Monotonous Parameter Estimation of One Class of Nonlinearly Parameterized Regressions without Overparameterization

In the majority of applications real technical systems have a limited number of significant physical parameters. At the same time, mathematical models of these systems, written in the state space or Euler-Lagrange form, are described by equations with overparameterization, i.e. with a large number of new virtual parameters that are nonlinearly related to the original ones [6], [12].

As far as classic methods of identification theory and adaptive control are concerned, each parameter of a mathematical model is considered to be unique and independent (decoupled) from the others. With increasing the system order and, as a result, the number of the above-mentioned virtual parameters, this leads to the well-known [6] shortcomings, which make it difficult to apply the basic estimation laws:

1. S1.

   Slower convergence and stringent excitation conditions due to the need to solve the identification problem in a larger parameter space.

2. S2.

   Necessity to apply projection operators for online estimation of the system physical parameters.

To overcome these problems, it has been proposed [1], [8], [9], [10] to take into account the relationship between the unknown parameters to design the estimation law. In [1] the dynamic regressor extension and mixing (DREM) technique is applied to “isolate the good mappings” from virtual to physical parameters and utilize the strong P-monotonicity property [11] to achieve consistent parameter estimation for nonlinearly parameterized regressions. In case the regressor is non-square integrable, the solution [1] ensures asymptotic convergence of the parameter identification error. The requirement of strong P-monotonicity has turned out to be strict enough for some applications, e.g. composite control of Euler-Lagrange systems [9], [12], adaptive observation of windmill power coefficient [2]. For some, mainly polynomial mappings, it is possible to relax this condition using a special monotonizability assumption [2], [8], [9]. The relaxation mechanism is based on the search for a bijective substitution such that the new nonlinear mapping satisfies the strong P-monotonicity condition. However, the solution from [8], [9] has several key problems:

1. P1.

   The convergence of the parametric error is guaranteed only for hardly validated non-square integrable regressors (Proposition 4 in [9]);

2. P2.

   Weak property of non-increasing norm of the parameter error vector is guaranteed, but not the elementwise monotonicity (Remark 8 in [9]).

3. P3.

   The estimation law requires a priori information about uncertainty parameters (for example, Lemma 2 in [9]).

4. P4.

   The calculation of the system physical parameters from the obtained estimates can lead to singularities and sometimes requires application of projection operator (for example, see definition ${{\mathcal{D}}^{I}}$ in Lemma 2 of [9]).

5. P5.

   Due to P2 and P4 the transient behavior of parametric error is unpredictable, singularity may occur if we want to use ${{\mathcal{D}}^{I}}$.

In a recent paper [10] a new estimation law has been proposed that solves P1 and ensures exponential convergence of the parametric error when a more realistic for some practical scenarios condition of regressor finite excitation is satisfied.

The motivation for this study is to solve all problems P1-P5 for one class of nonlinearly parametrized regression equations (NLPRE).

Notation and Definitions. Further the following notation is used: $\left|.\right|$ is the absolute value, $\left\|.\right\|$ is the suitable norm of $(.)$, ${I_{n\times n}}=I_{n}$ is an identity $n\times n$ matrix, ${0_{n\times n}}$ is a zero $n\times n$ matrix, $0_{n}$ stands for a zero vector of length $n$, ${\rm{det}}\{.\}$ stands for a matrix determinant, ${\rm{adj}}\{.\}$ represents an adjoint matrix. Denote $\mathfrak{H}\left[.\right]:=1\;/\left({p+k}\right)\left[.\right]$ as a stable operator ($k>0$ and $p:=d/dt$). For a mapping ${\mathcal{F}}{\rm{:\;}}{\mathbb{R}^{n}}\mapsto{\mathbb{R}^{n}}$ we denote its Jacobian by $\nabla_{x}{\mathcal{F}}\left(x\right)=\linebreak={\textstyle{{\partial{\mathcal{F}}}\over{\partial x}}}\left(x\right)$. We also use the fact that for all (possibly singular) ${n\times n}$ matrices $M$ the following holds: ${\rm{adj}}\{M\}M={\rm{det}}\{M\}I_{n\times n}$.

Definition. A regressor $\omega\left(t\right)\in{\mathbb{R}^{n\times m}}$ is finitely exciting $(\omega\left(t\right)\in{\rm{FE}})$ over a time range $\left[{t_{r}^{+}{\rm{,\;}}{t_{e}}}\right]{\rm{}}$ if there exist $t_{r}^{+}\geq 0$, ${t_{e}}>t_{r}^{+}$ and $\alpha$ such that the following inequality holds:

where $\alpha\!>\!0$ is the excitation level.

The following NLPRE is considered:

where $y\left(t\right)\in{\mathbb{R}^{n}}$, $\Omega\left(t\right)\in{\mathbb{R}^{n\times p}}$ are measurable regressand and regressor, respectively, $\theta\in{D_{\theta}}\subset{\mathbb{R}^{q}}$ is a vector of unknown time-invariant parameters, $\Theta{\rm{:\;}}{\mathbb{R}^{q}}\mapsto{\mathbb{R}^{p}}$ is a known mapping and $p>q$. The problem is to estimate parameters $\theta$ using $y(t)$ and $\Omega(t)$ such that:

where $\hat{\theta}\left(t\right)$ is an estimate of the unknown parameters, ${\tilde{\theta}_{i}}\left(t\right)$ is an estimation error of the i^th parameter from $\theta$, $\left(\rm{exp}\right)$ is an abbreviation for exponential rate of convergence.

The feasibility conditions for the problem (3a) are

1. FC1.

   $\Omega\left(t\right)\in{\rm{FE}}$, i.e. condition of identifiability of an overparametrized parameters $\Theta\left(\theta\right)$.

2. FC2.

   ${D_{\theta}}{\rm{:=}}\Big{\{}{\theta\in{\mathbb{R}^{{q}}}{\rm{:de}}{{\rm{t}}}\Big{\{}{{\nabla_{\theta}}{\psi}\left(\theta\right)}\Big{\}}\neq 0}\Big{\}}$, where ${\psi}\left(\theta\right)=\linebreak={{\mathcal{L}}}\Theta\left(\theta\right)\in{\mathcal{C}^{{k}}}{\rm{,}}$ i.e. existence of inverse mapping ${\mathcal{F}}{\rm{:\;}}{\mathbb{R}^{q}}\mapsto{\mathbb{R}^{q}}$ that reconstructs the unknown parameters $\theta={\mathcal{F}}\left(\psi\right)$ from a ”good” elements $\psi\left(\theta\right)$ handpicked by $\mathcal{L}\in\mathbb{R}^{q\times p}$ from $\Theta\left(\theta\right)$.

When FC1-FC2¹¹1It should be understood that, if the inverse function ${\mathcal{F}}\left(\psi\right)$ does not exist, then there is no way to obtain $\theta$ from $\Theta(\theta)$.are met, then the parameters $\Theta\left(\theta\right)$ can be obtained and recalculated into $\theta$ (possibly, only asymptotically). However the main contribution of this paper is to solve all problems P1-P5 and shortcomings S1-S2 of existing solutions and consequently ensure elementwise monotonicity (3b) and obtain $\theta$ without estimation of $\Theta\left(\theta\right)$ and substitution ${\mathcal{F}}\left({\mathcal{L}}\hat{\Theta}\left(t\right)\right)$.

To facilitate the proposed estimation design, in addition to FC1-FC2 a class of mappings $\Theta\left(\theta\right)$ and respective inverse functions ${\mathcal{F}}\left(\psi\right)$, to which we restrict our attention, is defined in the following linearizing assumption.

Assumption 1.There exist ${\mathcal{G}}{\rm{:\;}}{\mathbb{R}^{q}}\!\mapsto\!{\mathbb{R}^{q\times q}}$, ${\mathcal{S}}{\rm{:\;}}{\mathbb{R}^{q}}\!\mapsto\!{\mathbb{R}^{q}}{\rm{,\;}}$ ${\Pi_{\theta}}{\rm{:\;}}{\mathbb{R}}\mapsto{\mathbb{R}^{q\times q}}{\rm{,}}$ ${{\mathcal{T}}_{\mathcal{G}}}{\rm{:\;}}{\mathbb{R}^{{\Delta_{\mathcal{G}}}}}\mapsto{\mathbb{R}^{q\times q}}{\rm{,}}$ ${{\mathcal{T}}_{\mathcal{S}}}{\rm{:\;}}{\mathbb{R}^{{\Delta_{\mathcal{S}}}}}\mapsto{\mathbb{R}^{q}}{\rm{,}}$ ${{\Xi_{\mathcal{G}}}}{\rm{:\;}}{\mathbb{R}}\mapsto{\mathbb{R}^{{{\Delta_{\mathcal{G}}}}\times q}}$, ${{\Xi_{\mathcal{S}}}}{\rm{:\;}}{\mathbb{R}}\mapsto{\mathbb{R}^{{{\Delta_{\mathcal{S}}}}\times q}}$ such that for all $\Delta\left(t\right)\in\mathbb{R}$ the following holds:

where

and all above mentioned mappings are known²²2Assumption 1 is not restrictive and can be easily verified via direct inspection of mapping $\mathcal{F}(\psi)$..

Assumption 1 is met in case when polynomials in $\theta$ are used to form $\Theta\left(\theta\right)$ and consequently the inverse transform function ${\mathcal{F}}\left(\psi\right)$ can be computed using algebraic functions.

Example. For vector $\psi\left(\theta\right)=col\left\{{{\theta_{1}}{\theta_{2}}{\rm{+}}\theta_{1}^{2}{\rm{,\;}}{\theta_{2}}+{\theta_{1}}}\right\}$ the mappings from (4) take the form:

Assumption 1 sets the conditions to obtain the following linearly parameterized regression equation from $\psi(\theta)$:

Taking into consideration that the following equalities hold in accordance with Assumption 1:

equation (6) is rewritten as:

where ${{\mathcal{Y}}_{\psi}}\left(t\right)=\Delta\left(t\right)\psi\left(\theta\right)$ is the unmeasurable linear regression equation with respect to $\psi\left(\theta\right)$.

Example (remainder). For $\psi\left(\theta\right)=col\left\{{{\theta_{1}}{\theta_{2}}{\rm{+}}\theta_{1}^{2}{\rm{,\;}}{\theta_{2}}+{\theta_{1}}}\right\}$ the mappings from (8) take the form:

Thus, if Assumption 1 is satisfied, having equation for ${{\mathcal{Y}}_{\psi}}\left(t\right)$ and the known mappings from (4) at hand, the regression equation with nonlinear parameterization (2) can be transformed into the new one with linear parameterization (8). That is the reason why Assumption 1 is called “linearizing”.

Using (2), the regression equation with measurable ${{\mathcal{Y}}_{\psi}}\left(t\right){\rm{,\;}}\Delta\left(t\right)\geq 0$ could be obtained with the help of DREM procedure [1]. Towards this end, we introduce the following dynamic extension:

and apply a mixing procedure to $\overline{y}(t)$:

The following proposition has been proved in [4], [5] for the scalar regressor $\Delta\left(t\right)$ obtained by (9) and (10).

Proposition 1. If $\Omega\left(t\right)\in{\rm{FE}}$, then for all $t\geq{t_{e}}\;\Delta\left(t\right)\geq\linebreak\geq{\Delta_{LB}}>0.$

So, the signals ${{\mathcal{T}}_{\mathcal{S}}}\left({{{\overline{\Xi}}_{\mathcal{S}}}\left(\Delta\right){{\mathcal{Y}}_{\psi}}}\right){\rm{,\;}}{{\mathcal{T}}_{\mathcal{G}}}\left({{{\overline{\Xi}}_{\mathcal{G}}}\left(\Delta\right){{\mathcal{Y}}_{\psi}}}\right)$ can be computed through equations (9) and (10). Then the mixing procedure is applied a novo:

Having the linear regression equation (11) at hand, the estimation law to identify the unknown parameters is introduced based on standard gradient descent method:

where $\gamma>0$ is an adaptive gain, $\hat{\theta}(t_{0})=\hat{\theta}_{0}$ is an initial condition.

The properties of the law (12) are considered in the following theorem.

Theorem 1. If FC1-FC2 and Assumption 1 are met, then goals (3a) and (3b) are achieved.

Proof. The solution of the differential equation (12) for all $t\geq{t_{0}}$ is written as:

from which $\forall{{\mathop{t}\nolimits}_{a}}\geq{t_{b}}{\rm{,\;}}\forall i\in\left\{{1,{\rm{}}q}\right\}{\rm{}}\left|{{{\tilde{\theta}}_{i}}\left({{t_{a}}}\right)}\right|\leq\left|{{{\tilde{\theta}}_{i}}\left({{t_{b}}}\right)}\right|$.

Following Assumption 1, it holds that ${\rm{det}}\left\{{{\Pi_{\theta}}\left(\Delta\right)}\right\}\geq\linebreak\geq{\Delta^{{\ell_{\theta}}}}\left(t\right){\rm{,rank}}\left\{{{\mathcal{G}}\left(\psi\right)}\right\}=q$, and $\forall t\geq{t_{e}}{\rm{\;}}\Delta\left(t\right)\geq{\Delta_{LB}}>0$ also holds owing to Proposition 1, then for all $t\geq{t_{e}}$ we can write the following expression for ${\mathcal{M}}\left(t\right)$:

which, in its turn, allows one to rewrite the solution (13) for all $t\geq{t_{e}}$ as:

from which it follows that $\mathop{{\rm{lim}}}\limits_{t\to\infty}\left\|{\tilde{\theta}\left(t\right)}\right\|=0{\rm{}}\left({{\rm{exp}}}\right)$.

This completes the proof of Theorem 1.

Therefore, if the mapping ${\mathcal{F}}\left(\psi\right)$ satisfy the premises of Assumption 1, then, in accordance with the extension (9) and mixing procedures (10) and (11), the estimation law (12) can be designed ensuring that the goals (3a) and (3b) are achieved. Note that, in contrast to [10], in addition to properties (3a) and (3b) the proposed law does not use a priori information about low and upper bounds of parameters (P3) in design procedure³³3It should be noted that the proposed law requires only knowledge that $\theta$ lies in the safe domain $D_{\theta}$ from FC2. and does not include singularity causing division operations (P4-P5).

The unknown parameters estimation law for one class of NLPRE was proposed. In contrast to existing solutions, elementwise monotonicity of the parametric error was ensured. Necessary and sufficient implementability conditions for the developed law were: (i) the regressor finite excitation requirement, (ii) existence of inverse function from overparameterized parameters to physical ones, (iii) that only polynomials functions in $\theta$ were used to form overparametrization. The results can be applied to improve the solutions quality of adaptive control and observation problems from recent studies [2], [3], [7], [9].