Steady State Error (page 4)
Besides system type, the input function type is needed to determine steady state error. The three input types covered in Table 7.2 are step (u(t)), ramp (t*u(t)), and parabola (0.5*t^{2}*u(t)). The system type and the input function type are used in Table 7.2 to get the proper static error constant. There are three of these: K_{p} (position error constant), K_{v} (velocity error constant), and K_{a} (acceleration error constant). Once you have the proper static error constant, you can find e_{ss}.
The static error constants are found from the following formulae:
Now use Table 7.2 to find e_{ss}.
Table 7.2
Type 0 | Type 1 | Type 2 | |||||
Input |
e_{ss} | Static Error Constant |
e_{ss} | Static Error Constant |
e_{ss} | Static Error Constant |
e_{ss} |
u(t) | K_{p} = Constant | K_{p} = Infinity | 0 | K_{p} = Infinity | 0 | ||
t*u(t) | K_{v} = 0 | Infinity |
K_{v} = Constant |
K_{v} = Infinity | 0 | ||
0.5*t^{2}*u(t) | K_{a} = 0 | Infinity | K_{a} = 0 | Infinity | K_{a} = Constant |
Note that e_{ss} has one of three values: 0, a constant, infinity. Notice how these values are distributed in the table. Also note the aberration in the formula for e_{ss} using the position error constant. e_{ss} is not equal to 1/K_{p}.