Order and Type of System

 

Order and Type of System

In control systems, order and type are fundamental classifications that describe system dynamics and steady-state behavior.

1. Order of a System

• Definition: The order of a system is the highest power of $s$ in the denominator of its transfer function.

• General Transfer Function:

$$G(s)=\frac{N(s)}{D(s)}=\frac{b_m{s}^m+\cdots +b_{1}s+b_0}{a_n{s}^n+\cdots +a_{1}s+a_0}$$

where $n$ = order of the system.

• Examples:

  • First-order system:

$$G(s)=\frac{K}{\tau s+1}$$

Example: RC circuit.

• Second-order system:

$$G(s)=\frac{{\omega }_n^2}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}$$

Example: Mass-spring-damper system.

2. Type of a System

• Definition: The type of a system is determined by the number of poles at the origin (i.e., the number of pure integrators in the open-loop transfer function).

• General Transfer Function:

$$G(s)H(s)=\frac{K}{s^N}\cdot \frac{N(s)}{D(s)}$$

where $N$ = system type.

• Type 0 system: No pole at origin. Example:

• Type 1 system: One pole at origin.

• Example:

$$G(s)=\frac{K}{s(s+2)}$$

• Type 2 system: Two poles at origin. Example:

$$G(s)=\frac{K}{s^2(s+2)}$$