = K P + K I s. (a) (4 points) For a given constant

### Pole-Placement Design A Polynomial Approach

TU Berlin Discrete-Time Control Systems 1 Pole-Placement Design A Polynomial Approach Overview A Simple Design Problem The Diophantine Equation More Realistic Assumptions TU Berlin Discrete-Time Control

### Properties of Open-Loop Controllers

Properties of Open-Loop Controllers Sven Laur University of Tarty 1 Basics of Open-Loop Controller Design Two most common tasks in controller design is regulation and signal tracking. Regulating controllers

### Some solutions of the written exam of January 27th, 2014

TEORIA DEI SISTEMI Systems Theory) Prof. C. Manes, Prof. A. Germani Some solutions of the written exam of January 7th, 0 Problem. Consider a feedback control system with unit feedback gain, with the following

### Analysis and Synthesis of Single-Input Single-Output Control Systems

Lino Guzzella Analysis and Synthesis of Single-Input Single-Output Control Systems l+kja» \Uja>)W2(ja»\ um Contents 1 Definitions and Problem Formulations 1 1.1 Introduction 1 1.2 Definitions 1 1.2.1 Systems

### Course Outline. Closed Loop Stability. Stability. Amme 3500 : System Dynamics & Control. Nyquist Stability. Dr. Dunant Halim

Amme 3 : System Dynamics & Control Nyquist Stability Dr. Dunant Halim Course Outline Week Date Content Assignment Notes 1 5 Mar Introduction 2 12 Mar Frequency Domain Modelling 3 19 Mar System Response

### IC6501 CONTROL SYSTEMS

DHANALAKSHMI COLLEGE OF ENGINEERING CHENNAI DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING YEAR/SEMESTER: II/IV IC6501 CONTROL SYSTEMS UNIT I SYSTEMS AND THEIR REPRESENTATION 1. What is the mathematical

### FEL3210 Multivariable Feedback Control

FEL3210 Multivariable Feedback Control Lecture 5: Uncertainty and Robustness in SISO Systems [Ch.7-(8)] Elling W. Jacobsen, Automatic Control Lab, KTH Lecture 5:Uncertainty and Robustness () FEL3210 MIMO

### Systems Analysis and Control

Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 23: Drawing The Nyquist Plot Overview In this Lecture, you will learn: Review of Nyquist Drawing the Nyquist Plot Using the

### EEE582 Homework Problems

EEE582 Homework Problems HW. Write a state-space realization of the linearized model for the cruise control system around speeds v = 4 (Section.3, http://tsakalis.faculty.asu.edu/notes/models.pdf). Use

### Systems Analysis and Control

Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 23: Drawing The Nyquist Plot Overview In this Lecture, you will learn: Review of Nyquist Drawing the Nyquist Plot Using

### Andrea Zanchettin Automatic Control AUTOMATIC CONTROL. Andrea M. Zanchettin, PhD Spring Semester, Linear systems (frequency domain)

1 AUTOMATIC CONTROL Andrea M. Zanchettin, PhD Spring Semester, 2018 Linear systems (frequency domain) 2 Motivations Consider an LTI system Thanks to the Lagrange s formula we can compute the motion of

### Lecture 6 Classical Control Overview IV. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

Lecture 6 Classical Control Overview IV Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Lead Lag Compensator Design Dr. Radhakant Padhi Asst.

### Control Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Jacopo Tani. Institute for Dynamic Systems and Control D-MAVT ETH Zürich

Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 2, 2018 J. Tani, E. Frazzoli (ETH) Lecture 7:

### (Continued on next page)

(Continued on next page) 18.2 Roots of Stability Nyquist Criterion 87 e(s) 1 S(s) = =, r(s) 1 + P (s)c(s) where P (s) represents the plant transfer function, and C(s) the compensator. The closedloop characteristic

### Lecture 4 and 5 Controllability and Observability: Kalman decompositions

1 Lecture 4 and 5 Controllability and Observability: Kalman decompositions Spring 2013 - EE 194, Advanced Control (Prof. Khan) January 30 (Wed.) and Feb. 04 (Mon.), 2013 I. OBSERVABILITY OF DT LTI SYSTEMS

### MEM 355 Performance Enhancement of Dynamical Systems

MEM 355 Performance Enhancement of Dynamical Systems Frequency Domain Design Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University 5/8/25 Outline Closed Loop Transfer Functions

### Learn2Control Laboratory

Learn2Control Laboratory Version 3.2 Summer Term 2014 1 This Script is for use in the scope of the Process Control lab. It is in no way claimed to be in any scientific way complete or unique. Errors should

### Digital Control Systems State Feedback Control

Digital Control Systems State Feedback Control Illustrating the Effects of Closed-Loop Eigenvalue Location and Control Saturation for a Stable Open-Loop System Continuous-Time System Gs () Y() s 1 = =

### Software Engineering/Mechatronics 3DX4. Slides 6: Stability

Software Engineering/Mechatronics 3DX4 Slides 6: Stability Dr. Ryan Leduc Department of Computing and Software McMaster University Material based on lecture notes by P. Taylor and M. Lawford, and Control

### Ingegneria dell Automazione - Sistemi in Tempo Reale p.1/28

Ingegneria dell Automazione - Sistemi in Tempo Reale Selected topics on discrete-time and sampled-data systems Luigi Palopoli palopoli@sssup.it - Tel. 050/883444 Ingegneria dell Automazione - Sistemi in

### Chapter 6 State-Space Design

Chapter 6 State-Space Design wo steps. Assumption is made that we have all the states at our disposal for feedback purposes (in practice, we would not measure all these states). his allows us to implement

### Outline. Classical Control. Lecture 1

Outline Outline Outline 1 Introduction 2 Prerequisites Block diagram for system modeling Modeling Mechanical Electrical Outline Introduction Background Basic Systems Models/Transfers functions 1 Introduction

EC6405 - CONTROL SYSTEM ENGINEERING Questions and Answers Unit - I Control System Modeling Two marks 1. What is control system? A system consists of a number of components connected together to perform

### Control System Design

ELEC ENG 4CL4: Control System Design Notes for Lecture #11 Wednesday, January 28, 2004 Dr. Ian C. Bruce Room: CRL-229 Phone ext.: 26984 Email: ibruce@mail.ece.mcmaster.ca Relative Stability: Stability

### Robust Control 3 The Closed Loop

Robust Control 3 The Closed Loop Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University /2/2002 Outline Closed Loop Transfer Functions Traditional Performance Measures Time

### Chapter 2. Classical Control System Design. Dutch Institute of Systems and Control

Chapter 2 Classical Control System Design Overview Ch. 2. 2. Classical control system design Introduction Introduction Steady-state Steady-state errors errors Type Type k k systems systems Integral Integral

### Control 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

### Zeros and zero dynamics

CHAPTER 4 Zeros and zero dynamics 41 Zero dynamics for SISO systems Consider a linear system defined by a strictly proper scalar transfer function that does not have any common zero and pole: g(s) =α p(s)

### K(s +2) s +20 K (s + 10)(s +1) 2. (c) KG(s) = K(s + 10)(s +1) (s + 100)(s +5) 3. Solution : (a) KG(s) = s +20 = K s s

321 16. Determine the range of K for which each of the following systems is stable by making a Bode plot for K = 1 and imagining the magnitude plot sliding up or down until instability results. Verify

### Uncertainty and Robustness for SISO Systems

Uncertainty and Robustness for SISO Systems ELEC 571L Robust Multivariable Control prepared by: Greg Stewart Outline Nature of uncertainty (models and signals). Physical sources of model uncertainty. Mathematical

### Represent this system in terms of a block diagram consisting only of. g From Newton s law: 2 : θ sin θ 9 θ ` T

Exercise (Block diagram decomposition). Consider a system P that maps each input to the solutions of 9 4 ` 3 9 Represent this system in terms of a block diagram consisting only of integrator systems, represented

### Autonomous Mobile Robot Design

Autonomous Mobile Robot Design Topic: Guidance and Control Introduction and PID Loops Dr. Kostas Alexis (CSE) Autonomous Robot Challenges How do I control where to go? Autonomous Mobile Robot Design Topic:

### Analysis of SISO Control Loops

Chapter 5 Analysis of SISO Control Loops Topics to be covered For a given controller and plant connected in feedback we ask and answer the following questions: Is the loop stable? What are the sensitivities

### Control Systems Design

ELEC4410 Control Systems Design Lecture 13: Stability Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 13: Stability p.1/20 Outline Input-Output

### Richiami di Controlli Automatici

Richiami di Controlli Automatici Gianmaria De Tommasi 1 1 Università degli Studi di Napoli Federico II detommas@unina.it Ottobre 2012 Corsi AnsaldoBreda G. De Tommasi (UNINA) Richiami di Controlli Automatici

### Control Systems I Lecture 10: System Specifications

Control Systems I Lecture 10: System Specifications Readings: Guzzella, Chapter 10 Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 24, 2017 E. Frazzoli (ETH) Lecture

### Collocated versus non-collocated control [H04Q7]

Collocated versus non-collocated control [H04Q7] Jan Swevers September 2008 0-0 Contents Some concepts of structural dynamics Collocated versus non-collocated control Summary This lecture is based on parts

### Robust fixed-order H Controller Design for Spectral Models by Convex Optimization

Robust fixed-order H Controller Design for Spectral Models by Convex Optimization Alireza Karimi, Gorka Galdos and Roland Longchamp Abstract A new approach for robust fixed-order H controller design by

### The stability of linear time-invariant feedback systems

The stability of linear time-invariant feedbac systems A. Theory The system is atrictly stable if The degree of the numerator of H(s) (H(z)) the degree of the denominator of H(s) (H(z)) and/or The poles

### Module 9: State Feedback Control Design Lecture Note 1

Module 9: State Feedback Control Design Lecture Note 1 The design techniques described in the preceding lectures are based on the transfer function of a system. In this lecture we would discuss the state

### H(s) = s. a 2. H eq (z) = z z. G(s) a 2. G(s) A B. s 2 s(s + a) 2 s(s a) G(s) 1 a 1 a. } = (z s 1)( z. e ) ) (z. (z 1)(z e at )(z e at )

.7 Quiz Solutions Problem : a H(s) = s a a) Calculate the zero order hold equivalent H eq (z). H eq (z) = z z G(s) Z{ } s G(s) a Z{ } = Z{ s s(s a ) } G(s) A B Z{ } = Z{ + } s s(s + a) s(s a) G(s) a a

### ROOT LOCUS. Consider the system. Root locus presents the poles of the closed-loop system when the gain K changes from 0 to. H(s) H ( s) = ( s)

C1 ROOT LOCUS Consider the system R(s) E(s) C(s) + K G(s) - H(s) C(s) R(s) = K G(s) 1 + K G(s) H(s) Root locus presents the poles of the closed-loop system when the gain K changes from 0 to 1+ K G ( s)

### 1 An Overview and Brief History of Feedback Control 1. 2 Dynamic Models 23. Contents. Preface. xiii

Contents 1 An Overview and Brief History of Feedback Control 1 A Perspective on Feedback Control 1 Chapter Overview 2 1.1 A Simple Feedback System 3 1.2 A First Analysis of Feedback 6 1.3 Feedback System

### Laplace Transform Analysis of Signals and Systems

Laplace Transform Analysis of Signals and Systems Transfer Functions Transfer functions of CT systems can be found from analysis of Differential Equations Block Diagrams Circuit Diagrams 5/10/04 M. J.

### EC CONTROL SYSTEM UNIT I- CONTROL SYSTEM MODELING

EC 2255 - CONTROL SYSTEM UNIT I- CONTROL SYSTEM MODELING 1. What is meant by a system? It is an arrangement of physical components related in such a manner as to form an entire unit. 2. List the two types

### Compensator Design to Improve Transient Performance Using Root Locus

1 Compensator Design to Improve Transient Performance Using Root Locus Prof. Guy Beale Electrical and Computer Engineering Department George Mason University Fairfax, Virginia Correspondence concerning

### OPTIMAL CONTROL AND ESTIMATION

OPTIMAL CONTROL AND ESTIMATION Robert F. Stengel Department of Mechanical and Aerospace Engineering Princeton University, Princeton, New Jersey DOVER PUBLICATIONS, INC. New York CONTENTS 1. INTRODUCTION

### ECE317 : Feedback and Control

ECE317 : Feedback and Control Lecture : Routh-Hurwitz stability criterion Examples Dr. Richard Tymerski Dept. of Electrical and Computer Engineering Portland State University 1 Course roadmap Modeling

### x(t) = x(t h), x(t) 2 R ), where is the time delay, the transfer function for such a e s Figure 1: Simple Time Delay Block Diagram e i! =1 \e i!t =!

1 Time-Delay Systems 1.1 Introduction Recitation Notes: Time Delays and Nyquist Plots Review In control systems a challenging area is operating in the presence of delays. Delays can be attributed to acquiring

### João P. Hespanha. January 16, 2009

LINEAR SYSTEMS THEORY João P. Hespanha January 16, 2009 Disclaimer: This is a draft and probably contains a few typos. Comments and information about typos are welcome. Please contact the author at hespanha@ece.ucsb.edu.

### AA/EE/ME 548: Problem Session Notes #5

AA/EE/ME 548: Problem Session Notes #5 Review of Nyquist and Bode Plots. Nyquist Stability Criterion. LQG/LTR Method Tuesday, March 2, 203 Outline:. A review of Bode plots. 2. A review of Nyquist plots

### Frequency domain analysis

Automatic Control 2 Frequency domain analysis Prof. Alberto Bemporad University of Trento Academic year 2010-2011 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011

### Discrete and continuous dynamic systems

Discrete and continuous dynamic systems Bounded input bounded output (BIBO) and asymptotic stability Continuous and discrete time linear time-invariant systems Katalin Hangos University of Pannonia Faculty

### STABILITY OF CLOSED-LOOP CONTOL SYSTEMS

CHBE320 LECTURE X STABILITY OF CLOSED-LOOP CONTOL SYSTEMS Professor Dae Ryook Yang Spring 2018 Dept. of Chemical and Biological Engineering 10-1 Road Map of the Lecture X Stability of closed-loop control

### Automatic Control Systems theory overview (discrete time systems)

Automatic Control Systems theory overview (discrete time systems) Prof. Luca Bascetta (luca.bascetta@polimi.it) Politecnico di Milano Dipartimento di Elettronica, Informazione e Bioingegneria Motivations

### Stability of Feedback Control Systems: Absolute and Relative

Stability of Feedback Control Systems: Absolute and Relative Dr. Kevin Craig Greenheck Chair in Engineering Design & Professor of Mechanical Engineering Marquette University Stability: Absolute and Relative

### Linear System Theory

Linear System Theory Wonhee Kim Chapter 6: Controllability & Observability Chapter 7: Minimal Realizations May 2, 217 1 / 31 Recap State space equation Linear Algebra Solutions of LTI and LTV system Stability

### Control Systems Engineering ( Chapter 6. Stability ) Prof. Kwang-Chun Ho Tel: Fax:

Control Systems Engineering ( Chapter 6. Stability ) Prof. Kwang-Chun Ho kwangho@hansung.ac.kr Tel: 02-760-4253 Fax:02-760-4435 Introduction In this lesson, you will learn the following : How to determine

### The loop shaping paradigm. Lecture 7. Loop analysis of feedback systems (2) Essential specifications (2)

Lecture 7. Loop analysis of feedback systems (2). Loop shaping 2. Performance limitations The loop shaping paradigm. Estimate performance and robustness of the feedback system from the loop transfer L(jω)

### ECE 486 Control Systems

ECE 486 Control Systems Spring 208 Midterm #2 Information Issued: April 5, 208 Updated: April 8, 208 ˆ This document is an info sheet about the second exam of ECE 486, Spring 208. ˆ Please read the following

### Contents. 1 State-Space Linear Systems 5. 2 Linearization Causality, Time Invariance, and Linearity 31

Contents Preamble xiii Linear Systems I Basic Concepts 1 I System Representation 3 1 State-Space Linear Systems 5 1.1 State-Space Linear Systems 5 1.2 Block Diagrams 7 1.3 Exercises 11 2 Linearization

### Control System Design

ELEC4410 Control System Design Lecture 19: Feedback from Estimated States and Discrete-Time Control Design Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science

### Discrete Systems. Step response and pole locations. Mark Cannon. Hilary Term Lecture

Discrete Systems Mark Cannon Hilary Term 22 - Lecture 4 Step response and pole locations 4 - Review Definition of -transform: U() = Z{u k } = u k k k= Discrete transfer function: Y () U() = G() = Z{g k},

### CONTROL SYSTEM STABILITY. CHARACTERISTIC EQUATION: The overall transfer function for a. where A B X Y are polynomials. Substitution into the TF gives:

CONTROL SYSTEM STABILITY CHARACTERISTIC EQUATION: The overall transfer function for a feedback control system is: TF = G / [1+GH]. The G and H functions can be put into the form: G(S) = A(S) / B(S) H(S)

### Wind Turbine Control

Wind Turbine Control W. E. Leithead University of Strathclyde, Glasgow Supergen Student Workshop 1 Outline 1. Introduction 2. Control Basics 3. General Control Objectives 4. Constant Speed Pitch Regulated

### CDS Solutions to Final Exam

CDS 22 - Solutions to Final Exam Instructor: Danielle C Tarraf Fall 27 Problem (a) We will compute the H 2 norm of G using state-space methods (see Section 26 in DFT) We begin by finding a minimal state-space

### Systems and Control Theory Lecture Notes. Laura Giarré

Systems and Control Theory Lecture Notes Laura Giarré L. Giarré 2017-2018 Lesson 17: Model-based Controller Feedback Stabilization Observers Ackerman Formula Model-based Controller L. Giarré- Systems and

### Advanced Process Control Tutorial Problem Set 2 Development of Control Relevant Models through System Identification

Advanced Process Control Tutorial Problem Set 2 Development of Control Relevant Models through System Identification 1. Consider the time series x(k) = β 1 + β 2 k + w(k) where β 1 and β 2 are known constants

### Nyquist Criterion For Stability of Closed Loop System

Nyquist Criterion For Stability of Closed Loop System Prof. N. Puri ECE Department, Rutgers University Nyquist Theorem Given a closed loop system: r(t) + KG(s) = K N(s) c(t) H(s) = KG(s) +KG(s) = KN(s)

### Frequency Response Techniques

4th Edition T E N Frequency Response Techniques SOLUTION TO CASE STUDY CHALLENGE Antenna Control: Stability Design and Transient Performance First find the forward transfer function, G(s). Pot: K 1 = 10