Observability. It was the property in Lyapunov stability which allowed us to resolve that
|
|
- Felicity Ford
- 6 years ago
- Views:
Transcription
1 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 in Lyapunov stability which allowed us to resolve that if V (x) x(t) T C T Cx(t) 0 when ẋ(t) Ax(t) then x(t) 0 It is a dual property to controllability Many of the results are completely parallel Duality refers to the property that (formally only in LTI) [A, C] is observable, [A T,C T ] is controllable [A, C] is constructible, [A T,C T ] is reachable Observability - two equivalent definitions The future inputs and outputs on [t0,t1] uniquely determine the initial state x0 With the same input signal on [t0,t1], two distinct initial conditions yield distinct output signals of [t0,t1] The first definition is behind our first use of observability The second is behind the Lyapunov result 1
2 (Re)Constructibility Observability deals with recovering the initial state x0 from the i/o data Constructibility deals with recovering the current state from the i/o data Constructibility - two equivalent definitions The past input and output signals on [t0,t1] determine uniquely the current state x(t1) For the same input signal on [t0,t1], two distinct terminal states must have distinct output signals on [t0,t1] ẋ(t) A(t)x(t)+B(t)u(t) y(t) C(t)x(t)+D(t)u(t) ) y(t) C(t) (t, )x( )+ With known u(t), recovering x(t0) from y(t) involves just [A(t),C(t)] on [t0,t1] Likewise, for x(t1), although we still need care in discrete time as we shall see Observability and constructibility rely on [A(t),C(t)] on [t0,t1] Similarly to how controllability and reachability rely on [A(t),B(t)] on [t0,t1] Z t1 C(t) (t, )B( )u( ) d + D(t)u(t) 2
3 Unobservable and unconstructible Subspaces Definition 15.1 (Unobservable subspace) Given two times t1>t0, the unobservable subspace on [t0,t1] consists of all states x 0 2 R n for which C(t) (t, )x 0 0, 8t 2 [,t 1 ] Definition 15.2 (Observable system) Given two times t1>t0, the LTV system is observable if its unobservable subspace contains only the zero vector Definition 15.3 (Unconstructible subspace) Given two times t1>t0, the unconstructible subspace on [t0,t1] consists of all states for which C(t) (t, t 1 )x 1 0, 8t 2 [,t 1 ] [Sneaky When the x 1 2 R n (t, t 1 ) Definition 15.3 (Constructible system) has issues, the unconstructible subspace shrinks] Given two times t1>t0, the LTV system is constructible if its unconstructible subspace contains only the zero vector 3
4 Example of an unobservable system A parallel interconnection of systems Output appleẋ1 (t) ẋ 2 (t) y(t) C 1 apple A1 (t) 0 0 A 2 (t) apple x1 (t) x 2 (t) + apple B1 B 2 If C1(t)C2(t) and A1(t)A2(t) then Φ1(t,t0)Φ2(t1,t0) and we cannot separate the contributions from the two initial conditions x1(t0) and x2(t0) We shall see that this a practical problem u(t) C 2 x(t)+[d1 (t)+d 2 (t)]u(t) y(t) C 1 (t) 1 (t, )x 1 ( )+C 2 (t) 2 (t, )x 2 ( ) + Z t C 1 (t) 1 (t, )B 1 ( )u( ) d + +[D 1 (t)+d 2 (t)]u(t) In order to separately estimate two states from a single measurement we need to have their dynamics differ u Z t P1 P2 + C 2 (t) 2 (t, )B 2 ( )u( ) d y 4
5 Observability and Constructibility Gramians Corollary 15.1 Given two times t1>t0 a LTV system is observable if Given two times t1>t0 a LTV system is constructible if Theorem 15.2 Given two times t1>t0 define W O (,t 1 ) W Cn (,t 1 ) Z t1 Z t1 (i) when the system is observable (ii) when the system is constructible (, ) T C( ) T C( ) (, ) d (,t 1 ) T C( ) T C( ) (,t 1 ) d ỹ(t) y(t) Z t rank W O (,t 1 )n rank W Cn (,t 1 )n C(t) (t, )B( )u( ) d D(t)u(t) Z t1 x( )W O (,t 1 ) 1 (t, ) T C(t) T ỹ(t) dt Z t1 x(t 1 )W Cn (,t 1 ) 1 (t, t 1 ) T C(t) T ỹ(t) dt 5
6 Definitions 15.6 Discrete-time Observability and Constructibility x(t + 1) A(t)x(t) + B(t)u(t) y(t) C(t)x(t)+D(t)u(t) Given two times t1>t0, the unobservable subspace on [t0,t1] consists of all states x 0 2 R n for which C(t) (t, )x 0 0, 8t 2 [,t 1 ] The unconstructible subspace on [t0,t1] consists of all states which Gramians ) y(t) C(t) (t, )x 0 + C(t) (t, t 1 )x 1 0, 8t 2 [,t 1 ] W O (,t 1 ) W Cn (,t 1 ) t 1 X Xt 1 (, ) T C( ) T C( ) (, ) t 1 X (,t 1 ) T C( ) T C( ) (,t 1 ) C(t) (t, )B( )u( )+D(t)u(t) x 1 2 R n for 6
7 Reconstructions Define ỹ(t) y(t) If the system is observable then If the system is constructible then Discrete-time Continued tx x( )W O (,t 1 ) 1 x(t 1 )W Cn (,t 1 ) 1 C(t) (t, )B( )u( ) t 1 X t 1 X D(t)u(t) (t, ) T C(t) T ỹ(t) (t, t 1 ) T C(t) T ỹ(t) We have the same admonitions about conditions such as observability and constructibility as we did for controllability and reachability If the Gramian is almost singular, then the state reconstruction can amplify any noise in the signal measurements This is the basis behind Kalman filtering and is the dual of the reachability issue with possibly large control signals 7
8 Constructibility in Discrete Time We need to be careful about constructibility when the transition function can be singular - a peculiarity of discrete time Look at a variant of the earlier example x(0) apple x1 (0) x 2 (0) y(0) x 2 (0) + du(0) apple x2 (0) + b x(1) 1 u(0) b 2 u(0) y(1) b 2 u(0) + du(1) apple b2 u(0) + b x(2) 1 u(1) b 2 u(1) y(2) b 2 u(1) + du(2) x(t + 1) y(t) 0 apple x(t)+ apple b1 b 2 1 x(t)+du(t) u(t) We cannot calculate x1(0) from the data But we can compute exactly x(t) for t>0 given the input/output data up to time t-1 The zero-input response has already gone to zero since A 2 0 Even though we cannot construct the initial state it has no effect on the current state 8
9 Continuous-time W C (,t 1 ) W O (,t 1 ) Consider LTI systems W R (,t 1 ) W Cn (,t 1 ) Z t1 e A( ) BB T e A Z t1 Z t1 Duality in LTI Systems T ( ) d e A T ( ) C T Ce A( t0) d e A( t 1) BB T e A Z t1 T ( t 1 ) d e A T ( t 1 ) C T Ce A( t1) d ẋ(t) Ax(t)+Bu(t) y(t) Cx(t)+Du(t) W C (,t 1 ) W O (,t 1 ) and Discrete-time W R (,t 1 ) W Cn (,t 1 ) x(t) A T x(t)+c T ū(t) ȳ(t) B T x(t)+d T ū(t) System 1 will be controllable if and only if System 2 is observable System 1 will be reachable if and only if System 2 is constructible Note that some Matlab functions are only written for one test But duality allows us to test the transposes using the same function X t 1 1 A t0 1 BB T A T tx 1 1 A T tx C T CA 1 A t1 1 BB T A T tx 1 1 A T t 1 1 t 1 1 C T CA t 1 1 9
10 Tests for Observability, Constructibility, Detectability The observability matrix for the LTI system [A,B,C,D] is The LTI system is observable iff [PBH] The LTI system is observable iff The continuous-time LTI system is detectable iff The discrete-time LTI system is detectable iff The LTI system is constructible iff Test for controllability is rank rank rank apple A apple A How does this fit with duality? rank O n apple A C C C I I I Ker A n n, 8 2 C Range A n Range C O n, 8 2 C, Re( ) 0 n, 8 2 C, 1 Ker O C CA CA 2. CA n
11 Consider the LTI system Observers and State Estimation Let u-kx be a stabilizing linear state variable feedback When we cannot measure the state x directly then we must reconstruct it from available measurements {u,y} The most direct approach would be to copy the system ẋ Ax + Bu, y Cx + Du, x 2 R n,u2 R k, y 2 R m This leads to the following error system with ẋ(t) Ax(t)+Bu(t), x(0) ˆx(t) Aˆx(t)+Bu(t), ˆx(0) x(t) A x(t), x(0) x(0) ˆx(0) if the original system is stable, A is a stability matrix, x(t) 0 or ˆx(t) x(t) ˆx Aˆx + Bu x(t) x(t) ˆx(t) But what if A is not a stability matrix? Use the output to correct the estimate ˆx(t) Aˆx(t)+Bu(t)+L[y(t) C ˆx(t) Du(t)] x(t) [A LC] x(t) 11
12 Observers ˆx(t) Aˆx(t)+Bu(t)+L[y(t) C ˆx(t) Du(t)] x(t) [A If A-LC is a stability matrix then LC] x(t) x(t) 0 or ˆx(t) x(t) A-LC has the same eigenvalues as (A-LC) T A T -C T L T If [A T,C T ] is controllable then we can find an L T to place these eigenvalues arbitrarily Theorem 16.8/9 If [A,C] is an observable pair then it is possible to find an output injection gain L to place the eigenvalues of A-LC arbitrarily in the complex plane If [A,C] is a detectable pair then it is possible to find an output injection gain L to make A-LC a stability matrix We can use the matlab functions place or acker to do this Lplace(A,C,p) 12
13 Output Feedback Control When we combine linear state variable feedback control with an observer as state estimator we arrive at the following closed-loop system ẋ(t) Ax(t)+Bu(t), y(t) Cx(t)+Du(t), Let us now substitute for u x(0) ˆx(t) [A LC]ˆx(t)+Bu(t)+L[y(t) Du(t)], ˆx(0) u(t) K ˆx(t)+w(t) ẋ(t) Ax(t) y(t) Cx(t) BKˆx(t)+Bw(t) DKˆx(t)+Dw(t) ˆx(t) [A LC]ˆx(t) BKˆx(t)+LCx(t)+Bw(t) u(t) K ˆx(t)+w(t) appleẋ(t) ˆx(t) apple y(t) u(t) apple apple apple A BK x(t) B LC A BK LC ˆx(t) + B w(t) apple apple apple C DK x(t) D 0 K ˆx(t) + w(t) I 13
14 Output Feedback Control continued Rewrite in terms of estimation error appleẋ(t) x(t) apple y(t) u(t) apple x(t) x(t) appleẋ(t) ˆx(t) apple y(t) u(t) apple I 0 I I apple apple apple A BK x(t) B LC A BK LC ˆx(t) + B w(t) apple apple apple C DK x(t) D 0 K ˆx(t) + w(t) I apple x(t) ˆx(t) x(t) x(t) ˆx(t) apple x(t) ˆx(t) apple apple apple I 0 A BK I 0 I I LC A BK LC I I apple apple apple apple C DK I 0 x(t) D 0 K I I x(t) + w(t) I appleẋ(t) x(t) apple y(t) u(t) apple I 0 I I apple apple x(t) I 0 x(t) + I I apple apple apple A BK BK x(t) B 0 A LC x(t) + 0 w(t) apple apple apple C DK DK x(t) D K K x(t) + w(t) I apple x(t) x(t) apple B B w(t) 14
15 Output Feedback Control continued The closed-loop system matrix is block triangular It is stable if A-BK and A-LC are stable Since appleẋ(t) x(t) 0 x(t) apple y(t) u(t) we have The estimate error is uncontrollable but it is stabilizable The closed-loop eigenvalues fall into two groups Those due to LSVF and those due to the observer First the state estimate converges and then we stabilize This is referred to as the Separation Theorem of LTI output feedback control The output feedback controller is apple apple apple A BK BK x(t) B 0 A LC x(t) + 0 w(t) apple apple apple C DK DK x(t) D K K x(t) + w(t) I ˆx(t) x(t) 15
16 Output Feedback Controller The separation theorem deals with the behavior of the closed-loop system of the plant and the controller together The output feedback controller itself is Rearranging the terms ˆx(t) [A LC]ˆx(t)+B[ K ˆx(t)+w(t)] + L[y(t)+DKˆx(t) Dw(t)] u(t) [A LC (B LD)K]ˆx(t)+(B LD)w(t)+Ly(t) K ˆx(t)+w(t) ˆx(t) [A LC (B LD)K]ˆx(t)+ L (B LD) apple y(t) w(t) u(t) K ˆx(t)+ 0 I apple y(t) w(t) This system has two inputs y(t) and w(t) There is no particular reason to think that this system would be stable The direct feedthrough D of the plant does not interact with the direct feedthrough of the controller because the gain on y is zero So there is no algebraic loop here and the controller transfer function from y to u is strictly proper 16
17 Discrete-time Output Feedback Control All of the preceding algebra holds for discrete time as it does for continuous time We could easily replace the derivatives by time steps and the results would be identical, including the separation theorem and the controller state-space realization The positioning of the eigenvalues of A-BK and A-LC would need to be strictly inside the unit circle rather than the olhp Discrete-time observers have one critical distinction however It is possible to design them without time delay We then need to ensure that there is no direct feedthrough, D0 x(t + 1) Ax(t)+Bu(t) y(t) Cx(t) ˆ x(t + 1) Aˆ x(t)+bu(t)+l y(t + 1) C{Aˆ x(t)+bu(t)} [A LCA]ˆ x(t)+[b LCB]u(t)+ Ly(t + 1) ˆ x(t + 1) (I LC)Aˆ x(t)+(i LC)Bu(t)+Ly(t + 1) x(t + 1) [(I LC)A] x(t) 17
18 Discrete-Time Output Feedback Control continued Delay-free observer Convergence/stability depends on (I-LC)A eigenvalues This form has advantages when there is an uncontrollable random disturbance affecting the plant state In control systems parlance, it is called a state filter while the version with delay is called a state predictor Delay-free controller is ˆ x(t + 1) (I LC)Aˆ x(t)+(i LC)Bu(t)+Ly(t + 1) x(t + 1) [(I LC)A] x(t) ˆ x(t + 1) (I LC)Aˆ x(t) +(I LC)B[ K ˆ x(t) + w(t)] + Ly(t + 1) u(t) (I LC)(A BK)ˆ x(t)+(i LC)Bw(t)+ Ly(t + 1) K ˆ x(t)+w(t) Note that u(t) depends on y(t) without delay 18
Controllability, 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 informationLMIs for Observability and Observer Design
LMIs for Observability and Observer Design Matthew M. Peet Arizona State University Lecture 06: LMIs for Observability and Observer Design Observability Consider a system with no input: ẋ(t) = Ax(t), x(0)
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 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 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 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 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 informationLinear Systems. Linear systems?!? (Roughly) Systems which obey properties of superposition Input u(t) output
Linear Systems Linear systems?!? (Roughly) Systems which obey properties of superposition Input u(t) output Our interest is in dynamic systems Dynamic system means a system with memory of course including
More informationModern Control Systems
Modern Control Systems Matthew M. Peet Arizona State University Lecture 09: Observability Observability For Static Full-State Feedback, we assume knowledge of the Full-State. In reality, we only have measurements
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 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 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 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 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 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 informationSYSTEMTEORI - KALMAN FILTER VS LQ CONTROL
SYSTEMTEORI - KALMAN FILTER VS LQ CONTROL 1. Optimal regulator with noisy measurement Consider the following system: ẋ = Ax + Bu + w, x(0) = x 0 where w(t) is white noise with Ew(t) = 0, and x 0 is a stochastic
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 informationEL 625 Lecture 10. Pole Placement and Observer Design. ẋ = Ax (1)
EL 625 Lecture 0 EL 625 Lecture 0 Pole Placement and Observer Design Pole Placement Consider the system ẋ Ax () The solution to this system is x(t) e At x(0) (2) If the eigenvalues of A all lie in the
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 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 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 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 informationTopic # Feedback Control Systems
Topic #17 16.31 Feedback Control Systems Deterministic LQR Optimal control and the Riccati equation Weight Selection Fall 2007 16.31 17 1 Linear Quadratic Regulator (LQR) Have seen the solutions to the
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 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 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 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 : MULTIVARIABLE CONTROL SYSTEMS by A.
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Q-Parameterization 1 This lecture introduces the so-called
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 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 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 informationControl Systems (ECE411) Lectures 7 & 8
(ECE411) Lectures 7 & 8, Professor Department of Electrical and Computer Engineering Colorado State University Fall 2016 Signal Flow Graph Examples Example 3: Find y6 y 1 and y5 y 2. Part (a): Input: y
More informationChapter 3. LQ, LQG and Control System Design. Dutch Institute of Systems and Control
Chapter 3 LQ, LQG and Control System H 2 Design Overview LQ optimization state feedback LQG optimization output feedback H 2 optimization non-stochastic version of LQG Application to feedback system design
More information6 OUTPUT FEEDBACK DESIGN
6 OUTPUT FEEDBACK DESIGN When the whole sate vector is not available for feedback, i.e, we can measure only y = Cx. 6.1 Review of observer design Recall from the first class in linear systems that a simple
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 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
ELEC4410 Control Systems Design Lecture 18: State Feedback Tracking and State Estimation Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 18:
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 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 informationControl Systems Design, SC4026. SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft
Control Systems Design, SC426 SC426 Fall 2, dr A Abate, DCSC, TU Delft Lecture 5 Controllable Canonical and Observable Canonical Forms Stabilization by State Feedback State Estimation, Observer Design
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 informationto have roots with negative real parts, the necessary and sufficient conditions are that:
THE UNIVERSITY OF TEXAS AT SAN ANTONIO EE 543 LINEAR SYSTEMS AND CONTROL H O M E W O R K # 7 Sebastian A. Nugroho November 6, 7 Due date of the homework is: Sunday, November 6th @ :59pm.. The following
More informationTopic # Feedback Control Systems
Topic #15 16.31 Feedback Control Systems State-Space Systems Open-loop Estimators Closed-loop Estimators Observer Theory (no noise) Luenberger IEEE TAC Vol 16, No. 6, pp. 596 602, December 1971. Estimation
More informationBALANCING-RELATED MODEL REDUCTION FOR DATA-SPARSE SYSTEMS
BALANCING-RELATED Peter Benner Professur Mathematik in Industrie und Technik Fakultät für Mathematik Technische Universität Chemnitz Computational Methods with Applications Harrachov, 19 25 August 2007
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 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 informationTopic # Feedback Control. State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback
Topic #17 16.31 Feedback Control State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback Back to reality Copyright 21 by Jonathan How. All Rights reserved 1 Fall
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 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 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 informationPole placement control: state space and polynomial approaches Lecture 2
: state space and polynomial approaches Lecture 2 : a state O. Sename 1 1 Gipsa-lab, CNRS-INPG, FRANCE Olivier.Sename@gipsa-lab.fr www.gipsa-lab.fr/ o.sename -based November 21, 2017 Outline : a state
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 informationH 2 Optimal State Feedback Control Synthesis. Raktim Bhattacharya Aerospace Engineering, Texas A&M University
H 2 Optimal State Feedback Control Synthesis Raktim Bhattacharya Aerospace Engineering, Texas A&M University Motivation Motivation w(t) u(t) G K y(t) z(t) w(t) are exogenous signals reference, process
More informationThe norms can also be characterized in terms of Riccati inequalities.
9 Analysis of stability and H norms Consider the causal, linear, time-invariant system ẋ(t = Ax(t + Bu(t y(t = Cx(t Denote the transfer function G(s := C (si A 1 B. Theorem 85 The following statements
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 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 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 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 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 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 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 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 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 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 informationROBUST PASSIVE OBSERVER-BASED CONTROL FOR A CLASS OF SINGULAR SYSTEMS
INTERNATIONAL JOURNAL OF INFORMATON AND SYSTEMS SCIENCES Volume 5 Number 3-4 Pages 480 487 c 2009 Institute for Scientific Computing and Information ROBUST PASSIVE OBSERVER-BASED CONTROL FOR A CLASS OF
More informationTopics in control Tracking and regulation A. Astolfi
Topics in control Tracking and regulation A. Astolfi Contents 1 Introduction 1 2 The full information regulator problem 3 3 The FBI equations 5 4 The error feedback regulator problem 5 5 The internal model
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 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 informationUniversity of California Department of Mechanical Engineering ECE230A/ME243A Linear Systems Fall 1999 (B. Bamieh ) Lecture 3: Simulation/Realization 1
University of alifornia Department of Mechanical Engineering EE/ME Linear Systems Fall 999 ( amieh ) Lecture : Simulation/Realization Given an nthorder statespace description of the form _x(t) f (x(t)
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 informationSynthesis via State Space Methods
Chapter 18 Synthesis via State Space Methods Here, we will give a state space interpretation to many of the results described earlier. In a sense, this will duplicate the earlier work. Our reason for doing
More informationDESIGN OF OBSERVERS FOR SYSTEMS WITH SLOW AND FAST MODES
DESIGN OF OBSERVERS FOR SYSTEMS WITH SLOW AND FAST MODES by HEONJONG YOO A thesis submitted to the Graduate School-New Brunswick Rutgers, The State University of New Jersey In partial fulfillment of the
More informationEquilibrium points: continuous-time systems
Capitolo 0 INTRODUCTION 81 Equilibrium points: continuous-time systems Let us consider the following continuous-time linear system ẋ(t) Ax(t)+Bu(t) y(t) Cx(t)+Du(t) The equilibrium points x 0 of the system
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 informationSystems and Control Theory Lecture Notes. Laura Giarré
Systems and Control Theory Lecture Notes Laura Giarré L. Giarré 2017-2018 Lesson 5: State Space Systems State Dimension Infinite-Dimensional systems State-space model (nonlinear) LTI State Space model
More informationControl System Design
ELEC4410 Control System Design Lecture 19: Feedback from Estimated States and Discrete-Time Control Design Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science
More informationTopic # Feedback Control
Topic #7 16.31 Feedback Control State-Space Systems What are state-space models? Why should we use them? How are they related to the transfer functions used in classical control design and how do we develop
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 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 informationThe Generalized Laplace Transform: Applications to Adaptive Control*
The Transform: Applications to Adaptive * J.M. Davis 1, I.A. Gravagne 2, B.J. Jackson 1, R.J. Marks II 2, A.A. Ramos 1 1 Department of Mathematics 2 Department of Electrical Engineering Baylor University
More informationSpectral factorization and H 2 -model following
Spectral factorization and H 2 -model following Fabio Morbidi Department of Mechanical Engineering, Northwestern University, Evanston, IL, USA MTNS - July 5-9, 2010 F. Morbidi (Northwestern Univ.) MTNS
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 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 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 informationSampling of Linear Systems
Sampling of Linear Systems Real-Time Systems, Lecture 6 Karl-Erik Årzén January 26, 217 Lund University, Department of Automatic Control Lecture 6: Sampling of Linear Systems [IFAC PB Ch. 1, Ch. 2, and
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 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 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 informationTRACKING AND DISTURBANCE REJECTION
TRACKING AND DISTURBANCE REJECTION Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 13 General objective: The output to track a reference
More informationSupplementary chapters
The Essentials of Linear State-Space Systems Supplementary chapters J. Dwight Aplevich This document is copyright 26 2 J. D. Aplevich, and supplements the book The Ussentials of Linear State-Space Systems,
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 informationPrashant Mhaskar, Nael H. El-Farra & Panagiotis D. Christofides. Department of Chemical Engineering University of California, Los Angeles
HYBRID PREDICTIVE OUTPUT FEEDBACK STABILIZATION OF CONSTRAINED LINEAR SYSTEMS Prashant Mhaskar, Nael H. El-Farra & Panagiotis D. Christofides Department of Chemical Engineering University of California,
More informationChapter 3. Tohru Katayama
Subspace Methods for System Identification Chapter 3 Tohru Katayama Subspace Methods Reading Group UofA, Edmonton Barnabás Póczos May 14, 2009 Preliminaries before Linear Dynamical Systems Hidden Markov
More informationMin-Max Output Integral Sliding Mode Control for Multiplant Linear Uncertain Systems
Proceedings of the 27 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July -3, 27 FrC.4 Min-Max Output Integral Sliding Mode Control for Multiplant Linear Uncertain
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 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 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. Time response
Control Systems Time response L. Lanari outline zero-state solution matrix exponential total response (sum of zero-state and zero-input responses) Dirac impulse impulse response change of coordinates (state)
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 informationACM/CMS 107 Linear Analysis & Applications Fall 2016 Assignment 4: Linear ODEs and Control Theory Due: 5th December 2016
ACM/CMS 17 Linear Analysis & Applications Fall 216 Assignment 4: Linear ODEs and Control Theory Due: 5th December 216 Introduction Systems of ordinary differential equations (ODEs) can be used to describe
More information