Diagnosability analysis of hybrid systems cast in a discrete-event framework

Abstract

This paper addresses the problem of assessing the diagnosability of hybrid systems modeled by a hybrid automaton coupling methods from the continuous and the discrete event model-based diagnosis fields. The discrete states of the hybrid automaton represent the modes of operation of the system for which the continuous dynamics are specified. The diagnosability of the continuously-valued part of the model is first analyzed and the new concept of mode signature is shown to characterize mode diagnosability from continuous measurements. Continuous dynamics are then abstracted by defining a set of signature-events associated to mode signature changes, preserving this way mode diagnosability. The behavior of the abstract hybrid system is then modeled by a prefix-closed language over the original event alphabet enriched by these additional events. Based on this language, diagnosability analysis of the hybrid system is cast into a discrete-event framework and hybrid diagnosability conditions are provided. A case study based on the Attitude and Orbit Control System of a spacecraft illustrates the method.

Appendix

This appendix develops the parity space residual generation method for a mode q _i with a discrete time¹⁰ linear state-space model obtained from (Eq. 4) of the form:

$$ \label{eq:space:state:approach} \left \{ \begin{array}{rl} x_i(n+1) & = A_i x_i (n)+ B_iu(n) + E_{x_i} \epsilon(n) \\ y(n) & =C_ix_i(n)+D_iu(n) + E_{y_i} \epsilon(n) \end{array} \right. $$

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x _i (n), u(n), y(n) and ε(n) are the state, the input, the output and the noise vectors of dimensions $n_{x_i}$, n _u , n _y and n _ε respectively, considered at the sampling time n. A _i , B _i , C _i and D _i are constant dynamic, input, output and direct transmission matrices of appropriate dimensions. $E_{x_i}$ and $E_{y_i}$ are constant matrices of appropriate dimensions that capture the influence of the noise on state and output evolution, respectively.

Following the parity space approach, ARRs can be obtained by relating the inputs with the outputs over a time-window of p _i  + 1 samples. Selecting p _i appropriately (typically $p_i \leq n_{x_i}$) allows us to eliminate any dependency upon the system state x _i . This procedure can be summarized as follows.

Given a vector V, let us denote by V ^p the vector obtained by the concatenation of the vector values at every sampling instant (n − p + k), 0 ≤ k ≤ p, for a given p. Then V ^p(n) = [V ^T(n − p), ..., V ^T(n − p + k), ..., V ^T(n) ]^T. By iterating state-evolution and observation equations (Eq. 8), we obtain:

$$ \label{eq:parity:space:state:elimination} y^{p_i}(n) = O^{p_i}_ix_i(n-p_i)+L^{p_i}_i(A_i , B_i, C_i, D_i)u^{p_i} + L^{p_i}_i(A_i , E_{x_i}, C_i, E_{y_i})\epsilon^{p_i}(n) $$

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with:

$$ L^{p_i}_i(A_i , N, C_i, Q) = \begin{pmatrix} Q & 0 & ... & 0 \\ C_iN & Q & ... & ...\\ ... & ... & ... & 0 \\ C_iA_i^{(p_i-1)}N & ... & C_iN & Q \end{pmatrix} \;\text{and}\; N \in \{B_i, E_{x_i}\}, Q \in \{ D_i, E_{y_i}\} $$

$$ O^{p_i}_i = \begin{pmatrix} C_i \\ C_iA_i\\ ... \\ C_iA_i^{p_i} \end{pmatrix} $$

The state x _i (n − p _i ) in Eq. 9 can be eliminated through left-hand multiplication by an operator $\Omega^{p_i}_i$. We obtain ARRs that can be decomposed into a computational and an evaluation form denoted $\rho^{p_i}_{c_i}$ and $\rho^{p_i}_{e_i}$, respectively:

$$ \label{eq:calcul:form} \rho^{p_i}_{c_i}(n) = \Omega^{p_i}_i y^{p_i}(n) - \Omega^{p_i}_i L^{p_i}_i(A_i, B_i, C_i, D_i)u^{p_i}(n) $$

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$$ \rho^{p_i}_{e_i}(n) = \Omega^{p_i}_i L^{p_i}(A_i, E_{x_i}, C_i, E_{y_i})\epsilon^{p_i}(n) $$

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The Boolean-residual vector of mode q _i is denoted $R_{q_i}=[r_i^1, r_i^2, ..., r_i^{n_i}]^T$ and is obtained by checking whether $\rho^{p_i}_{c_i}(n) = \rho^{p_i}_{e_i}(n)$. Two cases are hence distinguished.

• Noise-free hypothesis: $\rho^{p_i}_{e_i}=0, \forall n \in \mathbb{N}$

  A threshold vector is defined as $\alpha_i=[\alpha_i^1, ..., \alpha_i^{n_i}]^T$. The threshold values take into account the computation precision and the relative order of magnitude of the different variables.

  $$ r_i^j = \left \{ \begin{array}{rl} 0 & \mbox{ if } \rho^{p_i}_{c_i}(n) \leq \alpha_i^j \\ 1 & \mbox{otherwise} \end{array} \right. $$

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• White–Gaussian–Noise hypothesis

  We have ε(n) ~N(0, σ ²), hence $\epsilon(n, n-p_i) \sim N(0, diag_{p_i+1}(\sigma^2))$, where σ ² denotes the variance and $diag_{p_i+1}(\sigma^2)$ denotes the diagonal matrix of dimension p _i  + 1 in which the diagonal values are equal to σ ². Consequently the probability density function of the evaluation form has a normal distribution:

  $$ \rho^{p_i}_{e_i} (n) \sim N(0, \Omega^{p_i}_iL^{p_i}(A_i, E_{x_i}, C_i, E_{y_i})diag(\sigma^2)(L^{p_i}(A_i, E_{x_i}, C_i, E_{y_i}))^T(\Omega^{p_i}_i)^T) $$
  $$ r_i^j = \left \{ \begin{array}{rl} 0 & \mbox{ if } \rho^{p_i}_{c_i}(n) \sim \rho^{p_i}_{e_{ij}} \\ 1 & \mbox{otherwise} \end{array} \right. $$

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  where $\rho^{p_i}_{e_{ij}}$ denotes the jth element of $\rho^{p_i}_{e_i}$.