# State space representation using a given nilppotent matrix

I want to build a question on state space using a given, nilpotent matrix, $$A$$.

$$q[n+1]=Aq[n]+Bx[n]$$ $$y[n]=Cq[n]+Dx[n]$$

Usually, I am using the canonical form so there is no problem extracting $$A,B,C,D$$ from the difference equation of the system. Given a nilpotent $$A$$, How do I choose reasonable $$B,C,D$$? can I simply go with the canonical form? Is there anything I should be aware of?

• There's not enough detail here. If A is nilpotent, then the system as a whole is deadbeat. That's a pretty specific behavior. So what's the specific system you're starting with? Or otherwise what's the further context of the question at hand? Commented Jan 21, 2021 at 15:08
• @TimWescott what do you mean by "deadbeat"? Do you mean that this is a redundant system? My goal is to give a HW question. This is not a system that is supposed to be functional in any way. I do am not starting with any system and I am not limited by the further context of the question. Commented Jan 25, 2021 at 6:29
• A deadbeat system is one that has a response that settles out in finite time. Commented Jan 25, 2021 at 15:52
• A nilpotent matrix can normally be reduced to a lower triangular matrix; which might speed up calcs and reduce the order to a usable one. Tim is right, an explicit example would be usefull. Commented Jan 26, 2021 at 22:20