12.2: Roots of Stability – Nyquist Criterion

We consider the SISO feedback system with reference trajectory \(r(s)\) and plant output \(y(s)\), as given previously. The tracking error signal is defined as \(e(s) = r(s)-y(s)\), thus forming the negative feedback loop. The sensitivity function is written as

\[ S(s) \, = \, \frac{e(s)}{r(s)} \, = \, \frac{1}{1 + P(s)C(s)}, \]

where \(P(s)\) represents the plant transfer function, and \(C(s)\) the compensator. The closed-loop characteristic equation, whose roots are the poles of the closed-loop system, is \(1+P(s)C(s) = 0\), equivalent to \(\underline{P}(s) \underline{C}(s) + \bar{P}(s) \bar{C}(s) = 0\), where the underline and overline denote the denominator and numerator, respectively. The Nyquist criterion allows us to assess the stability properties of a feedback system based on \(P(s)C(s)\) only. This method for design involves plotting the complex loci of \(P(s)C(s)\) for the range \(s = j \omega\), \(\omega = [-\infty, \, \infty]\). Remarkably, there is no explicit calculation of the closed-loop poles, and in this sense the design approach is quite different from the root-locus method (see Ogata, also the rlocus() command in MATLAB).

Mapping Theorem

To give some understanding of the Nyquist plot, we begin by imposing a reasonable assumption from the outset: The number of poles in \(P(s)C(s)\) exceeds the number of zeros. It is a reasonable constraint because otherwise the loop transfer function could pass signals with infinitely high frequency. In the case of a PID controller (two zeros) and a second-order zero-less plant, this constraint can be easily met by adding a high-frequency rolloff to the compensator, the equivalent of low-pass filtering the error signal.

Now let \(F(s) = 1 + P(s)C(s)\) (the denominator of \(S(s)\)). The heart of the Nyquist analysis is the mapping theorem, which answers the following question: How do paths in the complex \(s\)-plane map into paths in the complex \(F\)-plane? We limit ourselves to closed, clockwise (CW) paths in the \(s\)-plane, and the powerful result of the mapping theorem is:

Result of the mapping theorem:

Every zero of \(F(s)\) that is enclosed by a path in the \(s\)-plane generates exactly one CW encirclement of the origin in the \(F(s)\)-plane. Conversely, every pole of \(F(s)\) that is enclosed by a path in the \(s\)-plane generates exactly one CCW encirclement of the origin in the \(F(s)\)-plane. Since CW and CCW encirclements of the origin may cancel, the relation is often written \(Z-P = CW.\)

Thus, it will be possible to relate poles and zeros in the \(F(s)\)-plane to encirclements of the origin in the \(s\)-plane. Since we get to design the path in the \(s\)-plane, the trick is to enclose all unstable poles, i.e., the path encloses the entire right-half plane, moving up the imaginary axis, and then proceeding to the right at an arbitrarily large radius, back to the negative imaginary axis.

Since the zeros of \(F(s)\) are in fact the poles of the closed-loop transfer function, e.g., \(S(s)\), stability requires that there are no zeros of \(F(s)\) in the right-half \(s\)-plane. This leads to a slightly shorter form of the above relation:

\[ P \, = \, CCW. \]

In words, stability requires that the number of unstable poles in \(F(s)\) is equal to the number of CCW encirclements of the origin, as \(s\) sweeps around the entire right-half \(s\)-plane.

Nyquist Criterion

The Nyquist criterion now follows from one translation. Namely, encirclements of the origin by \(F(s)\) are equivalent to encirclements of the point \((-1 + 0j)\) by \(F(s)-1\), or \(P(s)C(s)\). Then the stability criterion can be cast in terms of the unstable poles of \(P(s)C(s)\), instead of those of \(F(s)\):

\[ P \, = \, CCW \,\, \longleftrightarrow \,\, \text{closed-loop stability} \]

This is in fact the complete Nyquist criterion for stability: It is a necessary and sufficient condition that the number of unstable poles in the loop transfer function \(P(s)C(s)\) must be matched by an equal number of CCW encirclements of the critical point \((-1 + 0j)\).

There are several details to keep in mind when making Nyquist plots:

• From the formula, if neither the plant nor the controller have unstable poles, then the loci of \(P(s)C(s)\) must not encircle the critical point at all, for closed-loop stability. If the plant and the controller comprise \(q\) unstable poles, then the loci of \(P(s)C(s)\) must encircle the critical point \(q\) times in the CCW direction.
• Because the path taken in the \(s\)-plane includes negative frequencies (i.e., the negative imaginary axis), the loci of \(P(s)C(s)\) occur as complex conjugates – the plot is symmetric about the real axis.
• The requirement that the number of poles in \(P(s)C(s)\) exceeds the number of zeros means that at high frequencies, \(P(s)C(s)\) always decays such that the loci go to the origin.
• For the multivariable (MIMO) case, the procedure of looking at individual Nyquist plots for each element of a transfer matrix is unreliable and outdated. Referring to the multivariable definition of \(S(s)\), we should count the encirclements for the function \([det(I + P(s)C(s)) - 1]\) instead of \(P(s)C(s)\). The use of gain and phase margin in design is similar to the SISO case.

Robustness on the Nyquist Plot

The question of robustness in the presence of modelling errors is central to control system design. There are two natural measures of robustness for the Nyquist plot, each having a very clear graphical representation. The loci need to stay away from the critical point \(P(s)C(s) = -1 = 1 \angle 180°\), and how close the loci come to it can be expressed in terms of magnitude and angle:

• When the angle of \(P(s)C(s)\) is \(-180°\), the magnitude \(|P(s)C(s)|\) should not be near one.
• When the magnitude \(|P(s)C(s)| = 1\), its angle should not be \(-180°\).

These notions lead to definition of the gain margin \(k_g\) and phase margin \(\gamma\) for a design. As the figure shows, the definition of \(k_g\) is different for stable and unstable \(P(s)C(s)\). Rules of thumb are as follows. For a stable plant, we desire \(k_g \geq 2\) and \(\gamma \geq 30°\); for an unstable plant, \(k_g \leq 0.5\) and \(\gamma \geq 30°\). As defined, these conditions will maintain stability even if the gain is increased by a factor of two for the stable open-loop system, or decreased by a factor of two for the unstable OL system. In both cases, the phase angle can be in error by thirty degrees without losing stability. Note that the system behavior in the closed-loop, while technically stable through these perturbations, might be very poor from the performance point of view. The following two sections outline how to manage robustness and performance simultaneously using the Nyquist plot.