CDS Solutions to the Midterm Exam
|
|
- Georgia Holt
- 6 years ago
- Views:
Transcription
1 CDS 22 - Solutions to the Midterm Exam Instructor: Danielle C. Tarraf November 6, 27 Problem (a) Recall that the H norm of a transfer function is time-delay invariant. Hence: ( ) Ĝ(s) = s + a = sup /2 w R jw + a = sup w R a 2 + w 2 = a (b) The transfer function of the system is Ĝ(s) =, with corresponding s + H 2 norm: Ĝ 2 2 = Ĝ (jw)ĝ(jw)dw = = jw + w 2 = arctan w = 2 + jw dw hence Ĝ 2 = 2 (c) Recall that: y G u where G(t) is the impulse response of the system. Here, G(t) = te t when
2 û û 2 s s s + 2 ˆx ˆx ŷ Figure : Figure for problem (d) t, with corresponding l norm: G = = G(t) dt te t dt = te t + = e t = and the amplitude of the output cannot exceed. e t dt (d) Consider the parallel interconnection of two first order systems as shown in the Figure above, and note by inspection that Ĝ is indeed the corresponding transfer matrix. Let x (t) = L (ˆx (s)) and x 2 (t) = L (ˆx 2 (s)). We have: ẋ 2 + 2x 2 = u 2 ẋ x = u y = x + x 2 Set x = x u, with ẋ = ẋ u = x = x + u. The state-space equations of the system are then: ] ] ] ] ] ẋ x u = + ẋ 2 2 x 2 u 2 y = ] ] x + ] ] u x 2 u 2 (e) For the autonomous system (i.e. input identically ), the state and output trajectories for t are given by: x(t) = e At x() y(t) = Ce At x() 2
3 where e At is defined by the infinite series: Note that here: Hence and e At = I + At + A2 t 2 A 2 = e At = x(t) = 2! + A3 t 3 3! +..., A k =, for k 3 t t 2 /2 t + 2t + t 2 /2 2 + t y(t) = + 2t + t 2 /2,, for t. (f) The transfer matrix is given by Ĝ(s) = C(sI A) B + D. Note that due to the structure of C and B, we only need to compute the upper right entry of (si A). Recall that We have: (si A) = si A = adj(si A) = adj(si A) det(si A) s s s, and Hence det(si A) = s s s + s = s3 Ĝ(s) = ] s 3 + = s3 + s 3 3
4 Problem 2 (a) FALSE. Consider a SISO system with A=B=, C=D=. A has an eigenvalue at while the transfer function is identically and hence has no poles - this is a very trivial example of an unobservable system. (b) TRUE. We have: Ĝ 2 2 = = = r Ĝ 2 2 Ĝ (rjw)ĝ(rjw)dw Ĝ (jv)ĝ(jv) r dv where the second equality follows by a change of the integration variable. (c) FALSE. Consider the case where M = ai for some scalar a <. Clearly, there exists a matrix D = I = D such that: DMD 2 = M 2 = a I 2 = a < However, the LMI: X a IXIa + X ] = X ( a 2 + )X ] < is not feasible, since it requires X to be both positive definite and negative definite. The correct statement will be derived later on in the class! (d) TRUE. One way of proving this is by using state-space methods to compute the H 2 norms of the relevant systems (see Section 2.6 in DFT). Possible state-space realizations of systems S and S 2 described by transfer functions Ĝ and Ĝ2 are given by: S = ( a ),S 2 = ( a2 System S, the cascade interconnection of S and S 2, with corresponding transfer function Ĝ(s) = Ĝ2(s)Ĝ(s), then has the following state space realization: ] ] a S = a 2 ] ] ) Recall that for a system ( A B S = C D ) 4
5 with A Hurwitz, the H 2 norm of the corresponding transfer function Ĝ can be computed as: Ĝ 2 2 = CLC where L is the solution to the Lyapunov equation: AL + LA = BB For first order systems S and S 2, we thus have: Ĝi 2 2 = c2 i b2 i 2a i For the second order system S, the relevant Lyapunov equation is: ] ] ] ] ] a l l l l + a ] = a 2 l l 2 l l 2 a 2 Solving the corresponding system of three equations in three unknowns: 2a l + = (a + a 2 )l + l = l a 2 l 2 = we get: l = 2a, l = Hence Ĝ 2 2 = l 2, and 2a (a + a 2 ), l 2 = Ĝ 2 Ĝ 2 Ĝ2 2 Ĝ 2 2 Ĝ 2 2 Ĝ a a 2 (a + a 2 ) 2a 2a 2 a a a a 2 (a + a 2 ) (e) TRUE. Note that I A being singular is equivalent to the existence of a v such that: (I A)v = Iv = Av v 2 = Av 2 2 A 2 v 2 2 = A 2 σ max (A) To show that this lower bound can be achieved, let UΣV be a singular value decomposition of A and consider the perturbation matrix = V ΣU 5
6 where Note that 2 = Σ = σ max (A) σ max (A) and that: I A = I V ΣU UΣV = I V ΣΣV... = V I V with (I A)v = ; hence I A is singular. 6
7 Problem 3 (a) Let x = y and x 2 = ẏ u. We then have: and ẋ 2 = ÿ u ẋ = ẏ = x 2 + u = aẏ by cy 2 + u = a(x 2 + u) bx cx 2 + u = bx ax 2 cx 2 + ( a)u The corresponding second order state space realization is: ẋ = x 2 + u ẋ 2 = bx ax 2 cx 2 + ( a)u y = x Note that this realization is not unique; it is simply one of many possible realizations. (b) When a = 3, b = 2 and c = 2, the state equations of the autonomous system reduce to: { ẋ = x 2 ẋ 2 = 2x 3x 2 2x 2 and the corresponding linearized (about the origin) dynamics are given by: ] ] ] δ δ = δ The eigenvalues of the linearized dynamics are λ = and λ 2 = 2. Since the linearized dynamics are asymptotically stable, and for f(x) = 2x 2, we have: f(δ) f() f (x) δ 2 2 δ 2 lim = lim = δ 2 δ 2 δ δ we conclude that the equilibrium point at the origin is locally asymptotically stable. Clearly, it cannot be globally stable as there exists another equilibrium point at (,). δ 2 7
8 (c) The dynamics of the autonomous nonlinear system are given by: where A = 2 3 ẋ = Ax + f(x) ] and f(x) = ] 2x 2 Let V (x) = x Px be a Lyapunov function for the linearized system; then P is a solution of the Lyapunov equation A P +PA = Q for some Q >, and along the trajectories of the nonlinear system, we have: V (x) = ẋ Px + x Pẋ = x A + f (x)]px + x PAx + f(x)] = x Qx + 2x Pf(x) λ min (Q) x 2 + 2λ max (P) f(x) 2 x 2 λ min (Q) + 2λ max (P)ɛ] x 2 whenever x ɛ/2 Thus, the function V : B ɛ R is a Lyapunov function for the nonlinear system in the neighborhood B ɛ = {x R 2 x 2 < ɛ/2} for ɛ < λ min(q), for any Q > and corresponding solution P to the 2λ max (P) Lyapunov equation; in particular, trajectories starting in this neighborhood will converge asymptotically to the origin. For instance, for Q = I with λ min (Q) =, we have P = ] 5 4 with λ max (P) = and the corresponding provable region of attraction: { } B = x R 2 x 2 < Ideally, we would like to choose Q > to maximize the ratio λ min(q) 2λ max (P), which is not an easy problem. What is straightforward though is computing an upper bound for this ratio, which establishes the limitations of this approach in finding a region of attraction. We have: 2λ max (P) A = 2 P A 2 PA A P + PA = Q λ min (Q) which implies: λ min (Q) 2λ max (P) A 2 8
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 information7.1 Linear Systems Stability Consider the Continuous-Time (CT) Linear Time-Invariant (LTI) system
7 Stability 7.1 Linear Systems Stability Consider the Continuous-Time (CT) Linear Time-Invariant (LTI) system ẋ(t) = A x(t), x(0) = x 0, A R n n, x 0 R n. (14) The origin x = 0 is a globally asymptotically
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 informationIntroduction to Nonlinear Control Lecture # 3 Time-Varying and Perturbed Systems
p. 1/5 Introduction to Nonlinear Control Lecture # 3 Time-Varying and Perturbed Systems p. 2/5 Time-varying Systems ẋ = f(t, x) f(t, x) is piecewise continuous in t and locally Lipschitz in x for all t
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 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 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 informationNonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems. p. 1/1
Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems p. 1/1 p. 2/1 Converse Lyapunov Theorem Exponential Stability Let x = 0 be an exponentially stable equilibrium
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 informationCDS Final Exam
CDS 22 - Final Exam Instructor: Danielle C. Tarraf December 4, 2007 INSTRUCTIONS : Please read carefully! () Description & duration of the exam: The exam consists of 6 problems. You have a total of 24
More informationProblem set 5 solutions 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.242, Fall 24: MODEL REDUCTION Problem set 5 solutions Problem 5. For each of the stetements below, state
More informationSolution of Additional Exercises for Chapter 4
1 1. (1) Try V (x) = 1 (x 1 + x ). Solution of Additional Exercises for Chapter 4 V (x) = x 1 ( x 1 + x ) x = x 1 x + x 1 x In the neighborhood of the origin, the term (x 1 + x ) dominates. Hence, the
More informationNonlinear Control. Nonlinear Control Lecture # 3 Stability of Equilibrium Points
Nonlinear Control Lecture # 3 Stability of Equilibrium Points The Invariance Principle Definitions Let x(t) be a solution of ẋ = f(x) A point p is a positive limit point of x(t) if there is a sequence
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 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 informationControl 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 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 informationNonlinear Control. Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems
Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems Time-varying Systems ẋ = f(t,x) f(t,x) is piecewise continuous in t and locally Lipschitz in x for all t 0 and all x D, (0 D). The origin
More informationMCE/EEC 647/747: Robot Dynamics and Control. Lecture 8: Basic Lyapunov Stability Theory
MCE/EEC 647/747: Robot Dynamics and Control Lecture 8: Basic Lyapunov Stability Theory Reading: SHV Appendix Mechanical Engineering Hanz Richter, PhD MCE503 p.1/17 Stability in the sense of Lyapunov A
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 informationProf. Krstic Nonlinear Systems MAE281A Homework set 1 Linearization & phase portrait
Prof. Krstic Nonlinear Systems MAE28A Homework set Linearization & phase portrait. For each of the following systems, find all equilibrium points and determine the type of each isolated equilibrium. Use
More informationL2 gains and system approximation quality 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.242, Fall 24: MODEL REDUCTION L2 gains and system approximation quality 1 This lecture discusses the utility
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 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 informationLyapunov Stability Theory
Lyapunov Stability Theory Peter Al Hokayem and Eduardo Gallestey March 16, 2015 1 Introduction In this lecture we consider the stability of equilibrium points of autonomous nonlinear systems, both in continuous
More information2 Lyapunov Stability. x(0) x 0 < δ x(t) x 0 < ɛ
1 2 Lyapunov Stability Whereas I/O stability is concerned with the effect of inputs on outputs, Lyapunov stability deals with unforced systems: ẋ = f(x, t) (1) where x R n, t R +, and f : R n R + R n.
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 informationTopic # /31 Feedback Control Systems. Analysis of Nonlinear Systems Lyapunov Stability Analysis
Topic # 16.30/31 Feedback Control Systems Analysis of Nonlinear Systems Lyapunov Stability Analysis Fall 010 16.30/31 Lyapunov Stability Analysis Very general method to prove (or disprove) stability of
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 informationEntrance Exam, Differential Equations April, (Solve exactly 6 out of the 8 problems) y + 2y + y cos(x 2 y) = 0, y(0) = 2, y (0) = 4.
Entrance Exam, Differential Equations April, 7 (Solve exactly 6 out of the 8 problems). Consider the following initial value problem: { y + y + y cos(x y) =, y() = y. Find all the values y such that the
More informationChapter III. Stability of Linear Systems
1 Chapter III Stability of Linear Systems 1. Stability and state transition matrix 2. Time-varying (non-autonomous) systems 3. Time-invariant systems 1 STABILITY AND STATE TRANSITION MATRIX 2 In this chapter,
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 informationProblem Set 4 Solution 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Problem Set 4 Solution Problem 4. For the SISO feedback
More informationIntroduction to Nonlinear Control Lecture # 4 Passivity
p. 1/6 Introduction to Nonlinear Control Lecture # 4 Passivity È p. 2/6 Memoryless Functions ¹ y È Ý Ù È È È È u (b) µ power inflow = uy Resistor is passive if uy 0 p. 3/6 y y y u u u (a) (b) (c) Passive
More informationSTRUCTURE MATTERS: Some Notes on High Gain Observer Design for Nonlinear Systems. Int. Conf. on Systems, Analysis and Automatic Control 2012
Faculty of Electrical and Computer Engineering Institute of Control Theory STRUCTURE MATTERS: Some Notes on High Gain Observer Design for Nonlinear Systems Klaus Röbenack Int. Conf. on Systems, Analysis
More information16.30/31, Fall 2010 Recitation # 13
16.30/31, Fall 2010 Recitation # 13 Brandon Luders December 6, 2010 In this recitation, we tie the ideas of Lyapunov stability analysis (LSA) back to previous ways we have demonstrated stability - but
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 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 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 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 informationQuadratic Stability of Dynamical Systems. Raktim Bhattacharya Aerospace Engineering, Texas A&M University
.. Quadratic Stability of Dynamical Systems Raktim Bhattacharya Aerospace Engineering, Texas A&M University Quadratic Lyapunov Functions Quadratic Stability Dynamical system is quadratically stable if
More informationHandout 2: Invariant Sets and Stability
Engineering Tripos Part IIB Nonlinear Systems and Control Module 4F2 1 Invariant Sets Handout 2: Invariant Sets and Stability Consider again the autonomous dynamical system ẋ = f(x), x() = x (1) with state
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 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 informationLecture 8. Chapter 5: Input-Output Stability Chapter 6: Passivity Chapter 14: Passivity-Based Control. Eugenio Schuster.
Lecture 8 Chapter 5: Input-Output Stability Chapter 6: Passivity Chapter 14: Passivity-Based Control Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture
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 informationIntro. Computer Control Systems: F9
Intro. Computer Control Systems: F9 State-feedback control and observers Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 21 dave.zachariah@it.uu.se F8: Quiz! 2 / 21 dave.zachariah@it.uu.se
More informationEG4321/EG7040. Nonlinear Control. Dr. Matt Turner
EG4321/EG7040 Nonlinear Control Dr. Matt Turner EG4321/EG7040 [An introduction to] Nonlinear Control Dr. Matt Turner EG4321/EG7040 [An introduction to] Nonlinear [System Analysis] and Control Dr. Matt
More informationDenis ARZELIER arzelier
COURSE ON LMI OPTIMIZATION WITH APPLICATIONS IN CONTROL PART II.2 LMIs IN SYSTEMS CONTROL STATE-SPACE METHODS PERFORMANCE ANALYSIS and SYNTHESIS Denis ARZELIER www.laas.fr/ arzelier arzelier@laas.fr 15
More informationModeling 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 informationLecture 10: Singular Perturbations and Averaging 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 10: Singular Perturbations and
More information3 rd Tutorial on EG4321/EG7040 Nonlinear Control
3 rd Tutorial on EG4321/EG7040 Nonlinear Control Lyapunov Stability Dr Angeliki Lekka 1 1 Control Systems Research Group Department of Engineering, University of Leicester arch 9, 2017 Dr Angeliki Lekka
More informationInput to state Stability
Input to state Stability Mini course, Universität Stuttgart, November 2004 Lars Grüne, Mathematisches Institut, Universität Bayreuth Part IV: Applications ISS Consider with solutions ϕ(t, x, w) ẋ(t) =
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 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 informationNonlinear Systems Theory
Nonlinear Systems Theory Matthew M. Peet Arizona State University Lecture 2: Nonlinear Systems Theory Overview Our next goal is to extend LMI s and optimization to nonlinear systems analysis. Today we
More informationNonlinear systems. Lyapunov stability theory. G. Ferrari Trecate
Nonlinear systems Lyapunov stability theory G. Ferrari Trecate Dipartimento di Ingegneria Industriale e dell Informazione Università degli Studi di Pavia Advanced automation and control Ferrari Trecate
More informationLMI Methods in Optimal and Robust Control
LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 4: LMIs for State-Space Internal Stability Solving the Equations Find the output given the input State-Space:
More informationLyapunov Stability Analysis: Open Loop
Copyright F.L. Lewis 008 All rights reserved Updated: hursday, August 8, 008 Lyapunov Stability Analysis: Open Loop We know that the stability of linear time-invariant (LI) dynamical systems can be determined
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 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 informationStability of Stochastic Differential Equations
Lyapunov stability theory for ODEs s Stability of Stochastic Differential Equations Part 1: Introduction Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH December 2010
More informationCalculating the domain of attraction: Zubov s method and extensions
Calculating the domain of attraction: Zubov s method and extensions Fabio Camilli 1 Lars Grüne 2 Fabian Wirth 3 1 University of L Aquila, Italy 2 University of Bayreuth, Germany 3 Hamilton Institute, NUI
More informationNonlinear Control. Nonlinear Control Lecture # 6 Passivity and Input-Output Stability
Nonlinear Control Lecture # 6 Passivity and Input-Output Stability Passivity: Memoryless Functions y y y u u u (a) (b) (c) Passive Passive Not passive y = h(t,u), h [0, ] Vector case: y = h(t,u), h T =
More informationNonlinear Control. Nonlinear Control Lecture # 2 Stability of Equilibrium Points
Nonlinear Control Lecture # 2 Stability of Equilibrium Points Basic Concepts ẋ = f(x) f is locally Lipschitz over a domain D R n Suppose x D is an equilibrium point; that is, f( x) = 0 Characterize and
More informationLecture 15: H Control Synthesis
c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 1/14 Lecture 15: H Control Synthesis Example c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control
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 informationSome solutions of the written exam of January 27th, 2014
TEORIA DEI SISTEMI Systems Theory) Prof. C. Manes, Prof. A. Germani Some solutions of the written exam of January 7th, 0 Problem. Consider a feedback control system with unit feedback gain, with the following
More information1 Lyapunov theory of stability
M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 1 Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability
More information6.241 Dynamic Systems and Control
6.241 Dynamic Systems and Control Lecture 12: I/O Stability Readings: DDV, Chapters 15, 16 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology March 14, 2011 E. Frazzoli
More informationGramians based model reduction for hybrid switched systems
Gramians based model reduction for hybrid switched systems Y. Chahlaoui Younes.Chahlaoui@manchester.ac.uk Centre for Interdisciplinary Computational and Dynamical Analysis (CICADA) School of Mathematics
More informationBalancing of Lossless and Passive Systems
Balancing of Lossless and Passive Systems Arjan van der Schaft Abstract Different balancing techniques are applied to lossless nonlinear systems, with open-loop balancing applied to their scattering representation.
More informationSemidefinite Programming Duality and Linear Time-invariant Systems
Semidefinite Programming Duality and Linear Time-invariant Systems Venkataramanan (Ragu) Balakrishnan School of ECE, Purdue University 2 July 2004 Workshop on Linear Matrix Inequalities in Control LAAS-CNRS,
More informationNonlinear Control Systems
Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 5. Input-Output Stability DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs2012/ 2012 1 Input-Output Stability y = Hu H denotes
More informationControl 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 informationLecture 4. Chapter 4: Lyapunov Stability. Eugenio Schuster. Mechanical Engineering and Mechanics Lehigh University.
Lecture 4 Chapter 4: Lyapunov Stability Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 4 p. 1/86 Autonomous Systems Consider the autonomous system ẋ
More informationApplied Math Qualifying Exam 11 October Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems.
Printed Name: Signature: Applied Math Qualifying Exam 11 October 2014 Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems. 2 Part 1 (1) Let Ω be an open subset of R
More informationSolution 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 informationDepartment of Mathematics IIT Guwahati
Stability of Linear Systems in R 2 Department of Mathematics IIT Guwahati A system of first order differential equations is called autonomous if the system can be written in the form dx 1 dt = g 1(x 1,
More informationCDS 101/110a: Lecture 2.1 Dynamic Behavior
CDS 11/11a: Lecture.1 Dynamic Behavior Richard M. Murray 6 October 8 Goals: Learn to use phase portraits to visualize behavior of dynamical systems Understand different types of stability for an equilibrium
More informationECEEN 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 informationLMI Methods in Optimal and Robust Control
LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 15: Nonlinear Systems and Lyapunov Functions Overview Our next goal is to extend LMI s and optimization to nonlinear
More informationLinear Matrix Inequality (LMI)
Linear Matrix Inequality (LMI) A linear matrix inequality is an expression of the form where F (x) F 0 + x 1 F 1 + + x m F m > 0 (1) x = (x 1,, x m ) R m, F 0,, F m are real symmetric matrices, and the
More informationMath 273, Final Exam Solutions
Math 273, Final Exam Solutions 1. Find the solution of the differential equation y = y +e x that satisfies the condition y(x) 0 as x +. SOLUTION: y = y H + y P where y H = ce x is a solution of the homogeneous
More informationSection 3.9. Matrix Norm
3.9. Matrix Norm 1 Section 3.9. Matrix Norm Note. We define several matrix norms, some similar to vector norms and some reflecting how multiplication by a matrix affects the norm of a vector. We use matrix
More informationCDS 101/110a: Lecture 2.1 Dynamic Behavior
CDS 11/11a: Lecture 2.1 Dynamic Behavior Richard M. Murray 6 October 28 Goals: Learn to use phase portraits to visualize behavior of dynamical systems Understand different types of stability for an equilibrium
More informationPoincaré Map, Floquet Theory, and Stability of Periodic Orbits
Poincaré Map, Floquet Theory, and Stability of Periodic Orbits CDS140A Lecturer: W.S. Koon Fall, 2006 1 Poincaré Maps Definition (Poincaré Map): Consider ẋ = f(x) with periodic solution x(t). Construct
More informationEE363 homework 7 solutions
EE363 Prof. S. Boyd EE363 homework 7 solutions 1. Gain margin for a linear quadratic regulator. Let K be the optimal state feedback gain for the LQR problem with system ẋ = Ax + Bu, state cost matrix Q,
More information7 Planar systems of linear ODE
7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution
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 informationInvariant Manifolds of Dynamical Systems and an application to Space Exploration
Invariant Manifolds of Dynamical Systems and an application to Space Exploration Mateo Wirth January 13, 2014 1 Abstract In this paper we go over the basics of stable and unstable manifolds associated
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 informationMEM 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 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 informationHankel Optimal Model Reduction 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.242, Fall 2004: MODEL REDUCTION Hankel Optimal Model Reduction 1 This lecture covers both the theory and
More informationHalf of Final Exam Name: Practice Problems October 28, 2014
Math 54. Treibergs Half of Final Exam Name: Practice Problems October 28, 24 Half of the final will be over material since the last midterm exam, such as the practice problems given here. The other half
More informationBIBO STABILITY AND ASYMPTOTIC STABILITY
BIBO STABILITY AND ASYMPTOTIC STABILITY FRANCESCO NORI Abstract. In this report with discuss the concepts of bounded-input boundedoutput stability (BIBO) and of Lyapunov stability. Examples are given to
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 information10 Transfer Matrix Models
MIT EECS 6.241 (FALL 26) LECTURE NOTES BY A. MEGRETSKI 1 Transfer Matrix Models So far, transfer matrices were introduced for finite order state space LTI models, in which case they serve as an important
More informationExam. 135 minutes, 15 minutes reading time
Exam August 6, 208 Control Systems II (5-0590-00) Dr. Jacopo Tani Exam Exam Duration: 35 minutes, 5 minutes reading time Number of Problems: 35 Number of Points: 47 Permitted aids: 0 pages (5 sheets) A4.
More information