Advanced Control Theory

Size: px
Start display at page:

Download "Advanced Control Theory"

Transcription

1 State Space Solution and Realization

2 Outline State space solution 2

3 Solution of state-space equations x t = Ax t + Bu t First, recall results for scalar equation: x t = a x t + b u t Laplace transforming: sx s x 0 = a X s + b U s (s a)x s = x(0) + b U s Solve for transformed variable: X s = x 0 s a + b s a U s 3

4 Solution of state-space equations Recall Laplace transform L 1 1 s a = eat L 1 F 1 s F 2 (s) = 0 t f 1 t τ f_2 τ dτ Taking inverse Laplace transform of x t = e at x(0) + zero input solution 0 t e a t τ bu τ dτ zero state solution 4

5 Solution of state-space equations Vector-matrix equation: x t = Ax t + Bu t Laplace transforming: sx s x 0 = A X s + B U s (si A)X s = x(0) + BU s X s = si A 1 x(0) + si A 1 BU s Introducing matrix exponential by analogy with scalar exponential function e at = at + 1 2! at ! at ! at 4 + e At = I At + 1 2! At ! At ! At 4 + 5

6 Ref) Matrix exponential e At matrix exponential If A is an (n n) square matrix, then e At is also an (n n) matrix From the following (verify!) 1 s a = 1 s + a s 2 + a2 s 3 + a3 s 4 + si A 1 = I s + A s 2 + A2 s 3 + A3 s 4 + Laplace transform of e At can be found L 1 1 s a = at + 1 2! at ! at ! at 4 + = e at L 1 si A 1 = I At + 1 2! At ! At ! At 4 + = e At 6

7 Solution of state-space equations Note that: L e At = si A 1 From the previous solution in Laplace domain X s = si A 1 x(0) + si A 1 BU s Taking inverse Laplace transform x t = e At x(0) + zero input solution 0 t e A t τ bu τ dτ zero state solution 7

8 State transition matrix State transition matrix: It describes the transition of the states from initial conditions x(0) to those at time t, when there is no input: x t = Φ t x(0) State transition matrix is used to find the solution of linear time varying system x t = A(t)x t + B(t)u t The matrix exponential e At is the state transition matrix for LTI system For LTI, Φ t satisfies: Φ 0 = e A 0 = I, Φ 1 t = e At = Φ t Other properties of Φ(t): Φ t 1 + t 2 = Φ t 2 Φ(t 1 ) 8

9 Computation of state transition matrix In principle, we can compute Φ t = e At = I At + 1 2! At ! At 3 + as a series, terminating when no significant change observed Very simple example: double-integrator plant control orientation of satellite with thrusters J States: θ = F d = u, Φ s U s = 1 Js 2 S-S: 9

10 Computation of state transition matrix Here: Hence: Then, solution to homogeneous equation x t = Ax(t) with initial conditions x 0 = θ 0 ω 0 T is In practice, more sophisticated numerical algorithms are employed to compute the state transition matrix 10

11 Ex. State transition matrix via Laplace transform State transition matrix can be obtained from Laplace transform solution To illustrate method, consider example plant with transfer function Y(s) U(s) = 1 s 2 + 3s + 2 = 1 (s + 1)(s + 2) A state-space realization is x 1 x 2 = x 1 x u For this system y = 1 0 x 1 x 2 si A = s = s s 11

12 Ex. State transition matrix via Laplace transform Hence Φ s = si A 1 = = s (s + 1)(s + 2) 2 (s + 1)(s + 2) s 1 2 s + 3 s 2 + 3s (s + 1)(s + 2) s + 3 (s + 1)(s + 2) = 1 s s s s s s s s + 1 Thus Φ t = L 1 si A 1 = e t + 2e 2t e t 2e 2t 2e t + 2e 2t 2e t e 2t 12

13 Ex. State transition matrix via Laplace transform Calculate with MATLAB (Symbolic math) >>A = [-3 1; -2 0]; >>syms t >>Phi = expm(a*t) Φ t = L 1 si A 1 = e t + 2e 2t e t 2e 2t 2e t + 2e 2t 2e t e 2t Phi = [ 2*exp(-2*t)-exp(-t), exp(-t)-exp(-2*t)] [ -2*exp(-t)+2*exp(-2*t), -exp(-2*t)+2*exp(-t)] can also compute numerical value with MATLAB fn >>Phi1=subs(Phi,t,1) Phi1 =

14 Ex. State transition matrix via Laplace transform Total system response to a unit step input u(t) = 1(t) [U(s) = 1/s] is applied to the system x 1 x 2 = x 1 x u Total response is: 14

15 Ex. State transition matrix via Laplace transform >>A=[-3 1;-2 0]; B=[0;1]; C=[1 0]; D=0; >>G = ss(a, B, C, D); >>ltiview(g) 15

16 Ex. State transition matrix via Laplace transform Zero input response: syms s, syms x10 x20 real xzi = Phi*[x10; x20] xzi = [ (2*exp(-2*t)-exp(-t))*x10+(exp(-t)-exp(-2*t))*x20] [ (-2*exp(-t)+2*exp(-2*t))*x10+(-exp(-2*t)+2*exp(-t))*x20] Zero state U=1/s; Response: xzs = ilaplace(inv(s*eye(2)-a)*b*u) xzs = [ 1/2-exp(-t)+1/2*exp(-2*t)] [ 3/2+1/2*exp(-2*t)-2*exp(-t)] 16

17 Outline Simulation diagram State space realization Controllable canonical form Observable canonical form Modal canonical form 17

18 Simulation diagrams Consider the problem of constructing an analog system to produce a given input-output behavior, using only integrators, gain elements and summing elements (as in an analog computer ) For example, input-output behavior defined by: Then: 18

19 Simulation diagrams Define states as outputs of integrators Now we can write the state-space equations: A state-space realization A matrix is in upper companion form 19

20 Simulation Diagram vs. Block Diagram Simulation diagram, in which: signals are in the time domain; e.g. u(t) the only system elements are integrators, gains and summers Block diagram, in which: the signals are in the Laplace domain; e.g. U(s) the system elements may be arbitrary transfer functions 20

21 Realization Realization: For a given LTI system, a state space model which is constructed from a given transfer function G s is called a realization of G. We were able to realize an ODE with no derivatives on the RHS (i.e., no zeros in the TF) with a simple chain of integrators (previous example) Consider SISO 3 rd order model y + a 2 y + a 1 y + a 0 y = b 2 u + b 1 u + b 0 u where a i, b i are arbitrary real numbers G s = Y(s) U(s) = b 2s 2 + b 1 s + b 0 s 3 + a 2 s 2 + a 1 s + a 0 21

22 Realization: Controllable canonical form How to deal with y + a 2 y + a 1 y + a 0 y = b 2 u + b 1 u + b 0 u i.e. G s = Y(s) = b 2s 2 +b 1 s+b 0 = B(s) U(s) s 3 +a 2 s 2 +a 1 s+a 0 A(s) Introduce a partial state ξ as an auxiliary variable, such that: ξ + a 2 ξ + a 1 ξ + a 0 ξ = u ; this can be realized with a chain of integrators as before Ξ(s) U(s) = 1 A(s) We can then construct the output thus: y = b 2 ξ + b 1 ξ + b 0 ξ; Y s = B s A s U(s) = B s Ξ(s) 22

23 Realization: Controllable canonical form Y(s) U(s) = b 2s 2 + b 1 s + b 0 s 3 + a 2 s 2 + a 1 s + a 0 ξ = u a 2 ξ a 1 ξ a 0 ξ b 2 b 1 y = b 2 ξ + b 1 ξ + b 0 ξ b 0 a 2 a 1 a 0 x 1 x 2 x 3 = a 2 a 1 a x 1 x 2 x u y = b 2 b 1 b 0 x 2 x 3 x 1 23

24 Realization: Observable canonical form Y(s) U(s) = b 2s 2 + b 1 s + b 0 s 3 + a 2 s 2 + a 1 s + a 0 To get another canonical form, rewrite the transfer function (s 3 + a 2 s 2 + a 1 s + a 0 )Y s = b 2 s 2 + b 1 s + b 0 U(s) Divide throughout by s 3 to obtain (1 + a 2 s + a 1 s 2 + a 0 s 3)Y s = b 0 s 3 + b 1 s 2 + b 0 s U(s) Rearrange term then, Y = b 0U a 0 Y s 3 + b 1U a 1 Y s 2 + b 2U a 2 Y s 24

25 Realization: Observable canonical form Y(s) U(s) = b 2s 2 + b 1 s + b 0 s 3 + a 2 s 2 + a 1 s + a 0 b 0 b 1 b 2 a 0 a 1 a 2 x 1 x 2 x 3 = a a a x 1 x 2 x 3 + b 2 b 1 b 0 u y = x 1 x 2 x 3 25

26 Relationships between CCF and OCF Observable canonical form Controllable canonical form 26

27 Realization: Modal canonical form If the poles of transfer function are all distinct, we can do the partial fraction expansion: Each term in the summation can be represented: 27

28 Realization: Modal canonical form Hence, parallel or diagonal realization: The modal states are decoupled Output is a linear combination of the system modes 28

29 Realization: Natural modes We saw previously that by performing a partial fraction expansion of the transfer function of a dynamic system we could obtain a parallel state-space realization for which the A matrix is diagonal so that the state-space equations are decoupled This modal canonical form is very fundamental: the 'natural modes' make independent contributions to the system output each mode has a characteristic 'natural frequency' (eigenvalue) and 'mode shape' (eigenvector) 29

30 Natural motions For 'natural' (unforced) motions (i.e., u 0): We know from before that the unforced response is a combination of simple exponential motions of the form The system is said to be moving in a natural mode when all the states have the same form of motion: The vector p (the mode shape) describes the relative amplitude of the common motion for each state 30

31 Eigenvalue problem If then But That is, We seek a solution for the eigenvalue (or mode shape) p. The condition for a non-trivial solution (i.e., p 0) is that The roots of the characteristic equation are: the "eigenvalues" λ i (i = 1, 2,..., n) of A the "natural frequencies" of the system the "poles" of the transfer function Y(s)/U(s) 31

32 Eigenvalue problem The eigenvalues are the values of λ for which the assumed motion x(t) = pe λt is possible without any external input u For each eigenvalue λ i, we can find the initial conditions x(0) = p which will excite this natural mode, as the solution of the auxiliary equation: The solution p (i) is: the "eigenvector" corresponding to the eigenvalue λ i the "mode shape" of the i-th mode Note that p (i) may be found easily as a vector proportional to any nonzero column of the adjugate of the auxiliary matrix F =[λ i I A] 32

33 Example Auxiliary matrix: Characteristic equation: Eigenvalues: 33

34 First mode: λ 1 = 2 Auxiliary equation: Solution: Note that the eigenvector, or mode shape, may be normalized in various ways set the first or last element to 1, as above set the vector "magnitude" to 1 (unit norm vector): 34

35 Rapid calculation of mode shape Auxiliary matrix: Adjoint: The columns of the adjoint of F are proportional to each other, and to the eigenvector It is thus only necessary to compute one column to find the mode shape 35

36 Motion in first natural mode The first mode may be excited by setting up initial conditions corresponding to the first mode shape: Then 36

37 Motion in first natural mode 37

38 MATLAB calculation of responses % Get controller canonical form >>[A, B, C, D] = tf2ss(1, [1 5 6]); >>G = ss(a, B, C, D); % Check >>G.a ans = % initial condition >>x0 = [-2 1]'; % compute response >>[y, t, x] = initial(g, x0); % plot response >>plot(t, x), grid >>legend('x_1', 'x_2') 38

39 Double check with transition matrix solution % Set up for symbolic computation >>syms t >>Phi = expm(a*t) % transition matrix >>Phi = [ 3*exp(-3*t)-2*exp(-2*t), -6*exp(-2*t)+6*exp(-3*t)] [ exp(-2*t)-exp(-3*t), -2*exp(-3*t)+3*exp(-2*t)] >>x0 = [-2 1]'; % initial condition = mode shape % Compute zero-input response >>X = Phi*x0 X = [ -2*exp(-2*t)] pure 1st mode [ exp(-2*t)] 39

40 Second mode: λ 2 = 3 Auxiliary equation: 40

41 Motion in second natural mode 41

42 Response to arbitrary initial conditions Example: Displace 1 unit and release from rest Construct initial condition as superposition of mode shapes: Then: Response is the sum of contributions from the independent natural modes 42

43 Response to initial displacement 43

44 Modal contributions to output y 44

45 Outline Similarity transform 45

46 Transformations between state variables We have seen that there is no unique set of state variables to realize a given input-output behaviour. That is, there are many s-s model for a given input-output model!! Given one realization, We can form another by a (nonsingular) linear transformation of variables: T = transformation matrix 46

47 Transformations between state variables Then: Hence where This is called a similarity transformation Two systems are called equivalent with each other 47

48 Transfer functions from state-space equations Until now, we have seen that there are many different state space realizations of a given transfer function The reverse process is also of interest; that is, SS TF Consider a linear time invariant state dynamics Start by taking the Laplace Transform of these equations 48

49 Transfer functions from state-space equations which gives And By definition G(s) = C(sI - A) -1 B + D is called the Transfer Function of the system. And C(sI - A) -1 x(0-) is the initial condition response. It is part of the response, but not part of the transfer function. 49

50 Transfer functions between equivalent models Let s get back to two equivalent models where Consider the two transfer functions: 50

51 Transfer functions between equivalent models The transfer function is not changed by putting the state-space model through a similarity transformation 51

52 Evaluation of transfer function matrix To evaluate transfer function from s-s model, we should calculate G(s) = C(sI - A) -1 B + D For a SISO system, G(s) may be evaluated using See Franklin et al. App C.6 For a SIMO system, G(s) may be evaluated using: The MATLAB function ss2tf uses this formulation 52

53 Example Choose Then, state-space equations are: Transfer function: s 5 6 det 1 s 0 1 s 5 det s s 2 1 5s 6 1 ( s 2)( s 3) 53

54 Transformation to modal state variables Let P [p (1) p (2) p (n) ] And modal matrix We have: assume eigenvalues are distinct Hence: That is: The modal matrix P is the similarity transformation matrix between the original state vector x and the modal state vector x d : 54

55 Ex Modal matrix: Similarity transformation: modal canonical form 55

Chap 4. State-Space Solutions 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

More information

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

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),

More information

Advanced Mechatronics Engineering

Advanced Mechatronics Engineering Advanced Mechatronics Engineering German University in Cairo 21 December, 2013 Outline Necessary conditions for optimal input Example Linear regulator problem Example Necessary conditions for optimal input

More information

Introduction to Modern Control MT 2016

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

More information

Advanced Control Theory

Advanced Control Theory State Feedback Control Design chibum@seoultech.ac.kr Outline State feedback control design Benefits of CCF 2 Conceptual steps in controller design We begin by considering the regulation problem the task

More information

Linear System Theory

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

More information

Modal Decomposition and the Time-Domain Response of Linear Systems 1

Modal Decomposition and the Time-Domain Response of Linear Systems 1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING.151 Advanced System Dynamics and Control Modal Decomposition and the Time-Domain Response of Linear Systems 1 In a previous handout

More information

State Variable Analysis of Linear Dynamical Systems

State Variable Analysis of Linear Dynamical Systems Chapter 6 State Variable Analysis of Linear Dynamical Systems 6 Preliminaries In state variable approach, a system is represented completely by a set of differential equations that govern the evolution

More information

21 Linear State-Space Representations

21 Linear State-Space Representations ME 132, Spring 25, UC Berkeley, A Packard 187 21 Linear State-Space Representations First, let s describe the most general type of dynamic system that we will consider/encounter in this class Systems may

More information

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

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

More information

Robust Control 2 Controllability, Observability & Transfer Functions

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

More information

MEM 355 Performance Enhancement of Dynamical Systems MIMO Introduction

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

More information

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 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:

More information

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 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 information

Control Systems. Laplace domain analysis

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

More information

Control Systems. Frequency domain analysis. L. Lanari

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

More information

Matrix Solutions to Linear Systems of ODEs

Matrix Solutions to Linear Systems of ODEs Matrix Solutions to Linear Systems of ODEs James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 3, 216 Outline 1 Symmetric Systems of

More information

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 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 information

Discrete and continuous dynamic systems

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

More information

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

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

More information

Control Systems Design

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

More information

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

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

More information

Autonomous system = system without inputs

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

More information

ELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky

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

More information

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 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/

More information

Identification Methods for Structural Systems

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

More information

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 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

More information

Multivariable Control. Lecture 03. Description of Linear Time Invariant Systems. John T. Wen. September 7, 2006

Multivariable Control. Lecture 03. Description of Linear Time Invariant Systems. John T. Wen. September 7, 2006 Multivariable Control Lecture 3 Description of Linear Time Invariant Systems John T. Wen September 7, 26 Outline Mathematical description of LTI Systems Ref: 3.1-3.4 of text September 7, 26Copyrighted

More information

Dr. Ian R. Manchester

Dr. Ian R. Manchester Dr Ian R. Manchester Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the s-plane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus

More information

Solution via Laplace transform and matrix exponential

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

More information

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

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

More information

CONTROL DESIGN FOR SET POINT TRACKING

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

More information

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

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 +

More information

Cayley-Hamilton Theorem

Cayley-Hamilton Theorem Cayley-Hamilton Theorem Massoud Malek In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n Let A be an n n matrix Although det (λ I n A

More information

Math Ordinary Differential Equations

Math Ordinary Differential Equations Math 411 - Ordinary Differential Equations Review Notes - 1 1 - Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x

More information

Linear dynamical systems with inputs & outputs

Linear 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 information

YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #1. July 11, 2013 Solutions

YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #1. July 11, 2013 Solutions YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 222 3. M Test # July, 23 Solutions. For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For

More information

Topic # /31 Feedback Control Systems

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

More information

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)

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

More information

Module 03 Linear Systems Theory: Necessary Background

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

More information

Problem 2 (Gaussian Elimination, Fundamental Spaces, Least Squares, Minimum Norm) Consider the following linear algebraic system of equations:

Problem 2 (Gaussian Elimination, Fundamental Spaces, Least Squares, Minimum Norm) Consider the following linear algebraic system of equations: EEE58 Exam, Fall 6 AA Rodriguez Rules: Closed notes/books, No calculators permitted, open minds GWC 35, 965-37 Problem (Dynamic Augmentation: State Space Representation) Consider a dynamical system consisting

More information

Linear ODEs. Types of systems. Linear ODEs. Definition (Linear ODE) Linear ODEs. Existence of solutions to linear IVPs.

Linear ODEs. Types of systems. Linear ODEs. Definition (Linear ODE) Linear ODEs. Existence of solutions to linear IVPs. Linear ODEs Linear ODEs Existence of solutions to linear IVPs Resolvent matrix Autonomous linear systems p. 1 Linear ODEs Types of systems Definition (Linear ODE) A linear ODE is a ifferential equation

More information

ME8281-Advanced Control Systems Design

ME8281-Advanced Control Systems Design ME8281 - Advanced Control Systems Design Spring 2016 Perry Y. Li Department of Mechanical Engineering University of Minnesota Spring 2016 Lecture 4 - Outline 1 Homework 1 to be posted by tonight 2 Transition

More information

Control Systems Design

Control Systems Design ELEC4410 Control Systems Design Lecture 14: Controllability Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 14: Controllability p.1/23 Outline

More information

Chapter 6: The Laplace Transform. Chih-Wei Liu

Chapter 6: The Laplace Transform. Chih-Wei Liu Chapter 6: The Laplace Transform Chih-Wei Liu Outline Introduction The Laplace Transform The Unilateral Laplace Transform Properties of the Unilateral Laplace Transform Inversion of the Unilateral Laplace

More information

Lecture Note 1: Background

Lecture Note 1: Background ECE5463: Introduction to Robotics Lecture Note 1: Background Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio, USA Spring 2018 Lecture 1 (ECE5463 Sp18)

More information

2.4 REALIZATION AND CANONICAL FORMS

2.4 REALIZATION AND CANONICAL FORMS Copyright F.L. Lewis 2008 All rights reserved Updated: Saturday, October 12, 2013 These notes are taken from F.L. Lewis, Applied Optimal Control and Estimation: Digital Design and Implementation, Prentice-Hall,

More information

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 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 information

EE263: Introduction to Linear Dynamical Systems Review Session 6

EE263: 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 information

Eigenvalues, Eigenvectors and the Jordan Form

Eigenvalues, Eigenvectors and the Jordan Form EE/ME 701: Advanced Linear Systems Eigenvalues, Eigenvectors and the Jordan Form Contents 1 Introduction 3 1.1 Review of basic facts about eigenvectors and eigenvalues..... 3 1.1.1 Looking at eigenvalues

More information

We use the overhead arrow to denote a column vector, i.e., a number with a direction. For example, in three-space, we write

We use the overhead arrow to denote a column vector, i.e., a number with a direction. For example, in three-space, we write 1 MATH FACTS 11 Vectors 111 Definition We use the overhead arrow to denote a column vector, ie, a number with a direction For example, in three-space, we write The elements of a vector have a graphical

More information

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

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

More information

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

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,

More information

Linear ODEs. Existence of solutions to linear IVPs. Resolvent matrix. Autonomous linear systems

Linear ODEs. Existence of solutions to linear IVPs. Resolvent matrix. Autonomous linear systems Linear ODEs p. 1 Linear ODEs Existence of solutions to linear IVPs Resolvent matrix Autonomous linear systems Linear ODEs Definition (Linear ODE) A linear ODE is a differential equation taking the form

More information

9.6: Matrix Exponential, Repeated Eigenvalues. Ex.: A = x 1 (t) = e t 2 F.M.: If we set

9.6: Matrix Exponential, Repeated Eigenvalues. Ex.: A = x 1 (t) = e t 2 F.M.: If we set 9.6: Matrix Exponential, Repeated Eigenvalues x Ax, A : n n (1) Def.: If x 1 (t),...,x n (t) is a fundamental set of solutions (F.S.S.) of (1), then X(t) x 1 (t),...,x n (t) (n n) is called a fundamental

More information

Intro. Computer Control Systems: F8

Intro. Computer Control Systems: F8 Intro. Computer Control Systems: F8 Properties of state-space descriptions and feedback Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 22 dave.zachariah@it.uu.se F7: Quiz! 2

More information

SYSTEMTEORI - ÖVNING 1. In this exercise, we will learn how to solve the following linear differential equation:

SYSTEMTEORI - ÖVNING 1. In this exercise, we will learn how to solve the following linear differential equation: SYSTEMTEORI - ÖVNING 1 GIANANTONIO BORTOLIN AND RYOZO NAGAMUNE In this exercise, we will learn how to solve the following linear differential equation: 01 ẋt Atxt, xt 0 x 0, xt R n, At R n n The equation

More information

Systems and Control Theory Lecture Notes. Laura Giarré

Systems and Control Theory Lecture Notes. Laura Giarré Systems and Control Theory Lecture Notes Laura Giarré L. Giarré 2017-2018 Lesson 7: Response of LTI systems in the transform domain Laplace Transform Transform-domain response (CT) Transfer function Zeta

More information

Math 240 Calculus III

Math 240 Calculus III Generalized Calculus III Summer 2015, Session II Thursday, July 23, 2015 Agenda 1. 2. 3. 4. Motivation Defective matrices cannot be diagonalized because they do not possess enough eigenvectors to make

More information

CDS Solutions to Final Exam

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

More information

EE221A Linear System Theory Final Exam

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,

More information

Identification Methods for Structural Systems. Prof. Dr. Eleni Chatzi Lecture March, 2016

Identification Methods for Structural Systems. Prof. Dr. Eleni Chatzi Lecture March, 2016 Prof. Dr. Eleni Chatzi Lecture 4-09. March, 2016 Fundamentals Overview Multiple DOF Systems State-space Formulation Eigenvalue Analysis The Mode Superposition Method The effect of Damping on Structural

More information

Linear Differential Equations. Problems

Linear Differential Equations. Problems Chapter 1 Linear Differential Equations. Problems 1.1 Introduction 1.1.1 Show that the function ϕ : R R, given by the expression ϕ(t) = 2e 3t for all t R, is a solution of the Initial Value Problem x =

More information

Fall 線性系統 Linear Systems. Chapter 08 State Feedback & State Estimators (SISO) Feng-Li Lian. NTU-EE Sep07 Jan08

Fall 線性系統 Linear Systems. Chapter 08 State Feedback & State Estimators (SISO) Feng-Li Lian. NTU-EE Sep07 Jan08 Fall 2007 線性系統 Linear Systems Chapter 08 State Feedback & State Estimators (SISO) Feng-Li Lian NTU-EE Sep07 Jan08 Materials used in these lecture notes are adopted from Linear System Theory & Design, 3rd.

More information

Math 331 Homework Assignment Chapter 7 Page 1 of 9

Math 331 Homework Assignment Chapter 7 Page 1 of 9 Math Homework Assignment Chapter 7 Page of 9 Instructions: Please make sure to demonstrate every step in your calculations. Return your answers including this homework sheet back to the instructor as a

More information

Linear Algebra Practice Problems

Linear Algebra Practice Problems Linear Algebra Practice Problems Math 24 Calculus III Summer 25, Session II. Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated,

More information

6 Linear Equation. 6.1 Equation with constant coefficients

6 Linear Equation. 6.1 Equation with constant coefficients 6 Linear Equation 6.1 Equation with constant coefficients Consider the equation ẋ = Ax, x R n. This equating has n independent solutions. If the eigenvalues are distinct then the solutions are c k e λ

More information

Dynamical system. The set of functions (signals) w : T W from T to W is denoted by W T. W variable space. T R time axis. W T trajectory space

Dynamical system. The set of functions (signals) w : T W from T to W is denoted by W T. W variable space. T R time axis. W T trajectory space Dynamical system The set of functions (signals) w : T W from T to W is denoted by W T. W variable space T R time axis W T trajectory space A dynamical system B W T is a set of trajectories (a behaviour).

More information

Lecture 2 and 3: Controllability of DT-LTI systems

Lecture 2 and 3: Controllability of DT-LTI systems 1 Lecture 2 and 3: Controllability of DT-LTI systems Spring 2013 - EE 194, Advanced Control (Prof Khan) January 23 (Wed) and 28 (Mon), 2013 I LTI SYSTEMS Recall that continuous-time LTI systems can be

More information

EE Control Systems LECTURE 9

EE Control Systems LECTURE 9 Updated: Sunday, February, 999 EE - Control Systems LECTURE 9 Copyright FL Lewis 998 All rights reserved STABILITY OF LINEAR SYSTEMS We discuss the stability of input/output systems and of state-space

More information

Lecture Notes for Math 524

Lecture Notes for Math 524 Lecture Notes for Math 524 Dr Michael Y Li October 19, 2009 These notes are based on the lecture notes of Professor James S Muldowney, the books of Hale, Copple, Coddington and Levinson, and Perko They

More information

Remark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.

Remark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero. Sec 6 Eigenvalues and Eigenvectors Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called an eigenvalue of A if there is a nontrivial

More information

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 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

More information

( ) ( = ) = ( ) ( ) ( )

( ) ( = ) = ( ) ( ) ( ) ( ) Vρ C st s T t 0 wc Ti s T s Q s (8) K T ( s) Q ( s) + Ti ( s) (0) τs+ τs+ V ρ K and τ wc w T (s)g (s)q (s) + G (s)t(s) i G and G are transfer functions and independent of the inputs, Q and T i. Note

More information

Module 07 Controllability and Controller Design of Dynamical LTI Systems

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

More information

Motivation. From SS to TF (review) Realization: From TF to SS. MECH468 Modern Control Engineering MECH550P Foundations in Control Engineering

Motivation. From SS to TF (review) Realization: From TF to SS. MECH468 Modern Control Engineering MECH550P Foundations in Control Engineering MECH468 Modern Control Engineering MECH550P Foundations in Control Engineering Realization Dr. Ryozo Nagamune Department of Mechanical Engineering University of British Columbia 2008/09 MECH468/550P 1

More information

Solution of Linear State-space Systems

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

More information

LS.5 Theory of Linear Systems

LS.5 Theory of Linear Systems LS.5 Theory of Linear Systems 1. General linear ODE systems and independent solutions. We have studied the homogeneous system of ODE s with constant coefficients, (1) x = Ax, where A is an n n matrix of

More information

Theory of Linear Systems Exercises. Luigi Palopoli and Daniele Fontanelli

Theory of Linear Systems Exercises. Luigi Palopoli and Daniele Fontanelli Theory of Linear Systems Exercises Luigi Palopoli and Daniele Fontanelli Dipartimento di Ingegneria e Scienza dell Informazione Università di Trento Contents Chapter. Exercises on the Laplace Transform

More information

Modeling and Analysis of Dynamic Systems

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

More information

Pseudospectra and Nonnormal Dynamical Systems

Pseudospectra and Nonnormal Dynamical Systems Pseudospectra and Nonnormal Dynamical Systems Mark Embree and Russell Carden Computational and Applied Mathematics Rice University Houston, Texas ELGERSBURG MARCH 1 Overview of the Course These lectures

More information

MODERN CONTROL DESIGN

MODERN CONTROL DESIGN CHAPTER 8 MODERN CONTROL DESIGN The classical design techniques of Chapters 6 and 7 are based on the root-locus and frequency response that utilize only the plant output for feedback with a dynamic controller

More information

Transfer function and linearization

Transfer function and linearization Transfer function and linearization Daniele Carnevale Dipartimento di Ing. Civile ed Ing. Informatica (DICII), University of Rome Tor Vergata Corso di Controlli Automatici, A.A. 24-25 Testo del corso:

More information

EEE582 Homework Problems

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

More information

Lecture 1: Pragmatic Introduction to Stochastic Differential Equations

Lecture 1: Pragmatic Introduction to Stochastic Differential Equations Lecture 1: Pragmatic Introduction to Stochastic Differential Equations Simo Särkkä Aalto University, Finland (visiting at Oxford University, UK) November 13, 2013 Simo Särkkä (Aalto) Lecture 1: Pragmatic

More information

MAE143A Signals & Systems - Homework 5, Winter 2013 due by the end of class Tuesday February 12, 2013.

MAE143A Signals & Systems - Homework 5, Winter 2013 due by the end of class Tuesday February 12, 2013. MAE43A Signals & Systems - Homework 5, Winter 23 due by the end of class Tuesday February 2, 23. If left under my door, then straight to the recycling bin with it. This week s homework will be a refresher

More information

Control Systems. Design of State Feedback Control.

Control Systems. Design of State Feedback Control. Control Systems Design of State Feedback Control chibum@seoultech.ac.kr Outline Design of State feedback control Dominant pole design Symmetric root locus (linear quadratic regulation) 2 Selection of closed-loop

More information

LTI Systems (Continuous & Discrete) - Basics

LTI 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 information

Lecture Notes of EE 714

Lecture Notes of EE 714 Lecture Notes of EE 714 Lecture 1 Motivation Systems theory that we have studied so far deals with the notion of specified input and output spaces. But there are systems which do not have a clear demarcation

More information

Laplace Transforms Chapter 3

Laplace Transforms Chapter 3 Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first. Laplace transforms play a key role in important

More information

EECS C128/ ME C134 Final Wed. Dec. 14, am. Closed book. One page, 2 sides of formula sheets. No calculators.

EECS C128/ ME C134 Final Wed. Dec. 14, am. Closed book. One page, 2 sides of formula sheets. No calculators. Name: SID: EECS C128/ ME C134 Final Wed. Dec. 14, 211 81-11 am Closed book. One page, 2 sides of formula sheets. No calculators. There are 8 problems worth 1 points total. Problem Points Score 1 16 2 12

More information

ECEEN 5448 Fall 2011 Homework #5 Solutions

ECEEN 5448 Fall 2011 Homework #5 Solutions ECEEN 5448 Fall 211 Homework #5 Solutions Professor David G. Meyer December 8, 211 1. Consider the 1-dimensional time-varying linear system ẋ t (u x) (a) Find the state-transition matrix, Φ(t, τ). Here

More information

Applied Differential Equation. November 30, 2012

Applied Differential Equation. November 30, 2012 Applied Differential Equation November 3, Contents 5 System of First Order Linear Equations 5 Introduction and Review of matrices 5 Systems of Linear Algebraic Equations, Linear Independence, Eigenvalues,

More information

ECE557 Systems Control

ECE557 Systems Control ECE557 Systems Control Bruce Francis Course notes, Version.0, September 008 Preface This is the second Engineering Science course on control. It assumes ECE56 as a prerequisite. If you didn t take ECE56,

More information

Linear Systems Theory

Linear Systems Theory ME 3253 Linear Systems Theory Review Class Overview and Introduction 1. How to build dynamic system model for physical system? 2. How to analyze the dynamic system? -- Time domain -- Frequency domain (Laplace

More information

Differential equations

Differential equations Differential equations Math 7 Spring Practice problems for April Exam Problem Use the method of elimination to find the x-component of the general solution of x y = 6x 9x + y = x 6y 9y Soln: The system

More information

Control Systems. State Estimation.

Control Systems. State Estimation. State Estimation chibum@seoultech.ac.kr Outline Dominant pole design Symmetric root locus State estimation We are able to place the CLPs arbitrarily by feeding back all the states: u = Kx. But these may

More information

MULTIVARIABLE ZEROS OF STATE-SPACE SYSTEMS

MULTIVARIABLE ZEROS OF STATE-SPACE SYSTEMS Copyright F.L. Lewis All rights reserved Updated: Monday, September 9, 8 MULIVARIABLE ZEROS OF SAE-SPACE SYSEMS If a system has more than one input or output, it is called multi-input/multi-output (MIMO)

More information

Eigenvalues, Eigenvectors. Eigenvalues and eigenvector will be fundamentally related to the nature of the solutions of state space systems.

Eigenvalues, Eigenvectors. Eigenvalues and eigenvector will be fundamentally related to the nature of the solutions of state space systems. Chapter 3 Linear Algebra In this Chapter we provide a review of some basic concepts from Linear Algebra which will be required in order to compute solutions of LTI systems in state space form, discuss

More information

Multi Degrees of Freedom Systems

Multi Degrees of Freedom Systems Multi Degrees of Freedom Systems MDOF s http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano March 9, 07 Outline, a System

More information