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Functional Observers for Nonlinear Systems
10th IFAC Symposium on Nonlinear Control Systems August 23-25, 2016. Monterey, California, USA 10th IFAC Symposium on Nonlinear Control Systems August 23-25, 2016. Monterey, California, USA 10th IFAC Symposium on Nonlinear Control Systems Available August 23-25, 2016. Monterey, California, USA online at www.sciencedirect.com August 23-25, 2016. Monterey, California, USA
ScienceDirect IFAC-PapersOnLine 49-18 (2016) 505–510 Functional Observers for Nonlinear Systems Functional Observers for Nonlinear Systems Functional for Functional Observers Observers for Nonlinear Nonlinear Systems Systems Costas Kravaris
Costas Kravaris Costas Costas Kravaris Kravaris Texas A&M University, College Station, TX 77843-3122 USA Texas(Tel: A&M979-458-4514; University, College Station, TX 77843-3122 USA e-mail: [email protected]). Texas A&M University, Station, TX e-mail: [email protected]). Texas(Tel: A&M979-458-4514; University, College College Station, TX 77843-3122 77843-3122 USA USA (Tel: 979-458-4514; e-mail: [email protected]). (Tel: 979-458-4514; e-mail: [email protected]). Abstract: In this work, the problem of designing observers for estimating a single nonlinear functional of Abstract: In this work,for thegeneral problem of designing observers forofestimating single nonlinear functional of the state is In formulated nonlinear systems. Notions functionalaaobserver linearization are also Abstract: this work, the problem of designing observers for estimating single nonlinear functional of the state is formulated for general nonlinear systems. Notions of functional observer linearization are also Abstract: In this work, the problem of designing observers for estimating a single nonlinear functional of formulated, in terms of achieving exactly systems. linear error dynamics in transformed coordinates and the state for general nonlinear Notions of observer linearization are with also formulated, in terms of exactly linearand error dynamics in transformed coordinates and with the state is is formulated formulated forachieving general nonlinear systems. Notions of functional functional observer linearization also prescribed rate of decay of the error. Necessary sufficient conditions for the existence of and aare lowerformulated, in terms of achieving exactly linear error dynamics in transformed coordinates with prescribed rate of decay of the error. Necessary and sufficient conditions for the existence of a lowerformulated, in terms of achieving exactly linear error dynamics in transformed coordinates and with order functional observer with linear dynamics and linear outputconditions map are derived. The resultsofprovide a prescribed rate decay of the error. Necessary and sufficient for the aa lowerorder functional with dynamics and linear outputobservers map are to derived. Thesystems. resultsofprovide a prescribed rate of ofobserver decay of thelinear error.linear Necessary and sufficient conditions for the existence existence lowerdirect functional generalization of Luenberger’s theory oflinear functional nonlinear order observer with linear dynamics and output map are derived. The results provide a direct functional generalization of Luenberger’s theory functional nonlinear order observer with linear linear dynamics andoflinear outputobservers map are to derived. Thesystems. results provide a direct generalization of Luenberger’s theory functional observers to © 2016, IFAC (International Federation of Automatic Hosting bychemical Elsevier Ltd. Allsystems. rights reserved. Keywords: Functional exactlinear linearization, observer design, process applications. direct generalization ofobservers, Luenberger’s linear theory of ofControl) functional observers to nonlinear nonlinear systems. Keywords: Functional observers, exact linearization, observer design, chemical process applications. Keywords: observer Keywords: Functional Functional observers, observers, exact exact linearization, linearization, observer design, design, chemical chemical process process applications. applications. In the present work, we will consider unforced nonlinear 1. INTRODUCTION In the present work, we will consider unforced nonlinear systems of the form 1. INTRODUCTION In the work, 1. systems of the form the present present work, we we will will consider consider unforced unforced nonlinear nonlinear The problem of estimating a function of the state vector, In 1. INTRODUCTION INTRODUCTION systems of the form dx The problem of estimating a function of the state vector, systems of the form without the need of estimatinga the entire state vector, arises in f (x) The problem of estimating function of state vector, dx without the need estimating entire state vector, arises in dt f (x) The problem of of estimating a the function of the the state vector, dx many applications. The design of output feedback controllers without the of The estimating the entire state vector, arises (1) dt dx many applications. designdesign of output feedback controllers yi ffh(x) without theaneed need estimating the entire vector, arises in in (x) i (x) , i 1, ,p dt based on stateoffeedback is astate classical example, (1) many applications. The design of output feedback controllers y h (x) , i 1, ,p dt i i (1) based onisaonly statethefeedback design is afeedback classical example, many The of output controllers zi q(x) y h (x) , i 1, ,p where applications. it statedesign feedback function that needs to be (1) i based on a state feedback design is a classical example, yzi q(x) hi (x) , i 1, ,p where it thethe state feedback function that needs to be based onisand aonly state feedback design is a classical example, estimated not entire state vector. Another important z q(x) where it is only the state feedback function that needs to be estimated and statetovector. Another important whereofit is onlynotthethestate feedback function thatofneeds to be where: z q(x) class applications isentire related the design inferential where: estimated and not the entire state vector. Another important class of applications is related to the design of inferential x nn is the system state estimated and not the entire state vector. Another important control systems, whereisone outputtoisthe measured and ainferential different where: where: class of applications related design of x the measured outputs control output measured and different class ofsystems, applications isone related toisthe of ainferential yi nn, is i the 1, system ,p arestate is the system x output, which iswhere unmeasured, needs todesign be regulated to set control systems, where one output is measured and a different the measured outputs is the system state x y , i 1, ,p arestate output, which is unmeasured, needs to be regulated to set i control systems, where one output is measured and a different are the outputs point. (scalar) output to be estimated y ,,isiithe 1, ,p z output, which is unmeasured, needs to be regulated to set i are the measured measured outputs y 1, ,p point. which is unmeasured, needs to be regulated to set is the (scalar) output to be estimated z output, i n n n n point. is the (scalar) output to be estimated z are smooth , h , qto : be These applications motivate the development of functional and f :zn is nthe point. (scalar) output estimated i: n n are smooth These applications motivate thereduction development of functional and f : n n , h i : n , q : n observers, the aim being a of dimensionality nonlinear functions. The objective is to construct a functional and are smooth f : , h : , q : n n n n These applications motivate the development of functional i : objective and are smooth f : , h , q : observers, the aim being a reduction of dimensionality These applications motivate the development of functional nonlinear functions. The is to construct a functional i relative to athe full-state observer. The notion of a functional observer of order ν < n, which generates an estimate of the observers, aim being a reduction of dimensionality nonlinear functions. The objective is to construct aa functional relative towas athe full-state observer. The notion of a functional observers, aimdefined being a Luenberger’s reduction ofpioneer dimensionality observer of order ν < n, which generates an estimate of nonlinear functions. The objective is to construct functional observer first in work on output z, driven by the output measurements y , i 1, , p .the relative to a full-state observer. The notion of a functional i of order < n, which generates an estimate of the observer first defined in Luenberger’s pioneer work on observer relative towas a full-state observer. The notion of a functional output z, driven byννthe output measurements observer of order < n, which generates an estimate of y , i 1, , p .the observers for linear multivariable systems. Luenberger (1966, i observer was first defined in Luenberger’s pioneer work on . output z, driven by the output measurements y , i 1, , p observers for linear multivariable systems. Luenberger (1966, i observer was first defined in Luenberger’s pioneer work on . a output by thethe output measurements , i 1, , pfor 1971) proved that multivariable it is feasiblesystems. to construct a functional Sectionz,2driven will define notion of functionaly iobserver observers for Luenberger (1966, 1971) proved that multivariable it is feasible to construct a functional observers for linear linear systems. Luenberger (1966, Section of 2 will define(1)the notion of functional observer for toa observer with number of states equal to observability index system the form in a completely analogous manner 1971) proved that to aa functional Section 2 define notion of observer for toaa observer with number of feasible states equal to observability index system 1971) proved that it it is is feasible to construct construct functional the form infor a completely analogous manner Section of 2 will will define(1)the the notion of functional functional observer minus one. Luenberger’s definition linear systems. Section 3 for will observer with number of states equal to observability index system of the form (1) in a completely analogous manner to minus one. observer with number of states equal to observability index system Luenberger’s definition linear systems. Section 3 will of problem the form (1)functional infor a completely analogous manner to4 pose the of observer design. Section minus one. Luenberger’s definition for linear systems. Section 3 will The basic minus one.theory of linear functional observers can be found will pose define the problem of functional observer design. Section 4 Luenberger’s definition for linear systems. Section 3 will notionsofoffunctional exact linearization for the functional pose the problem observer design. Section 4 The basic theory of lineartexts, functional observers can be found will in standard linear systems e.g. in Chen (1984). In recent define notions of exact linearization for the functional pose the problem of functional observer design. Section The basic theory of linear functional observers can be found observer problem. Section 5linearization will develop necessary and4 will define notions of exact for the functional in standard linear systems texts, e.g. in Chen (1984). In recent The basic theory of linear functional observers can be found years, therelinear has systems been atexts, renewed interest in functional observer 5linearization will of develop necessary and will defineproblem. notions Section of thelinearization functional in e.g. in Chen (1984). In sufficient conditions forexact the solution the for total observer Section 5 will develop necessary and years, there has systems been atexts, renewed inet functional in standard standard linear e.g. 1998; in interest Chen (1984). In recent recent observers for linear systems (Tsui, Trinh al., 2006; problem, sufficient conditions for the solution of the total linearization observer problem. problem. Section 5 will develop necessary and years, there has been a renewed interest in functional as well asfora the simple formula for the resulting observers for linear systems (Tsui, 1998; Trinh al., 2006; sufficient conditions solution of the total linearization years, there hasKorovin been aet renewed interest inet functional Darouach, 2000; al., 2008 and 2010; Fernando et problem, as well as a the simple formula the resulting sufficient conditions forSection solution of the for total linearization observers for linear systems (Tsui, 1998; Trinh et al., 2006; functional observer. In 6, the results of Section 5 will Darouach, 2000; Korovin et 2008 and Trinh 2010; Fernando et problem, as well as aa simple formula for the resulting observers systems (Tsui, et al., 2006; al., 2010),for thelinear goal being to al., find the1998; smallest possible order functional observer. In Section 6, the results of Section 5 will problem, as well as simple formula for theleading resulting Darouach, 2000; Korovin et al., 2008 and 2010; Fernando et be specialized to linear time-invariant systems, to al., 2010), the goal being to find the smallest possible order functional observer. In Section 6, the results of Section 5 Darouach, 2000; Korovin et al., 2008 and 2010; Fernando et of the linear functional observer. be specialized to linear time-invariant systems, leading to functional observer. In Section 6, the results of Section 5 will will al., 2010), the goal being to find the smallest possible order simple and easy-to-check conditions for the design of lowerbe specialized to linear time-invariant systems, leading to of the linearthe functional observer. al., 2010), goal being to find the smallest possible order simple and easy-to-check conditions for the design of lowerbe specialized to linear time-invariant systems, leading to of linear observer. functional observers.conditions for the design of lowersimple and easy-to-check Forthe systems, there have been significant order of the nonlinear linear functional functional observer. order functional observers. simple and easy-to-check conditions for the design of lowerFor nonlinearin systems, have observers, been significant functional developments the theory there of full-state with a order For nonlinear systems, there have been functional observers. observers. developments in the theory of full-state observers, with a order For nonlinear systems, there have been significant significant 2. DEFINITION OF A FUNCTIONALOBSERVER FOR A variety of methods and approaches. In particular, in the developments in the theory of full-state observers, with a 2. DEFINITION OF A FUNCTIONALOBSERVER FOR A variety of methods and approaches. In particular, in the developments in the theory of full-state observers, with a NONLINEAR DYNAMIC SYSTEM context of of exact linearization methods In(Krener and Isidori, 2. DEFINITION OF A FUNCTIONALOBSERVER FOR variety methods and approaches. particular, in the NONLINEAR DYNAMIC SYSTEM 2. DEFINITION OF A FUNCTIONALOBSERVER FOR A A contextKrener methods andKravaris, Isidori, variety of exact methods and approaches. In(Krener particular, in the 1983; and linearization Respondek, 1985; Kazantzis and NONLINEAR DYNAMIC SYSTEM context of exact linearization methods (Krener and Isidori, In complete analogy to Luenberger’s construction for the NONLINEAR DYNAMIC SYSTEM 1983; Krener and Respondek, 1985; Kazantzis and Kravaris, contextKazantzis of exact etlinearization methods (Krener and Isidori, 1998; al., 2000; Kreisselmeier and and Engel, 2003; In complete Luenberger’s construction for the 1983; Krener and Respondek, 1985; Kazantzis Kravaris, linear case, weanalogy seek forto a smooth mapping 1998; Kazantzis al., 2000; Kreisselmeier and and Engel, 2003; In 1983; Krener andet2002 Respondek, 1985; Kazantzis Kravaris, complete Luenberger’s construction Krener and Xiao, and 2005; Andrieu and Praly, 2006), linear case, weanalogy seek forto mapping In complete analogy toa smooth Luenberger’s construction for for the the 1998; Kazantzis et al., 2000; Kreisselmeier and Engel, 2003; Krener and Xiao, 2002 and 2005; Andrieu and Praly, 2006), linear case, we seek for a smooth mapping 1998; Kazantzis et al., 2000; Kreisselmeier and Engel, 2003; Luenberger theory2002 for full-state observers has been extended linear case, we seek for a smooth 1 (x) mapping Krener and Xiao, and 2005; Andrieu and Praly, 2006), Luenberger theory forinfull-state observers has extended Krener and Xiao, 2002 and 2005; Andrieu andbeen Praly, 2006), 1 (x) to nonlinear systems a direct and analogous manner. The Luenberger theory for full-state observers has been extended (x) (2) to nonlinear systems in a direct and analogous manner. The Luenberger theory for full-state observers has been extended 1 (x) goal of the present work is to develop a direct generalization (x) 1 (x) (2) to nonlinear systems in aaisdirect and analogous manner. The goal of the present work to develop a direct generalization to nonlinear systems in direct and analogous manner. The (x) (2) (x) of Luenberger’s functional observers toanonlinear systems. (x) goal of the present work is to develop direct generalization (x) (2) of Luenberger’s functional systems. goal of the present work is observers to developtoanonlinear direct generalization (x) of Luenberger’s functional observers to nonlinear systems. (x) of Luenberger’s functional observers to nonlinear systems.
Copyright © 2016 IFAC 517 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016, 2016 IFAC 517Hosting by Elsevier Ltd. All rights reserved. Peer review©under of International Federation of Automatic Copyright 2016 responsibility IFAC 517Control. Copyright © 2016 IFAC 517 10.1016/j.ifacol.2016.10.215
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from n to , to immerse system (1) to a ν-th order system (ν < n), with inputs y i , i 1, , p and output z: -----------------
d (, y1 , , y p ) dt z (, y1 , , y p )
p
which is condition (4) stated earlier. If the functional observer (6) is initialized consistently with the system (1), i.e. if ˆ (0) (x(0)) , then ˆ (t) (x(t)) , and therefore zˆ (t) ˆ (t), y1 (t), , y p (t)
(3)
p
where : , : , the aim being that system (3), driven by the measured outputs y i , i 1, , p of (1), can generate an estimate of unmeasured output z of (1).
q(x) (x), h1(x),
, h p (x)
, h p (x)
F, Hi, Q being n n, 1 n, 1 n matrices respectively, and a linear mapping (x) Tx is considered. Definition 1 tells us that for a linear time-invariant system dx Fx dt (8) y i H i x , i 1, , p
z Qx
(1) the system
p dˆ Aˆ B i y i dt i 1
where f : n n , h i : n , q : n are smooth nonlinear functions, y i , i 1, , p are the measured outputs and z is the output to be estimated , the system dˆ (ˆ , y1 , , y p ) (6) dt ˆz (ˆ , y1 , , y p )
zˆ Cˆ D i y i i 1
will be a functional observer if the following conditions are met: p
TF AT B i H i
p
Q CT D i H i
ˆ
(11)
i 1
for some n matrix T. These are exactly Luenberger’s conditions for a functional observer for a linear timeinvariant system (Luenberger, 1971).
, h p (x)
(7)
3. DESIGNING LOWER-ORDER FUNCTIONAL OBSERVERS
possesses an invariant manifold ˆ (x) with the property that q(x) (x), h 1 (x),
(10)
i 1
the overall dynamics
dx f (x) dt dˆ ˆ , h 1 (x), dt
(9)
p
where : p , : p ( n) , is called a functional observer for (1), if in the series connection x
At this point, it is important to examine the special case of a linear system, where f (x) Fx , h i (x) H i x , q(x) Qx with
The foregoing considerations lead to the following definition of a functional observer:
y1 y p
, h p (x(t)) q(x(t)) t 0 ,
In the presence of initialization errors, additional stability requirements will need to be imposed on the ˆ -dynamics, for the estimate zˆ (t) to asymptotically converge to z(t) .
(5)
Definition 1: Given a dynamic system dx f (x) dt y i h i (x) , i 1, , p z q(x)
which means that the functional observer will be able to exactly reproduce z(t).
(4)
(x(t)), h 1 (x(t)),
But in order for system (1) to be mapped to system (3) under the mapping (x), the following relations have to hold: (x)f (x) (x), h1(x), x
For the design of a functional observer, one must be able to 1 (x) find a continuously differentiable map (x) to (x)
, h p (x) .
In the above definition, the requirement that ˆ (x) is an invariant manifold of (7), i.e. that
satisfy conditions (4) and (5), i.e. such that j (x)f (x), j 1, , is a function of 1 (x), , (x), hi (x) x and q(x) is a function of 1 (x), , (x), hi (x) , i 1, , p .
ˆ (0) x(0) ˆ (t) x(t) t 0 ,
translates to (x) f (x) (x), h 1 (x), x
, h p (x) ,
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However, such scalar functions 1 (x), exist, if ν is too small.
, (x) may not
or, slightly more generally,
(, y1 ,
Moreover, even when they do exist, there is an additional very important requirement:
(x)f (x) A(x) B h1(x), x
, h p (x)
for some Hurwitz matrix A , and if in addition θ(x) satisfies condition (5), i.e. that q(x) can be expressed as a function of (x) and hi (x) , i 1, , p , then we have a stable functional
,p
observer with linear dynamics:
h
is a linearly observable system, i (x) , i 1, , p are linearly x independent and ν n p : Available design methods for reduced order state observers (e.g. observer error linearization methods in Kazantzis et al., 2000) will generate a functional observer of order ν n p , by modifying output map of the observer (so that the estimate of z is the observer output instead of the entire state vector). The question is whether a lower order ν n p would be feasible and how to go about designing the functional observer. This is not an easy question because we will be trying to impose too many requirements at the same time.
dˆ Aˆ B(y1 , , y p ) dt zˆ (ˆ , y1 , , y p ) This leads to the Functional Observer Linearization Problem. Note that stricter linearity requirements could be imposed, in particular, that the functions B() and ( , ) be linear as well. This leads to the Functional Observer Total Linearization Problem. A. Functional Observer Linearization Problem
One possible line of attack is to first try to identify functions j (x)f (x), j 1,..., and q(x) 1 (x), , (x) such that x can be expressed as functions of 1 (x), , (x), hi (x) and, as a second step, check stability of the error dynamics.
Given a system of the form (1), find a functional observer of the form dˆ Aˆ B(y1 , , y p ) dt (12) ˆ zˆ (, y1 , , y p ) where is a matrix with stable eigenvalues and
Alternatively, one could try to follow the opposite path. As a first step, try to enforce stability: given some desirable dˆ dynamics for the observer (ˆ , y1 , , y p ) , with ( , ) dt so as to guarantee stability and rapid decay of the error, find 1 (x) so that (x)f (x) (x), h (x), (x) 1 x (x)
B:
p
, :
p . Equivalently, find a n
continuously differentiable mapping :
(x)f (x) A(x) B h1(x), x
, h p (x)
such that
(13)
and
, h p (x) .
q(x) (x), h 1 (x),
The second step will then be to check if q(x) can be expressed as a function of (x), h1 (x), , h p (x) (condition (5)).
, h p (x)
(14)
Assuming that the above problem can be solved, the resulting error dynamics will be linear:
4. EXACT LINEARIZATION OF A FUNCTIONAL OBSERVER
d ˆ (x) A ˆ (x) dt zˆ z ˆ , h 1 (x), , h p (x) (x), h 1 (x),
Along the second line of attack of the functional observer design problem, the most natural - function to work with is the linear one:
(, y1 ,
, yp)
If we can find a continuously differentiable map (x) to satisfy the corresponding partial differential equation (4), i.e.,
All the above requirements can be satisfied when dx f (x) dt y h (x) , i 1, i i
, y p ) A B(y1,
It will then be the eigenvalues of the matrix A that will determine stability of the functional observer and the rate of decay of the error.
(x)f (x) (x), h1(x), , h p (x) will define the x right hand side of the functional observer’s dynamics, it must be such that the functional observer’s dynamics is stable and the decay of the error is sufficiently rapid. Since
507
, h p (x)
(15)
from which, for A having stable eigenvalues, the effect of the initial error ˆ (0) (x(0)) will die out, ˆ (t) will approach (x(t)) asymptotically, therefore zˆ (t) will approach z(t) .
p
, y p ) A B i y i i 1
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B. Functional Observer Total Linearization Problem
Proposition 1: Let f :
p
n
f 0 . Also, x
1. All the eigenvalues of A are non-resonant with F , n
i 1
i.e. no eigenvalue j of A is of the form j
where , Bi , C, D i are , 1, 1 , 1 1 matrices respectively, with having stable eigenvalues. Equivalently, n
find a continuously differentiable mapping : such that
i 1
with i F and m i nonnegative integers, not all zero.
Then the system of first-order partial differential equations
(17)
(x)f (x) A(x) (x) x
and (18)
solution (x) in a neighborhood of x 0 .
i 1
5. NECESSARY AND SUFFICIENT CONDITIONS FOR SOLVABILITY OF THE FUNCTIONAL OBSERVER TOTAL LINEARIZATION PROBLEM
Assuming that the above problem can be solved, the resulting error dynamics will be linear
The trial-and-error approach outlined in the previous section is far from being practical due to the computational effort involved in solving (13) or (17), which will be multiplied by the number of trials until (14) or (18) is satisfied.
(19)
(20)
with initial condition (0) 0 , admits a unique real analytic
p
q(x) C(x) D i h i (x)
d ˆ (x) A ˆ (x) dt zˆ z C ˆ (x)
mii ,
2. 0 does not lie in the convex hull of F .
p (x)f (x) A(x) B i h i (x) x i 1
, :
let A be a ν ν matrix. Assume: (16)
zˆ Cˆ D i y i
n
denote by F the set of eigenvalues of F
p
be real analytic functions with f 0 0, 0 0 and
Given a system of the form (1), find a functional observer of the form
dˆ Aˆ B i y i dt i 1
n
from which zˆ (t) z(t) Ce At ˆ (0) (x(0)) .
To be able to develop a practical approach for designing functional observers, it would be helpful to develop criteria to check if for a given set of ν eigenvalues, there exists a functional observer whose error dynamics is governed by these eigenvalues. This is done in the present Section for the Functional Observer Total Linearization Problem. The main result is as follows:
With the matrix A having stable eigenvalues, the effect of the initialization error ˆ (0) (x(0)) will die out, and zˆ (t) will approach z(t) asymptotically. In order to solve either version of the Functional Observer Linearization Problem, it is natural to first try to solve the system of partial differential equations (13) or (17) given some small-size matrix A with fast enough eigenvalues, and then check to see if q(x) can be expressed as a function of (x) and hi (x) , i 1, , p according to (14) or (18). If it can,
Proposition 2: Under the assumptions of Proposition 1, for a real analytic nonlinear system of the form (1), there exists a functional observer of the form p dˆ Aˆ B i y i dt i 1
we have a functional observer with linear error dynamics – if not, we could try a different matrix A with different eigenvalues and/or larger size, until both conditions can be met.
(16)
p
zˆ Cˆ D i y i i 1
The above approach will be feasible as long as the system of partial differential equations (13) or (17) is solvable. This is a singular system of partial differential equations because their principal part vanishes at the equilibrium point, and therefore standard existence/uniqueness results do not apply. However, we are covered by Lyapunov’s Auxiliary Theorem for singular partial differential equations (Lyapunov, 1992), as long as the nonlinear system (1) is analytic and in the Poincaré domain. The following Proposition is an immediate consequence of Lyapunov’s Auxiliary Theorem. (See also Kazantzis and Kravaris, 1998.)
with the eigenvalues of A being the roots of a given polynomial λν α1λν 1 α ν 1λ α ν , if and only if
Lνf q(x) α1Lνf1q(x) is a
α ν 1Lf q(x) α ν q(x)
-linear combination of
hi (x), Lf hi (x),
, Lνf hi (x),i 1,
, p , where L f n
denotes the Lie derivative operator L f
520
f j (x) x j . j1
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Proof of Proposition 2: i) Necessity
h i (x) L h (x) f i C C A,b i a i 1 1 L h (x) i f p
Suppose that there exists a functional observer of the form (16) for the system (1). Then, condition (17) will be satisfied 1 (x) . Using the Lie derivative operator for some (x) (x)
p D i L f h i (x) 1L f 1h i (x) i 1
notation, this condition may be written as ---------------------- L f 1 (x) 1 (x) p A B h (x) i i L f (x) (x) i 1 Applying the Lie derivative operator L f to each component
p
i
b i Lkf 1h i (x)
1
hi (x), Lf hi (x),
L
1 f 1L f
L
1 f 1L f
p
C A,b i i 1 I (x)
a
is
CA,b i b i
and
1 2 1 1
a
Ab i 2
1
3
1
1
0
0
0
1 0 0 0
, Lνf hi (x),i 1,
1 Lf q(x) 1L f q(x)
,p ,
, β νi
such that
1L f q(x) q(x)
1 0 Lf h i (x) 1 L h i (x) i i f
i 1
h i (x) i
(24)
Consider the PDE
1 0 0 0 S (x)f (x) x 0 0 α ν α ν 1
(22)
0 m 0 p 1i S(x) h i (x) i 1 1 m νi α1 (25) -------------------------------------------------------------
where
(23)
m1 i with
m ν β ν i i
a
β1 β0 α1 a 1 i i
β0i α ν
(26)
given by (23).
It can be verified that the unique solution of (25) is given by:
At the same time, the functions 1 (x), , (x) must satisfy condition (18), i.e. -------------------------------------------------1 (x) p D h (x) -------------- q(x) C , ----------- i i (x) i 1
S(x) p q(x) 0 i h i (x) i 1 p L f q(x) 0 i L f h i (x) m 1i h i (x) i 1 p 1 2 1 L f q(x) 0 i L f h i (x) m 1i L f h i (x) i 1
for some constants C, Di .
Then, applying the operator Lνf α1Lνf1 α ν 1Lf α ν I on both sides of the above condition and using (21), we obtain: 1 Lf q(x) 1L f q(x)
α ν 1Lf q(x) α ν q(x)
i.e. there exist β0i , β1i ,
h i (x) L h (x) f i 1 L h (x) i f
A 1 b i
,p ,
-linear combination of
hi (x), Lf hi (x),
(21) where
α ν 1Lf q(x) α ν q(x)
, Lνf hi (x),i 1,
Lνf q(x) α1Lνf1q(x)
p
I 1 (x)
ii) Sufficiency: Suppose that
and we can calculate:
1L f h i (x) h i (x)
This proves Lνf q(x) α1Lνf1q(x) is -linear combination of
in the above equation (k-1) times, we find that for k = 2,3,… , Lk (x) 1 (x) f 1 k A (x) Lkf (x)
A k 1b i h i (x) A k 2 b i L f h i (x)
509
1L f q(x) q(x)
m (1) h i (x) i
Therefore,
q(x) 1
521
0
0 S(x)
p
β0i hi (x) i 1
(27)
IFAC NOLCOS 2016 510 Costas Kravaris / IFAC-PapersOnLine 49-18 (2016) 505–510 August 23-25, 2016. Monterey, California, USA
The conclusion is that both conditions for the solution of the Functional Observer Total Linearization Problem are satisfied and 1 0 0 0 m 0 0 p 1i dξˆ ξˆ yi dt i 1 0 1 (28) 0 m νi α ν α ν 1 α1
has formulated notions of exact linearization for functional observer design and has derived a specific criterion for total linearization, including a simple formula for the resulting functional observer. REFERENCES
zˆ 1
0 ξˆ
0
Andrieu, V. and Praly, L. (2006). On the existence of a Kazantzis-Kravaris/ Luenberger observer, SIAM J. Control and Optimization, Vol. 45, 432-456. Chen, C.T. (1984). Linear System Theory and Design, Holt, Rinehart and Winston, New York. Darouach, M. (2000). Existence and design of functional observers for linear systems, IEEE Trans. Automat. Contr., Vol. 45, No.5, 940-943. Fernando, T.L., Trinh, H.M., and Jennings, L. (2010). Functional observability and the design of minimum order linear functional observers, IEEE Trans. Automat. Contr., Vol. 55, No.5, 1268-1273. Kazantzis, N. and Kravaris, C. (1998). Nonlinear observer design using Lyapunov's auxiliary theorem, Systems & Control Letters, Vol. 34, 241-247. Kazantzis, N., Kravaris, C., and Wright, R.A. (2000). Nonlinear observer design for process monitoring, Ind. Eng. Chem. Res., Vol. 39, 408-419. Korovin, S.K., Medvedev, I.S., and Fomichev, V.V. (2008). Minimum dimension of a functional observer with a given convergence rate, Doklady Mathematics, Vol. 78, No.3, 940-943. Korovin, S.K., Medvedev, I.S., and Fomichev, V.V. (2010). Minimum-order functional observers, Computational Mathematics and Modeling, Vol. 21, No.3, 275-296. Kreisselmeier, G. and Engel, R, (2003). Nonlinear observers for autonomous Lipschitz continuous systems, IEEE Trans. Automat. Contr., Vol. 48, No.3, 451-464. Krener, A.J. and Isidori, A. (1983). Linearization by output injection and nonlinear observers, Systems & Control Letters, 3, 47-52. Krener, A.J. and Respondek, W. (1985). Nonlinear observers with linearizable error dynamics, SIAM J. Control and Optimization, Vol. 23, 197-216. Krener, A.J. and Xiao, M. (2002). Nonlinear observer design in the Siegel domain, SIAM J. Control and Optimization, Vol. 41, No. 3, 932-953. Krener, A.J. and Xiao, M. (2005). Nonlinear observer design for smooth systems. In W. Perruquetti and J.-P. Barbot (ed.), Chaos in Automatic Control, 411-422, Marcel Dekker, New York. Luenberger, D.G. (1966). Observers for multivariable systems, IEEE Trans. Automat. Contr., Vol. AC-11, No.2, 190-197. Luenberger, D.G. (1971). An introduction to observers, IEEE Trans. Automat. Contr., Vol. AC-16, No.6, 596-602. Lyapunov, A.M. (1992). The General Problem of the Stability of Motion (English Translation by A. T. Fuller), Taylor & Francis, London. Trinh, H., Fernando, T., and Nahavandi, S. (2006). Partialstate observers for nonlinear systems, IEEE Trans. Automat. Contr., Vol. 51, No.11, 1808-1812. Tsui, C.-C. (1998). What is the minimum function observer order?, J. Franklin Inst., Vol. 335B, No.4, 623-628.
p
β0i yi i 1
is a functional observer. This completes the proof. 6. LOWER-ORDER FUNCTIONAL OBSERVERS FOR LINEAR SYSTEMS The results of the previous section can now be specialized to linear time-invariant systems. The following is a Corollary to Proposition 2. Proposition 3: For a linear time-invariant system of the form
dx Fx dt y i H i x , i 1, , p z Qx
(8)
there exists a functional observer of the form p dˆ Aˆ B i y i dt i 1
(16)
p
zˆ Cˆ D i y i i 1
with the eigenvalues of A being the roots of a given polynomial λν α1λν 1 α ν 1λ α ν , if and only if
(QFν α1QFν 1
α ν 1QF α ν Q)
span Hi , Hi F,
, Hi Fν , i 1,
,p .
The above Proposition provides a simple and easy-to-check feasibility criterion for a lower-order functional observer with a pre-specified set of eigenvalues governing the error dynamics. Moreover, an immediate consequence of the above Proposition is the following: Corollary: Consider a linear time-invariant system of the form (8) with observability index νo. Then, there exists a functional observer of the form (16) of order ν = νo – 1 and arbitrarily assigned eigenvalues. The result of the Corollary is a well-known result, first found by Luenberger (1966, 1971) following a different approach. 7. CONCLUSION The present work has developed a direct generalization of Luenberger’s functional observers to nonlinear systems. It 522
ScienceDirect IFAC-PapersOnLine 49-18 (2016) 505–510 Functional Observers for Nonlinear Systems Functional Observers for Nonlinear Systems Functional for Functional Observers Observers for Nonlinear Nonlinear Systems Systems Costas Kravaris
Costas Kravaris Costas Costas Kravaris Kravaris Texas A&M University, College Station, TX 77843-3122 USA Texas(Tel: A&M979-458-4514; University, College Station, TX 77843-3122 USA e-mail: [email protected]). Texas A&M University, Station, TX e-mail: [email protected]). Texas(Tel: A&M979-458-4514; University, College College Station, TX 77843-3122 77843-3122 USA USA (Tel: 979-458-4514; e-mail: [email protected]). (Tel: 979-458-4514; e-mail: [email protected]). Abstract: In this work, the problem of designing observers for estimating a single nonlinear functional of Abstract: In this work,for thegeneral problem of designing observers forofestimating single nonlinear functional of the state is In formulated nonlinear systems. Notions functionalaaobserver linearization are also Abstract: this work, the problem of designing observers for estimating single nonlinear functional of the state is formulated for general nonlinear systems. Notions of functional observer linearization are also Abstract: In this work, the problem of designing observers for estimating a single nonlinear functional of formulated, in terms of achieving exactly systems. linear error dynamics in transformed coordinates and the state for general nonlinear Notions of observer linearization are with also formulated, in terms of exactly linearand error dynamics in transformed coordinates and with the state is is formulated formulated forachieving general nonlinear systems. Notions of functional functional observer linearization also prescribed rate of decay of the error. Necessary sufficient conditions for the existence of and aare lowerformulated, in terms of achieving exactly linear error dynamics in transformed coordinates with prescribed rate of decay of the error. Necessary and sufficient conditions for the existence of a lowerformulated, in terms of achieving exactly linear error dynamics in transformed coordinates and with order functional observer with linear dynamics and linear outputconditions map are derived. The resultsofprovide a prescribed rate decay of the error. Necessary and sufficient for the aa lowerorder functional with dynamics and linear outputobservers map are to derived. Thesystems. resultsofprovide a prescribed rate of ofobserver decay of thelinear error.linear Necessary and sufficient conditions for the existence existence lowerdirect functional generalization of Luenberger’s theory oflinear functional nonlinear order observer with linear dynamics and output map are derived. The results provide a direct functional generalization of Luenberger’s theory functional nonlinear order observer with linear linear dynamics andoflinear outputobservers map are to derived. Thesystems. results provide a direct generalization of Luenberger’s theory functional observers to © 2016, IFAC (International Federation of Automatic Hosting bychemical Elsevier Ltd. Allsystems. rights reserved. Keywords: Functional exactlinear linearization, observer design, process applications. direct generalization ofobservers, Luenberger’s linear theory of ofControl) functional observers to nonlinear nonlinear systems. Keywords: Functional observers, exact linearization, observer design, chemical process applications. Keywords: observer Keywords: Functional Functional observers, observers, exact exact linearization, linearization, observer design, design, chemical chemical process process applications. applications. In the present work, we will consider unforced nonlinear 1. INTRODUCTION In the present work, we will consider unforced nonlinear systems of the form 1. INTRODUCTION In the work, 1. systems of the form the present present work, we we will will consider consider unforced unforced nonlinear nonlinear The problem of estimating a function of the state vector, In 1. INTRODUCTION INTRODUCTION systems of the form dx The problem of estimating a function of the state vector, systems of the form without the need of estimatinga the entire state vector, arises in f (x) The problem of estimating function of state vector, dx without the need estimating entire state vector, arises in dt f (x) The problem of of estimating a the function of the the state vector, dx many applications. The design of output feedback controllers without the of The estimating the entire state vector, arises (1) dt dx many applications. designdesign of output feedback controllers yi ffh(x) without theaneed need estimating the entire vector, arises in in (x) i (x) , i 1, ,p dt based on stateoffeedback is astate classical example, (1) many applications. The design of output feedback controllers y h (x) , i 1, ,p dt i i (1) based onisaonly statethefeedback design is afeedback classical example, many The of output controllers zi q(x) y h (x) , i 1, ,p where applications. it statedesign feedback function that needs to be (1) i based on a state feedback design is a classical example, yzi q(x) hi (x) , i 1, ,p where it thethe state feedback function that needs to be based onisand aonly state feedback design is a classical example, estimated not entire state vector. Another important z q(x) where it is only the state feedback function that needs to be estimated and statetovector. Another important whereofit is onlynotthethestate feedback function thatofneeds to be where: z q(x) class applications isentire related the design inferential where: estimated and not the entire state vector. Another important class of applications is related to the design of inferential x nn is the system state estimated and not the entire state vector. Another important control systems, whereisone outputtoisthe measured and ainferential different where: where: class of applications related design of x the measured outputs control output measured and different class ofsystems, applications isone related toisthe of ainferential yi nn, is i the 1, system ,p arestate is the system x output, which iswhere unmeasured, needs todesign be regulated to set control systems, where one output is measured and a different the measured outputs is the system state x y , i 1, ,p arestate output, which is unmeasured, needs to be regulated to set i control systems, where one output is measured and a different are the outputs point. (scalar) output to be estimated y ,,isiithe 1, ,p z output, which is unmeasured, needs to be regulated to set i are the measured measured outputs y 1, ,p point. which is unmeasured, needs to be regulated to set is the (scalar) output to be estimated z output, i n n n n point. is the (scalar) output to be estimated z are smooth , h , qto : be These applications motivate the development of functional and f :zn is nthe point. (scalar) output estimated i: n n are smooth These applications motivate thereduction development of functional and f : n n , h i : n , q : n observers, the aim being a of dimensionality nonlinear functions. The objective is to construct a functional and are smooth f : , h : , q : n n n n These applications motivate the development of functional i : objective and are smooth f : , h , q : observers, the aim being a reduction of dimensionality These applications motivate the development of functional nonlinear functions. The is to construct a functional i relative to athe full-state observer. The notion of a functional observer of order ν < n, which generates an estimate of the observers, aim being a reduction of dimensionality nonlinear functions. The objective is to construct aa functional relative towas athe full-state observer. The notion of a functional observers, aimdefined being a Luenberger’s reduction ofpioneer dimensionality observer of order ν < n, which generates an estimate of nonlinear functions. The objective is to construct functional observer first in work on output z, driven by the output measurements y , i 1, , p .the relative to a full-state observer. The notion of a functional i of order < n, which generates an estimate of the observer first defined in Luenberger’s pioneer work on observer relative towas a full-state observer. The notion of a functional output z, driven byννthe output measurements observer of order < n, which generates an estimate of y , i 1, , p .the observers for linear multivariable systems. Luenberger (1966, i observer was first defined in Luenberger’s pioneer work on . output z, driven by the output measurements y , i 1, , p observers for linear multivariable systems. Luenberger (1966, i observer was first defined in Luenberger’s pioneer work on . a output by thethe output measurements , i 1, , pfor 1971) proved that multivariable it is feasiblesystems. to construct a functional Sectionz,2driven will define notion of functionaly iobserver observers for Luenberger (1966, 1971) proved that multivariable it is feasible to construct a functional observers for linear linear systems. Luenberger (1966, Section of 2 will define(1)the notion of functional observer for toa observer with number of states equal to observability index system the form in a completely analogous manner 1971) proved that to aa functional Section 2 define notion of observer for toaa observer with number of feasible states equal to observability index system 1971) proved that it it is is feasible to construct construct functional the form infor a completely analogous manner Section of 2 will will define(1)the the notion of functional functional observer minus one. Luenberger’s definition linear systems. Section 3 for will observer with number of states equal to observability index system of the form (1) in a completely analogous manner to minus one. observer with number of states equal to observability index system Luenberger’s definition linear systems. Section 3 will of problem the form (1)functional infor a completely analogous manner to4 pose the of observer design. Section minus one. Luenberger’s definition for linear systems. Section 3 will The basic minus one.theory of linear functional observers can be found will pose define the problem of functional observer design. Section 4 Luenberger’s definition for linear systems. Section 3 will notionsofoffunctional exact linearization for the functional pose the problem observer design. Section 4 The basic theory of lineartexts, functional observers can be found will in standard linear systems e.g. in Chen (1984). In recent define notions of exact linearization for the functional pose the problem of functional observer design. Section The basic theory of linear functional observers can be found observer problem. Section 5linearization will develop necessary and4 will define notions of exact for the functional in standard linear systems texts, e.g. in Chen (1984). In recent The basic theory of linear functional observers can be found years, therelinear has systems been atexts, renewed interest in functional observer 5linearization will of develop necessary and will defineproblem. notions Section of thelinearization functional in e.g. in Chen (1984). In sufficient conditions forexact the solution the for total observer Section 5 will develop necessary and years, there has systems been atexts, renewed inet functional in standard standard linear e.g. 1998; in interest Chen (1984). In recent recent observers for linear systems (Tsui, Trinh al., 2006; problem, sufficient conditions for the solution of the total linearization observer problem. problem. Section 5 will develop necessary and years, there has been a renewed interest in functional as well asfora the simple formula for the resulting observers for linear systems (Tsui, 1998; Trinh al., 2006; sufficient conditions solution of the total linearization years, there hasKorovin been aet renewed interest inet functional Darouach, 2000; al., 2008 and 2010; Fernando et problem, as well as a the simple formula the resulting sufficient conditions forSection solution of the for total linearization observers for linear systems (Tsui, 1998; Trinh et al., 2006; functional observer. In 6, the results of Section 5 will Darouach, 2000; Korovin et 2008 and Trinh 2010; Fernando et problem, as well as aa simple formula for the resulting observers systems (Tsui, et al., 2006; al., 2010),for thelinear goal being to al., find the1998; smallest possible order functional observer. In Section 6, the results of Section 5 will problem, as well as simple formula for theleading resulting Darouach, 2000; Korovin et al., 2008 and 2010; Fernando et be specialized to linear time-invariant systems, to al., 2010), the goal being to find the smallest possible order functional observer. In Section 6, the results of Section 5 Darouach, 2000; Korovin et al., 2008 and 2010; Fernando et of the linear functional observer. be specialized to linear time-invariant systems, leading to functional observer. In Section 6, the results of Section 5 will will al., 2010), the goal being to find the smallest possible order simple and easy-to-check conditions for the design of lowerbe specialized to linear time-invariant systems, leading to of the linearthe functional observer. al., 2010), goal being to find the smallest possible order simple and easy-to-check conditions for the design of lowerbe specialized to linear time-invariant systems, leading to of linear observer. functional observers.conditions for the design of lowersimple and easy-to-check Forthe systems, there have been significant order of the nonlinear linear functional functional observer. order functional observers. simple and easy-to-check conditions for the design of lowerFor nonlinearin systems, have observers, been significant functional developments the theory there of full-state with a order For nonlinear systems, there have been functional observers. observers. developments in the theory of full-state observers, with a order For nonlinear systems, there have been significant significant 2. DEFINITION OF A FUNCTIONALOBSERVER FOR A variety of methods and approaches. In particular, in the developments in the theory of full-state observers, with a 2. DEFINITION OF A FUNCTIONALOBSERVER FOR A variety of methods and approaches. In particular, in the developments in the theory of full-state observers, with a NONLINEAR DYNAMIC SYSTEM context of of exact linearization methods In(Krener and Isidori, 2. DEFINITION OF A FUNCTIONALOBSERVER FOR variety methods and approaches. particular, in the NONLINEAR DYNAMIC SYSTEM 2. DEFINITION OF A FUNCTIONALOBSERVER FOR A A contextKrener methods andKravaris, Isidori, variety of exact methods and approaches. In(Krener particular, in the 1983; and linearization Respondek, 1985; Kazantzis and NONLINEAR DYNAMIC SYSTEM context of exact linearization methods (Krener and Isidori, In complete analogy to Luenberger’s construction for the NONLINEAR DYNAMIC SYSTEM 1983; Krener and Respondek, 1985; Kazantzis and Kravaris, contextKazantzis of exact etlinearization methods (Krener and Isidori, 1998; al., 2000; Kreisselmeier and and Engel, 2003; In complete Luenberger’s construction for the 1983; Krener and Respondek, 1985; Kazantzis Kravaris, linear case, weanalogy seek forto a smooth mapping 1998; Kazantzis al., 2000; Kreisselmeier and and Engel, 2003; In 1983; Krener andet2002 Respondek, 1985; Kazantzis Kravaris, complete Luenberger’s construction Krener and Xiao, and 2005; Andrieu and Praly, 2006), linear case, weanalogy seek forto mapping In complete analogy toa smooth Luenberger’s construction for for the the 1998; Kazantzis et al., 2000; Kreisselmeier and Engel, 2003; Krener and Xiao, 2002 and 2005; Andrieu and Praly, 2006), linear case, we seek for a smooth mapping 1998; Kazantzis et al., 2000; Kreisselmeier and Engel, 2003; Luenberger theory2002 for full-state observers has been extended linear case, we seek for a smooth 1 (x) mapping Krener and Xiao, and 2005; Andrieu and Praly, 2006), Luenberger theory forinfull-state observers has extended Krener and Xiao, 2002 and 2005; Andrieu andbeen Praly, 2006), 1 (x) to nonlinear systems a direct and analogous manner. The Luenberger theory for full-state observers has been extended (x) (2) to nonlinear systems in a direct and analogous manner. The Luenberger theory for full-state observers has been extended 1 (x) goal of the present work is to develop a direct generalization (x) 1 (x) (2) to nonlinear systems in aaisdirect and analogous manner. The goal of the present work to develop a direct generalization to nonlinear systems in direct and analogous manner. The (x) (2) (x) of Luenberger’s functional observers toanonlinear systems. (x) goal of the present work is to develop direct generalization (x) (2) of Luenberger’s functional systems. goal of the present work is observers to developtoanonlinear direct generalization (x) of Luenberger’s functional observers to nonlinear systems. (x) of Luenberger’s functional observers to nonlinear systems.
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from n to , to immerse system (1) to a ν-th order system (ν < n), with inputs y i , i 1, , p and output z: -----------------
d (, y1 , , y p ) dt z (, y1 , , y p )
p
which is condition (4) stated earlier. If the functional observer (6) is initialized consistently with the system (1), i.e. if ˆ (0) (x(0)) , then ˆ (t) (x(t)) , and therefore zˆ (t) ˆ (t), y1 (t), , y p (t)
(3)
p
where : , : , the aim being that system (3), driven by the measured outputs y i , i 1, , p of (1), can generate an estimate of unmeasured output z of (1).
q(x) (x), h1(x),
, h p (x)
, h p (x)
F, Hi, Q being n n, 1 n, 1 n matrices respectively, and a linear mapping (x) Tx is considered. Definition 1 tells us that for a linear time-invariant system dx Fx dt (8) y i H i x , i 1, , p
z Qx
(1) the system
p dˆ Aˆ B i y i dt i 1
where f : n n , h i : n , q : n are smooth nonlinear functions, y i , i 1, , p are the measured outputs and z is the output to be estimated , the system dˆ (ˆ , y1 , , y p ) (6) dt ˆz (ˆ , y1 , , y p )
zˆ Cˆ D i y i i 1
will be a functional observer if the following conditions are met: p
TF AT B i H i
p
Q CT D i H i
ˆ
(11)
i 1
for some n matrix T. These are exactly Luenberger’s conditions for a functional observer for a linear timeinvariant system (Luenberger, 1971).
, h p (x)
(7)
3. DESIGNING LOWER-ORDER FUNCTIONAL OBSERVERS
possesses an invariant manifold ˆ (x) with the property that q(x) (x), h 1 (x),
(10)
i 1
the overall dynamics
dx f (x) dt dˆ ˆ , h 1 (x), dt
(9)
p
where : p , : p ( n) , is called a functional observer for (1), if in the series connection x
At this point, it is important to examine the special case of a linear system, where f (x) Fx , h i (x) H i x , q(x) Qx with
The foregoing considerations lead to the following definition of a functional observer:
y1 y p
, h p (x(t)) q(x(t)) t 0 ,
In the presence of initialization errors, additional stability requirements will need to be imposed on the ˆ -dynamics, for the estimate zˆ (t) to asymptotically converge to z(t) .
(5)
Definition 1: Given a dynamic system dx f (x) dt y i h i (x) , i 1, , p z q(x)
which means that the functional observer will be able to exactly reproduce z(t).
(4)
(x(t)), h 1 (x(t)),
But in order for system (1) to be mapped to system (3) under the mapping (x), the following relations have to hold: (x)f (x) (x), h1(x), x
For the design of a functional observer, one must be able to 1 (x) find a continuously differentiable map (x) to (x)
, h p (x) .
In the above definition, the requirement that ˆ (x) is an invariant manifold of (7), i.e. that
satisfy conditions (4) and (5), i.e. such that j (x)f (x), j 1, , is a function of 1 (x), , (x), hi (x) x and q(x) is a function of 1 (x), , (x), hi (x) , i 1, , p .
ˆ (0) x(0) ˆ (t) x(t) t 0 ,
translates to (x) f (x) (x), h 1 (x), x
, h p (x) ,
518
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However, such scalar functions 1 (x), exist, if ν is too small.
, (x) may not
or, slightly more generally,
(, y1 ,
Moreover, even when they do exist, there is an additional very important requirement:
(x)f (x) A(x) B h1(x), x
, h p (x)
for some Hurwitz matrix A , and if in addition θ(x) satisfies condition (5), i.e. that q(x) can be expressed as a function of (x) and hi (x) , i 1, , p , then we have a stable functional
,p
observer with linear dynamics:
h
is a linearly observable system, i (x) , i 1, , p are linearly x independent and ν n p : Available design methods for reduced order state observers (e.g. observer error linearization methods in Kazantzis et al., 2000) will generate a functional observer of order ν n p , by modifying output map of the observer (so that the estimate of z is the observer output instead of the entire state vector). The question is whether a lower order ν n p would be feasible and how to go about designing the functional observer. This is not an easy question because we will be trying to impose too many requirements at the same time.
dˆ Aˆ B(y1 , , y p ) dt zˆ (ˆ , y1 , , y p ) This leads to the Functional Observer Linearization Problem. Note that stricter linearity requirements could be imposed, in particular, that the functions B() and ( , ) be linear as well. This leads to the Functional Observer Total Linearization Problem. A. Functional Observer Linearization Problem
One possible line of attack is to first try to identify functions j (x)f (x), j 1,..., and q(x) 1 (x), , (x) such that x can be expressed as functions of 1 (x), , (x), hi (x) and, as a second step, check stability of the error dynamics.
Given a system of the form (1), find a functional observer of the form dˆ Aˆ B(y1 , , y p ) dt (12) ˆ zˆ (, y1 , , y p ) where is a matrix with stable eigenvalues and
Alternatively, one could try to follow the opposite path. As a first step, try to enforce stability: given some desirable dˆ dynamics for the observer (ˆ , y1 , , y p ) , with ( , ) dt so as to guarantee stability and rapid decay of the error, find 1 (x) so that (x)f (x) (x), h (x), (x) 1 x (x)
B:
p
, :
p . Equivalently, find a n
continuously differentiable mapping :
(x)f (x) A(x) B h1(x), x
, h p (x)
such that
(13)
and
, h p (x) .
q(x) (x), h 1 (x),
The second step will then be to check if q(x) can be expressed as a function of (x), h1 (x), , h p (x) (condition (5)).
, h p (x)
(14)
Assuming that the above problem can be solved, the resulting error dynamics will be linear:
4. EXACT LINEARIZATION OF A FUNCTIONAL OBSERVER
d ˆ (x) A ˆ (x) dt zˆ z ˆ , h 1 (x), , h p (x) (x), h 1 (x),
Along the second line of attack of the functional observer design problem, the most natural - function to work with is the linear one:
(, y1 ,
, yp)
If we can find a continuously differentiable map (x) to satisfy the corresponding partial differential equation (4), i.e.,
All the above requirements can be satisfied when dx f (x) dt y h (x) , i 1, i i
, y p ) A B(y1,
It will then be the eigenvalues of the matrix A that will determine stability of the functional observer and the rate of decay of the error.
(x)f (x) (x), h1(x), , h p (x) will define the x right hand side of the functional observer’s dynamics, it must be such that the functional observer’s dynamics is stable and the decay of the error is sufficiently rapid. Since
507
, h p (x)
(15)
from which, for A having stable eigenvalues, the effect of the initial error ˆ (0) (x(0)) will die out, ˆ (t) will approach (x(t)) asymptotically, therefore zˆ (t) will approach z(t) .
p
, y p ) A B i y i i 1
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B. Functional Observer Total Linearization Problem
Proposition 1: Let f :
p
n
f 0 . Also, x
1. All the eigenvalues of A are non-resonant with F , n
i 1
i.e. no eigenvalue j of A is of the form j
where , Bi , C, D i are , 1, 1 , 1 1 matrices respectively, with having stable eigenvalues. Equivalently, n
find a continuously differentiable mapping : such that
i 1
with i F and m i nonnegative integers, not all zero.
Then the system of first-order partial differential equations
(17)
(x)f (x) A(x) (x) x
and (18)
solution (x) in a neighborhood of x 0 .
i 1
5. NECESSARY AND SUFFICIENT CONDITIONS FOR SOLVABILITY OF THE FUNCTIONAL OBSERVER TOTAL LINEARIZATION PROBLEM
Assuming that the above problem can be solved, the resulting error dynamics will be linear
The trial-and-error approach outlined in the previous section is far from being practical due to the computational effort involved in solving (13) or (17), which will be multiplied by the number of trials until (14) or (18) is satisfied.
(19)
(20)
with initial condition (0) 0 , admits a unique real analytic
p
q(x) C(x) D i h i (x)
d ˆ (x) A ˆ (x) dt zˆ z C ˆ (x)
mii ,
2. 0 does not lie in the convex hull of F .
p (x)f (x) A(x) B i h i (x) x i 1
, :
let A be a ν ν matrix. Assume: (16)
zˆ Cˆ D i y i
n
denote by F the set of eigenvalues of F
p
be real analytic functions with f 0 0, 0 0 and
Given a system of the form (1), find a functional observer of the form
dˆ Aˆ B i y i dt i 1
n
from which zˆ (t) z(t) Ce At ˆ (0) (x(0)) .
To be able to develop a practical approach for designing functional observers, it would be helpful to develop criteria to check if for a given set of ν eigenvalues, there exists a functional observer whose error dynamics is governed by these eigenvalues. This is done in the present Section for the Functional Observer Total Linearization Problem. The main result is as follows:
With the matrix A having stable eigenvalues, the effect of the initialization error ˆ (0) (x(0)) will die out, and zˆ (t) will approach z(t) asymptotically. In order to solve either version of the Functional Observer Linearization Problem, it is natural to first try to solve the system of partial differential equations (13) or (17) given some small-size matrix A with fast enough eigenvalues, and then check to see if q(x) can be expressed as a function of (x) and hi (x) , i 1, , p according to (14) or (18). If it can,
Proposition 2: Under the assumptions of Proposition 1, for a real analytic nonlinear system of the form (1), there exists a functional observer of the form p dˆ Aˆ B i y i dt i 1
we have a functional observer with linear error dynamics – if not, we could try a different matrix A with different eigenvalues and/or larger size, until both conditions can be met.
(16)
p
zˆ Cˆ D i y i i 1
The above approach will be feasible as long as the system of partial differential equations (13) or (17) is solvable. This is a singular system of partial differential equations because their principal part vanishes at the equilibrium point, and therefore standard existence/uniqueness results do not apply. However, we are covered by Lyapunov’s Auxiliary Theorem for singular partial differential equations (Lyapunov, 1992), as long as the nonlinear system (1) is analytic and in the Poincaré domain. The following Proposition is an immediate consequence of Lyapunov’s Auxiliary Theorem. (See also Kazantzis and Kravaris, 1998.)
with the eigenvalues of A being the roots of a given polynomial λν α1λν 1 α ν 1λ α ν , if and only if
Lνf q(x) α1Lνf1q(x) is a
α ν 1Lf q(x) α ν q(x)
-linear combination of
hi (x), Lf hi (x),
, Lνf hi (x),i 1,
, p , where L f n
denotes the Lie derivative operator L f
520
f j (x) x j . j1
IFAC NOLCOS 2016 August 23-25, 2016. Monterey, California, USA Costas Kravaris / IFAC-PapersOnLine 49-18 (2016) 505–510
Proof of Proposition 2: i) Necessity
h i (x) L h (x) f i C C A,b i a i 1 1 L h (x) i f p
Suppose that there exists a functional observer of the form (16) for the system (1). Then, condition (17) will be satisfied 1 (x) . Using the Lie derivative operator for some (x) (x)
p D i L f h i (x) 1L f 1h i (x) i 1
notation, this condition may be written as ---------------------- L f 1 (x) 1 (x) p A B h (x) i i L f (x) (x) i 1 Applying the Lie derivative operator L f to each component
p
i
b i Lkf 1h i (x)
1
hi (x), Lf hi (x),
L
1 f 1L f
L
1 f 1L f
p
C A,b i i 1 I (x)
a
is
CA,b i b i
and
1 2 1 1
a
Ab i 2
1
3
1
1
0
0
0
1 0 0 0
, Lνf hi (x),i 1,
1 Lf q(x) 1L f q(x)
,p ,
, β νi
such that
1L f q(x) q(x)
1 0 Lf h i (x) 1 L h i (x) i i f
i 1
h i (x) i
(24)
Consider the PDE
1 0 0 0 S (x)f (x) x 0 0 α ν α ν 1
(22)
0 m 0 p 1i S(x) h i (x) i 1 1 m νi α1 (25) -------------------------------------------------------------
where
(23)
m1 i with
m ν β ν i i
a
β1 β0 α1 a 1 i i
β0i α ν
(26)
given by (23).
It can be verified that the unique solution of (25) is given by:
At the same time, the functions 1 (x), , (x) must satisfy condition (18), i.e. -------------------------------------------------1 (x) p D h (x) -------------- q(x) C , ----------- i i (x) i 1
S(x) p q(x) 0 i h i (x) i 1 p L f q(x) 0 i L f h i (x) m 1i h i (x) i 1 p 1 2 1 L f q(x) 0 i L f h i (x) m 1i L f h i (x) i 1
for some constants C, Di .
Then, applying the operator Lνf α1Lνf1 α ν 1Lf α ν I on both sides of the above condition and using (21), we obtain: 1 Lf q(x) 1L f q(x)
α ν 1Lf q(x) α ν q(x)
i.e. there exist β0i , β1i ,
h i (x) L h (x) f i 1 L h (x) i f
A 1 b i
,p ,
-linear combination of
hi (x), Lf hi (x),
(21) where
α ν 1Lf q(x) α ν q(x)
, Lνf hi (x),i 1,
Lνf q(x) α1Lνf1q(x)
p
I 1 (x)
ii) Sufficiency: Suppose that
and we can calculate:
1L f h i (x) h i (x)
This proves Lνf q(x) α1Lνf1q(x) is -linear combination of
in the above equation (k-1) times, we find that for k = 2,3,… , Lk (x) 1 (x) f 1 k A (x) Lkf (x)
A k 1b i h i (x) A k 2 b i L f h i (x)
509
1L f q(x) q(x)
m (1) h i (x) i
Therefore,
q(x) 1
521
0
0 S(x)
p
β0i hi (x) i 1
(27)
IFAC NOLCOS 2016 510 Costas Kravaris / IFAC-PapersOnLine 49-18 (2016) 505–510 August 23-25, 2016. Monterey, California, USA
The conclusion is that both conditions for the solution of the Functional Observer Total Linearization Problem are satisfied and 1 0 0 0 m 0 0 p 1i dξˆ ξˆ yi dt i 1 0 1 (28) 0 m νi α ν α ν 1 α1
has formulated notions of exact linearization for functional observer design and has derived a specific criterion for total linearization, including a simple formula for the resulting functional observer. REFERENCES
zˆ 1
0 ξˆ
0
Andrieu, V. and Praly, L. (2006). On the existence of a Kazantzis-Kravaris/ Luenberger observer, SIAM J. Control and Optimization, Vol. 45, 432-456. Chen, C.T. (1984). Linear System Theory and Design, Holt, Rinehart and Winston, New York. Darouach, M. (2000). Existence and design of functional observers for linear systems, IEEE Trans. Automat. Contr., Vol. 45, No.5, 940-943. Fernando, T.L., Trinh, H.M., and Jennings, L. (2010). Functional observability and the design of minimum order linear functional observers, IEEE Trans. Automat. Contr., Vol. 55, No.5, 1268-1273. Kazantzis, N. and Kravaris, C. (1998). Nonlinear observer design using Lyapunov's auxiliary theorem, Systems & Control Letters, Vol. 34, 241-247. Kazantzis, N., Kravaris, C., and Wright, R.A. (2000). Nonlinear observer design for process monitoring, Ind. Eng. Chem. Res., Vol. 39, 408-419. Korovin, S.K., Medvedev, I.S., and Fomichev, V.V. (2008). Minimum dimension of a functional observer with a given convergence rate, Doklady Mathematics, Vol. 78, No.3, 940-943. Korovin, S.K., Medvedev, I.S., and Fomichev, V.V. (2010). Minimum-order functional observers, Computational Mathematics and Modeling, Vol. 21, No.3, 275-296. Kreisselmeier, G. and Engel, R, (2003). Nonlinear observers for autonomous Lipschitz continuous systems, IEEE Trans. Automat. Contr., Vol. 48, No.3, 451-464. Krener, A.J. and Isidori, A. (1983). Linearization by output injection and nonlinear observers, Systems & Control Letters, 3, 47-52. Krener, A.J. and Respondek, W. (1985). Nonlinear observers with linearizable error dynamics, SIAM J. Control and Optimization, Vol. 23, 197-216. Krener, A.J. and Xiao, M. (2002). Nonlinear observer design in the Siegel domain, SIAM J. Control and Optimization, Vol. 41, No. 3, 932-953. Krener, A.J. and Xiao, M. (2005). Nonlinear observer design for smooth systems. In W. Perruquetti and J.-P. Barbot (ed.), Chaos in Automatic Control, 411-422, Marcel Dekker, New York. Luenberger, D.G. (1966). Observers for multivariable systems, IEEE Trans. Automat. Contr., Vol. AC-11, No.2, 190-197. Luenberger, D.G. (1971). An introduction to observers, IEEE Trans. Automat. Contr., Vol. AC-16, No.6, 596-602. Lyapunov, A.M. (1992). The General Problem of the Stability of Motion (English Translation by A. T. Fuller), Taylor & Francis, London. Trinh, H., Fernando, T., and Nahavandi, S. (2006). Partialstate observers for nonlinear systems, IEEE Trans. Automat. Contr., Vol. 51, No.11, 1808-1812. Tsui, C.-C. (1998). What is the minimum function observer order?, J. Franklin Inst., Vol. 335B, No.4, 623-628.
p
β0i yi i 1
is a functional observer. This completes the proof. 6. LOWER-ORDER FUNCTIONAL OBSERVERS FOR LINEAR SYSTEMS The results of the previous section can now be specialized to linear time-invariant systems. The following is a Corollary to Proposition 2. Proposition 3: For a linear time-invariant system of the form
dx Fx dt y i H i x , i 1, , p z Qx
(8)
there exists a functional observer of the form p dˆ Aˆ B i y i dt i 1
(16)
p
zˆ Cˆ D i y i i 1
with the eigenvalues of A being the roots of a given polynomial λν α1λν 1 α ν 1λ α ν , if and only if
(QFν α1QFν 1
α ν 1QF α ν Q)
span Hi , Hi F,
, Hi Fν , i 1,
,p .
The above Proposition provides a simple and easy-to-check feasibility criterion for a lower-order functional observer with a pre-specified set of eigenvalues governing the error dynamics. Moreover, an immediate consequence of the above Proposition is the following: Corollary: Consider a linear time-invariant system of the form (8) with observability index νo. Then, there exists a functional observer of the form (16) of order ν = νo – 1 and arbitrarily assigned eigenvalues. The result of the Corollary is a well-known result, first found by Luenberger (1966, 1971) following a different approach. 7. CONCLUSION The present work has developed a direct generalization of Luenberger’s functional observers to nonlinear systems. It 522