Nyquist stability criterion

The stability criterion of Nyquist, also Strecker Nyquist criterion, after Harry Nyquist and Felix Strecker, is a term from the field of control engineering and systems theory. The Nyquist criterion describes the stability of a system with feedback, such as a control circuit. Examples of control circuits in everyday life are the cruise control in the car or the temperature control in the refrigerator.

Basics

We call the system as BIBO stable (bounded input, bounded output) when it responds to limited input variables with limited output variables. An unstable system, however, can already " out of hand " at low input disturbances. A bar on the fingertip, for example, an unstable system, which is stabilized by balancing.

Mathematically describes the properties of the systems in control engineering with a transfer function: Y is output transfer function G times the input W, formal.

Because the arithmetic operations be easy, W, G and Y are not given as a function of time, rather than depending on the (complex) frequency of Laplace. is a complex value which is related to the frequency by the formula.

In a typical control system, one can make four transfer functions:

The transfer functions are typically fractions of polynomials. Such Polynombrüche have anywhere where the denominator has a zero, a pole. The value of G is pursuing its infinity.

The Nyquist criterion, and provided the transfer functions are known to tell if a control system is stable or not. We distinguish two cases. Both are applicable only if the transfer function at very high frequencies tends to 0, so the degree of Nennerpolynomes is greater than that of the Zählerpolynomes.

Special Nyquist / "left -hand rule "

Do all complex poles of and a real part less than 0 ( with the exception of Poland at the origin ), so says the special Nyquist criterion that the entire (closed ) control system is asymptotically stable if (so only one subsystem) for 0 to the complex plane, the point -1 is not spinning. Such a representation is called locus.

The point -1 is therefore also called Nyquist point or critical point.

For simpler loci one can say alternatively that the curve must be the point -1 are left so that the closed loop is stable. This is necessary because the point corresponds to -1 on the real axis of the complex plane of a phase rotation about 180 °. A feedback signal, which is to act as a counter-coupling principle, has a phase shift of 180 ° relative to the input signal of a system. Then passes through another stage in the course of rotation of the continuous increase in the frequency of a phase shift by a further 180 °, then the system oscillates with certainty, if the feedback signal is greater than 1. It is then left of the point -1 in this locus on the real axis.

To calibrate the controller are still considered two parameters. Firstly, the gain margin (or gain margin ), which states the factor by which the controlled system must be strengthened in order to be still stable, and on the other the phase margin (or phase margin ) (important for systems with dead time). The phase margin is at that angle, by which the phase of the feedback signal can be further shifted to the positive feedback ( a total of 360 ° phase rotation ) occurs in the system. The phase margin is thus the angle between the line through the origin through the point on the locus, which is at distance one to the origin (design by intersection with the unit circle ), and the negative real axis.

General Nyquist criterion

First form

The general Nyquist criterion is also applicable to cases where the requirement for the special Nyquist criterion is not met. The requirements for application are weaker, it must apply only that there are no poles of the system and the controller are on the imaginary axis.

In contrast to the special Nyquist criterion one must also record negative for Omega here the course of depending on.

Now you can introduce the following designations:

• Is the number of poles of the open-loop unstable. Unstable poles are those with positive real part.

• Is the number of unstable poles of the entire control system.

• Is the number of revolutions of the frequency response curve of the open loop to the Nyquist point. This one drives into positive ω - direction and counts rounds counterclockwise positive, such clockwise negative.

The general Nyquist criterion states first that applies in every case.

Second, the control system is asymptotically stable if and only if, otherwise it is unstable.

Second Form

Another known form of the general Nyquistkriteriums is more extensively used. It is both unstable and also those poles on the imaginary axis allowed.

• Is the number of poles of the open-loop unstable. Unstable poles are those with positive real part.

• Is the number of poles of the open-loop to the imaginary axis.

• Is the total of the course of the frequency response curve of the open loop around the Nyquist swept angle. This one drives into positive ω - direction and one angle counterclockwise positive, such clockwise negative. It is used only over the course of the positive frequencies.

If the relationship is satisfied, the control system is stable, otherwise it is unstable.

Nyquist

The term Nyquist is sometimes used in the literature for the Nyquist frequency, which some confusion.

Nyquist criterion for multivariable systems

The Nyquist criterion can also be used for multivariable systems. The locus of the open line to be replaced by the curve of the determinant of the critical point and through the point 0 -1

Other criteria

The Hurwitzkriterium and the Routh criterion, alternative stability criteria in control engineering.