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Sensor location optimization for fault diagnosis with a comparison to linear programming approaches

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The critical importance of sustaining fault diagnosis, as a major system tool, is unquestionable if the high performance and reliability of increasingly complex engineering systems is to be sustained over time and across a wide operating range. However, it is quite difficult to retain the joint ability of fault detection and isolation as it requires a strong system architecture. That is why, before designing an industrial supervision system, the determination of a system’s monitoring ability based on technical specifications is important as finding the source of the failure is not trivial in systems with a large number of components and complex component relationships. This paper presents an efficient and cost-effective fault detection and isolation (FDI) scheme that evolved from an earlier work [1]. FDI specifications are translated into constraints of the optimization problem considering that the whole set of analytical redundancy relations has been generated, under the assumption that all candidate sensors are installed and later on tested by an optimization algorithm using binary and relaxed versions of linear and nonlinear programming. By doing so, critical information about the presence or absence of a fault is gained in the shortest possible time, with not only confirmation of the findings but also an accurate unfolding in time of the finer details of the fault, thus completing the overall diagnostic picture of the system under test. The proposed scheme is evaluated extensively on a two-tank process used in industry, exemplified by a benchmarked laboratory-scale coupled-tank system.

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Correspondence to Haris M. Khalid.



  1. (a)

    Nonlinear constraints

    $$ \matrix{ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_7} + {q_4} \times {q_5} \times {q_8} + {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_7} \times {q_8} \geqslant 1\;;} \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_7} + {q_4} \times {q_5} \times {q_8} + {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_7} \times {q_7} \geqslant 1\;;} \\ }<!end array> $$
    $$ \matrix{ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_7} + {q_4} \times {q_5} \times {q_8} + {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_7} \times {q_8} \geqslant 1\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_7} + {q_4} \times {q_5} \times {q_8} + {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_7} \times {q_8} \geqslant 1\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_7} + {q_4} \times {q_5} \times {q_8} + {q_4} \times {q_7} \times {q_8} \geqslant {q_4}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_7} + {q_4} \times {q_5} \times {q_8} + {q_5} \times {q_7} \times {q_8} \geqslant {q_5}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_7} + {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_7} \times {q_8} \geqslant {q_7}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_8} + {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_7} \times {q_8} \geqslant {q_8}\;;} \hfill \\ }<!end array> $$
    $$ \matrix{ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_5} \times {q_7} \times {q_8} \geqslant {q_4}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_5} \times {q_7} \times {q_8} \geqslant {q_4}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_7} \times {q_8} \geqslant {q_5}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_7} \times {q_8} \geqslant {q_5}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_8} \geqslant {q_7}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_8} \geqslant {q_7}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_7} \geqslant {q_8}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_7} \geqslant {q_8}\;;} \hfill \\ }<!end array> $$
    $$ \matrix{ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_5} \times {q_7} \times {q_8} \geqslant {q_4}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_5} \times {q_7} \times {q_8} \geqslant {q_4}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_7} \times {q_8} \geqslant {q_5}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_7} \times {q_8} \geqslant {q_5}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_8} \geqslant {q_7}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_8} \geqslant {q_7}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_7} \geqslant {q_8}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_7} \geqslant {q_8}\;;} \hfill \\ }<!end array> $$
    $$ \matrix{ {{q_4} \times {q_5} \times {q_7} + {q_5} \times {q_7} \times {q_8} \geqslant {q_4} \times {q_8}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} + {q_4} \times {q_5} \times {q_8} \geqslant {q_7} \times {q_8}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_7} + {q_4} \times {q_7} \times {q_8} \geqslant {q_5} \times {q_8}\;;} \hfill \\ {{q_5} \times {q_7} \times {q_8} + {q_4} \times {q_5} \times {q_7} \geqslant {q_4} \times {q_8}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_8} + {q_4} \times {q_7} \times {q_8} \geqslant {q_5} \times {q_7}\;;} \hfill \\ {{q_4} \times {q_5} \times {q_8} + {q_5} \times {q_7} \times {q_8} \geqslant {q_4} \times {q_7}\;;} \hfill \\ {{q_4} \times {q_7} \times {q_8} + {q_5} \times {q_7} \times {q_8} \geqslant {q_4} \times {q_5}\;;} \hfill \\ }<!end array> $$
  2. (b)

    Linear constraints

    $$ \matrix{ {{q_4} + {q_5} \leqslant {x_1} + 1\;;} \hfill \\ {{x_1} \leqslant {q_4}\;;} \hfill \\ {{x_1} \leqslant {q_5}\;;} \hfill \\ {{q_4} + {q_7} \leqslant {x_2} + 1\;;} \hfill \\ {{x_2} \leqslant {q_4}\;;} \hfill \\ }<!end array> $$
    $$ {x_2} \leqslant {q_7}\;; $$
    $$ {q_5} + {q_7} \leqslant {x_3} + 1\;; $$
    $$ {x_3} \leqslant {q_5}\;; $$
    $$ {x_3} \leqslant {q_7}\;; $$
    $$ {q_4} + {q_8} \leqslant {x_4} + 1\;; $$
    $$ {x_4} \leqslant {q_4}\;; $$
    $$ {x_4} \leqslant {q_8}\;; $$
    $$ {q_5} + {q_8} \leqslant {x_5} + 1\;; $$
    $$ {x_5} \leqslant {q_5}\;; $$
    $$ {x_5} \leqslant {q_8}\;; $$
    $$ {q_7} + {q_8} \leqslant {x_6} + 1\;; $$
    $$ {x_6} \leqslant {q_7}\;; $$
    $$ {x_6} \leqslant {q_8}\;; $$
    $$ {q_4} + {q_5} + {q_7} \leqslant {x_7} + 1 + 1\;; $$
    $$ {x_7} \leqslant {q_4}\;; $$
    $$ {x_7} \leqslant {q_5}\;; $$
    $$ {x_7} \leqslant {q_7}\;; $$
    $$ {q_4} + {q_5} + {q_8} \leqslant {x_8} + 1 + 1\;; $$
    $$ {x_8} \leqslant {q_4}\;; $$
    $$ {x_8} \leqslant {q_5}\;; $$
    $$ {x_8} \leqslant {q_8}\;; $$
    $$ {q_4} + {q_7} + {q_8} \leqslant {x_9} + 1 + 1\;; $$
    $$ {x_9} \leqslant {q_4}\;; $$
    $$ {x_9} \leqslant {q_7}\;; $$
    $$ {x_9} \leqslant {q_8}\;; $$
    $$ {q_5} + {q_7} + {q_8} \leqslant {x_{{10}}} + 1 + 1\;; $$
    $$ {x_{{10}}} \leqslant {q_5}\;; $$
    $$ {x_{{10}}} \leqslant {q_7}\;; $$
    $$ {x_{{10}}} \leqslant {q_8}\;; $$
    $$ {q_4} + {q_5} + {q_7} + {q_8} \leqslant {x_{{11}}} + 1 + 1 + 1\;; $$
    $$ {x_{{11}}} \leqslant {q_4}\;; $$
    $$ {x_{{11}}} \leqslant {q_5}\;; $$
    $$ {x_{{11}}} \leqslant {q_7}\;; $$
    $$ {x_{{11}}} \leqslant {q_8}\;; $$
    $$ {x_{{11}}} + {x_7} + {x_8} + {x_{{10}}} + {x_9} \geqslant 1\;; $$
    $$ {x_{{11}}} + {x_{{11}}} + {x_7} + {x_8} + {x_9} \geqslant {q_4}\;; $$
    $$ {x_{{11}}} + {x_{{11}}} + {x_7} + {x_8} + {x_{{10}}} \geqslant {q_5}\;; $$
    $$ {x_{{11}}} + {x_{{11}}} + {x_7} + {x_{{10}}} + {x_9} \geqslant {q_7}\;; $$
    $$ {x_{{11}}} + {x_{{11}}} + {x_8} + {x_{{10}}} + {x_9} \geqslant {q_8}\;; $$
    $$ {x_{{11}}} + {x_{{10}}} \geqslant {q_4}\;; $$
    $$ {x_{{11}}} + {x_9} \geqslant {q_5}\;; $$
    $$ {x_{{11}}} + {x_8} \geqslant {q_7}\;; $$
    $$ {x_{{11}}} + {x_7} \geqslant {q_8}\;; $$
    $$ {x_{{11}}} + {x_{{10}}} \geqslant {q_4}\;; $$
    $$ {x_{{11}}} + {x_9} \geqslant {q_5}\;; $$
    $$ {x_{{11}}} + {x_8} \geqslant {q_7}\;; $$
    $$ {x_{{11}}} + {x_7} \geqslant {q_8}\;; $$
    $$ {x_7} + {x_{{10}}} \geqslant {x_4}\;; $$
    $$ {x_7} + {x_8} \geqslant {x_6}\;; $$
    $$ {x_7} + {x_9} \geqslant {x_5}\;; $$
    $$ {x_{{10}}} + {x_7} \geqslant {x_4}\;; $$
    $$ {x_8} + {x_9} \geqslant {x_3}\;; $$
    $$ {x_8} + {x_{{10}}} \geqslant {x_2}\;; $$
    $$ {x_9} + {x_{{10}}} \geqslant {x_1}\;; $$

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Rahim, M.A., Khalid, H.M. & Akram, M. Sensor location optimization for fault diagnosis with a comparison to linear programming approaches. Int J Adv Manuf Technol 65, 1055–1065 (2013).

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