Chapter Four – Lecture 8

Steady-State Errors.

Consider the system shown in Figure .

The closed-loop transfer function is :

The transfer function between the error signal e(t) and

the input signal r(t) is

where the error e(t) is the difference between the input signal and the output signal. The final-value theorem provides a convenient way to find the steady-state performance of a stable system. Since E(s) is

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The term velocity error is used here to express the steady-state error for a ramp input. The dimension of the velocity error is the same as the system error. .That is, velocity error is not an error in velocity, but it is an error in position due to a ramp input.

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Lecture 9

Stability

Routh’s stability criterion

The most important problem in linear control systems concerns stability. That is, under what conditions will a system become unstable? If it is unstable, how should we stabilize the system?

We can state that a control system is stable if and only if all closed-loop poles lie in the

left-half s plane.

Most linear closed-loop systems have closed-loop transfer functions of the form:

where the a’s and b’s are constants and m n.A simple criterion, known as Routh’s stability criterion,enables us to determine the number of closed-loop poles that lie in the right-half s plane without having to factor the denominator polynomial.

If all coefficients are positive, arrange the coefficients of the polynomial in rows and columns according to the following pattern:

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Example:

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