Abstract
This chapter covers basic concepts related to feedback and its effect on the behaviour and stability of closed-loop systems. Four interactive tools are used to understand the basic concepts: Root locus, Nyquist stability criterion, stability margins, and influence of time delay on closed-loop stability.
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Notes
- 1.
Often unit feedback is used (\(H(s)=1\)), because the sensor dynamics are much faster than the process dynamics and can be neglected.
- 2.
\(F(s)=1\) if there is no need for reference filtering (pure error feedback).
- 3.
The controller always exhibits a certain degree of uncertainty, due to precision errors in the control algorithms and hardware (related to actuators, amplifiers and controllers). Moreover, this realization of perfect control could lead to very large control signals that would not be realizable by the actuator.
- 4.
If \(H(s) \ne 1\) the root locus is built with \(L(s)=C(s)G(s)H(s)\).
- 5.
There are variants of the method for systems with time delays that are not treated in this text. Some of them make use of polynomial approximations of the time delay.
- 6.
Any rational fraction has the same number of poles as zeros if one takes into account that it can have poles or zeros at infinity.
- 7.
Recall that the angles of the segments joining the open-loop poles and zeros with a test point s are measured counterclockwise from the positive real axis.
- 8.
The location of the closed-loop poles can be computed through a polynomial roots numerical solver for different values of K or using interactive tools like the one explained in this card. The advent of interactive control system design software tools, which easily draw the root locus, allows teachers to focus on explaining how the introduction of new poles/zeros in the open-loop transfer function affects the root locus, rather than on explaining its construction rules.
- 9.
If the transfer function considered is not causal, there will be poles at infinity.
- 10.
This rule (and others) does not apply when \(K<0\), positive feedback and cases of NMP systems. Some comments are included at the end of the section.
- 11.
When more than one parameter changes, the locus of the roots is called the root contour [5].
- 12.
In such cases, it is recommended to use another relative stability index called the stability margin [6], which is defined as the minimum distance to the critical point \((-1, 0)\) from any point on the frequency response curve.
- 13.
If \(PM>0\) then \(t_d>0\) and are both realizable. If \(PM<0\) then \(t_d<0\).
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Guzmán, J.L., Costa-Castelló, R., Berenguel, M., Dormido, S. (2023). Closed-Loop Systems and Stability. In: Automatic Control with Interactive Tools. Springer, Cham. https://doi.org/10.1007/978-3-031-09920-5_7
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