# Augmented Adaptive Observer for a Differentiable Regressor

Jian Wang<fnref n="1" />, Stanislav Aranovskiy<fnref n="2" />, and Denis Efimov<fnref n="3" />

**Abstract**— We show that constructing an augmented adaptive observer, where the extended state is a mix of the original states and their derivatives, we can enhance the transient performance and obtain robust estimation results for the original state and the unknown parameters. The convergence of the proposed augmented adaptive observer is shown under the persistence of excitation assumption by constructing a Lyapunov function and using the input-to-output stability framework. The performance of the design is illustrated via numerical simulations.

## I. INTRODUCTION

For many control engineering applications, which deal with regulation or monitoring of dynamical systems, the information about the actual values of the state and the parameters in the model are required [1], [2]. However, due to technical or economical reasons, usually, only a part of the state can be directly measured, while the parameters, even being identified in advance, are subjected by time-varying drifts (due to aging, variation of environmental conditions, etc.). Therefore, the problem of adaptive estimation (online estimation of the state vector and the parameters of a dynamical system) arises, which has many solutions [3], [4]. It is important to highlight that frequently the system model contains other different sources of uncertainty, as external disturbances or measurement noises, for instance, then its is required to find a robust estimation algorithm making the problem solution even more difficult.

There are two main groups of methods for adaptive estimation depending on the order of the derivative of the output where the unknown parameters are appearing. If the first derivative of the output depends explicitly on the vector of unknown parameters (the so-called case of relative degree one), then simple adaptive observer can be constructed [5], [6], which necessarily contains dynamics of the estimates of the state and the parameters only. In the case of high relative degree (when higher order derivative of output are dependent on the unknown parameters), the existing estimators [7]–[10] should include additional filters augmenting the complexity of design and analysis.

In order to guarantee the (robust) identification of values of unknown parameters, the system should contain a sufficient amount of excitation in the regressors, which is related with identifiability of the studied system. This requirement constitutes a principal restriction for application of adaptive observers, since it may be difficult to simultaneously guarantee the regulation to a steady state and excitation of some

functions of output and input. If such excitation is artificially added to a system via its input signal, it can compromise the quality of control. Therefore, there are various approaches oriented on relaxation of this condition, not only for adaptive observers design [11]–[15].

Addressing the mentioned above problems, in this note we propose an adaptive observer, where the regressor function (the one which brings excitation to the system) is augmented to include also the derivative of the original signal. This augmentation can enhance transient performance and excitation level at the price of assumption that regressor and some other components of the model are time-differentiable. In fact, these restrictions seem to be natural in many applications (if regressor is a linear function of the state, then it clearly has a well defined time derivative), while they were not explicitly analyzed in the literature. Moreover, our design just assumes existence of time derivatives of some functions in the model of the system, but we are not using them in the equations of the adaptive observer. We will demonstrate that such assumption allows us to get additional useful terms in the estimation error dynamics for the parameters, which can provide a better convergence rate and robustness. For stability analysis, we are using the input-to-state (input-to-output) stability framework [16], [17], and our findings are illustrated on simple examples with numeric experiments.

The rest of the paper is organized as follows. The problem statement is given in Section II. Section III introduces some preliminaries: the input-to-state stability notions and a strong Lyapunov function for persistently excited systems. The main result of the paper, which is the augmented adaptive observer design and analysis, is given in Section IV, and illustrative simulation results follow in Section V. Concluding remarks are given in Section VI together with a discussion of open questions and future research directions.

### Notation

* The real numbers are denoted by $\mathbb{R}$, $\mathbb{R}_+ = \{s \in \mathbb{R} : s \ge 0\}$.

* Euclidean norm for a vector $x \in \mathbb{R}^n$ is denoted as $\|x\|$, for a matrix $\| \cdot \|_2$ denotes the induced norm.

* For a (Lebesgue) measurable function $d: \mathbb{R}_+ \to \mathbb{R}^m$ and $[t_0, t_1) \subset \mathbb{R}_+$, we define the norm $\|d\|_{[t_0, t_1)} = \ess \sup_{t \in [t_0, t_1]} \|d(t)\|$. Then, $\|d\|_{\infty} = \|d\|_{[0, +\infty)}$ and the set of such functions $d$ satisfying the property $\|d\|_{\infty} < +\infty$ is further denoted as $\mathcal{L}_{\infty}^m$, the set of essentially bounded measurable functions.

* A continuous function $\sigma: \mathbb{R}_+ \to \mathbb{R}_+$ belongs to class $\mathcal{K}$ if it is strictly increasing and $\sigma(0) = 0$; it belongs to class $\mathcal{K}_{\infty}$ if it is also unbounded. A continuous function

<footnote>This work was supported by the China 111 Project D17019.</footnote>

<footnote n="1">¹Hangzhou Dianzi University, Hangzhou, China. Corresponding author, wangjian119@hdu.edu.cn</footnote>

<footnote n="2">²CentraleSupélec-IETR, 35576 Cesson-Sévigné, France.</footnote>

<footnote n="3">³Inria, Univ. Lille, CNRS, UMR 9189 - CRISTAL, F-59000 Lille, France.</footnote>
$\beta: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ belongs to class $KL$ if $\beta(\cdot, r) \in K$ and $\beta(r, \cdot)$ is a strictly decreasing to zero for any fixed $r > 0$.

* The $n \times n$-identity matrix is denoted by $I_n$.

## II. PROBLEM STATEMENT

Consider the system

$$ 
\begin{aligned}
\dot{x}(t) &= Ax(t) + \phi(t) + g(t)\theta + d(t) \\
y(t) &= Cx(t) + v(t)
\end{aligned}
\qquad (1) 
$$

where $x(t) \in \mathbb{R}^n$, $y(t) \in \mathbb{R}^l$, $d(t) \in \mathbb{R}^n$ and $v(t) \in \mathbb{R}^l$ are the state, the output, a disturbance and a measurement distortion, respectively, $\theta \in \mathbb{R}^p$ is the vector of unknown parameters. The constant matrices $A$ and $C$, the input function $\phi(t)$, and the regressor function $g(t)$ have appropriate dimensions and are known. Note that $\phi$ and $g$ can depend on the measured components of the state representing the system nonlinearities, but for brevity of notation such a dependence is not shown.

**The goal** is to estimate $x(t)$ and $\theta$ given the measurements $y(t)$ under the following assumptions.

**Assumption 1:** The functions $\phi: \mathbb{R}_+ \to \mathbb{R}^n$ and $g: \mathbb{R}_+ \to \mathbb{R}^{n \times p}$ are bounded and having bounded derivatives for almost all $t \ge 0$; $d(t)$ and $v(t)$ are bounded for all $t \ge 0$.

**Assumption 2:** The pair $(A, C)$ is observable.

Note that Assumption 1 implies only the existence of $\dot{\phi}$ and $\dot{g}$; these functions are not necessarily known or measured. We will use them for the analysis, but not for the implementation of adaptive observer.

An excitation condition should also be imposed on the regressor $g$, which will be done later. Thus, under all these restrictions, our objective is to design a new adaptive observer and, in the presence of disturbances and measurement noises, to guarantee input-to-output stability of the state estimation and parameter identification errors.

## III. PRELIMINARIES

### A. Input-to-state and Input-to-output Stability

Consider a nonlinear system:

$$ \dot{x}(t) = f(t, x(t), d(t)), \quad t \in \mathbb{R}, \qquad (2) $$

where $x(t) \in \mathbb{R}^n$ is the state, $d(t) \in \mathbb{R}^m$ is the external input, $d \in \mathcal{L}_\infty^m$, and $f: \mathbb{R}^{1+n+m} \to \mathbb{R}^n$ is a locally Lipschitz continuous function, $f(\cdot, 0, 0) = 0$. For an initial condition $x_0 \in \mathbb{R}^n$ at time $t_0 \in \mathbb{R}$ and input $d \in \mathcal{L}_\infty^m$, define the corresponding solution by $x(t, t_0, x_0, d)$ for any $t \ge t_0$ for which the solution exists.

In this work, we will be interested in the following stability property [16], [17]:

**Definition 1:** The system (2) is called input-to-state stable (ISS), if there are functions $\beta \in KL$ and $\gamma \in K$ such that for all $t \ge t_0$

$$ \|x(t, t_0, x_0, d)\| \le \beta(\|x(t_0)\|, t - t_0) + \gamma(\|d\|_{[t_0, t)}) $$

for any $t_0 \in \mathbb{R}$, $x(t_0) \in \mathbb{R}^n$ and $d \in \mathcal{L}_\infty^m$. The function $\gamma$ is called the nonlinear asymptotic gain.

Note that the above property implies that (2) is globally uniformly asymptotically stable at the origin for zero input $d$.

This robust stability property has the following characterizations in terms of existence of a Lyapunov function:

**Definition 2:** A smooth $V: \mathbb{R}^{1+n} \to \mathbb{R}_+$ is called ISS-Lyapunov function for the system (2) if there are $\alpha_1, \alpha_2, \alpha_3 \in \mathcal{K}_\infty$ and $\eta \in K$ such that

$$ \alpha_1(\|x\|) \le V(t, x) \le \alpha_2(\|x\|), \qquad (3a) $$

$$ \frac{\partial V(t, x)}{\partial t} + \frac{\partial V(t, x)}{\partial x} f(t, x, d) \le \eta(\|d\|) - \alpha_3(\|x\|) \quad (3b) $$

for all $x \in \mathbb{R}^n$ and all $d \in \mathbb{R}^m$.

The relation between this Lyapunov characterization and the robust stability property is given below:

**Theorem 1:** The system (2) is ISS if it admits an ISS-Lyapunov function.

If the system (2) is augmented by an output

$$ y(t) = h(x(t)) $$

where $h: \mathbb{R}^n \to \mathbb{R}^l$ is a continuous function, then replacing in Definition 1 and in (3) the variable $x$ by $y$ we get the formulaion of state-independent input-to-output stability (SIIOS) property and its sufficient Lyapunov characterization (under assumption that (2) is forward complete).

### B. Strong Lyapunov Functions for Persistently Excited Systems

We recall the definition of a persistently exciting (PE) signal.

**Definition 3:** A bounded signal $\psi: \mathbb{R}_+ \to \mathbb{R}^{p \times l}$ is PE if there exist $\mu > 0$ and $T > 0$ such that for all $t \ge 0$ it holds

$$ \int_t^{t+T} \psi(\tau) \psi^\top(\tau) d\tau \ge \mu I_p. $$

Consider the system

$$ \dot{x}(t) = -\gamma \psi(t) \psi^\top(t) x(t), \quad t \ge 0, \qquad (4) $$

with $x(t) \in \mathbb{R}^p$, $\gamma > 0$ and $\psi$ being PE. Whereas the global exponential stability (GES) of this system is a standard result in adaptive control [18], [19], a strong Lyapunov function design for linear time-varying systems driven by PE signals is more recent, see, e.g., [20]–[22]. The existence of a strong Lyapunov function for (4) is summarized in the following lemma, where proofs suggesting different approaches to construct such a function can be found in the references cited above.

**Lemma 1:** Let $\psi: \mathbb{R}_+ \to \mathbb{R}^{p \times l}$ be a PE signal and define $A(t) := -\gamma \psi(t) \psi^\top(t)$. Then there exist a time-varying matrix $P: \mathbb{R}_+ \to \mathbb{R}^{p \times p}$ and positive scalars $a_1, a_2, a_3$ such that for all $t \ge 0$ it holds

$$ a_1 I_p \le P(t) \le a_2 I_p, $$

$$ \dot{P}(t) + P(t)A(t) + A^\top(t)P(t) \le -a_3 I_p. $$

The GES of (4) then follows considering the Lyapunov function $x^\top P x$.
IV. OBSERVER DESIGN

For observer design, we first augment the state vector x with m selected components of the derivative dot x. To this end, let i₁, ..., iₘ be the indices selecting m, for 1 ≤ m ≤ n, components of dot x, and j₁, ..., jₙ₋ₘ be the indices of the remaining components of dot x. Define variables z₁₁ ∈ ℝⁿ, z₁₂ ∈ ℝⁿ₋ₘ and constant matrices T₁ ∈ ℝⁿ×ₙ and T₂ ∈ ℝ⁽ⁿ₋ₘ⁾×ₙ such that

$$
z_{11} := \begin{bmatrix} x_{i_1} \\ \vdots \\ x_{i_m} \end{bmatrix} = T_1 x, \quad z_{12} := \begin{bmatrix} x_{j_1} \\ \vdots \\ x_{j_{n-m}} \end{bmatrix} = T_2 x \quad (5)
$$

and denote $z_1 \in \mathbb{R}^n$ as

$$
z_1 := \begin{bmatrix} z_{11} \\ z_{12} \end{bmatrix} = \begin{bmatrix} T_1 \\ T_2 \end{bmatrix} \quad x = Tx.
$$

Obviously, the matrix *T* can be obtained from the identity
*I*<sub>*n*</sub> matrix by a permutation of rows and is invertible, *x* =
*T*<sup>-1</sup>*z*<sub>1</sub>.

Next, define the variable $z_2 \in \mathbb{R}^m$ as the mentioned part of the derivative of $\dot{x}$,

$$
z_2 := T_1 (\dot{x} - d)
$$

and denote $z := [z_1^\top \quad z_2^\top]^\top \in \mathbb{R}^{n+m}$.

The dynamic of *z* obeys the differential equation

$$
\dot{z}(t) = Rz(t) + F(t) + G(t)\theta + \delta(t)
$$

where

$$
\begin{align}
F(t) &:= B_0 \phi(t) + B_1 \dot{\phi}(t) \nonumber \\
G(t) &:= B_0 g(t) + B_1 \dot{g}(t) \tag{6}
\end{align}
$$

and

$$
R := \begin{bmatrix} 0_{m \times n} & I_m \\ T_2 A T^{-1} & 0_{(n-m) \times m} \\ T_1 A^2 T^{-1} & 0_{m \times m} \end{bmatrix} \quad (7a)
$$

$$
B_0 := \begin{bmatrix} 0 \\ T_2 \\ T_1 A \end{bmatrix}, \quad B_1 := \begin{bmatrix} 0 \\ 0 \\ T_1 \end{bmatrix}, \quad \delta := \begin{bmatrix} T_1 d \\ T_2 d \\ T_1 Ad \end{bmatrix}. \quad (7b)
$$

The output $y$ is then

$$
y = \begin{bmatrix} CT^{-1} & 0 \end{bmatrix} \begin{bmatrix} z_1 \\ z_2 \end{bmatrix} + v = Hz + v. \quad (8)
$$

To proceed, we need to assume the observability of the
the augmented dynamics.

*Assumption 3:* The matrix *T*₁ is such that the pair (*R*, *H*) is observable.

Remark 1: Note that under Assumption 2, the state z₁ is observable through the output y, and hence, the state z admits the same property (z is an extension of z₁ by a part of its derivatives). However, the particular realization of the augmented dynamics chosen in (7) may loose the observability. For the general case of arbitrary matrices A, C, and T₁, the observability of (R, H) is introduced as an assumption.

Now we are in the position to present the augmented
adaptive observer. Let the gain matrix $L \in \mathbb{R}^{(n+m) \times \ell}$ be
such that $R - LH$ is Hurwitz, which is always feasible
under Assumption 3. Then the augmented adaptive observer
is given by

$$
\begin{equation}
\begin{split}
\dot{\zeta} &= R\hat{z} + B_0 (\phi + g\hat{\theta}) + L(y - H\hat{z}) \\
&\quad - (\Omega + B_1 g) \dot{\theta}
\end{split}
\tag{9a}
\end{equation}
$$

$$
\hat{z} = \zeta + B_1 (\phi + g\hat{\theta}) \quad (9b)
$$

$$
\dot{\Theta} = (R - LH) \Omega - B_0 g \quad (9c)
$$

$$
\Omega = \Theta - B_1 g \tag{9d}
$$

$$
\begin{equation}
\begin{split}
\dot{\theta} &= -\gamma \Omega^{\top} H^{\top} (y - H\hat{z}) \\
&\quad + \kappa g^{\top} T_1^{\top} \left( \hat{z}_2 - T_1 \left( A T^{-1} \hat{z}_1 + \phi(t) + g\hat{\theta} \right) \right)
\end{split}
\tag{9e}
\end{equation}
$$

$$
\hat{x} = [T^{-1} \quad 0_{m \times n}] \hat{z} \quad (9f)
$$

where $\zeta \in \mathbb{R}^{n+m}$ and $\Theta, \Omega \in \mathbb{R}^{(n+m) \times p}$ are auxiliary fil-
tered versions of the state $z$ and the regressor $G$, respectively;
$\hat{z} \in \mathbb{R}^{n+m}$ and $\hat{x} \in \mathbb{R}^n$ are the estimates of $z$ and $x$,
correspondingly; $\hat{\theta} \in \mathbb{R}^p$ is the estimate of the unknown
parameter vector $\theta$; $\gamma > 0$ and $\kappa > 0$ are tuning gains of the
adaptation algorithm.

As we can conclude from the analysis of (9), its dynamics
do not depend on the derivatives of *g* and *φ* (its existence was
assumed), hence, it is an implementable estimator. However,
if we derive the differential equations governing the behavior
of the variables *ẑ* and *Ω*, these derivatives influence their
dynamics. Therefore, it could be the case that the original
regressor is not exciting enough, while the new one in *G*
demonstrates this feature. Moreover, due to these manipu-
lations, additional negative feedback in *θ̂* (amplified by *κ*)
appears in the adaptation algorithm (9e), which can improve
the convergence rates and robustness under the restrictions
presented later in this paper.

Next, we introduce an assumption related to the excitation
of the augmented regressor *G* defined in (6) and allowing
for estimation of *θ*. To this end, note that *Ω* defined by (9d)
obeys the differential equation:

$$
\dot{\Omega}(t) = (R - LH)\Omega(t) - G(t). \quad (10)
$$

Since the matrix $R - LH$ is Hurwitz and $g, \dot{g}$ are assumed to be bounded by Assumption 1, the variables $\Theta$ and $\Omega$ are also bounded. Define the signal $\psi: \mathbb{R}_{+} \to \mathbb{R}^{p \times \ell}$ as

$$
\psi(t) := (H\Omega(t))^{\top} \qquad (11)
$$

where *H* is defined in (8).

*Assumption 4:* The matrices *R* and *H*, the gain *L*, and the augmented regressor function *G* are such that the signal *ψ* defined in (11) is PE.

The applicability of the proposed augmented observer un-
der the introduced assumptions is formulated in the following
theorem.

*Theorem 2:* Consider system (1) under Assumption 1 and Assumption 2. Choose *m* ≤ *n* distinct indices *i*<sub>*j*</sub>, 1 ≤ *j* ≤
m, and construct matrices $T_1$ and $T_2$ as in (5). Construct the matrices $R$, $B_0$, $B_1$, and $H$ as in (7) and (8). Under Assumption 3, choose the gain matrix $L$ such that $R-LH$ is Hurwitz. Construct the adaptive augmented observer (9) for $\hat{x}$ and $\hat{\theta}$ and define the estimation errors $\tilde{\theta} = \theta - \hat{\theta}$ and $\tilde{x} = x - \hat{x}$. If Assumption 4 is satisfied, then there exists $\kappa^* > 0$ such that for all $\kappa < \kappa^*$, the variable $\Theta$ is bounded, the whole system is forward complete and SIIOS with respect to the outputs $\tilde{x}$, $\tilde{\theta}$ and the inputs $d$ and $v$.

The complete proof is omitted due to the lack of space. The proof is based on defining the error $\tilde{z} := z - \hat{z}$ and an auxiliary variable $\varepsilon = \tilde{z} + \Omega\tilde{\theta}$. Then, due to the choice of the gain matrix $L$, Assumption 4, and Lemma 1, there exist time-varying positive-definite matrix $P_{11}(t)$ and constant positive-definite matrix $P_{22}$, such that for the Lyapunov function

$$V(t, \tilde{\theta}, \varepsilon) = \begin{bmatrix} \tilde{\theta} \\ \varepsilon \end{bmatrix}^{\top} \begin{bmatrix} P_{11}(t) & 0 \\ 0 & cP_{22} \end{bmatrix} \begin{bmatrix} \tilde{\theta} \\ \varepsilon \end{bmatrix}$$

for some $c > 0$ it can be shown that

$$\dot{V} \leq -c_1 (\|\tilde{\theta}\|^2 + \|\varepsilon\|^2) + c_2 (\|\tilde{\delta}\|^2 + \|v\|^2).$$

for some positive constants $c_1$ and $c_2$. Thus, $\dot{V}$ satisfies (3b) and $V$ is an SIIOS-Lyapunov function for the dynamics of $\tilde{\theta}$ and $\varepsilon$. Finally, we apply Theorem 1 and note that due to the definition of $\varepsilon$, the SIIOS of $\tilde{\theta}$ and $\varepsilon$ implies also the SIIOS of $\tilde{z}$.

The following observations are in order:

* the constant $\kappa^*$ can be estimated;

* Assumption 4 is used for the construction of the Lyapunov function (through Lemma 1), but it may be not required for the functioning of the proposed observer (9), since the identification error differential equation involves the term $-g^\top T_1^\top T_1 g \tilde{\theta}$ which, under PE of $g$, can ensure the stability of the own dynamics of $\tilde{\theta}$.

## V. ILLUSTRATIVE EXAMPLE

### A. Considered System

For the illustrative example, we consider the system (1) with $n = 3$, $p = 2$, and $\ell = 1$ with the following matrices

$$A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -3 & -5 & -3 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}.$$

The signals $\phi$ and $d$ are

$$\phi(t) = \begin{bmatrix} -1 \\ 0 \\ 2 \end{bmatrix} u(t), \quad d(t) = \begin{bmatrix} 0 \\ 1 \\ -1 \end{bmatrix} \bar{d}(t)$$

with $u(t) = \sin(t)$ and $\bar{d}(t) = 0.1 \sin(10t)$. The regressor $g$ and the vector of unknown parameters $\theta$ are chosen as

$$g(t) = \begin{bmatrix} 0 & 1 \\ 1 & \sin(t) \\ \cos(t) & 0 \end{bmatrix}, \quad \theta = \begin{bmatrix} 1 \\ 2 \end{bmatrix}.$$

It is straightforward to verify that $g$ is PE in the sense of Definition 3 with $T = 2\pi$ and $\mu = 3\pi$. The measurement noise $v$ is a random signal uniformly distributed in the interval $[-0.01, 0.01]$.

### B. Nominal Adaptive Observer

We compare the proposed augmented adaptive observer with the nominal high-order adaptive observer [10] for the system (1) given by:

$$\begin{align} \dot{\hat{x}}_n(t) &= A\hat{x}_n(t) + \phi(t) + g(t)\hat{\theta}_n(t) \tag{12a} \\ &\quad + (L_n + \gamma_n \Upsilon(t)\Upsilon^\top(t)C^\top)(y(t) - C\hat{x}_n(t)), \nonumber \end{align}$$

$$\dot{\Upsilon}(t) = (A - L_n C) \Upsilon(t) + g(t), \quad (12b)$$

$$\dot{\hat{\theta}}_n = \gamma_n \Upsilon^\top(t) C^\top(y(t) - C\hat{x}_n(t)), \quad (12c)$$

where $\hat{x}_n(t) \in \mathbb{R}^n$ and $\hat{\theta}_n(t) \in \mathbb{R}^p$ are the estimates of $x(t)$ and $\theta$, respectively, $\Upsilon(t) \in \mathbb{R}^{n \times p}$ is an internal variable, the gain $L_n \in \mathbb{R}^{n \times \ell}$ is such that $A - L_n C$ is Hurwitz (which is always feasible under Assumption 2), and $\gamma_n > 0$ is a scalar adaptation gain. Similarly to Assumption 4, the nominal adaptive estimator (12) imposes the following assumption on the excitation of the regressor $g$.

**Assumption 5:** The matrices $A$, $C$, the gain $L_n$, and the regressor $g$ are such that the signal $\psi_n: \mathbb{R}_+ \to \mathbb{R}^{p \times \ell}$

$$\psi_n(t) := (C\Upsilon(t))^\top \quad (13)$$

is PE.

It is worth noting that the dimension of $\psi_n$ is the same as the dimension of $\psi$ defined in (11) for the augmented adaptive observer. However, the nominal signal $\psi_n$ depends only on the regressor $g$, whereas the signal $\psi$ depends on the augmented regressor $G$ involving the derivative $\dot{g}$.

### C. Augmented Adaptive Observer and Gains Tuning

The design of the augmented adaptive observer (9) is performed as follows. First, we choose $m = 2$ distinct indices $i_1 = 1$ and $i_2 = 3$ of the derivatives used for the state augmentation. This choice yields

$$T_1 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \quad T_2 = \begin{bmatrix} 0 & 1 & 0 \end{bmatrix}.$$

The matrices $R$, $B_0$, $B_1$ and $H$ are then computed as in (7) and (8), and it is straightforward to verify that Assumption 3 is satisfied.

The gain matrices $L$ in (9) and $L_n$ in (12) are chosen as solutions of corresponding LQ optimal control problems with identity weighting matrices; such choice provides approximately equal convergence rate for autonomous systems with state matrices $A - L_n C$ and $R - LH$. The scalar adaptation gains $\gamma$ in (9) and $\gamma_n$ in (12) are chosen equal, $\gamma = \gamma_n = 5$. Finally, for the augmented observer (9) we choose $\kappa = 0.02$. All initial conditions for both observers are set to zeros.

### D. Simulation Results

Simulation results for parameter $\theta$ estimation are given in Figures 1 and 2 for $\theta_1$ and $\theta_2$, respectively. Despite higher peak values, the augmented adaptive observer illustrates faster convergence and superior steady-state behavior.

The estimates of the state $x$ are compared in Figures 3–5 for the three components of $x$, respectively. Similarly to the estimation of $\theta$, the augmented adaptive observer provides higher initial peaking but faster convergence.
<figure><img src="image_1.png" /><figcaption>Fig. 1: Estimates of the parameter $\theta_1$: $\hat{\theta}_{n,1}$ by the nominal adaptive observer (12) and $\hat{\theta}_1$ by the augmented adaptive observer (9).</figcaption></figure>

<figure><img src="image_2.png" /><figcaption>Fig. 2: Estimates of the parameter $\theta_1$: $\hat{\theta}_{n,2}$ by the nominal adaptive observer (12) and $\hat{\theta}_2$ by the augmented adaptive observer (9).</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 3: Estimate of the state $x_1$: $\hat{x}_{n,1}$ by the nominal adaptive observer (12) and $\hat{x}_1$ by the augmented adaptive observer (9).</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 4: Estimate of the state $x_2$: $\hat{x}_{n,2}$ by the nominal adaptive observer (12) and $\hat{x}_2$ by the augmented adaptive observer (9).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 5: Estimate of the state $x_3$: $\hat{x}_{n,3}$ by the nominal adaptive observer (12) and $\hat{x}_3$ by the augmented adaptive observer (9).</figcaption></figure>

## VI. CONCLUSION

The papers studied the problem of adaptive observer design for a system with differentiable regressor. It is shown that by augmentation of the state vector with a set of the state derivatives, one can design an augmented state observer. The augmented regressor matrix $G$ involves both the original regressor $g$ and its derivative $\dot{g}$. Moreover, our stability analysis shows that the augmented error dynamics include an additional negative feedback for the estimation of the vector of unknown parameters $\theta$ proportional to the tuning gain $\kappa$. Simulation results illustrate superior parameter estimation performance of the augmented adaptive observer.

Our future research is oriented towards two open question. The first question is related to Assumption 3: under which constrains on the matrix $T_1$ we can guarantee the observability of the pair $(R, H)$. The second open question deals with the additional excitation coming from $\dot{g}$. More precisely, we are interested for what class of regressors $g$ Assumption 5 can be violated while Assumption 4 remains satisfied.

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