ANALYSIS OF SISO CONTROL LOOPS
5.7 Nominal Stability using Frequency Response
A classical and lasting tool that can be used to assess the stability of a feedback loop is Nyquist stability theory. In this approach, stability of the closed loop is predicted using the open loop frequency response of the system. This is achieved by plotting a polar diagram of the productGo(s)C(s) and then counting the number of encirclements of the (−1,0) point. We show how this works below.
We shall first consider an arbitrary transfer functionF(s) (not necessarily related to closed loop control).
Nyquist stability theory depends upon mappings between two complex planes:
• The independent variables
• The dependent variableF
The basic idea of Nyquist stability analysis is as follows:
Assume you have a closed oriented curveCsin s 2which encirclesZ zeros and P poles of the functionF(s). We assume that there are no poleson Cs .
If we move along the curve Cs in a defined direction, then the function F(s) mapsCsinto another oriented closed curve,CF in F .
We will show below that the number of times thatCF encircles the origin in the complex plane F , is determined by the difference betweenP andZ. IfGo(s)C(s)
2 s and F denote the s- and the F- planes respectively.
Section 5.7. Nominal Stability using Frequency Response 137
(s-c)
(s-c)
s s
c
a) b)
c
Cs Cs
Figure 5.4. Single zero function and Nyquist analysis
is modified, we can observe how the encirclement scenery changes (or how close it is to changing).
In the sequel, it will be helpful to recall that every clockwise (counterclockwise) encirclement of the origin by a variable in the complex plane implies that the angle of the variable has changed by−2π[rad] (2π[rad]).
We first analyze the case of a simple functionF(s) =s−c with clying inside the region enclosed byCs. This is illustrated in Figure 5.4, part a).
We see that as s moves clockwise along Cs, the angle of F(s) changes by
−2π[rad], i.e. the curve CF will enclose the origin in F once in the clockwise direction.
In Figure 5.4, part b), we illustrate the case when c lies outside the region enclosed byCs. In this case, the angle ofF(s) does not change ass moves around the curveCsand thus, no encirclement of the origin in F occurs.
Using similar reasoning, we see that for the functionF(s) = (s−p)−1(where the polepliesinside the region determined byCs) there is an angle change of +2π[rad]
when s moves clockwise alongCs. This is equivalent to saying that the curve CF
encloses the origin in F once in the counterclockwise direction. The phase change is zero ifpisoutside the region, leading again to no encirclements of the origin in
F .
Consider now the case whereF(s) takes the more general form:
F(s) =K m
i=1(s−ci) n
k=1(s−pk) (5.7.1)
Then any net change in the angle ofF(s) results from the sum of changes of the
angle due to the factors (s−ci) minus the sum of changes in the angle due to the factors (s−pk). This leads to the following result
Consider a function F(s) as in (5.7.1) and a closed curveCs in s. Assume thatF(s) hasZ zeros andP poles inside the region enclosed by Cs. Then ass moves clockwise alongCs, the resulting curveCF encircles the origin in F Z-P times in a clockwise direction.
So far this result seems rather abstract. However, as hinted earlier it has a direct connection to the issue of closed loop stability. Specifically, to apply this result, we consider a special functionF(s), related to the open loop transfer function of plant, Goand a controllerC(s) by the simple relationship:
F(s) = 1 +Go(s)C(s) (5.7.2)
We note that the zeros of F(s) are the closed loop poles in a unity feedback control system. Also the poles ofF(s) are the open loop poles of plant and controller.
We assume thatGo(s)C(s) is strictly proper so thatF(s) in (5.7.2) satisfies
|slim|→∞F(s) = 1 (5.7.3) In the context of stability assessment we are particularly interested in the number of closed loop poles (if any) that lie in the right half plane. Towards this end, we introduce a particular curve Cs which completely encircles the right half plane (RHP) in s in theclockwise direction.
This curve combines the imaginary axisCiand the return curveCr(a semi circle of infinite radius) as shown in Figure 5.5. This choice ofCsis known as the Nyquist path.
We now determine the closed curveCF in the complex plane F which results from evaluating F(s) for every s ∈ Cs. S ince F(s) satisfies equation (5.7.3), then the whole mapping ofCrcollapses to the point (1,0) in F . Thus only the mapping ofCi has to be computed, i.e. we need only to plot the frequency response F(jω),
∀ω ∈ (−∞,∞) in the complex plane F . This is a polar plot of the frequency response, known as the Nyquist plot.
Since we have chosenF(s) = 1 +Go(s)C(s), then the zeros ofF(s) correspond to the closed loop poles. Moreover, we see thatF(s) andGo(s)C(s) share exactly the same poles, (the open loop poles). We also see that the origin of the complex plane F corresponds to the point (−1,0) in the complex plane GoC . Thus the Nyquist plot for F can be substituted by that of GoC by simply counting encirclements about the−1 point. The basic Nyquist theorem, which derives from the previous analysis, is then given by
Section 5.7. Nominal Stability using Frequency Response 139
s
Cr
Ci
r→ ∞
Figure 5.5. Nyquist path
Theorem 5.1. If a proper open loop transfer functionGo(s)C(s)has P poles in the open RHP,and none on the imaginary axis,then the closed loop has Z poles in the open RHP if and only if the polar plot Go(jω)C(jω)encircles the point (−1,0) clockwise N=Z-P times.
From this theorem we conclude that
• If the system is open loop stable, then for the closed loop to be internally stable it is necessary and sufficient that no unstable cancellations occur and that the Nyquist plot ofGo(s)C(s)does not encircle the point (−1,0)
• If the system is open loop unstable, withP poles in the open RHP, then for the closed loop to be internally stable it is necessary and sufficient that no unstable cancellations occur and that the Nyquist plot of Go(s)C(s) encircles the point (−1,0)P times counterclockwise
• If the Nyquist plot ofGo(s)C(s) passes exactly through the point (−1,0), there exists an ωo ∈ R such that F(jωo) = 0, i.e. the closed loop has poles located exactly on the imaginary axis. This situation is known as a critical stability condition.
There is one important remaining issue, namely, how to apply Nyquist theory when there are open loop poles exactly on the imaginary axis. The main difficulty in this case can be seen from Figure 5.4 on page 137, part a). Ifcis located exactly on the curve Cs, then the change in the angle of the vector s−c is impossible to determine. To deal with this problem, amodified Nyquist path is employed as shown
in Figure 5.6. The modification is illustrated for the simple case when there is one open loop pole at the origin
The Nyquist path Cs is now composed of three curves: Cr, Ca and Cb. The resulting closed curveCF will only differ from the previous case whensmoves along Cb. This is a semi circle of radius&, where & is an infinitesimal quantity (in that way the encircled region is still the whole RHP, except for an infinitesimal area).
Using the analysis developed above, we see that we must compute the change in the angle of the vector (s−p) whenstravels along Cb withp= 0. This turns out to be +π[rad], i.e. the factors−1maps the curveCbinto a semi-circle with infinite radius and clockwise direction.
To compute the number and direction of encirclements, these infinite radius semi-circles must be considered, since they are effectively part of the Nyquist dia- gram.
This analysis can be extended to include any finite collection of poles ofGo(s)C(s) onCs.
The modified form of the Nyquist theorem to accommodate the above changes is shown in Figure 5.6.
s
Cr
Ca
Cb
Figure 5.6. Modified Nyquist path
The modified form of Theorem 5.1 is
Theorem 5.2 (Nyquist theorem). Given a proper open loop transfer func- tion Go(s)C(s)withP poles in the open RHP,then the closed loop hasZ poles in the open RHP if and only if the plot ofGo(s)C(s)encircles the point(−1,0) clockwise N=Z-P times when stravels along the modified Nyquist path.
Remark 5.1. To use the Nyquist theorem to evaluate internal stability,we need to have the additional knowledge that no cancellation of unstable poles occurs between
Section 5.8. Relative Stability: Stability Margins and Sensitivity Peaks 141 C(s)andGo(s). This follows from the fact that the Nyquist theorem applies only to the product Go(s)C(s),whereas internal stability depends also on the fact that no unstable pole-zero cancellation occurs (see Lemma 5.1 on page 126).