Existence of an observation window of finite width for continuous-time autonomous nonlinear systems
aa r X i v : . [ m a t h . O C ] A p r Existence of an observation window of finitewidth for continuous-time autonomous nonlinearsystems ∗ Shigeru Hanba † April 19, 2018
Abstract
In this note, the relationship between notions of observability forcontinuous-time nonlinear system related to distinguishability, observabil-ity rank condition and K-function has been investigated. It is proved thatan autonomous nonlinear system that is observable in both distinguisha-bility and rank condition sense permits an observation window of finitewidth, and it is possible to construct a K-function related to observabilityfor such system.
The state estimation problem is one of the most fundamental problems in controlsystem theory. The intrinsic property of the system that makes the state esti-mation problem feasible is observability, and this property has been extensivelystudied in past decades.For linear systems, the notion of observability is firmly established[12]. Con-trary, for nonlinear systems, several non-equivalent definitions of observabilityhave been proposed, and by using these definitions, the state estimation prob-lem of nonlinear systems have been extensively studied [1, 3, 4, 6, 9, 10, 13].However, the relations between several different notions of observability havenot been fully understood (although there are several established facts[3, 6, 9].) ∗ This work was supported by the Japan Society for the Promotion of Science un-der Grant-in-Aid for Scientific Research (C) 23560535. This document is the ac-cepted version of the manuscript published in Automatica in Volume 75 with DOI:https://doi.org/10.1016/j.automatica.2016.08.005 and is a replacement of the prepara-tory version given in arXiv, which the author uploaded at the time instant of itssubmission. If possible, please use the official version instead of this file. c (cid:13) † Department of Electrical and Electronics Engineering, University of the Ryukyus, 1Senbaru Nishihara, Nakagami-gun, Okinawa 903-0213, Japan; email: [email protected] K -function related toobservability[8]. This note is an attempt to establish corresponding results forcontinuous-time systems.The scope of this note is limited to autonomous nonlinear systems, and wedeal with three typical definitions of observability for nonlinear systems whichare related to distinguishability, rank condition and K -function, respectively(precise definitions are given later.) Roughly speaking, we prove that distin-guishability together with the observability rank condition implies that thereis a ‘observation window’ (the sequence of past output as a function of time)of finite width which determines the initial state uniquely, and it is possible toconstruct a K -function related to observability.A preliminary version of this manuscript is available in arXiv[14]. In this note, we consider an autonomous nonlinear system of the form˙ x = f ( x ) ,y = h ( x ) , (1)where x ∈ R n is the state and y ∈ R p is the output. The functions f ( x ) and h ( x ) are assumed to be of compatible dimensions, and smooth up to requiredorder. The solution of (1) is assumed to be unique, and the solution initializedat t = 0 by x is denoted by ϕ ( t, , x ), which is assumed to be a continuousfunction of x . The set of permissible initial conditions of (1) is assumed to becompact, and is denoted by Ω. Note that the state space itself is not necessarilycompact.Next, we introduce the definitions of observability considered in this note.Unfortunately, there is no general agreement on how to name these properties.Hence, we temporally call them ‘D-observability’, ’R-observability’, and ‘K-observability’ (to be defined below) for brevity. Definition 1 [3] A pair of initial states ( x , x ) of (1) with x = x is said tobe an indistinguishable pair if ∀ t ≥ , h ( ϕ ( t, , x )) = h ( ϕ ( t, , x )) . Definition 2 [3] The system (1) is said to be D-observable (with respect to Ω )if there is no indistinguishable pair in the set Ω . Definition 3 [3] The system (1) is said to be R-observable (with respect to Ω )if ∃ N > , the Jacobian of the map H ( x ) = ( h ( x ) , L f h ( x ) , . . . , L N − f h ( x )) is offull rank on Ω , where L f h = ∂h∂x f , and L kf h = L f (cid:16) L k − f h (cid:17) . efinition 4 [5, 11] A function α : D → [0 , ∞ ) (where D is either [0 , ∞ ) , [0 , a ) or [0 , a ] with a > ) is said to be a K -function if it is continuous, α (0) = 0 , andis strictly increasing. Definition 5 [1] The system (1) is said to be K-observable (with respect to Ω )if ∃ T > , ∀ x , x ∈ Ω , Z T | h ( ϕ ( t, , x )) − h ( ϕ ( t, , x )) | dt ≥ α ( | x − x | ) , (2) where α ( · ) is a K -function and | · | denotes the Euclid norm of a vector. Remark 6
Each of above definitions require smoothness of f ( x ) and h ( x ) indifferent level. • For Definition 2, the only requirement is that the system (1) has a uniquesolution. A finite escape time is allowed, as far as the state distinction isachievable before the arrival of the finite escape time. There is no restric-tion to h ( x ) . • For Definition 3, h ( x ) should be N − times continuously differentiable,and f ( x ) should be N − times continuously differentiable, but the valueof N cannot be specified (although it is finite). • For Definition 5, the requirements are that (1) has a unique solution, thesolution of (1) is defined for t ∈ [0 , T ] , and h ( ϕ ( t, , x )) is integrable foreach x ∈ Ω . It is a known fact that, if a system is R-observable at a point x , then itis D-observable on a neighborhood of x [3, 9], but it is not always possible toextend the result to the whole of Ω. On the other hand, D-observability doesnot imply R-observability, as the following example shows. Example 1
Consider a 1-dimensional system ˙ x = − x,y = h ( x ) = x (3) This system is D-observable because it is possible to directly calculate x from y ( x = y / ), but is not R-observable at x = 0 , because h ( x ) = x , L f h ( x ) = − x ,and inductively, L kf h ( x ) = ( − k k x , and hence their derivatives vanish at x = 0 . It is desirable that the width of the ‘observation window’ (the time intervalthat the output of the system is stored in order to determine the initial stateuniquely) is finite. In this sense, K-observability is convenient, and has beenwidely adopted in works on moving horizon state estimation[1, 2]. If the sys-tem (1) is K-observable, then for x , x ∈ Ω with x = x , ∃ t : 0 ≤ t ≤ T ,3 ( ϕ ( t, , x )) = h ( ϕ ( t, , x )), hence (1) is D-observable. Then, a natural ques-tion arises: do systems that are D-observable always permit an observationwindow of finite width? Unfortunately, the answer is negative, which is givenin the following example. Example 2
Consider a 1-dimensional system ˙ x = xy = h ( x ) = ( x < Mx − M x ≥ M, (4) where M is a positive constant. If the initial condition is zero, then the outputis identically zero. For an initial condition x > , x ( t ) = exp[ t ] x , and hencethe output is identically zero for t < ln( M/x ) and is exp[ t ] x − M for t ≥ ln( M/x ) . Hence, the zero initial condition and x cannot be distinguished until t = ln( M/x ) , and hence as the initial condition gets smaller, the required widthof the observation window tends to infinity.One may argue that the reason for making the width of the observation win-dow infinite is the non-differentiability of the output function, but this is not thecase. For example, by replacing the output function h ( x ) of (4) with h ( x ) = ( x ≤ M exp[ − / ( x − M )] x ≥ M , a similar conclusion holds.
Thus far, we have seen that there are gaps between D-observability, R-observability and K-observability, and a D-observable system does not alwayspermit an observation window of finite width. In the following, we show that, if(1) is D-observable as well as R-observable, then there exists an observation win-dow of finite width, and it is possible to construct a K -function correspondingto Definition 5, and hence (1) is K-observable. Proposition 7
If (1) is D-observable as well as R-observable for the initialcondition set Ω , then there is a finite T > such that ∀ x , x ∈ Ω with x = x , ∃ t : 0 ≤ t ≤ T , h ( ϕ ( t, , x )) = h ( ϕ ( t, , x )) . Proof.
We first prove that ∀ x ∈ Ω , ∃N ( x ) , ∀ z , z ∈ N ( x ) with z = z , ∀ T > , ∃ t : 0 ≤ t ≤ T, h ( ϕ ( t, , z )) = h ( ϕ ( t, , z )) (5)by contradiction, where N ( x ) denotes an open neighborhood of x . Suppose that(5) is false, that is, ∃ x ∈ Ω , ∀N ( x ) , ∃ z , z ∈ N ( x ) with z = z , ∃ T > , ∀ t : 0 ≤ t ≤ T, h ( ϕ ( t, , z )) = h ( ϕ ( t, , z )) . (6)4hen, h ( ϕ ( t, , z )) − h ( ϕ ( t, , z )) is identically zero as a function of t . Hence, forall k ≥ d k dt k h ( ϕ ( t, , z )) = d k dt k h ( ϕ ( t, , z )), hence H ( ϕ ( t, , z )) = H ( ϕ ( t, , z )),and by letting t = 0, H ( z ) = H ( z ). On the other hand, because the Jaco-bian of H is of full rank, there is a neighborhood of x in which H is injective.By choosing such neighborhood N ( x ) (recall that N ( x ) is arbitrary), it followsthat H ( z ) = H ( z ) because z = z , hence a contradiction has been obtained.Therefore, (6) is false and hence (5) is true.Next, fix x ∈ Ω, and let N loc ( x ) be an open neighborhood of x in which ∀ z , z ∈ N loc ( x ) with z = z , ∀ T > , ∃ t : 0 ≤ t ≤ T, h ( ϕ ( t, , z )) = h ( ϕ ( t, , z )) (7)holds. Because Ω is compact and N loc ( x ) is open, Ω \ N loc ( x ) is compact. Foreach z ∈ Ω \ N loc ( x ), by D-observability, ∃ t z > , h ( ϕ ( t z , , x )) = h ( ϕ ( t z , , z )).Because h and ϕ are continuous, there is an open neighborhood O z of x and anopen neighborhood G z of z for which h ( ϕ ( t z , , O z )) ∩ h ( ϕ ( t z , , G z )) = ∅ . Let W z = ( z, t z , O z , G z ) be the tuple satisfying this condition, and consider the set { W z : z ∈ Ω \ N loc ( x ) } . (8)Because { G z : z ∈ Ω \ N loc ( x ) } covers Ω \ N loc ( x ) and Ω \ N loc ( x ) is compact,for some L >
0, there is a finite subcollection { W (1) z , . . . , W ( L ) z } of (8) (let us rewrite W ( i ) z = ( z ( i ) , t ( i ) z , O ( i ) z , G ( i ) z )) for which { G (1) z , . . . , G ( L ) z } covers Ω \ N loc ( x ). Let V x = (cid:16) ∩ Li =1 O ( i ) z (cid:17) ∩ N loc ( x ), T x = max { t (1) z , . . . , t ( L ) z } . (9) V x is an open neighborhood of x , because it is a finite intersection of opensets containing x . Let M x = ( x, V x , T x ) be the tuple corresponding to aboveconstruction, and consider the set { M x : x ∈ Ω } . (10)Because Ω is compact, for some J >
0, there is a finite subcollection { M (1) x , . . . , M ( J ) x } of (10) (let us rewrite M ( j ) x = ( x ( j ) , V ( j ) x , T ( j ) x )) for which { V (1) x , . . . , V ( J ) x } coversΩ. Let T = max { T (1) x , . . . , T ( J ) x } . (11)Then, all pairs of initial conditions ( x , x ) with x = x are distinguishablefor some t with 0 ≤ t ≤ T . For, by construction, there is a V ( j ) x such that x ∈ V ( j ) x . If x is in N loc ( x ( j ) ), then by (7), x and x are distinguishable.Otherwise, z ∈ Ω \N loc ( x ( j ) ). Because Ω \N loc ( x ( j ) ) is covered by corresponding { G (1) z , . . . , G ( L ) z } , by (9) and (11), x and x are distinguishable at some t with0 ≤ t ≤ T . (cid:3) emark 8 In Proposition 7, it has been implicitly assumed that (1) has a well-defined solution for t ∈ [0 , T ] . Next, we prove that a system that is D-observable as well as R-observableon Ω is K-observable.
Proposition 9
If the system (1) is D-observable as well as R-observable on Ω ,then it is K-observable on Ω . Proof.
Instead of constructing a K -function that satisfies (2) directly, we con-struct an increasing function of | x − x | that is positive if | x − x | 6 = 0.We assume that T of (2) is sufficiently large and ∀ x , x ∈ Ω with x = x , ∃ t : 0 ≤ t ≤ T , h ( ϕ ( t, , x )) = h ( ϕ ( t, , x )). The existence of such T is assuredby Proposition 7. By using this T , let θ ( x , x ) = Z T | h ( ϕ ( t, , x )) − h ( ϕ ( t, , x )) | dt. Note that θ ( x , x ) is a continuous function of ( x , x ). By the construction of T , if x = x , then ∃ t : 0 ≤ t ≤ T with h ( ϕ ( t, , x )) = h ( ϕ ( t, , x )), and hence θ ( x , x ) >
0. For r >
0, let D r = { ( x , x ) ∈ Ω × Ω : | x − x | ≥ r } , which isa compact set. Let α ( r ) = min ( x ,x ) ∈ Ξ( r ) θ ( x , x ). Because D r is compact,the minimum is well defined and because ( x , x ) ∈ Ξ( r ) implies that x = x ,the minimum is positive. If r < r , then D r ⊃ D r , hence α ( r ) ≤ α ( r ).Therefore, α ( r ) is a monotone nondecreasing function of r and its value ispositive. For definiteness, let α (0) = 0. Note that α ( r ) may not be strictlyincreasing, and may be discontinuous. However, by Lemma 5 of [8], it is possibleto construct a K -function that satisfy α ( r ) ≤ α ( r ). Hence, Z T | h ( ϕ ( t, , x )) − h ( ϕ ( t, , x )) | dt ≥ α ( | x − x | ) ≥ α ( | x − x | ) , whence (1) is K-observable. (cid:3) Remark 10
If all of f ( x ) , h ( x ) and ϕ ( t, , x ) are real analytic functions oftheir arguments and the system is forward complete, it can be proved that D-observability implies K-observability without assuming R-observability, and thewidth of the observation window of Proposition 9 may be arbitrarily small. Theproof is as follows. First, if h ( ϕ ( t, , x )) is real analytic and h ( ϕ ( t, , x )) ≡ h ( ϕ ( t, , x )) on an interval [0 , T ) for a pair ( x , x ) with x = x , then h ( ϕ ( t, , x )) ≡ h ( ϕ ( t, , x )) for any t due to real analyticity. Therefore, if (1) is D-observable,then ∀ T > , ∀ x , x with x = x , ∃ t : 0 ≤ t ≤ T , h ( ϕ ( t, , x )) = h ( ϕ ( t, , x )) .Hence, for real analytic systems, the existence of the finite observation windowof Proposition 7 is established without assuming R-observability, and its widthmay be arbitrarily small. For any T > , the construction of a K-function isidentical to the proof of Proposition 9. Hence, we have proved that, if (1) is -observable and forward complete, and all of f ( x ) , h ( x ) and ϕ ( t, , x ) are realanalytic functions of their arguments, then (1) is K-observable for any observa-tion window of positive length. Because a K-observable system is automaticallyD-observable (without assuming real analyticity), we have shown that, for aforward complete and real analytic system, the notions of D-observability andK-observability are equivalent. In this note, we have shown that an autonomous nonlinear system which isD-observable and R-observable always permits an observation window of finitewidth, and it is actually K-observable as well. A theoretical construction ofcorresponding K -function has been provided as well. In many researches relatedto nonlinear observability, the existence of an observation window of finite widthand a K -function are assumed a priori . Contrary, in this note, it has beenproved that they are consequences of D-observability and R-observability. It isalso to be noted that our result is purely existential, and no practical methodof obtaining the width of the observation window and the K -function have beenprovided. References [1] M. Alamir, Nonlinear moving horizon observers: theory and real-time im-plementation. In Besan¸con G. (Ed.),
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