# Intro. Computer Control Systems: F8

Size: px
Start display at page:

Transcription

1 Intro. Computer Control Systems: F8 Properties of state-space descriptions and feedback Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 22

2 F7: Quiz! 2 / 22

3 F7: Quiz! 1) The state-space description of a system is a not unique b unique c stable 2 / 22

4 F7: Quiz! 1) The state-space description of a system is a not unique b unique c stable 2) The eigenvalues of the system matrix A reveals something about a poles b zeros c the closed-loop system 2 / 22

5 F7: Quiz! 1) The state-space description of a system is a not unique b unique c stable 2) The eigenvalues of the system matrix A reveals something about a poles b zeros c the closed-loop system 3) Solution to ẋ = Ax + Bu with initial condition x 0 is obtained using a a linear system of equations b the matrix exponential c the Nyquist contour 2 / 22

6 Nonlinear time-invariant systems 3 / 22

7 Nonlinear systems and states Most systems are nonlinear! Nonlinear differential equations: ẋ = f(x, u) y = h(x, u) Linearize around operating point x 0, u 0. Typically use a stationary point: ẋ = f(x 0, u 0 )=0 4 / 22

8 Nonlinear systems and states Nonlinear differential equations: ẋ = f(x, u) y = h(x, u) Taylor series expansion around stationary point x 0, u 0 with y 0 = h(x 0, u 0 ) results in linear deviation model: x = A x + B u y = C x + D u Linear state-space description of the deviations around the operating point of system. Matrices A, B, C and D given by derivatives of f(x, u) and h(x, u) with respect to x and u. See ch. 8.4 G&L. 4 / 22

9 Feedback control using states 5 / 22

10 State-feedback control State space description of linear time-invariant system ẋ = Ax + Bu y = Cx Y (s) = G(s)U(s) G u x y (si A) 1 B C 6 / 22

11 State-feedback control State space description of linear time-invariant system ẋ = Ax + Bu y = Cx G(s) = C(sI A) 1 B u x y (si A) 1 B C 6 / 22

12 State-feedback control Idea: Feedback control using states u = Lx + l 0 r, where L and l 0 are design parameters. rl 0 u x y (si A) 1 B C + L ẋ = Ax + B ( Lx + l 0 r) }{{} =u 6 / 22

13 State-feedback control rl 0 u x y (si A) 1 B C + L Closed-loop system from r to y comes: ẋ = Ax + B ( Lx + l 0 r) = (A BL)x + Bl 0 r y = Cx Is it possible to control the system to all states x in R n? design the closed-loop system s poles? (estimate the state x(t)?) 6 / 22

14 Controllability 7 / 22

15 Controllability A sought state x is controllable if some input u(t) can move the system from x(0) = 0 to x(t ) = x x(t) x 8 / 22

16 Controllability For x 0 = 0, we can compute the state at t = T T x(t ) = e At x 0 + e Aτ Bu(T τ)dτ 0 8 / 22

17 Controllability Med x 0 = 0 är tillståndet vid t = T Therefore: x(t ) = T 0 e Aτ Bu(T τ)dτ = via Cayley-Hamiltons theorem = Bγ 0 + ABγ A n 1 Bγ n 1 x(t ) is a linear combination of B, AB,..., A n 1 B. A state x is controllable if it can be expressed as such a linear combination, i.e., if x is in the column space of S [B AB A n 1 B] 8 / 22

18 Controllability x(t ) x Figur : Example column space of S and non-controllable state x. Controllable system All states x are controllable S:s columns are linearly independent Note: rank(s) = n or det(s) 0 8 / 22

19 Controllability x(t ) x Figur : Example column space of S and non-controllable state x. Controllable canonical form System is controllable It can be written on controllable canonical form 8 / 22

20 Observability 9 / 22

21 Observability Assume u(t) 0. A state x 0 is unobservable if the output y(t) 0 when system starts at x(0) = x. y(t) = Cx(t) x t 10 / 22

22 Observability When u(t) 0 we obtain y(t) = Cx(t) = Ce At x +0 When y(t) 0 we do not observe any changes in the output: That is, d k dt k y(t) t=0 = CA k x =0. Cx = 0, CAx = 0,..., CA n 1 x = 0 10 / 22

23 Observability When u(t) 0 and y(t) 0 we observe no changes: Cx = 0, CAx = 0,..., CA n 1 x = 0 or where O Ox = 0 C CA. CA n 1 Therefore: A state x 0 is unobservable if it belongs to the null space of O. 10 / 22

24 Observability y(t) = Cx(t) x t Figur : Example null space of O and unobservable state x. Observable system All states x are observable O:s columns are linearly independent Note: rank(o) = n or det(o) 0 10 / 22

25 Observability y(t) = Cx(t) x t Figur : Example null space of O and unobservable state x. Observable canonical form System is observable It can be written on observable canonical form 10 / 22

26 Build intuition 11 / 22

27 Build intuition from simple systems Example: controllable system System on controllable canonical form: [ 2 1 ẋ(t) = 1 0 Transfer function: y(t) = [ 1 G(s) = C(sI A) 1 B = 1 ] x(t) ] x(t) + [ ] 1 u(t) 0 s + 1 s 2 + 2s + 1 = s + 1 (s + 1) 2 = 1 s + 1 [Board: investigate observability using O] 12 / 22

28 Build intuition from simple systems Example: controllable system System on controllable canonical form: [ 2 1 ẋ(t) = 1 0 Transfer function: y(t) = [ 1 G(s) = C(sI A) 1 B = 1 ] x(t) ] x(t) + [ ] 1 u(t) 0 s + 1 s 2 + 2s + 1 = s + 1 (s + 1) 2 = 1 s + 1 [Board: investigate observability using O] [ ] 1 1 O = det O = 0 unonbservable / 22

29 Build intuition from simple systems Example: observable system System on observable canonical form: [ ] 2 1 ẋ(t) = x(t) Transfer function: y(t) = [ 1 G(s) = C(sI A) 1 B = 0 ] x(t) [ ] 1 u(t) 1 s + 1 s 2 + 2s + 1 = s + 1 (s + 1) 2 = 1 s + 1 [Board: investigate controllability using S] 13 / 22

30 Build intuition from simple systems Example: observable system System on observable canonical form: [ ] 2 1 ẋ(t) = x(t) Transfer function: y(t) = [ 1 G(s) = C(sI A) 1 B = 0 ] x(t) [ ] 1 u(t) 1 s + 1 s 2 + 2s + 1 = s + 1 (s + 1) 2 = 1 s + 1 [Board: investigate controllability using S] [ ] 1 1 S = det S = 0 non-controllable / 22

31 Build intuition from simple systems Exemple: controllable and observable system Systems in previous examples have the same transfer function G(s) = 1 s + 1. Can also be written in state-space form ẋ(t) = x(t) + u(t), y(t) = x(t). where x(t) is a scalar. [Board: investigate S and O] 14 / 22

32 Build intuition from simple systems Exemple: controllable and observable system Systems in previous examples have the same transfer function G(s) = 1 s + 1. Can also be written in state-space form ẋ(t) = x(t) + u(t), y(t) = x(t). where x(t) is a scalar. [Board: investigate S and O] S = 1 O = 1 det S = 1 det O = 1 controllable and observable (1) Note: we eliminated invisible states 14 / 22

33 Minimal realization 15 / 22

34 Minimal realization System with transfer function G(s) and state-space form ẋ = Ax + Bu y = Cx u x y (si A) 1 B C Definition 8.2 G&L State-space form of G(s) is a minimal realization if vector x has the smallest possible dimension. 16 / 22

35 Minimal realization System with transfer function G(s) and state-space form ẋ = Ax + Bu y = Cx u x y (si A) 1 B C Definition 8.2 G&L State-space form of G(s) is a minimal realization if vector x has the smallest possible dimension. Result 8.11(+8.12) G&L A state-space form is minimal realization controllable and observable A:s eigenvalues = G(s):s poles 16 / 22

36 Design of state-feedback control 17 / 22

37 State-feedback control State-space model with controller u = Lx + l 0 r where L = [ l 1 l 2 l n ] 18 / 22

38 State-feedback control State-space model with controller u = Lx + l 0 r where L = [ l 1 l 2 l n ] Closed-loop system ẋ = (A BL)x + Bl 0 r y = Cx 18 / 22

39 State-feedback control State-space model with controller u = Lx + l 0 r where L = [ l 1 l 2 l n ] Closed-loop system as a transfer function Output is Y (s) = G c (s)r(s), where G c (s) = C(sI A + BL) 1 Bl 0 18 / 22

40 State-feedback control State-space model with controller u = Lx + l 0 r where L = [ l 1 l 2 l n ] System matrix of closed-loop system: (A BL) Eigenvalues/poles given by polynomial equation det(si A + BL) = 0 which we can design via L! 18 / 22

41 State-feedback control Design of the gain l 0 Y (s) = G c (s)r(s) where G c (s) = C(sI A + BL) 1 Bl 0. It is desirable to have at least G c (0) = 1 19 / 22

42 State-feedback control Design of the gain l 0 Y (s) = G c (s)r(s) where G c (s) = C(sI A + BL) 1 Bl 0. It is desirable to have at least G c (0) = 1 G c (0) = C( A + BL) 1 Bl 0 = 1 and so l 0 = 1 C( A + BL) 1 B 19 / 22

43 State-feedback control Design of the gain l 0 Y (s) = G c (s)r(s) where G c (s) = C(sI A + BL) 1 Bl 0. It is desirable to have at least G c (0) = 1 G c (0) = C( A + BL) 1 Bl 0 = 1 and so l 0 = 1 C( A + BL) 1 B More generally, replace l 0 r with F r (s)r(s) How to design L? 19 / 22

44 Build intuition from simple systems Exemple: state-vector in R 2 y u Figur : Force u(t) and position y(t). State-space form: [ ] [ ] ẋ = x + u k/m 0 1/m y = [ 1 0 ] x [Board: design L so that closed-loop system has poles -2 and -3] 20 / 22

45 Pole placement State-feedback control rl 0 u x y (si A) 1 B C + L Result 9.1 State-space form is controllable L can be designed to yield arbitrarily placed poles (real and complex-conjugated) of the closed-loop system 21 / 22

46 Pole placement State-feedback control rl 0 u x y (si A) 1 B C + L Result 9.1 State-space form is controllable L can be designed to yield arbitrarily placed poles (real and complex-conjugated) of the closed-loop system L solved by det(si A + BL) = 0 with desired roots L very simple to solve for system on controllable canonical form 21 / 22

47 Pole placement State-feedback control rl 0 u x y (si A) 1 B C + L Result 9.1 State-space form is controllable L can be designed to yield arbitrarily placed poles (real and complex-conjugated) of the closed-loop system L solved by det(si A + BL) = 0 with desired roots L very simple to solve for system on controllable canonical form What to do when we can t measure x directly? 21 / 22

48 Summary and recap Linearization of nonlinear system models Properties: Controllable Observable Minimal realization State-feedback control Pole placement for the closed-loop system 22 / 22

### Intro. Computer Control Systems: F9

Intro. Computer Control Systems: F9 State-feedback control and observers Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 21 dave.zachariah@it.uu.se F8: Quiz! 2 / 21 dave.zachariah@it.uu.se

### Lec 6: State Feedback, Controllability, Integral Action

Lec 6: State Feedback, Controllability, Integral Action November 22, 2017 Lund University, Department of Automatic Control Controllability and Observability Example of Kalman decomposition 1 s 1 x 10 x

### Control 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)

### Topic # Feedback Control

Topic #11 16.31 Feedback Control State-Space Systems State-space model features Observability Controllability Minimal Realizations Copyright 21 by Jonathan How. 1 Fall 21 16.31 11 1 State-Space Model Features

### CONTROL DESIGN FOR SET POINT TRACKING

Chapter 5 CONTROL DESIGN FOR SET POINT TRACKING In this chapter, we extend the pole placement, observer-based output feedback design to solve tracking problems. By tracking we mean that the output is commanded

### Control 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

### 1. Find the solution of the following uncontrolled linear system. 2 α 1 1

Appendix B Revision Problems 1. Find the solution of the following uncontrolled linear system 0 1 1 ẋ = x, x(0) =. 2 3 1 Class test, August 1998 2. Given the linear system described by 2 α 1 1 ẋ = x +

### 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 Design, SC4026. SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft

Control Systems Design, SC4026 SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft Lecture 4 Controllability (a.k.a. Reachability) vs Observability Algebraic Tests (Kalman rank condition & Hautus test) A few

### Balanced Truncation 1

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.242, Fall 2004: MODEL REDUCTION Balanced Truncation This lecture introduces balanced truncation for LTI

### Solution of Linear State-space Systems

Solution of Linear State-space Systems Homogeneous (u=0) LTV systems first Theorem (Peano-Baker series) The unique solution to x(t) = (t, )x 0 where The matrix function is given by is called the state

### Control Systems Design, SC4026. SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft

Control Systems Design, SC4026 SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft Lecture 4 Controllability (a.k.a. Reachability) and Observability Algebraic Tests (Kalman rank condition & Hautus test) A few

### Observability. Dynamic Systems. Lecture 2 Observability. Observability, continuous time: Observability, discrete time: = h (2) (x, u, u)

Observability Dynamic Systems Lecture 2 Observability Continuous time model: Discrete time model: ẋ(t) = f (x(t), u(t)), y(t) = h(x(t), u(t)) x(t + 1) = f (x(t), u(t)), y(t) = h(x(t)) Reglerteknik, ISY,

### Module 03 Linear Systems Theory: Necessary Background

Module 03 Linear Systems Theory: Necessary Background Ahmad F. Taha EE 5243: Introduction to Cyber-Physical Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha/index.html September

### ECE 388 Automatic Control

Controllability and State Feedback Control Associate Prof. Dr. of Mechatronics Engineeering Çankaya University Compulsory Course in Electronic and Communication Engineering Credits (2/2/3) Course Webpage:

### Control Systems. Laplace domain analysis

Control Systems Laplace domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic equations define an Input/Output

### ECEN 605 LINEAR SYSTEMS. Lecture 7 Solution of State Equations 1/77

1/77 ECEN 605 LINEAR SYSTEMS Lecture 7 Solution of State Equations Solution of State Space Equations Recall from the previous Lecture note, for a system: ẋ(t) = A x(t) + B u(t) y(t) = C x(t) + D u(t),

### The disturbance decoupling problem (DDP)

CHAPTER 3 The disturbance decoupling problem (DDP) Consider the system 3.1. Geometric formulation { ẋ = Ax + Bu + Ew y = Cx. Problem 3.1 (Disturbance decoupling). Find a state feedback u = Fx+ v such that

### ẋ n = f n (x 1,...,x n,u 1,...,u m ) (5) y 1 = g 1 (x 1,...,x n,u 1,...,u m ) (6) y p = g p (x 1,...,x n,u 1,...,u m ) (7)

EEE582 Topical Outline A.A. Rodriguez Fall 2007 GWC 352, 965-3712 The following represents a detailed topical outline of the course. It attempts to highlight most of the key concepts to be covered and

### Module 09 From s-domain to time-domain From ODEs, TFs to State-Space Modern Control

Module 09 From s-domain to time-domain From ODEs, TFs to State-Space Modern Control Ahmad F. Taha EE 3413: Analysis and Desgin of Control Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/

### 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

### EE C128 / ME C134 Fall 2014 HW 9 Solutions. HW 9 Solutions. 10(s + 3) s(s + 2)(s + 5) G(s) =

1. Pole Placement Given the following open-loop plant, HW 9 Solutions G(s) = 1(s + 3) s(s + 2)(s + 5) design the state-variable feedback controller u = Kx + r, where K = [k 1 k 2 k 3 ] is the feedback

### 16.31 Fall 2005 Lecture Presentation Mon 31-Oct-05 ver 1.1

16.31 Fall 2005 Lecture Presentation Mon 31-Oct-05 ver 1.1 Charles P. Coleman October 31, 2005 1 / 40 : Controllability Tests Observability Tests LEARNING OUTCOMES: Perform controllability tests Perform

### EE221A Linear System Theory Final Exam

EE221A Linear System Theory Final Exam Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2016 12/16/16, 8-11am Your answers must be supported by analysis,

### 5. Observer-based Controller Design

EE635 - Control System Theory 5. Observer-based Controller Design Jitkomut Songsiri state feedback pole-placement design regulation and tracking state observer feedback observer design LQR and LQG 5-1

### ECEN 605 LINEAR SYSTEMS. Lecture 8 Invariant Subspaces 1/26

1/26 ECEN 605 LINEAR SYSTEMS Lecture 8 Invariant Subspaces Subspaces Let ẋ(t) = A x(t) + B u(t) y(t) = C x(t) (1a) (1b) denote a dynamic system where X, U and Y denote n, r and m dimensional vector spaces,

### POLE PLACEMENT. Sadegh Bolouki. Lecture slides for ECE 515. University of Illinois, Urbana-Champaign. Fall S. Bolouki (UIUC) 1 / 19

POLE PLACEMENT Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 19 Outline 1 State Feedback 2 Observer 3 Observer Feedback 4 Reduced Order

### Control 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:

### 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

### CDS Solutions to the Midterm Exam

CDS 22 - Solutions to the Midterm Exam Instructor: Danielle C. Tarraf November 6, 27 Problem (a) Recall that the H norm of a transfer function is time-delay invariant. Hence: ( ) Ĝ(s) = s + a = sup /2

### Identification Methods for Structural Systems

Prof. Dr. Eleni Chatzi System Stability Fundamentals Overview System Stability Assume given a dynamic system with input u(t) and output x(t). The stability property of a dynamic system can be defined from

### State will have dimension 5. One possible choice is given by y and its derivatives up to y (4)

A Exercise State will have dimension 5. One possible choice is given by y and its derivatives up to y (4 x T (t [ y(t y ( (t y (2 (t y (3 (t y (4 (t ] T With this choice we obtain A B C [ ] D 2 3 4 To

### State Feedback and State Estimators Linear System Theory and Design, Chapter 8.

1 Linear System Theory and Design, http://zitompul.wordpress.com 2 0 1 4 State Estimator In previous section, we have discussed the state feedback, based on the assumption that all state variables are

### ACM/CMS 107 Linear Analysis & Applications Fall 2016 Assignment 4: Linear ODEs and Control Theory Due: 5th December 2016

ACM/CMS 17 Linear Analysis & Applications Fall 216 Assignment 4: Linear ODEs and Control Theory Due: 5th December 216 Introduction Systems of ordinary differential equations (ODEs) can be used to describe

### Perspective. ECE 3640 Lecture 11 State-Space Analysis. To learn about state-space analysis for continuous and discrete-time. Objective: systems

ECE 3640 Lecture State-Space Analysis Objective: systems To learn about state-space analysis for continuous and discrete-time Perspective Transfer functions provide only an input/output perspective of

### Autonomous system = system without inputs

Autonomous system = system without inputs State space representation B(A,C) = {y there is x, such that σx = Ax, y = Cx } x is the state, n := dim(x) is the state dimension, y is the output Polynomial representation

### Module 07 Controllability and Controller Design of Dynamical LTI Systems

Module 07 Controllability and Controller Design of Dynamical LTI Systems Ahmad F. Taha EE 5143: Linear Systems and Control Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ataha October

### ECE504: Lecture 9. D. Richard Brown III. Worcester Polytechnic Institute. 04-Nov-2008

ECE504: Lecture 9 D. Richard Brown III Worcester Polytechnic Institute 04-Nov-2008 Worcester Polytechnic Institute D. Richard Brown III 04-Nov-2008 1 / 38 Lecture 9 Major Topics ECE504: Lecture 9 We are

### SYSTEMTEORI - ÖVNING 5: FEEDBACK, POLE ASSIGNMENT AND OBSERVER

SYSTEMTEORI - ÖVNING 5: FEEDBACK, POLE ASSIGNMENT AND OBSERVER Exercise 54 Consider the system: ẍ aẋ bx u where u is the input and x the output signal (a): Determine a state space realization (b): Is the

### ELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky

ELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky Equilibrium points and linearization Eigenvalue decomposition and modal form State transition matrix and matrix exponential Stability ELEC 3035 (Part

### ECEN 420 LINEAR CONTROL SYSTEMS. Lecture 6 Mathematical Representation of Physical Systems II 1/67

1/67 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 6 Mathematical Representation of Physical Systems II State Variable Models for Dynamic Systems u 1 u 2 u ṙ. Internal Variables x 1, x 2 x n y 1 y 2. y m Figure

### Full State Feedback for State Space Approach

Full State Feedback for State Space Approach State Space Equations Using Cramer s rule it can be shown that the characteristic equation of the system is : det[ si A] 0 Roots (for s) of the resulting polynomial

### ECE 602 Exam 2 Solutions, 3/23/2011.

NAME: ECE 62 Exam 2 Solutions, 3/23/211. You can use any books or paper notes you wish to bring. No electronic devices of any kind are allowed. You can only use materials that you bring yourself. You are

### Introduction to Nonlinear Control Lecture # 4 Passivity

p. 1/6 Introduction to Nonlinear Control Lecture # 4 Passivity È p. 2/6 Memoryless Functions ¹ y È Ý Ù È È È È u (b) µ power inflow = uy Resistor is passive if uy 0 p. 3/6 y y y u u u (a) (b) (c) Passive

### Observability and state estimation

EE263 Autumn 2015 S Boyd and S Lall Observability and state estimation state estimation discrete-time observability observability controllability duality observers for noiseless case continuous-time observability

### EL 625 Lecture 10. Pole Placement and Observer Design. ẋ = Ax (1)

EL 625 Lecture 0 EL 625 Lecture 0 Pole Placement and Observer Design Pole Placement Consider the system ẋ Ax () The solution to this system is x(t) e At x(0) (2) If the eigenvalues of A all lie in the

### ECEEN 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

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science : Dynamic Systems Spring 2011

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.4: Dynamic Systems Spring Homework Solutions Exercise 3. a) We are given the single input LTI system: [

### Solution for Homework 5

Solution for Homework 5 ME243A/ECE23A Fall 27 Exercise 1 The computation of the reachable subspace in continuous time can be handled easily introducing the concepts of inner product, orthogonal complement

### Topics in control Tracking and regulation A. Astolfi

Topics in control Tracking and regulation A. Astolfi Contents 1 Introduction 1 2 The full information regulator problem 3 3 The FBI equations 5 4 The error feedback regulator problem 5 5 The internal model

### PARAMETERIZATION OF STATE FEEDBACK GAINS FOR POLE PLACEMENT

PARAMETERIZATION OF STATE FEEDBACK GAINS FOR POLE PLACEMENT Hans Norlander Systems and Control, Department of Information Technology Uppsala University P O Box 337 SE 75105 UPPSALA, Sweden HansNorlander@ituuse

### Lecture 19 Observability and state estimation

EE263 Autumn 2007-08 Stephen Boyd Lecture 19 Observability and state estimation state estimation discrete-time observability observability controllability duality observers for noiseless case continuous-time

### Kalman Decomposition B 2. z = T 1 x, where C = ( C. z + u (7) T 1, and. where B = T, and

Kalman Decomposition Controllable / uncontrollable decomposition Suppose that the controllability matrix C R n n of a system has rank n 1

### 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)

### Linear Systems. Linear systems?!? (Roughly) Systems which obey properties of superposition Input u(t) output

Linear Systems Linear systems?!? (Roughly) Systems which obey properties of superposition Input u(t) output Our interest is in dynamic systems Dynamic system means a system with memory of course including

### EC Control Engineering Quiz II IIT Madras

EC34 - Control Engineering Quiz II IIT Madras Linear algebra Find the eigenvalues and eigenvectors of A, A, A and A + 4I Find the eigenvalues and eigenvectors of the following matrices: (a) cos θ sin θ

### Chap. 3. Controlled Systems, Controllability

Chap. 3. Controlled Systems, Controllability 1. Controllability of Linear Systems 1.1. Kalman s Criterion Consider the linear system ẋ = Ax + Bu where x R n : state vector and u R m : input vector. A :

### Introduction to Modern Control MT 2016

CDT Autonomous and Intelligent Machines & Systems Introduction to Modern Control MT 2016 Alessandro Abate Lecture 2 First-order ordinary differential equations (ODE) Solution of a linear ODE Hints to nonlinear

### Control, Stabilization and Numerics for Partial Differential Equations

Paris-Sud, Orsay, December 06 Control, Stabilization and Numerics for Partial Differential Equations Enrique Zuazua Universidad Autónoma 28049 Madrid, Spain enrique.zuazua@uam.es http://www.uam.es/enrique.zuazua

### 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

### Nonlinear Observers. Jaime A. Moreno. Eléctrica y Computación Instituto de Ingeniería Universidad Nacional Autónoma de México

Nonlinear Observers Jaime A. Moreno JMorenoP@ii.unam.mx Eléctrica y Computación Instituto de Ingeniería Universidad Nacional Autónoma de México XVI Congreso Latinoamericano de Control Automático October

### Introduction to Computer Control Systems

Introduction to Computer Control Systems Lecture 1: Introduction Dave Zachariah Div. Systems and Control, Dept. Information Technology, Uppsala University October 28, 2014 (UU/Info Technology/SysCon) Intro.

### Topic # Feedback Control

Topic #5 6.3 Feedback Control State-Space Systems Full-state Feedback Control How do we change the poles of the state-space system? Or,evenifwecanchangethepolelocations. Where do we put the poles? Linear

### Topic # Feedback Control. State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback

Topic #17 16.31 Feedback Control State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback Back to reality Copyright 21 by Jonathan How. All Rights reserved 1 Fall

### MEM 355 Performance Enhancement of Dynamical Systems MIMO Introduction

MEM 355 Performance Enhancement of Dynamical Systems MIMO Introduction Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University 11/2/214 Outline Solving State Equations Variation

### Exam 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

### 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

### Nonlinear Control Systems

Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 7. Feedback Linearization IST-DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs1/ 1 1 Feedback Linearization Given a nonlinear

### Course Outline. Higher Order Poles: Example. Higher Order Poles. Amme 3500 : System Dynamics & Control. State Space Design. 1 G(s) = s(s + 2)(s +10)

Amme 35 : System Dynamics Control State Space Design Course Outline Week Date Content Assignment Notes 1 1 Mar Introduction 2 8 Mar Frequency Domain Modelling 3 15 Mar Transient Performance and the s-plane

### 1 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

### Nonlinear Control. Nonlinear Control Lecture # 6 Passivity and Input-Output Stability

Nonlinear Control Lecture # 6 Passivity and Input-Output Stability Passivity: Memoryless Functions y y y u u u (a) (b) (c) Passive Passive Not passive y = h(t,u), h [0, ] Vector case: y = h(t,u), h T =

### Nonlinear Control Systems

Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 5. Input-Output Stability DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs2012/ 2012 1 Input-Output Stability y = Hu H denotes

### Control Systems. Frequency domain analysis. L. Lanari

Control Systems m i l e r p r a in r e v y n is o Frequency domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic

### 4F3 - Predictive Control

4F3 Predictive Control - Discrete-time systems p. 1/30 4F3 - Predictive Control Discrete-time State Space Control Theory For reference only Jan Maciejowski jmm@eng.cam.ac.uk 4F3 Predictive Control - Discrete-time

### State Regulator. Advanced Control. design of controllers using pole placement and LQ design rules

Advanced Control State Regulator Scope design of controllers using pole placement and LQ design rules Keywords pole placement, optimal control, LQ regulator, weighting matrixes Prerequisites Contact state

### Control engineering sample exam paper - Model answers

Question Control engineering sample exam paper - Model answers a) By a direct computation we obtain x() =, x(2) =, x(3) =, x(4) = = x(). This trajectory is sketched in Figure (left). Note that A 2 = I

### Control Systems Design, SC4026. SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft

Control Systems Design, SC426 SC426 Fall 2, dr A Abate, DCSC, TU Delft Lecture 5 Controllable Canonical and Observable Canonical Forms Stabilization by State Feedback State Estimation, Observer Design

### There are none. Abstract for Gauranteed Margins for LQG Regulators, John Doyle, 1978 [Doy78].

Chapter 7 Output Feedback There are none. Abstract for Gauranteed Margins for LQG Regulators, John Doyle, 1978 [Doy78]. In the last chapter we considered the use of state feedback to modify the dynamics

### Topic # /31 Feedback Control Systems

Topic #7 16.30/31 Feedback Control Systems State-Space Systems What are the basic properties of a state-space model, and how do we analyze these? Time Domain Interpretations System Modes Fall 2010 16.30/31

### Modeling and Analysis of Dynamic Systems

Modeling and Analysis of Dynamic Systems Dr. Guillaume Ducard Fall 2017 Institute for Dynamic Systems and Control ETH Zurich, Switzerland G. Ducard c 1 / 57 Outline 1 Lecture 13: Linear System - Stability

### ECE504: Lecture 10. D. Richard Brown III. Worcester Polytechnic Institute. 11-Nov-2008

ECE504: Lecture 10 D. Richard Brown III Worcester Polytechnic Institute 11-Nov-2008 Worcester Polytechnic Institute D. Richard Brown III 11-Nov-2008 1 / 25 Lecture 10 Major Topics We are finishing up Part

### ME 234, Lyapunov and Riccati Problems. 1. This problem is to recall some facts and formulae you already know. e Aτ BB e A τ dτ

ME 234, Lyapunov and Riccati Problems. This problem is to recall some facts and formulae you already know. (a) Let A and B be matrices of appropriate dimension. Show that (A, B) is controllable if and

### Chap 4. State-Space Solutions and

Chap 4. State-Space Solutions and Realizations Outlines 1. Introduction 2. Solution of LTI State Equation 3. Equivalent State Equations 4. Realizations 5. Solution of Linear Time-Varying (LTV) Equations

### Solution via Laplace transform and matrix exponential

EE263 Autumn 2015 S. Boyd and S. Lall Solution via Laplace transform and matrix exponential Laplace transform solving ẋ = Ax via Laplace transform state transition matrix matrix exponential qualitative

### Robust Control 2 Controllability, Observability & Transfer Functions

Robust Control 2 Controllability, Observability & Transfer Functions Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University /26/24 Outline Reachable Controllability Distinguishable

### 1. The Transition Matrix (Hint: Recall that the solution to the linear equation ẋ = Ax + Bu is

ECE 55, Fall 2007 Problem Set #4 Solution The Transition Matrix (Hint: Recall that the solution to the linear equation ẋ Ax + Bu is x(t) e A(t ) x( ) + e A(t τ) Bu(τ)dτ () This formula is extremely important

### Linear System Theory

Linear System Theory Wonhee Kim Lecture 3 Mar. 21, 2017 1 / 38 Overview Recap Nonlinear systems: existence and uniqueness of a solution of differential equations Preliminaries Fields and Vector Spaces

### Linearization problem. The simplest example

Linear Systems Lecture 3 1 problem Consider a non-linear time-invariant system of the form ( ẋ(t f x(t u(t y(t g ( x(t u(t (1 such that x R n u R m y R p and Slide 1 A: f(xu f(xu g(xu and g(xu exist and

### 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

### Controllability. Chapter Reachable States. This chapter develops the fundamental results about controllability and pole assignment.

Chapter Controllability This chapter develops the fundamental results about controllability and pole assignment Reachable States We study the linear system ẋ = Ax + Bu, t, where x(t) R n and u(t) R m Thus

### Chapter 9 Observers, Model-based Controllers 9. Introduction In here we deal with the general case where only a subset of the states, or linear combin

Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter 9 Observers,

### State Feedback and State Estimators Linear System Theory and Design, Chapter 8.

1 Linear System Theory and Design, http://zitompul.wordpress.com 2 0 1 4 2 Homework 7: State Estimators (a) For the same system as discussed in previous slides, design another closed-loop state estimator,

### Control Systems I. Lecture 2: Modeling and Linearization. Suggested Readings: Åström & Murray Ch Jacopo Tani

Control Systems I Lecture 2: Modeling and Linearization Suggested Readings: Åström & Murray Ch. 2-3 Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 28, 2018 J. Tani, E.

### Module 08 Observability and State Estimator Design of Dynamical LTI Systems

Module 08 Observability and State Estimator Design of Dynamical LTI Systems Ahmad F. Taha EE 5143: Linear Systems and Control Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ataha November

### Pole placement control: state space and polynomial approaches Lecture 2

: state space and polynomial approaches Lecture 2 : a state O. Sename 1 1 Gipsa-lab, CNRS-INPG, FRANCE Olivier.Sename@gipsa-lab.fr www.gipsa-lab.fr/ o.sename -based November 21, 2017 Outline : a state

### 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

### Stability of Parameter Adaptation Algorithms. Big picture

ME5895, UConn, Fall 215 Prof. Xu Chen Big picture For ˆθ (k + 1) = ˆθ (k) + [correction term] we haven t talked about whether ˆθ(k) will converge to the true value θ if k. We haven t even talked about

### Control Systems I. Lecture 2: Modeling. Suggested Readings: Åström & Murray Ch. 2-3, Guzzella Ch Emilio Frazzoli

Control Systems I Lecture 2: Modeling Suggested Readings: Åström & Murray Ch. 2-3, Guzzella Ch. 2-3 Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 29, 2017 E. Frazzoli