# What will be minimum number of states / state variables for these control systems in state-space model?

Consider the systems as LTI and SISO with transfer functions given below:

I have read that minimum number of states for an LTI SISO system required to represent in state space analysis for any transfer function is maximum number of power in denominator of transfer function.I am confused that can I violate this rule.For correct me for state-space diagram for first transfer function

I am getting 4 states but if I see degree of numerator is 15 then accordingly number of states should be minimum 15

Also please mention State matrix for each case.

• So you’re asking us to solve the four homework questions. – Chu Jul 26 at 11:15
• @Chu I want to ask you for favour what makes you think that this is a homework question. I mean this is one of comment I faced frequently earlier it'll be helpful to consider your suggestion in my next question.And If you say that I simply posted question as in objective form so if I would have posted my question in theoretical form like I have posted in some questions,I never got the answers satisfactorily so I asked to the point and will draw interference from answers or comment I get to refine my concept. – SUNITA GUPTA Jul 26 at 12:48
• @Chu and yes many of my questions were put on hold when I tried to ask questions giving theory so its better even if it's called homework and I get answer.... – SUNITA GUPTA Jul 26 at 12:50
• Generally, you just post the homework question without a substantial attempt at a solution. This is not a homework site. In this particular question you are asking for us to provide you with the state matrix for four separate cases; so, where are your attempts at the solutions ? – Chu Jul 26 at 13:15
• The violation you mentioned can be obtained when "pole-zero" cancellation occursbetween numerator and denominator. Please, see my related answer also involving degeneration:electronics.stackexchange.com/questions/394083/… – Dirceu Rodrigues Jr Jul 26 at 13:15