Generation of Alternative Connections of Series-Connected Subsystems in a Redundant Equipment Complex

Abstract

An analytical approach is developed to integrate redundant equipment complexes (ECs) with linear stationary models. A general technique for managing the redundancy of the complex at the design stage is formulated in the case of ensuring the invariance of the given set of transfer functions. A formalism is obtained for solving the integration problem of a complex when it is configured as a sequence of related subsystems. The formalism includes both the conditions for the admissibility of a preselected configuration and the formula for the entire set of alternative relationships between components that ensure the fulfillment of the given objective function of the complex. An example of a simple navigation system consisting of measuring and indicator subsystems is given.

APPENDIX

Proof of the theorem. The solution to the problem is related to the resolution of the equation

$${{\Phi }_{{{\text{req}}}}}(z) = \left[ {\begin{array}{*{20}{c}} 0&{\beta ''} \end{array}} \right]\underbrace {\left[ {\begin{array}{*{20}{c}} {D'}&0 \\ 0&{D''} \end{array}} \right]}_D{{\underbrace {\left[ {\begin{array}{*{20}{c}} {zI' - A'}&0 \\ {B''C_{{{\text{in}}}}^{{''}}{{E}^{{21}}}(z)C_{{{\text{out}}}}^{'}D'}&{zI'' - A''} \end{array}} \right]}_{z{{I}_{n}} - A - B{{Q}^{{21}}}(z)D}}^{{ - 1}}}\underbrace {\left[ {\begin{array}{*{20}{c}} {G'} \\ {G''} \end{array}} \right]}_G\alpha ,$$

(A.1)

where the left side is determined by formula (3.5) with the nominal values of the corresponding matrices relative to the matrix $E{{}^{{21}}}(z)$. We use the lemma on the inverse of block matrices [15], according to which the following equality is justified:

$${{\left[ {\begin{array}{*{20}{c}} {zI' - A'}&0 \\ {B''{{Q}^{{21}}}(z)D'}&{zI'' - A''} \end{array}} \right]}^{{ - 1}}} = \left[ {\begin{array}{*{20}{c}} {{{{(zI' - A')}}^{{ - 1}}}}&0 \\ { - {{{(zI'' - A'')}}^{{ - 1}}}B''{{Q}^{{21}}}(z)D'{{{(zI' - A')}}^{{ - 1}}}}&{{{{(zI'' - A'')}}^{{ - 1}}}} \end{array}} \right].$$

As a result, by introducing the notation ${{\Phi }_{{{\text{req}}}}}(z) = [\begin{array}{*{20}{c}} 0&{\beta ''W_{{y{\text{nom}}}}^{{v}}(z)\alpha } \end{array}]$, we get the equation

$$\begin{gathered} \text{[}\begin{array}{*{20}{c}} 0&{\beta ''W_{{y{\text{nom}}}}^{{v}}(z)\alpha } \end{array}] \\ = \left[ {\begin{array}{*{20}{c}} 0&{\beta ''} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {D'}&0 \\ 0&{D''} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{{(zI' - A')}}^{{ - 1}}}}&0 \\ { - {{{(zI'' - A'')}}^{{ - 1}}}B''{{Q}^{{21}}}(z)D'{{{(zI' - A')}}^{{ - 1}}}}&{{{{(zI'' - A'')}}^{{ - 1}}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {G'} \\ {G''} \end{array}} \right]\alpha , \\ \end{gathered} $$

which after fulfilling the matrix products takes the form

$$\beta ''W_{{y{\text{nom}}}}^{v}(z)\alpha = \beta ''D''{{(zI'' - A'')}^{{ - 1}}}G''\alpha - \beta ''D''{{(zI'' - A'')}^{{ - 1}}}B''{{Q}^{{21}}}(z)D'{{(zI' - A')}^{{ - 1}}}G'\alpha ,$$

where, taking into account the notation introduced in (1.11), the following value is used:

$$W_{{y{\text{nom}}}}^{{v}}(z) = D''{{(zI'' - A'')}^{{ - 1}}}G'' - D''{{(zI'' - A'')}^{{ - 1}}}B''C_{{{\text{in}}}}^{{''}}{{E}^{{21}}}(z)C_{{{\text{out}}}}^{'}D'{{(zI' - A')}^{{ - 1}}}G.$$

(A.2)

Taking into account notation (4.1) and (4.2), we obtain the equation

$$\beta ''W''(z)C_{{{\text{in}}}}^{{''}}E_{{{\text{nom}}}}^{{21}}(z)C_{{{\text{out}}}}^{'}W'(z)\alpha = \beta ''(W''(z) - W_{{y{\text{nom}}}}^{\user1{v}}(z))\alpha ,$$

which  after  substitution (A.2) takes the form of a two-sided linear equation with respect to the matrix E²¹(z):

$$\beta ''W''(z)C_{{{\text{in}}}}^{{''}}{{E}^{{21}}}(z)C_{{{\text{out}}}}^{'}W'(z)\alpha = \beta ''W''(z)C_{{{\text{in}}}}^{{{\text{nom}''}}}E_{{{\text{nom}}}}^{{21}}(z)C_{{{\text{out}}}}^{{{\text{nom}'}}}W'(z)\alpha .$$

The solvability conditions for this equation appear in [13] as equalities (4.3) and (4.4), and the solution is determined by formulas (4.5) and (4.6). The theorem is proved.