Recurrent High Order Neural Observer for Discrete-Time Non-Linear Systems with Unknown Time-Delay

Abstract

This work proposes a discrete-time non-linear neural observer based on a recurrent high order neural network in parallel model trained with an algorithm based on the extended Kalman filter for discrete-time multiple input multiple output non-linear systems with unknown dynamics and unknown time-delay. To prove the semi-globally uniformly ultimately boundedness of the proposed neural observer the stability analysis based on the Lyapunov approach is included. Applicability of the proposed observer is shown via simulation and experimental results.

Appendix

Proof of Theorem 1

Consider the candidate Lyapunov function

$$\begin{aligned} {V}_{i}(k)={\tilde{w}^\top }_{i}(k){P}_{i}(k){\tilde{w}}_{i}(k) + {\tilde{x}^\top }_{i}(k){P}_{i}(k){\tilde{x}}_{i}(k) \end{aligned}$$

(25)

whose first increment is defined as

$$\begin{aligned} {\varDelta V}_{i}(k)= & {} V(k+1) - V(k) \nonumber \\ {\varDelta V}_{i}(k)= & {} {\tilde{w}^\top }_{i}(k+1){P}_{i}(k+1){\tilde{w}_{i}(k+1)} + {\tilde{x}^\top }_{i}(k+1){P}_{i}(k+1){\tilde{x}}_{i}(k+1) \nonumber \\&-\, {\tilde{w}^\top }_{i}(k){P}_{i}(k){\tilde{w}}_{i}(k) - {\tilde{x}^\top }_{i}(k){P}_{i}(k){\tilde{x}}_{i}(k) \end{aligned}$$

(26)

Using (9), (13) and (16) in (26)

$$\begin{aligned} {\varDelta V}_{i}(k)= & {} [{\tilde{w}}_{i}(k)+ {\eta }_{i}{K}_{i}(k)C\tilde{x}(k)]^\top [{A}_{i}(k)][{\tilde{w}}_{i}(k)+ {\eta }_{i}{K}_{i}(k)C\tilde{x}(k)] \nonumber \\&+\, [f(k)-{g}_{i}C\tilde{x}(k)]^\top [{A}_{i}(k)][f(k)-{g}_{i}C\tilde{x}(k)] \nonumber \\&-\, {\tilde{w}^\top }_{i}(k){P}_{i}(k){\tilde{w}}_{i}(k) - {\tilde{x}^\top }_{i}(k){P}_{i}(k){\tilde{x}}_{i}(k) \end{aligned}$$

(27)

$$\begin{aligned} {\varDelta V}_{i}(k)= & {} -\,{\tilde{w}^\top }_{i}(k){B}_{i}(k){\tilde{w}}_{i}(k) - \eta _{i}{\tilde{w}^\top }_{i}(k){A}_{i}(k){K}_{i}(k)C\tilde{x}(k) \nonumber \\&-\, \eta _{i}\tilde{x}^{\top }(k)C^{\top }{K}^{\top }_{i}(k){A}_{i}(k)\tilde{w}_{i}(k)+{\eta }^{2}_{i}\tilde{x}^{\top }(k)C^{\top }{K}^{\top }_{i}(k){A}_{i}(k){K}_{i}(k)C\tilde{x}(k) \nonumber \\&+\, {f}^{\top }(k){A}_{i}(k){f}(k) - {f}^{\top }(k){A}_{i}(k){g}_{i}C\tilde{x}(k) - \tilde{x}^{\top }(k)C^{\top }{g}^{\top }_{i}{A}_{i}(k){f}(k) \nonumber \\&+\, \tilde{x}^{\top }(k)C^{\top }{g}^{\top }_{i}{A}_{i}(k){g}_{i}C\tilde{x}(k) -\, {\tilde{x}^\top }_{i}(k){P}_{i}(k){\tilde{x}}_{i}(k) \end{aligned}$$

(28)

with

$$\begin{aligned} {A}_{i}(k)= & {} {P}_{i}(k) - {B}_{i}(k)\\ {B}_{i}(k)= & {} {K}_{i}(k){H}^{\top }_{i}(k){P}_{i}(k)+{Q}_{i}(k)\\ f(k)= & {} {\tilde{w}^\top }_{i}(k)z(\hat{x}(k),u(k))+\epsilon _{{z}_{i}}^{\prime } \end{aligned}$$

Using the inequalities

$$\begin{aligned} X^{\top }X + Y^{\top }Y\ge & {} 2{X}^{\top }Y \\ X^{\top }X + Y^{\top }Y\ge & {} -2{X}^{\top }Y \\ -\lambda \text {min}(P)X^2\ge & {} -{X}^{\top }{P}{X} \ge -\lambda \text {max}(P)X^2 \end{aligned}$$

which are valid $\forall X,Y \in \mathfrak {R}^n,\forall P \in \mathfrak {R}^{n \times n}, P = P^\top > 0$ then (28) can be rewritten as

$$\begin{aligned} {\varDelta V}_{i}(k)\le & {} -\,{\tilde{w}^\top }_{i}(k){B}_{i}(k){\tilde{w}}_{i}(k) + {\eta }^{2}_{i}\tilde{x}^{\top }(k)C^{\top }{K}^{\top }_{i}(k){A}_{i}(k){K}_{i}(k)C\tilde{x}(k) \nonumber \\&+\, {\tilde{w}^\top }_{i}(k){\tilde{w}}_{i}(k) + {\eta }^{2}_{i}\tilde{x}^{\top }(k)C^{\top }{K}^{\top }_{i}(k){A}^{\top }_{i}(k){A}_{i}(k){K}_{i}(k)C\tilde{x}(k) \nonumber \\&+\, {f}^{\top }(k){A}_{i}(k){f}(k) + \tilde{x}^{\top }(k)C^{\top }{g}^{\top }_{i}{A}_{i}(k){g}_{i}C\tilde{x}(k) \nonumber \\&+\, {f}^{\top }(k){f}(k) + \tilde{x}^{\top }(k)C^{\top }{g}^{\top }_{i}{A}^{\top }_{i}(k){A}_{i}(k){g}_{i}C\tilde{x}(k) \nonumber \\&- \,{\tilde{x}^\top }_{i}(k){P}_{i}(k){\tilde{x}}_{i}(k) \end{aligned}$$

(29)

then

$$\begin{aligned} {\varDelta V}_{i}(k)\le & {} - \Vert {\tilde{w}}_{i}(k) \Vert ^2 \lambda \text {min}({B}_{i}(k)) + {\eta }^{2}_{i} \Vert {\tilde{x}}(k) \Vert ^2 \Vert C{K}_{i}(k) \Vert ^2 \lambda \text {max}({A}_{i}(k)) \nonumber \\&+\, \Vert {\tilde{w}}_{i}(k) \Vert ^2 + {\eta }^{2}_{i} \Vert {\tilde{x}}(k) \Vert ^2 \Vert C{K}_{i}(k) \Vert ^2 \lambda ^2 \text {max}({A}_{i}(k)) \nonumber \\&+\, \Vert f(k) \Vert ^2 \lambda \text {max}({A}_{i}(k)) + \Vert \tilde{x}(k) \Vert ^2 \Vert C{g}_{i} \Vert ^2 \lambda \text {max}({A}_{i}(k)) \nonumber \\&+\, \Vert f(k) \Vert ^2 + \Vert \tilde{x}(k) \Vert ^2 \Vert C{g}_{i} \Vert ^2 \lambda ^2 \text {max}({A}_{i}(k)) - \Vert \tilde{x}(k) \Vert ^2 \lambda \text {min}({P}_{i}(k)) \nonumber \\ \end{aligned}$$

(30)

substituting f(k), then

$$\begin{aligned} {\varDelta V}_{i}(k)\le & {} - \Vert {\tilde{w}}_{i}(k) \Vert ^2 \lambda \text {min}({B}_{i}(k)) + {\eta }^{2}_{i} \Vert {\tilde{x}}(k) \Vert ^2 \Vert C{K}_{i}(k) \Vert ^2 \lambda \text {max}({A}_{i}(k)) \nonumber \\&+\, \Vert {\tilde{w}}_{i}(k) \Vert ^2 + {\eta }^{2}_{i} \Vert {\tilde{x}}(k) \Vert ^2 \Vert C{K}_{i}(k) \Vert ^2 \lambda ^2 \text {max}({A}_{i}(k)) \nonumber \\&+\, \Vert {\tilde{w}}_{i}(k) \Vert ^2 \Vert z(\hat{x}(k),u(k)) \Vert ^2 \lambda \text {max}({A}_{i}(k)) + \mid \epsilon _{{z}_{i}}^{\prime } \mid ^2 \lambda \text {max}({A}_{i}(k)) \nonumber \\&+\, \Vert \tilde{x}(k) \Vert ^2 \Vert C{g}_{i} \Vert ^2 \lambda \text {max}({A}_{i}(k)) + \Vert {\tilde{w}}_{i}(k) \Vert ^2 \Vert z(\hat{x}(k),u(k)) \Vert ^2 \nonumber \\&+ \,\mid \epsilon _{{z}_{i}}^{\prime } \mid ^2 + \Vert \tilde{x}(k) \Vert ^2 \Vert C{g}_{i} \Vert ^2 \lambda ^2 \text {max}({A}_{i}(k)) - \Vert \tilde{x}(k) \Vert ^2 \lambda \text {min}({P}_{i}(k)) \end{aligned}$$

(31)

Defining

$$\begin{aligned} {E}_{i}(k)= & {} -\eta ^2 \Vert C{K}_{i}(k) \Vert ^2 \lambda \text {max}({A}_{i}(k))-\eta ^2 \Vert C{K}_{i}(k) \Vert ^2 \lambda \text {max}({A}^{2}_{i}(k)) \nonumber \\&-\,\Vert C{g}_{i} \Vert ^2 \lambda \text {max}({A}_{i}(k))-\Vert C{g}_{i} \Vert ^2 \lambda \text {max}({A}^{2}_{i}(k)) + \lambda \text {min}({P}_{i}(k)) \nonumber \\ {F}_{i}(k)= & {} -1 + \lambda \text {min}({B}_{i}(k)) - \Vert z(\hat{x}(k),u(k)) \Vert ^2 \lambda \text {max}({A}_{i}(k)) \nonumber \\&- \,\Vert z(\hat{x}(k),u(k)) \Vert ^2 \end{aligned}$$

(32)

then (31) can be express as:

$$\begin{aligned} {\varDelta V}_{i}(k)\le & {} -\Vert \tilde{x}(k) \Vert ^2 {E}_{i}(k) - \Vert \tilde{w}_{i}(k) \Vert ^2 {F}_{i}(k) \nonumber \\&+\, \mid \epsilon _{{z}_{i}}^{\prime } \mid ^2 (1 + \lambda \text {max}({A}_{i}(k))) \end{aligned}$$

(33)

Hence, ${\varDelta V}_{i}(k) < 0$ when

$$\begin{aligned} \Vert {\tilde{x}}(k) \Vert > \sqrt{\frac{\mid \epsilon _{{z}_{i}}^{\prime } \mid ^2 (1 + \lambda \text {max}({A}_{i}(k)))}{{E}_{i}(k)}}\equiv \kappa _{1} \end{aligned}$$

(34)

or

$$\begin{aligned} \Vert {\tilde{w}}_{i}(k) \Vert > \sqrt{\frac{\mid \epsilon _{{z}_{i}}^{\prime } \mid ^2 (1 + \lambda \text {max}({A}_{i}(k)))}{{F}_{i}(k)}} \equiv \kappa _{2} \end{aligned}$$

(35)

Therefore, the solution of (19) and (13) is stable; hence the estimation error and the RHONN weights are SGUUB. Considering (15) and (12), it is easy to see that the output error has an algebraic relation with $\tilde{x}(k)$, for that reason. if $\tilde{x}(k)$ is bounded, e(k) is bounded too.

$$\begin{aligned} e(k)= & {} C\tilde{x}(k) \\ \Vert e(k) \Vert= & {} \Vert C \Vert \Vert \tilde{x}(k) \Vert \end{aligned}$$

$\square$

Remark 1

Considering Theorem 1 and its proof, it can be easily shown that the result can be extended to systems with multiple delays like $x\left( k-l_{i}\right)$ with $i=1,2,\ldots$, instead of $x\left( k-l\right)$ in (2) and/or for time-variying delays $x\left( k-l_{i}\left( k\right) \right)$ with $l_{i}\left( k\right)$ bounded by $l_{i}\left( k\right) \le l$.