New Results on Parameter Estimation via Dynamic Regressor Extension and Mixing: Continuous and Discrete-time Cases

TL;DR

This paper advances parameter estimation techniques in linear regression models by unifying continuous and discrete-time approaches, introducing new regressor matrices, and ensuring finite-time convergence with improved transient performance.

Contribution

It provides a unified framework for continuous and discrete-time estimators, introduces two novel regressor matrices, and offers an estimator for finite-time parameter convergence.

Findings

Unified treatment of continuous and discrete-time cases

Introduction of two new extended regressor matrices

Finite-time parameter estimation with tracking of time-varying parameters

Abstract

We present some new results on the dynamic regressor extension and mixing parameter estimators for linear regression models recently proposed in the literature. This technique has proven instrumental in the solution of several open problems in system identification and adaptive control. The new results include: (i) a unified treatment of the continuous and the discrete-time cases; (ii) the proposal of two new extended regressor matrices, one which guarantees a quantifiable transient performance improvement, and the other exponential convergence under conditions that are strictly weaker than regressor persistence of excitation; and (iii) an alternative estimator ensuring parameter estimation in finite-time that retains its alertness to track time-varying parameters. Simulations that illustrate our results are also presented.

Equations

$$y={\varphi }^{\top }\theta +{\epsilon }_t$$

$${\int }_t^{t+T}\varphi (\tau ){\varphi }^{\top }(\tau )d\tau \ge \alpha I_m,\forall t\in {\mathbb{R}}_{\ge 0},$$

$$\sum _{j=k+1}^{k+K}\varphi (j){\varphi }^{\top }(j)\ge \alpha I_m,\forall k\in {\mathbb{Z}}_{\ge 0},$$

$$\dot{\hat{\theta }}(t)=\gamma \varphi (t)[y(t)-{\varphi }^{\top }(t)\hat{\theta }(t)],$$

$$|\tilde{\theta }(t_b)|\le |\tilde{\theta }(t_a)|,\forall t_b\ge t_a\in {\mathbb{R}}_{\ge 0}.$$

$$\dot{\tilde{\theta }}(t)=-\gamma \varphi (t){\varphi }^{\top }(t)\tilde{\theta }(t),$$

$$\hat{\theta }(k)=\hat{\theta }(k-1)+\frac{\varphi (k)}{\gamma +|\varphi (k){|}^2}[y(k)-{\varphi }^{\top }(k)\hat{\theta }(k-1)],$$

$$|\tilde{\theta }(k_b)|\le |\tilde{\theta }(k_a)|,\forall k_b\ge k_a\in {\mathbb{Z}}_{\ge 0}.$$

$$\tilde{\theta}(k)=\bigg{[}I_{m}-{1\over\gamma+|\phi(k)|^{2}}\phi(k)\phi^{\top}(k)\bigg{]}\tilde{\theta}(k-1),$$

$$Y$$

$$Y=\Phi \theta .$$

$${\mathcal{Y}}_i=\Delta {\theta }_i,i\in \{1,2,\dots ,m\}$$

$$\Delta :=\det \{\Phi \},$$

$$\mathcal{Y}:=\text{mbox}adj\{\Phi \}Y.$$

$${\dot{\hat{\theta }}}_i(t)={\gamma }_i\Delta (t)[{\mathcal{Y}}_i(t)-\Delta (t){\hat{\theta }}_i(t)],$$

$${\dot{\tilde{\theta }}}_i(t)=-{\gamma }_i{\Delta }^2(t){\tilde{\theta }}_i(t).$$

$$|{\tilde{\theta }}_i(t_b)|\le |{\tilde{\theta }}_i(t_a)|,\forall t_b\ge t_a\in {\mathbb{R}}_{\ge 0}.$$

$$\underset{t\to \infty }{\lim }{\tilde{\theta }}_i(t)=0\iff \Delta (t)\notin {\mathcal{L}}_2,$$

$${\hat{\theta }}_i(k)={\hat{\theta }}_i(k-1)+\frac{\Delta (k)}{{\gamma }_i+{\Delta }^2(k)}[{\mathcal{Y}}_i(k)-\Delta (k){\hat{\theta }}_i(k-1)],$$

$${\tilde{\theta }}_i(k)=\frac{1}{1+\frac{{\Delta }^2(k)}{{\gamma }_i}}{\tilde{\theta }}_i(k-1).$$

$$|{\tilde{\theta }}_i(k_b)|\le |{\tilde{\theta }}_i(k_a)|,\forall k_b\ge k_a\in {\mathbb{Z}}_{\ge 0}.$$

$$\underset{t\to \infty }{\lim }{\tilde{\theta }}_i(k)=0\iff \Delta (k)\notin {\ell }_2,$$

$$\mathcal{H}:=\left[\begin{array}{ccc}\varphi (k-1)& \varphi (k-2)& \cdots \varphi (k-\bar{K})\end{array}\right]\left[\begin{array}{c}{q}^{-1}\\ q^{-2}\\ ⋮\\ q^{-\bar{K}}\end{array}\right].$$

$$\Delta (k)\in PE[K\ge 2]$$

$$\Phi (k)=\sum _{j=k+1}^{k+\bar{K}}\varphi (j-(1+\bar{K})){\varphi }^{\top }(j-(1+\bar{K})).$$

$${\hat{\theta }}_i^{\mathtt{FTC}-\mathtt{D}}(t):=\frac{1}{1-w_i^{\mathtt{D}}(t)}[{\hat{\theta }}_i(t)-w_i^{\mathtt{D}}(t){\hat{\theta }}_i(0)],$$

$$\dot{\hat{\theta }}(t)=\gamma \Delta (t)[y(t)-\Delta (t)\hat{\theta }(t)],$$

$$\theta (t)=\{\begin{array}{l}10\text{ for }0\le t<10,\\ 15\text{ for }10\le t<20,\\ 15-0.5(t-20)\text{ for }20\le t<30,\\ 10\text{ for }t>30,\end{array}$$

Full text

I Introduction

Estimation of the parameters that describe an underlying physical setting is one of the central problems in control and systems theory that has attracted the attention of many researchers for several years. A typical scenario, which appears in system identification and adaptive control [9, 10, 16, 17, 21], is when the unknown parameters and the measured data are linearly related in a so-called linear regression equation (LRE). Classical solutions for this problem are gradient and least-squares (LS) estimators. The main drawback of these schemes is that convergence of the parameter estimates relies on the availability of signal excitation, a feature that is codified in the restrictive assumption of persistency of excitation (PE) of the regressor vector. Moreover, their transient performance is highly unpredictable and only a weak monotonicity property of the estimation errors can be guaranteed.

To overcome these two problems a new parameter estimation procedure, called dynamic regressor extension and mixing (DREM), has recently been proposed in [2] for continuous-time (CT) and in [5] for discrete-time (DT) systems. The construction of DREM estimators proceeds in two steps, first, the inclusion of a free linear operator that creates an extended, matrix LRE. Second, a nonlinear manipulation of the data that allow generating, out of an $m$-dimensional LRE, $m$ scalar, and independent, LRE. DREM estimators have been successfully applied in a variety of identification and adaptive control problems. Interestingly, it has been shown in [18] that DREM can be reformulated as a functional Luenberger observer.

DREM estimators outperform classical gradient or LS estimators in the following precise aspects: independently of the excitation conditions, DREM guarantees monotonicity of each element of the parameter error vector that is much stronger than monotonicity of the vector norm, which is ensured with classical estimators. Moreover, parameter convergence in DREM is established without the PE condition. Instead of PE a non-square integrability condition on the determinant of a designer-dependent extended regressor matrix is imposed. A final interesting property of DREM that has been established in [8] is that it can be used to generate estimates with finite-time convergence (FTC), under interval excitation assumption.

The following new results on DREM are presented here:

(i) The unified treatment of the CT and the DT cases.

(ii) The definition of new linear operators that:

$\bullet\;$ensure parameter error convergence under excitation conditions that are strictly weaker than regressor PE;

$\bullet\;$guarantee a transient performance improvement;

$\bullet\;$show that DREM contains, as a particular case, the extended LRE proposed in [11], which is used also in the adaptive controllers recently proposed in [6, 7, 20].

(iii) An alternative estimator, ensuring FTC, that retains its alertness to track time-varying parameters.

The remainder of the paper is organized as follows. To set up the notation a brief description of gradient and DREM estimators is given in Section II. In Section III we present the new version of DREM that ensures convergence under excitation conditions that are strictly weaker than regressor PE. In Section LABEL:sec4 a general form of the free operator used in DREM is proposed to, on one hand, re-derive the extended regressor of [11] and, on the other hand, prove that transient performance is—quantifiably—improved. Section LABEL:sec5 presents a new DREM-based estimator with FTC. Simulation results are presented in Section LABEL:sec6. The paper is wrapped-up with future research in Section VII.

Notation. $I_{n}$ is the $n\times n$ identity matrix. $\mathbb{R}_{>0}$, $\mathbb{R}_{\geq 0}$, $\mathbb{Z}_{>0}$ and $\mathbb{Z}_{\geq 0}$ denote the positive and non-negative real and integer numbers, respectively. For $x\in\mathbb{R}^{n}$, we denote $|x|^{2}:=x^{\top}x$. Continuous-time (CT) signals $s:\mathbb{R}_{\geq 0}\to\mathbb{R}$ are denoted $s(t)$, while for discrete-time (DT) sequences $s:\mathbb{Z}_{\geq 0}\to\mathbb{R}$ we use $s(k):=s(kT_{s})$, with $T_{s}\in\mathbb{R}_{>0}$ the sampling time. The action of an operator $\mathcal{H}:{\cal L}_{\infty}\to{\cal L}_{\infty}$ on a CT signal $u(t)$ is denoted $\mathcal{H}[u](t)$, while for an operator ${\cal H}:\ell_{\infty}\to\ell_{\infty}$ and a sequence $u(k)$ we use $\mathcal{H}[u](k)$. When a formula is applicable to CT signals and DT sequences the time argument is omitted.

II Background Material

We deal with the problem of on-line estimation of the unknown, constant parameters $\theta\in\mathbb{R}^{m}$ appearing in a LRE of the form

[TABLE]

where $y\in\mathbb{R}$ and $\phi\in\mathbb{R}^{m}$ are measurable CT or DT signals and $\varepsilon_{t}$ is a (generic) exponentially decaying signal.111This signal may be stemming from the effect of the initial conditions of various filters used to generate the LRE. It is well-known that the availability of a LRE of the form (1) is instrumental for the development of most system identifiers and adaptive controllers [21]. Following standard practice, throughout the paper, the term $\varepsilon_{t}$ is omitted.

II-A Gradient estimator and the PE condition

In this subsection we recall the well-known gradient estimator, derive its parameter error equation (PEE) and recall its stability properties. Although this material is very well-known, it is included to make the document self-contained and set up the notation. First, we introduce the following.

Definition 1**.**

A bounded signal $\phi\in\mathbb{R}^{m}$ is PE (denoted $\phi\in PE$) if there exist $\alpha\in\mathbb{R}_{>0}$ such that

[TABLE]

for some $T\in\mathbb{R}_{>0}$ in CT or

[TABLE]

for some $K\in\mathbb{Z}_{>0}$, with $K\geq m$, in DT. $\Box\Box\Box$

The following proposition is a milestone for systems theory and may be found in all identification and adaptive control textbooks, e.g., [21].

Proposition 1**.**

Consider the LRE (1).

(CT) The CT gradient-descent estimator

[TABLE]

with $\gamma>0$ ensures the following.

$\bullet\;$The norm of the parameter error vector $\tilde{\theta}:=\hat{\theta}-\theta$ is monotonically non-increasing, that is,

[TABLE]

$\bullet\;$The CT PEE is given by

[TABLE]

and its zero equilibrium is globally exponentially stable (GES) if and only if $\phi(t)\in PE$. Moreover, there exist an optimal value of $\gamma$ for which the rate of convergence is maximum.

(DT) The DT gradient-descent estimator

[TABLE]

ensures the following.

$\bullet\;$The norm of the parameter error vector verifies

[TABLE]

$\bullet\;$The DT PEE is given by

[TABLE]

and its zero equilibrium is GES if and only if $\phi(k)\in PE$.

$\Box\Box\Box$

In most applications, PE is an extremely restrictive condition, hence the interest of relaxing it. See [19] for a recent review of new estimators relaxing the PE condition, which include the ones reported in [6, 7, 20].

II-B Generation of $m$ scalar LRE via DREM

To overcome the limitation imposed by the PE condition and improve the transient performance of the estimator the DREM procedure, introduced in [2, 5], generates $m$ new, one–dimensional, LRE to independently estimate each of the parameters. The first step in DREM is to introduce a linear, single-input $m$-output, bounded-input bounded-output (BIBO)–stable operator ${\cal H}$ and define the vector $Y\in\mathbb{R}^{m}$ and the matrix $\Phi\in\mathbb{R}^{m\times m}$ as

[TABLE]

Clearly, because of linearity and BIBO stability, these signals satisfy

[TABLE]

At this point the key step of regressor “mixing” of the DREM procedure is used to obtain a set of $m$ scalar equations as follows. First, recall that, for any (possibly singular) $m\times m$ matrix $M$ we have [12] $\mbox{adj}\{M\}M=\det\{M\}I_{m}$, where $\mbox{adj}\{\cdot\}$ is the adjunct (also called “adjugate”) matrix. Now, multiplying from the left the vector equation (6) by the adjunct matrix of $\Phi$, we get

[TABLE]

where we have defined the scalar function $\Delta\in\mathbb{R}$

[TABLE]

and the vector ${\cal Y}\in\mathbb{R}^{m}$

[TABLE]

Remark 1**.**

In [13] an extended regressor like (6) has been constructed in CT using linear time-invariant (LTI) filters in the operator ${\cal H}$ used in (5)—see also [11], where this modification is also discussed. Unfortunately, besides some simulation evidence, no quantitative advantage—with respect to the gradient estimation—has been established for it.

II-C Properties of gradient parameter estimators in DREM

The availability of the scalar LREs (7) is the main feature of DREM that distinguishes it with respect to all other estimators. Indeed, as shown in the propostion below—the proof of which may be found in [2, 5]—it allows obtaining significantly stronger results using simple gradient estimators.

Proposition 2**.**

Consider the scalar LREs (7).

(CT) The CT gradient-descent estimators222In the sequel, the quantifier $i\in\{1,2,\dots,m\}$ is omitted for brevity.

[TABLE]

with $\gamma_{i}\in\mathbb{R}_{>0}$ ensures the following.

$\bullet\;$The CT PEEs are given by

[TABLE]

$\bullet\;$The individual parameter errors are monotonically non-increasing, that is,

[TABLE]

$\bullet\;$The following equivalence holds

[TABLE]

and convergence can be made arbitrarily fast increasing $\gamma_{i}$.

$\bullet\;$If $\Delta(t)\in PE$, the convergence is exponential.

(DT) The DT gradient-descent estimator

[TABLE]

ensures the following.

$\bullet\;$The DT PEEs are given by

[TABLE]

$\bullet\;$The elements of the parameter error vector verify

[TABLE]

$\bullet\;$The following equivalence holds

[TABLE]

and convergence can be made arbitrarily fast decreasing $\gamma_{i}$.

$\bullet\;$If $\Delta(k)\in PE$, the convergence is exponential. $\Box\Box\Box$

There are three important advantages of DREM over the standard gradient estimator.

P1 As shown in (14) the individual parameter errors are monotonically non-increasing, a property that is strictly stronger than monotonicity of their norm indicated in (3) and (4).

P2 Parameter convergence is established without the restrictive PE assumption—being replaced, instead, by a non square-integrability/summability assumption.

P3 Convergence rates of DREM can be made arbitrarily fast simply increasing $\gamma_{i}$ in CT (or decreasing it in DT).

Remark 2**.**

Regarding the property P2, in [2] the relationship in CT between the conditions $\phi(t)\in PE$ and $\Delta(t)\notin{\cal L}_{2}$ is thoroughly discussed. In particular, in [2] it has been shown that, for arbitrary regressor vectors $\phi(t)$, these conditions are unrelated. On the other hand, for the case of identification of LTI systems, it has been shown in [4] that $\phi(t)\in PE$ if and only if $\Delta(t)\in PE$ for almost all LTI operators ${\cal H}$.

III A DREM Estimator with Strictly Weaker Convergence Conditions

In this section we present a particular version of DREM for which it is possible to show that its convergence conditions are strictly weaker than $\phi\in PE$. Since the construction, and the results, are very similar for CT and DT estimators, for brevity, we consider the latter case only.

Proposition 3**.**

Consider the DT version of the LRE (1). Fix an integer $\bar{K}\geq m$ and define (5) using the LTV operator

[TABLE]

Assume $\phi(k)\in PE$ and $\bar{K}\geq K$, with $K$ the size of the window given in Definition 1. The scalar, gradient-descent DT estimators (12), with $\Delta(k)$ and ${\cal Y}(k)$ defined in (8) and (9), ensure the following additional properties.

$\bullet\;$The condition for parameter convergence of DREM, i.e., $\Delta(k)\not\in\ell_{2}$, is strictly weaker than $\phi(k)\in PE$. More precisely, the following implications hold:

[TABLE]

$\bullet\;$The condition for exponential parameter convergence of DREM, i.e., $\Delta(k)\in PE$, is also weaker than $\phi(k)\in PE$ in the following precise sense

[TABLE]

Proof.

To prove the claims we make the key observation that

[TABLE]

The implications (LABEL:phipeimpdelnotl2) and (LABEL:phipeimpdelpe) follow using the identity (17), Definition 1 and noting the obvious fact that if $\phi(k)\in PE$ in a window of size $K$, then it is also PE for any window of size $\bar{K}\geq K$.

The proof of (3) is established with the following scalar counterexample: $\phi(k)=(k+1)^{-\frac{1}{4}}$ with $\bar{K}=1$. Since $\phi(k)$ tends to zero it is not PE, however, $\Delta(k)=(k+1)^{-{1\over 2}}\notin\ell_{2}.$

Finally, the proof of (3) is established with the following chain of implications:

Figure 1: Transients of the parameter estimation errors for different estimators and the control input $u(t)=15\sin(2.5t+1)$.

Figure 2: Transients of the parameter estimation errors for different estimators and the control input $u(t)=15$.

VI-B Alertness preserving DREM with FTC of Proposition LABEL:pro6

In this subsection we compare the two FTC DREMs presented in Section LABEL:sec5. Namely, the FTC DREM of Proposition LABEL:pro60, defined by (LABEL:hatthew), (LABEL:wi), and the new FTC DREM of Proposition LABEL:pro6 given by (LABEL:dotwd) and

[TABLE]

which is computed as soon as $w^{\tt D}_{i}(t)<\mu_{i}$. The objective of the simulation is to prove that the new FTC DREM is able to track time-varying parameters when new excitation arrives. This is in contrast with the old FTC DREM estimator that, since $w(t)\to 0$, converges to the gradient estimator and loses its FTC alertness property.

We consider the simplest case of a scalar system $y(t)=\Delta(t)\theta$ and simulate the gradient estimator (10), that is,

[TABLE]

together with (LABEL:wi) and (LABEL:dotwd), which are computed for $t\geq t_{c}$, with $t_{c}$ defined via the interval excitation criteria (LABEL:inttc) and (LABEL:inttcd), respectively.

We consider two scenarios: with and without excitation in $\Delta(t)$. For the first case we consider the PE signal $\Delta(t)=\sin(2\pi t)$, and for the second one $\Delta(t)=\frac{1}{t+1}$. Note that in the second case $\Delta(t)\to 0$, hence it is not PE. However, $\Delta(t)\not\in\mathcal{L}_{2}$, hence it satisfies the conditions for convergence of the DREM estimator.

For simulations we set $\gamma=2$, $\mu=0.98$, and $T_{\tt D}=0.2$. These parameters have been chosen such that the transients of both FTC estimators coincide in the ideal case when $\theta$ is constant and the system is excited. To illustrate the FTC tracking capabilities of the estimators the unknown parameter $\theta$ is time-varying and given by

[TABLE]

i.e., it starts at $10$, jumps to $15$ at $t=10$, and then linearly returns to $10$.

The transient of the estimators for $t\in[0,3]$ and $\Delta(t)=\sin(2\pi t)$ are given in Fig. 3, where we plot the gradient estimate $\hat{\theta}(t)$, as well as the old and the new FTC estimates $\hat{\theta}^{\tt FTC}(t)$ and $\hat{\theta}^{\tt FTC-D}(t)$. We observe that, as expected, both FTC estimators are overlapped and converge in finite time, while the gradient converges only asymptotically.

The behavior of the estimators for $t\in[9,40]$ is shown in Figure 4, where we also plot the time-varying parameter $\theta(t)$. As predicted by the theory, the old FTC behaves as the gradient estimator and their trajectories coincide. On the other hand, the new estimator preserves FTC alertness after the first parameter jump and achieves fast tracking of the linearly time-varying $\theta(t)$.

For the non-PE case of $\Delta(t)=\frac{1}{t+1}$, the transients of the estimators are given in Fig. 5. We observe that both FTC estimators, again, essentially coincide in the first few seconds and converge in finite time, while the gradient does it only asymptotically. After the first parameter change at $t=10$ the old FTC and the gradient coincide, while the new FTC manages to track in finite time the parameter jump. However, during the ramp parameter change—because of the lack of excitation—neither one of the estimators can track the parameter variation but the new FTC estimator performs much better.

VII Future Work

Current research is under way to derive some of the new results presented only for the CT time case, to the practically important, DT case. Moreover, in the spirit of [4], we are further exploring the role of the operator ${\cal H}$ on the determinant of the extended regressor matrix $\Phi$ and we plan to study the effect of an additive signal in the LRE (1), to study its input-to-state stability properties.

A widely open, long-term research topic is how to deal with nonlinear parameterizations, that is, the case in which (1) is replaced by $y=F(\phi,\theta)$, where $F(\cdot,\cdot)$ is a nonlinear function. Some preliminary results exploiting convexity, concavity or monotonicity may be found in [1, 14, 15]. As pointed out in [2], DREM is directly applicable—without overparameterization—in the simplest case of separable nonlinearities, that is, when the regression is of the form $y=F_{\phi}(\phi)F_{\theta}(\theta)$. The more general case is a challenging open problem.

Acknowledgment

The authors would like to thank Vladimir Nikiforov and Dmitry Gerasimov for many useful discussions that helped us to improve the quality of our contribution.

This paper is partly supported by by Government of Russian Federation (GOSZADANIE 2.8878.2017/8.9, grant 08-08), the European Union’s Horizon 2020 Research and Innovation Programme under Grant 739551 (KIOS CoE).