Final answer:
The ask is to determine the system type, steady-state error constants Kp, Kv, and Ka, and calculate the steady-state errors for step, ramp, and parabolic inputs of a unity feedback system with a given transfer function.
Step-by-step explanation:
The student is asked to evaluate system type, steady-state error constants (Kp, Kv, and Ka), and determine the steady-state errors for standard step, ramp, and parabolic inputs using the forward transfer function G(s) = 1000/(s(s + 8)(s + 7)(s + 9)) of a unity feedback system.
The system type is determined by the number of poles at the origin of the transfer function (Type 0, 1, or 2, etc.), which directly affects the steady-state errors for different types of inputs. In this case, there is one pole at the origin (s=0), which indicates a Type 1 system. The steady-state error constants are obtained by the limits as s approaches 0 of s*G(s) for Kp, s^2*G(s) for Kv, and s^3*G(s) for Ka. These constants are used in the formulas for steady-state errors for step (1/Kp), ramp (1/Kv), and parabolic (1/Ka) inputs respectively.
Steady-State Error Calculation:
- Position Error Constant, Kp = lim s->0 [s*G(s)]
- Velocity Error Constant, Kv = lim s->0 [s^2*G(s)]
- Acceleration Error Constant, Ka = lim s->0 [s^3*G(s)]
Once the constants are calculated, the corresponding steady-state errors for a step input (Ess_step), a ramp input (Ess_ramp), and a parabolic input (Ess_parabolic) can be evaluated using the following formulas:
- Ess_step = 1/Kp
- Ess_ramp = 1/Kv
- Ess_parabolic = 1/Ka