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How high loop gains reduce steady-state errors in feedback systems. It covers the concept of steady-state errors, the use of the final value theorem, and the definitions of gain, type number, position error constant, velocity error constant, and acceleration error constant. The document also includes a table of steady-state errors for different system types and inputs, and a discussion on pi control.
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Recall from the last lecture that high loop gains reduce the sensitivity to parameter variations and disturbance inputs. Today, we will see that they also reduce steady-state errors in feedback systems.
Consider the unity feedback system: How do we determine the steady-state error, ess? 2
Use the final value theorem: ess = = (1) For G(s) the following general form is assumed: G(s) = (2)
(a) Gain (b) Type number 3
Using equation (3), we can construct a handy table: e.g. type 2 system, unit ramp input 5
Type 0, step input: If c is constant, then ess must be This means: 6
Type 1, step input:
This section includes some extra reading for you. This should give you some hints as to how the concepts we have seen so far are important for controller design. We will revisit this example in more detail in later lectures. So far we have seen that integrators seem like good things to put in our controller... (a) Suppose you have a plant G(s) = K s(τ s+1) and^ you^ add^ an^ integrator^ to make it type 2. First draw the block diagram: The closed-loop transfer function will be: K C (^) s^2 (τ s+1) K = = R (^) 1 + (^) s (^2) (τ sK+1)^ τ s^3 + s^2 + K and the characteristic equation of the system will be τ s^3 + s^2 + K = 0 9
