Decision Making in Nonlinear Dynamical System Diagnosis by a Nonparametric Method

Abstract

The problem of diagnosing dynamical systems described by models in the form of nonlinear differential equations with unknown coefficients (system parameters) is considered. To solve the problem, a nonparametric diagnostic method is used, which permits one to exclude the influence of unknown coefficients of the equations on the diagnosis results. The essence of the nonparametric method is presented. A new method for making decisions about the presence and type of a fault is proposed. The method takes into account model errors, uncontrolled disturbances, and measurement noise. A distinctive feature of the method is that it involves a complex combination of decision making by threshold logic with the comparison of the error (residual) signal against the fault signatures formed during the diagnosis process.

APPENDIX

Consider the problem of transforming the model (1) into the model (2), (3) taking into account (4) and (5). Differentiating both parts of relation (4) with respect to time, we write

$$ \dot x^{(i)}=\left (\partial \varphi ^{(i)}(x)/\partial x\right ) f(x, u, a),\quad 1\leqslant i\leqslant N. $$

(A.1)

Performing substitutions according to (2) into the left-hand side of Eqs. (A.1) and taking into account (1) and (4), we obtain

$$ \begin {aligned} f^{(1)}(h(x), u, a)&=\left (\partial \varphi ^{(1)}(x)/\partial x\right ) f(x, u, a), \\ f^{(2)}\left (\varphi ^{(1)}(x), h(x), u, a\right )&=\left (\partial \varphi ^{(2)}(x)/\partial x\right ) f(x, u, a),\\ &\ \,\vdots \\ f^{(N)}\left (\varphi ^{(1)}(x),\dots ,\varphi ^{(N-1)}(x), h(x), u, a\right )&=\left (\partial \varphi ^{(N)}(x)/\partial x\right ) f(x, u, a). \end {aligned}$$

(A.2)

Further, simultaneously considering the second equation in (1) and Eqs. (3) and (5), we write

$$ h_*\left (\varphi ^{(1)}(x), \varphi ^{(2)}(x), \dots , \varphi ^{(N)}(x)\right )=\psi \big (h(x, a)\big ). $$

(A.3)

Equations (A.2) and (A.3) can be used directly for finding the functions $f^{(i)}$, $1\leqslant i\leqslant N$, and the $h_*$-transformed model (2), (3) for the predetermined functions $\varphi ^{(i)}$, $1\leqslant i\leqslant N$, and $\psi$.

Using the language of differential geometry [12], let us describe a way for finding the functions $\varphi ^{(i)}$, $1\leqslant i\leqslant N$, ensuring the solvability of Eqs. (A.2). Let $\Lambda _i$ denote the distribution introduced for the function $\varphi ^{(i)}$: $\Lambda _i =\mathrm {span}\, \{\lambda \, | \, (\partial \varphi ^{(i)}/\partial x)\lambda =0\}$, where the symbol “$\mathrm {span}$” denotes the linear span of a system of vectors (covectors). Let also $\Omega _i=\Lambda _i^{\bot }$ be the corresponding codistribution, where the symbol “$\bot$” designates an annihilator, i.e., $\Lambda _i^{\bot }=\{\omega \in R^n \, | \, \langle \omega , \lambda \rangle =0\; \forall \lambda \in \Lambda _i\}$, and the brackets

$$ \langle \omega , \lambda \rangle =\sum _s \omega _{s} \lambda _{s}$$

have been introduced to denote the inner product.

Set $f_{ij}(x)=f(x, u_i, a_j)$, where $u_i$, $1\leqslant i\leqslant v_1$, $a_j$, $1\leqslant j\leqslant v_2$ are some fixed values of the vectors of input $u$ and parameters $a$, while $v_1$ and $v_2$ are final values such that the vector function $f(x, u, a)$ is linearly expressed via the family of vector functions (fields) $F=\{f_{ij}(x), 1\leqslant i\leqslant v_1, 1\leqslant j\leqslant v_2\}$ for any admissible values of the vectors $u$ and $a$.

Consider three differentiation operations. The first operation (Lie differentiation) is introduced for a scalar function $\alpha$ and a vector field $\gamma \in \mathrm {span}\, \, F$ as follows:

$$ L_{\gamma }\alpha =(\partial \alpha /\partial x) \gamma .$$

The second operation (Lie bracket) is introduced for two vector fields $\lambda _1\in \Lambda$ and $\lambda _2\in \Lambda$, where $\Lambda$ is some distribution,

$$ [\lambda _1, \lambda _2]=(\partial \lambda _2 / \partial x) \lambda _1-(\partial \lambda _1 / \partial x) \lambda _2.$$

The third operation is introduced for a covector field $\omega \in \Omega$ and a vector field $\lambda \in \Lambda$, where $\Lambda$ and $\Omega$ are some distribution and codistribution, respectively; the result is a covector field defined as follows:

$$ L_{\lambda } \omega = \lambda ^{\mathrm {T}}(\partial \omega ^{\mathrm {T}}/\partial x)^{\mathrm {T}} + \omega (\partial \lambda /\partial x).$$

In accordance with the definition of the distribution $\Lambda _i$ , the vector function $\varphi ^{(i)}$ can be found by integrating the homogeneous partial differential equations

$$ \left (\partial \varphi ^{(i)}/\partial x\right )\lambda =0\quad \forall \lambda \in \Lambda _i.$$

(A.4)

Necessary and sufficient conditions for the solvability of Eqs. (A.4) are determined by the Frobenius theorem (see [12, p. 23]). In accordance with this theorem, a regular (i.e., having a constant rank over the entire domain) distribution is integrable if it is involutive (closed) with respect to the operation of taking Lie brackets. The procedure for integrating Eqs. (A.4) can also be found in [12].

Further, according to the properties of the above operations of differentiation (see [10, p. 10]), the following relation holds for the covector field $\omega \in \Omega _i$ and the vector field

$$ \lambda \in \Lambda _i=\bigcap _{j=1}^{i-1}\Lambda _j\cap \ker (\partial h/\partial x), \quad i\geqslant 2$$

(A.5)

(for $i=1$ we take $\Lambda = \ker \, (\partial h/\partial x)$):

$$ L_{\gamma }\langle \omega , \lambda \rangle = \langle L_{\gamma }\omega , \lambda \rangle + \big \langle \omega , [\gamma , \lambda ]\big \rangle , \quad \gamma \in G.$$

(A.6)

By the construction of the codistribution $\Omega _i$, for $\lambda \in \Lambda _i$ and $\omega \in \Omega _i$ one has $\langle \omega ,\lambda \rangle =0$ and, as a consequence,

$$ L_{\gamma }\langle \omega , \lambda \rangle =0.$$

In addition, we set

$$ \big \langle \omega , [\gamma , \lambda ]\big \rangle =0 $$

(A.7)

in (A.6), which leads to

$$ \langle L_{\gamma }\omega , \lambda \rangle =0. $$

(A.8)

Relation (A.8) is equivalent to the inclusion

$$ L_{\gamma } \omega \in \sum _{j=1}^{i-1} \Omega _j +\mathrm {span}\, (\partial h /\partial x).$$

(A.9)

For $\partial \varphi ^{(i)}_s/\partial x=\omega$, where the subscript “$s$” indicates the number of the component of the vector function $\varphi ^{(i)}$, this implies that the corresponding component of the vector function $(\partial \varphi ^{(i)}/\partial x) f(*)$ can be expressed in terms of the component of the vector functions $h(x)$, $\varphi ^{(1)}(x), \ldots , \varphi ^{(i-1)}(x)$ and vectors $u$, $a$. Since this should hold true for all components of the vector function $(\partial \varphi ^{(i)}/\partial x) f(*)$ for which there exist “their own” covectors $\omega$ satisfying (A.9), it follows that there exist some vector functions $f^{(i)}(*)$, $1\leqslant i \leqslant N$, satisfying Eqs. (A.2). In this case, relation (A.7) can be viewed as a sufficient condition for the solvability of Eqs. (A.2). Based on the above, we give an algorithm for finding the functions $\varphi ^{(i)}(x)$, $1\leqslant i\leqslant N$.

Algorithm 3 (finding the functions $\varphi ^{(i)}(x), \; 1\leqslant i\leqslant N$).

• 1. Set $i=1$ and form the distribution $\Lambda _1^{\prime }= \ker \, (\partial h/\partial x)$.

• 2. Form the distribution $\Lambda _i=\Lambda _i^{\prime }\,\cap \, \mathrm {span}\,\{[\gamma , \lambda ]\, |\; \gamma \in G, \lambda \in \Lambda _i^{\prime }\}$.

• 3. If $i\geqslant 2$ and $\Lambda _i \supseteq \Lambda _{i-1}$, then set $N=i$ and go to step 5.

• 4. Set $i=i+1$ and find the distribution $\Lambda _i^{\prime }$ in accordance with (A.5). Go to step 2.

• 5. Find vector functions $\varphi ^{(i)}$, $1\leqslant i\leqslant N$, by integrating Eqs. (A.4).

• 6. To lower the dimension of the resulting model, in the functions $\varphi ^{(i)}$, $i\geqslant 2$, eliminate the components that functionally depend on the components of the vector functions $\varphi ^{(j)}$, $1\leqslant j \leqslant i-1$. Stop.

Note the following. Steps 1, 2, and 4 of the algorithm above make it possible to form distributions $\Lambda _i$ that simultaneously satisfy relation (A.5) and constraint (A.7). The functional dependence (the last algorithm step) can be verified using the analysis of the rank of the corresponding Jacobi matrices (see [15, p. 112 of the Russian translation]).

A method for finding the functions $\psi (h(x))$ that ensures the solvability of Eqs. (A.3) presumes the integration of the closed distribution ${\Lambda _{\psi }} {\supseteq \ker \,(\partial h/\partial x) + \bigcap _{i=1}^N \Lambda _i}$.

An example of transforming the model (25) into the model (26) can be found in [7, pp. 104–114].