Non-linear estimation is easy
Non-linear state estimation and some related topics, like parametric estimation, fault diagnosis, and perturbation attenuation, are tackled here via a new methodology in numerical differentiation. The corresponding basic system theoretic definitions and properties are presented within the framework of differential algebra, which permits to handle system variables and their derivatives of any order. Several academic examples and their computer simulations, with on-line estimations, are illustrating our viewpoint.
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Since fifteen years non-linear flatness-based control (Fliess, Lévine, Martin & Rouchon (1995, 1999)) has been quite effective in many concrete and industrial applications (see also Lamnabhi-Lagarrigue & Rouchon (2002b); Rudolph (2003); Sira-Ramírez & Agrawal (2004)). On the other hand, most of the problems pertaining to non-linear state estimation, and to related topics, like
fault diagnosis and fault tolerant control,
remain largely open in spite of a huge literature111See,
e.g., the surveys and encyclopedia edited by
Aström, Blanke, Isidori,
Schaufelberger & Sanz (2001); Lamnabhi-Lagarrigue & Rouchon (2002a, b); Levine (1996); Menini, Zaccarian & Abdallah (2006); Nijmeijer & Fossen (1999); Zinober & Owens (2002), and the
references therein.. This paper aims at providing simple and
effective design methods for such questions. This is made possible
by the following facts:
According to the definition given by Diop & Fliess (1991a, b), a
non-linear input-output system is observable if, and only if,
any system variable, a state variable for instance, is a differential function of the control and output variables, i.e., a
function of those variables and their derivatives up to some finite
order. This definition is easily generalized to parametric
identifiability and fault isolability. We will say more generally
that an unknown quantity may be determined if, and only if, it is
expressible as a differential function of the control and output
It follows from this conceptually simple and natural viewpoint that non-linear estimation boils down to numerical differentiation, i.e., to the derivatives estimations of noisy time signals222The origin of flatness-based control may also be traced back to a fresh look at controllability (Fliess (2000)).. This classic ill-posed mathematical problem has been already attacked by numerous means333For some recent references in the control literature, see, e.g., Braci & Diop (2001); Busvelle & Gauthier (2003); Chitour (2002); Dabroom & Khalil (1999); Diop, Fromion & Grizzle (2001); Diop, Grizzle & Chaplais (2000); Diop, Grizzle, Moraal & Stefanopoulou (1994); Duncan, Madl & Pasik-Duncan (1996); Ibrir (2003, 2004); Ibrir & Diop (2004); Kelly, Ortega, Ailon & Loria (1994); Levant (1998, 2003); Su, Zheng, Mueller & Duan (2006). The literature on numerical differentiation might be even larger in signal processing and in other fields of engineering and applied mathematics.. We follow here another thread, which started in Fliess & Sira-Ramírez (2004b) and Fliess, Join, Mboup & Sira-Ramírez (2004, 2005): derivatives estimates are obtained via integrations. This is the explanation of the quite provocative title of this paper444There are of course situations, for instance with a very strong corrupting noise, where the present state of our techniques may be insufficient. See also Remark 2.5. where non-linear asymptotic estimators are replaced by differentiators, which are easy to implement555Other authors like Slotine (1991) had already noticed that “good” numerical differentiators would greatly simplify control synthesis..
This approach to non-linear estimation should be regarded as an extension of techniques for linear closed-loop parametric estimation (Fliess & Sira-Ramírez (2003, 2007)). Those techniques gave as a byproduct linear closed-loop fault diagnosis (Fliess, Join & Sira-Ramírez (2004)), and linear state reconstructors (Fliess & Sira-Ramírez (2004a)), which offer a promising alternative to linear asymptotic observers and to Kalman’s filtering.
Let us start with the first degree polynomial time function , , . Rewrite thanks to classic operational calculus (see, e.g., Yosida (1984)) as . Multiply both sides by :
Take the derivative of both sides with respect to , which corresponds in the time domain to the multiplication by :
The coefficients are obtained via the triangular system of equations (Non-linear estimation is easy)-(Non-linear estimation is easy). We get rid of the time derivatives, i.e., of , , and , by multiplying both sides of Equations (Non-linear estimation is easy)-(Non-linear estimation is easy) by , . The corresponding iterated time integrals are low pass filters which attenuate the corrupting noises, which are viewed as highly fluctuating phenomena (cf. Fliess (2006)). A quite short time window is sufficient for obtaining accurate values of , .
The extension to polynomial functions of higher degree is straightforward. For derivatives estimates up to some finite order of a given smooth function , take a suitable truncated Taylor expansion around a given time instant , and apply the previous computations. Resetting and utilizing sliding time windows permit to estimate derivatives of various orders at any sampled time instant.
Note that our differentiators are not of asymptotic nature, and do not require any statistical knowledge of the corrupting noises. Those two fundamental features remain therefore valid for our non-linear estimation666They are also valid for the linear estimation questions listed in Remark 1.1.. This is a change of paradigms when compared to most of today’s approaches777See, e.g., Schweppe (1973); Jaulin, Kiefer, Didrit & Walter (2001), and the references therein, for other non-statistical approaches..
Our paper is organized as follows. Section Non-linear estimation is easy deals with the differential algebraic setting for nonlinear systems, which was introduced in Fliess (1989, 1990). When compared to those expositions and to other ones like Fliess, Lévine, Martin & Rouchon (1995); Delaleau (2002); Rudolph (2003); Sira-Ramírez & Agrawal (2004), the novelty lies in the two following points:
We provide simple and natural definitions related to non-linear diagnosis such as detectability, isolability, parity equations, and residuals, which are straightforward extensions of the module-theoretic approach in Fliess, Join & Sira-Ramírez (2004) for linear systems.
The main reason if not the only one for utilizing differential algebra is the absolute necessity of considering derivatives of arbitrary order of the system variables. Note that this could have been also achieved with the differential geometric language of infinite order prolongations (see, e.g., Fliess, Lévine, Martin & Rouchon (1997, 1999))888The choice between the algebraic and geometric languages is a delicate matter. The formalism of differential algebra is perhaps suppler and more elegant, whereas infinite prolongations permit to take advantage of the integration of partial differential equations. This last point plays a crucial rôle in the theoretical study of flatness (see, e.g., Chetverikov (2004); Martin & Rouchon (1994, 1995); van Nieuwstadt, Rathinam & Murray (1998); Pomet (1997); Sastry (1999), and the references therein) but seems to be unimportant here. Differential algebra on the other hand permitted to introduce quasi-static state feedbacks (Delaleau & Pereira da Silva (1998a, b)), which are quite helpful in feedback synthesis (see also Delaleau & Rudolph (1998); Rudolph & Delaleau (1998)). The connection of differential algebra with constructive and computer algebra might be useful in control (see, e.g., Diop (1991, 1992); Glad (2006), and the references therein)..
Section Non-linear estimation is easy details Subsection 1.1 on numerical differentiation.
Illustrations are provided by several academic examples999These examples happen to be flat, although our estimation techniques are not at all restricted to such systems. We could have examined as well uncontrolled systems and/or non-flat systems. The control of non-flat systems, which is much more delicate (see, e.g., Fliess, Lévine, Martin & Rouchon (1995); Sira-Ramírez & Agrawal (2004), and the references therein), is beyond the scope of this article. and their numerical simulations101010Any interested reader may ask C. Join for the corresponding computer programs (Cedric.Join@cran.uhp-nancy.fr). which we wrote in a such a style that they are easy to grasp without understanding the algebraic subtleties of Section Non-linear estimation is easy:
Closed-loop parametric identification is achieved in Section 9.
Section 13 deals with closed-loop fault diagnosis and fault tolerant control.
Perturbation attenuation is presented in Section Non-linear estimation is easy, via linear and non-linear case-studies.
Commutative algebra, which is mainly concerned with the study of commutative rings and fields, provides the right tools for understanding algebraic equations (see, e.g., Hartshorne (1977); Eisenbud (1995)). Differential algebra, which was mainly founded by Ritt (1950) and Kolchin (1973), extends to differential equations concepts and results from commutative algebra111111Algebraic equations are differential equations of order ..
A differential ring , or, more precisely, an ordinary differential ring, (see, e.g., Kolchin (1973) and Chambert-Loir (2005)) will be here a commutative ring121212See, e.g., Atiyah & Macdonald (1969); Chambert-Loir (2005) for basic notions in commutative algebra. which is equipped with a single derivation such that, for any ,
where , , . A differential field, or, more precisely, an ordinary differential field, is a differential ring which is a field. A constant of is an element such that . A (differential) ring (resp. field) of constants is a differential ring (resp. field) which only contains constants. The set of all constant elements of is a subring (resp. subfield), which is called the subring (resp.subfield) of constants.
A differential ring (resp. field) extension is given by two differential rings (resp. fields) , , such that , and qthe derivation of is the restriction to of the derivation of .
Notation Let be a subset of . Write (resp. ) the differential subring (resp. subfield) of
generated by and .
Notation Let be a differential field and a set of differential
indeterminates, i.e., of indeterminates and their derivatives of
any order. Write the differential ring of differential polynomials, i.e., of polynomials belonging to . Any
differential polynomial is of the form , .
Notation If and are differential
fields, the corresponding field extension is often written .
A differential ideal of is an ideal which is also a differential subring. It is said to be prime if, and only if, is prime in the usual sense.
All fields are assumed to be of characteristic zero. Assume also that the differential field extension is finitely generated, i.e., there exists a finite subset such that . An element of is said to be differentially algebraic over if, and only if, it satisfies an algebraic differential equation with coefficients in : there exists a non-zero polynomial over , in several indeterminates, such that . It is said to be differentially transcendental over if, and only if, it is not differentially algebraic. The extension is said to be differentially algebraic if, and only if, any element of is differentially algebraic over . An extension which is not differentially algebraic is said to be differentially transcendental.
The following result is playing an important rôle:
The extension is differentially algebraic if, and only if, its transcendence degree is finite.
A set of elements in is said to be differentially algebraically independent over if, and only if, the set of derivatives of any order is algebraically independent over . If a set is not differentially algebraically independent over , it is differentially algebraically dependent over . An independent set which is maximal with respect to inclusion is called a differential transcendence basis. The cardinalities, i.e., the numbers of elements, of two such bases are equal. This cardinality is the differential transcendence degree of the extension ; it is written . Note that this degree is if, and only if, is differentially algebraic.
Kähler differentials (see, e.g., Hartshorne (1977); Eisenbud (1995)) provide a kind of analogue of infinitesimal calculus in commutative algebra. They have been extended to differential algebra by Johnson (1969). Consider again the extension . Denote by
the set of linear differential operators , , which is a left and right principal ideal ring (see, e.g., McConnell & Robson (2000));
the left -module of Kähler differentials of the extension ;
the (Kähler) differential of .
The next two properties are equivalent:
The set is differentially algebraically dependent (resp. independent) over .
The set is -linearly dependent (resp. independent).
The module satisfies the following properties:
The rank131313See, e.g., McConnell & Robson (2000). of is equal to the differential transcendence degree of .
is torsion141414See, e.g., McConnell & Robson (2000). if, and only if, is differentially algebraic.
. It is therefore finite if, and only if, is differentially algebraic.
if, and only if, is algebraic.
Let be a given differential ground field. A (nonlinear) (input-output) system is a finitely generated differential extension . Set where
is a finite set of system variables, which contains the sets and of control and output variables,
denotes the fault variables,
denotes the perturbation, or disturbance, variables.
They satisfy the following properties:
The control, fault and perturbation variables do not “interact”, i.e., the differential extensions , and are linearly disjoint151515See, e.g., Eisenbud (1995)..
The control (resp. fault) variables are assumed to be independent, i.e., (resp. W) is a differential transcendence basis of (resp. ).
The extension is differentially algebraic.
Assume that the differential ideal generated by is prime161616Any reader with a good algebraic background will notice a connection with the notion of differential specialization (see, e.g., Kolchin (1973)).. Write
the quotient differential ring, where the nominal system and fault variables , are the canonical images of , W. To those nominal variables corresponds the nominal system171717Let us explain those algebraic manipulations in plain words. Ignoring the perturbation variables in the original system yields the nominal system. , where is the quotient field of , which is an integral domain, i.e., without zero divisors. The extension is differentially algebraic.
Assume as above that the differential ideal generated by is prime. Write
where the pure system variables are the canonical images of . To those pure variables corresponds the pure system181818Ignoring as above the fault variables in the nominal system yields the pure system. , where is the quotient field of . The extension is differentially algebraic.
We make moreover the following natural assumptions: ,
Remember that differential algebra considers algebraic differential equations, i.e., differential equations which only contain polynomial functions of the variables and their derivatives up to some finite order. This is of course not always the case in practice. In the example of Section 3.4, for instance, appears the transcendental function . As already noted in Fliess, Lévine, Martin & Rouchon (1995), we recover algebraic differential equations by introducing .
We know, from proposition 2.1, that the transcendence degree of the extension is finite, say . Let be a transcendence basis. Any derivative , , and any output variable , , are algebraically dependent over on :
where , , i.e., the coefficients of the polynomials , depend on the control, fault and perturbation variables and on their derivatives up to some finite order.
Eq. ( ‣ Non-linear estimation is easy) becomes for the nominal system
where , , i.e., the coefficients of and depend on the nominal control and fault variables and their derivatives and no more on the perturbation variables and their derivatives.
We get for the pure system
where , , i.e., the coefficients of and depend only on the pure control variables and their derivatives.
The derivatives of the control variables in the equations of the dynamics cannot be in general removed (see Delaleau & Respondek (1995)).
Call (resp. , ) the variational, or linearized, system (resp. nominal system, pure system) of system . Proposition 2.2 yields for pure systems
is of full rank,
The pure transfer matrix191919See Fliess (1994) for more details on transfer matrices of time-varying linear systems, and, more generally, Fliess, Join & Sira-Ramírez (2004), Bourlès (2006) for the module-theoretic approach to linear systems. is the matrix , where , , is the skew quotient field202020See, e.g., McConnell & Robson (2000). of .
The system is said to be (differentially) flat if, and only if, the pure system is (differentially) flat (Fliess, Lévine, Martin & Rouchon (1995)): the algebraic closure of is equal to the algebraic closure of a purely differentially transcendental extension of . It means in other words that there exists a finite subset of such that
are differentially algebraically independent over ,
are algebraic over ,
any pure system variable is algebraic over .
is a (pure) flat, or linearizing, output. For a flat dynamics, it is known that the number of its elements is equal to the number of independent control variables.
Take a system with control and output .
This new definition212121See Fliess & Rudolph (1997) for a definition via infinite prolongations. of observability is “roughly” equivalent (see Diop & Fliess (1991a, b) for details222222The differential algebraic and the differential geometric languages are not equivalent. We cannot therefore hope for a “one-to-one bijection” between definitions and results which are expressed in those two settings.) to its usual differential geometric counterpart due to Hermann & Krener (1977) (see also Conte, Moog & Perdon (1999); Gauthier & Kupka (2001); Isidori (1995); Nijmeijer & van der Schaft (1990); Sontag (1998)).
Set , where is a differential field and a finite set of unknown parameters, which might not be constant. According to Diop & Fliess (1991a, b), a parameter , , is said to be algebraically (resp. rationally) identifiable if, and only if, it is algebraic over (resp. belongs to) :
is rationally identifiable if, and only if, it is equal to a differential rational function over of the variables , , i.e., to a rational function of , and their derivatives up to some finite order, with coefficients in ;
is algebraically identifiable if, and only if, it satisfies an algebraic equation with coefficients in .
More generally, a variable is said to be rationally (resp. algebraically) determinable if, and only if, belongs to (resp. is algebraic over) . A system variable is then said to be rationally (resp. algebraically) observable if, and only if, belongs to (resp. is algebraic over) .
In the case of algebraic determinability, the corresponding algebraic equation might possess several roots which are not easily discriminated (see, e.g., Li, Chiasson, Bodson & Tolbert (2006) for a concrete example).
The fault variable , , is said to be detectable if, and only if, the field extension , where , is differentially transcendental. It means that is indeed “influencing” the output. When considering the variational nominal system, formula ( ‣ Non-linear estimation is easy) yields
where , . Call the fault transfer matrix. The next result is clear:
The fault variable is detectable if, and only if, the column of the fault transfer matrix is non-zero.
A subset of the set W of fault variables is said to be
Differentially algebraically isolable if, and only if, the extension is differentially algebraic. It means that any component of satisfies a parity differential equation, i.e., an algebraic differential equations where the coefficients belong to .
Algebraically isolable if, and only if, the extension is algebraic. It means that the parity differential equation is of order , i.e., it is an algebraic equation with coefficients .
Rationally isolable if, and only if, belongs to . It means that the parity equation is a linear algebraic equation, i.e., any component of may be expressed as a rational function over in the variables , and their derivatives up to some finite order.
The next property is obvious:
Rational isolability algebraic isolability differentially algebraic isolability.
When we will say for short that fault variables are isolable, it will mean that they are differentially algebraically isolable.
Assume that the fault variables belonging to are isolable. Then .
The differential transcendence degree of the extension (resp. ) is equal to (resp. is less than or equal to ). The equality of those two degrees implies our result thanks to the Remark 2.1. ∎
Consider the real-valued polynomial function , , of degree . Rewrite it in the well known notations of operational calculus:
We know utilize , which is sometimes called the algebraic derivative (cf. Mikusinski (1983); Mikusinski & Boehme (1987)). Multiply both sides by , . The quantities , are given by the triangular system of linear equations232323Following Fliess & Sira-Ramírez (2003, 2007), those quantities are said to be linearly identifiable.:
The time derivatives, i.e., , , , are removed by multiplying both sides of Eq. ( ‣ Non-linear estimation is easy) by , .
Consider a real-valued analytic time function defined by the convergent power series , where . Introduce its truncated Taylor expansion
Approximate in the interval , , by its truncated Taylor expansion of order . Introduce the operational analogue of , i.e., , which is an operationally convergent series in the sense of Mikusinski (1983); Mikusinski & Boehme (1987). Denote by , , the numerical estimate of , which is obtained by replacing by in Eq. ( ‣ Non-linear estimation is easy). The next result, which is elementary from an analytic standpoint, provides a mathematical justification for the computer implementations:
Following ( ‣ Non-linear estimation is easy) replace by . The quantity becomes negligible if or . ∎
Assume that our signals are corrupted by additive noises. Those noises are viewed here as highly fluctuating, or oscillatory, phenomena. They may be therefore attenuated by low-pass filters, like iterated time integrals. Remember that those iterated time integrals do occur in Eq. ( ‣ Non-linear estimation is easy) after multiplying both sides by , for large enough.
The estimated value of , which is obtained along those lines, should be viewed as a denoising of the corresponding signal.
See Fliess (2006) for a precise mathematical foundation, which is based on nonstandard analysis. A highly fluctuating function of zero mean is then defined by the following property: its integral over a finite time interval is infinitesimal, i.e., “very small”. Let us emphasize that this approach242424This approach applies as well to multiplicative noises (see Fliess (2006)). The assumption on the noises being only additive is therefore unnecessary., which has been confirmed by numerous computer simulations and several laboratory experiments in control and in signal processing252525For numerical simulations in signal processing, see Fliess, Join, Mboup & Sira-Ramírez (2004, 2005); Fliess, Join, Mboup & Sedoglavic (2005). Some of them are dealing with multiplicative noises., is independent of any probabilistic setting. No knowledge of the statistical properties of the noises is required.
and represent respectively the angular deviation of the motor shaft and the angular position of the inverted pendulum,
, , , , , , and are physical parameters which are assumed to be constant and known.
System ( ‣ Non-linear estimation is easy), which is linearizable by static state feedback, is flat; is a flat output.
Tracking of a given smooth reference trajectory is achieved via the linearizing feedback controller
The subscript “”denotes the estimated value. The design parameters , …, are chosen so that the resulting characteristic polynomial is Hurwitz.
We might nevertheless be interested in obtaining an estimate of the unmeasured state :
The physical parameters have the same numerical values as in Fan & Arcak (2003): , , , , , . The numerical simulations are presented in Figures 2 - 9. Robustness has been tested with an additive white Gaussian noise N(0; 0.01) on the output . Note that the off-line estimations of and , where a “small” delay is allowed, are better than the on-line estimation of .
Consider the fully actuated rigid body, depicted in Figure 10, which is given by the Euler equations
where , , are the measured angular velocities, , , the applied control input torques, , , the constant moments of inertia, which are poorly known. System ( ‣ Non-linear estimation is easy) is stabilized around the origin, for suitably chosen design parameters , , , by the feedback controller, which is an obvious extension of the familiar proportional-integral (PI) regulators,
Write Eq. ( ‣ Non-linear estimation is easy) in the following matrix form:
The output measurements are corrupted by an additive Gaussian white noise . Figure 11 shows an excellent on-line estimation of the three moments of inertia. Set for the design parameters in the controllers ( ‣ Non-linear estimation is easy) and (0) , , , where , . The stabilization with the above estimated values in Figure 12 is quite better than in Figure 13 where the following false values where utilized: , and .
Its mathematical description is given by
The constant and the area of the tank’s bottom are known parameters.
The perturbation is constant but unknown,
The actuator failure , , is constant but unknown. It starts at some unknown time which is not “small”.
Only the output is available for measurement.
The corresponding pure system, where we are ignoring the fault and perturbation variables (cf. Section 2.1),
is flat. Its flat output is . The state variable and control variable are given by
It is desired that the output tracks a given smooth reference trajectory . Rewrite Formulae ( ‣ Non-linear estimation is easy)-( ‣ Non-linear estimation is easy) by taking into account the perturbation variable and the actuator failure :
With reliable on-line estimates and of the failure signal and of the perturbation , we design a failure accommodating linearizing feedback controller. It incorporates a classical robustifying integral action:
This is a generalized proportional integral (GPI) controller (cf. Fliess, Marquez, Delaleau & Sira-Ramírez (2002)) where
denotes the convolution product,
the transfer function of is
is the on-line denoised estimate of (cf. Remark 3.3),
is the on-line estimated value of