Robust Control 2 Controllability, Observability & Transfer Functions


 Harvey Robertson
 2 years ago
 Views:
Transcription
1 Robust Control 2 Controllability, Observability & Transfer Functions Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University /26/24
2 Outline Reachable Controllability Distinguishable Observability Zeros Transfer Functions Poles Interconnections Stability
3 System x = Ax+ Bu y= Cx x R, u R, y R n m p x(t; x,u) e x e = + At t A(ts) Bu(s)ds
4 Reachability Definition: A state x R is reachable from x if there n exists a finite time t > and a piecewise continuous control ( ) u such that x(t;x,u) = x. R x denotes the set of states reachable from x.
5 Reachability: Properties If x is reachable from x in some time t >, it is reachable in every time t. To see this simply rescale s: t t A(ts) A(t st  t ) = st t st e Bu(s)ds e Bu( )d = t ( t ) ( t ) A(t st t ) A(tt) st st t e Be u( )d A(t t) Thus, we have the replacement us ( ) e u( ). Notice that x is reachable from x if and only if x e x At is reachable from the origin for any < t < t t At A(ts) At A(ts) = + e x = e Bu(s)ds x e x e Bu(s)ds x st t
6 Some Geometry, Consider two linear vector spaces X, X with inner products 2 2 { y X y = Ax, x X } 2 2, and,,respectively, and a mapping A: X X  the range or image ( Im ) of A is the set of points in X  the null space or kernel ( ker ) of A is the set of points in X { x X Ax= } *  the adjoint mapping A : X2 X is defined by yax *, = Ayx, 2 2
7 Some Geometry, 2 * If then is called selfadjoint A= A A ( A) * * * A = A A = A It is always true that and * * If then is called normal AA= AA A ( ) The following decompositions of finite dimensional linear vector spaces are true: X 2 = ker Im ( * A ) ( * ) Im ( ) X = ker A A These are orthogonal decompositions, i.e., ( ) ( * ) x ker A, y Im A x, y =
8 Some Geometry, 3 x 3 y = Ax, A= ker A y 2 x 2 y x Im A T T T x = A y, A =
9 Reachability Condition Let U denote the linear vector space of control functions u( τ ), τ [, t ], ( ) ( ) n and X R the space of states x t. The map A : U X is defined by At ( τ ) xt ( ) = e Bu( τ) dτ x * * R x ImA x ImA note : Proposition: t [ t ] the time interval, is T τ T At ( ) A ( t τ) e BB e d A AA X X The set of states reachable from the origin over * ImAA = Im t τ
10 Controllability Definition: The system or the matrix pair ( AB, ) is said to be (completely) controllable if any state x is reachable from any other state x in finite time. Proposition: only if where The system is completely controllable if and rank G ( t ) = n C T At ( τ) A ( t ) G ( t ) e BB T τ e d C = is the controllability Grammian. t τ
11 Controllability Main Result B : = Im( B) B : = B+ B+ + B= Im n n A A A BAB A B Theorem: R ( ) = A B If the system is completely controllable there is a unique control T T A ( t t () ) ( ) ut = Be G t x C that steers the origin to x in precisely time t.
12 Distinguishable Definition: A state x R is indistinguishable from x n if for every finite time t and piecewise continuous control ( ) ut ( ), ytx ( ;, u) = ytx ( ;, u). I x denotes the set of states 2 indistinguishable from x. C n N : ( i CA = ker CA ) = ker i= CA I() = N Theorem: n
13 Observability Definition: The system or the matrix pair ( C, A) is said to be (completely) observable if knowledge of ut ( ) and yt ( ) on a finite time interval determines the state trajectory on that interval. Theorem: The system or the matrix pair ( C, A) is (completely) observable if and only if I () C CA rank = n n CA =, i.e.
14 Summary: Controllability/Observability Controllability rank = C CA Observability rank = n CA Kalman Decomposition, x z such that d dt n B AB A B n z A A2 A3 A4 z B z z 2 A22 A 24 z 2 B 2 z = 2 + u, y = [ C 2 C ] 4 z 3 A33 A 34 z 3 z 3 z4 A44 z4 z 4 z, z controllable z, z observable Notice that the substate z 2 is both controllable and observable
15 Example 2 2 x = x u, y [ ] x 4 + = 4 C = [ B AB] = 4 s s + 2 G s = C si A B= + + ( ) [ ] [ ] ( s 2 )( s 4 ) ( s + 2) = = ( s+ 2)( s+ 4) s+ 4
16 Some MATLAB Functions Function canon ctrb ctrbf gram obsv obsvf ss2ss ssbal minreal canonical ststespace realizations controllability matrix controllability staircase form controllability and observability gramians observability matrix observability staircase form state coordinate transformation diagonal balancing of statespace realizations returns a minimal realization
17 Example, Continued >> A=[22;,4]; >> B=[;]; >> C=[ ]; >> sys=ss(a,b,c,); >> ctrb(sys) ans = 44 >> gram(sys,'c') ans = >> tf(sys) Transfer function: s s^2 + 6 s + 8 >> tf(minreal(sys)) state removed. Transfer function: s + 4
18 System Poles & Zeros Two descriptions of linear timeinvariant systems state space and transfer function. x = Ax+ Bu y = Cx+ Du ( ) [ ] G s = C si A B+ D Assumption: G is a complete characterization of A, B, C, D or, equivalently, ABCD,,, is a minimal realization of G. Defining poles via state space is very easy: the poles of G are the eigenvalues of A. Defining zeros is more complicated. We do it via state space in the followin g.
19 SISO System Zeros recall: x = Ax+ Bu n x R, u R, y R y = Cx+ Du { } [ ] [ ] Y() s = C si A x + C si A B+ D U() s suppose: λt () { [ ] } ns () u t = e, G( s): = C si A B + D = k, d ( s) = si A ds () [ ] [ ] = + k = + + ds () ds () s λ ds () ds () s λ C si A x ns () C si A x ns () G( λ) (), Y s
20 SISO System Zeros, Cont d x can always be chosen so that ( ) [ ] C si A x ns () + ds () d() s in which case ( λ ) G Y() s =, if λ is a zero of G( λ), then Y ( s) =, y( t) = s λ In summary: if λ is a system zero, there exists x such that x t = x λt and ut ( ) = e y t ()
21 MIMO System Zeros x = Ax+ Bu y = Cx+ Du x R, u R, y R n m p m n Does there exist g R and x R such that λt λt ut () = ge xt () = xe and yt ()? The assumed solution must satisfy λt λt λt λxe = Axe + Bge λi A B x = Cx e Dge C D g λt λt = +
22 MIMO System Zeros, Cont d This represents n+ p equations in n+ m unknowns. Suppose λi A B r = rank C D r = n+ min( m, p) max r< n+ min( m, p) nontrivial sol'ns p< m always nontrivial sol'ns r= r = n+ p m p max max independent sol'ns p> m and r= r = n+ m there are no nontrivial sol'ns
23 Square MIMO Systems (p=m) Nondegenerate case: If for typical λ, λi A B r = rank n m C D = + Those specific values of λ for which r < n+ m are called invariant zeros. Invariant zeros consist of input decoupling zeros (uncontrollable modes), λ satisfies rank [ λ ] I A B < n output decoupling zeros (unobservable modes), λ satisfies λi A rank n C < transmission zeros, all other invariant zeros.
24 Square MIMO Systems, Cont d degenerate case: ( λ ) For typical λ, λi A B r = rank < n+ m C D G insufficient independent controls,rank B< m insufficient independent outputs, rank C < p
25 Transfer Functions x = Ax+ Bu y = Cx ( ) = [ ] = [ ] x R, u R, y R n m p G s C si A B C si A B where A, B, C are those of the Kalman decomposition , i.e., parameters of a minimal realization. So, only the controllable and observable part of the system is characterized by its transfer function. ( ) Definition: G s is called a complete characterization of the system if the system is completely observable and controllabl e.
26 Poles & Zeros from Transfer Functions /26/24
27 Numbers: Prime & Coprime A prime number (or integer) is a positive integer p > that has no positive integer divisors other than and itself. Two integers are relatively prime or coprime if they share no positive integer factors (divisors) other than  i.e., their greatest common divisor is. Bezout's identity: exist integers x and y such that (, ) GCD a b = ax + by If a and b are integers not both zero, then there If a and b are coprime then there exist integers x and y such that = ax + by
28 Polynomials These ideas have been extended to polynomials, matrices with polynomial elements matrices with rational elements Two polynomials ( ) ( ) n s = a s + a s + + a, a m m m m m d s = b s + b s + + b, b n n n n are coprime if their greatest common divisor is a nonzero constant, i.e., they have no common factors.
29 Example F = s 4 + 2s 3 + s + 2 = (s + ) (s + 2) (s 2  s + ) F 2 = s 5 + s 4 + 2s 3 + 3s 2 + 3s + 2 = (s + ) (s 2  s + 2) (s 2 + s + ) gcd(f,f 2 ) = (s + ) Bezout relation: (5/24s 3 + /2s 2 + /4s + 5/24) F + (5/24s 27/24s + 7/24) F 2 = s +
30 Polynomial Matrices Two matrix polynomials m m ( ) = n i ( ) = n + n + + right common divisor R( s) R ( ) = ( ) ( ), ( ) = ( ) ( ) ( ) D( s) right coprime R( s) N s A s A s A m m pa, q pb, q A R, Bi R D s B s B s B qq, have a if N s N s R s D s D s R s N s, are if the only right common divisors are unimodular, i.e., det = c. Similary, left coprimeness can be defined for polynomial matrices with the same number of rows.
31 Poles ( ) Suppose the q m transfer matrix G s is a complete characterization of x = Ax+ Bu, y = Cx+ Du ( ) can always be factored into G( s) = D ( s) N ( s) = N ( s) D ( s) G s l l r r where D, N and D, N are coprime pairs of polynomial matrices. N l l r r and D, D are called numerator, denominator matrices, respectively. Theorem: [ si A] = D s = D ( s) det α det ( ) α det α, α are constants. Definition: Poles are the roots of: det r r l 2 r 2 [ si A], or det D ( s), or det D ( s) = = = l r r, N l
32 Poles & Zeros from Transfer Functions Assume G(s) is a complete characterization. Theorem: The pole polynomial (s) is the least common denominator of all nonidenticallyzero minors of all orders of G(s). Theorem: The zero polynomial is the greatest common divisor of all numerators of all orderr minors of G(s), where r is the generic rank of G(s), provided that these minors have been adjusted to have (s) as there denominator.
33 Example Recall a minor of a matrix is the determinant of a matrix obtained by deleting rows and columns. Consider the transfer function: s 4 G( s) = ( s s + ) To determine poles we need all minors of all orders. The 4 minors of order are s s,,,2 s+ 2 s+ 2 s+ 2 s+ 2 ( ) rank G s =2. ( ) The single minor of order 2 is det G s = 2 ( s) ( ) pole polynomial: φ = s+ 2 zero polynomial: z s = s 4 s 4 ( s + 2)
34 Multivariable Interconnections & Feedback Loops /26/24
35 WellPosed Loops: Example G( s) s G( s) = s 2 s+ s+ s+ s Gcl ( s) = G( s) I + G( s) = s Theorem: Let GH, be proper rational transfer matrices. Then cl [ ] G = G I + HG ( ) ( ) is proper and rational iff I + H G is nonsingular.
36 Poles of Closed Loops G( s) H ( s) cl [ ] G = G I + HG It might be anticipated that the poles of G are the roots of det [ I + HG] s s s s+ G( s) =, H( s) = I2 2 s + 2s s s+ s+ Gcl ( s) =, det [ I + HG] s 2 s obviously, has poles at s =±.. Not True!! cl
37 Poles, Cont d Theore m: ( ) If GH, are proper, reational matrices and ( ) ( ) det I + H G, then the poles of Gcl are the roots of the polynomial G s ( s) ( s) ( s) det I H( s) G( s) = G H + Example: + = s +.5 ( s.5)( s.5) ( s. 5) ( s.5)( s+.5 ) ( s.5), H = I 2
38 Poles: Example Cont d s +.5 det I + G( s) = s.5 = + = = + + G G cl ( s) ( s.5)( s.5 ), ( s) ( s) ( s.5)( s.5) ( s) ( s.5) = ( s+.5) ( s+.5) s +.5 = ( s+.5)( s+.5) s+.5 s+.5 H
39 System Interconnections G Systems G, G are complete characterizations, with 2 G = D N = N D, i =, 2 coprime fractions. i li li ri ri parallel connection G G 2 G 2 controllable D, D left coprime r r2 observable D, D right coprime series connection l l2 G controllable D, N, or D D, N or D, N N are left coprime r2 r l r2 l l2 l2 r G 2 observable D, N, or D D, N or D, N N right coprime ( ) ( ) det I + G2 G. Then controllable GG controllable l r2 l r2 r2 r l2 r 2 observable G 2 G observable
40 Stability x = Ax+ Bu y = Cx+ Du ( ) [ ] x R, u R, y R G s = C si A B+ D n m p The basic idea is that stable system responds to a perturbation by remaining within a neighborhood of its equilibrium point. a state perturbation with zero input (Lyapunov/Asymptotic) an input perturbation with zero state (BIBO) simultaneous state and input perturbation (Total) ( A) Lyapunov: Reλ, eigenvalues with Reλ = have full set of eigenvectors. BIBO: ( A) ( ) Asymptotic: Reλ < poles of G s < ( ) Total: Lyapunov + poles of G s <
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 informationControllability, Observability, Full State Feedback, Observer Based Control
Multivariable Control Lecture 4 Controllability, Observability, Full State Feedback, Observer Based Control John T. Wen September 13, 24 Ref: 3.23.4 of Text Controllability ẋ = Ax + Bu; x() = x. At time
More informationJoão 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.
More informationFull 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
More informationMULTIVARIABLE ZEROS OF STATESPACE SYSTEMS
Copyright F.L. Lewis All rights reserved Updated: Monday, September 9, 8 MULIVARIABLE ZEROS OF SAESPACE SYSEMS If a system has more than one input or output, it is called multiinput/multioutput (MIMO)
More informationMultivariable Control. Lecture 05. Multivariable Poles and Zeros. John T. Wen. September 14, 2006
Multivariable Control Lecture 05 Multivariable Poles and Zeros John T. Wen September 4, 2006 SISO poles/zeros SISO transfer function: G(s) = n(s) d(s) (no common factors between n(s) and d(s)). Poles:
More informationControl 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
More informationMultivariable 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: 3.13.4 of text September 7, 26Copyrighted
More informationModule 03 Linear Systems Theory: Necessary Background
Module 03 Linear Systems Theory: Necessary Background Ahmad F. Taha EE 5243: Introduction to CyberPhysical Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha/index.html September
More informationControl 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 informationMASSACHUSETTS 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: [
More informationControl 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
More informationZeros 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)
More information16.31 Fall 2005 Lecture Presentation Mon 31Oct05 ver 1.1
16.31 Fall 2005 Lecture Presentation Mon 31Oct05 ver 1.1 Charles P. Coleman October 31, 2005 1 / 40 : Controllability Tests Observability Tests LEARNING OUTCOMES: Perform controllability tests Perform
More informationContents. 1 StateSpace 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 StateSpace Linear Systems 5 1.1 StateSpace Linear Systems 5 1.2 Block Diagrams 7 1.3 Exercises 11 2 Linearization
More informationModule 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 informationKalman 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
More informationLinear 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 informationControl 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 informationGrammians. Matthew M. Peet. Lecture 20: Grammians. Illinois Institute of Technology
Grammians Matthew M. Peet Illinois Institute of Technology Lecture 2: Grammians Lyapunov Equations Proposition 1. Suppose A is Hurwitz and Q is a square matrix. Then X = e AT s Qe As ds is the unique solution
More informationSolution 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
More informationCONTROL DESIGN FOR SET POINT TRACKING
Chapter 5 CONTROL DESIGN FOR SET POINT TRACKING In this chapter, we extend the pole placement, observerbased output feedback design to solve tracking problems. By tracking we mean that the output is commanded
More informationEL2520 Control Theory and Practice
So far EL2520 Control Theory and Practice r Fr wu u G w z n Lecture 5: Multivariable systems Fy Mikael Johansson School of Electrical Engineering KTH, Stockholm, Sweden SISO control revisited: Signal
More informationECE 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:
More informationSMITH MCMILLAN FORMS
Appendix B SMITH MCMILLAN FORMS B. Introduction Smith McMillan forms correspond to the underlying structures of natural MIMO transferfunction matrices. The key ideas are summarized below. B.2 Polynomial
More informationControl 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 informationModule 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
More informationEE 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 statespace
More informationEEE582 Homework Problems
EEE582 Homework Problems HW. Write a statespace 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 informationLetting be a field, e.g., of the real numbers, the complex numbers, the rational numbers, the rational functions W(s) of a complex variable s, etc.
1 Polynomial Matrices 1.1 Polynomials Letting be a field, e.g., of the real numbers, the complex numbers, the rational numbers, the rational functions W(s) of a complex variable s, etc., n ws ( ) as a
More informationAdvanced Control Theory
State Space Solution and Realization chibum@seoultech.ac.kr Outline State space solution 2 Solution of statespace equations x t = Ax t + Bu t First, recall results for scalar equation: x t = a x t + b
More informationẋ 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, 9653712 The following represents a detailed topical outline of the course. It attempts to highlight most of the key concepts to be covered and
More informationControl Systems I. Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback. Readings: Emilio Frazzoli
Control Systems I Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback Readings: Emilio Frazzoli Institute for Dynamic Systems and Control DMAVT ETH Zürich October 13, 2017 E. Frazzoli (ETH)
More informationStability, Pole Placement, Observers and Stabilization
Stability, Pole Placement, Observers and Stabilization 1 1, The Netherlands DISC Course Mathematical Models of Systems Outline 1 Stability of autonomous systems 2 The pole placement problem 3 Stabilization
More informationRaktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Dynamic Response
.. AERO 422: Active Controls for Aerospace Vehicles Dynamic Response Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. . Previous Class...........
More informationLecture 7 (Weeks 1314)
Lecture 7 (Weeks 1314) Introduction to Multivariable Control (SP  Chapters 3 & 4) Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 7 (Weeks 1314) p.
More informationModel reduction for linear systems by balancing
Model reduction for linear systems by balancing Bart Besselink Jan C. Willems Center for Systems and Control Johann Bernoulli Institute for Mathematics and Computer Science University of Groningen, Groningen,
More informationTheorem 1. ẋ = Ax is globally exponentially stable (GES) iff A is Hurwitz (i.e., max(re(σ(a))) < 0).
Linear Systems Notes Lecture Proposition. A M n (R) is positive definite iff all nested minors are greater than or equal to zero. n Proof. ( ): Positive definite iff λ i >. Let det(a) = λj and H = {x D
More information7 Minimal realization and coprime fraction
7 Minimal realization and coprime fraction 7 Introduction If a tranfer function i realizable, what i the mallet poible dimenion? Realization with the mallet poible dimenion are called minimaldimenional
More informationFEL3210 Multivariable Feedback Control
FEL3210 Multivariable Feedback Control Lecture 8: Youla parametrization, LMIs, Model Reduction and Summary [Ch. 1112] Elling W. Jacobsen, Automatic Control Lab, KTH Lecture 8: Youla, LMIs, Model Reduction
More informationFall 線性系統 Linear Systems. Chapter 08 State Feedback & State Estimators (SISO) FengLi Lian. NTUEE Sep07 Jan08
Fall 2007 線性系統 Linear Systems Chapter 08 State Feedback & State Estimators (SISO) FengLi Lian NTUEE Sep07 Jan08 Materials used in these lecture notes are adopted from Linear System Theory & Design, 3rd.
More informationChapter 6 Controllability and Obervability
Chapter 6 Controllability and Obervability Controllability: whether or not the statespace equation can be controlled from input. Observability: whether or not the initial state can be observed from output.
More informationZero controllability in discretetime structured systems
1 Zero controllability in discretetime structured systems Jacob van der Woude arxiv:173.8394v1 [math.oc] 24 Mar 217 Abstract In this paper we consider complex dynamical networks modeled by means of state
More informationTheory of Robust Control
Theory of Robust Control Carsten Scherer Mathematical Systems Theory Department of Mathematics University of Stuttgart Germany Contents 1 Introduction to Basic Concepts 6 1.1 Systems and Signals..............................
More informationState 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 informationECEN 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 informationChap 3. Linear Algebra
Chap 3. Linear Algebra Outlines 1. Introduction 2. Basis, Representation, and Orthonormalization 3. Linear Algebraic Equations 4. Similarity Transformation 5. Diagonal Form and Jordan Form 6. Functions
More informationDynamical Systems & Lyapunov Stability
Dynamical Systems & Lyapunov Stability Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline Ordinary Differential Equations Existence & uniqueness Continuous dependence
More informationStationary trajectories, singular Hamiltonian systems and illposed Interconnection
Stationary trajectories, singular Hamiltonian systems and illposed Interconnection S.C. Jugade, Debasattam Pal, Rachel K. Kalaimani and Madhu N. Belur Department of Electrical Engineering Indian Institute
More informationSYSTEMTEORI  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
More informationME 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
More information1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det
What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix
More informationObservability. 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,
More informationRaktim Bhattacharya. . AERO 632: Design of Advance Flight Control System. Norms for Signals and Systems
. AERO 632: Design of Advance Flight Control System Norms for Signals and. Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. Norms for Signals ...
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More informationRepresent 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 informationCDS 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 timedelay invariant. Hence: ( ) Ĝ(s) = s + a = sup /2
More informationLinear System Fundamentals
Linear Sytem Fundamental MEM 355 Performance Enhancement of Dynamical Sytem Harry G. Kwatny Department of Mechanical Engineering & Mechanic Drexel Univerity Content Sytem Repreentation Stability Concept
More informationLecture plan: Control Systems II, IDSC, 2017
Control Systems II MAVT, IDSC, Lecture 8 28/04/2017 G. Ducard Lecture plan: Control Systems II, IDSC, 2017 SISO Control Design 24.02 Lecture 1 Recalls, Introductory case study 03.03 Lecture 2 Cascaded
More informationEE 380. Linear Control Systems. Lecture 10
EE 380 Linear Control Systems Lecture 10 Professor Jeffrey Schiano Department of Electrical Engineering Lecture 10. 1 Lecture 10 Topics Stability Definitions Methods for Determining Stability Lecture 10.
More informationLecture 3. Chapter 4: Elements of Linear System Theory. Eugenio Schuster. Mechanical Engineering and Mechanics Lehigh University.
Lecture 3 Chapter 4: Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 3 p. 1/77 3.1 System Descriptions [4.1] Let f(u) be a liner operator, u 1 and u
More informationIntro. Computer Control Systems: F8
Intro. Computer Control Systems: F8 Properties of statespace descriptions and feedback Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 22 dave.zachariah@it.uu.se F7: Quiz! 2
More informationUniversity of Toronto Department of Electrical and Computer Engineering ECE410F Control Systems Problem Set #3 Solutions = Q o = CA.
University of Toronto Department of Electrical and Computer Engineering ECE41F Control Systems Problem Set #3 Solutions 1. The observability matrix is Q o C CA 5 6 3 34. Since det(q o ), the matrix is
More information1. 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 informationLinear State Feedback Controller Design
Assignment For EE5101  Linear Systems Sem I AY2010/2011 Linear State Feedback Controller Design Phang Swee King A0033585A Email: king@nus.edu.sg NGS/ECE Dept. Faculty of Engineering National University
More informationECEEN 5448 Fall 2011 Homework #4 Solutions
ECEEN 5448 Fall 2 Homework #4 Solutions Professor David G. Meyer Novemeber 29, 2. The statespace realization is A = [ [ ; b = ; c = [ which describes, of course, a free mass (in normalized units) with
More informationChap. 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 :
More informationOn some interpolation problems
On some interpolation problems A. Gombani Gy. Michaletzky LADSEBCNR Eötvös Loránd University Corso Stati Uniti 4 H1111 Pázmány Péter sétány 1/C, 35127 Padova, Italy Computer and Automation Institute
More informationOutline. Control systems. Lecture4 Stability. V. Sankaranarayanan. V. Sankaranarayanan Control system
Outline Control systems Lecture4 Stability V. Sankaranarayanan Outline Outline 1 Outline Outline 1 2 Concept of Stability Zero State Response: The zerostate response is due to the input only; all the
More informationModule 09 From sdomain to timedomain From ODEs, TFs to StateSpace Modern Control
Module 09 From sdomain to timedomain From ODEs, TFs to StateSpace Modern Control Ahmad F. Taha EE 3413: Analysis and Desgin of Control Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/
More information1 Continuoustime Systems
Observability Completely controllable systems can be restructured by means of state feedback to have many desirable properties. But what if the state is not available for feedback? What if only the output
More informationTopic # Feedback Control
Topic #11 16.31 Feedback Control StateSpace Systems Statespace model features Observability Controllability Minimal Realizations Copyright 21 by Jonathan How. 1 Fall 21 16.31 11 1 StateSpace Model Features
More informationEmpirical Gramians and Balanced Truncation for Model Reduction of Nonlinear Systems
Empirical Gramians and Balanced Truncation for Model Reduction of Nonlinear Systems Antoni Ras Departament de Matemàtica Aplicada 4 Universitat Politècnica de Catalunya Lecture goals To review the basic
More informationState Space Control D R. T A R E K A. T U T U N J I
State Space Control D R. T A R E K A. T U T U N J I A D V A N C E D C O N T R O L S Y S T E M S M E C H A T R O N I C S E N G I N E E R I N G D E P A R T M E N T P H I L A D E L P H I A U N I V E R S I
More informationDISTANCE BETWEEN BEHAVIORS AND RATIONAL REPRESENTATIONS
DISTANCE BETWEEN BEHAVIORS AND RATIONAL REPRESENTATIONS H.L. TRENTELMAN AND S.V. GOTTIMUKKALA Abstract. In this paper we study notions of distance between behaviors of linear differential systems. We introduce
More informationControl Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control DMAVT ETH Zürich
Control Systems I Lecture 6: Poles and Zeros Readings: Emilio Frazzoli Institute for Dynamic Systems and Control DMAVT ETH Zürich October 27, 2017 E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/2017
More informationBalanced 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
More informationPOLE PLACEMENT. Sadegh Bolouki. Lecture slides for ECE 515. University of Illinois, UrbanaChampaign. Fall S. Bolouki (UIUC) 1 / 19
POLE PLACEMENT Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, UrbanaChampaign Fall 2016 S. Bolouki (UIUC) 1 / 19 Outline 1 State Feedback 2 Observer 3 Observer Feedback 4 Reduced Order
More informationECEN 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,
More informationOPTIMAL CONTROL SYSTEMS
SYSTEMS MINMAX Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University OUTLINE MINMAX CONTROL Problem Definition HJB Equation Example GAME THEORY Differential Games Isaacs
More informationThe Behavioral Approach to Systems Theory
The Behavioral Approach to Systems Theory Paolo Rapisarda, Un. of Southampton, U.K. & Jan C. Willems, K.U. Leuven, Belgium MTNS 2006 Kyoto, Japan, July 24 28, 2006 Lecture 2: Representations and annihilators
More informationRobust 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
More informationMASSACHUSETTS 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 6.24: Dynamic Systems Spring 20 Homework 9 Solutions Exercise 2. We can use additive perturbation model with
More informationProblem 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, 96537 Problem (Dynamic Augmentation: State Space Representation) Consider a dynamical system consisting
More informationMEM 355 Performance Enhancement of Dynamical Systems
MEM 355 Performance Enhancement of Dynamical Systems State Space Design Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University 11/8/2016 Outline State space techniques emerged
More informationCANONICAL LOSSLESS STATESPACE SYSTEMS: STAIRCASE FORMS AND THE SCHUR ALGORITHM
CANONICAL LOSSLESS STATESPACE SYSTEMS: STAIRCASE FORMS AND THE SCHUR ALGORITHM Ralf L.M. Peeters Bernard Hanzon Martine Olivi Dept. Mathematics, Universiteit Maastricht, P.O. Box 616, 6200 MD Maastricht,
More informationDiscrete and continuous dynamic systems
Discrete and continuous dynamic systems Bounded input bounded output (BIBO) and asymptotic stability Continuous and discrete time linear timeinvariant systems Katalin Hangos University of Pannonia Faculty
More informationCME 345: MODEL REDUCTION
CME 345: MODEL REDUCTION Balanced Truncation Charbel Farhat & David Amsallem Stanford University cfarhat@stanford.edu These slides are based on the recommended textbook: A.C. Antoulas, Approximation of
More informationECE557 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 informationState Space Design. MEM 355 Performance Enhancement of Dynamical Systems
State Space Design MEM 355 Performance Enhancement of Dynamical Systems Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline State space techniques emerged around
More informationObjective Type Questions
DISTANCE EDUCATION, UNIVERSITY OF CALICUT NUMBER THEORY AND LINEARALGEBRA Objective Type Questions Shyama M.P. Assistant Professor Department of Mathematics Malabar Christian College, Calicut 7/3/2014
More informationChap 4. StateSpace Solutions and
Chap 4. StateSpace Solutions and Realizations Outlines 1. Introduction 2. Solution of LTI State Equation 3. Equivalent State Equations 4. Realizations 5. Solution of Linear TimeVarying (LTV) Equations
More informationMath 4153 Exam 3 Review. The syllabus for Exam 3 is Chapter 6 (pages ), Chapter 7 through page 137, and Chapter 8 through page 182 in Axler.
Math 453 Exam 3 Review The syllabus for Exam 3 is Chapter 6 (pages 2), Chapter 7 through page 37, and Chapter 8 through page 82 in Axler.. You should be sure to know precise definition of the terms we
More informationCanonical lossless statespace systems: staircase forms and the Schur algorithm
Canonical lossless statespace systems: staircase forms and the Schur algorithm Ralf L.M. Peeters Bernard Hanzon Martine Olivi Dept. Mathematics School of Mathematical Sciences Projet APICS Universiteit
More informationECE504: Lecture 9. D. Richard Brown III. Worcester Polytechnic Institute. 04Nov2008
ECE504: Lecture 9 D. Richard Brown III Worcester Polytechnic Institute 04Nov2008 Worcester Polytechnic Institute D. Richard Brown III 04Nov2008 1 / 38 Lecture 9 Major Topics ECE504: Lecture 9 We are
More informationThe Important State Coordinates of a Nonlinear System
The Important State Coordinates of a Nonlinear System Arthur J. Krener 1 University of California, Davis, CA and Naval Postgraduate School, Monterey, CA ajkrener@ucdavis.edu Summary. We offer an alternative
More informationWe are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists. International authors and editors
We are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists 3,500 08,000.7 M Open access books available International authors and editors Downloads Our authors
More informationLinear Algebra Exercises
9. 8.03 Linear Algebra Exercises 9A. Matrix Multiplication, Rank, Echelon Form 9A. Which of the following matrices is in rowechelon form? 2 0 0 5 0 (i) (ii) (iii) (iv) 0 0 0 (v) [ 0 ] 0 0 0 0 0 0 0 9A2.
More informationPole placement control: state space and polynomial approaches Lecture 1
: state space and polynomial approaches Lecture 1 dynamical O. Sename 1 1 Gipsalab, CNRSINPG, FRANCE Olivier.Sename@gipsalab.fr www.gipsalab.fr/ o.sename November 7, 2017 Outline dynamical dynamical
More informationChapter 30 Minimality and Stability of Interconnected Systems 30.1 Introduction: Relating I/O and StateSpace Properties We have already seen in Chapt
Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology 1 1 c Chapter
More information