Robust Delay-Derivative-Dependent Sliding Mode Observer for Fault Reconstruction : A Diesel Engine System Application

Abstract

In this paper, a new delay-derivative-dependent sliding mode observer (SMO) design for a class of linear uncertain time-varying delay systems is presented. Based on this observer, a robust actuator fault reconstruction method is developed. In the meantime, the considered uncertainty is bounded and the time-delay is varying and affects the state system. Besides, the dynamic properties of the observer are analyzed and the reachability condition is satisfied. Applying the developed SMO, the $H_\infty$ concept and a delay-derivative-dependent bounded real lemma (BRL), a robust actuator fault reconstruction is obtained wherein the effect of the uncertainty is minimized. Also, both the SMO and the BRL are delay-derivative-dependent which reduces the time-varying delay conservatism on the state estimation and on the fault reconstruction. A diesel engine system is included to illustrate the validity and the applicability of the proposed approaches.

Appendices

Appendices

1.1 Appendix 1

The proof of Lemma 1. Extending a proof given by [36] we get:

$$\begin{aligned}&\mathrm{rank} \left( \left[ {\begin{array}{c@{\quad }c} {sI_{n} - A - A_h {e}^{{ - h_{m}s}} } &{} F\\ C &{} 0_{p\times q} \\ \end{array}} \right] \right) \nonumber \\&\quad =\mathrm{rank} \left( \left[ {\begin{array}{c} {sI_{n} - A - A_h {e}^{{ - h_{m}s}} }\\ C \\ \end{array}} \right] \right) + q; \end{aligned}$$

(56)

however, the matrix F is full column rank and then for all complex s with $Re(s)\ge 0$, the system $(A + A_h e^{ - h_{m}s} ,\;\,F,\;\,C)$ is minimum phase if and only if

$$\begin{aligned} \mathrm{rank}\left( \left[ {\begin{array}{c} {sI_{n} - A - A_h {e}^{{ - h_{m}s}} }\\ C \\ \end{array}} \right] \right) =n, \end{aligned}$$

(57)

which is equivalent to $(A + A_h e^{ - h_{m}s} ,\;\,C)$ is detectable. Also the invariant zeros of $(A + A_h e^{ - h_{m}s} ,\;\,F,\;\,C)$ are the unobservable modes of $(A + A_h e^{ - h_{m}s} ,\;\,C)$ and lie in $\mathbb {C}_-$. So $(A + A_h e^{ - h_{m}s} ,\;\,F,\;\,C)$ is minimum phase if and only if the pair $(A + A_h e^{ - h_{m}s} ,\;\,C)$ is detectable.

1.2 Appendix 2

The proof of Corollary 1. From the fact that $\dot{\bar{e}}_y(t) = \bar{C}\dot{\bar{e}}(t)$ and using the Eq. (8) we can obtain

$$\begin{aligned} \dot{ \bar{e}}_y (t)= & {} \bar{C}(\bar{A} - \bar{K} \bar{C})\bar{e}(t) + \bar{C}\bar{A}_h \bar{e}(t - h(t)) + \bar{C}\bar{G} \nu (t) - \bar{C}\bar{M}\xi (t,x(t)) \nonumber \\&\,- \bar{C}\bar{F}f(t), \end{aligned}$$

(58)

also, during the sliding $e_y (t) = \dot{e}_y (t) = 0$ where $\det (\bar{C}\bar{G} ) \ne 0$ , then, using (58), the equivalent output error injection is

$$\begin{aligned} \nu _{eq}(t)\cong & {} - (\bar{C}\bar{G} )^{ - 1} \left[ \bar{C}(\bar{A} - \bar{K} \bar{C})\bar{e}(t) + \bar{C}\,\bar{A}_h \bar{e}(t - h(t)) - \bar{C}\,\bar{M}\xi (t,x(t)) \nonumber \right. \\&\left. - \bar{C}\,\bar{F}f(t)\right] \end{aligned}$$

(59)

and during the sliding, the estimation error (8) will be

$$\begin{aligned} \dot{\bar{e}}(t)= & {} (\bar{A} - \bar{K} \bar{C})\bar{e}(t) + \bar{A}_h \bar{e}(t - h(t)) + \bar{G} \nu _{eq}(t) - \bar{M}\xi (t,x(t)) \,- \bar{F}f(t).\quad \quad \end{aligned}$$

(60)

Substituting (59) in (60), we obtain:

$$\begin{aligned} \dot{\bar{e}}(t)= & {} [I_n - \bar{G} (\bar{C}\bar{G} )^{ - 1} \bar{C}]((\bar{A} - \bar{K} \bar{C})\bar{e}\,(t) + \bar{A}_h \bar{e}(t - h(t)) - \bar{M}\xi (t,x(t)) \nonumber \\&\,- \bar{F}f(t)), \end{aligned}$$

(61)

to be insensitive to the uncertainty the last equation must verify

$$\begin{aligned} \left[ I_n - \bar{G} (\bar{C}\bar{G} )^{ - 1} \bar{C}\right] \bar{M} = 0. \end{aligned}$$

(62)

For the remainder of this proof, we need to calculate the matrix $[I_n - \bar{G} (\bar{C}\bar{G} )^{ - 1} \bar{C}]$, so

$$\begin{aligned} {I_n - \bar{G} (\bar{C}\bar{G} )^{ - 1} \bar{C}}= & {} I_n - \left[ {\begin{array}{cc} { - L } \\ {I_p} \\ \end{array} } \right] \bar{C}{_2}^T\left( {\left[ {\begin{array}{c@{\quad }c} 0_{p \times (n-p)} &{} \bar{C}{_2} \\ \end{array} } \right] \left[ {\begin{array}{cc} { - L } \\ {I_p} \\ \end{array} } \right] } \right) ^{ - 1} \nonumber \\&\,\times \bar{C}{_2}^T \left[ {\begin{array}{c@{\quad }c} 0_{p \times (n-p)} &{} \bar{C}{_2} \\ \end{array} } \right] . \end{aligned}$$

(63)

Since the matrix $\bar{C}{_2}$ is an orthogonal matrix, then $\left( {\bar{C}{_2}\bar{C}{_2}^T } \right) = I_p$; therefore,

$$\begin{aligned} {I_n- \bar{G} (\bar{C}\bar{G} )^{ - 1} \bar{C}}= & {} I_n - \left[ {\begin{array}{cc} { - L } \\ {I_p } \\ \end{array} } \right] \,\bar{C}{_2}^T\,\left[ {\begin{array}{c@{\quad }c} 0_{p \times (n-p)} &{} \bar{C}{_2} \\ \end{array} } \right] \nonumber \\= & {} I_n - \left[ {\begin{array}{c@{\quad }c} 0_{(n-p) \times (n-p)} &{} { - L} \\ 0_{p \times (n-p)} &{} I_p \\ \end{array} } \right] \nonumber \\= & {} \left[ {\begin{array}{c@{\quad }c} I_{n-p} &{} L \\ 0_{p \times (n-p)} &{} 0_{p \times p} \\ \end{array} } \right] , \end{aligned}$$

(64)

then

$$\begin{aligned} (I_n - \bar{G} (\bar{C}\bar{G} )^{ - 1} \bar{C}) \bar{F}= & {} \left[ {\begin{array}{c@{\quad }c} I_{n-p}&{}{\underbrace{\left[ {\begin{array}{c@{\quad }c} {L_q } &{} 0_{(n-p) \times q} \\ \end{array} }\right] }_L} \\ 0_{p \times (n-p)} &{} 0_{p \times p} \end{array} } \right] \left[ {\begin{array}{cc} 0_{(n-p) \times q} \\ {\underbrace{\left[ {\begin{array}{cc} 0_{(p-q) \times q} \\ {\bar{F}_q } \\ \end{array} } \right] }_{\bar{F}_2 }} \\ \end{array} } \right] \nonumber \\= & {} 0, \end{aligned}$$

(65)

where this development gives the importance of the dimension condition $q<p<n$. Then from (65), we get

$$\begin{aligned} \bar{F} = \bar{G} (\bar{C}\bar{G} )^{ - 1} \bar{C}\bar{F}. \end{aligned}$$

(66)

So it is clear that rank $\left( {\bar{C}\bar{F}} \right)$ must be equal to rank $\left( {\bar{F}} \right)$ where the assumption A1 appears. Also $0<h(t)\le h_m$ then replacing the time-varying delay by $e^{-h_{m}s}$ and using the Eqs. (62) and (65) then the dynamic of the error (61) is assured by:

$$\begin{aligned} \left[ {I_n - {\bar{G}} ({\bar{C}}{\bar{G}} )^{ - 1}{\bar{C}}} \right] \left[ {\bar{A}} - {\bar{K}} {\bar{C}} +{\bar{A}}_{h}e^{-h_{m}s} \right] = \left[ {\begin{array}{c@{\quad }c} {\maltese _1 } &{} {\maltese _2} \\ 0_{p \times (n-p)} &{} 0_{p \times p} \\ \end{array} } \right] , \end{aligned}$$

(67)

where

$$\begin{aligned} \maltese _1 = \left( \bar{A}_{11}+ \bar{A}_{h11} e^{-h_{m}s}\right) + L_q \left( \bar{A}_{211}+\bar{A}_{h211} e^{-h_{m}s}\right) \end{aligned}$$

and

$$\begin{aligned} \maltese _2 =\left( \bar{A}_{12} +\bar{A}_{h12} e^{-h_{m}s}\right) + \bar{L}\left( \bar{A}_{22}+ \bar{A}_{h22} e^{-h_{m}s}\right) . \end{aligned}$$

Consequently, it is clear that from the Eq.(67), the sliding dynamic is governed by the radii matrix $\maltese _1$ which is stable, and then, the sliding surface $S_g$ is taken in finite time.