Linear System Theory
|
|
- Donald Neal
- 5 years ago
- Views:
Transcription
1 Linear System Theory Wonhee Kim Chapter 6: Controllability & Observability Chapter 7: Minimal Realizations May 2, / 31
2 Recap State space equation Linear Algebra Solutions of LTI and LTV system Stability We will study Controllability & Observability Kalman Decomposition Minimal realizations 2 / 31
3 Controllability & Observability Controllability (informal): we want to know whether the state of the system is controllable or not from the input Analyze the system structure from the input With the input, we want to move the state to the desired point in a finite time. Observability (informal): we want to observe the initial state of the system from the output and input to quantify the behavior of the system State: position, velocity, acceleration, etc Sensors are required to measure the state. We are not able to use many sensors in real applications. Controllability & Observability Important concepts in control, estimation, and filtering problems Optimal control (LQG, Kalman filtering, etc.) 3 / 31
4 Controllability & Observability Example ẋ = ( ) 1 x + 3 ( b1 b 2 ) u, x = (x 1 x 2 ) T (b 1, b 2 ) T = ( 1, 1) T : can move both eigenvalues can control the state x 1 and x 2 (b 1, b 2 ) T = (1, ) T : cannot move the eigenvalue 3 cannot control state x 2 (b 1, b 2 ) T = (1, ) T : No matter input, x 2 diverges we cannot control x 2 4 / 31
5 Controllability & Observability Example ẋ = ( ) 1 x, y = ( ) c 3 1 c 2 x (c 1, c 2 ) = (1, 1): can observe the state x 1 and x 2 (c 1, c 2 ) = (1, ): cannot observe the state x 2 (c 1, c 2 ) = (1, ): Output is always stable, but the system is internally unstable 5 / 31
6 Controllability ẋ(t) = Ax(t) + Bu(t), x R n, u R m Definition (Definition 6.1) The state equation with the pair (A, B) is said to be controllable if for any initial state x() = x, any final state x 1, there exists an input that transfers x to x 1 in a finite time. Equivalent Definition: A system is controllable at time t if there exists a finite time t f such that for any initial condition x, and any final state x f, there is a control input u defined on [t, t f ] such that x(t f ) = x f. We need an input u to transfer the state from the initial to the final state Given initial and finial state conditions in R n, is it possible to steer x(t) to the final state by choosing an appropriate input u(t)? 6 / 31
7 Controllability: A Preview Discrete-time LTI system x(k + 1) = Ax(k) + Bu(k), x() =, x R n, u R m x(1) = Bu() x(2) = Ax(1) + Bu(1) = ABu() + Bu(1) x(3) = A 2 Bu() + ABu(1) + Bu(2). x(r) = A r 1 Bu() + A r 1 Bu(1) + + Bu(r 1) u(r 1) x(r) = ( B AB A r 1 B ) u(r 2). u() 7 / 31
8 Controllability: A Preview u(r 1) x(r) = ( B AB A r 1 B ) u(r 2). u() R((B AB A r 1 B)) = {z R n, z = (B AB A r 1 B)p, p R nm } If x f R((B AB A r 1 B)), then x f is reachable This implies that we can reach arbitrary x f R n at time t f = r if and only if R((B AB A r 1 B)) = R n that is equivalent to rank((b AB A r 1 B)) = n Rank of (B AB A r 1 B)) By C-H theorem, A k is a linear combination of {I, A,..., A n 1 } For r n, the rank of (B AB A r 1 B)) cannot increase Hence, if rank((b AB A n 1 B)) = n, then we can find u for an arbitrary x f R n for any finite time 8 / 31
9 Controllability ẋ(t) = Ax(t) + Bu(t), x R n x(t) = e At x() + e A(t τ) Bu(τ)dτ The system is controllable (page 213 of the textbook) for any x, there exists u(t) on [t, t f ] that transfers x to the origin at t f (controllability to the origin) there exists u(t) on [t, t f ] that transfers state from the origin to any final state x f at t f (reachability) Proof: Exercise!! (note that e A(t t) is always invertible!) Basically, we need the surjectivity of ea(t τ) Bu(τ)dτ 9 / 31
10 Controllability ẋ(t) = Ax(t) + Bu(t), x R n, u R m, x = Set of reachable state for a fixed time t: R t = {ξ R n, there exists u such that x(t) = ξ} Note that R t is a subspace of R n Controllability matrix and controllability subspace C AB = {ξ R n : ξ = ( B AB A n 1 B ) z, z R nm } C AB : range space of C, where C = (B AB A n 1 B) R n nm Controllability Gramian W t = e A(t τ) BB T e AT (t τ) dτ = e Aτ BB T e AT τ dτ R(W t ): the range space of W t, W t is a symmetric positive semi-definite matrix 1 / 31
11 Controllability Theorem: Controllability (Theorem 6.1 of the textbook) For each time t >, the following set equality holds: R t = C AB = R(W t ). C = (B AB A n 1 B): controllability matrix Hence if dim C AB = rank((b AB A n 1 B)) = n, the system is controllable Due to C AB, the controllability is independent of the time If the system is controllable, then R t = R n, all the states are reachable by an appropriate choice of the control u We will show that R t C AB, C AB R(W t ), R(W t ) R t Required tools C-H theorem (Chapter 3), R(A T ) = (N(A)) : Problem 1 in HW3 11 / 31
12 Controllability Theorem: R t C AB Proof: Fix t >, and choose any reachable state ξ R t. We need to show that ξ R t implies ξ C AB. We have ξ R t, which implies ξ = ea(t τ) Bu(τ)dτ. Then by C-H theorem, e At = β (t)i + + β n 1 (t)a n 1 (β i (t): scalar function); hence, ξ =B β (t τ)u(τ)dτ + + A n 1 B β n 1 (t τ)u(τ)dτ Hence, ξ C AB = ( B AB A n 1 B ) β (t τ)u(τ)dτ. } β n 1(t τ)u(τ)dτ {{ } R nm 12 / 31
13 Controllability Theorem: C AB R(W t ) Proof: Since C AB R(W t ) is equivalent to C AB R(W t) (proof: exercise!!), we will show that C AB R(W t). From Problem 1 in HW3, R(W t ) = (N(W t )), which is equivalent to (R(W t )) = N(W t ), and similarly, C AB = N((B AB An 1 B)). Hence we need to show that if ξ N(W t ), then ξ N((B AB A n 1 B)). Let ξ N(W t ), then W t ξ = R n, which also implies ξ T W t ξ = R. Then = ξ T e Aτ BB T e AT τ dτξ = Since y(τ) = ξ T e Aτ B =, τ [, t], we have B T e AT τ ξ 2 dτ B T e AT τ ξ =, τ [, t] ξ T ( d k dτ k eaτ ) τ= B = ξ T A k B =, k ξ T ( B AB A n 1 B ) = ξ N((B AB A n 1 B)) = C AB 13 / 31
14 Controllability Theorem: R(W t ) R t Proof: Let ξ R(W t ). Then there exists v R n such that ξ = W t v = Define u(τ) = B T e AT (t τ) v, τ [, t] e Aτ BB T e AT τ vdτ Then, since ẋ = Ax + Bu with x() =, we have x(t) = = e A(t τ) Bu(τ)dτ e A(t τ) BB T e AT (t τ) vdτ = W t v = ξ This means that ξ R t, since we have found the control u that steers the state to ξ from the origin. 14 / 31
15 Controllability Theorem (Theorem 6.1 of the textbook) If (A, B) is controllable, and A is stable (eigenvalues of A have negative real parts), then there exists a unique solution of where P = e Aτ BB T e AT τ dτ > Note that BB T AP + PA T = BB T, In Chapter 5, AP + PA T = Q where Q > Theorem (Theorem 6.1 of the textbook) (A, B) is controllable if and only if rank((a λi B)) = n for all eigenvalues, λ, of A. Hautus-Rosenbrock test 15 / 31
16 Controllability Theorem (Theorem 6.1 of the textbook) (A, B) is controllable if and only if W t >, that is, the controllability Gramian is non-singular Theorem (Theorem 6.2 of the textbook) Let Ā = PAP 1 and B = PB. Then (A, B) is controllable if and only if (Ā, B) is controllable Controllability is invariant under the similarity transformation Fact: The state space equation with the controllable canonical form is always controllable. 16 / 31
17 Controllability ẋ = Ax + Bu, G(s) = X (s) U(s) = (si A) 1 B Kalman Decomposition Theorem (Theorem 6.6 of the textbook): Suppose that C AB = r < n. Let P = (v 1,..., v r, v r+1,..., v n ) where v i, i = 1, 2,..., r is eigenvectors of C, and v r+1,..., v n are arbitrary vectors that guarantees P being nonsingular. Let z = Px. Then ż = PAP 1 z + PBu Ā = PAP 1 = (Ā11 Ā 12 Ā 22 Ā 11 R r r, B1 R r m ) ) ( B1, B = PB = Also, (Ā11, B 1 ) is controllable, and G(s) = (si Ā11) 1 B1. 17 / 31
18 Observability ẋ = Ax + Bu, y = Cx, x R n, y R p Definition (Definition 6.O1) The state-space equation is said to be observable if for any unknown initial condition, there exists a finite t 1 such that the knowledge of the input and the output over [, t 1 ] is suffices to determine uniquely the initial condition x(). W.L.G., u =, (since u is completely known) Note that y(t) = Ce At x() Hence, if N(Ce At ) =, i.e., dim(n(ce At )) = nullity(ce At ) =, then the system is observable. N(Ce At ): unobservable subspace 18 / 31
19 Observability Let O = C CA. CA n 1 O: Observability matrix, O R pn n Theorem: N(Ce At ) = N(O) Proof: We will show that N(Ce At ) N(O) and N(Ce At ) N(O). If x N(Ce At ), then ( d = Ce At x = C dt eat) t= x = CA k x, k Hence, x N(O). If x N(O). then x N(Ce At ), since by C-H Theorem, we have Ce At = Cβ (t)i + + CA n 1 β n 1 (t) 19 / 31
20 Observability If N(Ce At ) =, i.e., dim(n(ce At )) = nullity(ce At ) =, then the system is observable. We need N(Ce At ) = N(O) = Hence, by the rank-nullity theorem, the system is observable if rank(o) = n We say that the system is observable if and only if the pair (C, A) is observable Observability also does not depend on the time (by C-H Theorem) 2 / 31
21 Observability Duality Theorem (Theorem 6.5 of the textbook) The following are equivalent: (C, A) is observable (A T, C T ) is controllable Proof: (A T, C T ) is controllable if and only if O T = (C T A T C T (A T ) n 1 C T ) rank(o T ) = n = rank(o) C CA O =. CA n 1 21 / 31
22 Observability Theorem (Theorem 6.O1) If (A, C) is observable, and A is stable, then there exists a unique solution of A T P + PA = C T C, where P = e AT τ C T Ce Aτ dτ >. Theorem (Theorem 6.O1) (C, A) is observable if and only if ( ) C rank = n A λi 22 / 31
23 Observability Theorem (Theorem 6.O1) (C, A) is observable if and only if the observability Gramian Q t = e AT τ C T Ce Aτ dτ > Theorem (Theorem 6.O3) Let Ā = PAP 1 and C = CP 1. Then (C, A) is observable if and only if ( C, Ā) is observable. 23 / 31
24 Observability ẋ = Ax + Bu, y = Cx, G(s) = Y (s) U(s) = C(sI A) 1 B Kalman Decomposition Theorem (Theorem 6.O6 of the textbook): Suppose that rank(o) = q < n. Let P = v 1. v q v q+1. v n, v 1,..., v q : eigenvectors. Let z = Px. Then ż = PAP 1 z + PBu, y(t) = CP 1 z, and ) ) (Ā11 Ā = PAP 1 ( B1 ( =, B = PB =, C = C Ā 21 Ā 22 B 1 ) 2 Ā 11 R q q, C 1 R p q Also, ( C 1, Ā 11 ) is observable, and G(s) = C 1 (si A 11 ) 1 B / 31
25 Kalman Decomposition Theorem Theorem (Theorem 6.7 of the textbook) We can extract the state that is controllable and observable. Fact: The state space equation with the observable canonical form is always observable Discrete-time LTI system x(k + 1) = Ax(k) + Bu(k), y(k) = Cx(k) (A, B) is controllable if and only if rank(c) = n (C, A) is observable if and only if rank(o) = n 25 / 31
26 Minimum Energy Control (page 189) ẋ = Ax + Bu, x() = x, x(t 1 ) = x f, (A, B) controllable Let W t1 = 1 e Aτ BB T e AT τ dτ >, invertible u (t) = B T e AT (t 1 t) Wt 1 1 (e At1 x x f ) ( 1 ) x(t 1 ) = e At1 x e A(t1 τ) BB T e AT (t 1 τ) dτ Wt 1 1 (e At1 x x f ) = x f } {{ } W t1 We can show that the controller u is the minimum energy controller in the sense that for any controller u that transfers the state from x to x f, we have 1 u(t) 2 dt 1 u (t) 2 dt, u 26 / 31
27 Stabilizability & Detectability Weaker notions of controllability and observability A system is stabilizable if and only if Ā22 is stable and (Ā11, B 1 ) is controllable A system is detectable if and only if Ā22 is stable and ( C 1, Ā11) is observable How about the example on pages 4-5. Is it stabilizable? Is it detectable? 27 / 31
28 Controllability & Observability: LTV system ẋ(t) = A(t)x(t) + B(t)u(t), y(t) = C(t)x(t) W t = Q t = Φ(t, τ)b(τ)b T (τ)φ T (t, τ)dτ, t Φ T (t, τ)c T (τ)c(τ)φ(t, τ)dτ, t The LTV system is is controllable if and only if there exists t f > such that W tf > is observable if and only if there exists t f > such that Q tf > W t : controllability Graminan Q t : observability Graminan 28 / 31
29 Minimal Realizations We have seen that the realization of the state-space equation is not unique. ẋ = Ax + Bu, y = Cx + Du, x() = ẋ = A 1 x + B 1 u, y = C 1 x + D 1 u, x() = y(t) = C Lemmas (not in the textbook) e A(t τ) Bu(τ)dτ = C 1 e A1(t τ) B 1 u(τ)dτ Two system realizations (A, B, C, D) and (A 1, B 1, C 1, D 1 ) are equivalent if and only if D = D 1 and Ce At B = C 1 e A1t B 1, t Two system realizations (A, B, C, D) and (A 1, B 1, C 1, D 1 ) are equivalent if and only if D = D 1 and CA k B = C 1 A k 1B, k 29 / 31
30 Minimal Realizations In view of the Kalman decomposition, we have the following result: Suppose (A, B, C, D) is a system realization. If either (C, A) is not observable or (A, B) is not controllable, then there exists a lower-order realization (A 1, B 1, C 1, D 1 ) for the system Definition (page 233 of the textbook) Realizations with the smallest possible dimension are called minimal realizations Theorem (Theorem 7.M2 (page 254)) (A, B, C, D) is a minimial realization of the transfer function G(s) if and only if (A, B) is controllable and (C, A) is observable If the system is not controllable or not observable (or not controllable and observable), then there are pole-zero cancellations in a transfer function. 3 / 31
31 MATLAB Commands controllability matrix: ctrb(a, B) observability matrix: ctrb(a T, C T ) minimal realization: minreal(a, B, C, D) reduce the system order that has only controllable and observable state Mostly, we use the balanced realization (Chapter 7.4) related to controllability and observability Gramians (robust control, advanced control topic) 31 / 31
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 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 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 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 informationModule 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 information1 Continuous-time Systems
Observability Completely controllable systems can be restructured by means of state feedback to have many desirable properties. But what if the state is not available for feedback? What if only the output
More information16.31 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 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. Wonhee Kim Lecture 1. March 7, 2018
Linear System Theory Wonhee Kim Lecture 1 March 7, 2018 1 / 22 Overview Course Information Prerequisites Course Outline What is Control Engineering? Examples of Control Systems Structure of Control Systems
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 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 informationME Fall 2001, Fall 2002, Spring I/O Stability. Preliminaries: Vector and function norms
I/O Stability Preliminaries: Vector and function norms 1. Sup norms are used for vectors for simplicity: x = max i x i. Other norms are also okay 2. Induced matrix norms: let A R n n, (i stands for induced)
More informationEEE582 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 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: 3.2-3.4 of Text Controllability ẋ = Ax + Bu; x() = x. At time
More information1 Similarity transform 2. 2 Controllability The PBH test for controllability Observability The PBH test for observability...
Contents 1 Similarity transform 2 2 Controllability 3 21 The PBH test for controllability 5 3 Observability 6 31 The PBH test for observability 7 4 Example ([1, pp121) 9 5 Subspace decomposition 11 51
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 informationChapter 6 Controllability and Obervability
Chapter 6 Controllability and Obervability Controllability: whether or not the state-space equation can be controlled from input. Observability: whether or not the initial state can be observed from output.
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, 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 information3 Gramians and Balanced Realizations
3 Gramians and Balanced Realizations In this lecture, we use an optimization approach to find suitable realizations for truncation and singular perturbation of G. It turns out that the recommended realizations
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, 965-37 Problem (Dynamic Augmentation: State Space Representation) Consider a dynamical system consisting
More information6.241 Dynamic Systems and Control
6.241 Dynamic Systems and Control Lecture 24: H2 Synthesis Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology May 4, 2011 E. Frazzoli (MIT) Lecture 24: H 2 Synthesis May
More informationIntro. 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 informationCONTROL 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 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 informationDiscrete 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 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: 3.1-3.4 of text September 7, 26Copyrighted
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 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 informationLecture 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
More informationControl Systems I. Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback. Readings: Emilio Frazzoli
Control Systems I Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 13, 2017 E. Frazzoli (ETH)
More informationSolution 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 informationContents. 1 State-Space Linear Systems 5. 2 Linearization Causality, Time Invariance, and Linearity 31
Contents Preamble xiii Linear Systems I Basic Concepts 1 I System Representation 3 1 State-Space Linear Systems 5 1.1 State-Space Linear Systems 5 1.2 Block Diagrams 7 1.3 Exercises 11 2 Linearization
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 informationModule 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 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 informationSYSTEMTEORI - 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
More information6.241 Dynamic Systems and Control
6.241 Dynamic Systems and Control Lecture 8: Solutions of State-space Models Readings: DDV, Chapters 10, 11, 12 (skip the parts on transform methods) Emilio Frazzoli Aeronautics and Astronautics Massachusetts
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 informationRobust 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 informationLinear System Theory
Linear System Theory Wonhee Kim Lecture 4 Apr. 4, 2018 1 / 40 Recap Vector space, linear space, linear vector space Subspace Linearly independence and dependence Dimension, Basis, Change of Basis 2 / 40
More informationPerspective. 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 informationObservability. It was the property in Lyapunov stability which allowed us to resolve that
Observability We have seen observability twice already It was the property which permitted us to retrieve the initial state from the initial data {u(0),y(0),u(1),y(1),...,u(n 1),y(n 1)} It was the property
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 informationSystems and Control Theory Lecture Notes. Laura Giarré
Systems and Control Theory Lecture Notes Laura Giarré L. Giarré 2018-2019 Lesson 14: Rechability Reachability (DT) Reachability theorem (DT) Reachability properties (DT) Reachability gramian (DT) Reachability
More informationAdvanced Control Theory
State Space Solution and Realization chibum@seoultech.ac.kr Outline State space solution 2 Solution of state-space equations x t = Ax t + Bu t First, recall results for scalar equation: x t = a x t + b
More informationObservability and Constructability
Capitolo. INTRODUCTION 5. Observability and Constructability Observability problem: compute the initial state x(t ) using the information associated to the input and output functions u(t) and y(t) of the
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 informationPOLE 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
More informationObservability 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
More information1 Controllability and Observability
1 Controllability and Observability 1.1 Linear Time-Invariant (LTI) Systems State-space: Dimensions: Notation Transfer function: ẋ = Ax+Bu, x() = x, y = Cx+Du. x R n, u R m, y R p. Note that H(s) is always
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 informationNonlinear 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
More informationTopic # 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
More informationLecture 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
More informationControllability. 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
More informationLinear 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
More informationChap 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 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 informationLQR, Kalman Filter, and LQG. Postgraduate Course, M.Sc. Electrical Engineering Department College of Engineering University of Salahaddin
LQR, Kalman Filter, and LQG Postgraduate Course, M.Sc. Electrical Engineering Department College of Engineering University of Salahaddin May 2015 Linear Quadratic Regulator (LQR) Consider a linear system
More informationLecture 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 informationFEL3210 Multivariable Feedback Control
FEL3210 Multivariable Feedback Control Lecture 8: Youla parametrization, LMIs, Model Reduction and Summary [Ch. 11-12] Elling W. Jacobsen, Automatic Control Lab, KTH Lecture 8: Youla, LMIs, Model Reduction
More information1 Some Facts on Symmetric Matrices
1 Some Facts on Symmetric Matrices Definition: Matrix A is symmetric if A = A T. Theorem: Any symmetric matrix 1) has only real eigenvalues; 2) is always iagonalizable; 3) has orthogonal eigenvectors.
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 informationECEEN 5448 Fall 2011 Homework #4 Solutions
ECEEN 5448 Fall 2 Homework #4 Solutions Professor David G. Meyer Novemeber 29, 2. The state-space realization is A = [ [ ; b = ; c = [ which describes, of course, a free mass (in normalized units) with
More 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 informationControl 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
More informationECE504: 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
More informationControl, 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
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 information4F3 - 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
More informationLinear Systems. Manfred Morari Melanie Zeilinger. Institut für Automatik, ETH Zürich Institute for Dynamic Systems and Control, ETH Zürich
Linear Systems Manfred Morari Melanie Zeilinger Institut für Automatik, ETH Zürich Institute for Dynamic Systems and Control, ETH Zürich Spring Semester 2016 Linear Systems M. Morari, M. Zeilinger - Spring
More informationIntroduction 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 information6.241 Dynamic Systems and Control
6.241 Dynamic Systems and Control Lecture 7: State-space Models Readings: DDV, Chapters 7,8 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology February 25, 2011 E. Frazzoli
More informationLecture 2. Linear Systems
Lecture 2. Linear Systems Ivan Papusha CDS270 2: Mathematical Methods in Control and System Engineering April 6, 2015 1 / 31 Logistics hw1 due this Wed, Apr 8 hw due every Wed in class, or my mailbox on
More informationModule 02 CPS Background: Linear Systems Preliminaries
Module 02 CPS Background: Linear Systems Preliminaries Ahmad F. Taha EE 5243: Introduction to Cyber-Physical Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha/index.html August
More information16.30 Estimation and Control of Aerospace Systems
16.30 Estimation and Control of Aerospace Systems Topic 5 addendum: Signals and Systems Aeronautics and Astronautics Massachusetts Institute of Technology Fall 2010 (MIT) Topic 5 addendum: Signals, Systems
More informationEE221A 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 informationEE451/551: Digital Control. Chapter 8: Properties of State Space Models
EE451/551: Digital Control Chapter 8: Properties of State Space Models Equilibrium State Definition 8.1: An equilibrium point or state is an initial state from which the system nevers departs unless perturbed
More information1. 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 informationControl Systems Design
ELEC4410 Control Systems Design Lectre 16: Controllability and Observability Canonical Decompositions Jlio H. Braslavsky jlio@ee.newcastle.ed.a School of Electrical Engineering and Compter Science Lectre
More informationRobust Multivariable Control
Lecture 2 Anders Helmersson anders.helmersson@liu.se ISY/Reglerteknik Linköpings universitet Today s topics Today s topics Norms Today s topics Norms Representation of dynamic systems Today s topics Norms
More informationReachability, Observability and Minimality for a Class of 2D Continuous-Discrete Systems
Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation, Athens, Greece, August 24-26, 27 Reachability, Observability and Minimality for a Class of 2D Continuous-Discrete
More informationECE504: 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
More informationControllability, Observability and Realizability
Georgia Southern University Digital Commons@Georgia Southern Electronic Theses and Dissertations Graduate Studies, Jack N. Averitt College of Fall 2005 Controllability, Observability and Realizability
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 informationOptimal Sensor and Actuator Location for Descriptor Systems using Generalized Gramians and Balanced Realizations
Optimal Sensor and Actuator Location for Descriptor Systems using Generalized Gramians and Balanced Realizations B. MARX D. KOENIG D. GEORGES Laboratoire d Automatique de Grenoble (UMR CNRS-INPG-UJF B.P.
More informationStabilisation of Linear Time-Invariant Systems subject to Output Saturation
Stabilisation of Linear Time-Invariant Systems subject to Output Saturation by Gijs Hilhorst, BSc A Thesis in Partial Fulfillment of the Requirement for the Degree of Master of Science at the Department
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 informationProblem Set 5 Solutions 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Problem Set 5 Solutions The problem set deals with Hankel
More informationLinear dynamical systems with inputs & outputs
EE263 Autumn 215 S. Boyd and S. Lall Linear dynamical systems with inputs & outputs inputs & outputs: interpretations transfer function impulse and step responses examples 1 Inputs & outputs recall continuous-time
More informationRECURSIVE ESTIMATION AND KALMAN FILTERING
Chapter 3 RECURSIVE ESTIMATION AND KALMAN FILTERING 3. The Discrete Time Kalman Filter Consider the following estimation problem. Given the stochastic system with x k+ = Ax k + Gw k (3.) y k = Cx k + Hv
More informationSYSTEMTEORI - 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 informationSteady State Kalman Filter
Steady State Kalman Filter Infinite Horizon LQ Control: ẋ = Ax + Bu R positive definite, Q = Q T 2Q 1 2. (A, B) stabilizable, (A, Q 1 2) detectable. Solve for the positive (semi-) definite P in the ARE:
More informationSolving Dynamic Equations: The State Transition Matrix II
Reading the Text Solving Dynamic Equations: The State Transition Matrix II EGR 326 February 27, 2017 Just a reminder to read the text Read through longer passages, to see what is connected to class topics.
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 information5. 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
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 informationECEN 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 informationHomogeneous and particular LTI solutions
Homogeneous and particular LTI solutions Daniele Carnevale Dipartimento di Ing. Civile ed Ing. Informatica (DICII), University of Rome Tor Vergata Fondamenti di Automatica e Controlli Automatici A.A. 2014-2015
More informationControl Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich
Control Systems I Lecture 6: Poles and Zeros Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 27, 2017 E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/2017
More information