state transition matrix for this system

Question

is there any concept of state transition matrix for this discrete system?

\begin{equation}N(k)x(k+1)=x(k)+B(k)u(k)\dots(1)\end{equation}

$N(k)$ are nilpotent matrices for $k=0,1,2\dots$

I know state transition matrix for the system $x(k+1)=A(k)x(k)+B(k)u(k)$

is given by $\phi(k,k_0)=A(k-1)\times \dots \times A(k_0)$

Thanks for helping.

$ ax(k+1)=x(k)+bu(k)$ where $a$ is an nilpotent matrix, I am given that its solution is $x(k)=-\sum_{i=1}^{q-1} a^ibu(k+i)$ where $q$ is the degree of nilpotency of $a$.

Answer 1

You can extend the definition of a discrete (and continues) state space model to

$$ \begin{array}{rl} E\,x[k+1] \!\!\!\!&= A\,x[k] + B\,u[k] \\ y[k] \!\!\!\!&= C\,x[k] + D\,u[k] \end{array} $$

which is also known as a descriptor state space model.

This will also allow you to represent improper transfer functions when $E$ is not invertible. This is the case when $E$ is a nilpotent matrix. So your equivalent transfer function and thus also your state space model is noncausal. So one can only define the state transition matrix $\Phi(k,\kappa)$ for $\kappa\geq k$ with

$$ \Phi(k,\kappa) = N(k)\,N(k+1)\,\cdots\,N(\kappa-1). $$