# What is the type number of a discrete time system given $H(z)$?

Given a continuous time impulse response $$h(t)$$, if I take the Laplace transform and count the no. of poles at origin, that gives the type number of the system. For e.g., $$H(s) = \frac{2}{s(s+2)}$$ Here, there is one pole at origin and hence it is type 1 system. This also implies that the number of integrators in the system is 1. I'm having difficulty in determining the system type when it comes to discrete time systems. For e.g., given the $$Z$$ transform of impulse response, $$H(z) = \frac1z+\frac1{z^6}$$ how to deduce the system type? Is it 6 because there are six poles at origin? Also how does it relate to no. of integrators in the system given that it is discrete time. Does it correspond to no. of accumulators in any way? I came across this while searching but it talks about the order of the input polynomial as the type of the system which makes the type number independent of the impulse response. Is this correct?

## 1 Answer

I couldn't find a definition of system type for discrete-time control systems, but in analogy with continuous-time systems you should count the number of poles at $$z=1$$, not at $$z=0$$, because in discrete time, $$z=1$$ is the DC point, just like $$s=0$$ in continuous time.

A discrete-time integrator (an accumulator), has a pole at $$z=1$$. Its transfer function is given by

$$H(z)=\frac{z}{z-1}\tag{1}$$

• Thanks for the response! I do not understand what you mean by DC point here. In s-plane, every point on real axis is dc point because omega is zero on this line. Similarly, in z-plane, every point on positive real axis should be a dc point. Could you please clarify this? – Navin Apr 29 '19 at 12:47
• @Navin: What people usually mean by DC is the point $s=0$ and $z=1$. Anyway, what is important is that these two points correspond to each other, unlike the points $s=0$ and $z=0$. – Matt L. Apr 29 '19 at 17:41
• Okay, I get it. Thanks for the help!! – Navin Apr 30 '19 at 5:07