Observability for deterministic systems and high-gain observers
|
|
- Ann Kennedy
- 6 years ago
- Views:
Transcription
1 Observability for deterministic systems and high-gain observers design. Part 1. March 29, 2011
2 Introduction and problem description Definition of observability Consequences of instantaneous observability ms
3 Systems considered We consider a model described by a differential equation in the form of ẋ = f (x, t), y = h(x, t), with x in R n, y the measured output in R p. For all x 0 in a given open set denoted O of R n, a unique solution starting from x 0 at t = t 0 denoted X (x 0, t 0, t). The time domain in which this solution is in O is an open time interval containing t 0 and is denoted ]σ O,t 0 (x 0 ), σ + O,t 0 (x 0 )[.
4 Some other cases We may consider also: 1. The time invariant case ẋ = f (x), y = h(x). 2. The controlled case (in Part 2) ẋ = f (x, u(t)), y = h(x, u(t)).
5 An induced output mapping For all initial point, we have an output trajectory P O,t0 : O C 0 ([0, σ + t (x 0,O 0)[, R p ) x 0 (t h(x (x 0, t 0, t)))
6 The estimation problem Construct an inverse to this mapping! For all t > 0, we wish to construct a map: ( y[0,t] (s), u [t0,t](s) ) x 0 R n. Note that by uniqueness of solutions, once x 0 is known, the entire trajectory can be found by integration. Consequently, it is equivalent with knowing X (x 0, t 0, t).
7 State observer An observer is an Algorithm! Knowing the existence of a solution is good, but we need to COMPUTE it. Exact estimation for all t is too difficult! We can compute an estimate ˆX (t) such that ˆX (t) X (x 0, t 0, t) goes to zero as long as the trajectory exists and for all initial conditions (x 0, t 0 ). In this lecture we give a particular solution based on an observability property.
8 Definition of observability Consequences of instantaneous observability Introduction and problem description Definition of observability Consequences of instantaneous observability ms
9 Definition of observability Consequences of instantaneous observability Distinguishability definition Definition (Distinguishability) Two initial states x a and x b both in O an open set of R n are said to be indistinguishable in O at t 0 if the output time functions of the solution starting from these points at t = t 0 are the same as long as both solutions remain { in O. In other words, } if we denote σ min = min σ + O,t 0 (x a ), σ + O,t 0 (x b ) then we have h(x (x a, t 0, t)) = h(x (x b, t 0, t)), t [t 0, σ min ). Otherwise, they are said to be distinguishable in O at t 0.
10 Definition of observability Consequences of instantaneous observability Observability definition Definition (Observability) The system is said to be observable in O if all x a and x b in O are distinguishable at t 0 in O for all t 0 in R. When O is a neighborhood of a certain point x 0 in R n then the system is said to be locally observable at x 0. When O = R n, the system is said to be globally observable.
11 Definition of observability Consequences of instantaneous observability Instantaneous observability Definition (Instantaneous observability) The system is locally instantaneously observable at x 0 if there exists O an open neighborhood such that for all open subset I O containing x 0 the system is observable in I. If for all x 0 in R n the system is instantaneously observable at x 0, the system is locally instantaneously observable. If O = R n then the system is globally instantaneously observable. x a and x b in { O with x a x b then } for all t 0 and all t 0 < δ < min σ + t (x 0,O a), σ + t (x 0,O b) ) there exists t 0 < t d < δ such that h(x (x a, t d, t 0 )) h(x (x b, t d, t 0 )).
12 Definition of observability Consequences of instantaneous observability Globally instantaneously observable Globally observable Locally instantaneously observable Locally observable
13 Definition of observability Consequences of instantaneous observability Introduction and problem description Definition of observability Consequences of instantaneous observability ms
14 Definition of observability Consequences of instantaneous observability Assume the system is analytic and time invariant. The time function y(x, t) = h(x (x, t)), is an analytic function of the time. Consequently, for all x 0 there exists an open set O 0 such that for all t in (σ O 0 (x 0 ), σ + O 0 (x 0 )) and all x in O 0, y(x, t) = h 1 (x) + + j=1 h j (x) tj j! with, h 1 (x) = h(x), h j+1 (x) = j y (x, 0). tj
15 Definition of observability Consequences of instantaneous observability Let x a and x b two initial points in O 0 giving rise to the same output function in O 0. Then we have for small t, Hence we have, h 1 (x a ) h 1 (x b ) + + j=1 [h j (x a ) h j (x b )] tj j! = 0. h j (x a ) h j (x b ) = 0, j. We see that to check observability, we need to check injectivity of a mapping denoted: x H (x) = (h 1 (x), h 2 (x),..., ) The state is living in an n-dimensional space! we may have some hopes that injectivity of the map may be obtained without checking to all coefficients.
16 Definition of observability Consequences of instantaneous observability Successive time derivative of the output Given an integer k, we introduce the mapping H k : R n R m which to a point x in R n gives the output and its first k 1 successive time derivatives H k (x) = h 1 (x). h k (x), h 1 (x) = h(x), h j (x) = L f h j 1 (x). Definition (Differential observability of order k) The system satisfies the differential observability property of order k in O if the mapping H k is injective in O.
17 Definition of observability Consequences of instantaneous observability Rank condition Lemma (Sufficient condition for instantaneous observability) Given x 0 in R n, if there exists k > 0 such that the rank of the matrix H k x (x 0) is n (the dimension of x) then the system is locally differentially observable of order k and consequently is locally instantaneously observable at x 0.
18 Definition of observability Consequences of instantaneous observability From local to global Lemma (Global differential observability property) Assume the functions f and h are analytic and p = 1 (only one output). Let O be a connected open subset of R n such that 1. The system is analytic and observable in O; 2. The matrix Hn x (x) is full rank for all x in O. Then the function H n is injective in O and is instantaneously observable in O.
19 Definition of observability Consequences of instantaneous observability Necessity Theorem (Necessity of the rank condition almost everywhere) If the functions f and h are C and if the system is instantaneously observable in an open set O then the matrix Hn x (x 0) has its rank equal to n almost everywhere 1 in O. From this analysis: As we have seen the existence of a positive integer k such that the mapping H k is injective is a sufficient and an almost necessary condition for the instantaneous observability! 1 In an open and dense subset of O.
20 Definition of observability Consequences of instantaneous observability In Conclusion It gives some hint on how to solve the estimation problem: y(t) Estimate the k 1 first time derivatives of the output at each time: Ĥ k (x) = ŷ(t), ẏ(t) ÿ(t)..., y k 1 (t) Apply τ such that τ(h k (x)) = x ) ˆx(t) = τ (ŷ(t), ẏ(t) ÿ(t)..., y k 1 (t)
21 Definition of observability Consequences of instantaneous observability Some comment on this approach However each step of this estimation strategy give rise to some difficulties: 1. Computing the left inverse of a given map H k may be difficult and most of the time rely on optimization procedure which may give some local minima. Indeed, a general expression for a left inverse can simply be given as ˆx = τ(ξ) = Argmin x O ξ H k (x) 2. However, in some example, this approach can give explicite solution. 2. It is very hard to compute the successive time derivatives of the output (especially when there are disturbances in the output).
22 Introduction and problem description Definition of observability Consequences of instantaneous observability ms
23 The framework 1. Only one output. 2. We assume also that there exists an integer m 0 such that the mapping H m is an injective function from an open subset O of R n toward H m (O).
24 Model of the output derivatives Along the solution of the system we have, { { H m (x) = A H m (x) + BL f h m (x) with A = , B =. 0 1.
25 Model of the output derivatives Injectivity in x of the mapping H m means the existence of a a function ϕ m : H m (O) R such that ϕ m (H m (x)) = L f h m (x), x O. Assume we are able to continuously extend ϕ m outside H m (O). ξ = H m (x) is an invariant manifold of the system ẋ = f (x), ξ = Aξ + Bϕm (ξ), Also, ξ 1 = h(x) = y.
26 Canonical observable systems We can focus on estimating the state of the system based on the knowledge of ξ 1. ξ = Aξ + Bϕ m (ξ), y = ξ 1, Given ξ in R m, we denote the solutions of this system initiated from ξ at time t = 0 by Ξ(ξ, t).
27 Introduction and problem description Definition of observability Consequences of instantaneous observability ms
28 Theorem (High-Observer for OCF systems ) Consider system in canonical observability form. Assume that ϕ m (ξ a ) ϕ m (ξ b ) c L ξ a ξ b (ξ a, ξ b ) R m. Then there exists a vector K in( R m and two positive) real number L min > 1 and c 0 such that the solutions Ξ(ξ, t), ˆΞ((ξ, ˆξ), t) of the system { ξ = Aξ + Bϕm (ξ), ˆξ = Aˆξ + Bϕ m (ˆξ) + L(L)K(ˆξ 1 ξ 1 ). where L(L) = Diag(L,..., L m ) are complete in positive time and satisfy for all (ξ 0, ˆξ 0 ) in R 2m and L > L min, Ξ(ξ, t) ˆΞ((ξ, ˆξ), t) 2 c 0 exp( (L L min )(t))l 2m 2 ξ ˆξ 2, t 0.
29 Exact estimation but only asymptotically. However, if ξ is in a known bounded set, then ɛ and t e > 0, we can select L such that Ξ(ξ, t) ˆΞ((ξ, ˆξ), t) ɛ, t > t e. If we know a bounded set in which the trajectory Ξ(ξ, t) remains, it is always possible to modify the model outside this set in order to guarantee a global Lipschitz property. as L increases, the bound on the estimation we get around t 0 goes to infinity. This is the well known picking phenomena.
30 Introduction and problem description Definition of observability Consequences of instantaneous observability ms
31 The system is : ξ = A ξ + Bϕ m (ξ) with A = Which can be rewritten : , B = ξ = A ξ }{{} Chain of integrator part + B ϕ(ξ) }{{} Nonlinearities = Disturbances The idea of the high-gain design can be decomposed into two steps: 1. In a first step we synthesize a robust observer for a linear system. 2. Amplify the convergence and robustness to deal with ϕ m (ξ).
32 Part1: Design of an observer for the linear part Consider ξ = Aξ + v, y = Cξ + w, with C = (1, 0..., 0) and where v and w are disturbances. The couple (A, C) is observable. There exists K such that (A + KC) P + P(A + KC) I d.
33 Part1: Design of an observer for the linear part Lemma (Robust observer for the linear system) There exists positive real numbers c 1, c 2, c 3, c 4, c 5 and c 6 such that for all ξ and ˆξ, if we denote e(t) = Ξ(t) ˆΞ(t) where (Ξ(t), ˆΞ(t)) is a solution of the system ξ = Aξ, ˆξ = Aˆξ + K(C ˆξ Cx) the following two inequalities are satisfied: and { { e(t) Pe(t) c 1 e (t)pe(t) + c 2 w(t) 2 + c 3 v(t) 2 e(t) 2 c 4 exp( c 1 t) e(0) + 1 c 1 { max c5 w(s) 2 + c 6 v(s) 2} 0 s t
34 Part2: Amplification of the robustness and convergence Given L we introduce the matrix L(L) defined as L(L) = Diag(L,..., L m ) With this matrix the idea is to modify the observer as follows ˆξ = Aˆξ + L(L)K(C ˆξ y),
35 Lemma (High-gain Amplification) Assume L > 1, for all ξ and ˆξ both in R m, the time function e(t) = Ξ(t) ˆΞ(t) where (Ξ(t), ˆΞ(t)) is a solution of the system ξ = Aξ, ˆξ = Aˆξ + L(L)K(C ˆξ Cx) satisfies the two inequalities, { { ε (t)pε(t) (L c 1 ) ε (t)pε(t) + c 2 L(L) 1 v(t) 2 + c 3 w(t) 2, where ε = L(L) 1 e and which gives, e(t) 2 c 4 exp( (L c 1 )t)l 2m 2 e(0) 2 + L 2m L c 1 [ max c4 w(s) 2 + c 6 L(L) 1 v(s) 2] s [0,t]
36 Increasing L makes the error going faster toward zero. This inequality highlights the fact that model uncertainties and measurement noises act in a very different manner on the quality of the estimate. 1. About the measurement noises w : As L increase, the term which multiplies w is in L 2m 1 which goes to infinity. high-gain observer behaves very badly with respect to measurement noises. 2. About the model uncertainties v : In this case, is we assume the disturbance acts only on the last component. Then it yields that the term which multiplies v is in 1 L which goes to zero.
37 Introduction and problem description Definition of observability Consequences of instantaneous observability ms
38 It is possible to asymptotically estimates the successive time derivatives of the measured output if 1. The system is differentially observable of order m in O. Hence, the evolution of the successive time derivatives of the output can be described by a system in canonical normal form. 2. The model of the successive output derivative satisfies a global Lipschitz property.
39 About differential observability: This injectivity property says that the map H m is left invertible. With the high gain observer we have H m (X (x, t)) ˆΞ(ξ, ˆξ, t) 0 With injectivity there exists τ such that τ(h m (x)) = x. To get we need τ uniformly continuous. X (x, t) τ(ˆξ(ξ, ˆξ, t)) 0
40 About the global Lipschitz property: The function ϕ m is only precisely defined on the set H m (O). ϕ m (ξ) = L f h m (τ(ξ))), ξ H m (O). Outside the set H m (O) this function has to be synthesized in order to guarantee the global Lipschitz property. If we have: L m f h(x a ) L m f h(x b ) c L H m (x a ) H m (x b ), (x a, x b ) cl(o) 2. then ϕ m can be computed as a Lipschitz extension of the function ϕ m (τ(ξ)).
41 back to the system ẋ = f (x), y = h(x) Theorem (High-gain estimation) If there exist an open set O and a positive real number m such that 1. The mapping H m is uniformly injective in O. In other words, there exists a class K function ρ such that we have the inequality: x a x b 2 ρ( H m (x a ) H m (x b ) 2 ), (x a, x b ) cl(o) 2 2. There exists a positive real number c L such that for all x a and x b in O, L m f h(x a ) L m f h(x b ) c L H m (x a ) H m (x b ), (x a, x b ) cl(o) 2.
42 Then there exist a mapping τ, a function ϕ m, a matrix K in R m, two positive real number L min > 1 and c 0 and a class K function ρ such that the solutions of the system with the observer ẋ = f (x), y = h(x), ˆx = τ(ˆξ), ˆξ = Aˆξ + Bϕ m (ˆξ) + L(L)K(ˆξ 1 y) where L(L) = Diag(L,..., L m ), L > L min satisfy for all (x, ˆξ) in O R m, L > L min, and 0 < t < σ + O (x) X (x, t) τ(ˆξ((x, ˆξ), t)) 2 ρ ( c 0 exp( (L L min )t)l 2m 2 H m (x) ˆξ 2).
43 To construct the observer we need to construct τ the left inverse. Giving an explicit formulation to this left inverse may not be an easy task. In some specific case, this crucial step may be avoided. Indeed, when m = n and if the mapping H n defines a diffeomorphism from the open set O toward the set H n (O) another representation of the high-gain observer is simply given as ( ) 1 Hn ˆx = f (ˆx) + x (ˆx) L(L)K(h(ˆx) y), in which τ don t appear anymore.
44 To get asymptotic convergence of a solution initiated from x, we need σ + O (x) = +. This is obtained for instance if O is invariant. Due to the presence of the class K function ρ it is not possible to say that the observer converges exponentially toward the state of the system. In order to obtain this further property on the estimate, we need the matrix Hm x (x) to be full rank in cl(o) and O to be bounded.
45 The high gain observer recipe Given a system ẋ = f (x), y = h(x) 1. Find an order m and a BOUNDED open set O such that H m is injective in its closure. 2. Construct and Extend a left inverse τ. 3. Construct and Extend the function ϕ m.
High gain observer for a class of implicit systems
High gain observer for a class of implicit systems Hassan HAMMOURI Laboratoire d Automatique et de Génie des Procédés, UCB-Lyon 1, 43 bd du 11 Novembre 1918, 69622 Villeurbanne, France E-mail: hammouri@lagepuniv-lyon1fr
More informationNonlinear Control Systems
Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 3. Fundamental properties IST-DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs2012/ 2012 1 Example Consider the system ẋ = f
More informationHigh-Gain Observers in Nonlinear Feedback Control
High-Gain Observers in Nonlinear Feedback Control Lecture # 1 Introduction & Stabilization High-Gain ObserversinNonlinear Feedback ControlLecture # 1Introduction & Stabilization p. 1/4 Brief History Linear
More informationDETERMINISTIC AND STOCHASTIC SELECTION DYNAMICS
DETERMINISTIC AND STOCHASTIC SELECTION DYNAMICS Jörgen Weibull March 23, 2010 1 The multi-population replicator dynamic Domain of analysis: finite games in normal form, G =(N, S, π), with mixed-strategy
More informationFeedback Linearization
Feedback Linearization Peter Al Hokayem and Eduardo Gallestey May 14, 2015 1 Introduction Consider a class o single-input-single-output (SISO) nonlinear systems o the orm ẋ = (x) + g(x)u (1) y = h(x) (2)
More informationConvergence Rate of Nonlinear Switched Systems
Convergence Rate of Nonlinear Switched Systems Philippe JOUAN and Saïd NACIRI arxiv:1511.01737v1 [math.oc] 5 Nov 2015 January 23, 2018 Abstract This paper is concerned with the convergence rate of the
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 informationMinimum-Phase Property of Nonlinear Systems in Terms of a Dissipation Inequality
Minimum-Phase Property of Nonlinear Systems in Terms of a Dissipation Inequality Christian Ebenbauer Institute for Systems Theory in Engineering, University of Stuttgart, 70550 Stuttgart, Germany ce@ist.uni-stuttgart.de
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 informationThe Convergence of the Minimum Energy Estimator
The Convergence of the Minimum Energy Estimator Arthur J. Krener Department of Mathematics, University of California, Davis, CA 95616-8633, USA, ajkrener@ucdavis.edu Summary. We show that under suitable
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 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 informationRobust Stabilization of Non-Minimum Phase Nonlinear Systems Using Extended High Gain Observers
28 American Control Conference Westin Seattle Hotel, Seattle, Washington, USA June 11-13, 28 WeC15.1 Robust Stabilization of Non-Minimum Phase Nonlinear Systems Using Extended High Gain Observers Shahid
More informationHigh-Gain Observers in Nonlinear Feedback Control. Lecture # 2 Separation Principle
High-Gain Observers in Nonlinear Feedback Control Lecture # 2 Separation Principle High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 1/4 The Class of Systems ẋ = Ax + Bφ(x,
More information1 The Observability Canonical Form
NONLINEAR OBSERVERS AND SEPARATION PRINCIPLE 1 The Observability Canonical Form In this Chapter we discuss the design of observers for nonlinear systems modelled by equations of the form ẋ = f(x, u) (1)
More informationNonlinear Control Lecture # 1 Introduction. Nonlinear Control
Nonlinear Control Lecture # 1 Introduction Nonlinear State Model ẋ 1 = f 1 (t,x 1,...,x n,u 1,...,u m ) ẋ 2 = f 2 (t,x 1,...,x n,u 1,...,u m ).. ẋ n = f n (t,x 1,...,x n,u 1,...,u m ) ẋ i denotes the derivative
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 informationChap. 1. Some Differential Geometric Tools
Chap. 1. Some Differential Geometric Tools 1. Manifold, Diffeomorphism 1.1. The Implicit Function Theorem ϕ : U R n R n p (0 p < n), of class C k (k 1) x 0 U such that ϕ(x 0 ) = 0 rank Dϕ(x) = n p x U
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 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 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 informationLarge Deviations for Small-Noise Stochastic Differential Equations
Chapter 22 Large Deviations for Small-Noise Stochastic Differential Equations This lecture is at once the end of our main consideration of diffusions and stochastic calculus, and a first taste of large
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 informationLarge Deviations for Small-Noise Stochastic Differential Equations
Chapter 21 Large Deviations for Small-Noise Stochastic Differential Equations This lecture is at once the end of our main consideration of diffusions and stochastic calculus, and a first taste of large
More informationNonlinear Control Systems
Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 7. Feedback Linearization IST-DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs1/ 1 1 Feedback Linearization Given a nonlinear
More informationLecture 21 Representations of Martingales
Lecture 21: Representations of Martingales 1 of 11 Course: Theory of Probability II Term: Spring 215 Instructor: Gordan Zitkovic Lecture 21 Representations of Martingales Right-continuous inverses Let
More informationNonlinear Dynamical Systems Eighth Class
Nonlinear Dynamical Systems Eighth Class Alexandre Nolasco de Carvalho September 19, 2017 Now we exhibit a Morse decomposition for a dynamically gradient semigroup and use it to prove that a dynamically
More informationHigh-Gain Observers in Nonlinear Feedback Control. Lecture # 3 Regulation
High-Gain Observers in Nonlinear Feedback Control Lecture # 3 Regulation High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 1/5 Internal Model Principle d r Servo- Stabilizing u y
More informationStability theory is a fundamental topic in mathematics and engineering, that include every
Stability Theory Stability theory is a fundamental topic in mathematics and engineering, that include every branches of control theory. For a control system, the least requirement is that the system is
More informationTarget Localization and Circumnavigation Using Bearing Measurements in 2D
Target Localization and Circumnavigation Using Bearing Measurements in D Mohammad Deghat, Iman Shames, Brian D. O. Anderson and Changbin Yu Abstract This paper considers the problem of localization and
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 informationConverse Lyapunov theorem and Input-to-State Stability
Converse Lyapunov theorem and Input-to-State Stability April 6, 2014 1 Converse Lyapunov theorem In the previous lecture, we have discussed few examples of nonlinear control systems and stability concepts
More informationLimits at Infinity. Horizontal Asymptotes. Definition (Limits at Infinity) Horizontal Asymptotes
Limits at Infinity If a function f has a domain that is unbounded, that is, one of the endpoints of its domain is ±, we can determine the long term behavior of the function using a it at infinity. Definition
More informationOften, in this class, we will analyze a closed-loop feedback control system, and end up with an equation of the form
ME 32, Spring 25, UC Berkeley, A. Packard 55 7 Review of SLODEs Throughout this section, if y denotes a function (of time, say), then y [k or y (k) denotes the k th derivative of the function y, y [k =
More informationOn the Exponential Stability of Moving Horizon Observer for Globally N-Detectable Nonlinear Systems
Asian Journal of Control, Vol 00, No 0, pp 1 6, Month 008 Published online in Wiley InterScience (wwwintersciencewileycom) DOI: 10100/asjc0000 - On the Exponential Stability of Moving Horizon Observer
More informationOutput Regulation of Uncertain Nonlinear Systems with Nonlinear Exosystems
Output Regulation of Uncertain Nonlinear Systems with Nonlinear Exosystems Zhengtao Ding Manchester School of Engineering, University of Manchester Oxford Road, Manchester M3 9PL, United Kingdom zhengtaoding@manacuk
More informationANALYSIS OF THE USE OF LOW-PASS FILTERS WITH HIGH-GAIN OBSERVERS. Stephanie Priess
ANALYSIS OF THE USE OF LOW-PASS FILTERS WITH HIGH-GAIN OBSERVERS By Stephanie Priess A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Electrical
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 informationOn Characterizations of Input-to-State Stability with Respect to Compact Sets
On Characterizations of Input-to-State Stability with Respect to Compact Sets Eduardo D. Sontag and Yuan Wang Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA Department of Mathematics,
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 informationON OPTIMAL ESTIMATION PROBLEMS FOR NONLINEAR SYSTEMS AND THEIR APPROXIMATE SOLUTION. A. Alessandri C. Cervellera A.F. Grassia M.
ON OPTIMAL ESTIMATION PROBLEMS FOR NONLINEAR SYSTEMS AND THEIR APPROXIMATE SOLUTION A Alessandri C Cervellera AF Grassia M Sanguineti ISSIA-CNR National Research Council of Italy Via De Marini 6, 16149
More informationNonlinear and robust MPC with applications in robotics
Nonlinear and robust MPC with applications in robotics Boris Houska, Mario Villanueva, Benoît Chachuat ShanghaiTech, Texas A&M, Imperial College London 1 Overview Introduction to Robust MPC Min-Max Differential
More informationL 2 -induced Gains of Switched Systems and Classes of Switching Signals
L 2 -induced Gains of Switched Systems and Classes of Switching Signals Kenji Hirata and João P. Hespanha Abstract This paper addresses the L 2-induced gain analysis for switched linear systems. We exploit
More informationIntelligent Control. Module I- Neural Networks Lecture 7 Adaptive Learning Rate. Laxmidhar Behera
Intelligent Control Module I- Neural Networks Lecture 7 Adaptive Learning Rate Laxmidhar Behera Department of Electrical Engineering Indian Institute of Technology, Kanpur Recurrent Networks p.1/40 Subjects
More informationRobust Stability. Robust stability against time-invariant and time-varying uncertainties. Parameter dependent Lyapunov functions
Robust Stability Robust stability against time-invariant and time-varying uncertainties Parameter dependent Lyapunov functions Semi-infinite LMI problems From nominal to robust performance 1/24 Time-Invariant
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 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 informationTakens embedding theorem for infinite-dimensional dynamical systems
Takens embedding theorem for infinite-dimensional dynamical systems James C. Robinson Mathematics Institute, University of Warwick, Coventry, CV4 7AL, U.K. E-mail: jcr@maths.warwick.ac.uk Abstract. Takens
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 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 informationNonlinear Systems and Control Lecture # 19 Perturbed Systems & Input-to-State Stability
p. 1/1 Nonlinear Systems and Control Lecture # 19 Perturbed Systems & Input-to-State Stability p. 2/1 Perturbed Systems: Nonvanishing Perturbation Nominal System: Perturbed System: ẋ = f(x), f(0) = 0 ẋ
More informationLecture 2: Controllability of nonlinear systems
DISC Systems and Control Theory of Nonlinear Systems 1 Lecture 2: Controllability of nonlinear systems Nonlinear Dynamical Control Systems, Chapter 3 See www.math.rug.nl/ arjan (under teaching) for info
More informationMATH 409 Advanced Calculus I Lecture 12: Uniform continuity. Exponential functions.
MATH 409 Advanced Calculus I Lecture 12: Uniform continuity. Exponential functions. Uniform continuity Definition. A function f : E R defined on a set E R is called uniformly continuous on E if for every
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 informationAn introduction to Mathematical Theory of Control
An introduction to Mathematical Theory of Control Vasile Staicu University of Aveiro UNICA, May 2018 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018
More informationCLASSIFICATIONS OF THE FLOWS OF LINEAR ODE
CLASSIFICATIONS OF THE FLOWS OF LINEAR ODE PETER ROBICHEAUX Abstract. The goal of this paper is to examine characterizations of linear differential equations. We define the flow of an equation and examine
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 information21 Linear State-Space Representations
ME 132, Spring 25, UC Berkeley, A Packard 187 21 Linear State-Space Representations First, let s describe the most general type of dynamic system that we will consider/encounter in this class Systems may
More information0.1 Complex Analogues 1
0.1 Complex Analogues 1 Abstract In complex geometry Kodaira s theorem tells us that on a Kähler manifold sufficiently high powers of positive line bundles admit global holomorphic sections. Donaldson
More information2tdt 1 y = t2 + C y = which implies C = 1 and the solution is y = 1
Lectures - Week 11 General First Order ODEs & Numerical Methods for IVPs In general, nonlinear problems are much more difficult to solve than linear ones. Unfortunately many phenomena exhibit nonlinear
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 information4 Countability axioms
4 COUNTABILITY AXIOMS 4 Countability axioms Definition 4.1. Let X be a topological space X is said to be first countable if for any x X, there is a countable basis for the neighborhoods of x. X is said
More informationPattern generation, topology, and non-holonomic systems
Systems & Control Letters ( www.elsevier.com/locate/sysconle Pattern generation, topology, and non-holonomic systems Abdol-Reza Mansouri Division of Engineering and Applied Sciences, Harvard University,
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 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 informationON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS
Bendikov, A. and Saloff-Coste, L. Osaka J. Math. 4 (5), 677 7 ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS ALEXANDER BENDIKOV and LAURENT SALOFF-COSTE (Received March 4, 4)
More informationState estimation of uncertain multiple model with unknown inputs
State estimation of uncertain multiple model with unknown inputs Abdelkader Akhenak, Mohammed Chadli, Didier Maquin and José Ragot Centre de Recherche en Automatique de Nancy, CNRS UMR 79 Institut National
More informationAUTOMATIC CONTROL. Andrea M. Zanchettin, PhD Spring Semester, Introduction to Automatic Control & Linear systems (time domain)
1 AUTOMATIC CONTROL Andrea M. Zanchettin, PhD Spring Semester, 2018 Introduction to Automatic Control & Linear systems (time domain) 2 What is automatic control? From Wikipedia Control theory is an interdisciplinary
More informationSupplemental Material for KERNEL-BASED INFERENCE IN TIME-VARYING COEFFICIENT COINTEGRATING REGRESSION. September 2017
Supplemental Material for KERNEL-BASED INFERENCE IN TIME-VARYING COEFFICIENT COINTEGRATING REGRESSION By Degui Li, Peter C. B. Phillips, and Jiti Gao September 017 COWLES FOUNDATION DISCUSSION PAPER NO.
More informationPutzer s Algorithm. Norman Lebovitz. September 8, 2016
Putzer s Algorithm Norman Lebovitz September 8, 2016 1 Putzer s algorithm The differential equation dx = Ax, (1) dt where A is an n n matrix of constants, possesses the fundamental matrix solution exp(at),
More informationEntrance Exam, Real Analysis September 1, 2017 Solve exactly 6 out of the 8 problems
September, 27 Solve exactly 6 out of the 8 problems. Prove by denition (in ɛ δ language) that f(x) = + x 2 is uniformly continuous in (, ). Is f(x) uniformly continuous in (, )? Prove your conclusion.
More informationI. D. Landau, A. Karimi: A Course on Adaptive Control Adaptive Control. Part 9: Adaptive Control with Multiple Models and Switching
I. D. Landau, A. Karimi: A Course on Adaptive Control - 5 1 Adaptive Control Part 9: Adaptive Control with Multiple Models and Switching I. D. Landau, A. Karimi: A Course on Adaptive Control - 5 2 Outline
More informationObserver design for a general class of triangular systems
1st International Symposium on Mathematical Theory of Networks and Systems July 7-11, 014. Observer design for a general class of triangular systems Dimitris Boskos 1 John Tsinias Abstract The paper deals
More informationTopics in Harmonic Analysis Lecture 6: Pseudodifferential calculus and almost orthogonality
Topics in Harmonic Analysis Lecture 6: Pseudodifferential calculus and almost orthogonality Po-Lam Yung The Chinese University of Hong Kong Introduction While multiplier operators are very useful in studying
More informationThe ϵ-capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems
IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 11, Issue 3 Ver. IV (May - Jun. 2015), PP 52-62 www.iosrjournals.org The ϵ-capacity of a gain matrix and tolerable disturbances:
More informationHadamard and Perron JWR. October 18, On page 23 of his famous monograph [2], D. V. Anosov writes
Hadamard and Perron JWR October 18, 1999 On page 23 of his famous monograph [2], D. V. Anosov writes Every five years or so, if not more often, someone discovers the theorem of Hadamard and Perron proving
More informationThe Liapunov Method for Determining Stability (DRAFT)
44 The Liapunov Method for Determining Stability (DRAFT) 44.1 The Liapunov Method, Naively Developed In the last chapter, we discussed describing trajectories of a 2 2 autonomous system x = F(x) as level
More informationPenalty and Barrier Methods General classical constrained minimization problem minimize f(x) subject to g(x) 0 h(x) =0 Penalty methods are motivated by the desire to use unconstrained optimization techniques
More informationGlobal Analysis of Piecewise Linear Systems Using Impact Maps and Quadratic Surface Lyapunov Functions
Global Analysis of Piecewise Linear Systems Using Impact Maps and Quadratic Surface Lyapunov Functions Jorge M. Gonçalves, Alexandre Megretski, Munther A. Dahleh Department of EECS, Room 35-41 MIT, Cambridge,
More informationLecture 3 (Limits and Derivatives)
Lecture 3 (Limits and Derivatives) Continuity In the previous lecture we saw that very often the limit of a function as is just. When this is the case we say that is continuous at a. Definition: A function
More informationThe theory of manifolds Lecture 2
The theory of manifolds Lecture 2 Let X be a subset of R N, Y a subset of R n and f : X Y a continuous map. We recall Definition 1. f is a C map if for every p X, there exists a neighborhood, U p, of p
More informationLecture 1. Stochastic Optimization: Introduction. January 8, 2018
Lecture 1 Stochastic Optimization: Introduction January 8, 2018 Optimization Concerned with mininmization/maximization of mathematical functions Often subject to constraints Euler (1707-1783): Nothing
More informationOBSERVATION AND IDENTIFICATION TOOLS FOR NONLINEAR SYSTEMS. APPLICATION TO A FLUID CATALYTIC CRACKER.
OBSERVATION AND IDENTIFICATION TOOLS FOR NONLINEAR SYSTEMS. APPLICATION TO A FLUID CATALYTIC CRACKER. ERIC BUSVELLE, JEAN-PAUL GAUTHIER Abstract. In this paper, we recall general methodologies we developed
More informationDesign of Observer-based Adaptive Controller for Nonlinear Systems with Unmodeled Dynamics and Actuator Dead-zone
International Journal of Automation and Computing 8), May, -8 DOI:.7/s633--574-4 Design of Observer-based Adaptive Controller for Nonlinear Systems with Unmodeled Dynamics and Actuator Dead-zone Xue-Li
More informationA framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models
1 A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models D.Nešić and A.R.Teel Abstract A unified framework for design of stabilizing controllers
More informationGlobal Stability and Asymptotic Gain Imply Input-to-State Stability for State-Dependent Switched Systems
2018 IEEE Conference on Decision and Control (CDC) Miami Beach, FL, USA, Dec. 17-19, 2018 Global Stability and Asymptotic Gain Imply Input-to-State Stability for State-Dependent Switched Systems Shenyu
More informationNonlinear Dynamical Systems Ninth Class
Nonlinear Dynamical Systems Ninth Class Alexandre Nolasco de Carvalho September 21, 2017 Lemma Let {T (t) : t 0} be a dynamically gradient semigroup in a metric space X, with a global attractor A and a
More information1 Relative degree and local normal forms
THE ZERO DYNAMICS OF A NONLINEAR SYSTEM 1 Relative degree and local normal orms The purpose o this Section is to show how single-input single-output nonlinear systems can be locally given, by means o a
More informationFirst-order optimality conditions for mathematical programs with second-order cone complementarity constraints
First-order optimality conditions for mathematical programs with second-order cone complementarity constraints Jane J. Ye Jinchuan Zhou Abstract In this paper we consider a mathematical program with second-order
More informationRobust Output Feedback Stabilization of a Class of Nonminimum Phase Nonlinear Systems
Proceedings of the 26 American Control Conference Minneapolis, Minnesota, USA, June 14-16, 26 FrB3.2 Robust Output Feedback Stabilization of a Class of Nonminimum Phase Nonlinear Systems Bo Xie and Bin
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 informationQuasi-ISS Reduced-Order Observers and Quantized Output Feedback
Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009 FrA11.5 Quasi-ISS Reduced-Order Observers and Quantized Output Feedback
More informationLecture 15: Expanders
CS 710: Complexity Theory 10/7/011 Lecture 15: Expanders Instructor: Dieter van Melkebeek Scribe: Li-Hsiang Kuo In the last lecture we introduced randomized computation in terms of machines that have access
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 informationObserver Design for a Class of Takagi-Sugeno Descriptor Systems with Lipschitz Constraints
Applied Mathematical Sciences, Vol. 10, 2016, no. 43, 2105-2120 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.64142 Observer Design for a Class of Takagi-Sugeno Descriptor Systems with
More informationA. LEVANT School of Mathematical Sciences, Tel-Aviv University, Israel,
Chapter 1 INTRODUCTION TO HIGH-ORDER SLIDING MODES A. LEVANT School of Mathematical Sciences, Tel-Aviv University, Israel, 2002-2003 1.1 Introduction One of the most important control problems is control
More informationAPPPHYS217 Tuesday 25 May 2010
APPPHYS7 Tuesday 5 May Our aim today is to take a brief tour of some topics in nonlinear dynamics. Some good references include: [Perko] Lawrence Perko Differential Equations and Dynamical Systems (Springer-Verlag
More informationh(x) lim H(x) = lim Since h is nondecreasing then h(x) 0 for all x, and if h is discontinuous at a point x then H(x) > 0. Denote
Real Variables, Fall 4 Problem set 4 Solution suggestions Exercise. Let f be of bounded variation on [a, b]. Show that for each c (a, b), lim x c f(x) and lim x c f(x) exist. Prove that a monotone function
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 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 information