Identification of Disturbed Control Systems

TL;DR

This paper develops criteria and algorithms for identifying nonlinear control systems from experiments, ensuring practical applicability with known convergence rates.

Contribution

It introduces new identifiability criteria and an implementable algorithm for nonlinear control system identification, with explicit convergence rate analysis.

Findings

Established several identifiability criteria.

Proposed an implementable identification algorithm.

Explicitly computed convergence rates.

Abstract

This paper studies the identification of nonlinearly parameterized control systems in given experiments. Several identifiability criteria are established and an implementable algorithm is proposed for practicality with the convergence rate explicitly computed.

Equations

$$\left\{\begin{array}[]{l}x_{t}=f(\theta,u_{t-1},\chi_{t-1})+w_{t}\\ y_{t}=h(\theta,x_{t})+v_{t}\end{array},\quad t\geq 1,\right.$$

$${\inf }_{z\in \mathcal{W}\times \mathcal{V}}P((w_1,v_1)\in B(z,\delta ))>0,\forall \delta >0.$$

$$\mathcal{E}\triangleq \{({\chi }_0,\{u_t{\}}_{t\ge 0}):\Vert u_t\Vert \le C_u,t\ge 1\}\text{mbox}for{C}_u>0.$$

$$\displaystyle\left\{\begin{array}[]{l}\bar{h}(x,\chi,\psi)\triangleq(h(x,z^{\prime\prime}_{1})+z_{1}^{\prime},\ldots,h(x,z^{\prime\prime}_{m})+z^{\prime}_{m})\\ \bar{h}^{-1}(x,\varphi,\psi)\triangleq(h^{-1}(x,z_{1}-z^{\prime}_{1}),\ldots,h^{-1}(x,z_{m}-z^{\prime}_{m}))\\ h^{\prime}(x,u,\chi,w,v)\triangleq h(x,f(x,u,\chi)+w)+v\\ \hat{h}(x,u,\varphi,\psi,w,v)\triangleq\bigcup_{\chi\in\bar{h}^{-1}(x,\varphi,\psi)}h^{\prime}(x,u,\chi,w,v)\end{array}.\right.$$

$$\text{mbox}Im(h_{x,u,\chi }^{\prime })\triangleq h^{\prime }\left(x,u,\chi ,\mathcal{W},\mathcal{V}\right)\text{mbox}and\text{mbox}Im({\hat{h}}_{x,u,\phi })\triangleq \hat{h}\left(x,u,\phi ,{\mathcal{V}}^m,\mathcal{W},\mathcal{V}\right),$$

$$\mathcal{B}{\zeta }_1({\check{z}}_1)\ne \mathcal{B}{\zeta }_2({\check{z}}_2)\text{mbox}if{\zeta }_1\left(Z_1\right)\cap {\zeta }_2\left(Z_2\right)=\varnothing .$$

$$\displaystyle\left\{\begin{array}[]{l}\mbox{Im}(\mathcal{B}\bar{h})^{\epsilon}_{\check{x},\check{\chi}}\triangleq\mathcal{B}\bar{h}\left(\check{x},\check{\chi},\mathcal{\check{V}}^{m}_{\epsilon}\right),\quad\mbox{Im}(\mathcal{B}h^{\prime})^{\epsilon}_{\check{x},\check{u},\check{\chi}}\triangleq\mathcal{B}h^{\prime}\left(\check{x},\check{u},\check{\chi},\check{\Pi}_{\epsilon}\right)\\ \mbox{Im}(\mathcal{B}\hat{h})^{\epsilon}_{\check{x},\check{u},\check{\varphi}}\triangleq\mathcal{B}\hat{h}\left(\check{x},\check{u},\check{\varphi},\mathcal{\check{V}}^{m}_{\epsilon},\check{\Pi}_{\epsilon}\right)\end{array}.\right.$$

$$y_t=f(\theta ,{\phi }_{t-1})+f^{\prime }(u_{t-1},{\phi }_{t-1})+w_t,t\ge 1,$$

$$\underset{z\in \mathcal{W}}{\inf }P(w_1\in B(z,\delta ))>0,\forall \delta >0.$$

$$f_i\triangleq \Vert f(\theta ,{\phi }_i)+f^{\prime }(u_i,{\phi }_i)\Vert \le C_0,i=t,\dots ,t+m-1,$$

$${\mathcal{P}}_{\alpha }\triangleq \{\eta \in {\mathbb{R}}^{pm}:\exists \beta \in \text{mbox}Im({\bar{h}}_{x,\eta })\text{mbox}s.t.Im(h_{x,u,\eta }^{\prime })⊈\text{mbox}Im({\hat{h}}_{x^{\prime },u,\beta })\},$$

$$\alpha \in A_0\triangleq \{(x,x^{\prime })\in \Theta \times \Theta :x\ne x^{\prime }\}.$$

$$\text{mbox}Im(h_{x,u,\eta }^{\prime })⊈\text{mbox}Im({\hat{h}}_{x^{\prime },u,\beta })\iff f(x,\beta )\ne f(x^{\prime },\beta ).$$

$$\underline{d}(Z^{\prime }|Z)=\frac{1}{{\sup }_{z\in Z}\inf \{d>0:B(z,d)\cap Z^{\prime }\ne \varnothing \}}.$$

$${\underline{d}}^m(Z^{\prime }|Z)\triangleq \underset{{\prod }_{j=1}^m{E}_j\subset Z^{\prime },E_j\in {\mathbb{R}}^l}{\sup }({\min }_{j\in [1,m]}\underline{d}(E_j|Z_j)).$$

$$\displaystyle f(x,y_{1},y_{2})=\left\{\begin{array}[]{ll}0,&y_{1}\in[-C_{w},C_{w}]\\ x(y_{1}-C_{w}),&y_{1}>C_{w}\\ x(y_{1}+C_{w}),&y_{1}<-C_{w}\end{array},\quad x\in[1,2].\right.$$

$${\mathcal{P}}_{\alpha }(ϵ)\triangleq \{\eta \in {\mathbb{R}}^{pm}:\exists \check{\beta }\in \text{mbox}Im(\mathcal{B}\bar{h}{)}_{{\check{x}}_{ϵ},{\check{\eta }}_{ϵ}}^{ϵ}\text{mbox}s.t.$$

$$A_k\triangleq \left\{(x,x^{\prime })\in {\Theta }_{k-1}\times {\Theta }_{k-1}:\Vert x-x^{\prime }\Vert \ge c_k\right\},k\ge 1,$$

$$\underline{d}(E_j^{ik}|\overline{B(0,C_0)})>1/(C_w-{\sigma }_k),{\sigma }_k\in (0,C_w)\text{mbox}and|{\Delta }_i^k|={\hat{n}}_k,$$

$$h^{\prime }(x,u,\chi ,w,v)\notin ({\bigcup }_{z\in N_{i,k}^{\prime }}\text{mbox}Im({\hat{h}}_{z,u,\phi }))\text{mbox}with\phi =\bar{h}(x,\chi ,\psi ).$$

$$\overline{B(0,C_0)}\subset {\bigcup }_{e_j^{\alpha }\in E_j^{\alpha }}B(e_j^{\alpha },C_w-2{\sigma }_{\alpha }).$$

$$\underline{d}(E_j^{\alpha }|\overline{B(0,C_0)})>1/(C_w-{\sigma }_{\alpha }).$$

$$\displaystyle\left\{\begin{array}[]{l}\beta_{\alpha s}=\bar{h}(x,\eta_{\alpha s},\psi_{\alpha s})\\ h^{\prime}(x,u,\eta_{\alpha s},w^{*}_{\alpha s,u},v^{*}_{\alpha s,u})\notin\mbox{Im}(\hat{h}_{x^{\prime},u,\beta_{\alpha s}})\end{array}.\right.$$

$$h^{\prime }(B_x^{\alpha },u,{\bar{D}}_{\alpha s,l})\cap ({\bigcup }_{z\in B_{x^{\prime }}^{\alpha },\phi \in B({\beta }_{\alpha s},{\epsilon }_{\alpha })}\text{mbox}Im({\hat{h}}_{z,u,\phi }))=\varnothing ,1\le l\le {\hat{n}}_{\alpha s},$$

$$\bar{h}(B_x^{\alpha },B({\eta }_{\alpha s},d_{\alpha }),B({\psi }_{\alpha s},d_{\alpha })\subset B({\beta }_{\alpha s},{\epsilon }_{\alpha }),\forall s\in [1,{\hat{n}}_{\alpha }].$$

$$\bar{h}(B_{i,k},B({\eta }_{is},d_k),B({\psi }_{is},d_k))\subset B({\beta }_{is},{\epsilon }_k)$$

$$\left(h^{\prime }(x,u,B({\eta }_{is},d_k),B((w_{is,l}^{\ast },v_{is,l}^{\ast }),d_k))\right)\cap ({\bigcup }_{z\in B_{i,k}^{\prime }}\text{mbox}Im({\hat{h}}_{z,u,\phi }))=\varnothing .$$

$$(x,\chi ,\psi ,w,v)\in B_{i,k}\times B({\eta }_{is},d_k)\times B({\psi }_{is},d_k)\times B((w_{is,l}^{\ast },v_{is,l}^{\ast }),d_k),$$

$$\{{\bar{N}}_{i,k}\times {\bar{N}}_{ij,k}^{\prime },1\le j\le m_{k,i}\},m_{k-1}\le i<m_k(m_0=1).$$

$$J_{it}^k\triangleq \{j\in [1,m_{k,i}]:y_t\notin {\bigcup }_{z\in {\bar{N}}_{ij,k}^{\prime }}\text{mbox}Im({\bar{h}}_{z,u_{t-1},{\phi }_{t-1}})\},m_{k-1}\le i<m_k.$$

Full text

1 Introduction

Consider the nonlinearly parameterized control system

[TABLE]

where $x_{t},y_{t},u_{t}$ and $(w_{t},v_{t})$ represent the $p\times 1$ state vector, $q\times 1$ output vector, $r\times 1$ input vector and $(p+q)\times 1$ noise vector, respectively. Denote $\chi_{t}\triangleq(x_{t},\ldots,x_{t-m+1})$ as the state regressor. Unknown parameter $\theta$ is non-random and belongs to a known nondegenerate compact hyperrectangle $\Theta\subset\mathbb{R}^{n}$. Moreover, $f:\mathbb{R}^{n}\times\mathbb{R}^{r}\times\mathbb{R}^{pm}\to\mathbb{R}^{p}$ and $h:\mathbb{R}^{n}\times\mathbb{R}^{p}\to\mathbb{R}^{q}$ are two known functions. Let $h^{-1}:\mathbb{R}^{n}\times\mathbb{R}^{q}\rightarrow 2^{\mathbb{R}^{p}}$ be a set-valued function that $h^{-1}(x,y)\triangleq\{z:h(x,z)=y\}$, then assume

A1

The noises $\{w_{t}\}$ and $\{v_{t}\}$ are two i.i.d sequences satisfying:

(i) $\{w_{t}\}$ is independent of $\{v_{t}\}$;

(ii) for each $t\geq 1$, $(w_{t},v_{t})$ is independent of $\chi_{0}$ and $\{u_{i}\}_{0\leq i\leq t-1}$;

(iii) $\|w_{1}\|\leq C_{w}$ and $\|v_{1}\|\leq C_{v}$ for some $C_{w}>0$ and $C_{v}\geq 0$. In addition,

[TABLE]

where $\mathcal{W}\triangleq\overline{B(0,C_{w})}\subset\mathbb{R}^{p}$ and $\mathcal{V}\triangleq\overline{B(0,C_{v})}\subset\mathbb{R}^{q}$.

A2

$f$ and $h$ are continuous; $h^{-1}$ is bounded-valued and upper semicontinuous*** A set-valued function $\zeta:X\rightarrow 2^{Y}$ is bounded-valued if for $\forall x\in X$, $\zeta(x)$ is bounded. $\zeta$ is said to be upper semicontinuous if for any $x\in X$ with $\zeta(x)\neq\emptyset$ and any neighborhood $U$ of $\zeta(x)$, there is a $d_{x}>0$ such that $\zeta(B(x,d_{x}))\subset U$. .

An important issue in system identification is to solve the identifiability of system (1) in an experiment $(\chi_{0},\{u_{t}\})\in\mathcal{E}$, where $\mathcal{E}$ is the set of all admissible experiments defined by

[TABLE]

This direction arises from numerous engineering applications where identification has to be performed in control processes, especially with feedbacks inherent [1], [2], [3], [6], [10], [11], [14]. Unlike identification operating in open loop, a prominent feature of closed-loop identification is that there is no design level on data in parameter estimation, once a feedback law is chosen. In this paper, we assume that the experiment is designed in advance for control purposes. Then, outputs $y_{t}$ will be produced by control system (1) automatically. We aim to identify parameter $\theta$ in the running process of the control system.

Historically, identification of noise-free systems from input-output data has been well addressed. Literatures on this topic have also shed some light on the determining factor of identifiablity for disturbed control systems. As stated by [7], parameter identification is in nature a procedure of distinguishing output trajectories of different parameters. From this viewpoint, the critical criterion, in some sense, on linear system structure was deduced by [7]. Nonlinear systems with noises absent were treated therein as well. Considering noises, however, different observations might be produced by the same parameter. We thus introduce the definition of identifiability for disturbed control systems as following.

Definition 1**.**

System (1) is identifiable under experiment $(\chi_{0},\{u_{t}\})\in\mathcal{E}$, if there is an estimator such that the unknown parameter $\theta$ in $\Theta$ can be uniquely determined by the data set $Z^{\infty}\triangleq\{y_{t+1},u_{t}\}_{t\geq 0}$ with probability $1$.

Examining output trajectories to check identifiability is not straightforward in most circumstances. So, interesting move to derive some simple identifiability criteria. This is exactly the first part of the paper, where it is argued in Section 2 that the excitation points of control system (1) are crucial for identifiability. In fact, given any experiment in $\mathcal{E}$, the identifiability of system (1) is ensured if the excitation point set is sufficiently dense. A lower bound of the required density is computed accordingly. On the other hand, if the density of the excitation points is smaller than the lower bound, the identification may possibly fail. Generally speaking, this structure condition for identifiability is weaker than that for the noise-free case. This is because noises $\{w_{t}\}$ in the state equation are advantageous in identification, as suggested by the results.

Since the estimator studied for the proofs of Theorems 4 and 5 is only of theoretical interest, the second part of the paper is intended to introduce an implementable algorithm for the sake of practicality. The proposed estimator is called the grid searching (GS) estimator and has its origins in the nonlinear least-squares (NLS) method, whose asymptotic behaviours and approximation algorithms have been explored for decades [4], [5], [12], [13], [16]. By modifying the NLS method in Section 3, the GS estimator is proved to be strong consistent for a basic class of disturbed control systems under some appropriate conditions. This estimator can also cope with the situation where the noise variances are unknown.

2 Identifiability for Control Systems

We shall establish some identifiability criteria for system (1) on the basis of experiment data.

2.1 Notations

Throughout this paper, we consider the probability measure space $(\Omega,\mathcal{F},P)$. The notations and definitions used in this section are introduced here. Let diam$(x,A)\triangleq\sup_{x^{\prime}\in A}d(x,x^{\prime})$, where $d(\cdot,\cdot)$ denotes the distance between two points. Denote $\varphi\triangleq(z_{1},\ldots,z_{m})$, $\psi\triangleq(z^{\prime}_{1},\ldots,z^{\prime}_{m})$ and $\chi\triangleq(z^{\prime\prime}_{1},\ldots,z^{\prime\prime}_{m})$ with $z_{j},z^{\prime}_{j}\in\mathbb{R}^{q},z^{\prime\prime}_{j}\in\mathbb{R}^{p},j\in[1,m]$. Then, for $x\in\mathbb{R}^{n},w\in\mathbb{R}^{p},v\in\mathbb{R}^{q}$, define

[TABLE]

So, $\bar{h}^{-1}$ and $\hat{h}$ are set-valued functions. Denote the images of $\bar{h}$, $h^{\prime}$ and $\hat{h}$ at fixed points $(x,\chi)$, $(x,u,\chi)$ and $(x,u,\varphi)$, respectively, by $\mbox{Im}(\bar{h}_{x,\chi})\triangleq\bar{h}(x,\chi,\mathcal{V}^{m})$,

[TABLE]

Now, let $k,l\in\mathbb{N}^{+}$. View set $Z\subset\mathbb{R}^{k}$ as a point $\check{z}\in 2^{\mathbb{R}^{k}}$ and by a slight abuse of notation, we write $\check{z}=Z$. Now, for function (respectively, set-valued function) $\zeta:\mathbb{R}^{k}\rightarrow\mathbb{R}^{l}$ (respectively, $2^{\mathbb{R}^{l}}$), define $\mathcal{B}\zeta:2^{\mathbb{R}^{k}}\rightarrow 2^{\mathbb{R}^{l}}$ by $\mathcal{B}\zeta(\check{z})=\zeta\left(Z\right).$ Let $\zeta_{i}$ be two functions and $\check{z}_{i}=Z_{i},i=1,2$. We say

[TABLE]

Given $\epsilon>0$, for any $z\in\mathbb{R}^{k}$, denote $\check{z}_{\epsilon}=B(z,\epsilon)\in 2^{\mathbb{R}^{k}}$. Then, let $\mathcal{\check{V}}^{m}_{\epsilon}\triangleq\bigcup_{\psi\in\mathcal{V}^{m}}\check{\psi}_{\epsilon}$ and $\check{\Pi}_{\epsilon}\triangleq\bigcup_{\pi\in\mathcal{W}\times\mathcal{V}}\check{\pi}_{\epsilon}$. Define the images of $\mathcal{B}\bar{h}$, $\mathcal{B}h^{\prime}$ and $\mathcal{B}\hat{h}$ at points $(\check{x},\check{\chi})\in 2^{\mathbb{R}^{n}}\times 2^{\mathbb{R}^{pm}}$, $(\check{x},\check{u},\check{\chi})\in 2^{\mathbb{R}^{n}}\times 2^{\mathbb{R}^{r}}\times 2^{\mathbb{R}^{pm}}$ and $(\check{x},\check{u},\check{\varphi})\in 2^{\mathbb{R}^{n}}\times 2^{\mathbb{R}^{r}}\times 2^{\mathbb{R}^{qm}}$ by

[TABLE]

2.2 Motivations and Excitation Points

Let us first look at a simple system

[TABLE]

where $\varphi_{t}\triangleq(y_{t},\ldots,y_{t-m+1})$ is an observable $pm\times 1$ vector. The experiment thus becomes $(\varphi_{0},\{u_{t}\})$ in $\mathcal{E}$ and Assumptions A1–A2 degenerate to

A1’

$\{w_{t}\}$ is an i.i.d sequence satisfying

(i) for each $t\geq 1$, $w_{t}$ is independent of $\chi_{0}$ and $\{u_{i}\}_{0\leq i\leq t-1}$;

(ii) $\|w_{1}\|\leq C_{w}$ for some finite $C_{w}>0$ and

[TABLE]

A2’

$f$ and $f^{\prime}$ are continuous.

The most familiar experiments are the ones that casue $\|\varphi_{t}\|\leq C$, $\forall t\geq 1$ almost surely for some $C>0$. Apparently, if $\|\varphi_{t}\|\leq C$, by (1), (3) and Assumption A2’, it is easy to compute a $C_{0}>0$ that

[TABLE]

and hence $(f_{t},\ldots,f_{t+m-1})\in\mathcal{S}\triangleq\prod_{i=1}^{m}\overline{B(0,C_{0})}\subset\mathbb{R}^{pm}$. So, the following result is not suprising.

Theorem 2**.**

Under Assumptions A1’–A2’, let $(\varphi_{0},\{u_{t}\})\in\mathcal{E}$ be an experiment such that $P\{\|\varphi_{t}\|\leq C,i.o.\}=1$ for some $C>0$. Then, control system (13) is identifiable if for each pair $x,x^{\prime}\in\Theta$ with $x\neq x^{\prime}$, there are sufficiently dense points $\beta\in\mathcal{S}$ such that $f(x,\beta)\neq f(x^{\prime},\beta)$.

This theorem is a direct consequence of Theorem 4 appearing in a later section. The observation of the above theorem enlightens us to introduce set

[TABLE]

where $u$ is restricted to $\overline{B(0,C_{u})}$ and

[TABLE]

We call $\eta\in\mathcal{P}_{\alpha}$ an excitation point of $\alpha\in A_{0}$ for system (1). If a system $(f,h)$ has sufficiently dense excitation points of $\alpha$, then states $\chi_{t}$ are very likely to fall in $\mathcal{P}_{\alpha}$. This means it is relatively easy to distinguish $x$ and $x^{\prime}$.

Example 2.1**.**

Consider system (13), in which case $\eta=\beta$ and

[TABLE]

Heuristically, $\mathcal{P}_{\alpha}=\{\beta:f(x,\beta)\neq f(x^{\prime},\beta)\}$ is composed of the points where different parameters give rise to different values of $f$.

Theorem 2 suggests that the identifiability of a control system depends on the density of $P_{\alpha},\alpha\in A_{0}$. More precisely, for two sets $Z,Z^{\prime}\in\mathbb{R}^{l},l\geq 1$, we define the lower density of $Z^{\prime}$ in $Z$ by

[TABLE]

Further, when $Z=\prod_{j=1}^{m}Z_{j},Z_{j}\subset\mathbb{R}^{l},l\geq 1$ and $Z^{\prime}\subset\mathbb{R}^{lm}$, the $m$-symmetric lower density of $Z^{\prime}$ in $Z$ is defined by

[TABLE]

Clearly, $\underline{d}^{1}(Z^{\prime}|Z)=\underline{d}(Z^{\prime}|Z)$. To identify parameter $\theta$, the density of $\mathcal{P}_{\alpha}$ for control system (1) is deduced in the next subsection.

2.3 Identifiability Criteria

The criteria are presented in two cases.

2.3.1 Criterion for $C$-Recurrence

System states are usually constrained in a bounded area in practice. It is a special case of $C$-recurrence defined below:

Definition 3**.**

*An experiment $(\chi_{0},\{u_{t}\})\in\mathcal{E}$ is said to be $C$-recurrent for some $C>0$, if the corresponding states satisfy $P\{\|\chi_{t}\|\leq C,i.o.\}=1.$

The main result of this section is stated as follows.

Theorem 4**.**

Under Assumptions A1–A2, control system (1) is identifiable for any $C$-recurrent experiment $(\chi_{0},\{u_{t}\})\in\mathcal{E}$ if $\underline{d}^{m}(\mathcal{P}_{\alpha}|\mathcal{S})>1/C_{w}$ for each $\alpha\in A_{0}$.

Remark 2.1**.**

General speaking, the lower bound $1/C_{w}$ in Theorem 4 cannot be further relaxed. For example, consider system (13) with $[1,2]\subset\Theta\subset\mathbb{R}$, $m=2,p=1$. Assume $f^{\prime}(u,y_{1},y_{2})=uy_{2}$ and

[TABLE]

It is evident that $\underline{d}^{2}(\mathcal{P}_{(1,2)}|\mathcal{S})=1/C_{w}$. Moreover, $\theta$ cannot be identified in experiment $((0,0),\{0\})$, which is $C$-recurrent for any given $C>0$.

Remark 2.2**.**

To some extent, noises $\{w_{t}\}$ in the state equation are advantageous in the closed-loop identification, whereas $\{v_{t}\}$ in the observation equation play an opposite role. This observation becomes clear during the proof of Theorem 4.

2.3.2 Criterion for General Case

Generally, given an $\alpha\in A_{0}$, the excitation points of $\alpha$ are expected in the following set for some $\epsilon>0$:

[TABLE]

where $u$ is only need to be considered in $\overline{B(0,C_{u})}$.

Theorem 5**.**

Under Assumptions A1–A2, control system (1) is identifiable for any experiment $(\chi_{0},\{u_{t}\})\in\mathcal{E}$ if for each $\alpha\in A_{0}$, there exists some $\epsilon>0$ such that $\underline{d}^{m}(\mathcal{P}_{\alpha}(\epsilon)|\mathbb{R}^{pm})>1/C_{w}$.

We have thus far solved the identifiability issue. Later, an implementable algorithm will be provided in Section 3 with the convergence rates explicitly computed.

2.4 Proofs of Theorems 4 and 5

The proofs of Theorems 4 and 5 are similar, so we only give the detailed proof of Theorem 4.

2.4.1 Theoretical Nonlinear Estimator

To design an estimator competent for the identification task, we need a simple result on functions $f$ and $h$. For this, let $\{\Theta_{k}\subset\mathbb{R}^{n},k\geq 0\}$ be a series of sets with $\Theta_{0}\triangleq\Theta$ and $\Theta_{k}\subset\Theta_{k-1}$. Define

[TABLE]

where $c_{k}\leq\frac{1}{k}$ is properly small such that $(\{x\}\times\Theta_{k-1})\cap A_{k}\neq\emptyset$ for all $x\in\Theta_{k-1}$.

Lemma 6**.**

*Let Assumption A2 hold and $\underline{d}^{m}(\mathcal{P}_{\alpha}|\mathcal{S})>1/C_{w},\forall\alpha\in A_{0}$. Then, for each $k\geq 1$, a finite covering of $A_{k}$ in the form $\{N_{i,k}\times N^{\prime}_{i,k}\}_{n_{k-1}\leq i<n_{k}},n_{0}=1$ exists. Moreover, for every $i\in[n_{k-1},n_{k})$, the following two statements hold:

(i) $N_{i,k}\cap N_{j,k}=\emptyset,N^{\prime}_{i,k}\cap N^{\prime}_{j,k}=\emptyset$ for $j\neq i$ and $(N_{i,k}\times N^{\prime}_{i,k})\cap A_{k}\neq\emptyset$;

(ii) there exists a finite set of points $\Delta^{k}_{i}=\prod_{j=1}^{m}E^{ik}_{j}\subset\mathcal{P}_{\alpha}$ for some $\alpha\in A_{0}$ with*

[TABLE]

a sequence $\{\psi_{is}\in\mathcal{V}^{m},(w^{*}_{is,l},v^{*}_{is,l})\in\mathcal{W}\times\mathcal{V},\mathcal{U}^{is}_{l}\subset\mathbb{R}^{r}\}_{1\leq s\leq\hat{n}_{k},1\leq l\leq\hat{n}_{is}}$ with $\{\mathcal{U}^{is}_{l}\}$ being some mutually disjoint sets that $\overline{B(0,C_{u})}=\sum_{l=1}^{\hat{n}_{is}}\mathcal{U}^{is}_{l}$ and a number $d_{k}\in(0,\sqrt{m}\sigma_{k})$, where $\hat{n}_{k}$, $\sigma_{k}$, $d_{k}$ depend only on $k$ and $\hat{n}_{is}$ depends on $i,s$, such that for every $s\in[1,\hat{n}_{k}]$, if $(x,\chi,\psi,w,v,u)\in N_{i,k}\times B(\eta_{is},d_{k})\times B(\psi_{is},d_{k})\times B((w^{*}_{is,l},v^{*}_{is,l}),d_{k})\times\mathcal{U}^{is}_{l}$, $l\in[1,\hat{n}_{is}]$, then

[TABLE]

Proof.

Note that $\underline{d}^{m}(\mathcal{P}_{\alpha}|\mathcal{S})>1/C_{w}$ for each $\alpha\in A_{0}$. Take a $\sigma_{\alpha}\in(0,C_{w})$ and a sequence $\{\bar{E}^{\alpha}_{j}\}_{1\leq j\leq m}$ such that $\prod_{j=1}^{m}\bar{E}^{\alpha}_{j}\subset\mathcal{P}_{\alpha}$ and $\min_{j\in[1,m]}\underline{d}(\bar{E}^{\alpha}_{j}|\overline{B(0,C_{0})})>1/(C_{w}-2\sigma_{\alpha})$. Then, for every $j\in[1,m]$ and $z\in\overline{B(0,C_{0})}$, $z\in B(e^{\alpha}_{j},C_{w}-2\sigma_{\alpha})$ for some $e^{\alpha}_{j}\in\bar{E}^{\alpha}_{j}$. Consequently, by the compactness of $\overline{B(0,C_{0})}$, there exists a finite set $E^{\alpha}_{j}=\{e^{\alpha}_{j}\in\bar{E}^{\alpha}_{j}\}$ such that

[TABLE]

So, for every $\alpha\in A_{0}$ and $j\in[1,m]$,

[TABLE]

Clearly, $\Delta_{\alpha}\triangleq\prod_{j=1}^{m}E^{\alpha}_{j}\subset\mathcal{P}_{\alpha}$ and $\hat{n}_{\alpha}\triangleq|\Delta_{\alpha}|$ is finite as well.

Now, fix $k\geq 1$. Given $\alpha=(x,x^{\prime})\in A_{k}$, (16) shows that for any $\eta_{\alpha s}\in\Delta_{\alpha}$ and $u\in\overline{B(0,C_{u})}$, there are some $\psi_{\alpha s}\in\mathcal{V}^{m}$ and $(w^{*}_{\alpha s,u},v^{*}_{\alpha s,u})\in\mathcal{W}\times\mathcal{V}$ such that

[TABLE]

In view of Assumption A2, $h^{\prime},\bar{h}$ are continuous and $\hat{h}$ is upper semicontinuous and bounded-valued. By the compactness of $\mathcal{W}$ and $\mathcal{V}$, $\mbox{Im}(\hat{h}_{x,u,\varphi})$ is upper semicontinuous and bounded-valued at $(x,u,\varphi)$ as well. Because $\overline{B(0,C_{u})}$ is compact, for each $s\in[1,\hat{n}_{\alpha}]$, there exist some mutually disjoint sets $\{\mathcal{U}^{\alpha s}_{l}\}_{1\leq l\leq\hat{n}_{\alpha s}}$ with $\overline{B(0,C_{u})}=\sum_{l=1}^{\hat{n}_{\alpha s}}\mathcal{U}^{\alpha s}_{l}$, some points $\{(w^{*}_{\alpha s,l},v^{*}_{\alpha s,l})\in\mathcal{W}\times\mathcal{V}\}_{1\leq l\leq\hat{n}_{\alpha s}}$, some neighbourhoods $B^{\alpha}_{x}$ and $B^{\alpha}_{x^{\prime}}$ of $x$ and $x^{\prime}$ respectively and a number $\varepsilon_{\alpha}>0$, such that if $u\in\mathcal{U}^{\alpha s}_{l}$, then

[TABLE]

where $\bar{D}_{\alpha s,l}\triangleq B(\eta_{\alpha s},\varepsilon_{\alpha})\times B((w^{*}_{\alpha s,l},v^{*}_{\alpha s,l}),\varepsilon_{\alpha})$. Note that $B^{\alpha}_{x},B^{\alpha}_{x^{\prime}}$ and $\varepsilon_{\alpha}$ are taken independent of $s\in[1,\hat{n}_{\alpha}]$. In addition, as long as $B^{\alpha}_{x}$ is sufficiently small, there is a $d_{\alpha}\in(0,\varepsilon_{\alpha})$ satisfying

[TABLE]

Now, for any $(x,x^{\prime})\in A_{k}$, we find an open set $B^{\alpha}_{x}\times B^{\alpha}_{x^{\prime}}$ fulfilling (24) and (25) for all $\eta_{\alpha s}\in\Delta_{\alpha},s\in[1,\hat{n}_{\alpha}]$. Therefore, the compact set $A_{k}$ can be covered by some finite open sets $\{B_{i,k}\times B^{\prime}_{i,k},\bar{n}_{k-1}\leq i<\bar{n}_{k}\}$ ($\bar{n}_{0}=1$), where for each $i\in[\bar{n}_{k-1},\bar{n}_{k})$, it corresponds to a set $\Delta^{k}_{i}=\prod_{j=1}^{m}E^{\alpha}_{j}=\Delta_{\alpha}$ for some $\alpha\in A_{k}$, a sequence $\{(\psi_{is},w^{*}_{is,l},v^{*}_{is,l})\in\mathcal{V}^{m}\times\mathcal{W}\times\mathcal{V},\mathcal{U}^{is}_{l}\subset\mathbb{R}^{r}\}_{1\leq s\leq\hat{n}_{k},1\leq l\leq\hat{n}_{is}}$ with $\overline{B(0,C_{u})}=\sum_{l=1}^{\hat{n}_{is}}\mathcal{U}^{is}_{l}$ and some numbers $\sigma_{k},d_{k},\varepsilon_{k}$ with $0<d_{k}<\min\{\varepsilon_{k},\sqrt{m}\sigma_{k}\}$, such that for any $(x,\varphi)\in B_{i,k}\times B(\beta_{is},\varepsilon_{k})$ with $\beta_{is}=\bar{h}(x,\eta_{is},\psi_{is})$ and $\eta_{is}\in\Delta^{k}_{i},s\in[1,\hat{n}_{k}]$,

[TABLE]

and when $u\in\mathcal{U}^{is}_{l}$, $1\leq l\leq\hat{n}_{is},$

[TABLE]

So, if for some $s\in[1,\hat{n}_{k}],l\in[1,\hat{n}_{is}]$, $\varphi=\bar{h}(x,\chi,\psi)$, $u\in\mathcal{U}^{is}_{l}$ and

[TABLE]

then by (26)–(27), $h^{\prime}(x,u,\chi,w,v)\notin(\bigcup_{z\in B^{\prime}_{i,k}}\mbox{Im}(\bar{h}_{z,u,\varphi}))$.

Finally, let $\{N_{i,k}\}$ and $\{N^{\prime}_{i,k}\}$ be a series of refined sets of $\{B_{j,k}\}$ and $\{B^{\prime}_{j,k}\}$, respectively, such that $N_{i,k}\cap N_{i^{\prime},k}=\emptyset$ and $N^{\prime}_{i,k}\cap N^{\prime}_{i^{\prime},k}=\emptyset,i^{\prime}\neq i$. Clearly, $\{N_{i,k}\times N^{\prime}_{i,k}\}_{i\in[n_{k-1},n_{k})}$ is a finite covering of $A_{k}$. Without loss of generality, let $(N_{i}\times N^{\prime}_{i})\cap A_{k}\neq\emptyset,i\in[n_{k-1},n_{k})$. Since every $N_{i,k}\times N^{\prime}_{i,k}\subset B_{j,k}\times B^{\prime}_{j,k}$ for some $j\in[\bar{n}_{k-1},\bar{n}_{k})$, (21) follows immediately. Besides, (20) holds by (22). $\Box$

We now provide a theoretical estimator to identify parameter $\theta$. Rewrite the finite covering of $A_{k},k\geq 1$ in Lemma 6 by

[TABLE]

So, $\bar{N}_{i,k}\cap\bar{N}_{j,k}=\emptyset,\forall i\neq j$ and $\sum_{i=m_{k-1}}^{m_{k}-1}m_{k,i}=n_{k}-n_{k-1}$. Let $\theta_{0}$ be the center of $\Theta$.

Algorithm:

Step 1

Let $t_{0}=0$, $\hat{\theta}_{0}=\theta_{0}$ and $\hat{\Theta}_{i,0}=\bar{N}_{i,1}\times\Theta_{0}$ for all $i=1,\ldots,m_{1}-1$.

Step 2

For $t>t_{k-1},k\geq 1$, if $\hat{\Theta}_{i,{t-1}}\neq(\hat{\Theta}_{i,{t_{k-1}}}\setminus A_{k})$ for all $i\in[m_{k-1},m_{k})$, denote

[TABLE]

Let $\hat{\Theta}_{i,t}=\hat{\Theta}_{i,t-1}\setminus((\bar{N}_{i,k}\times\bigcup_{j\in J^{k}_{it}}\bar{N}^{\prime}_{ij,k})\cap A_{k})$, where $i\in[m_{k-1},m_{k}).$ If for all $i\in[m_{k-1},m_{k})$, $\hat{\Theta}_{i,t}\neq(\hat{\Theta}_{i,{t_{k-1}}}\setminus A_{k})$, set

[TABLE]

Step 3

For $t>t_{k-1},k\geq 1$, if $\hat{\Theta}_{i,t}=(\hat{\Theta}_{i,{t_{k-1}}}\setminus A_{k})$ for some $i\in[m_{k-1},m_{k})$, take a point $(x,x)\in\hat{\Theta}_{i,t}$ and set

[TABLE]

Set $\Theta_{k}=\overline{B(\hat{\theta}_{t},c_{k})}$, $\hat{\Theta}_{i,t}=\bar{N}_{i,k+1}\times\Theta_{k}$ for $i=m_{k},\ldots,m_{k+1}-1$, and $t_{k}=t$.

Remark 2.3**.**

If $t_{k-1}<\infty$ for some $k\geq 1$, then the algorithm implies that $\Theta_{k-1}$, $A_{k}$ and $\{\bar{N}_{i,k}\times\bar{N}_{ij,k}\}_{i\in[1,m_{k}),j\in[1,m_{k,i}]}$ are well defined. As a result, in Lemma 6, $\{(\eta_{is},\psi_{is},w^{*}_{is,l},v^{*}_{is,l}),\mathcal{U}^{is}_{l}\}_{i\in[1,n_{k}),s\in[1,\hat{n}_{k}],l\in[1,\hat{n}_{is}]}$ are also well defined.

For each $k\geq 1$ and $i\in[n_{k-1},n_{k})$, denote $\Gamma^{k}_{i}\triangleq\bigcup_{s=1}^{\hat{n}_{k}}\bigcup_{l=1}^{\hat{n}_{is}}D^{k}_{is,l}$ with

[TABLE]

where $\eta_{is},\psi_{is},w^{*}_{is,l},v^{*}_{is,l},\mathcal{U}^{is}_{l}$ and $d_{k}>0$ are defined in Lemma 6.

Lemma 7**.**

Let $(\varphi_{0},\{u_{t}\})\in\mathcal{E}$ be an experiment designed that for each $k\geq 1$, if $t_{k-1}<\infty$ almost surely and $t_{k}=\infty$ on a set $D$ with $P(D)>0$, then

[TABLE]

*will hold almost surely on $D$ for all $i\in[n_{k-1},n_{k})$ satisfying $\theta\in N_{i,k}$, where $\psi_{t-1}\triangleq(v_{t-1},\ldots,v_{t-m})^{T}$. Then, under the conditions of Lemma 6, the nonlinear estimator constructed by (29)–(30) satisfies $\lim_{t\rightarrow\infty}\hat{\theta}_{t}=\theta$ almost surely. *

Proof.

We first show that under an experiment $(\varphi_{0},\{u_{t}\})\in\mathcal{E}$ designed in this lemma, the nonlinear algorithm will fulfill $t_{k}<\infty$ and $\theta\in\Theta_{k}$ for all $k\geq 0$ almost surely (this also means $\Theta_{k}$ are well defined for all $k$ almost surely). Since $t_{0}=0$ and $\Theta_{0}=\Theta$, suppose for some $k\geq 1$, $t_{i}<\infty$ and $\theta\in\Theta_{i}$ for all $i\in[0,k-1]$ almost surely. We claim that $t_{k}<\infty$ a.s. for this $k$. Otherwise, there is a set $D$ with $P(D)>0$ such that $t_{k}=\infty$ on $D$. Now, $t_{k-1}<\infty$, by Remark 2.3, $\Theta_{k-1}$ and $A_{k}$ are well defined. Note that $(\{\theta\}\times\Theta_{k-1})\cap A_{k}\neq\emptyset$ and hence $\theta\in\bar{N}_{\varsigma,k}$ for some $\varsigma\in[m_{k-1},m_{k})$. Let

[TABLE]

So, $|I_{k}|=m_{k,\varsigma}>0$. The experiment ensures $T^{k}_{i}\neq\emptyset$ for all $i\in I_{k}$ on $D$ almost surely. Consequently, for each $j\in[1,m_{k,\varsigma}]$ which corresponds to an integer $i(j)\in I_{k}$, there exist some random integers $t(j),s(j),l(j)$ taking values in $T^{k}_{i(j)},[1,\hat{n}_{k}]$ and $[1,\hat{n}_{i(j)s(j)}]$ respectively such that

[TABLE]

Considering $\theta\in\bar{N}_{\varsigma,k}=N_{i(j),k}$, by statement (ii) of Lemma 6,

[TABLE]

holds almost surely on $D$, where $\varphi_{t(j)-1}=\bar{h}(\theta,\chi_{t(j)-1},\psi_{t(j)-1})$.

Now, by Step 3 of the algorithm, it is clear that for each $i\in[m_{k-1},m_{k}-1]$,

[TABLE]

Since $t_{k}=\infty$ on $D$, $\hat{\Theta}_{\varsigma,{t}}\neq(\hat{\Theta}_{\varsigma,{t_{k-1}}}\setminus A_{k})$ for all $t\geq t_{k-1}$ on $D$. Denote $\bar{t}_{k-1}\triangleq\max_{1\leq j\leq m_{k,\varsigma}}t(j)$, then (33) yields $J^{k}_{\varsigma\bar{t}_{k-1}}=\{1,\ldots,m_{k,\varsigma}\}$ a.s. on D. So, by Step 2,

[TABLE]

on $D$ almost surely, which leads to a contradiction. Therefore, $t_{k}<\infty$ almost surely. Moreover, Step 3 implies that $\Theta_{k}$ is well defined almost surely.

The remainder is devoted to verifying $\theta\in\Theta_{k}$ on $\{t_{k}<\infty\}$. Take a trajectory on which $t_{k}<\infty$. The follow-up arguments are restricted on this trajectory. Denote

[TABLE]

which means for each $i\in I^{\prime}_{k}$, there is a point $x\in\bar{N}_{i,k}$ such that $d(x,\theta)>c_{k}$. Recall that $\theta\in\Theta_{k-1}$, then $(x,\theta)\in A_{k}$ and thus $\theta\in\bar{N}^{\prime}_{ij,k}$ for some $j\in[1,m_{k,i}]$ due to

[TABLE]

So, for all $t>t_{k-1}$, $y_{t}\in\bigcup_{z\in\bar{N}^{\prime}_{ij,k}}\mbox{Im}(\hat{h}_{z,u_{t-1},\varphi_{t-1}})$, which implies $((\bar{N}_{i,k}\times\bar{N}^{\prime}_{ij,k})\cap A_{k})\neq\emptyset$ belongs to $\hat{\Theta}_{i,t}$. Consequently, $\hat{\Theta}_{i,t}\neq(\hat{\Theta}_{i,{t_{k-1}}}\setminus A_{k})$ for all $t>t_{k-1}$ whenever $i\in I^{\prime}_{k}$. Now, $t_{k}<\infty$, so any index $\varsigma$ causes $\hat{\Theta}_{\varsigma,t_{k}}=(\hat{\Theta}_{\varsigma,{t_{k-1}}}\setminus A_{k})$ at Step 3 must satisfy $\varsigma\in[m_{k-1},m_{k})\backslash I^{\prime}_{k}$. Hence, diam$(\theta,\bar{N}_{\varsigma,k})\leq c_{k}.$ Moreover, because of (34), $\hat{\theta}_{t_{k}}$ in (30) is well defined at Step 3 and $\hat{\theta}_{t_{k}}\in\bar{N}_{\varsigma,k}$. As a result,

[TABLE]

which immediately yields that $\theta\in\Theta_{k}=\overline{B(\hat{\theta}_{t_{k}},c_{k})}$ on the fixed trajectory.

Therefore, we have verified that $t_{k}<\infty$ and $\theta\in\Theta_{k}$ for all $k\geq 0$ almost surely and hence (35) holds for all $k\geq 1$ accordingly. Since Step 2 in the algorithm implies that for each $k\geq 1$,

[TABLE]

the lemma is thus proved by letting $k\rightarrow\infty$. $\Box$

2.4.2 Proofs of the Theorems

Some notations are needed in the sequel. For each $t\geq 0$, denote $f_{t}\triangleq f(\theta,u_{t},\chi_{t})$ and $\Omega_{t}\triangleq\{\|\chi_{t}\|\leq C\}$. Let

[TABLE]

Write $E^{ik}_{j}=\{e^{ik}_{s,j}\}_{1\leq s\leq|E^{ik}_{j}|},j\in[1,m]$ in Lemma 6. Clearly, $\prod_{j=1}^{m}|E^{ik}_{j}|=\hat{n}_{k}$. In addition, by Assumption A2,

[TABLE]

Lemma 8**.**

*Let $t_{k-1}<\infty,k\geq 1$ and $i\in[n_{k-1},n_{k})$. If $\underline{d}^{m}(\mathcal{P}_{\alpha}|\mathcal{S})>1/C_{w}$ for all $\alpha\in A_{0}$ and Assumption A2 holds, then for each $t\geq m$, there are some random integers $\{s_{t_{j}}\in\mathcal{F}_{t-j}\}_{j\in[1,m]}$ taking values in $\mathcal{N}_{k,j}=\{1,\ldots,|E^{ik}_{j}|\}$ on $\Omega_{t-m}$ such that *

[TABLE]

where $B^{ik}_{j,t}\triangleq B(e^{ik}_{s_{t_{j}},j},C_{w}-\sigma_{k}),j\in[1,m]$ and $\sigma_{k}$ is defined in Lemma 6.

Proof.

Since $t_{k-1}<\infty,k\geq 1$, by the algorithm and Lemma 6, all the quantities appearing in the lemma are well defined. Fix $i\in[n_{k-1},n_{k})$. Note that by (39), $(f_{t-1},\ldots,f_{t-m})\in\mathcal{S}$ on $\Omega_{t-m}$, then for $j=1,\ldots,m$, define

[TABLE]

Random sequence $\{s_{t_{j}}\}_{j\in[1,m]}$ is well defined on $\Omega_{t-m}$ since $\underline{d}^{m}(\Delta^{k}_{i}|\mathcal{S})>1/(C_{w}-\sigma_{k})$ by Lemma 6. So, $s_{t_{j}}\in\mathcal{F}_{t-j}$ and (40) follows immediately. $\Box$

Lemma 9**.**

Let $(\chi_{0},\{u_{t}\})\in\mathcal{E}$ be a $C$-recurrent experiment and $\underline{d}^{m}(\mathcal{P}_{\alpha}|\mathcal{S})>1/C_{w}$ for all $\alpha\in A_{0}$. Then, under Assumptions A1–A2, for each $k\geq 1$, (32) holds for all $i\in[n_{k-1},n_{k})$ a.s. whenever $t_{k-1}<\infty$ a.s..

Proof.

Fix a $k\geq 1$ that $t_{k-1}<\infty$ a.s. and take an integer $i\in[n_{k-1},n_{k})$. Let

[TABLE]

and $D^{k}_{is,l}(2)\triangleq B((w^{*}_{is,l},v^{*}_{is,l}),d_{k})\times\mathcal{U}^{is}_{l}$. Recall that $\{\mathcal{U}^{is}_{l}\}$ are mutually disjoint, then $\{D^{k}_{is}(1)\times D^{k}_{is,l}(2)\}_{s\in[1,\hat{n}_{k}],l\in[1,\hat{n}_{is}]}$ are mutually disjoint as well. As a result, for $t\geq m$,

[TABLE]

By Assumption A1, for each $s\in[1,\hat{n}_{k}]$ and $l\in[1,\hat{n}_{is}]$, there is a $\rho_{k,1}>0$ such that

[TABLE]

and hence, by the independence of $\chi_{t}$ and $\psi_{t}$, (2) indicates that for some $\rho_{k,2}>0$,

[TABLE]

where $\mathcal{F}^{\chi}_{t}\triangleq\sigma\{\mathcal{F}_{t-m}\cup\sigma\{\chi_{t}\}\},t\geq m$. So, $\mathcal{F}_{t-m}\subset\mathcal{F}^{\chi}_{t}$.

Now, at time $t\geq m$, take $\{s_{t_{j}}\}_{j\in[1,m]}$ in Lemma 8, which corresponds to some random index $s_{t}$ and point $\eta^{k}_{is_{t}}=(e^{ik}_{s_{t_{1}},1},\ldots,e^{ik}_{s_{t_{m}},m})^{T}$ taking values in $\{1,\ldots,\hat{n}_{k}\}$ and $\Delta^{k}_{i}$ on set $\Omega_{t-m}$, respectively. Let $\bar{d}_{k}=d_{k}/\sqrt{m}<\sigma_{k}$ and

[TABLE]

According to Lemma 8, $\Omega^{ik}_{t,j}$ is $\mathcal{F}_{t-j}$ measurable, $j\in[1,m]$. So, by Assumption A1 and Lemma 8, for any $t\geq m$, there is a $\rho_{k,3}>0$ such that

[TABLE]

where the third inequality follows from (40). So, in view of (47), for each $t\geq m$,

[TABLE]

Now, for $t\geq 1$ and $l\in[0,m]$, denote $\zeta_{t,l}\triangleq\chi_{(m+1)(t-1)+l}$ and

[TABLE]

Clearly, $\{(\zeta_{t,l},\zeta^{\prime}_{t,l})\}_{t\geq 1}$ is adapted to the filtration $\{\mathcal{F}^{\prime}_{t,l}\}_{t\geq 1}$ with $\mathcal{F}^{\prime}_{t,l}\triangleq\mathcal{F}_{(m+1)t+l}$. Since the experiment is $C$-recurrent, $\sum_{t=m}^{\infty}I_{\Omega_{t-m}}=\infty$ almost surely. So, by (51),

[TABLE]

which means there at least exists some $l\in[0,m]$ such that $P_{l}=\infty$ a.s.. According to the Borel-Cantelli-L$\acute{e}$vy theorem,

[TABLE]

Since $t_{k-1}<\infty$ almost surely, it is obvious that for every $i\in[n_{k-1},n_{k})$,

[TABLE]

The result follows immediately. $\Box$

Proof of Theorem 4: It is a direct result of Lemmas 7 and 9.

Proof of Theorem 5: Given $\alpha\in A_{0}$, since $\underline{d}^{m}(\mathcal{P}_{\alpha}(\epsilon)|\mathbb{R}^{pm})>1/C_{w}$ for some $\epsilon>0$, a countable set $\Delta_{\alpha}=\prod_{j=1}^{m}E^{\alpha}_{j}\subset\mathcal{P}_{\alpha}(\epsilon)$ exists ($|\Delta_{\alpha}|=\aleph_{0}$) and $\underline{d}(E^{\alpha}_{j}|\mathbb{R}^{p})>1/(C_{w}-\sigma_{\alpha}),\sigma_{\alpha}\in(0,C_{w})$. If $\eta\in\Delta_{\alpha}$, by (19), for any $u\in\overline{B(0,C_{u})}$, there are some $\psi\in\mathcal{V}^{m}$ and $\pi(u)\in\mathcal{W}\times\mathcal{V}$ such that

[TABLE]

As a result, by (9) and (12),

[TABLE]

Moreover, $\mbox{Im}(\hat{h}_{\check{x}^{\prime}_{\epsilon},\check{u}_{\epsilon},\check{\beta}})\subset\bigcup_{\psi\in\mathcal{V}^{m},\pi\in\mathcal{W}\times\mathcal{V}}\hat{h}(\check{x}^{\prime}_{\epsilon},\check{u}_{\epsilon},\check{\beta},\check{\psi}_{\epsilon},\check{\pi}_{\epsilon})$, then it yields

[TABLE]

So, a similar proof of Lemma 6 shows that Lemma 6 holds with $\mathcal{P}_{\alpha}$ replaced by $\mathcal{P}_{\alpha}(\epsilon)$, $\hat{n}_{k}=\aleph_{0}$ and $\mathcal{S}=\mathbb{R}^{mp}$ ($C_{0}=\infty$). Now, since any $(\chi_{0},\{u_{t}\})$ can be viewed as a $C$-recurrent experiment with $C=\infty$ and Lemmas 8–9 are still true for $C=\infty$, the result follows from Lemma 7.

3 Implementable Algorithm

The estimator in Section 2.4.1 is only theoretical valid, so we are going to develop an implementable nonlinear estimator here. For simplicity, study the following basic control system

[TABLE]

in an experiment $(\chi_{0},\{u_{t}\})\in\mathcal{E}$, where $\mathcal{E}$ is defined by (3), $\theta\in\Theta\subset\mathbb{R}^{n}$, $u_{t},y_{t},w_{t}$ are scalars and $\varphi_{t}=(y_{t},\ldots,y_{t-m+1})^{T}$. Moreover, $f(x,z):\mathbb{R}^{n}\times\mathbb{R}^{m}\to\mathbb{R}$ is known and $\frac{\partial f(x,z)}{\partial x}$ exists. Both the above two functions are continuous. Assume

B1

$\{w_{t}\}$ is an i.i.d sequence with $Ew_{1}=0$ and $E|w_{1}|^{\kappa}<\infty,\kappa>4$. In addition,

(i) if $C_{w}<\infty$, $w_{1}$ satisfies (14);

(ii) if $C_{w}=\infty$, then for every $C^{\prime}>0$,

[TABLE]

Remark 3.1**.**

Assumption B1 includes a large class of familiar distributions, such as uniform distribution $U(-C_{w},C_{w})$ for finite $C_{w}$, as well as Gaussian distributions and t-distributions for $C_{w}=\infty$.

3.1 Grid Searching Estimator

Assumption B1 implies that $Ew^{2}_{1}$ exists. Denote $\sigma^{2}_{w}\triangleq Ew^{2}_{1}$ and $\bar{\sigma}^{2}_{w}\triangleq E(w^{2}_{1}-\sigma^{2}_{w})^{2}$. Recall that $\Omega_{i}=\{\|\varphi_{i}\|\leq C\}$ for some given $C>0$ ($C$ can be taken $\infty$). Let $\gamma>0$ and define

[TABLE]

Let $\eta_{t}(\gamma)\triangleq\sum_{i=1}^{t}\Omega_{i}(\gamma,C)$. At time $t\geq 2$, the grid searching estimator is designed according to function $\hat{G}_{t}:\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}$ defined below:

[TABLE]

Moreover, we remark that the knowledge of $\sigma^{2}_{\omega}$ can be described by one of the following three scenarios:

(i) $\sigma^{2}_{\omega}$ is known. Let $\Sigma^{0}_{t}\equiv\{\sigma^{2}_{w}\}$, $t\geq 1$.

(ii) $\sigma^{2}_{\omega}$ is unknown without any prior information. Let $\Sigma^{0}_{t}=[0,t]$, $t\geq 1$.

(iii) $\sigma^{2}_{\omega}$ is unknown but bounded by a known constant $\sigma>0$, i.e., $\sigma^{2}_{w}\leq\sigma$. Let $\Sigma^{0}_{t}\equiv[0,\sigma]$, $t\geq 1$.

Let $\lambda,\gamma,C>0$ be some adjustable parameters and let

[TABLE]

Algorithm

Step 1: At time $t=0$, denote $o_{0}$ and $\sigma^{2}_{0}$ as the center points of sets $\Theta$ and $\Sigma^{0}_{0}$, respectively. Set

[TABLE]

Step 2: At time $t\geq 1$, equally divide $\Theta$ and $\Sigma^{0}_{t}$ into two finite sequences of small boxes $\{\Theta_{ti}\}$ and $\{\Sigma_{tj}\}$ that $\Theta=\bigcup_{i}\Theta_{ti}$ and $\Sigma^{0}_{t}=\bigcup_{j}\Sigma_{tj}$, where the side lengthes of $\Theta_{ti}$ and $\Sigma_{tj}$ are less than $1/(t^{\frac{1}{4}-\frac{1}{2\kappa}-\lambda})$ and $1/(t^{\frac{1}{2}-\frac{1}{\kappa}-\lambda})$, respectively. Let $o_{ti}$ and $\sigma^{2}_{tj}$ be the center points of $\Theta_{ti}$ and $\Sigma_{tj}$. If

[TABLE]

set

[TABLE]

Otherwise, for $\mathcal{J}_{t}\neq\emptyset$, take an arbitrary $(i^{*},j^{*})\in\mathcal{J}_{t}$ satisfying

[TABLE]

Set

[TABLE]

3.2 Strong Consistency

For $1\leq k\leq n$, let $x^{(k)},\bar{x}^{(k)}\in\mathbb{R}^{2^{k-1}n}$, $y^{(k)},\bar{y}^{(k)}\in\mathbb{R}^{2^{k-1}m}$ and $z^{(k)}=\mbox{col}\{x^{(k)},y^{(k)}\}$. Write $x^{(1)}=(x^{(1)}_{1},\ldots,x^{(1)}_{n})$. Now, recursively define a sequence of functions $\{g^{(k)}_{j},1\leq k\leq j\leq n\}$ for system (52) as follows:

[TABLE]

Example 3.1**.**

In system (52) with $n=1$, $g^{1}_{1}(x,y)=\frac{\partial f(x,y)}{\partial x}$. For $n=2$,

[TABLE]

The convergence of estimates $\hat{\theta}_{t}$ is related to the density of set

[TABLE]

in $\mathbb{R}^{2^{n-1}m}$ for $C_{w}=\infty$ or in $\mathcal{S}=\overline{B(0,C_{0})}\subset\mathbb{R}^{2^{n-1}m}$ for $C_{w}<\infty$, where $C_{0}$ is defined similarly as that in (15). This claim is verified for the case where the closed-loop system is stable, i.e.,

[TABLE]

Theorem 10**.**

Under Assumption B1, let the closed-loop system (52) be stable. If for each $x\in\Theta$, either $\underline{d}^{m}(\mathcal{P}^{\prime}|\mathcal{S}^{2^{n-1}m})>1/C_{w}$ for $C_{w}<\infty$ or $\mathcal{P}^{\prime}\neq\emptyset$ for $C_{w}=\infty$, then by choosing parameter $\gamma$ sufficiently large and parameter $\lambda\in(0,\frac{1}{4}-\frac{1}{2\kappa})$, the grid searching estimator satisfies

[TABLE]

Example 3.2**.**

Let us consider system (52) with $f(x_{1},x_{2},y)=x_{1}y^{b_{1}}+x_{2}y^{b_{2}},$ where $x_{1},x_{2},y\in\mathbb{R}$ and $b_{1}\neq b_{2}$. By Example 3.1, $g^{2}_{2}(x_{1},x_{2},\bar{x}_{1},\bar{x}_{2};y,\bar{y})=y^{b_{1}}\bar{y}^{b_{2}}-\bar{y}^{b_{1}}y^{b_{2}},$ which causes $\mathcal{P}^{\prime}$ dense in $\mathbb{R}^{2}$.

Example 3.3**.**

If $C_{w}=\infty$, the only requirement on $\mathcal{P}^{\prime}$ for parameter identifiability is $\mathcal{P}^{\prime}\neq\emptyset$. This applies to a lot of control systems. For instance, in system (52), let $f(x,y)=\mbox{sin}\,(xy)$ for $x,y\in\mathbb{R}$ and $\Theta=[0,2\pi]$. Example 3.1 shows $g^{n}_{n}(x,y)=\cos(xy)$. If $y=1/8$, then $\cos(xy)\in[\sqrt{2}/2,1]$ for all $x\in[0,2\pi]$. Thus, $1/8\in\mathcal{P}^{\prime}$.

3.3 Proof of Theorem 10

We first introduce some notations. For two vectors $p=(p_{i})_{i=1}^{l},q=(q_{i})_{i=1}^{l},l\geq 1$, we say $p\prec q$ if there is an index $j\in[1,l)$ such that $p_{i}=q_{i},1\leq i\leq j$ and $p_{j+1}<q_{j+1}$. Define a series of sets $\{\mathcal{H}^{t}_{k}\}$ by

[TABLE]

Lemma 11**.**

Let $\alpha_{i}\triangleq(a_{i,1},\ldots,a_{i,n})^{T}$, $i\in[1,t]$ for some fixed $t\geq 1$ and $n\geq 1$. Denote $M(k)$ as the $k$th order leading principal minor of $\det(\sum_{i=1}^{t}\alpha_{i}\alpha^{T}_{i})$ for $k\in[1,n]$ and $M^{\prime}(k,k)$ as the $k,k$ cofactor of $M(k+1)$ for $k\in[1,n-1]$. If $\sum_{i=1}^{t}a^{2}_{i,j}\neq 0$ for all $j\in[1,n]$, then there is a sequence $\{\mu_{h}(k),\nu_{h}(k),h\in\mathcal{H}^{t}_{k},k\in[1,n]\}$ such that each $M(k)$ and $M^{\prime}(k,k)$ can be written as †††$\prod_{j=1}^{0}(\sum_{h\in\mathcal{H}^{t}_{j}}\mu^{2}_{h}(j))^{-1}\triangleq 1.$

[TABLE]

where, for each $h=(p,q)\in\mathcal{H}^{t}_{k},k\in[2,n]$,

[TABLE]

and there is a function $\zeta_{h,k}(\cdot):\mathbb{R}^{tk}\rightarrow\mathbb{R}$ independent of $\alpha_{i},i\in[1,t]$ such that

[TABLE]

Proof.

Let $n=2$. Clearly, $M(1)=\sum_{i=1}^{t}a^{2}_{i,1},M^{\prime}(1,1)=\sum_{i=1}^{t}a^{2}_{i,2}$ and

[TABLE]

Similarly, $M^{\prime}(2,2)=\sum_{(p,q)\in\mathcal{H}^{t}_{2}}\left(a_{p,1}a_{q,3}-a_{q,1}a_{p,3}\right)^{2}.$ Hence, the lemma is true when $n=2$ with $\mu_{h}(1)=a_{h,1},\nu_{h}(1)=a_{h,2},h\in\mathcal{H}^{t}_{1}$ and

[TABLE]

Now, let $n\geq 3$. Suppose for some integer $l\in[2,n-1]$, there is a sequence $\{\mu_{h}(k),\nu_{h}(k),h\in\mathcal{H}^{t}_{k},k\in[1,l]\}$ satisfying (65)–(67), then we will show the existence of $\{\mu_{h}(k),\nu_{h}(k),h\in\mathcal{H}^{t}_{k},k\in[1,l+1]\}$ such that (65)–(67) hold.

For $k=l+1$, write $M(k)$ as a block matrix by

[TABLE]

where

[TABLE]

Since $\sum_{i=1}^{t}a^{2}_{i,1}\neq 0$, then

[TABLE]

Note that the $(j,s)$ entry of $M_{2}(k)\left(\sum_{i=1}^{t}a^{2}_{i,1}\right)-M_{1}(k)M^{T}_{1}(k)$ is

[TABLE]

Let $\alpha^{\prime}_{p,q}(k)\triangleq(a_{p,1}a_{q,2}-a_{q,1}a_{p,2},\ldots,a_{p,1}a_{q,k}-a_{q,1}a_{p,k})^{T}$, then

[TABLE]

Observe that matrix $\sum_{(p,q)\in\mathcal{H}^{t}_{2}}\alpha^{\prime}_{p,q}(l+1)(\alpha^{\prime}_{p,q}(l+1))^{T}$ has the same form of $M(l)$, which is of dimension $l$. Moreover, $a_{i,j},j\in[1,l]$ can be taken any values in $M(l)$, so by the assumption and (68),

[TABLE]

holds for some $\{\mu^{\prime}_{h}(k),\nu^{\prime}_{h}(k),h\in\mathcal{H}^{t}_{k+1},k\in[1,l]\}$ satisfying

[TABLE]

In addition, there is a sequence of $\{\zeta^{\prime}_{h,k}(\cdot),h\in\mathcal{H}^{t}_{k+1},k\in[1,l]\}$ such that

[TABLE]

Considering (76), if $l=2$, then for all $k\in[1,l-1]$,

[TABLE]

For $l>2$, suppose there is an $s\in[1,l-2]$ such that (80) holds for all $k\in[1,s]$. Since $s+2\leq l$, then by (66) and (76), for any $h=(p,q)\in\mathcal{H}^{t}_{s+2}$,

[TABLE]

This, together with (67) and (79), infers

[TABLE]

Note that $\zeta_{h,s+2}$ and $\zeta^{\prime}_{h,s+1}$ are independent of the values of $\alpha_{i},i\in[1,t]$, then

[TABLE]

Or equivalently, $\nu^{\prime}_{h}(s+1)=\nu_{h}(s+2)$. Therefore, $\mu^{\prime}_{h}(k)=\mu_{h}(k+1)$ and $\nu^{\prime}_{h}(k)=\nu_{h}(k+1)$ for all $k\in[1,l-1]$.

Define $\mu_{h}(l+1)\triangleq\mu^{\prime}_{h}(l)$ for all $h\in\mathcal{H}^{t}_{l+1}$, then

[TABLE]

Since $\sum_{i=1}^{t}a^{2}_{i,1}=\sum_{h\in\mathcal{H}^{t}_{1}}\mu^{2}_{h}(1),$ combining (72) and (73) leads to the first formula of (65) immediately for $k=l+1$. If $l<n-1$, also let $\nu_{h}(l+1)\triangleq\nu^{\prime}_{h}(l),h\in\mathcal{H}^{t}_{l+1}$. Note that

[TABLE]

Finally, for each $h\in\mathcal{H}^{t}_{l+1}$, there is a $\zeta_{h,l+1}$ independent of $\alpha_{i},i\in[1,t]$ such that

[TABLE]

So, with $\{\mu_{h}(l+1),\nu_{h}(l+1),h\in\mathcal{H}^{t}_{l+1}\}$ defined above, (65)–(67) hold for $k=l+1$, which completes the proof by induction. $\Box$

Lemma 12**.**

Let the conditions of Lemma 11 hold and denote $\lambda_{\min}(\sum_{i=1}^{t}\alpha_{i}\alpha^{T}_{i})$ as the minimal eigenvalue of matrix $\sum_{i=1}^{t}\alpha_{i}\alpha^{T}_{i}$. If there is a number $\epsilon>0$ such that for each $k\in[1,n-1]$ and $s\in[k+1,n]$,

[TABLE]

where $\nu_{h,s}(k)\triangleq\zeta_{h,k}(a_{i,j},i=1,\ldots,t;j=1,\ldots,k-1,s),h\in\mathcal{H}^{t}_{k},s\in[k,n]$‡‡‡$\nu_{h,k}(k)=\mu_{h}(k)$, then

[TABLE]

Proof.

Let $\pi(n-1)$ be the set of the $(n-1)$-permutations of $\{1,2,\ldots,n\}$. For $p=(i_{1},\ldots,i_{n-1})\in\pi(n-1)$, define $\alpha_{i,p}\triangleq(a_{i,i_{1}},\ldots,a_{i,i_{n-1}})^{T}$ and denote the $n$ eigenvalues of $\sum_{i=1}^{t}\alpha_{i}\alpha^{T}_{i}$ by $\lambda_{i},1\leq i\leq n$ with $\lambda_{i}\geq\lambda_{i+1},1\leq i\leq n-1$. According to the Vieta’s formulas, one has

[TABLE]

Note that reordering the $n$ elements $a_{i,1},\ldots,a_{i,n}$ of vector $\alpha_{i},i\in[1,t]$ does not change the minimal eigenvalue of $\sum_{i=1}^{t}\alpha_{i}\alpha^{T}_{i}$. So, without loss of generality, for $p_{1}=(1,2,\ldots,n-1)$, assume

[TABLE]

Therefore,

[TABLE]

Consequently, by Lemma 11 and (81),

[TABLE]

where $\nu_{q}(n-1)=\nu_{q,n}(n-1)$ for $q\in\mathcal{H}^{t}_{n-1}$.

Now, Lemma 11 implies that for any $k\in[1,n-2]$ and $h=(p,q)\in\mathcal{H}^{t}_{k+1}$,

[TABLE]

As a result, (81) yields

[TABLE]

which, by (85), leads to $\lambda_{n}\geq\frac{\epsilon^{n-1}}{n}\left(\sum_{i=1}^{t}a^{2}_{i,n}\right).$ The lemma thus follows. $\Box$

Lemma 13**.**

*Assume either $\underline{d}^{m}(\mathcal{P}^{\prime}|\mathcal{S}^{2^{n-1}m})>1/C_{w}$ for $C_{w}<\infty$ or $\mathcal{P}^{\prime}\neq\emptyset$ for $C_{w}=\infty$. Then, the following two statements hold:

(i) there is a sequence of sets $\mathcal{B}_{jl}\triangleq\{b_{jl,s}\}_{s\in[1,N_{jl}]},j\in[1,2^{n-1}],l\in[1,m]$ with integers $N_{jl}\geq 1$ such that $\prod_{j=1}^{2^{n-1}}\prod_{l=1}^{m}\mathcal{B}_{jl}\subset\mathcal{P}^{\prime}$, and if $C_{w}<\infty$,*

[TABLE]

(ii) there is a number $d>0$ such that

[TABLE]

where $\mathcal{D}\triangleq\prod\nolimits_{j=1}^{2^{n-1}}D_{j}$ and

[TABLE]

Proof.

Since either $\underline{d}^{m}(\mathcal{P}^{\prime}|\mathcal{S}^{2^{n-1}m})>1/C_{w}$ for $C_{w}<\infty$ or $\mathcal{P}^{\prime}\neq\emptyset$ for $C_{w}=\infty$, statement (i) is straightforward ($N_{jl}\equiv 1$ for $C_{w}=\infty$). To show statement (ii), note that $g^{n}_{n}(x,y)$ is continuous, then for each $x\in\Theta^{2^{n-1}}$, there is a number $d_{x}>0$ and a neighbourhood $B_{x}$ of $x$ such that for $D_{j}(x)\triangleq\prod\nolimits_{l=1}^{m}(\bigcup\nolimits_{s\in[1,N_{jl}]}[b_{jl,s}-d_{x},b_{jl,s}+d_{x}]),$

[TABLE]

Now, $\Theta^{2^{n-1}}$ is compact, by the finite covering theorem, there is a sequence $\{x(i)\in\Theta^{2^{n-1}}\}_{i\in[1,N]}$ for some $N\in\mathbb{N}^{+}$ such that $\Theta^{2^{n-1}}\subset\bigcup_{i\in[1,N]}B_{x(i)}$. So, (88) holds by letting $d=\min_{1\leq i\leq N}d_{x(i)}$. $\Box$

To state the next lemma, denote $\eta_{t}\triangleq\sum_{i=1}^{t}I_{i-m}$, $\Omega_{\eta}\triangleq\left\{\omega:\lim_{t\rightarrow\infty}\eta_{t}=\infty\right\}$ and let $D_{j},j\in[1,2^{n-1}]$ be defined by (89).

Lemma 14**.**

Under the conditions of Theorem 10, for all sufficiently large $t$,

[TABLE]

where $C_{D}>0$ is a number independent of $t$.

Proof.

Let filtration $\{\mathcal{F}_{h}\}$ be defined by (38). Fix $j\in[1,2^{n-1}]$. Observe that for each $l\in[0,m-1]$, $\{I_{\{\varphi_{hm+l}\in D_{j}\}}-P(\varphi_{hm+l}\in D_{j}|\mathcal{F}_{(h-1)m+l}),\mathcal{F}_{hm+l}\}_{h\geq 0}$ is a martingale difference sequence, then for all sufficiently large $t$,

[TABLE]

For $h\geq m$, we compute $P(\varphi_{h}\in D_{j}|\mathcal{F}_{h-m})I_{\Omega_{h-m}}$ by the following two cases:

(i) $C_{w}<\infty$. In this case, $f_{h-l}=(f(\theta,\varphi_{h-l})+u_{h-l})$ falls in $\mathcal{S}$ for all $l\in[1,m]$ on set $\Omega_{h-m}$. So, (87) yields

[TABLE]

For $h\geq m$ and $l\in[1,m]$, denote

[TABLE]

So, by Assumption B1 and (92), there is a $C_{d}>0$ such that

[TABLE]

holds for all $h\geq m$ and $l\in[1,m]$. By virtue of (93),

[TABLE]

(ii) $C_{w}=\infty$. Note that $\{D_{j}\}_{j\in[1,2^{n-1}]}$ are bounded, then there is a $C_{f}>0$ such that

[TABLE]

Since $N_{jl}\equiv 1$, by Assumption B1, for any $h\geq m$ and $l\in[1,m]$,

[TABLE]

where $C_{d}$ is a positive number. So, (94) also holds for this case.

Combined with (91), both the two cases indicate that for all sufficiently large $t$,

[TABLE]

Then, (90) follows from (95) by noting that $j$ is finite. $\Box$

Now, at time $t\geq 1$, for any $k\in[1,n]$ and $h=(h_{1},h_{2},\ldots,h_{2^{k-1}})\in\mathcal{H}^{t}_{k}$, denote

[TABLE]

Take $\gamma$ sufficiently large that for each $h\geq 1$,

[TABLE]

For $t\geq 1$, let $\{\theta_{t,h}\}_{h\in[1,t]}$ be a sequence of random variables taking values in $\Theta$ and define $\vartheta_{t,h},h\in\mathcal{H}^{t}_{k},k\in[1,n]$ by $\vartheta_{t,h}\triangleq\mbox{col}\{\theta_{t,h_{1}},\theta_{t,h_{2}},\ldots,\theta_{t,h_{2^{k-1}}}\}$.

Lemma 15**.**

Under the conditions of Theorem 10, there are some $C_{g},C_{g,\eta}>0$ such that for all $k\in[1,n],s\in[k,n]$ and all sufficiently large $t$,

[TABLE]

Proof.

First, in view of (90) and (98), for all sufficiently large $t$,

[TABLE]

Moreover, considering Lemma 13, let $C^{g}_{n}\triangleq\min_{x\in\Theta^{2^{n-1}}}\min_{y\in\mathcal{D}}|g^{n}_{n}(x,y)|>0,$ then

[TABLE]

whenever $t$ is sufficiently large.

Now, recursively define $C^{g}_{k-1}\triangleq C^{g}_{k}/(2\bar{C}_{g}),k=n,\ldots,2$, where for $\overline{B(0,\gamma)}\subset\mathbb{R}^{m}$,

[TABLE]

Because of (100), suppose there is an integer $k\in[2,n]$ such that for all $s\in[k,n]$ and all sufficiently large $t$,

[TABLE]

Let $s\in[k,n]$ and $h=(p,q)$ with $p=(p_{j})_{j=1}^{2^{k-2}},q=(q_{j})_{j=1}^{2^{k-2}}\in\mathcal{H}^{t}_{k-1}$. By (61), on set $(\bigcap\nolimits_{j=1}^{2^{k-2}}\Omega_{p_{j}}(\gamma,C))\cap(\bigcap\nolimits_{j=1}^{2^{k-2}}\Omega_{q_{j}}(\gamma,C))$, it is evident that for $r=k-1$ and $s$,

[TABLE]

As a result, both $r=k-1$ and $s$ lead to

[TABLE]

or equivalently, by (101) and $\sum\nolimits_{q\in\mathcal{H}^{t}_{k-1}}I_{\Omega_{q_{j}}(\gamma,C)}\leq\eta^{2^{k-2}}_{t}$,

[TABLE]

This implies that (101) is also true for $k-1$. The lemma is thus proved by taking $C_{g}=\min_{1\leq i\leq n}C^{g}_{i}$ and $C_{g,\eta}=\frac{C_{D}^{2^{n-1}}}{2^{n}}$. $\Box$

Lemma 16**.**

For $t\geq 1$, let $\{\theta_{t,h}\}_{h\in[1,t]}$ be a sequence of random variables taking values in $\Theta$. In Lemmas 11 and 12, set

[TABLE]

where $\gamma$ is a positive number. Then, under the conditions of Theorem 10, there is a $C_{P}>0$ independent of $t$ such that for all be sufficiently large $t$,

[TABLE]

*In addition, taking $\gamma$ appropriately large, there is a number $\epsilon>0$ such that for all be sufficiently large $t$, (81) holds a.s. on $\Omega_{\eta}$ for each $k\in[1,n-1]$ and $s\in[k+1,n]$. *

Proof.

First, by (61), (66), (86) and (102), it is easy to verify that for each $k\in[1,n-1]$, $h=(p,q)$, $p,q\in\mathcal{H}^{t}_{k}$ with $p=(p_{j})_{j=1}^{2^{k-1}},q=(q_{j})_{j=1}^{2^{k-1}}$ and $s\in[k+1,n]$,

[TABLE]

As a consequence, by Lemma 15 and (107), for each $s\in[1,n]$,

[TABLE]

Hence, (103) holds by letting $C_{P}=C^{2}_{g}C_{g,\eta}$. Furthermore, for each $k\in[1,n-1]$,

[TABLE]

On the other hand, for every $s\in[k+1,n]$,

[TABLE]

where

[TABLE]

This with (108) completes the proof by letting $\epsilon=(C^{2}_{g}C_{g,\eta})/(2C^{\prime}_{g})$ in (81). $\Box$

Let $t\geq 1$. For each $i\in[1,t]$, denote $\phi_{i}(x)\triangleq\frac{\partial f(x,\varphi_{i})}{\partial x}$ and define

[TABLE]

where $x=\mbox{col}\{x_{1},\ldots,x_{t}\},x_{i}\in\Theta$. The next lemma is straightforward.

Lemma 17**.**

*Under the conditions of Theorem 10, let $\theta_{t}\triangleq\mbox{col}\{\theta_{t,1},\ldots,\theta_{t,t}\}$, where $\{\theta_{t,h},h\in[1,t]\}$ is a sequence of random variables taking values in $\Theta$. Then,

(i) for all sufficiently large $t$ and $\gamma$, there is a random positive number $C_{1}$ such that*

[TABLE]

(ii) let $C_{2}=\max_{x\in\Theta,\|z\|\leq\gamma}\|\frac{\partial f(x,z)}{\partial x}\|^{2}$ with $\gamma>0$ sufficiently large, then

[TABLE]

Proof.

Note that $P^{-1}_{t+1}(\theta_{t})\geq\sum_{i=1}^{t}\alpha_{i}\alpha^{T}_{i}$, where $\alpha_{i}$ is defined by (102). In view of Lemma 16, there is a number $\epsilon>0$ such that (81) holds almost surely on $\Omega_{\eta}$ for each $k\in[1,n-1]$ and $s\in[k+1,n]$, and hence Lemma 12 yields

[TABLE]

where (110) follows directly from (103) in Lemma 16.

Next, we show (111). By (53), if $C_{w}=\infty$, it is clear that

[TABLE]

When $C_{w}<\infty$, without loss of generality, assume $\sup_{i\geq 1}\|\varphi_{i}\|I_{\Omega_{i-m}}<\gamma$. Hence (111) follows as well. $\Box$

Lemma 18**.**

Under Assumption B1, for any $\varepsilon\in(0,1)$,

[TABLE]

Proof.

By Assumption B1, $m^{\prime}\triangleq E|w_{1}|^{\tau}$ exists for any $\tau\in(2,\sqrt{\kappa}]$. Observe that $\tau^{2}\in(1,\kappa]$, employing the Minkowski inequality and the Lyapunov inequality yields

[TABLE]

Since $\{|w_{i}|^{\tau}-m^{\prime},\mathcal{F}_{i}\}$ is a martingale difference sequence with

[TABLE]

[9, Lemma 2(iii)] shows that

[TABLE]

and hence, as $t\rightarrow\infty$,

[TABLE]

Note that

[TABLE]

which, together with (113) and $\tau>2$, implies that as $i\rightarrow\infty$,

[TABLE]

Applying [15, Corollary 5.4.2] to the martingale difference sequence $\{I_{\Omega_{i-1}(\gamma,C)}(w^{2}_{i}-\sigma^{2}_{w}),\mathcal{F}_{i}\}$ yields

[TABLE]

which leads to (112) immediately. $\Box$

Lemma 19**.**

Under Assumption B1, if (110) and (111) hold for every $\theta_{t}$ defined in Lemma 17, then for any $\lambda\in(0,\frac{1}{4}-\frac{1}{2\kappa})$,

[TABLE]

where $\Omega^{\prime}_{\eta}\triangleq\{t/\eta_{t}=O(1)\}$.

Proof.

Fix $\lambda\in(0,\frac{1}{4}-\frac{1}{2\kappa})$ in the algorithm and let $\theta$ be the true value of the parameter. It is clear that for every sufficiently large $t$, $(\theta,\sigma^{2}_{w})\in\Theta\times\Sigma_{t}.$ Therefore, there are two points $o_{ti_{t}}\in\Theta$ and $\sigma^{2}_{tj_{t}}\in\Sigma_{t}$ such that

[TABLE]

Since $\Theta$ is convex, (54) shows

[TABLE]

where $\theta_{ts},s\in[1,t]$ are $t$ random variables taking values in $\Theta$ and $\theta_{t}=\mbox{col}\{\theta_{t1},\ldots,\theta_{tt}\}.$

We now estimate the three terms in (115). First, by (111), (114) and $\eta_{t}\leq t$,

[TABLE]

Next, for each $t\geq 1$, denote the number of points $o_{ti}$ in the algorithm by $N_{o}(t)$. Evidently, $N_{o}(t)=O(t^{n})$. We estimate the following sequences

[TABLE]

To this end, note that by Assumption B1 and the Borel-Cantelli theorem,

[TABLE]

Moreover, $2(f(o_{ti},\varphi_{s})-f(\theta,\varphi_{s}))I_{\Omega_{s}(\gamma,C)}$ is $\mathcal{F}_{s}$-measurable for all $o_{ti},i\in[1,N_{o}(t)],t\geq 1$. Then, in view of [8, Lemma 3.2], it yields

[TABLE]

and hence

[TABLE]

At last, for all sufficiently large $t$, Lemma 18 implies

[TABLE]

which, together with (114), yields

[TABLE]

So, combining (116), (118) and (120), for all sufficiently large $t$,

[TABLE]

Note that $C_{\phi}=C_{2}n/4+1$, then $\mathcal{J}_{t}\neq\emptyset$. By (57) and (114), for every sufficiently large $t$, there is a point $(o_{ti^{*}},\sigma^{2}_{tj^{*}})\in\mathcal{J}_{t}$ satisfying

[TABLE]

We claim that $\|o_{ti^{*}}-\theta\|^{2}=O\left(\frac{1}{t^{\frac{1}{2}-\frac{1}{\kappa}-2\lambda}}\right)$ on $\Omega^{\prime}_{\eta}$ almost surely. Otherwise, there is a set $\Omega^{\prime\prime}_{\eta}\subset\Omega^{\prime}_{\eta}$ with $P(\Omega^{\prime\prime}_{\eta})>0$ such that

[TABLE]

By (110), (117), (119), (121) and the fact $\eta_{t-1}(\gamma)\leq t$, a random $\theta_{t,ti^{*}}$ exists that for all sufficiently large $t$,

[TABLE]

which contradicts to the fact that $(i^{*},j^{*})\in\mathcal{J}_{t}$. Consequently, by (58),

[TABLE]

as desired in (63). $\Box$

Proof of Theorem 10: When the closed-loop system is stable, $P(\Omega^{\prime}_{\eta})=1$ and hence the theorem is a direct result of Lemmas 11–19.