On functional observers for polynomial systems with nonlinear oscillations

In this contribution we discuss the design of functional observers for polynomial systems. Our approach is based on a high gain design employing an embedded observer. The functional to be estimated is generated from the high gain observer's state. The restriction to polynomial systems allows the representation of the system's and the observer's dynamics as polynomial ideals. For the calculations we utilize methods from algebraic geometry. Our approach is illustrated on a chaotic system.


OBSERVABILITY AND OBSERVER DESIGN
Lie derivatives and observability.
We consider a nonlinear state space system ẋ = () (system dynamics)  = ℎ() (measured output)  = () (functional output) (1) with analytic functions , ℎ, .The system dynamics is described by the vector field  ∶ ℝ  → ℝ  .The full state  is not known, but mapped via the function ℎ ∶ ℝ  → ℝ to the measured output .The functional  is defined as a function of the state by the map  ∶ ℝ  → ℝ. Observers are used for the deterministic estimation of quantities that are not measured directly.State observers are used to obtain an estimate x of the state .In this contribution we will design a functional observer to compute an estimate v of the functional  [2].
The Lie derivatives of a function ℎ along a vector field  are defined by  +1  ℎ() =      ℎ() with   ℎ() = ℎ  ()() and  0  ℎ() = ℎ(). ( The Lie derivatives of the output map ℎ correspond to the time derivatives of the output : Since the maps  and ℎ are assumed to be analytic, the output curve generated by system (1) with the initial value (0) ∈ ℝ  can be expanded into a convergent Taylor series where the Taylor coefficients are Lie derivatives.Therefore, the series (3) is also called a Lie series [7].Roughly speaking, the property of observability describes, whether the state () at a time  can be obtained from the output signal (⋅).The series expansion (3) shows that all information of the output signal is contained in the sequence (   ℎ()) ∈ℕ 0 .We combine the first  Lie derivatives in the observability map   ∶ ℝ  → ℝ  defined by In the literature [3], the observability of nonlinear systems is axiomatically introduced with the concept of indistinguishability.This indistinguishability of states can be investigated based on the observability map (4).The following definition is consistent with the global observability from the literature [3] and the local observability from the literature [4,5]: Definition 2.1 (State observability).System (1) is said to be (locally) state observable if the observability map is (locally) injective for some  ∈ ℕ.
For topological reasons, the map (4) can only be injective for  ≥ .In case of linear time-invariant systems it is sufficient to examine the observability map for  = .In nonlinear systems it can happen that the map (4) only becomes injective for  > , see the literature [8][9][10] and references cited there.For polynomial system, the maximum number  for which the observability map has to be analyzed can be computed with methods from algebraic geometry [11][12][13][14].To the authors' knowledge, no bound of  for general nonlinear systems is known.

Embedding into observability canonical form.
Assume that the number  has been chosen such that the observability map ( 4) is (at least locally) injective.Then, the mapping embeds system (1) into the observability canonical form [15] with the Brunovsky triple (, ,   ) see the literature [16].The system dynamics is described by the (nonlinear) map Figure 1 shows the structure of the observability canonical form.The system consists of a chain of integrators with the map  at the beginning and the output  at the end of this chain.The difficulty in the computation of the function  is the inversion of the map   in (8).

Observer design in canonical form coordinates.
Let us consider a system in observability canonical form The state  should be estimated by a state observer in Luenberger structure correction term (10) with the observer gain  ∈ ℝ  .The observer consists of a copy of model ( 9) (simulation term).The output error, that is, the difference between the measured output  and the simulated output   ẑ is injected into the observer dynamics with the observer gain  (correction term).We want to design the observer such that the observer state ẑ() converges to the state () of system (9) for  → ∞.To describe the relation between system (9) and its observer we introduce the observer or observation error  =  − ẑ.The observation error is governed by the error dynamics ė = ( −    )  linear +  (() − (ẑ)) nonlinear (11) consisting of a linear and a nonlinear part.The eigenvalues of the matrix  −    can be placed arbitrarily by the observer gain  because the pair (,   ) defined in (7) fulfills the Kalman observability rank condition [17].If the nonlinear map  fulfills a Lipschitz condition, the observer gain can be chosen such that the linear part dominates the nonlinear part and achieves convergence [18][19][20]: Theorem 2.2 (High gain observer design).Assume that the map  is (locally) Lipschitz continuous.Then, the observer gain  can be chosen such that the equilibrium point  = 0 of the error dynamics ( 11) is (locally) asymptotically stable.

POLYNOMIALS, IDEALS AND POLYNOMIAL SYSTEMS
Polynomial ideals.
We want to remind the reader of some notions of algebra [21,22].Here, ℝ[] = ℝ[ 1 , … ,   ] denotes the ring of real polynomials in the variables  1 , … ,   .An important structure in a ring is an ideal: An ideal usually consists of an infinite number of elements.However, every polynomial ideal is finitely generated: Theorem 3.3 (Hilbert's Basis Theorem).For every ideal  ⊆ ℝ[] there exists a finite set of polynomials For a given ideal, the generating set (basis) is not unique.Gröbner bases prove to be helpful for many problems, but depend on the term order of the polynomial ring.

Observer design for polynomial systems.
Assume that (1) is a polynomial system, that is, the maps , ℎ,  consist of polynomials.Then, all Lie derivatives (2) are polynomials as well.We want to employ algebraic methods to obtain a functional observer.For the high gain observer (10) we need the map  of the observability canonical form (6). The coordinates  1 , … ,   of the canonical form are defined by Lie derivatives of order 0, … ,  − 1, see (4) and (5).We introduce an additional variable  +1 for the map  given by the Lie derivative of order , see (8).This approach yields the equations F I G U R E 2 State space system with the functional observer.
The equations in ( 12) can be reformulated as polynomials in the variables  1 , … ,   and  1 , … ,  +1 that generate an ideal.
To obtain a representation of the system in the transformed coordinates alone we introduce the elimination ideal which is represented by a Gröbner basis with the polynomials  1 , … ,   ∈ ℝ[ 1 , … ,  +1 ].On the image of the observability map ( 4) with ( 5), these polynomial equations are fulfilled.We collect all polynomials of the form   =   +1 + ⋯ with  ∈ ℝ[ 1 , … ,   ] and obtain a possibly overdetermined but always solvable linear system of equations to determine  +1 = ( 1 , … ,   ) as suggested in the literature [23].
In addition, we want to express the functional in the transformed coordinates for an estimation based on the state of the high gain observer.Similar as above we introduce the elimination ideal with the Gröbner basis  1 , … ,   .Collecting all polynomials of the form   =   + ⋯ with  ∈ ℝ[ 1 , … ,   ] results in a linear system of equations to determine the functional .The structure of the system with the designed functional observer is shown in Figure 2.

EXAMPLE
The Rössler system is a well-known nonlinear system that may exhibit chaotic behavior for appropriate parameter values [24].The system is polynomial with the parameters , , .From the measurement of the first state component  1 we want to estimate the second state component  2 .The first Lie derivatives of the output map are For higher order Lie derivatives we obtain larges expressions.The algebraic calculations were carried out with the open source computer algebra system SageMath [25] using polynomial rings over the rational numbers ℚ with degree lexicographic order (deglex).
F I G U R E 3 Simulation of the Rössler system (15) and its functional observer with  = Embedding into observability canonical form with  = 3.
The elimination ideal ( 14) for the functional is also generated by a single polynomial.In the coordinates of the observability canonical form we obtain Clearly, the rational expressions ( 16) and ( 17) have a pole for  1 −  −  = 0.In the implementation of the functional observer we can suppress this singularity using a saturation function.The numerical simulations were carried out with Scilab 6.1.1 [26].We used the initial condition (0) = (1, 1, 1)  for system (15) and ẑ(0) = (1, 0, 0)  for the observer (10).The observer eigenvalues were placed at −10 resulting in an observer gain  = (30, 300, 1000)  .The nonlinearity ( 16) was saturated at ±1000, the functional (17) at ±20.The simulation results are shown in Figure 3.Although the original functional  and the estimated functional v agree most of the time, we see some spikes due to the singularities in ( 16) and (17).Embedding into observability canonical form with  = 4.

A C K N O W L E D G M E N T S
This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -project number 417698841.Open access funding enabled and organized by Projekt DEAL.

F I G U R E 1
Structure of the observability canonical form(6).

F I G U R E 4
Simulation of the Rössler system (15) and its functional observer with  = 4.