An embedding observer for nonlinear dynamical systems with global convergence

State observers for nonlinear systems are often designed for a canonical form of this system. However, this form may possess singular points, where the vector field is not defined or a Lipschitz condition is not fulfilled. This unpleasant behavior can possibly be avoided using an embedding into a higher dimensional space. A construction of such an embedding and the corresponding inverse map is discussed for polynomial systems using methods from algebraic geometry.


INTRODUCTION
Dynamical systems usually possess an internal state, which cannot be measured directly.Thus, only a projection of the systems state is available as measurement.If an estimate of this state is required, an observer is constructed for this purpose, which is a dynamical system of its own that generates a state estimate from the systems output trajectory.There are many well-known strategies to design such an observer for linear systems, for example, the Luenberger observer [1,2], the Kalman filter [3], or the Kalman-Bucy filter [4].
For nonlinear systems, observer design is more complicated.One approach is to transform the system into its observer normal form, where linear observer error dynamics can be enforced like in the linear case.However, this normal form does not exist for all systems.A wider class of systems allows the transformation into the observability normal form.In this form, high-gain observers can be constructed systematically [5][6][7].As a precondition for the observers convergence, the vector field in the normal form must be globally Lipschitz continuous.This is also not the case for all systems.However, for those with a global attractor, the vector field can be extended in a global Lipschitz continuous way outside the attractor [5].
Nonetheless, there are nonlinear systems, where the vector field in the observability normal form is not defined everywhere in the global attractor or fails to obey a local Lipschitz condition.This can sometimes be mitigated by embedding the observability normal form into a higher dimensional space.If this is possible, the observer can be designed based on the embedded system ignoring the embedding using established methods.Then, a projection is required to map the higher dimensional observer state to an estimate for the systems state.This contribution shows how to systematically construct such an embedding and projection for polynomial system.This article is structured as follows: Since the construction relies on polynomial ideals, the basic concepts are recapped in Section 2. A short introduction into nonlinear observability follows in Section 3. Based in these ideas, the construction of the observability normal form as well as the projection are handled in Sections 4 and 5, respectively.Section 6 shows this construction for an example system.

POLYNOMIAL IDEALS AND ELIMINATION
As the methods used herein to obtain a representation for the system dynamics in a special form are based on polynomial algebra, a short introduction into the basic concepts is given.The reader is referred to [8] for a more detailed explanation.
To this end, polynomials in several intermediates  1 , … ,   over the field of reals ℝ are considered, 1 that is, polynomials in the ring is a special subset of the ring, which is closed under addition and multiplication with any polynomial: Any polynomial ideal  is finitely generated, that is, there exists a finite set { 1 , … ,   } ⊂ ℝ[] of generator polynomial such that This is known as Hilbert's basis theorem [8, pp. 76].The set of generator polynomials is referred by the generating set of basis of the ideal.The existence of a finite generating set allows the (finite) representation of an ideal.There are different generating sets for the same ideal.
An ideal  can be identified with a set of polynomial equations  1 () = 0, … ,   () = 0. Thus, the sum of polynomials in an ideal or any multiple must also evaluate to zero, which coincides with the definition.The condition 0 ∈  resembles the fact that the equation 0 = 0 is also true.

Elimination of variables
In contrast to the univariable case, there are different term orders for multivariable polynomials.Such a term order is a total ordering for the monomials and obeys the rule for arbitrary monomials , , .The lexicographical ordering compares the monomials by their exponent of   first, then in case of a tie by the exponent in  −1 , and so on.The sorting of the individual variables may be arbitrarily chosen.A graduated lexicographical ordering compares the terms by the total degree || first, then, in case of a tie lexicographical.With a fixed ordering, there is a well-defined leading term of a polynomial, namely, that one that is greater than all other terms.The Gröbner basis of an ideal  is a special generating set with the property that the leading terms of the basis polynomials generate the ideal of leading terms of the polynomials in .Gröbner bases are thus especially important for the decision of the ideal membership problem and, therefore, also for comparison of ideals.Another useful property is that a Gröbner basis with respect to a lexicographical ordering  1 < ⋯ <   resembles a triangular system of polynomial equations.Furthermore, the subset of basis polynomials depending only on  1 ,  2 , … ,   corresponds to all equations in these variables.Thus, such a Gröbner basis with respect to a lexicographical ordering can be used to eliminate variables from a polynomial system of equations.The ideal generated by polynomials in the variables  1 , … ,   only is called the elimination ideal and denoted by  ∩ ℝ[ 1 , … ,   ]. rational coefficients.These numbers allow an exact representation in the computer, which is crucial for the correctness of the algorithms.Nonetheless, the basic concepts can easily be discussed using the real field.
Usually, computing with graduated term orders usually behaves computational better, because Gröbner bases tend to have less basis polynomials with fewer terms.Thus, for the purpose of eliminating some variables, one prefers to use a block ordering such that  1 , … ,   <  +1 , … ,   , that is, the terms are sorted first lexicographically blockwise, then by a graded ordering for each block second.

Real radicals
The real variety  ⊆ ℝ  of an ideal  ⊆ ℝ[] is the set of real solutions of all polynomial equations in .There are in general different ideals with the same real variety.All are subsets of one ideal, the real radical ℝ √  of .An ideal, which equals its real radical, is called real.
Real ideals are useful to compare systems of polynomial equations regarding their solutions.In addition, a basis of the real radical is a simpler form of the polynomial equations.

OBSERVABILITY OF DYNAMICAL SYSTEMS
To this end, autonomous dynamical systems in the form with an analytic vector field  and an analytic scalar field ℎ are considered.By the Picard-Lindelöff theorem, the solution of this system is locally analytic, that is, analytic in some interval [−, ] for each initial value  0 ∈ .Thus, the corresponding output trajectory  ∶ [−, ] → ℝ,  ↦ () is locally analytic as well.This allows to expand the output trajectory on an interval [-T,T] in a convergent Maclaurin series The coefficients of this series are the th-order Lie derivatives, which are defined recursively by [9] For this reason, the series (2) is also referred as the Lie series.Since the series is convergent, there is a one-to-one correspondence between the output trajectory and the series coefficients, that is, the Lie derivatives evaluated at the initial state.These Lie derivatives are collected in the so-called observability map ) .
Two states , x ∈  of system (1) are called indistinguishable in an interval [−, ], if the output trajectories corresponding to these initial values coincide.For systems with analytical fields, the indistinguishability on a sufficiently small interval is equivalent with the same image under the observability map, that is, the states are indistinguishable, if () = ( x).
A system of the form (1) is called globally observable, if there are no two different indistinguishable states, that is, In other words, the system is globally observable, if the observability map is injective.If this is the case, it is possible to uniquely obtain the initial state from the output trajectory.Similarly, the system is called locally observable at a point  0 ∈ , if the observability map is locally injective in a neighborhood of  0 .

Polynomial systems
For linear systems ẋ = ,  =  T  with  ∈ ℝ × ,  ∈ ℝ  ,  ∈ ℝ  , the observability map  is a linear map, where () =  and is the observability matrix.Therefore, global and local observability are equivalent and are identical to Kalman's rank criterion.By the Cayley Hamilton theorem, the powers  0 ,  1 , 2 , … ,   of matrix  are linear dependent, because they obey   () = 0, where   is the characteristic polynomial of .Therefore, all higher powers can be written as a linear combination of  0 , … ,  −1 and the observability of a linear system can be determined by the first  rows of the observability matrix alone.A similar result does not hold for nonlinear systems of some state dimension .In general, more output derivatives than the state dimension are required in order to uniquely restore the state, see [10] for examples.In this contribution, the injectivity of the observability map, which was truncated after the first  ≥  components, is tested for injectivity using quantifier elimination on (3).If this truncated map is injective, so is the untruncated one.By increasing the number  of considered components of , the test may eventually yield a positive result.This results also in a lower bound  of required output derivatives for a global observer.However, no statement can be made for not globally observable systems using this approach.
A similar approach based on polynomial ideals can be used to decide global observability of a polynomial system algorithmically [11,12].Also, the local observability is decidable and the algorithm in [13] also yields the set of not locally observable points.Using these methods, the required number  of Lie derivatives for (local) injectivity of the truncated map is computed, too.
The reader is referred to these publications for the purpose of the system's observability properties.In the sequel, the system will be embedded in a space of dimension , where  is chosen such that the truncated observability map   , which contains the first  components of , is injective.This gives a lower bound for the dimension  of the embedding.However, it will become apparent that this number is not sufficient for a converging observer.

OBSERVABILITY CANONICAL FORM
If the truncated observability map   is injective for some  ∈ ℕ, it induces a state transformation  →   (),  =   ().In these coordinates, the system takes the form where the output map is simply given by  =  1 .This special form is called the observability normal form and can be written as with the Brunovski triple (, ,  T ).In general, the number  of coordinates introduced is larger than the dimension of the state space , in which case we have an embedding in the topological sense. 2 This means that the (redundant) coordinates  may be algebraically dependent.Thus, there are different representations for the function  in terms of the redundant coordinates .
The function  ∶   () → ℝ occurring in (4) may be computed by Although such a function defined for every  ∈   () exists, it may not be Lipschitz continuous, a condition required for the observer based on this form.Thus, a construction of a canonical form (4) with Lipschitz continuous function  will be required.
For polynomial systems, it is not required to invert the observability map.Instead, from the defining equations for the map and the derivative ż , one eliminates the (original) state variables  by computing the elimination ideal A Gröbner basis of this ideal with an elimination ordering ż >  1 , … ,   can be shown [14] to contain generators of the form
Considering the polynomials of degree 1 in ż of   , their leading coefficients (as viewed as a polynomial in ℝ[][ ż ] in the variable ż ) generate an ideal   ⊆ ℝ[].At least one of these polynomial equations can be solved for ż , unless  solves all polynomial equations in   .Due to the polynomial character of the coefficients, the map  ↦ ż is locally Lipschitz continuous in the neighborhood of points  not in the real variety of .Note also that all these equations are equivalent, apart from these singularities, that is, there is a tuple ( 1 , … ,   , ż ) with  ∈ () that solves all of them simultaneously.Problematic are those points  in the real variety of   .Not all of them, however, are in the image of the observability map.These are found by substituting the variables  in the ideal   using the observability map   .This results in an ideal   ⊆ ℝ[], whose real variety is the point set in the original state space, at which the so-introduced observability canonical form (4) in dimension  is not defined.If this variety contains no real points, that is, the real radical fulfills ℝ √   = ⟨1⟩, one can construct a map  that is everywhere local Lipschitz continuous on ().It may be required to increase the dimension  of the embedding beyond the bound for an injective observability map in order to get this property.
Once a sufficient large dimension  for the embedding has been found, it remains to compute the nonlinearity  in (4).Consider the generator polynomials of degree 1 in ż in a Gröbner basis of   , which should read  1 () ż +  1 (), … ,   () ż +   ().
Since the polynomials  1 , … ,   do not vanish simultaneously for  ∈   (), so will the polynomial with positive weights  1 , … ,   (which may also depend on , but must be positive for all  ∈ ℝ  ).Thus, a global solution of this usually redundant system of equations is given by ż = () = −  1  1 () 1 () + ⋯ +     ()  () which completes the observability canonical form.

OBSERVERS DESIGN AND INVERSE MAP
Based on the canonical form (4), the observer can be designed in the transformed coordinates.This may be accomplished by stabilizing the observer dynamics by a linear observer error feedback: If the function  is globally Lipschitz continuous, there exists a gain vector  = ( −1 , … ,  0 ) T ∈ ℝ  that ensures asymptotic stability of the observer error  − ẑ1 [5,6].There are different methods to choose such a gain, for example, using linear matrix inequalities [15] based on a Lipschitz constant of the function .The approach in [16] uses a different observer structure with higher dimensional observer state, but achieves a lower observer gain .Not all systems possess a nonlinearity  with a global Lipschitz bound.However, the systems state is usually bounded, or converges to a bounded positive invariant set.Thus, for the observer dynamics, the function  can simply be exchanged by one that is identical within the region of interest, and is smoothly extended in the other region such that a Lipschitz bound exists.
In contrast to the transformed systems state, the observer state ẑ is not restricted to the image under the observability map   , but resides in the higher dimension euclidean space ℝ  due to the corrective term.This requires a projection of the observer state onto the original state space in order to obtain an estimate x for the systems state.Such a projection can also be computed algebraically.For this reason, a Gröbner basis is computed using an lexicographical ordering   >  −1 > ⋯ >  1 >  for the state .This leads to a triangular system.If the algebraic equations that only depend on the variables , and thus, enforce  ∈ (), are omitted from the triangular system, the remaining equations can be solved for  given an observer state ẑ ∈ ℝ  ⊇ ().

EXAMPLE: THE RöSSLER ATTRACTOR
The Rössler system [17] given by the differential equations which is often studied using the parameters  =  = .For these parameters, the system dynamics exhibit chaotic behavior albeit the solution converges to a bounded attractor.The output map  =  1 = ℎ() leads to the observability map .
We will herein omit the observability analysis and directly try to construct the normal form.All computations were carried out using the computer algebra system Singular [18] using SageMath [19] as a front end.
Starting with dimension  = 3 for the embedding, that is, the same dimension as the systems state space, one has equations ( 1 ,  2 ,  3 , ż3 ) T =  4 () for the state transformation.These equations generate the ideal ⟩ defines a two-dimensional hyperplane in the five-dimensional space.However, we are, in particular, interested in the points in the image of  5 .Substituting the coordinates  =  5 (), one gets  5 = ⟨1⟩ corresponding to no real point.Thus, at any point  ∈  5 (ℝ 3 ), one of the equations given by the first three generators of  5 can be uniquely solved for ż5 .To construct a smooth function  ∶  5 (ℝ 3 ) → ℝ, a weighted sum of these equations is used according to (5).Finally, the observer is constructed according to (6) by selecting a suitable gain.
For the construction of the inverse map, the state variables are not eliminated.This ideal (7) contains all the polynomials in  5 that only depend on , again.In addition, the Gröber basis with respect to the order  1 >  2 >  3 >  contains the polynomials These are the ones that were eliminated by computing  5 .Due to the lexicographical ordering, the last three polynomials depend only on  3 .The leading coefficients are the same as before, and do not vanish simultaneously.Thus, by equating these to zero, they can be solved for  3 the same way as was done for the nonlinear map .The second polynomial depends only on  2 and  3 and the corresponding equation can trivially be solved for  2 .Finally,  1 is given by the first polynomial.
As could be seen from the example, it might be beneficial for observer design to embed the system into a higher dimensional space, although a normal form exists in lower dimension.Here, all singularities could be avoided due to the embedding.

A C K N O W L E D G M E N T S
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