Conditions for Convergence of Dynamic Regressor Extension and Mixing Parameter Estimators Using LTI Filters

A key component of all the new modified estimators is the construction of an extended LRE (ELRE). This idea was first reported in [16] within the context of system identification and later used in [14] for adaptive observers and in [15] for adaptive controller designs. In both cases, the ELRE is created applying stable, linear filters to the LRE (1). Namely, we introduce a linear, bounded-input-bounded-output (BIBO), single-input $\ell$-output operator ${\cal H}:{\cal L}_{\infty}\to{\cal L}_{\infty}^{\ell}$ to define the ELRE

$$Y(t)=\Phi(t)\theta,$$

(5)

where $Y(t):={\cal H}[y](t)\in\mathbb{R}^{\ell}$ and the extended regressor matrix is defined as

$$\Phi(t):=[{\cal H}[\phi_{1}](t)~{}|~{}\ldots~{}|{\cal H}[\phi_{q}](t)]\in\mathbb{R}^{\ell\times q}.$$

(6)

Applying a gradient-descent estimation to the ELRE (5) yields

$$\dot{\hat{\theta}}(t)=\gamma\Phi^{\top}(t)[Y(t)-\Phi(t)\hat{\theta}(t)],\;\gamma>0,$$

whose corresponding parameter estimation error equation is

$$\dot{\tilde{\theta}}(t)=-\gamma\Phi^{\top}(t)\Phi(t)\tilde{\theta}(t).$$

(7)

Notice that, in contrast to (2), the matrix $\Phi^{\top}(t)\Phi(t)\in\mathbb{R}^{q\times q}$ is not necessarily of rank one. This is the central property that motivates the extension of the regressor. However, it is well known [17, Subsection 6.5.3(a.iv)] and [21, Proposition 2], that the provable stability properties of the error equation (7) are the same as the ones of (2), and still require the PE condition for GES.¹¹1The only provable advantage of the new estimator is that the convergence speed can be improved increasing $\gamma$. However, as discussed in [21, Remark 6], the interest of increasing the gain in adaptive systems is highly questionable.

Two different ways to generate the ELRE have been studied in the literature. In [16], they used $\ell=q$ and the BIBO operator ${\cal H}$ is obtained with the stable, linear time-invariant (LTI) filters

$${\cal H}_{i}(p)={\lambda_{i}\over p+\lambda_{i}},\;i\in\bar{q}:=\{1,\ldots,q\},$$

(8)

with the differential operator $p:={d\over dt}$, $\lambda_{i}>0$ and $\lambda_{i}\neq\lambda_{j}$ for all $i\neq j$. We refer in the sequel to the ELRE as Lion’s (L-ELRE). Its state space realization can be written as

	$\displaystyle\dot{\Phi}$	$\displaystyle=-\Lambda\Phi+\Lambda\begin{bmatrix}\phi^{\top}\\ \vdots\\ \phi^{\top}\end{bmatrix},$		(9)
	$\displaystyle\dot{Y}$	$\displaystyle=-\Lambda Y+\Lambda\mbox{col}(y,\ldots,y),$

with $\Lambda=\mbox{diag}(\lambda_{1},\ldots,\lambda_{\ell})$.

In [14, 15] they also consider $\ell=q$ and the operator ${\cal H}$ is LTV of the form²²2It is clear that ${\cal H}_{i}$ is an LTV operator of the form $\dot{x}_{i}=A_{i}x_{i}+b_{i}(t)u$ with $A_{i}=-\alpha$ and $b_{i}(t)=\phi_{i}$.

$${\cal H}_{i}(p,t)={1\over p+\alpha}\phi_{i}(t),\;i\in\bar{q},$$

(10)

with $\alpha>0$. A state space realization of Kreisselmeier’s ELRE—called K-ELRE—is given by

	$\displaystyle\dot{\Phi}(t)$	$\displaystyle=-\alpha\Phi(t)+\phi(t)\phi^{\top}(t)$		(11)
	$\displaystyle\dot{Y}(t)$	$\displaystyle=-\alpha Y(t)+\phi(t)y(t).$

In recent years, ELREs have been used to ease the PE requirement in novel estimator schemes—see the recent survey in [21] for more details and a complete list of references. Two examples of these estimators are the concurrent [10] and composite learning [22]. Within these methodologies, a dynamic data stack is built to discretely record online historical data, and the convergence of parameter estimation is managed monitoring the excitation over an interval. That is, the PE condition (3) is replaced by the strictly weaker assumption that the regressor is IE, whose definition is given as follows.

The IE condition is called “exciting over a finite time interval” in [25, Definition 3.1, pp. 108], where it was used to analyze some convergence properties of the estimation error in the gradient algorithm.

A new estimator that has attracted a lot of attention, and has proven to be very successful to solve many theoretical and practical open problems is DREM, first proposed in [ARAetaltac17] and recently reviewed in [19]. The main idea of DREM is to generate, out of the ELRE (5), $q$ scalar LREs. Towards this end, we also fix $\ell=q$ and then introduce the key mixing step of multiplying from the left (5) by the adjugate of the (square) matrix $\Phi(t)$, denoted $\mbox{adj}\{\Phi(t)\}$, to get

$${\cal Y}_{i}(t)=\Delta(t)\theta_{i},\;i\in\bar{q},$$

(13)

where ${\cal Y}(t):=\mbox{adj}\{\Phi(t)\}Y(t)$ and

The DREM design is completed with the $q$ scalar estimators

$$\dot{\hat{\theta}}_{i}(t)=\gamma_{i}\Delta(t)[{\cal Y}_{i}(t)-\Delta(t)\hat{\theta}_{i}(t)],\;\gamma_{i}>0,\;i\in\bar{q},$$

(14)

with associated error equations

$$\dot{\tilde{\theta}}_{i}(t)=-\gamma_{i}\Delta^{2}(t)\tilde{\theta}_{i}(t),\;i\in\bar{q},$$

(15)

for which the following proposition can be easily proved [ARAetaltac17, 19].

Moreover, a variation of DREM that converges in finite time under the weaker IE assumption (12) has been recently reported in [20] and, as discussed in [19], it has proven instrumental to solve many practical and theoretical open problems.

Note that in DREM we have replaced the convergence condition on the vector $\phi$ being PE by a condition—either non-square integrability for GAS or PE for GES—on a scalar quantity $\Delta(t)$, which is the determinant of the extended regressor matrix (2.2). A natural question that arises is the relationship between the excitation properties of the original regressor $\phi(t)$ and the new scalar regressor $\Delta(t)$. This question has been recently answered in [ARAetaltac20] for the case of K-ELRE where the following results are proven.

The results above prove that, in a scenario with suitable excitation, DREM with K-ELRE has the same convergence properties as the standard gradient, with the following additional advantages:

1. (i)

   GAS under the non-square integrability condition of the scalar regressor $\Delta$ that, as shown in [19, Proposition 3] is strictly weaker than PE of the regressor $\phi$;

2. (ii)

   element-by-element monotonicity of the parameter errors, which is strictly stronger than (4);

3. (iii)

   ability to tune, via $\gamma_{i}$, the convergence rate of each parameter error, in an independent way;

4. (iv)

   possibility to ensure finite convergence time under IE [20, Proposition 3] without the injection of high-gain.

The main objective of this paper is to prove a similar result for DREM with L-ELRE.³³3In [ARAetalijacsp18] this question was studied for the particular case of systems identification, when the regressor is generated via LTI filtering of a sum of sinusoidal signals. In this way we conclusively establish the superiority of DREM—in either one of its forms, K- or L-ELRE—over classical estimators. Instrumental to establish our results is to adopt a Kazantzis-Kravaris-Luenberger (KKL) observer perspective of the DREM estimator, as done in [18]. In this way, we can invoke a fundamental result on injectivity of the key mapping of KKL observers for nonautonomous nonlinear systems recently reported in [6]. This result extends to the nonautonomous case the previous results of [ANDPRA] for autonomous systems, allowing then to include the study at hand.