A High-Gain Observer for Embedded Polynomial Dynamical Systems

Abstract

This article deals with the construction of high-gain observers for autonomous polynomial dynamical systems. In contrast to the usual approach, the system’s state is embedded into a higher dimensional Euclidean space. The observer state will be contained in said Euclidean space, which has usually higher dimension than the system’s state space. Due to this embedding it is possible to avoid singularities in the observation matrix. For some systems this even allows constructing global observers in a structured way, which would not be possible in the lower-dimensional case. Finally, the state estimate in the original coordinates can be obtained by a projection. The proposed method is applied on some example systems.

1. Introduction

In many cases the complete system’s state cannot be measured directly. This is sometimes not a problem if one wishes to stabilize or control the system and it is output-feedback controllable. However, state feedback control requires the knowledge of the systems state. In addition, one may want to estimate the non directly measurable state of a dynamical system even in absence of a control. This state estimation is done using observers.

There are many different approaches to design and implement these observers. For linear systems, the probably most well-known is the Luenberger observer [1,2]. Very similar to the Luenberger observer is the Kalman–Bucy filter [3,4], designed for stochastic systems. There is even a counterpart to the linear-quadratic regulator, the deterministic Kalman filter [5].

As expected, observer design for nonlinear systems is usually more challenging. One approach is to transform the system into a suitable normal form. If the system can be transformed into observer normal form, a straight-forward design results in exactly linear error dynamics [6,7]. However, not all systems possess this property. It may be necessary to immerse the system into a higher dimensional space [8,9,10].

An alternative procedure is to transform the system into its observability normal form. This approach is subject to weaker conditions. If the observability matrix is regular, a high-gain observer [11,12,13] can be designed. This typically results in large observer gains, which is why these are called high-gain observers. Recently, there are approaches to mitigate the undesirable high observer gain while relying on the same normal form [14,15,16]. All these approaches assume that the vector field in the normal form is Lipschitz continuous [17]. However, the convergence of the observer may be subject to additional structural conditions [18,19].

This article addresses systems, where the observability matrix is not required to be regular at every point in the state space, i.e., the observability map is not a diffeomorphism. Thus, those systems cannot be immersed into the observability normal form, where the observer design becomes trivial [20]. Nonetheless, there exists an embedding into a higher-dimensional space due to the observability map, such that this map is (locally) invertible [10,21]. By possibly increasing the dimension of the image of this embedding, the vector field in the normal form may be locally Lipschitz continuous. This allows us to construct a high-gain observer for the embedded system. Finally, the observer state can in return be projected to the original state space. Thus, this contribution makes use of a topological embedding of the systems state, contrary to an immersion.

This article is structured as follows: First, the system class considered in the article as well as the observability map, which plays a crucial role, is introduced in Section 2. A short introduction to polynomial ideals and real varieties follows, as the used observability criterion as well as the construction of the observer is based on these concepts. Said observability criterion is recapped and the observability normal form as well as its construction is shown in Section 3. Finally, in Section 4, the high-gain observer, which will be used to observe the embedded state, and the projection to the original state space are discussed. Said observer is constructed for four example systems with different properties and the results are discussed.

2. Preliminaries

2.1. Dynamical Systems and the Observability Map

Consider a real-smooth manifold $\mathcal{M}\subseteq {\mathbb{R}}^n$ and an autonomous system of the form

$$\begin{array}{cc}\hfill \dot{x}& =f\left(x\right)\hfill \end{array}$$

(1a)

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{y}_i& =h_i\left(x\right),i=1,\dots ,m\hfill \end{array}$$

(1b)

with real-analytic vector field $f:\mathcal{M}\to \mathrm{T}\mathcal{M}$ and output maps $h_i:\mathcal{M}\to \mathbb{R}$. The general solution ${\phi }_t\left(x_0\right)$ of (1a) is locally analytic as well. Thus, the output trajectories $y_i\left(t\right)$ for some initial state $x_0\in \mathcal{M}$ can be expanded into its Lie series

$$y_i\left(t\right)=\sum _{k=0}^{\infty }\frac{t^k}{k!}{L}_f^k{h}_i\left(x_0\right),$$

(2)

which is at least locally convergent. This means that the output trajectories are (locally) uniquely determined by the series coefficients, i.e., the Lie derivatives $L_f^{k}h$. Here, $L_f^{k}h$ denotes the k-th Lie derivative, which is defined recursively [22] by

$$L_f^{k}h=L_f{L}_f^{k-1}h,L_{f}h\left(x\right)=L_f^{1}h\left(x\right)=\frac{\partial h}{\partial x}\left(x\right)f\left(x\right),L_f^{0}h=h.$$

All these Lie derivatives form the observability map

$$q\left(x\right)=\left(\begin{array}{c}{h}_1\left(x\right)\\ ⋮\\ h_m\left(x\right)\\ L_f{h}_1\left(x\right)\\ ⋮\\ L_f{h}_m\left(x\right)\\ L_f^2{h}_1\left(x\right)\\ ⋮\end{array}\right).$$

In the following, we will restrict the vector fields f as well as the output maps $h_i$ to polynomials in the state variables x, i.e., $f\in \mathbb{R}{\left[x\right]}^n$ and $h_i\in \mathbb{R}\left[x\right]$. In addition, the manifold $\mathcal{M}$ is the real variety of a polynomial ideal ${\mathfrak{M}}_x⊊\mathbb{R}\left[x\right]$.

2.2. Polynomial Ideals and Real Varieties

A polynomial ideal is a special subset of a polynomial ring. To this end, here only the ring $\mathbb{R}\left[x\right]$ over the reals in the variables $x=(x_1,\dots ,x_n)$ is considered. A polynomial ideal $I\subseteq \mathbb{R}\left[x\right]$ must have the following properties:

• $0\in I$,
• $a,b\in I\Rightarrow a+b\in I$,
• $a\in I,c\in \mathbb{R}\left[x\right]\Rightarrow ac\in I$.

Apart from the trivial ideal $\left\{0\right\}$, which consists only of the zero polynomial, ideals contain in general an infinite numbers of elements. However, ideals over Noetherian rings are finitely generated. This is known as Hilbert’s basis theorem [23], p. 76. Thus, for any polynomial ideal $I\subseteq \mathbb{R}\left[x\right]$ there exists a finite set $G=\left\{g_1,\dots ,g_s\right\}\subset \mathbb{R}\left[x\right]$ such that I can be written as

$$I=\left\{a_1{g}_1+\cdots +a_s{g}_s|a_1,\dots ,a_s\in \mathbb{R}\left[x\right]\right\},$$

which is also denoted by

$$I=〈g_1,\dots ,g_s〉.$$

The set G is called a generating set or basis for the ideal I. There might be different generating sets, possibly with a different number of elements, that generate the same ideal.

In the multivariable case the reduction of a polynomial w.r.t. another polynomial or a set of those is not unique. This makes the decision of the ideal membership problem a little more complicated as in the univariable case. One addresses this problem by fixing a monomial ordering and computes a special basis, the Gröbner basis, which has the property that the leading term of every polynomial in the ideal is a multiple of the leading term of some of its generators. This ensures that every polynomial that is contained in the ideal will reduce to zero by reduction with the polynomials in the Gröbner basis. There are different Gröbner bases for the same ideal. However, for a fixed monomial ordering there is a unique reduced Gröber basis.

The real variety $V={\mathbf{var}}^{\mathbb{R}}\left(I\right)\subseteq {\mathbb{R}}^n$ of an ideal $I\subseteq \mathbb{R}[x_1,\dots ,x_n]$ is the common real zero set of all polynomials in the ideal. This equals the common zero set of all generators of some basis. There are different ideals with the same real variety, e.g., consider the ideal

$$I=〈x^2(1+x^2)〉\subset \mathbb{R}\left[x\right]$$

over the univariable ring. The polynomial generating the ideal has a double root at zero and a pair of purely imaginary roots. Therefore, the real variety equals

$${\mathbf{var}}^{\mathbb{R}}\left(I\right)=\left\{0\right\}\subseteq \mathbb{R}.$$

However, this is also the real variety of a greater ideal $〈x〉$. Therefore, the inverse operation Id that assigns the largest ideal $I=\mathrm{Id}\left(V\right)$ to the variety V with ${\mathbf{var}}^{\mathbb{R}}\left(I\right)=V$ is introduced. The ideal $\mathrm{Id}\left({\mathbf{var}}^{\mathbb{R}}\left(I\right)\right)={\mathbf{rad}}^{\mathbb{R}}\left(I\right)$ is called the real radical of the ideal I. An ideal I for that ${\mathbf{rad}}^{\mathbb{R}}\left(I\right)=I$ holds is called real. Between real ideals and real varieties there is a one-to-one correspondence by the operations ${\mathbf{var}}^{\mathbb{R}}$ and Id, which allows studying the varieties by algebraic means.

Using this one-to-one correspondence, geometric operations on the varieties have an algebraic counterpart for the corresponding ideals: The intersection ${\mathbf{var}}^{\mathbb{R}}\left(I\right)\cap {\mathbf{var}}^{\mathbb{R}}\left(J\right)$ of two varieties is the variety of the ideal sum $I+J$, which is the union of the ideals, i.e., the ideal that contains all polynomials in I or J. Conversely, the union ${\mathbf{var}}^{\mathbb{R}}\left(I\right)\cup {\mathbf{var}}^{\mathbb{R}}\left(J\right)$ of the varieties is the variety of the ideal intersection $I\cap J$, i.e., the ideal that contains the polynomials in both I and J.

The difference set $W={\mathbf{var}}^{\mathbb{R}}\left(I\right)\backslash {\mathbf{var}}^{\mathbb{R}}\left(J\right)$ of two varieties is in general not a variety. However, the Zariski closure $\overline{W}$, which is the smallest variety containing W, has an algebraic counterpart, namely the saturation ideal $I:J^{\infty }$.

Similarly, the projection of a real variety ${\mathbf{var}}^{\mathbb{R}}\left(I\right)$, $I\in \mathbb{R}[x,z]$, onto a coordinate subspace, say the subspace spanned by variables x, corresponds to the elimination ideal $I\cap \mathbb{R}\left[x\right]$. This elimination ideal contains only those polynomials independent of the variables z. In the context of polynomial dynamical systems the elimination ideal is often used to eliminate state variables in order to obtain an input-output relation [24,25].

3. Observability

Observability of nonlinear system is based on the concept of distinguishability [26]. Two states $x,\overline{x}\in \mathcal{M}$ of the system (1a) and (1b) are called indistinguishable if the output trajectories $h_i\left(x\left(t\right)\right)$, $i=1,\dots ,m$ with initial conditions x and $\overline{x}$, respectively, are identical on some interval $[0,T]\ni t$. Since the output trajectories can be expanded into their Lie series (2), the states x and $\overline{x}$ are indistinguishable if

$$q\left(x\right)=q\left(\overline{x}\right)$$

holds. We write

$$\mathcal{I}=\left\{(x,\overline{x})\in {\mathcal{M}}^2|q\left(x\right)=q\left(\overline{x}\right)\right\}$$

for the set of indistinguishable pairs and

$$\mathcal{E}=\left\{(x,\overline{x})\in {\mathcal{M}}^2|x=\overline{x}\right\}$$

(3)

for the set of equal pairs.

A system of the form (1a) and (1b) is locally observable at $x_0\in \mathcal{M}$, if there is a neighbourhood $U_{x_0}\subset \mathcal{M}$ of $x_0$ such that

$$\mathcal{I}\cap U_{x_0}^2=\mathcal{E}\cap U_{x_0}^2.$$

This definition follows [27,28] and differs from that given in [26]. The system is called locally observable, if it is locally observable at each point $x_0\in \mathcal{M}$. The system is called globally observable, if

$$\mathcal{I}=\mathcal{E}.$$

As the sets $\mathcal{I}$ and $\mathcal{E}$ are real varieties, the system’s observability can be tested by algebraic means. In addition, the not locally observable points also form a real variety $\mathcal{N}\subseteq \mathcal{M}$ as can be seen in the following section.

3.1. Observability Criteria

The observability properties can be decided for polynomial systems [29,30,31,32], and the basic concept is recalled: Consider the real variety ${\mathcal{I}}_N\subset {\mathcal{M}}^2$ given by

$$q_N\left(x\right)=q_N\left(\overline{x}\right),x,\overline{x}\in \mathcal{M},$$

where $q_N$ is the truncated observability map including only the first N Lie derivatives up to order $N-1$ of the output map h. With increasing N these varieties form a descending chain

$${\mathcal{I}}_1\supseteq {\mathcal{I}}_2\supseteq {\mathcal{I}}_3\supseteq \cdots .$$

(4)

These are the varieties of the ideals

$${\mathfrak{I}}_k={\mathfrak{M}}_x+{\mathfrak{M}}_{\overline{x}}+〈h\left(x\right)-h\left(\overline{x}\right),\dots ,L_f^{k-1}h\left(x\right)-L_f^{k-1}h\left(\overline{x}\right)〉$$

in the ring $\mathbb{R}[x,\overline{x}]$, where the ideal ${\mathfrak{M}}_x\subset \mathbb{R}[x,\overline{x}]$ contains the equations that govern $x\in \mathcal{M}$ and ${\mathfrak{M}}_{\overline{x}}$ likewise. The ideals ${\mathfrak{I}}_k$ form an ascending chain, which stabilizes by the ascending chain condition [23], p. 80. The same holds for the chain

$${\mathbf{rad}}^{\mathbb{R}}\left({\mathfrak{I}}_1\right)\subseteq {\mathbf{rad}}^{\mathbb{R}}\left({\mathfrak{I}}_2\right)\subseteq \cdots$$

(5)

formed by the real radicals, which eventually stabilizes with some number N of Lie derivatives included. At this point the set $\mathcal{I}$ of indistinguishable pairs equals ${\mathcal{I}}_N$. Thus, the global observability of the system can be decided by comparison with the variety (3), or, algebraically, by comparing the ideals ${\mathbf{rad}}^{\mathbb{R}}\left({\mathfrak{I}}_N\right)$ with

$$\mathbf{Id}\left(\mathcal{E}\right)=\mathfrak{E}={\mathfrak{M}}_x+〈x_1-{\overline{x}}_1,\dots ,x_n-{\overline{x}}_n〉.$$

Similarly, those points where the system is not locally observable can be identified by comparing the varieties $\mathcal{I}$ and $\mathcal{E}$ locally. As equal states are also indistinguishable, i.e., $\mathcal{E}\subseteq \mathcal{I}$, the difference set $\mathcal{I}\backslash \mathcal{E}$ contains points in any neighbourhood of copies $\left(x_0,x_0\right)$ of not locally observable points $x_0$. This means that the Zariski closure of the difference set contains these pairs $\left(x_0,x_0\right)$. Thus, the not locally observable points are precisely the real variety

$$\mathcal{N}={proj}_x\left(\overline{\mathcal{I}\backslash \mathcal{E}}+\mathcal{E}\right)$$

projected onto the state space $\mathcal{M}$. The reader is referred to [32,33] for a more detailed explanation including the proof.

One may also construct a descending chain for the set of not locally observable points using the varieties

$${\mathcal{N}}_k={proj}_x\left(\overline{{\mathcal{I}}_k\backslash \mathcal{E}}+\mathcal{E}\right).$$

This chain will stabilize with the not locally observable points.

3.2. Observability Normal Form

First consider a system (1a) and (1b) with scalar output $y\in \mathbb{R}$ that is locally observable somewhere. For this system we have the corresponding truncated observability map

$$q_N\left(x\right)=\left(\begin{array}{c}h\left(x\right)\\ L_{f}h\left(x\right)\\ L_f^{2}h\left(x\right)\\ ⋮\\ L_f^{N-1}h\left(x\right)\end{array}\right).$$

If N is chosen sufficiently large, the map $q_N$ is locally invertible in a neighbourhood of every locally observable point. Thus, $q_N$ induces a local change of coordinates $z=q_N\left(x\right)$. From Section 3.1 we know a minimum value for the number N of Lie derivatives required. As the map $q_N$ is (locally) invertible, the spaces $\mathcal{M}$ and ${\mathcal{Z}}_N=q_N\left(\mathcal{M}\right)$ must have equal dimension for each point $x\in \mathcal{M}\backslash \mathcal{N}$ and $q_N\left(x\right)$, respectively. (Note that, although $\mathcal{M}$ is a real manifold, $q_N\left(\mathcal{M}\right)$ is not necessarily, as can be seen in the example $\mathcal{M}=\mathbb{R}$, $q_N\left(x\right)=x^2$.) Thus, the coordinates obey some algebraic equations, which can be obtained by eliminating the state variables x from the ideal

$${\mathfrak{M}}_x+〈z_1-h\left(x\right),z_2-L_{f}h\left(x\right),\dots ,z_N-L_f^{N-1}h\left(x\right)〉.$$

The real variety of the elimination ideal is the Zariski closure of ${\mathcal{Z}}_N$.

The coordinates z obey the differential equation

$$\begin{array}{cc}\hfill {\dot{z}}_1& =z_2\hfill \\ \hfill & ⋮\hfill \\ \hfill {\dot{z}}_{N-1}& =z_N\hfill \\ \hfill {\dot{z}}_N& =L_f^{N}h\left(x\right)=L_f^{N}h\left(q_N^{-1}\left(z\right)\right)=\alpha \left(z\right),\hfill \end{array}$$

(6)

which is the observability normal form, possibly including more output derivatives as the state dimension. The function $\alpha :q_N\left(\mathcal{M}\right)\to \mathbb{R}$ is not necessarily local Lipschitz continuous as can be seen from the following example:

Example 1. 

Consider the system

$$\dot{x}=1,x\in \mathbb{R},y=h\left(x\right)=x^3.$$

The output map can be directly inverted to obtain the systems state, thus the system is globally observable, and we may set $N=1$. The polynomial equations

$$\begin{array}{cc}\hfill z_1& =L_f^{0}h\left(x\right)=x^3\hfill \\ \hfill {\dot{z}}_1& =L_f^{1}h\left(x\right)=3x^3\hfill \end{array}$$

generate an ideal over the ring $\mathbb{R}[x,z_1,{\dot{z}}_1]$, where $z_1$ and ${\dot{z}}_1$ are treated as independent variables. The elimination of x yields the differential equation

$${\dot{z}}_1^3-27z_1^2=0,$$

which could be solved for

$${\dot{z}}_1=3z_1^{\frac{2}{3}}=\alpha \left(z_1\right),$$

where the inverse of the odd power function is extended for negative real arguments in such a way that $z^{\frac{1}{n}}=-{(-z)}^{\frac{1}{n}}.$ However, α is not Lipschitz continuous at $z_1=0$. Indeed, for the initial condition $z_1=0$ one has two solutions

$$\begin{array}{cc}\hfill z_1\left(t\right)& =0\hfill \\ \hfill z_1\left(t\right)& =t^3\hfill \end{array}$$

while only the second one equals $q_1\left(x\left(t\right)\right)$.

The unboundedness of the derivative of $\alpha$ is undesirable for observer design. One can, however, mitigate this problem by embedding the transformed state z into an even higher-dimensional space by considering a larger number N of Lie derivatives in the observability map.

Example 1 

(continued). If we set $N=3$ instead, one obtains the map

$$z_1=x^3,z_2=3x^2,z_3=6x.$$

These (redundant) coordinates follow the differential equation

$$\begin{array}{cc}\hfill {\dot{z}}_1& =z_2\hfill \\ \hfill {\dot{z}}_2& =z_3\hfill \\ \hfill {\dot{z}}_3& =6=\alpha \left(z\right).\hfill \end{array}$$

Obviously, the function α is now Lipschitz continuous. Since we have embedded a one-dimensional state into a three-dimensional space ${\mathbb{R}}^3$, the redundant coordinates z must obey the equations

$$216z_1-z_3^3=0,12z_2-z_3^2=0$$

obtained from the elimination ideal

$$〈z_1-x^3,z_2-3x^2,z_3-6x〉\cap \mathbb{R}\left[z\right].$$

Consider the elimination ideals

$${\mathfrak{F}}_k=\left({\mathfrak{M}}_x+〈z_1-h\left(x\right),z_2-L_{f}h\left(x\right),\dots ,z_k-L_f^{k-1}h\left(x\right)〉\right)\cap \mathbb{R}[z_1,\dots ,z_k]$$

(7)

that define the algebraic equations for the embedded state variables z for $k\le N$. Clearly, ${\mathfrak{F}}_k\subseteq {\mathfrak{F}}_{k+1}$, considering both in the ring $\mathbb{R}[z_1,\dots ,z_{k+1}]$. Since ${\dot{z}}_k=z_{k+1}$ for $k<N$, and for any polynomial $g\left(z\right)\in {\mathfrak{F}}_k$ also

$$\frac{\mathrm{d}}{\mathrm{d}t}g\left(z\right)=\sum _{i=1}^k\frac{\partial g}{\partial z_i}\left(z\right){\dot{z}}_i=0$$

holds, the polynomial is contained in ${\mathfrak{F}}_{k+1}$:

$$\sum _{i=1}^k\frac{\partial g}{\partial z_i}\left(z\right)z_{i+1}\in {\mathfrak{F}}_{k+1}.$$

(8)

In general, the function $\alpha$ can be found in the following way: One computes a reduced Gröbner basis of the ideal ${\mathfrak{F}}_{N+1}$ for any block ordering where $x>w>z$ with $w=z_{N+1}$ and $z=z_1,\dots ,z_N$. All polynomials in the basis that are independent of the state variables x generate the elimination ideal

$${\mathfrak{F}}_{N+1}\cap \mathbb{R}[z,w].$$

(9)

Further eliminating w results in the ideal defining the Zariski closure of ${\mathcal{Z}}_N$. Consider the polynomials in the reduced Gröbner basis that explicitly depend on w. Without loss of generality, it is assumed that the degree of these polynomials in the variable w is exactly one. Otherwise, increasing N will reduce the degree due to the inclusion (8). Then the equations in the elimination ideal (9) can be solved for the variable w, which is in general a fractional expression in the variables z. The exception to this property are those values of z that are in the real variety of the ideal generated by the leading coefficients (the coefficients of w) of the polynomials with degree one in w. The function $\alpha :z\mapsto w$ is not defined at these points.

With increasing N the variety with the problematic points decreases. However, in contrast to the observability problem, we do not know a bound or a termination criterion for the number of Lie derivatives to consider, where no further singularities are removed, yet. Still, if the leading coefficient is constant, from the (sole) polynomial including w in ${\mathfrak{F}}_{N+1}$ the polynomial function $\alpha$ can be obtained directly, which is locally Lipschitzian and defined everywhere in ${\mathcal{Z}}_N$.

4. The Embedding Observer

4.1. The High-Gain Observer

For a system in observability normal form (6) with the state $z\in {\mathcal{Z}}_N=q_N\left(\mathcal{M}\right)\subseteq {\mathbb{R}}^N$ and a local Lipschitz continuous function $\alpha :{\mathbb{R}}^N\to \mathbb{R}$ the high-gain observer reads [11]

$$\begin{array}{cc}\hfill {\dot{\widehat{z}}}_1& ={\widehat{z}}_2+{\theta }^1{\ell }_0\left(y-\widehat{y}\right)\hfill \\ \hfill & ⋮\hfill \\ \hfill {\dot{\widehat{z}}}_{N-1}& ={\widehat{z}}_N+{\theta }^{N-1}{\ell }_{N-2}\left(y-\widehat{y}\right)\hfill \\ \hfill {\dot{\widehat{z}}}_N& =\alpha \left(\widehat{z}\right)+{\theta }^N{\ell }_{N-1}\left(y-\widehat{y}\right)\hfill \\ \hfill \widehat{y}& ={\widehat{z}}_1,\hfill \end{array}$$

(10)

where the observer state $\widehat{z}\in {\mathbb{R}}^N$ is not restricted to the subspace ${\mathcal{Z}}_N$ like the systems state z. The coefficients $l_0,\dots ,l_{N-1}$ are the coefficients of a Hurwitz polynomial

$$s^N+l_{N-1}{s}^{N-1}+\cdots +l_0$$

and $\theta$ is a scaling parameter, which must take a minimum value dependent on the Lipschitz constant of the function $\alpha :{\mathbb{R}}^N\to \mathbb{R}$.

For the systems discussed herein, the vector fields are polynomials, which are in general not global Lipschitzian. Still, we are able to show local convergence of the observer, if the (redundant) systems state z is contained in a compact, positively invariant set $\Omega$:

Let further the observer error $\tilde{z}=z-\widehat{z}$ be contained in a compact set $\Gamma$. Then, the map $\alpha$ is Lipschitzian in the sense of

$$\forall z\in \Omega ,\tilde{z}\in \Gamma :\left|\alpha \left(z\right)-\alpha (z-\tilde{z})\right|\le \gamma {∥\tilde{z}∥}_2$$

with Lipschitz constant $\gamma$. Dependent on this Lipschitz constant, there is a sufficiently large value $\theta$ such that the observer error dynamic is asymptotically stable [11]. However, the observer error must lie in a subset of $\Gamma$ whose size deceases very rapidly with increasing $\Theta$, even faster for higher dimensions N of the observer state space. As this is undesirable, one may follow the approach already proposed in [11]: The function $\alpha$ restricted to a ball $\Omega$ defined by

$$\Omega =\left\{z\in {\mathbb{R}}^N:\parallel z\parallel \le R\right\},$$

where $\parallel ·\parallel$ is a norm of the form

$$\parallel z\parallel =\sqrt{z^{\mathrm{T}}Pz},P\succ 0,$$

is extended smoothly to the whole ${\mathbb{R}}^N$ such that the resulting function is globally Lipschitzian. This can be accomplished by introducing a smooth scaling function, e.g.,

$$s_b\left(x\right)=\left\{\begin{array}{cc}1\hfill & x\le 1\hfill \\ \frac{exp\left(\frac{1}{x-b}\right)}{exp\left(\frac{1}{x-b}\right)+exp\left(\frac{1}{1-x}\right)}\hfill & 1<x<b\hfill \\ 0\hfill & x\ge b\hfill \end{array}\right.$$

and using the function

$$\overline{\alpha }\left(z\right)=s_{{\rho }^2}\left({\parallel z\parallel }^2/R^2\right)\alpha \left(z\right)$$

(11)

instead. Note that $\overline{\alpha }$ has a compact support, namely the domain given by $\parallel z\parallel \le \rho R$ such that an upper bound for the global Lipschitz constant can be easily computed. If the nonlinearity is scaled this way, the observer will converge globally for sufficiently large gain $\theta$.

A different approach to compute a suitable observer gain is presented in [19]. Given a bound for the Lipschitz constant, a linear matrix inequality has to be solved, which yields the observer gain. Using this approach, one can expect to get a smaller stabilizing gain.

It is noted that a large observer gain results in a large transient response, which is undesirable. For practical applications the gain cannot be arbitrary high, as measurement noise is amplified as well. This so called peaking phenomenon gets more impact with increasing embedding dimension N. There are, however, approaches to address this problem [15,16].

In general, once the system has be transformed to its observability normal form (6), any observer for the system (6) can be put to use. This includes, but is not limited to, advanced high-gain observers.

4.2. Projection to the Original State Space

To this end, the observer was designed in the embedded state space. It remains to obtain an estimate $\widehat{x}$ for the systems state x given an estimate $\widehat{z}$ for $q_N\left(x\right)$. Since $q_N$ is in general not a diffeomorphism, the observer cannot be simply formulated in the original coordinates. Furthermore, the inverse mapping $q_N^{-1}:{\mathcal{Z}}_N\to \mathcal{M}$ cannot be directly applied as the observer state is contained in a higher-dimensional space ${\mathbb{R}}^N$.

One approach is to use a projection onto the subspace ${\mathcal{Z}}_N$ by solving the minimum problem

$$\widehat{x}=\underset{x}{\arg \min }∥\widehat{z}-q_N\left(x\right)∥.$$

(12)

Given an initial estimate $\widehat{x}\left(0\right)$ one would like to initialize the embedding observer with state $q_N\left(\widehat{x}\left(0\right)\right)$. Thus, an initial guess $\widehat{x}\left(0\right)$ for the projection is already known. Although the initial observer state $\widehat{z}\left(0\right)$ is contained in the subspace ${\mathcal{Z}}_N$, which is the image of $q_N$, the distance $∥\widehat{z}-q_N\left(\widehat{x}\right)∥$ will nonetheless diverge from zero if there is an initial observer error. This may cause the projection $\widehat{x}$ to be discontinuous. If, however, the observer error is comparably small, a gradient descent method will be able to keep track of the projection.

Another very interesting approach is that in [34], pp. 465–467. This gives an explicit formula for solving (12), at least if the system is globally observable and $\mathcal{M}={\mathbb{R}}^n$. If the original system is also embedded, i.e., $\mathcal{M}⊊{\mathbb{R}}^n$, one can still use this approach to project the observer state to a lower-dimensional space ${\mathbb{R}}^n$ in the first place.

In general there is a problem if the state passes through a not locally observable point. This means that there are multiple state trajectories starting from such a point, which yield the same output trajectories. Consequently, the observer cannot distinguish them and the minimum problem (12) is ill conditioned.

5. Examples

5.1. The Duffing Oscillator

We consider a particular instance of the undamped and unexcited Duffing oscillator [35]

$$\begin{array}{cc}\hfill {\dot{x}}_1& =x_2\hfill \\ \hfill {\dot{x}}_2& =x_1-x_1^3\hfill \\ \hfill y& =x_2,\hfill \end{array}$$

(13)

whose trajectories in the phase space can be seen in Figure 1. The chain (5) of radical ideals stabilizes after including the third Lie derivative of the residuum map. The same number of Lie derivatives are required if no radicals are computed. One computes an empty set $\mathcal{N}$ of not locally observable points showing local observability. However, since

$$\begin{array}{cc}\hfill {\mathfrak{I}}_4={\mathfrak{I}}_{\infty }=\mathfrak{E}& \cap 〈x_1+1,{\overline{x}}_1\phantom{+1},x_2,{\overline{x}}_2〉\hfill \\ \hfill & \cap 〈x_1-1,{\overline{x}}_1\phantom{+1},x_2,{\overline{x}}_2〉\hfill \\ \hfill & \cap 〈x_1\phantom{+1},{\overline{x}}_1+1,x_2,{\overline{x}}_2〉\hfill \\ \hfill & \cap 〈x_1-1,{\overline{x}}_1+1,x_2,{\overline{x}}_2〉\hfill \\ \hfill & \cap 〈x_1+1,{\overline{x}}_1-1,x_2,{\overline{x}}_2〉\hfill \\ \hfill & \cap 〈x_1\phantom{+1},{\overline{x}}_1-1,x_2,{\overline{x}}_2〉\hfill \end{array}$$

is different from $\mathfrak{E}=\mathbf{Id}\left(\mathcal{E}\right)$, the system is not globally observable. From this prime decomposition one identifies the three equilibria $\left\{(-1,0),(0,0),(1,0)\right\}$ of the system, where pairs of these cannot be distinguished. This is precisely what is encoded in the six prime components apart from $\mathfrak{E}$.

As the chain has stabilized after including four Lie derivatives, we will have to embed the observers state in a space of dimension at least four using the observability map

$$z=\left(\begin{array}{c}{z}_1\\ z_2\\ z_3\\ z_4\end{array}\right)=q_4\left(x\right)=\left(\begin{array}{c}{x}_2\\ x_1-x_1^3\\ x_2-3x_1^2{x}_2\\ x_1-4x_1^3+3x_1^5-6x_1{x}_2^2\end{array}\right).$$

These dependencies are computed using the ideal ${\mathfrak{F}}_4$ in (7). Of the first three components of z one gets

$$z_3^3+3z_1{z}_3^2+27z_1^3{z}_2^2-4z_1^3=0.$$

(14a)

There are six further equations

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em} z_4^3+3z_2{z}_4^2-36z_1^4{z}_4+36z_1^2{z}_2{z}_3^2+54z_1{z}_2^3{z}_3+36z_1^3{z}_2{z}_3+\hfill & \hfill \\ \hfill 27z_2^5-4z_2^3-216z_1^6{z}_2-36z_1^4{z}_2& =0\hfill \end{array}$$

(14b)

$$\begin{array}{cc}\hfill z_3{z}_4^2+3z_2{z}_3{z}_4+12z_1^3{z}_3^2-\frac{4}{3}{z}_1{z}_3^2+36z_1^2{z}_2^2{z}_3-12z_1^4{z}_3-\frac{4}{3}{z}_1^2{z}_3+& \hfill \\ \hfill 27z_1{z}_2^4-18z_1^3{z}_2^2-4z_1{z}_2^2+\frac{8}{3}{z}_1^3& =0\hfill \end{array}$$

(14c)

$$\begin{array}{cc}\hfill z_1{z}_4^2-z_2{z}_3{z}_4+\frac{4}{3}{z}_1{z}_3^2+12z_1^4{z}_3+\frac{4}{3}{z}_1^2{z}_3+& \hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} 18z_1^3{z}_2^2-12z_1^5-\frac{8}{3}{z}_1^3\hfill & =0\hfill \end{array}$$

(14d)

$$\begin{array}{cc}\hfill z_3^2{z}_4-4z_1^2{z}_4+3z_2{z}_3^2+18z_1^3{z}_2{z}_3+4z_1{z}_2{z}_3+& \hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} 27z_1^2{z}_2^3-36z_1^4{z}_2-4z_1^2{z}_2\hfill & =0\hfill \end{array}$$

(14e)

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}ʠz_1{z}_3{z}_4+2z_1^2{z}_4-z_2{z}_3^2-2z_1{z}_2{z}_3+18z_1^4{z}_2\hfill & =0\hfill \end{array}$$

(14f)

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} z_1{z}_2{z}_4-\frac{2}{3}{z}_1{z}_3^2-z_2^2{z}_3-\frac{2}{3}{z}_1^2{z}_3+\frac{4}{3}{z}_1^3\hfill & =0\hfill \end{array}$$

(14g)

obtained from the reduced Gröbner basis using lexicographical ordering. Although there are altogether seven equations, these still describe a two-dimensional subspace in ${\mathbb{R}}^4$.

In order to get a differential equation for $z_4$, the last component of the integrator chain, the ideal ${\mathfrak{F}}_5$ is considered (with ${\dot{z}}_4=z_5$). Again computing a reduced Gröbner basis w.r.t. lexicographical ordering gives Equations (14a)–(14g) as well as

$$\begin{array}{cc}\hfill z_4{z}_5+7z_3{z}_4+6z_1^3{z}_4+10z_1{z}_4+54z_1^2{z}_2{z}_3-6z_2{z}_3+81z_1{z}_2^3+54z_1^3{z}_2-12z_1{z}_2& =0\hfill \end{array}$$

(14h)

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} z_3{z}_5+7z_3^2+6z_1^3{z}_3+4z_1{z}_3+81z_1^2{z}_2^2-12z_1^2\hfill & =0\hfill \end{array}$$

(14i)

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} z_2{z}_5-3z_3{z}_4-6z_1{z}_4+4z_2{z}_3-48z_1^3{z}_2+4z_1{z}_2\hfill & =0\hfill \end{array}$$

(14j)

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{z}_1{z}_5-3z_3^2-2z_1{z}_3+6z_1^4+4z_1^2\hfill & =0\hfill \end{array}$$

(14k)

depending on $z_5$. As can be seen, the leading terms already have degree one in $z_5$. However, none of these equations can be (uniquely) solved for $z_5$, if $z_1$ to $z_4$ equal zero (The value of $z_5$ is well-defined for $z_1,\dots ,z_4\to 0$ and $z=q_4\left(x\right)$ for some x. This is still not useful for the observer to be designed). Indeed, the three equilibria map to $z=(0,0,0,0)$ under $q_4$.

If the state z is embedded in a space of even higher dimension, one finally arrives at

$$z_7+9z_5+126z_1^2{z}_3+6z_3+441z_1{z}_2^2-6z_1^3-16z_1\in {\mathfrak{F}}_7.$$

The leading coefficient, namely that of $z_7$, is constant. This means that for $z=q_6\left(x\right)$ one has the differential equations

$$\begin{array}{cc}\hfill {\dot{z}}_1& =z_2\hfill \\ \hfill & ⋮\hfill \\ \hfill {\dot{z}}_5& =z_6\hfill \\ \hfill {\dot{z}}_6& =-9z_5-126z_1^2{z}_3-6z_3-441z_1{z}_2^2+6z_1^3+16z_1,\hfill \end{array}$$

which are well-defined and locally Lipschitz everywhere in ${\mathbb{R}}^6$.

The trajectories for the embedded state can be seen in Figure 2. These were computed as the image of the solution of (13) under the observability map $q_6$.

For the observer the nonlinear function $\alpha$ is scaled as in (11), such that the observer can only be applied to bounded orbits. The domain was chosen as the orbit that passes through the point $(1.5,0)$ and its interior. A very conservative bound for the embedded state is given by ${\parallel z\parallel }^2=z^{\mathrm{T}}Pz\le 1$ with

$$\sqrt{P}=diag\left(0.8,0.145,0.103,0.0158,4.07·{10}^{-3},5.26·{10}^{-4}\right).$$

The smooth transition is between $1\le \parallel z\parallel \le \sqrt{2}$, i.e., $R=1$ and $\rho =\sqrt{2}$. As can be seen, the observer converges, if a sufficiently large gain, i.e., a sufficiently small value $\theta$ is chosen, see Figure 3. Here, all eigenvalues of the linear error dynamics were placed at $-1$ and a gain of 10, i.e., $\theta \le 0.1$ seems to be required.

For this system, there are different state trajectories dependent on the initial condition: Initial values with sufficient distance from the origin let the state trace out a handle-shaped path around the origin, which contains all equilibria in its interior. For initial values near the equilibria $(\pm 1,0)$ one gets solutions encircling them, respectively. As the distance of the initial value to the equilibria decreases, the output trajectories look more and more similar. In the limit, the output trajectories cannot be distinguished. Therefore, due to the symmetry of this system, a particularly adverse initialization for the observer is $\widehat{z}\left(0\right)=q_6\left((0,0)\right)$, while the state is near one of the other two equilibria. This scenario was chosen for the simulation, whose result is shown as the orange (light) curve in Figure 2. As can be seen, the observer converges nonetheless. The projection (Figure 4) possesses some discontinuities. These arise from an ill conditioned local minimum problem at these points. Note that with the current implementation it is only guaranteed that the observer converges to the embedded state, and no assertions can be made on the projection, yet.

5.2. An Oscillator with Nonlinear Output Map

Consider the system

$$\begin{array}{cc}\hfill {\dot{x}}_1& =x_2\hfill \\ \hfill {\dot{x}}_2& =-x_1\hfill \\ \hfill y& =x_1^2+x_1\hfill \end{array}$$

(15)

from [36]. In order to prove global observability, the first four Lie derivatives of the output map are required, as was already shown in [37]. Thus, the system has to be embedded in a space of dimension at least four. The ideal ${\mathfrak{F}}_5$ is generated by the polynomials

$$\begin{array}{cc}\hfill & z_5-\frac{2}{3}{z}_4^2-\frac{16}{3}{z}_2{z}_4+7z_3-\frac{32}{3}{z}_2^2+6z_1,\hfill \\ \hfill & z_4^3-\frac{9}{2}{z}_3{z}_4-48z_2^2{z}_4-45z_1{z}_4-\frac{45}{4}{z}_4+36z_2{z}_3-128z_2^3-72z_1{z}_2-\frac{45}{4}{z}_2,\hfill \\ \hfill & z_3{z}_4^2+8z_2{z}_3{z}_4+\frac{3}{2}{z}_2{z}_4-\frac{9}{2}{z}_3^2+16z_2^2{z}_3-18z_1{z}_3-\frac{9}{2}{z}_3+2\frac{1}{2}{z}_2^2-18z_1^2-\frac{9}{2}{z}_1,\hfill \\ \hfill & z_2{z}_4^2+8z_2^2{z}_4+3z_1{z}_4+\frac{3}{4}{z}_4-\frac{9}{2}{z}_2{z}_3+16z_2^3+3z_1{z}_2+\frac{3}{4}{z}_2,\hfill \\ \hfill & z_1{z}_4^2+\frac{1}{4}{z}_4^2+24z_1{z}_3^2+6z_3^2-12z_2^2{z}_3+96z_1^2{z}_3+48z_1{z}_3+\hfill \\ & 6z_3-40z_1{z}_2^2-\frac{49}{4}{z}_2^2+96z_1^3+48z_1^2+6z_1,\hfill \\ \hfill & z_1{z}_3{z}_4+\frac{1}{4}{z}_3{z}_4-\frac{1}{2}{z}_2^2{z}_4+7z_1{z}_2{z}_3+\frac{7}{4}{z}_2{z}_3-\frac{7}{2}{z}_2^3+6z_1^2{z}_2+\frac{3}{2}{z}_1{z}_2,\hfill \\ \hfill & z_2^3{z}_4-6z_1{z}_3^3-\frac{3}{2}{z}_3^3+3z_2^2{z}_3^2-24z_1{z}_2^2{z}_3-6z_2^2{z}_3+72z_1^3{z}_3+60z_1^2{z}_3+3\frac{3}{2}{z}_1{z}_3+\hfill \\ & \frac{3}{2}{z}_3+13z_2^4-60z_1^2{z}_2^2-27z_1{z}_2^2-3z_2^2+96z_1^4+72z_1^3+18z_1^2+\frac{3}{2}{z}_1,\hfill \\ \hfill & z_1{z}_2{z}_4+\frac{1}{4}{z}_2{z}_4-3z_1{z}_3^2-\frac{3}{4}{z}_3^2+\frac{3}{2}{z}_2^2{z}_3-12z_1^2{z}_3-6z_1{z}_3-\frac{3}{4}{z}_3+7z_1{z}_2^2+\hfill \\ & \frac{7}{4}{z}_2^2-12z_1^3-6z_1^2-\frac{3}{4}{z}_1,\hfill \\ \hfill & z_1^2{z}_4+\frac{1}{2}{z}_1{z}_4+\frac{1}{16}{z}_4-\frac{3}{2}{z}_1{z}_2{z}_3-\frac{3}{8}{z}_2{z}_3+\frac{3}{4}{z}_2^3+z_1^2{z}_2+\frac{1}{2}{z}_1{z}_2+\frac{1}{16}{z}_2,\hfill \\ \hfill & z_1^2{z}_3^2+\frac{1}{2}{z}_1{z}_3^2+\frac{1}{16}{z}_3^2-z_1{z}_2^2{z}_3-\frac{1}{4}{z}_2^2{z}_3+4z_1^3{z}_3+3z_1^2{z}_3+\frac{3}{4}{z}_1{z}_3+\frac{1}{16}{z}_3+\hfill \\ & \frac{1}{4}{z}_2^4-2z_1^2{z}_2^2-z_1{z}_2^2-\frac{1}{8}{z}_2^2+4z_1^4+3z_1^3+\frac{3}{4}{z}_1^2+\frac{1}{16}{z}_1\hfill \end{array}$$

forming a reduced Gröbner basis w.r.t. lexicographical order. The first of these as they are listed has a constant coefficient as a polynomial in $z_5$. Thus, the observability normal form has a locally Lipschitz continuous vector field

$$\begin{array}{cc}\hfill {\dot{z}}_1& =z_2\hfill \\ \hfill {\dot{z}}_2& =z_3\hfill \\ \hfill {\dot{z}}_3& =z_4\hfill \\ \hfill {\dot{z}}_4& =\frac{2}{3}{z}_4^2+\frac{16}{3}{z}_2{z}_4-7z_3+\frac{32}{3}{z}_2^2-6z_1\hfill \end{array}$$

even embedded in ${\mathbb{R}}^4$.

A nice property of this system is that the observability map

$$q_4\left(x\right)=\left(\begin{array}{c}{x}_1^2+x_1\\ 2x_1{x}_2+x_2\\ 2x_2^2-2x_1^2-x_1\\ -8x_1{x}_2-x_2\end{array}\right)$$

can be inverted using the globally defined map ${\mathcal{Z}}_4\to {\mathbb{R}}^2$

$$x=q_4^{-1}\left(z\right)=\left(\begin{array}{c}-\frac{2}{9}{z}_4^2-\frac{16}{9}{z}_2{z}_4-\frac{32}{9}{z}_2^2+z_3+2z_1\\ \frac{4}{3}{z}_2+\frac{1}{3}{z}_4\end{array}\right).$$

The preimage of this inverse map can be extended to the whole ${\mathbb{R}}^4$, resulting in a projection to the original state space.

In Figure 5, the trajectories of the embedded state as well as the observer state are shown. As initial condition for the systems state the point $x\left(0\right)=(-\frac{1}{2},0)$ has been chosen, which results in the most flat output trajectory. Albeit the relative small initial observer error $(-0.2,0)$, relatively large deviations result in the first place. This is due to the low sensitivity of the output w.r.t. the error at this state, such that a much larger error can accumulate. However, the observer converges finally.

For this simulation the embedded state was measured using

$$\sqrt{P}=diag\left(0.594,0.421,0.266,0.153\right)$$

such that the sought system trajectory lies definitely within the ellipsoid given by the inequality $\parallel z\parallel =\sqrt{z^{\mathrm{T}}Pz}\le 1$. This bound was obtained using ${\parallel x\parallel }_{\infty }\le 0.7$. Again, all poles were placed at $-1$ and a gain parameter $\theta =0.3$ was chosen. As soon as the linear dynamics dominates, the observer error decreases with about this slope (see Figure 6).

The projection of the observer state using the numerical solution of the optimization problem (12) can be seen in Figure 7. Again, this projection possesses some discontinuities.

5.3. The Lorenz Attractor

The Lorenz Attractor [38] given by the differential equations

$$\begin{array}{cc}\hfill {\dot{x}}_1& =\sigma \left(x_2-x_1\right)\hfill \\ \hfill {\dot{x}}_2& =\rho x_1-x_2-x_1{x}_3\hfill \\ \hfill {\dot{x}}_3& =x_1{x}_2-\beta x_3\hfill \end{array}$$

(16)

is chaotic for the parameter set $(\sigma ,\rho ,\beta )=(10,28,\frac{8}{3})$, but possesses well-known bounded positively invariant sets [39,40,41,42]. We consider the system with output map $y=h\left(x\right)=x_1$. In this case the system can be shown to be locally observable everywhere except at the $x_3$-axis. In order to distinguish all points not on the $x_3$-axis, output derivatives including order three are required, i.e., the truncated observability map

$$q_4\left(x\right)=\left(\begin{array}{c}{x}_1\\ -10x_1+10x_2\\ -10x_1{x}_3+380x_1-110x_2\\ -10x_1^2{x}_2+\frac{710}{3}{x}_1{x}_3-100x_2{x}_3-6880x_1+3910x_2\end{array}\right)$$

is locally invertible at all locally observable points. It can indeed be easily seen from the system dynamics (16) that the $x_3$-axis is an invariant set, which maps to the constant trajectory $y\left(t\right)=0$.

In order to compute the observability normal form, we start with the ideal ${\mathfrak{F}}_5$, whose reduced Gröbner basis reads

$$\begin{array}{cc}\hfill & z_2{z}_5-z_3{z}_4+\frac{49}{3}{z}_2{z}_4-\frac{41}{3}{z}_3^2+\frac{328}{9}{z}_2{z}_3-10z_1^3{z}_3+720z_1{z}_3+3z_1{z}_2^3+\hfill \\ \hfill & \frac{128}{3}{z}_1^2{z}_2^2-\frac{12256}{9}{z}_2^2+\frac{80}{3}{z}_1^3{z}_2-1920z_1{z}_2,\hfill \\ \hfill & z_1{z}_5-z_3^2+\frac{16}{3}{z}_2{z}_3+z_1^3{z}_3-\frac{1417}{9}{z}_1{z}_3+3z_1^2{z}_2^2+\frac{539}{3}{z}_2^2+\hfill \\ \hfill & \frac{79}{3}{z}_1^3{z}_2-\frac{16568}{9}{z}_1{z}_2-\frac{410}{3}{z}_1^4+9840z_1^2,\hfill \\ \hfill & z_1{z}_4-z_2{z}_3+\frac{41}{3}{z}_1{z}_3-11z_2^2+z_1^3{z}_2+\frac{88}{3}{z}_1{z}_2+10z_1^4-720z_1^2.\hfill \end{array}$$

The first two of these polynomials already have degree one in the highest variable $z_5$. However, the coefficients are not constant. This unpleasant behaviour seems to persist, even if the state is embedded in a space of higher dimension. At least did we not succeed in finding a sufficient dimension and had to abort the search due to the increasing computational effort.

To solve the equations for $z_5={\dot{z}}_4$ in a way such that the resulting field is smoothly embedded in ${\mathbb{R}}^4$, the first two generators of ${\mathfrak{F}}_5$ can be multiplied by $z_2$ and $z_1$, respectively. The leading coefficient of their sum does not vanish, except at the not locally observable points. Thus, we compute

$$\begin{array}{c}\hfill {\dot{z}}_4=\alpha \left(z\right)=\frac{1}{z_1^2+z_2^2}(-z_2{z}_3{z}_4+\frac{49}{3}{z}_2^2{z}_4-\frac{41}{3}{z}_2{z}_3^2-z_1{z}_3^2+\frac{328}{9}{z}_2^2{z}_3-10z_1^3{z}_2{z}_3+\\ \hfill \frac{2176}{3}{z}_1{z}_2{z}_3+z_1^4{z}_3-\frac{1417}{9}{z}_1^2{z}_3+3z_1{z}_2^4+\frac{128}{3}{z}_1^2{z}_2^3-\frac{12256}{9}{z}_2^3+\\ \hfill \frac{89}{3}{z}_1^3{z}_2^2-\frac{5221}{3}{z}_1{z}_2^2+\frac{79}{3}{z}_1^4{z}_2-\frac{16568}{9}{z}_1^2{z}_2-\frac{410}{3}{z}_1^5+9840z_1^3)\end{array}$$

(17)

for the nonlinear function $\alpha$. Note that for $z\in {\mathcal{Z}}_4$ the function $\alpha \left(z\right)$ has the same limit for $z\to 0$. However, $\alpha$ is unbounded, if $z\in {\mathbb{R}}^4$ is not restricted to ${\mathcal{Z}}_4$. This makes it more complicated to find a Lipschitz constant for $\alpha$, since the system trajectory may come arbitrary close to the $x_3$-axis. However, since the observer cannot observe the system at this set, one could exclude a neighbourhood from the convergence considerations and regularize the function (17). In contrast to arbitrary introduced singularities, the non-observability is a system property.

5.4. The Van Der Pol Oscillator

As a final example we consider an instance of the Van der Pol oscillator

$$\begin{array}{cc}\hfill {\dot{x}}_1& =x_2\hfill \\ \hfill {\dot{x}}_2& =-x_1-x_2\left(x_1^2-1\right)\hfill \end{array}$$

with output map $y=h\left(x\right)=x_2$. An observer analysis shows that this system is locally observable everywhere. Four Lie derivatives of the output map are required for the local invertibility of the observability map $q_4$. If an additional output derivative is included, the map $q_5$ is even injective. Thus, the system is globally observable.

Although one would expect that a globally smooth vector field for the observability normal form exists in some dimension, we were not able to compute such a one, yet. The ideal ${\mathfrak{F}}_6$ has the form

$$〈z_4{z}_6+\cdots ,z_3{z}_6+\cdots ,z_2{z}_6+\cdots ,z_1^2{z}_6+\cdots ,\dots 〉$$

while computing ${\mathcal{F}}_7$ is still not yet feasible. The problematic point when solving for $z_6={\dot{z}}_5$ is only the origin $(x_1,x_2)=(0,0)$ (or its image under the observability map). This should not be a big issue for observing the Van der Pol oscillator. Indeed, all trajectories starting in the open circle $x_1^2+x_2^2<1$ will leave this circle, as can be shown using the time-reversed system and the Lyapunov function $V\left(x\right)=-x_1^2-x_2^2$. Thus, the Van der Pol oscillator possesses a positively invariant set $\Omega$ where the observer has a Lipschitz continuous vector field. The vector field, especially the nonlinearity $\alpha$, for the observer can be constructed similar as for the Lorenz attractor. However, the nonlinearity $\alpha$ has to be scaled for large magnitudes of the observer state $\widehat{z}$ and for small magnitudes as well.

For this particular system very high observer gains were required in order to achieve convergence. A better bound for the image $q_5\left(\Omega \right)$ of the positively invariant set would possibly allow a much smaller gain.

6. Numerical Implementation

In order to verify the results presented in this article, the observers are simulated on a digital processor. Given initial conditions $x_0=x\left(0\right)$ and ${\widehat{z}}_0=q\left({\widehat{x}}_0\right)$ for the systems and the observers state, respectively, the corresponding differential Equations (1a), (1b) and (10) can be numerically integrated. Here, the system can be represented in the original coordinates. The numerical integration is carried out simultaneous using the compound state $(x,\widehat{z})$ of the system and observer. For moderate observer gains explicit integration schemes such as explicit Runge–Kutta methods [43] can be used. As the observer gain is increased, the differential equations become more and more stiff such that implicit methods like implicit Runge–Kutta methods may be better suited. One may also exploit explicit exponential Rosenbrock integrators [44], which perform well over wide ranges of observer gains.

The vector field for the observer is adopted, such that it is globally Lipschitzian by scaling the nonlinear part $\alpha$. One wants on one hand to preserve the behaviour of the system in the region of interest. On the other hand, large Lipschitz constants require large observer gains, which is undesirable. Thus, the nonlinear part should drop relatively fast beyond the region of interest. In order to compute a suitable scaling (11), a symmetric box

$$|x_i|\le X_i>0,i=1,\dots ,n$$

for the systems state is chosen first, such that it contains the positively invariant set with some margin. This box transforms under the observability map $q_N$ to a subset of ${\mathbb{R}}^N$. Again, a superset of this image is computed, which takes the same form

$$|z_j|\le Z_j,j=1,\dots ,N$$

as before. Since the observability map is a polynomial, an upper bound is easily found by considering the absolute value of each polynomial term. Thus, the embedded systems state will be contained in the box. As norm for (11) we choose

$$\parallel z\parallel =\sqrt{\left(\sum _{i=1}^N\frac{z_i^2}{Z_i^2}\right)}$$

such that $\parallel z\parallel \le \sqrt{N}$ holds for the positively invariant set. The nonlinearity can be damped for grater magnitudes.

It remains to project the solution $t\mapsto \widehat{z}$ of the embedded observer to the original state space. This was done by solving the optimization problem

$$\widehat{x}\left(t\right)=\underset{\widehat{x}}{\arg \min }∥q\left(\widehat{x}\right)-\widehat{z}\left(t\right)∥$$

(18)

point-wise. An initial guess ${\widehat{x}}_0$ for $t=0$ is already known, and this value also minimizes the objective function. For other samples $\widehat{x}\left(t_k\right)$ we simply use the solution $\widehat{x}\left(t_{k-1}\right)$ of the previous time step of the numerical integrator as initial guess. This should be no problem for small discrete time steps. If the integrator uses large time steps, the solution is interpolated such that the time step $t_k-t_{k-1}$ remains bounded. On the other hand (1a) and (1b) can be integrated to predict $\widehat{x}\left(t_k\right)$ from the previous solution. Then the solution of the optimization problem (18) using this initial guess can be regarded as a correction. This technique will work well once the observer error has sufficiently decreased.

This proposed numerical solution of the projection has still a flaw, especially for larger observer errors. Since the observer state is not restricted to the embedded set ${\mathcal{Z}}_N\subseteq {\mathbb{R}}^N$, it may diverge from ${\mathcal{Z}}_N$ and eventually converge again. However, the (possibly multivalued) projection $q\left(\widehat{x}\left(t\right)\right)$ onto ${\mathcal{Z}}_N$ with $\widehat{x}$ given by (18) is not unique. A gradient descent method will always only find a local minimum, which may be located far from the global one. During the trajectory there are therefore some points where a local minimum vanishes, a local minimum splits into multiple ones, or multiple minima merge. Near these regions the Hessian is ill-conditioned and the minimum may be discontinuous, as can be observed at the presented examples.

In a practical application all these numerical schemes have to be implemented in a robust way. Further, the integration of the differential equations and the projection have to be done within a limited number of computation steps. While this can be relatively easy guaranteed when using an explicit and stable integration scheme, there are still some problems to be solved for the projection.

7. Conclusions and Outlook

In this contribution an algebraic approach was presented to embed a polynomial system into its observability form with different dimension of the embedded space. With increasing dimension, the set of points where this map fails to be locally injective is reduced. That dimension, where this point set equals the set of intrinsically not locally observable points can be computed at first.

The examples show that some systems already possess an observability normal form with Lipschitzian vector field. Others required the embedding into an ever higher-dimensional space until the vector field is Lipschitz continuous. There are also systems for that we did not succeed in finding a Lipschitzian vector field, even for globally observable systems. It would be desirable to have an algorithm available that computes an observability normal form with the fewest points at that the vector field fails to be locally Lipschitzian, similar to the observability criterion.

Once the observability normal form has been found, the observer design is straight forward. A more sophisticated approach is yet required for the projection step in order to transform the observer state back to the original coordinates. Especially for implementations in practical systems a numerically robust method will be required.

This article only considers polynomial systems to this end. However, many systems can be transformed such that the vector field is in polynomial form, possibly through an embedding again. This holds in general for rational systems [45], p. 18.

Observers can also be used to identify system parameters or to detect faults. In order to show identifiability of the parameters, higher order derivatives of the measurable outputs may be required as well [46]. The proposed observer can be easily adopted to estimate the parameters by augmenting the systems state with the parameters.