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Could you please help me to understand what is a standard (multinomial) logistic link function between $X_{t}$ and the entries of $Q_{t}$ used in 'Modeling Portfolio Defaults using Hidden Markov Models with Covariates', which were written by Konrad Banachewicz and Aad van der Vaart, André Lucas. The entries of $Q_{t}$ are
$$q_{ij,t}=frac{expleft(Phi_{ij}^{'}X_{t}+eta_{ij}right)}{sumlimits_{j=1}^{N}expleft(Phi_{ij}^{'}X_{t}+eta_{ij}right)}.$$

I do not know why $q_{ij,t}$ is defined in this way and what are the variables $Phi_{ij}$ and $eta_{ij}$...

Also I wrote the program in C++, which should check if the procedure described in this paper is fine - Expectation-Maximization (EM). In program at the begining the observation sequence, hidden states sequence, transition matrix $Q_{t}$ together with the distribution of the initial state $P(W_{1} = i) = pi_{i}$ and matrix $B$ are generated by pseudorandom numbers generator. Then the initial parameters of the model are generated again and
and procedure changing parameters is performed until we get the convergence with the original parameters - here we using the observation sequence generated at the begining - it is fixed. I do know why the program does not find at the end
similar parameters to the original ones.... Can it be that the model is not convergent or I am doing something wrong?

Thank you for your help!