By Patrick Anderson
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Extra resources for Control Systems - Classical Controls
These are all different names for the same mathematical space and they all may be used interchangably in this book and in other texts on the subject. The Transform can only be applied under the following conditions: 1. 2. 3. 4. The system or signal in question is analog. The system or signal in question is Linear. The system or signal in question is Time-Invariant. The system or signal in question is causal. The transform is defined as such: [Laplace Transform] Laplace transform results have been tabulated extensively.
We define the position error constant as follows: [Position Error Constant] Where G(s) is the transfer function of our system. Velocity Error The velocity error is the amount of steady state error when the system is stimulated with a ramp input. We define the velocity error constant as such: [Velocity Error Constant] Acceleration Error The acceleration error is the amount of steady-state error when the system is stimulated with a parabolic input. We define the acceleration error constant to be: [Acceleration Error Constant] Now, this table will show briefly the relationship between the system type, the kind of input (step, ramp, parabolic), and the steady state error of the system: Unit System Input Type, M Au(t) Ar(t) Ap(t) 0 1 ess = 0 2 ess = 0 ess = 0 >2 ess = 0 ess = 0 ess = 0 Z-Domain Type Likewise, we can show that the system order can be found from the following generalized transfer function in the Z domain: Where the constant M is the order of the digital system.
Example: Second-Order System If we have a given equation in the S-domain, we can expand it into several smaller fractions as such: This looks impossible, because we have a single equation with 3 unknowns (s, A, B), but in reality s can take any arbitrary value, and we can "plug in" values for s to solve for A and B, without needing other equations. For instance, in the above equation, we can multiply through by the denominator, and cancel terms: Now, when we set s → -2, the A term disappears, and we are left with B → 3.
Control Systems - Classical Controls by Patrick Anderson