Nyquist stability criterion

The Nyquist stability criterion , also called Strecker-Nyquist criterion , according to 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 , e.g. B. a control loop . Examples of control loops in everyday life are the cruise control in the car or the temperature control in a radiator.

Basics

The Nyquist locus for .

{\ displaystyle G (s) = {\ frac {1} {s ^ {2} + s + 1}}}

The system is called BIBO-stable (bounded input, bounded output) if it reacts to limited input variables with limited output variables. An unstable system, on the other hand, can "get out of hand" even with minor input interference. A stick on the fingertip is e.g. B. an unstable system which is stabilized by balancing.

Mathematically, the properties of the systems in control engineering are described with a transfer function : output Y equals transfer function G times input W, formally . ${\ displaystyle Y = G \ cdot W}$

Because the arithmetic operations are thereby easier, F, G and Y are not a function of time, rather than from the (complex) frequency dependent Laplace indicated: . is a complex value, which is related to the frequency via the formula . ${\ displaystyle W = W (s), \, G = G (s), \, Y = Y (s)}$ ${\ displaystyle s}$ ${\ displaystyle s = \ sigma + \ omega \ cdot i}$

Four transfer functions can be identified in a typical control system:

The transfer function of the controlled system (stick to finger, car with gas pedal, refrigerator with motor) is mentioned . ${\ displaystyle G_ {S} (s)}$
The transfer function of the controller (balancing person, cruise control, refrigerator electronics) is mentioned . ${\ displaystyle G_ {R} (s)}$
The multiplication of 1 and 2 is transfer function of the open called control loop . ${\ displaystyle G_ {R} (s) \ cdot G_ {S} (s) = G_ {0} (s)}$
1 and 2 together form the transfer function of the control system as a whole ( closed control loop). For our examples, these are the stick on the fingertip of a balancing person, a car with cruise control switched on or a functioning refrigerator. ${\ displaystyle G (s)}$ ${\ displaystyle G _ {\ text {regulated}} = {\ frac {G_ {0} (s)} {1 + G_ {0} (s)}}}$

The transfer functions are typically fractions of polynomials . Such polynomial fractions have a pole wherever the denominator polynomial has a zero . The value of G tends towards infinity there.

If the transfer functions are known, the Nyquist criterion can say whether a control system is stable or not. There are two cases. Both can only be used if the transfer function tends towards 0 at very high frequencies, i.e. the degree of the denominator polynomial is greater than that of the numerator polynomial. ${\ displaystyle G_ {0} (i \ omega)}$

Special Nyquist criterion / "left-hand rule"

Special Nyquist criterion
red: unstable
green: stable with amplitude and phase reserve

If all complex poles of and have a real part smaller than 0 (with the exception of a maximum of 2 poles in the origin), the special Nyquist criterion says that the entire (closed) control system is asymptotically stable if (i.e. only one subsystem) for from 0 to does not revolve around point −1 in the complex plane . Such a representation is called a locus . ${\ displaystyle G_ {R} (s)}$ ${\ displaystyle G_ {S} (s)}$ ${\ displaystyle G_ {0} (i \ omega)}$ ${\ displaystyle \ omega}$ ${\ displaystyle \ infty}$

The point −1 is therefore also called the Nyquist point or critical point.

For simpler locus curves, one can alternatively say that the curve must leave point −1 on the left so that the closed circle is stable. This is necessary because the point −1 on the real axis of the complex plane corresponds to a phase rotation of 180 °. A feedback signal that is intended to act as a negative feedback basically has a phase shift of 180 ° to the input signal of a system. If a phase shift of a further 180 ° occurs due to further phase rotation in the course of the constant increase in frequency, then the system will definitely oscillate if the feedback signal is greater than 1. It is then to the left of point −1 in this locus on the real axis.

To adjust the controller, two parameters must be observed. On the one hand, the amplitude reserve (or amplitude margin ), which states by which factor the controlled system may be amplified in order to remain stable, and on the other hand, the phase reserve (or phase margin ) (important for systems with dead time ). The phase reserve indicates the angle by which the phase position of the feedback signal can be shifted even further until positive feedback (a total of 360 ° phase rotation) occurs in the system. The phase reserve is therefore the angle between the straight line through the origin through the point on the locus, which has the distance 1 to the origin (construction by intersection with unit circle), and the negative real axis.

The Bode diagram , which is easier to implement, contains the same information as the locus curve, except that the amplitude response and phase response are shown separately for the frequency response. There, too, the amplitude and phase edge are the most important results of the observation. ${\ displaystyle G_ {o} (i \ omega)}$

Transition to the general Nyquist criterion

The following example has poles in origin. It is an integral term, followed by two proportional-integral terms (I-term and double PI-term).

{\ displaystyle G_ {o} (i \ omega) = {\ frac {\ omega _ {As}} {i \ omega}} \ cdot \ left (1 + {\ frac {\ omega _ {PI}} {i \ omega}} \ right) ^ {2}}

A closed system that has such a function as the product of all transfer functions that lie in the open loop is considered stable, and meaningful amplitude and phase margins can be set. ${\ displaystyle \ omega _ {As}> \ omega _ {PI}}$

Special feature of the example: There is a frequency at which the phase is rotated by 180 ° (a total of 360 ° phase rotation) at a point with a very high gain. Although this looks like positive feedback, the behavior of the closed system does not show the slightest defect. Knowledge of this paradox is part of basic training. Quote: "The interpretation of the simplified Nyquist criterion, which is common in the literature, that harmonic signals fed in at the interface may not generate larger signals of the same phase position at the interface if the closed system is to be stable, is misleading and incorrect." As a mathematical proof of stability, the technical control literature mentioned below provides information on the looping of the locus around the critical point −1. This is a subject of function theory . There, when calculating residuals , the closing conditions of the locus curves at infinity also play a role.

The Bode diagram is sufficient for practice. Bode diagrams can be conveniently displayed with a mathematics tool, for example also via the spreadsheet of an office package, if the commands available there are used for complex calculations. Bode diagrams are also easy to generate for transfer functions with a closed loop.

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 prerequisites for application are weaker, it only has to apply that no poles of the system and of the controller lie on the imaginary axis . ${\ displaystyle \ Re = 0}$

In contrast to the special Nyquist criterion, the course of in dependence on must also be recorded for negative omega. ${\ displaystyle G_ {0} (i \ omega)}$ ${\ displaystyle i \ omega}$

Now the following terms can be introduced:

${\ displaystyle P}$ is the number of unstable poles of the open loop . Unstable poles are those with a positive real part. ${\ displaystyle G_ {0}}$

${\ displaystyle N}$ is the number of unstable poles of the entire control system . ${\ displaystyle G}$

${\ displaystyle U}$ is the number of revolutions of the open-loop frequency response curve around the Nyquist point. One drives in the positive ω-direction and counts positive revolutions in the counterclockwise direction, negative ones in the clockwise direction. ${\ displaystyle G_ {0} (i \ omega)}$

The general Nyquist criterion states, first, that applies in every case . ${\ displaystyle N = PU}$

Second, the control system is asymptotically stable if it holds, otherwise it is unstable. ${\ displaystyle P = U}$

Second form

Another known form of the general Nyquist criterion is even more extensively useful. Both unstable poles and those on the imaginary axis are allowed.

${\ displaystyle r_ {k}}$ is the number of unstable poles of the open loop . Unstable poles are those with a positive real part. ${\ displaystyle G_ {0}}$

${\ displaystyle i_ {k}}$ is the number of poles of the open loop on the imaginary axis. ${\ displaystyle G_ {0}}$

${\ displaystyle \ Delta \ varphi}$ is the total angle swept by the frequency response curve of the open-loop control around the Nyquist point. One drives in the positive ω-direction and counts angles in the counterclockwise direction as positive, those in the clockwise direction as negative. Only the course over positive frequencies is used. ${\ displaystyle G_ {0} (i \ omega)}$

If the relationship is fulfilled, the rule system is stable, otherwise it is unstable. ${\ displaystyle \ Delta \ varphi = i_ {k} \ cdot \ pi / 2 + r_ {k} \ cdot \ pi}$

Nyquist point

The term Nyquist point is also occasionally used in the literature for the Nyquist frequency , which causes some confusion.

Nyquist criterion for multivariable systems

The Nyquist criterion can also be used for multi-variable systems. The locus of the open section must be replaced by the curve of the determinant of and the critical point −1 by the point 0. ${\ displaystyle 1 + G_ {0}}$

Other criteria

The Routh-Hurwitz criterion is an alternative stability criterion in control engineering.

literature

Geering: Control engineering, Springer 2001, ISBN 3-540-41264-6

Web links

Java applet for the Nyquist criterion

Individual evidence

↑ Helmut Schwarz: Frequency response and root locus method . Bibliographisches Institut AG, Vol. 193 / 193a, 1968, pp. 65 .

[1] Helmut Schwarz: Frequency response and root locus method . Bibliographisches Institut AG, Vol. 193 / 193a, 1968, pp. 65 .