Closed loop is stable if the open-loop frequency response does not encircle point [ -1, j0 ].  By passing somewhere between the origin and point [ -1, j0 ] it exerts some ASM and PHSM, as shown in fig. 12.1.  “Closed loop is stable” means that the transients converge to zero after a time.  However, an existence of ASM and PHSM provides no information on the transients damping.

Such information can be reached by a family of  α_R = const circles, (where α_R denotes resonant gain). It is sometimes referred to as Hall diagram.  The coordinate of the resonant gain centers is defined as:

Each circle radius is:

Looking at it as a curves pure mathematical parametric description with 0 < α_R < ∞ being a parameter, it presents a family of circles shown in fig. 12.2.

Figure 12.2 Family of α_R = const circles in the region 0 < α_R < ∞

In most control problems the closed-loop transfer function has one or more pairs of complex poles.  Even in a case when closed loop is stable, the transients would most probably be oscillatory.  That brings up a phenomenon of resonance.  What we are interested in is the range of α_R > 1.  It means that the left-hand side of the plane in fig. 12.2 is what is relevant only, as is shown in fig. 12.3.

Figure 12.3  Closed loop α_R = const circles in the open-loop Nyquist coordinates.

The second order oscillator is detailed in Chapter  10.6.  The dynamics of a 2nd order stable system that is characterized by a single pair of complex poles has been presented there.  The meaning of damping ratio and the resonance was explained as well.  There is a relationship between resonant gain α_R and  damping ratio ζ :

There is also an expression for the opposite direction, expressing ζ as a function of α_R :

As per the above relationship unity resonance gain α_R  = 1 corresponds to the damping ratio ζ  ≈ 0.7.  That is considered as the optimal value for the closed-loop dynamics in most control applications.  It offers the fastest transients fading without much risk of the resonance.

To get some idea about closed-loop transients damping there is a presentation in  fig. 12.4.  It shows the step responses for several characteristic values of damping ratio ζ and corresponding resonance gain α_R .

Figure 12.4 Step responses for several characteristic values of damping ratio ζ and corresponding resonance gain α_R

Let us examine a hypothetical control loop whose open-loop frequency response in Nyquist diagram is presented by the red line in fig. 12.5.  The open-loop frequency response does not encircle point [ -1, j0 ] and exerts positive stability margins ASM and PHSM.  It means that the subject closed loop is stable.

Additionally it touches the circle α_R  = 1 .3.  It would mean that the closed-loop transients will be damped at the damping ratio ζ ≈ 0.43 and would look as presented in fig. 12.4.

Figure 12.5 Open-loop frequency response of a hypothetical control loop (red line) in Nyquist diagram with ASM and PHSM points and  α_R = const circles

There is a question hanging at the end of this chapter:  What could be a benefit of bothering with the open loop frequency response in Nyquist diagram when the information of the closed loop dynamics is obtainable?

The answer is simple. Due to the fact that the open loop structure is commonly characterized by a serial blocks connection, the influence of each blocks to the overall open loop frequency response is much more transparent than its influence to the closed loop behavior.